dep_langelaar_20061215.

dep_langelaar_20061215.
Design Optimization
of Shape Memory Alloy Structures
Matthijs Langelaar
Design Optimization
of Shape Memory Alloy Structures
PROEFSCHRIFT
ter verkrijging van de graad van doctor
aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus Prof. dr. ir. J.T. Fokkema,
voorzitter van het College voor Promoties,
in het openbaar te verdedigen
op vrijdag 15 december 2006 om 10.00 uur
door
Matthijs LANGELAAR
werktuigkundig ingenieur
geboren te Apeldoorn.
Dit proefschrift is goedgekeurd door de promotor:
Prof. dr. ir. A. van Keulen
Samenstelling promotiecommissie:
Rector Magnificus
Prof. dr. ir. A. van Keulen
Prof. dr. Y.Y. Kim
Prof. dr. V.V. Toropov
Prof. dr. ir. A. de Boer
Prof. dr. Z. Gürdal
Prof. dr. ir. S. van der Zwaag
Prof. ir. R.H. Munnig Schmidt
voorzitter
Technische Universiteit Delft, promotor
Seoul National University
University of Leeds
Universiteit Twente
Technische Universiteit Delft
Technische Universiteit Delft
Technische Universiteit Delft
c 2006 by M. Langelaar
Copyright All rights reserved. No part of the material protected by this copyright notice
may be reproduced or utilized in any form or by any means, electronic or
mechanical, including photocopying, recording or by any information storage
and retrieval system, without written permission of the copyright owner.
Printed in the Netherlands by PrintPartners Ipskamp
ISBN-13 978-90-9021347-7
ISBN-10 90-9021347-3
Cover:
Rendered three-dimensional model of the mechanical structure of a shape memory alloy active catheter, designed for
controlled bending. Derived from an actual optimized finite
element model (see Chapter 3 and Chapter 9).
Contents
1 Introduction
1.1
1.2
1.3
1.4
Motivation . . . . . . . . . . . . . . . . . . . .
1.1.1 The vision: microsubmarines . . . . . .
1.1.2 The reality: minimally invasive therapy
1.1.3 A challenge: the active catheter . . . . .
Background . . . . . . . . . . . . . . . . . . . .
1.2.1 Shape memory alloys . . . . . . . . . . .
1.2.2 Design optimization . . . . . . . . . . .
Problem statement . . . . . . . . . . . . . . . .
1.3.1 Aim . . . . . . . . . . . . . . . . . . . .
1.3.2 Scope . . . . . . . . . . . . . . . . . . .
Outline . . . . . . . . . . . . . . . . . . . . . .
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1
1
1
3
5
8
8
14
17
17
18
19
2 A Simple R-Phase Transformation Model
2.1
2.2
2.3
2.4
2.5
2.6
Introduction . . . . . . . . . . . . .
Design optimization considerations
R-phase transformation modeling .
Generalization to 3-D . . . . . . .
Example . . . . . . . . . . . . . . .
Concluding remarks . . . . . . . .
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21
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23
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29
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31
32
33
33
35
3 Modeling of a Shape Memory Alloy Active Catheter
3.1
3.2
Introduction . . . . . . . .
Active catheter . . . . . .
3.2.1 Design concept . .
3.2.2 Material selection .
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ii
CONTENTS
3.3
3.4
3.5
Shape memory alloy constitutive modeling . . . . . . .
3.3.1 One-dimensional case . . . . . . . . . . . . . .
3.3.2 Three-dimensional case . . . . . . . . . . . . .
3.3.3 Plane stress case . . . . . . . . . . . . . . . . .
3.3.4 Discussion . . . . . . . . . . . . . . . . . . . . .
Finite element model . . . . . . . . . . . . . . . . . . .
3.4.1 Electro-thermo-mechanical simulation . . . . .
3.4.2 Load case, boundary conditions and symmetry
3.4.3 Solution process . . . . . . . . . . . . . . . . .
3.4.4 Results . . . . . . . . . . . . . . . . . . . . . .
Conclusions . . . . . . . . . . . . . . . . . . . . . . . .
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36
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50
53
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55
55
56
58
60
64
67
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.1 Optimization involving bounded-but-unknown uncertainties
5.1.2 Cycle-based approaches . . . . . . . . . . . . . . . . . . . .
5.1.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Multipoint Approximation Method . . . . . . . . . . . . . . . . . .
5.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.2 Optimization problem formulation . . . . . . . . . . . . . .
Bounded-But-Unknown uncertainty . . . . . . . . . . . . . . . . . .
Uncertainty-based optimization using Anti-optimization . . . . . .
5.4.1 Anti-optimization . . . . . . . . . . . . . . . . . . . . . . .
5.4.2 Lombardi-Haftka Alternating anti-optimization . . . . . . .
5.4.3 Cycle-based Alternating anti-optimization . . . . . . . . . .
5.4.4 Combined Cycle-based Alternating and Asymptotic method
5.4.5 Parallel computing . . . . . . . . . . . . . . . . . . . . . . .
Test case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
SMA Microgripper optimization under uncertainty . . . . . . . . .
5.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6.2 Microgripper model . . . . . . . . . . . . . . . . . . . . . .
5.6.3 Optimization problem formulation . . . . . . . . . . . . . .
5.6.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
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99
4 Shape Optimization of an SMA Gripper
4.1
4.2
4.3
4.4
4.5
4.6
Introduction . . . . . . . . .
Design optimization of SMA
Constitutive modeling . . .
Gripper design case . . . . .
Results . . . . . . . . . . . .
Conclusion and outlook . .
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structures
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5 Shape Optimization under Uncertainty
5.1
5.2
5.3
5.4
5.5
5.6
iii
CONTENTS
5.7
Discussion and conclusions . . . . . . . . . . . . . . . . . . . . . . . 104
6 Topology Optimization of SMA Actuators
6.1
6.2
6.3
6.4
6.5
6.6
6.7
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1.1 Shape memory alloys . . . . . . . . . . . . . . . . . . .
6.1.2 Topology optimization . . . . . . . . . . . . . . . . . .
6.1.3 SMA material modeling . . . . . . . . . . . . . . . . .
6.1.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . .
SMA modeling . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.1 Design optimization considerations . . . . . . . . . . .
6.2.2 One-dimensional R-phase transformation modeling . .
6.2.3 Three-dimensional modeling . . . . . . . . . . . . . . .
6.2.4 Plane stress case . . . . . . . . . . . . . . . . . . . . .
6.2.5 Verification and discussion . . . . . . . . . . . . . . . .
6.2.6 Robust analysis techniques for optimization . . . . . .
Element Connectivity Parameterization method . . . . . . . .
6.3.1 Difficulties in conventional method (density approach)
6.3.2 Basic idea and mesh layout . . . . . . . . . . . . . . .
6.3.3 Governing equations . . . . . . . . . . . . . . . . . . .
6.3.4 Sensitivity analysis . . . . . . . . . . . . . . . . . . . .
6.3.5 Interpolation function . . . . . . . . . . . . . . . . . .
Problem formulation and regularization . . . . . . . . . . . .
6.4.1 Problems considered . . . . . . . . . . . . . . . . . . .
6.4.2 Numerical artifacts in topology optimization . . . . .
6.4.3 Effect of filtering . . . . . . . . . . . . . . . . . . . . .
6.4.4 Nodal design variables . . . . . . . . . . . . . . . . . .
SMA topology optimization results . . . . . . . . . . . . . . .
6.5.1 Effect of load magnitude . . . . . . . . . . . . . . . . .
6.5.2 Effect of mesh refinement . . . . . . . . . . . . . . . .
6.5.3 Effect of load direction . . . . . . . . . . . . . . . . . .
6.5.4 Design improvement from a baseline design . . . . . .
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . .
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107
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151
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159
7 SMA Sensitivity Analysis
7.1
7.2
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Sensitivity analysis approaches . . . . . . . . . . . . . . . . . .
7.2.1 Variational approaches . . . . . . . . . . . . . . . . . . .
7.2.2 Discrete approaches: finite differences . . . . . . . . . .
7.2.3 Discrete approaches: semi-analytical design sensitivities
7.2.4 Discrete approaches: sensitivity code generation . . . .
7.2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . .
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iv
CONTENTS
7.3
7.4
7.5
Sensitivity analysis of SMA finite element model
7.3.1 Electrical and thermal case . . . . . . . .
7.3.2 Mechanical case . . . . . . . . . . . . . .
7.3.3 Effective strain sensitivity . . . . . . . . .
Evaluation . . . . . . . . . . . . . . . . . . . . . .
7.4.1 Numerical results . . . . . . . . . . . . . .
7.4.2 Discussion . . . . . . . . . . . . . . . . . .
Conclusions . . . . . . . . . . . . . . . . . . . . .
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160
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173
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
SMA miniature gripper . . . . . . . . . . . . . . . . . . . . . . .
8.2.1 Concept and modeling . . . . . . . . . . . . . . . . . . . .
8.2.2 Optimization problem . . . . . . . . . . . . . . . . . . . .
Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3.2 Handling of interdisciplinary coupling . . . . . . . . . . .
8.3.3 Handling of the implicit SMA material model . . . . . . .
8.3.4 Comparative evaluation of sensitivity analysis techniques
8.3.5 Impact of numerical noise on design sensitivities . . . . .
Gripper design optimization . . . . . . . . . . . . . . . . . . . . .
8.4.1 Optimization methods . . . . . . . . . . . . . . . . . . . .
8.4.2 Optimization results . . . . . . . . . . . . . . . . . . . . .
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . .
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217
8 Gradient-based Shape Optimization of an SMA Gripper
8.1
8.2
8.3
8.4
8.5
8.6
9 Gradient-based Optimization of SMA Active Catheters
9.1
9.2
9.3
9.4
9.5
Introduction . . . . . . . . . . . . . . . .
Problem formulation . . . . . . . . . . .
9.2.1 Active catheter design concept .
9.2.2 Active catheter modeling . . . .
9.2.3 Design optimization formulation
Sensitivity analysis . . . . . . . . . . . .
9.3.1 Method . . . . . . . . . . . . . .
9.3.2 Implementation and validation .
Optimization results . . . . . . . . . . .
Discussion and conclusions . . . . . . . .
10 Conclusions and Future Directions
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219
10.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
10.1.1 SMA modeling . . . . . . . . . . . . . . . . . . . . . . . . . 219
10.1.2 Direct shape optimization . . . . . . . . . . . . . . . . . . . 221
v
CONTENTS
10.1.3 Topology optimization . . . . . . .
10.1.4 Gradient-based shape optimization
10.1.5 Overall conclusion . . . . . . . . .
10.2 Future directions . . . . . . . . . . . . . .
10.2.1 Validation and realization . . . . .
10.2.2 Extensions . . . . . . . . . . . . .
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222
223
225
226
226
227
A Derivation of 3-D Tangent Operator
231
B Shear Modulus/Effective Strain Relation in Plane Stress Case
233
C Plane Stress Constitutive Tangent Operator
235
C.1 Method 1: based on the 3-D tangent operator . . . . . . . . . . . . 235
C.2 Method 2: based on the plane stress equations . . . . . . . . . . . 238
D Jacobian and Hessian of Symmetry Plane Constraint
241
E Derivatives of Material Parameters Ci and Di
243
F Maximum Effective Strain Values Occur at Outer Layers
245
G Full Color Illustrations
249
Bibliography
255
Summary
275
Samenvatting
279
List of Publications
283
Acknowledgements
287
About the author
289
vi
CONTENTS
Chapter
1
Introduction
1.1
Motivation
This thesis considers design optimization of shape memory alloy (SMA) structures. This subject is motivated by the promising opportunities for miniature
SMA actuators in medical applications.
1.1.1
The vision: microsubmarines
CARTER: “And take a little trip with them . . . ”
GRANT (bewildered): “Trip? Where to?”
CARTER (matter-of-fact): “Well, the only way to reach that clot is
from inside the Brain. So we’ve decided to put a Surgical Team
and a Crew into a submarine – reduce it way down in size, and
inject it into an Artery – ”
GRANT (jolted): “You mean I’m going along?”
CARTER: “As part of the Crew.”
GRANT: “Wait a minute! They can’t shrink me!”
CARTER (assuringly): “Grant, our Miniaturizer can shrink anything.”
These lines are quoted from the 1965 screenplay for the movie Fantastic Voyage
written by Harry Kleiner and directed by Richard Fleischer. In this sciencefiction movie, miniaturization is accomplished by a technique to shrink objects
temporarily by shrinking their atoms. The technique is not perfect, as objects
1
2
INTRODUCTION
1.1
revert to their original size after merely one hour. A scientist called Benes has
found a way to make the shrinkage permanent, but his life is in danger as he suffers
from a bloodclot in his brain. Brain surgery is impossible, as it would destroy too
much delicate brain tissue. In a desperate attempt to save his life, a crew of five
attempts to remove the clot from the inside, by miniaturizing themselves and their
submarine named Proteus and traveling to the brain via the bloodstream. While
their time runs out, they make a “fantastic voyage” inside Benes’ bloodvessels,
fighting off attacks from white blood cells and antibodies.
Figure 1.1: Artist impressions of a “Fantastic Voyage” micro-submarine in a blood
vessel (left, by David Morgan-Mar, with permission) and of nanorobots cleaning plaque
from arterial walls (right, by Tim Fonseca, with permission).
In spite of several strong arguments that render this concept of a microsubmarine impossible, the vision itself continues to inspire (Figure 1.1). The idea
for this movie may in fact stem from another source often cited by authors addressing the prospects of miniaturization: in his famous lecture, Feynman (1960,
1992) also brings up the concept of a device that can operate on patients from
inside their arteries:
“A friend of mine suggests a very interesting possibility for relatively small machines. He says that, although it is a very wild idea, it
would be interesting in surgery if you could swallow the surgeon. You
put the mechanical surgeon inside the blood vessel and it goes into the
heart and looks around. It finds out which valve is the faulty one and
takes a little knife and slices it out.” - R. Feynman, 1960.
That these inspiring visions actually lead to very real prototypes is illustrated
by Figure 1.2, which shows a realization of a micro-submarine (hull dimensions 4
mm by 650 µm) presented by the German company microTEC in 1997 and then
at the EXPO 2000 in Hannover (Moore, 2001). Next to the fact that this microsubmarine illustrated the capabilities of a highly precise UV curing additive fabrication technique for fabrication of complex three-dimensional microcomponents
1.1
MOTIVATION
3
(Reinhardt and Götzen, 2006), it also showed the possibilities of this technique for
integration of various components of different materials at the microscale. This is
illustrated by the fact that the propeller, which has a diameter of 600 µm and is
attached to an axis with a diameter of merely 10 µm, can actually rotate freely.
Note, however, that this prototype did not include sensors, actuators and control
systems and a power source necessary to make it function autonomously, and the
integration of those remains an enormous challenge. Still this micro-submarine
received a lot of attention, and it even holds the world record for smallest medical
submarine (Guinness, 2005).
Figure 1.2: Conceptual colored scanning electron micrograph (SEM) of a microsubmarine in a human artery, made by German company MicroTEC ( www.microtecd.com), picture made by Oliver Meckes, Eye of Science ( www.eyeofscience.com). Copyright: microTEC Germany, reprinted with permission.
Micro-submarines in this form, however, will probably never be feasible for
medical interventions such as actively hunting clots in bloodvessels. Scaling laws
render propulsion in liquids increasingly inefficient, and speeds needed to overcome the blood flow velocity seem out of reach (see e.g. Avron et al., 2004, Dario
et al., 1992, Edd et al., 2003, Kosa et al., 2005, Power, 1995, Vogel, 1994, Wautelet,
2001). Still, the potential uses for miniaturization in medicine are many, as discussed below.
1.1.2
The reality: minimally invasive therapy
Although miniaturization has been and is a strong trend in many industries nowadays, it is not accomplished by shrinking atoms, as suggested by Asimov (1966).
Still, the idea of being able to operate from inside the body may have been the
inspiration for many instruments currently used in so-called minimally invasive
procedures (e.g. Fuchs, 2002, Mack, 2001, Rattner, 1999). An example is shown
in Figure 1.3, which shows a comparison of conventional open heart surgery to a
4
INTRODUCTION
1.1
minimally invasive version of the same procedure. Technical advancements have
enabled invasive diagnostic studies or surgery to be carried out with minimal damage to healthy tissue, through the use of specialized instruments. This continuing
trend from open surgery towards minimally invasive procedures offers clear advantages: less discomfort and complications, faster recovery times, shorter hospital
stays, reduced treatment costs and higher survival rates.
Figure 1.3: Conventional (left) and minimally invasive (right) approaches to cardiac
surgery. Image courtesy of the Department of Cardiothoracic Surgery of the University
of Southern California, reprinted with permission.
However, while the improvements for patients are clear, minimally invasive
techniques make the task of physicians more difficult. Procedures must be carried
out in an indirect way, which leads to hand-eye coordination problems and requires
more extensive training. The maneuverability and dexterity of the instruments
is often very limited. Moreover, much of the sensory feedback present in the
“hands-on” approach in open surgery is lacking. Improved instruments can help
to overcome these difficulties (Agrawal and Erdman, 2005, Mack, 2001, Tendick
et al., 1998).
Clearly, there is an incentive to make instruments used for minimally invasive
therapy as small as possible, to achieve maximum access and minimal damage.
But at the same time, as much relevant functionality and dexterity as possible
should be implemented, to reduce the limitations imposed on the physicians. This
can only be accomplished by extensive use of miniaturization. For example, diagnostics and sensory feedback can be enhanced by miniature sensors manufactured
by microfabrication techniques adopted from IC processing. In addition, for the
mechanical functionality of the instruments and mechanical interaction with tissues, micromanipulation and microactuation techniques are required (Fujimasa,
1996).
1.1
1.1.3
MOTIVATION
5
A challenge: the active catheter
An instrument that combines many aspects of instrument development for minimally invasive procedures is the active catheter. Catheters are long, thin, flexible
tube-like instruments that are inserted in the groin, arm or neck, and advanced
through bloodvessels to the site that requires investigation or treatment. Typical
diameters range from 0.3 to 3 mm, depending on the target vessel. These instruments play an important role in the treatment of circulatory diseases, which are
the leading cause of death in the European Union, accounting for over 40% of all
deaths (Eurostat, 2005).
Figure 1.4: Illustration of cardiac catheterization.
Although existing catheters in the hands of experienced physicians already
save many lives today, they still suffer from clear limitations. The inability to
reach the target vessel due to navigation and maneuvering difficulties is the most
common problem in cardiovascular catheter treatments, and difficult cases associated with tortuous or difficult anatomical obstacles still challenge even the most
experienced physicians (Geske et al., 2005). The placement of the catheter tip
must be controlled from sometimes more than 1 meter away, by manipulating
the distal end of the catheter, which often is a very challenging task. In addition, in tortuous venous anatomy, there is a risk of vessel wall penetration or
other trauma inflicted by the catheter. Even slight damage can cause inflammation (vasculitis), which in turn can lead to potentially life-threatening vessel
weakening (aneurysms) or blockage.
Enhanced catheters with steerable tips aim to alleviate the steering problem,
but since these are operated by puller-wires, conventional steerable catheters require a high bending stiffness of the shaft to prevent buckling (Haga and Esashi,
2004). This high stiffness increases the risk of damaging the vessel wall in bent
sections significantly. Recognizing this difficulty, research efforts are underway
6
INTRODUCTION
1.1
Figure 1.5: Schematic illustration of an active catheter in use (left) and a design
drawing of an active catheter prototype by Park and Esashi (1999b) (right).
to integrate the capability of controlled steering into catheters, by exploiting microactuation techniques and “steer-by-wire” approaches. This is the essence of
active catheters, as illustrated in Figure 1.5, which also shows the design of an
active catheter by Park and Esashi (1999b). An overview of prototypes that have
been realized in the past 10–15 years is given in Table 1.1. Note that many of
these make use of shape memory alloy (SMA) microactuators, which are discussed
in more detail in Section 1.2.1. A catheter that can be controlled to bend in any
direction would also allow easier insertion, because it can follow the shape of the
vessel precisely. Some groups have even managed to integrate tactile sensors along
with the microactuators, that are able to detect contact with the vessel wall (e.g.
Kaneko et al., 1996). Although many prototypes have already been developed,
active catheter design still remains challenging. Further design improvements are
needed to achieve the required bending performance, rigidity and reliability, to
avoid overheating and to reduce assembly complexity, before active catheters are
technically and economically feasible.
The active catheter case illustrates the possibilities of miniaturization, microactuation and microsensors for medical applications. Note that this application shows a remote resemblance to the vision of microsubmarines discussed in
Section 1.1.1, but in a more practical form. Controlled bending does not only
allow steering and easier insertion, but also enables autonomous propulsion by
means of crawling along vessel walls. This makes far more sense than swimming
through the bloodstream. In fact, related concepts have been demonstrated already on a larger scale, for devices crawling on the outer surface of a beating heart
(Patronik et al., 2004) and for controlled motion of endoscopes through intestines
(e.g. Dario et al., 1997, 1999, Kim et al., 2005, Peirs et al., 2001). Conceptually,
an active catheter could also be equipped with additional microactuated tools,
e.g. grippers to take biopsies. Therefore, miniaturization could actually make
Feynman’s vision of a “surgeon in a vessel” a reality (Rebello, 2004).
1.1
MOTIVATION
7
Table 1.1: Realized active catheter prototypes and related technologies for catheter steering/propulsion. In addition, large number of patents on catheter steering or propulsion
has been filed (see e.g. Bakker, 2000, for an overview), but this table only includes demonstrated prototypes. Note that the existence of a patent does not imply that the described
invention actually works.
Publication
Outer
diameter
Actuation principle
Remarks
Mineta et al. (2001)
1 mm
Etched thin film actuators.
Haga et al. (2000)
1.5 mm
Kaneko et al. (1996),
Ohta (2001), Takizawa
et al. (1999)
Fukuda et al. (1994a,b)
1.5 mm
Shape memory
alloy
Shape memory
alloy
Shape memory
alloy
1.7 mm
2 mm
Shape memory
alloy
Shape memory
alloy
Shape memory
alloy
Shape memory
alloy
Shape memory
alloy
Conducting polymer actuator
Park and Esashi
(1999a,b)
Chang et al. (2002)
2.1 mm
Dario et al. (1991)
2.4 mm
Lim et al. (1995, 1996)
2.5 mm
Santa et al. (1996),
Sewa et al. (1998)
0.8 mm
Guo et al. (1996)
2 mm
Polymer actuator
Ruzzu et al. (1998)
Ikuta et al. (2002,
2003)
Ernst et al. (2005),
Faddis et al. (2002),
Schiemann et al. (2004)
Sendoh et al. (1999)
2.5 mm
3 mm
Pneumatic
Hydraulic
-
Magnetic
0.6 mm
Magnetic
Ishiyama et al. (2001)
2 mm
Magnetic
Behkam and Sitti
(2004)
5 mm
Electromagnetic
Coil actuators.
Equipped with tactile sensors
to avoid wall contact.
Wire actuators embedded in
polymer matrix.
Coil actuators, integrated
control circuits.
Combined with ultrasound
probe at tip.
Uses shape memory alloy
wires.
Uses externally heated shape
memory alloy coils.
Biocompatibility unknown,
requires ion exchange with
environment.
Biocompatibility unknown,
requires ion exchange with
environment.
Tip positioning by balloons.
Bends only in one direction.
Magnetic catheter guidance
by permanent magnet tip and
external magnetic field.
Propulsion concept: external
rotating magnetic field rotates spiral.
Propulsion concept: external
rotating magnetic field rotates screw through gel and
beef.
Propulsion concept: swimming microrobot using flagellum (similar to E. Coli bacterium).
8
1.2
INTRODUCTION
1.2
Background
In this section, preliminaries on two essential topics for this thesis will be briefly
introduced. Section 1.2.1 discusses shape memory alloys, followed by an overview
of design optimization in Section 1.2.2.
1.2.1
Shape memory alloys
Microactuation
The dominance of viscous and frictional forces at the microscale (Wautelet, 2001)
makes that direct actuation principles are preferred for microactuation (Fearing,
1998). This means that actuator forces are directly transferred to their point
of application, without using complex leverage or transformation mechanisms.
Direct actuation can be achieved by the use of active materials such as piezoceramics or shape memory alloys (SMAs), and thermal actuation based on phase
transformation or thermal expansion. Also electrostatic actuation is frequently
used in microsystem applications (Fatikow and Rembold, 1997). Characteristic
properties of various options for microactuation are listed in Table 1.2.
For active catheters (Table 1.1) and similar micromanipulator applications
that interact with relatively large external forces of the macroscopic environment,
SMA microactuators are generally preferred (see e.g. Stevens and Buckner, 2005).
The scale of the forces involved requires powerful actuators, and considering the
limited available volume, the energy density of the actuation is of prime importance (Fujita and Gabriel, 1991). Furthermore, relatively large deformations are
required for steerable catheters, and SMAs offer a practical amount of actuation strain (Table 1.2). In addition, the typical low-frequency operation of active
catheters and related minimally invasive tools (< 10 Hz) is compatible with SMA
actuation. SMA response speed will improve further with miniaturization and
the presence of active cooling by liquids (blood flow). Moreover, their excellent
biocompatibility (Kapanen et al., 2002) also makes SMAs excellent candidates for
in vivo applications. The limited efficiency of SMA actuators is not a critical disadvantage here, as tethered operation is preferred for safety reasons, and therefore
power supply is not a problem (Dario et al., 1992). Finally, the relative mechanical simplicity of SMA actuation is a large advantage over approaches that require
various additional components, such as relatively complex hydraulic/pneumatic
solutions or complex and fragile linear actuators based on displacement accumulation (e.g. De Boer et al., 2004, Shutov et al., 2004, Yeh et al., 2002).
Shape memory alloy characteristics
Many alloys exhibit shape memory behavior. The phenomenon was first discovered 1932, in a AgCd-alloy. Shape memory alloys started to be applied in
a variety of applications after the discovery of the effect in NiTi alloys in 1959
(Kauffman and Mayo, 1997). In near-equiatomic composition, NiTi alloys exhibit
1.36
5–34
32
2
0.2
> 40
50
0.2
< 50
0.1
Piezoelectric (PZT)
Natural muscle (human)
Electromagnetic
Magnetostrictive (Terfenol-D)
Electrostatics (comb drive)
Piezo-polymer (PVDF)
Electrostatics (force array)
4.8
0.008
70
0.1
110
0.35
15
1
4
100
Thermal, solid-liquid phase
change
Pneumatic (macroscale)
70
78
100
Hydraulic (macroscale)
200–500
1
1–8
Shape memory alloy (NiTi)
Stress
[MPa]
Thermal expansion
(Al, ∆T = 100 K)
Electrostrictive polymer
(P(VDF-TrFE)
Electrostatics (ideal)
Dielectric elastomer (Silicone)
Conducting polymer
Strain
[%]
Actuator type
1–1000
Fast
Fast
Fast
5
>1000
-
Fast
>1000
Fast
Fast
Fast
4·105
3·105
2·105
2·105
1·105
1·105
7·104
2.5·104
2.5·104
3·103
2.4·103
7·102
106
4.7·106
Fast
108
High
High
60
>90
>90
40
High
90
Low
-
< 10
High
High voltages.
Voltage 100 V, gap 0.5 µm.
High voltages.
Low voltages. Encapsulation typically
needed.
High voltages. Brittle in tension.
Not an engineering material yet (with
few exceptions (e.g Herr and Dennis,
2004)).
Increasingly less attractive at smaller
scales.
Requires additional means to locally
change magnetic field.
High pressure source, channels, valves
required.
Low voltages. Faster at smaller scales.
Large temperature changes.
High voltages.
Low voltages. Hysteresis. Gradual
degradation at high strains (> 4%).
Faster at smaller scales.
High pressure source, channels, valves
required.
Modest temperature changes.
1–100
6 · 106 –> 108
<5
Response Efficiency Comments
speed
[%]
[Hz]
Energy density [J/m3 ]
Table 1.2: Comparison of various microactuation principles, sorted on energy density. Properties are indicative and often depend
on actual actuator realization and scale. Data is collected from Dario et al. (1992), Huber et al. (1997), Jacobson et al. (1995),
James and Rizzoni (2000), Madden et al. (2004), Otsuka and Wayman (1998), Pelrine et al. (2000).
1.2
BACKGROUND
9
10
1.2
INTRODUCTION
very pronounced and relatively stable shape memory behavior, which is suited for
practical applications. Also a variety of less expensive Cu-based SMAs exists, but
their actuator properties are generally inferior to those of NiTi.
The actuator properties of SMAs originate from the diffusionless solid-state
phase transformation that can occur in these materials (Funakubo, 1987, Otsuka
and Wayman, 1998). In diffusionless transformations, no atomic bonds are broken
or formed, so this transformation merely consists of a distortion of the atomic
lattice. A transformation strain is associated with this distortion, which can be
used for actuation. The transformation is triggered by changes in temperature,
stress state and – in some alloys – magnetic field. In SMAs, one of the phases
is a highly symmetric phase (cubic), that can only exist in one configuration.
This phase is called the austenite phase, and is stable at higher temperatures.
The other phase has lower symmetry, and can therefore exist in several variants.
This martensite phase is stable at lower temperatures. Figure 1.6 schematically
illustrates the transformation between these two phases. Note that martensite,
due to its lower symmetry, can occur in a twinned state and a detwinned state.
Decreasing temperature,
increasing stress
Increasing temperature,
decreasing stress
Martensite (detwinned)
Austenite
Increasing stress
Decreasing temperature
Increasing temperature
Martensite (twinned)
Austenite
Figure 1.6: Illustration of martensite-austenite forward and reverse transformations,
schematically showing lattices as they typically occur in shape memory alloys.
The characteristics of the temperature-induced phase transformation are shown
in Figure 1.7. Upon heating from the martensite state, the material starts to
transform locally into austenite at the austenite starting temperature AS , and
this process continues until at the austenite finish temperature AF , all material
has transformed to austenite. When cooling down, the reverse process takes place,
and austenite transforms back to martensite. The temperatures where this trans-
1.2
BACKGROUND
11
formation starts and finishes are denoted by MS and MF , respectively. Note that
the forward and reverse transformations do not occur at the same temperatures.
The difference or hysteresis depends on the alloy and its composition, but typical
values are 30–50 K. This hysteresis complicates the use of SMAs as continuously
controlled actuators, and they are therefore most often used as discrete (on/off)
actuators. However, in nickel-rich NiTi alloys, next to the austenite and martensite states, also a third rhombohedral phase occurs, which is called the R-phase.
The phase transformation between austenite and this R-phase is known to have a
very small hysteresis of approximately 2 K and an excellent resistance to fatigue
(Otsuka and Wayman, 1998, Tobushi et al., 1996). These properties make this
transformation very attractive for actuation applications.
Martensite phase fraction
100 %
Heating
Cooling
0 %
MF
MS
AS
AF
Temperature
Figure 1.7: Transformations in shape memory alloys induced by temperature changes,
in a stress-free setting. MS and AS indicate the temperatures at which the transformations to martensite respectively austenite start, and likewise MF and AF indicate the
temperatures at which they finish.
Shape memory alloys have obtained their name from the curious property
that these materials appear to “remember” their original shape. After a seemingly
permanent deformation of an SMA structure, a temporary increase in temperature
is sufficient to restore it back to its original state. This effect is called the one-way
shape memory effect, and is illustrated in Figure 1.8 by the curve A-B-C-D-E-A.
This behavior is observed when the material is initially in the twinned martensite
state. The loading causes detwinning of the initially twinned martensite, which
leads to a relatively flat section (B-C) in the loading stress-strain curve. When the
load is decreased after further elastic deformation of the twinned martensite (CD), the unloading curve (D-E) differs from the loading curve (A-B-C-D). Hence,
also in the stress-strain space hysteresis is generally observed. After removing
12
1.2
INTRODUCTION
the load, an apparently permanent deformation remains (E), as the detwinned
martensite does not revert to the twinned state by itself, but remains in a new
equilibrium configuration. By subsequent heating to a temperature above AF (EA), the phase transformation to austenite is triggered. As austenite has a cubic
lattice which has no variants, this causes the material to recover its initial shape
(A). This shape is maintained after returning to the (twinned) martensite state
after cooling.
G
Stress
A
H
Unloading
C
Loading
B
Heating > AF
> AF
D Temperature < AS
Loading
F
Temperature Unloading
E
Strain
Figure 1.8: Schematic loading and unloading stress-strain curves for a typical shape
memory alloy, at two different temperatures. The one-way shape memory effect is shown
by the sequence A-B-C-D-E-A, and the superelastic effect by A-F-G-H-A.
Another effect, based on the same underlying phase transformation, is observed when the alloy is loaded in the austenite phase. The material in that case
is able to elastically recover from quite large deformations (up to 8%). Normally,
the elastic strain range for metals is nearly two orders less, at 0.1–0.2%. This
remarkable effect is known as superelasticity, or more correctly transformation
pseudo-elasticity. Pseudo-elastic behavior is also depicted in Figure 1.8, by the
curve A-F-G-H-A. The stressing of the material in its austenite state triggers the
stress-induced phase transformation (F), which shows a similar plateau in the
stress-strain curve as observed before. At a certain load, all material has transformed into the detwinned martensite state (G). Upon unloading, however, the
material transforms back to the austenite phase when the load has sufficiently decreased (H), and finally returns to its original shape when the load is completely
removed (A). This ability to elastically recover large strains is often used in appli-
1.2
BACKGROUND
13
cations, next to the use of SMAs as actuator materials by the temperature-induced
transformation. See for example Duerig et al. (1999) for an overview of medical
applications of SMA superelasticity. An overview of non-medical applications is
given by Van Humbeeck (1999). Upcoming applications also include the use of
SMA actuators in microsystems, using thin films produced by sputtering techniques. An extensive overview of recent developments in this direction is given
by Fu et al. (2004).
Shape memory alloy design
Designing SMA structures can be challenging, particularly in a two- or threedimensional setting. The complex stress-strain-temperature behavior (Figure 1.7,
Figure 1.8) of these materials makes design based on intuition alone very difficult.
Moreover, SMA actuators are often controlled by direct resistive heating. This
means that in the design process, electrical, thermal and mechanical aspects of
the structure have to be considered simultaneously. For one-dimensional configurations (SMA wires), the design problem is more or less manageable, and various
approaches have been proposed (Liang and Rogers, 1992a, 1997, Troisfontaine
et al., 1999). For more complex devices, numerical modeling is generally necessary to provide insight in the performance of a design. To this end, many models
have been proposed to simulate the behavior of SMA materials. This is still an
active field of research, and since the last review by Birman (1997), many improved or refined models have been proposed. A recent development is the use
of micro-macro formulations, where the behavior of individual grains is considered and aspects such as crystallographic texture can be included (see e.g. Jung
et al., 2004, Nae et al., 2003). These sophisticated models are however less useful
for design purposes, as their complexity and the required computational effort is
currently not suited for practical design iterations.
The ability to simulate SMA devices is of great importance to the design
process of more complex SMA structures. By a cyclic process of model building/adaptation, simulation and evaluation, computational models can be of assistance in designing SMA structures. However, for an effective design process,
manual and trial-and-error based iterative design adaptation (e.g. Büttgenbach
et al., 2001) is not the most suitable approach. Instead, use can be made of design optimization techniques. These techniques are discussed in more detail in
the following section. Although design optimization has proven its value already
in a wide variety of structural design problems (see e.g. Maute et al., 1999, for
an overview), application to the design of SMA structures has so far been limited
to a few cases. Lu et al. (2001) have reported design optimization of a flexural
actuator equipped with SMA wires, based on analytical modeling, and Birman
(1996) optimized a panel with embedded SMA fibers. However, these approaches
focus on one-dimensional SMA configurations and cannot deal with more general
structures. Kohl and coworkers have considered peak stress reduction of twodimensional SMA microactuators (e.g. Kohl et al., 2000b, Kohl and Skrobanek,
14
1.2
INTRODUCTION
1998, Skrobanek et al., 1997), using finite element modeling. Although their approach bears some resemblance to an optimization procedure, it is not nearly as
versatile as actual design optimization approaches based on the formulation of a
formal optimization problem. Clearly, design optimization of SMA structures still
is largely unexplored, although it has great potential for this application considering the challenging design problem. The aim of this thesis is therefore to develop
and explore design optimization of SMA structures. However, before further discussion of the problem statement of this thesis, the following section first gives a
brief overview of existing design optimization techniques.
1.2.2
Design optimization
Optimization-based design process
Design optimization provides a systematic approach to solve design problems
(Arora, 2004). With the availability of affordable computer power and capable analysis software, this structured model-based approach has recently gained
widespread acceptance in many branches of product development (Saitou et al.,
2005). The process starts by formulating the design problem that is considered,
defining the model to evaluate designs and the parameters that will be considered as design variables, x, and their domain X . Generally, the model will be a
numerical model. Subsequently, the design problem can be cast in the form of a
formal optimization problem as follows:
min
f (x)
subject to
gi (x) ≤ 0 i = 1 . . . n
hj (x) = 0 j = 1 . . . m
x∈X
x
(1.1)
This problem consists of at least one objective f , i.e. the quantity that is to
be minimized or maximized, and optionally a number of constraints g and h,
i.e. conditions that have to be satisfied. The objective and constraint values are
functions of the design variables, and are collectively called the responses of the
model. Next, explorative studies could be carried out, to gain more insight in
the problem, and to possibly even adapt the initial problem formulation based
on the findings. A well-posed design optimization problem often includes contradicting requirements between which the optimization process has to find the best
compromise.
In order to solve the problem defined by Equation 1.1, an optimization algorithm is used. This algorithm is coupled to the computational model in order
to find the optimal design, as illustrated in Figure 1.9. This results in an iterative process that usually requires many evaluations of the model. The efficiency
of the process can usually be improved significantly when also design sensitivities, i.e. derivatives of the model responses with respect to the design variables,
can be computed. The availability of sensitivity information allows the use of
1.2
BACKGROUND
Optimization
algorithm
Design variables x
15
Model
Responses f(x), gi(x), hj(x)
df dgi dhj
Sensitivities , ,
dx dx dx
Figure 1.9: Layout of the cyclic process for solving design optimization problems.
gradient-based optimization algorithms, that generally have superior convergence
characteristics compared to direct methods, that only use the values of the responses. For this reason, design sensitivity analysis has a central role in design
optimization research (Haftka and Gürdal, 1992). In case of complicated, timeconsuming models, optimization might not be feasible simply because it takes too
long. In that case, more efficient simplified models can be used in the optimization process (Papalambros and Wilde, 2000). After an optimal design is obtained
from the optimization process, a detailed analysis can be done.
The outlined process allows designers to ask specific design questions, and
explore different design concepts in a systematic way. In finding the best design,
this approach is far more effective than the traditional “trial-and-error” approach.
It is both powerful and versatile, and can be used to simply find a design that
“works”, or to optimize performance to gain a competitive edge.
Structural optimization
The field of structural optimization covers the application of optimization techniques to the design of structures. Extensive information on structural optimization can, for example, be found in the textbooks by Haftka and Gürdal (1992),
Arora (1997) and the review by Maute et al. (1999). Among other classifications, structural optimization approaches can be categorized in three categories
of increasing versatility: sizing, shape and topology optimization.
Sizing optimization is generally applied to structures composed of members
with certain cross-sectional properties, such as trusses, beams, plates or shells.
The cross-sectional properties are used as design variables, for example the crosssectional areas of trusses in a truss structure. The structural layout itself is not
modified, only the “sizes” of the members are. Sizing optimization is illustrated
in Figure 1.10(a).
16
1.3
INTRODUCTION
Sizing optimization
(a)
Shape optimization
(b)
Figure 1.10: Schematic illustration of sizing (a) and shape optimization (b). In sizing
optimization, the cross-sectional areas of the trusses are used as design variables. In
shape optimization, the coordinates of the Bézier control points and the dimensions of
the elliptic hole are potential design variables.
A more general approach is given by shape optimization, where the optimal
shape of a design is determined. This method requires a geometrical description
of the shape of the design, and parameters of this description are used as design
variables. For example, for a design defined by Bézier curves, as depicted in
Figure 1.10(b), the coordinates of the control points defining the curves could
be used as design variables. Although shape optimization in principle allows
more extensive modifications of the design than sizing optimization, the layout
(topology) of the design remains unchanged. New holes or connections that were
not present in the original parameterization can not be introduced during the
optimization process, simply because such features are not described by the used
design variables.
These limitations of shape optimization are overcome by topology optimization. This is a relatively recent technique, that finds its origin in the pioneering
work by Bendsoe and Kikuchi (1988). In topology optimization, no a priori assumptions are made about the topological structure of the design. The design
variables parameterize the discretized material distribution in a given design domain. This allows practically any conceivable arrangement of material, in contrast
to the limited shapes achievable through shape optimization based on predefined
geometrical features. Extensive overviews of topology optimization techniques can
be found in the reviews by Eschenauer and Olhoff (2001), Hassani and Hinton
(1998a,b,c) and the book by Bendsoe and Sigmund (2003).
Topology optimization is regarded to be particularly useful in the initial stages
of a design process, since this technique is able to generate promising layouts
without any prior assumptions. The accuracy that can be achieved in topology
optimization is generally less than that of shape optimization. Constraints on
stresses and strains are also easier to incorporate in shape optimization. Therefore, after an initial design has been suggested by topology optimization, shape
optimization could be applied to the generated layout in a subsequent detailed
design phase.
1.3
PROBLEM STATEMENT
Problem formulation
17
Topology optimization result
Design domain
Figure 1.11: Topology optimization problem definition including boundary and loading
conditions (left) and the corresponding result (right) for the stiffest design using 40% of
the design domain.
1.3
Problem statement
After the preceding discussion on the underlying motivation as well as preliminaries on SMAs and design optimization, this section states the objective and scope
of the research presented in this thesis.
1.3.1
Aim
The objective of this research is to develop design optimization techniques for the
design of SMA structures. This objective is motivated by the challenging nature
of SMA design problems, the lack of existing work on SMA design optimization
and thirdly the expected benefit of structured and systematic design approaches.
In particular, this work is aimed at emerging applications of SMAs as microor miniature actuators for minimally invasive applications, such as the active
catheter case discussed in Section 1.1.3. The focus is however on the more generic
design approaches, rather than on the specific applications themselves. The latter
merely serve as carriers that give guidance to the design techniques that are
developed. The main question that this research aims to answer is how and
to what extent design optimization techniques can support the design of SMA
structures in this context.
Sizing optimization has been demonstrated already by several studies on wirebased SMA active structures (e.g. Birman, 1996, Troisfontaine et al., 1999). Therefore, the focus of this thesis is on shape and topology optimization, which so far
have not been considered yet for SMA structures. In addition, SMA shape optimization involving uncertainties and sensitivity analysis of SMA computational
models is studied.
The approach taken to investigate the merit of design optimization approaches
for SMA design problems is to apply the mentioned techniques to representative
practical cases. A first requirement therefore is the availability of an analysis
model to evaluate the performance of SMA structures. After development of a
18
INTRODUCTION
1.3
suitable model, the various optimization techniques are explored using an SMA
active catheter and a miniature SMA gripper as carrier applications.
1.3.2
Scope
As mentioned, potential SMA applications in the field of medical instrumentation
for minimally invasive procedures are used as relevant cases to achieve the necessary focus in the development and study of SMA design optimization techniques.
The economic feasilibity, or an assessment of chances on clinical acceptance of
these applications, such as active catheters, are outside the scope of this research.
Moreover, this research is not intended to focus on technicalities, detailed manufacturing procedures and specific issues of a certain design concept. The applications merely serve as means to provide guidance to the design optimization
techniques that are developed. In addition, the emphasis of the design problems
is on the mechanical performance of the SMA structures, because this aspect is
critical for their mechanical functionality.
The second limitation concerns the SMA material considered in this thesis.
Because of its suitability for actuation applications and in vivo biomedical use,
the focus is on the R-phase transformation. As mentioned earlier, typical of this
R-phase transformation is its small hysteresis and its excellent resistance to fatigue
(Otsuka and Wayman, 1998, Tobushi et al., 1996). The small hysteresis makes
that this transformation can be used for actuation in a narrow temperature range,
making it suitable for in vivo applications.
Furthermore, for SMAs, many computational models have been proposed,
targeted at various applications, describing various aspects of SMA behavior and
having various levels of detail. In the modeling of SMA behavior in this thesis,
the emphasis is on simplification, since design optimization itself is a computationally expensive process. The prime objective is to create a practical model
that is suited for performing design optimization. This implies that it should
be computationally efficient, sufficiently accurate, and able to capture the main
characteristics of the material behavior. Subtleties and second-order effects, such
as the detailed description of multi-axial and non-proportional loading, are not
considered at this stage, for three reasons. Firstly, the existing models that allow
inclusion of these effects currently are not suited for design optimization due to
their complexity and computational cost. Secondly, the experimental validation
of such models, particularly in the case of the R-phase transformation, is currently
not conclusive (Raniecki et al., 1999). Thirdly, it turns out that in practical designs, the influence of such higher-order effects is hardly relevant (e.g. Rejzner
et al., 2002). Therefore, although these aspects of SMA constitutive behavior are
highly interesting, they are outside the scope of this thesis. In addition, since
the considered applications generally are not critical with respect to the speed
of operation, dynamic aspects are not taken into account either. This also helps
to further simplify the modeling. The low requirements the medical applications
impose on the dynamic behavior (1 Hz is expected to be sufficient) and the fact
1.4
OUTLINE
19
that downscaling increases the bandwidth of SMA actuators justify ignoring the
transient behavior in this thesis.
1.4
Outline
The majority of the chapters in this thesis are reproduced from journal or conference articles. This inevitably results in a certain degree of redundancy, but has the
advantage that each chapter is self-contained and can be read independently. The
structure of this thesis is illustrated by Figure 1.12. The underlying motivation,
some preliminaries and the problem statement of this research have been given
in the present chapter. Chapters 2 and 3 focus on the modeling of SMA material
and devices. A simplified SMA material model, suited for design optimization
purposes and aimed at (in vivo) SMA actuator applications, is described first in
Chapter 2. Subsequently, Chapter 3 introduces a refined version of this model,
and discusses the modeling of a steerable catheter equipped with integrated SMA
microactuators.
1. Introduction
2. SMA material model
3. SMA active catheter model
4. SMA shape optimization
5. SMA shape optimization
under uncertainty
6. SMA topology optimization
7. SMA sensitivity analysis
8. Gradient­based SMA gripper shape optimization
9. Gradient­based SMA active
catheter shape optimization
10. Conclusions
Figure 1.12: Organization of this thesis.
The models introduced in Chapters 2 and 3 form the foundation for the chapters dedicated to various aspects of design optimization that follow. A second
20
INTRODUCTION
1.4
pair of chapters, Chapters 4 and 5, focus on the shape optimization of an SMA
miniature gripper, which also potentially could be used in surgical procedures.
Chapter 4 describes the shape optimization of two versions of this gripper. In
Chapter 5, this work is combined with anti-optimization techniques developed by
Gurav et al. (2003), which allow the efficient consideration of uncertainties in the
design optimization.
Although shape optimization is a powerful design tool, it has its limitations
due to the fixed layout of the designs that can be generated. The application of the
more versatile topology optimization approach to the design of SMA actuators is
studied in Chapter 6. This chapter adopts the parameterization technique recently
proposed by Yoon and Kim (2005), which helps to overcome various difficulties
in the topology optimization of SMA structures.
The 7th chapter discusses various options for the sensitivity analysis of SMA
computational models, and the model introduced in Chapter 3 in particular. The
efficient computation of design sensitivities can dramatically reduce the computational expenses required for design optimization procedures. The availability of
sensitivities allows the use of gradient-based optimization algorithms, which are
in most cases superior to direct methods in terms of convergence and efficiency.
The structure of the SMA material model and the fact that actuator models
generally involve coupled electrical, thermal and mechanical simulations leads to
several complications in the computation of the design sensitivities, which are
resolved in this chapter. Chapter 8 subsequently demonstrates the effectiveness
of gradient-based design optimization, using again the miniature gripper design
problem considered before in Chapter 4 and Chapter 5. Chapter 9 does the same
for the active catheter model introduced in Chapter 3 and adresses some specific
challenges encountered in that case. Finally, conclusions, recommendations and
future directions are given in Chapter 10.
Based on: Langelaar, M. and van Keulen, F (2004c). A simple R-phase transformation
model for engineering purposes. Materials Science and Engineering: A - Structural
Materials: Properties, Microstructure and Processing., 378(1–2) pp. 507–512.
Chapter
2
A Simple R-Phase Transformation
Model
For in vivo biomedical applications of shape memory alloys, actuators based on the Rphase/austenite transformation as present in certain TiNi alloys are particularly suited.
Numerical modeling and optimization can improve the design of such actuators. In this
chapter, a one- and three-dimensional constitutive model for this R-phase/austenite
transformation is presented. In contrast to previous work, its formulation is particularly
directed towards the efficiency in finite element based design optimization, next to its
agreement with experimental data. Important features of this model are its simplicity,
an explicit stress strain relationship and its path-independence.
2.1
Introduction
The development of medical instrumentation continues to offer opportunities for
the application of shape memory alloys. Structures with integrated shape memory alloy actuators can enable further miniaturization of medical instruments and
extension of their functionality. The application that motivated the present research is the use of shape memory alloy actuators in catheters. This significantly
enhances the ability to position and steer a catheter, as has been demonstrated
by various prototypes (Mineta et al., 2002, Park and Esashi, 1999b). A more integrated design can further improve the reliability and manufacturability of these
active catheters, but requires the capability to simulate the behavior of two- and
three-dimensional structures with integrated actuators.
21
22
A SIMPLE R-PHASE TRANSFORMATION MODEL
2.2
The R-phase/austenite transformation of TiNi alloys is particularly suited for
this application, for three reasons: TiNi alloys are biocompatible, the narrow
thermal operating range of the phase transformation does not violate restrictions
for in vivo applications and its cyclic stability is excellent (Tobushi et al., 1996).
However, compared to the martensite/austenite transformation in TiNi, the recoverable strain of approximately 0.5% is clearly smaller.
Because the transformation strain is limited, it is important to exploit it as
much as possible. In a multidimensional setting, this is not a trivial task. The
combination of computational models with optimization techniques provides a
structured process to find the best design. The application of this methodology
on the design of structures with integrated shape memory alloy actuators holds
the promise for further improvement of existing shape memory alloy devices as
well as the discovery of new innovative designs.
The constitutive model may affect the efficiency of design optimization procedures to a large extent. This will be discussed in more detail in Section 2.2.
With regard to this aspect, existing practical one-dimensional models (Ikuta et al.,
1991, Leclercq et al., 1994, Lexcellent et al., 1994, Tobushi et al., 1992) for the
R-phase/austenite transformation have been studied. In Section 2.3 a new model
is proposed, which combines a simpler formulation with a better agreement with
previously published experimental data, and which is well suited for design optimization. The generalization of this model to a three-dimensional setting is also
affected by an emphasis on efficiency, which is why the approach proposed in Section 2.4 differs from the conventional one based on the Von Mises stress (Briggs
and Ostrowski, 2002). An example illustrating the practical use of this model is
discussed in Section 2.5, followed by some concluding remarks.
2.2
Design optimization considerations
In structural optimization, the best design is searched for by combining mathematical optimization techniques with a computational model parameterized by a
set of design variables. This process typically requires a large number of evaluations of the model. The focus of the present work is initially directed towards a
finite element model based on shell elements, which is computationally less costly
than one based on fully three-dimensional elements. Still, due to the nonlinearities caused by the constitutive behavior and the expected large rotations, an
incremental-iterative strategy is required to obtain a solution, which significantly
increases the time of each design evaluation. Because the practicality and feasibility of the proposed design optimization approach depends on its computational
costs, a computationally cheap constitutive model is preferred, since it directly
affects the total cost of the procedure.
A second way the choice of the constitutive model influences the cost of design optimization is through its mathematical nature. Efficient optimization algorithms for computationally intensive models make use of gradient information,
2.3
R-PHASE TRANSFORMATION MODELING
23
i.e. derivatives of the objective and constraint functions to the design variables, to
minimize the number of required evaluations. The computational effort to obtain
these derivatives, also known as design sensitivities, depends on various factors.
An important factor is whether the model is history-dependent. Models that make
use of internal state variables belong to this category. In that case, computation
of design sensitivities is considerably more involved (Kleiber et al., 1997). Thus,
from a design optimization point of view, next to their computational cost clearly
the history-dependence of constitutive models is to be considered.
2.3
R-phase transformation modeling
For the intended applications, the Ti-55.3wt%Ni alloy used in an experimental
study by Tobushi et al. (1992), which exhibits the R-phase/austenite transformation, is selected as a suitable material. Stress-strain curves from those experiments
are shown in Figure 2.1. The focus will be particularly on the pseudo-elastic behavior observed for temperatures between 328 and 343 K, shown in Figure 2.1
by thick solid lines. The loading and unloading stress-strain curves in this temperature range are very similar, which allows approximating them by a single
curve, i.e. neglecting the hysteresis. These temperatures are too high for in vivo
600
σ [MPa]
500
343 K
338 K
400
333 K
328 K
300
200
100
ε [%]
0
0.2
0.4
0.6
0.8
1.0
Figure 2.1: Stress-strain curves at various temperatures from Tobushi et al. (1992) of
a TiNi alloy. Thin lines are unloading curves.
24
A SIMPLE R-PHASE TRANSFORMATION MODEL
2.3
applications, but based on the fact that composition and heat treatment can influence transformation temperatures (Sawada et al., 1993, Todoroki, 1990), it is
reasonable to assume that shifting of this temperature range is possible.
Neglecting hysteresis is a common assumption in the models that have been
published on the R-phase transformation. It turns out not only to be a convenient simplification, but due to this assumption the model also becomes historyindependent. This can be understood by the fact that the phase fraction of
martensite (or R-phase), which is an internal variable in Tanaka-based phenomenological models (Brinson and Huang, 1996, Tanaka et al., 1986), becomes
simply a function of the stress σ and temperature T due to this assumption:
ξ = ξ(σ, T ). The simplifications discussed by Brinson and Huang (1996) result in
the following general structure of the one-dimensional constitutive equation:
σ = E(ξ)(ε − εmax
tr ξ).
(2.1)
is the maxHere ε represents the strain, E(ξ) is the effective stiffness and εmax
tr
imum possible amount of transformation strain. The effective stiffness depends
on the phase fraction because the material is generally a mixture of two phases
with different properties. The actual transformation strain is assumed to be proportional to the phase fraction ξ. When hysteresis is ignored and ξ = ξ(σ, T ),
an equation remains from which for a given strain and temperature the stress
can be obtained, as required in a finite element model. However, generally the
stress can not be expressed explicitly in terms of strain and temperature, and
the resulting nonlinear equation has to be solved by a relatively costly iterative
numerical procedure.
An example of this situation is found in the model proposed by Ikuta et al.
(1991), which includes martensite, R-phase and austenite phases. Its basic structure is similar to the described Tanaka-based approach. Hysteresis is neglected
for the R-phase transformation, but because its phase fraction depends on stress
and temperature through a logistic function, in this model the stress cannot be
solved directly.
In contrast, the model proposed by Lexcellent et al. (1994) and Leclercq et al.
(1994) results in an explicit equation to obtain the stress. This model is based on
a thermodynamics framework, and has a rather complex mathematical structure.
However, with the parameters given by Leclercq et al. (1994), this model closely
resembles a piecewise linear relationship between stress and strain, at a given
temperature. This suggests that as an approximation to this model, in order to
further simplify the formulation, piecewise linear functions can give good results.
The model given by Tobushi et al. (1992) also results in an explicit piecewise
linear stress-strain relationship. However, both these models assume that Young’s
moduli of austenite and R-phase material are equal, although the experimental
data suggests otherwise.
In order to improve the agreement with the experimental data and at the
same time simplify the model as much as possible, the following piecewise linear
2.3
R-PHASE TRANSFORMATION MODELING
25
stress-strain-temperature relationship is proposed to describe the one-dimensional
constitutive behavior:

: σ0 = EA ε,
 ε ≤ ε1
ε1 < ε ≤ ε2 : σ1 = ET (ε − ε1 ) + σ0 (ε1 ),
σ=
(2.2)

ε > ε2
: σ2 = ER (ε − ε2 ) + σ1 (ε2 ).
Here EA and ER are constant parameters, and ε1 , ε2 and ET are linear functions
of temperature:
ε1 (T ) = Kε (T − T0 ) + ε0 ,
ε2 (T ) = ε1 (T ) + ∆,
(2.3)
ET (T ) = KE (T − T0 ) + E0 .
In total this model contains eight parameters, Kε , KE , ε0 , ∆, E0 , T0 , EA and ER ,
which can be determined by curve fitting. Introducing the parameter T0 is convenient for shifting the family of curves along the temperature axis. We chose to let
ε2 differ from ε1 only by a constant, since the experimental data indicated that
these parameters depend similarly on temperature.
The stress-strain relations are shown for the case of positive strains, but similar
expressions can be formulated for negative strains. Although tension-compression
asymmetry (TCA), the phenomenon that the magnitude of the stress inducing the
phase transformation is different in tension and compression, has not so clearly
been observed in the R-phase/austenite case (Raniecki et al., 1999), in principle such an effect could be incorporated in this model by using different values
for Kε , KE , ε0 , ∆ and E0 in compression, fitted to the appropriate experimental
stress-strain curves.
In the following, TCA is not considered. By least squares curve fitting the
model to the four experimental stress-strain curves from Tobushi et al. (1992)
of T = 328–343 K, the eight constants could be determined. They are given in
Table 2.1. The resulting curves generated by the model are shown together with
the experimental data in Figure 2.2. It shows that with the proposed model a
good representation of the experimental data can be constructed.
Note that the existing one-dimensional models (Ikuta et al., 1991, Leclercq
et al., 1994, Lexcellent et al., 1994, Tobushi et al., 1992) aim to describe a broader
range of phenomena than just the transformation pseudo-elasticity treated here.
Table 2.1: Parameter values for the proposed model found by curve fitting to experimental data.
Kε
ε0
E0
EA
2.55×10−4 K−1
5.71×10−4
20.0 GPa
68.9 GPa
KE
∆
T0
ER
619 MPa K−1
54.2×10−4
328 K
45.6 GPa
26
2.4
A SIMPLE R-PHASE TRANSFORMATION MODEL
500
σ [MPa]
400
343 K
338 K
300
333 K
200
328 K
100
ε [%]
0
0.2
0.4
0.6
0.8
1.0
Figure 2.2: Stress-strain curves at various temperatures from Tobushi et al. (1992) of a
TiNi alloy (dashed lines) together with curves given by the proposed model (solid lines).
Focusing on representing this particular aspect allows the proposed model to be
both simple and more accurate. It is possible to extend this model to fit the
experimental data even better by using additional terms and parameters, but in
the current design optimization setting this simple piecewise linear formulation is
a good compromise between accuracy and efficiency. It is emphasized that it is
only meant to represent the experimentally observed behavior, although it could
also be interpreted as a first order approximation of a more complex and general
constitutive relation.
2.4
Generalization to 3-D
To generalize the proposed piecewise linear elastic model for transformation pseudoelasticity to the three-dimensional case, a generalized version of the power-law
nonlinear elastic model described by Pedersen and Taylor (1993) and Pedersen
(1998) will be used. The fact that strains and particularly the transformation
strains remain small allows the assumption that a linear elastic model with a
varying Young’s modulus can approximate the constitutive behavior. The model
2.4
GENERALIZATION TO 3-D
27
uses a scalar effective strain based on the strain energy, defined by:
ε2e = {ε}T [C]{ε}.
(2.4)
Using this effective strain as a scalar strain measure, the secant stress-strain relation is given by:
f (εe )
{σ} =
E[C]{ε}.
(2.5)
εe
Here {σ} = {σ11 , σ22 , σ33 , σ12 , σ23 , σ31 }T and {ε} = {ε11 , ε22 , ε33 , γ12 , γ23 , γ31 }T
are energetically conjugated stress and strain tensors in vector notation, [C] is the
associated dimensionless, constant, symmetric and positive definite constitutive
matrix, and E is a reference Young’s modulus. The function f (εe ) should be
chosen such that f (εe )E represents the one-dimensional material behavior. For
instance, in power-law nonlinear elasticity as discussed by Pedersen (1998), Pedersen and Taylor (1993), it is given by εpe . In this way, the Young’s modulus
effectively depends on the value of the scalar strain measure. For f = εe this
model reduces to classical linear elasticity. Note that the way the stress and
strain components relate to each other is fixed and controlled by the content of
the matrix [C], and that only their relative magnitudes are influenced by the
term f (εe )/εe . Also note that in implementations, the situation at εe = 0 should
clearly be dealt with properly. In the isotropic case [C] is given by the well-known
expression:


1−ν
ν
ν
0
0
0
 ν
1−ν
ν
0
0
0 


 ν
1
ν
1−ν
0
0
0 
 , (2.6)

[C] =
1−2ν
0
0
0
0 
(1 − 2ν)(1 + ν) 
2

 0
1−2ν
 0
0
0
0
0 
2
1−2ν
0
0
0
0
0
2
where ν is the Poisson’s ratio of the material. For TiNi, ν = 0.33 is a commonly
used value (Birman, 1997, Boyd and Lagoudas, 1994).
Differentiation and combination of Equation 2.4 and Equation 2.5 yields the
following symmetric constitutive tangent matrix:
E
[KT ] =
[C] Ψ{ε}{ε}T [C] + f [I] ,
(2.7)
εe
where [I] is the unity matrix and Ψ is given by:
Ψ=
1 0
(f εe − f )
ε2e
with f 0 =
df
.
dεe
(2.8)
When the strain vector is equal to zero, the tangent matrix reduces to f 0 E[C].
To generalize the proposed one-dimensional model, for f the discussed piecewise linear stress-strain relation is taken, divided by a reference Young’s modulus,
28
A SIMPLE R-PHASE TRANSFORMATION MODEL
2.5
for which the austenite Young’s modulus is used (E = EA ). In addition, the onedimensional strain is replaced by the effective strain εe , which gives:

 εe ≤ ε 1
ε1 < εe ≤ ε2
f=

εe > ε 2
: f0 = εe ,
ET
(εe − ε1 ) + f0 (ε1 ),
: f1 = E
A
ER
: f2 = EA (εe − ε2 ) + f1 (ε2 ).
(2.9)
Note, using EA as a reference does not mean that the change in stiffness
between austenite and R-phase is ignored in the 3-D setting. When f , as given
here, is substituted in the stress-strain relation given in Equation 2.2, it is clearly
seen that, e.g. when εe > ε2 , the reference value EA cancels out and ER remains.
Another often used approach to generalize one-dimensional constitutive laws
is to use the Von Mises stress as the equivalent stress (Briggs and Ostrowski,
2002), and this results in formulations which conveniently are quite similar to J2 plasticity models. These can be further simplified by adopting a Hencky’s stressstrain relation (Liang and Rogers, 1992b), justified considering the rather small
strains in the current case. That combined with negligence of the hysteresis leads
to a model without internal variables, with the associated favorable properties for
design sensitivity analysis. However, it turns out that such a formulation leads
to a model with a non-symmetric tangent operator and an implicit stress-strain
relation, making it computationally much more expensive.
Regarding the multi-dimensional limit stress states at which the stress-induced
transformation begins, not much experimental data was available for the Rphase/austenite case until recently. Torsion-tension (compression) tests have been
carried out and analyzed by Raniecki et al. (1999), and their results seem to indicate higher limit stresses in pure shear loading compared to the Von Mises criterion. The presented model qualitatively agrees with this, giving a 15% higher limit
stress in pure shear in comparison to the Von Mises equivalent stress. Neither a
Von Mises based approach nor the proposed model includes TCA, but as mentioned previously, Raniecki et al. did not clearly observe TCA for this material.
In spite of their efforts, the total picture of the transformation limit stress states
for the R-phase/austenite transformation is still not completely clear. Moreover,
regarding non-proportional loading, no results have been published for this material at all, to the authors’ knowledge. A complete validation of the proposed
three-dimensional model is therefore currently not possible.
On the other hand, not all aspects of the constitutive behavior are equally important, and second-order effects can often be ignored. For example, Rejzner et al.
(2002) recently showed that even very pronounced TCA in martensite/austenite
TiNi does not significantly affect the moment-curvature relation in bending. This
means that for structures that are mainly loaded in bending, as the example presented in the following section, the effect of possible TCA can be neglected. In the
design optimization context, aspects that do not significantly change the behavior
of a structure are not likely to have an influence on the optimal design either.
2.5
2.5
EXAMPLE
29
Example
The proposed model has been implemented in a thin shell element as described
in detail by Van Keulen and Booij (1996). For the present nonlinear material
model numerical integration is used to evaluate the tangential stress resultants
and couples and the tangent stiffness.
To demonstrate the proposed material model and to illustrate the opportunity
for design optimization, a structure that could act as a gripper will be discussed,
modeled by the material model described in this chapter, and using the parameter
values as given in Table 2.1. It is a structure made from TiNi plate with a thickness
of 1 mm. The undeformed configuration of the top half of this gripper is shown
in Figure 2.3. The bottom half is similar, but mirrored in the x, y-plane. The
purpose of the active structure is to grasp certain objects by moving the tip C of
the gripper in the z-direction.
To induce an internal stress field, which is essential for the working principle of
this device, the clamped ends A and B are pinched together. Then, it is assumed
that by external heating the temperature of either the top plate (I) or bottom plate
(II) can be raised uniformly from the initial 328 K to 338 K. Because the stressstrain relation of the material is temperature-dependent, different equilibrium
positions are found, as shown in Figure 2.4. These configurations were found by
an incremental-iterative nonlinear finite element analysis of a model consisting of
about 1600 shell elements. Because of symmetry, only half the structure needed
to be considered. In the closed configuration, a concentrated clamping force of
10 N was added at the tip C, acting in the positive z-direction. In both cases,
the stresses and strains in the structure remained in the range covered by the
constitutive model. The difference of the tip location in z-direction in the two
configurations was 4.5 mm.
Figure 2.3: Gripper geometry in undeformed configuration. The plate thickness is 1
mm.
30
2.6
A SIMPLE R-PHASE TRANSFORMATION MODEL
The geometry of this gripper was initially chosen randomly, and its range of
motion was increased by adjusting the shape of Plates I and II by a trial-anderror approach. It is expected that the performance of this device could be further
improved by applying a more structured approach, based on design optimization
techniques. This topic will be explored further in Chapters 4, 5 and 8.
10
z [mm]
5
a
0
338 K
−5
−10
b
0
10
20
30
40
x [mm]
50
Figure 2.4: Side view of gripper in open (a) and closed configuration (b) with a clamping
force of 10 N.
2.6
Concluding remarks
In this chapter, a constitutive model for transformation pseudo-elasticity involving
the R-phase transformation is proposed. It adequately represents one-dimensional
experimental data using a simple, computationally efficient formulation. An important feature of this material model is the absence of internal variables, which
considerably reduces the effort required for design sensitivity computations.
Its generalization to a three-dimensional setting has been presented as well.
Also here the aspects of computational efficiency and simplicity were given priority. When suitable experimental data becomes available, the accuracy of this
three-dimensional version of the present model can be evaluated.
Utilization of structural optimization techniques in the design of shape memory alloy actuators has the potential to produce new, innovative solutions. The
present model was developed with specifically the needs of design optimization in
mind, and is therefore particularly suitable for exploration of this promising field.
Based on: Langelaar, M. and van Keulen, F. (2004b). Modeling of a shape memory
alloy active catheter. In 12th AIAA/ASME/AHS Adaptive Structures Conference,
Palm Springs, CA.
Chapter
3
Modeling of a Shape Memory Alloy
Active Catheter
Shape memory alloys (SMAs) have interesting properties for application in adaptive
structures, and many researchers have already explored their possibilities. However,
the complex behavior of the material makes the development of SMA adaptive structures a challenging task. It is generally accepted that systematic, model-based design
approaches and design optimization techniques can be of great assistance in this case.
Although some studies on design optimization of relatively simple SMA structures have
been published, formal design optimization of more complex SMA devices still requires
further exploration.
By considering a typical example, i.e. an active catheter, this chapter aims to provide
new insights into and solutions for the problems encountered in the practical application of model-based design approaches to SMA adaptive structures. Active catheters are
equipped with integrated micro-actuators that enable controlled bending, which yields
enhanced maneuverability compared to conventional catheters. Next to a detailed discussion of an SMA active catheter finite element model, a novel SMA constitutive model
is introduced. This model combines an adequate representation of the experimentally
observed behavior with computational efficiency. Moreover, its history-independent nature significantly simplifies sensitivity analysis. Due to these features, application of
optimization techniques to shape memory alloy adaptive structures becomes a realistic
possibility.
31
32
MODELING OF A SHAPE MEMORY ALLOY ACTIVE CATHETER
3.1
Introduction
3.1
Due to their ability to generate relatively large strains and high stresses, shape
memory alloys (SMAs) (Duerig et al., 1990, Otsuka and Wayman, 1998) are important materials for adaptive structure design (Chopra, 2002). They can be used
both as actuator and as structural material, which allows for designs with complex functions, and yet with a relatively simple structure. However, an obstacle
for the widespread application of SMAs is given by the complexity of the behavior
of the material. Realistic structures often do not allow a design approach based
on intuition. Structured design approaches based on numerical models, combined
with design optimization techniques, offer a way to overcome this difficulty.
While the constitutive modeling of shape memory alloys has received and continues to receive much attention, little has been reported on the subject of design
optimization of SMA structures. Particularly, detailed finite element models of
realistic complexity are rare in this context. Some work has been done on shape
optimization of SMA structures through minimization of Von Mises stresses (Fischer et al., 1999, Kohl and Skrobanek, 1998, Skrobanek et al., 1997), but full
design optimization based on detailed finite element models including electrical,
thermal and mechanical effects has not been reported yet.
The aim of the present research is to fill this gap by exploring this topic. The
final goal is to successfully apply gradient-based design optimization techniques
to SMA adaptive structure design. In the present work, the focus is on the first
step of this approach: the construction of a numerical simulation model suited for
subsequent optimization. An active catheter design problem is taken as a typical
example of a realistic shape memory alloy adaptive structure.
The new design concept for the active catheter considered in this chapter is
introduced in Section 3.2.1. The design is aimed at using a small number of
individual parts. In this concept, the shape memory alloy is not used for just
one-dimensional actuation. This makes that it serves as a good example of a
complicated, highly integrated shape memory alloy adaptive structure. The SMA
material chosen for this application is a nickel-rich Ni-Ti alloy, which exhibits the
R-phase transformation. Section 3.2.2 will elaborate on the motivation for this
material selection, which is closely linked to restrictions imposed by the in vivo
operating conditions of the catheter.
The third section focuses on the constitutive modeling of the considered material, and a new material model is introduced. Next to the sufficiently accurate
representation of experimental data, its formulation is particularly aimed at its
suitability for design optimization, as highlighted in the final part of this section.
The subject of the fourth section of this chapter is the finite element modeling
of the active catheter structure. In this section, various aspects of the multidisciplinary finite element analysis, which consists of an electrical, thermal and
mechanical component, are considered. Also, a subsection is devoted to the way
symmetries in the design are exploited, which leads to unconventional constraints
in the mechanical analysis. Subsequently, the techniques used to solve the re-
3.2
ACTIVE CATHETER
33
sulting nonlinear problem robustly and efficiently are discussed, followed by the
presentation and discussion of results of the finite element simulation.
In the fifth and final section, conclusions are formulated and applications and
extensions of the presented work are discussed. Some rather detailed but nonetheless important derivations related to the constitutive modeling and the symmetry
constraint equations can be found in Appendices A to D at the end of this thesis.
3.2
Active catheter
3.2.1
Design concept
The continuing popularity of minimally invasive procedures in modern medicine
leads to a growing demand for miniaturized medical instruments with added functionality. This trend is exemplified by efforts to develop active catheters, catheters
equipped with actuators that enable bending to improve their maneuverability.
In recent years, technologically advanced prototypes of such active catheters have
been created by a number of researchers (Chang et al., 2000, Haga et al., 2000,
Park and Esashi, 1999b), demonstrating the feasibility of the concept. All of them
make use of SMA actuation elements, because these can deliver large actuation
strains and/or stresses while the designs remain relatively simple. However, the
prototypes presented in literature typically consist of a large number of individual
parts that are connected together. This increases the complexity of the device
and makes the assembly process involved and costly. Moreover, the large number
of interconnections is likely to decrease reliability. In contrast to the published
prototypes, the new active catheter design concept presented in this chapter aims
at minimizing the number of individual parts, to overcome these disadvantages.
The concept itself originates from manufacturability considerations: the proposed
manufacturing method is to laser-cut or etch a pattern out of small-diameter shape
memory alloy tubing. This procedure is currently routinely used for stents (e.g.
Dickson et al., 2002), but can be used for other structures as well. The pattern
used in the present concept enables active bending in two orthogonal directions,
and by combining these, the catheter can bend in any direction. The resulting
structure is then subjected to a macroscopic strain, for example by applying a
tensile axial load. This gives a deformation that allows the placement of spacers,
which keep the structure in a partly deformed state. The internal stresses that
are induced by these spacers are essential for actuation by means of the shape
memory effect. A schematic illustration of part of the structure together with the
spacers is shown in Figure 3.1. The fabrication procedure based on laser cutting
is illustrated in Figure 3.2.
Finally, electrodes are attached to specific locations on the structure, for example by wrapping it with a thin flexible substrate with integrated wiring. It
is even conceivable to integrate the spacers in this wrapping, thereby combining
functions of wrapping fixation and actuator prestraining. By applying voltages to
34
MODELING OF A SHAPE MEMORY ALLOY ACTIVE CATHETER
3.2
Figure 3.1: Illustration of a segment of the proposed active catheter concept based
on a small-diameter SMA tube patterned by laser cutting. The triangles schematically
represent spacers. Tube wall thickness is not shown in the figure.
Figure 3.2: Proposed fabrication of the new active catheter concept using laser cutting
from a small diameter SMA tube (a). After laser cutting (b–c), the structure is stretched
(d) and spacers are applied (e), in order to generate internal stresses. A full color version
is given by Figure G.2 on page 250.
3.2
ACTIVE CATHETER
35
the electrodes, electrical currents are generated in the structure, which produce
heat due to the Joule effect. This generated heat leads to a local change in temperature, that locally influences the constitutive behavior of the SMA. This results
in a redistribution of strains and a different equilibrium configuration. Given the
right design, an active catheter can be constructed in this way, that bends locally
under the influence of an applied voltage.
Although many other interesting sub-problems of active catheter design can
be identified (e.g. sensor integration, control, isolation), the focus in this work is
primarily on the basic structure of this conceptual active catheter. Note that the
present design no longer consists of many individual parts with a single function
that can be considered separately. Instead, many functions are integrated in a
single structure, and the electrical, thermal and mechanical properties of the material combined with the specific geometry all simultaneously determine the total
performance. Design of such a structure using intuition alone is very difficult,
because of its integrated nature. Therefore, effective model-based design methods are essential for the development and realization of this concept. Results of
finite element calculations, as presented in Section 3.4.4, will further illustrate the
functioning of this design concept.
3.2.2
Material selection
Not all shape memory alloys are suited for use in active catheter applications
as considered in this chapter. Since the catheter is inserted into blood vessels,
an important restriction is that only a limited temperature range can be used,
because most tissues are damaged by extended exposure to elevated temperatures.
An exact upper limit on the working temperature is difficult to specify, because
the resulting damage depends on many factors, such as the amount of active
convection provided by the blood flow, the type of tissue, and the duration and
power of the exposure. A certain amount of local tissue degradation can also be
tolerated, since due to the self-healing capabilities of the body it will not have
any long-term effect.
In literature, various maximum tolerable temperatures are mentioned for the
situation without active convection. Most publications are concerned with burning of skin tissue. Because the outer layer of skin, the epidermis, is composed
of the same so-called squamous epithelial cells as the lining of blood vessels, it
is assumed here, that these published findings are relevant for an assessment of
blood vessel wall damage by high temperatures as well. Herzog et al. (1989) give
52◦ C as the contact temperature above which burns occur. Ng and Chua (2002a)
mention 44◦ C as a starting temperature at which tissue degradation occurs with
increasing severity. But to place this remark in some perspective, the severity
of the damage should be considered as well. In another paper (Ng and Chua,
2002b), Ng and Chua also compare five published damage models. From these,
it can be derived that for tissues held at 50◦ C, over 7 minutes are required to
36
MODELING OF A SHAPE MEMORY ALLOY ACTIVE CATHETER
3.3
inflict a first degree burn. A first degree burn causes swelling and redness of skin,
and is comparable to a sunburn. It heals within a few days. At 49◦ C, this period
required for a first degree burn is over 8 days, whereas at 60◦ C, with some variation, the models predict times around half a second. Based on these indications,
in the present work an upper limit on the local temperature of the active catheter
structure of 49◦ C (322 K) is used. Note that in these considerations, the effect of
active convection through blood flow is not accounted for. This makes that this
upper working temperature is still a conservative temperature limit.
The lower operating temperature limit of the catheter is imposed by the body
temperature. This gives a temperature range of 37 to 49◦ C (310 to 322 K), which
is too small for most shape memory alloys to be useful. The reason is that most
SMAs exhibit hysteresis, meaning that the transformation temperatures for the
forward and reverse transformation are different. Typically this hysteresis is 30
K, often even more. In practice this would mean that even when starting from
37◦ C a forward transformation still might be possible upon heating, a reverse
transformation would not occur when the material returns to body temperature.
This behavior is clearly not desirable for an active catheter.
A solution for this problem is provided by the R-phase transformation, which
is a specific type of martensite transformation that occurs in certain nickel-rich
Ni-Ti alloys. It is characterized by a very low hysteresis (typically 2 K) and
has excellent cyclic stability (Tobushi et al., 1996) and good biocompatibility
(Duerig, 2002). However, the standard martensite/austenite transformation in
Ni-Ti typically allows for 5 to 7% transformation strain (for a few cycles). This
R-phase transformation in contrast only offers approximately 0.8%. Because the
transformation strain is not very large in the R-phase case, it is important to
exploit it as much as possible by carefully designing the structure of the actuator.
This is possible by adopting a model-based design approach as proposed in this
chapter. A key component in a computation model for SMA structures is the
material model, and computational modeling of the R-phase transformation is
the subject of the next section.
3.3
Shape memory alloy constitutive modeling
The formulation of a new constitutive model for R-phase transformation pseudoelasticity in Ni-Ti is discussed here. First, the one-dimensional case is considered.
Subsequently, the generalization of this model to the three-dimensional and plane
stress case is presented.
3.3.1
One-dimensional case
For the intended active catheter and other in vivo biomedical applications, the
Ti-55.3wt%Ni alloy used in an experimental study by Tobushi et al. (1992), which
exhibits the R-phase/austenite transformation, is selected as a suitable material.
3.3
SHAPE MEMORY ALLOY CONSTITUTIVE MODELING
37
Stress-strain curves from those experiments are shown in Figure 3.3. The focus
will be particularly on the pseudo-elastic behavior observed for temperatures between 328 and 343 K, shown in Figure 3.3 by thick solid lines. The loading and
unloading stress-strain curves in this temperature range are very similar, which
allows approximating them by a single curve, i.e. neglecting the hysteresis. This
temperature range is too high for in vivo applications, but because transformation temperatures can be influenced by composition and heat treatment (Sawada
et al., 1993, Todoroki, 1990), shifting them to lower temperatures is possible.
600
Nominal stress, MPa
500
343 K
338 K
400
333 K
328 K
300
200
100
Engineering
strain, %
0
0.2
0.4
0.6
0.8
1.0
Figure 3.3: Stress-strain curves at various temperatures from Tobushi et al. (1992) of
a Ni-Ti alloy. Thin lines are unloading curves.
For a review of R-phase transformation models previously presented in literature, and a more elaborate argumentation for the present one-dimensional model,
see Chapter 2. An approximation used in the formulation of this model is that
in the temperature range of interest, the hysteresis is neglected, since the small
hysteresis of the R-phase transformation allows for this simplification. As a result,
the model no longer requires internal variables and becomes history-independent.
This significantly simplifies sensitivity analysis (Kleiber et al., 1997), which increases the practical applicability of the intended model-based design optimization
approach.
It was found in Chapter 2, that the following piecewise linear relation can
38
MODELING OF A SHAPE MEMORY ALLOY ACTIVE CATHETER
represent the experimental stress-strain curves sufficiently accurate:

: σ0 = EA ε,
 ε ≤ ε1
ε1 < ε ≤ ε2 : σ1 = ET (ε − ε1 ) + σ0 (ε1 ),
σ=

ε > ε2
: σ2 = ER (ε − ε2 ) + σ1 (ε2 ),
3.3
(3.1)
where ε1 , ε2 and ET are linear functions of temperature:
ε1 (T ) = Kε (T − T0 ) + ε0 ,
ε2 (T ) = ε1 (T ) + ∆,
ET (T ) = KE (T − T0 ) + E0 ,
(3.2)
and EA and ER are constant parameters. These relations are only valid for
positive strains, but similar relations can be formulated for negative strains. ε1
and ε2 are the transition strains at which the R-phase transformation starts and
finishes. ET is the apparent Young’s modulus dσ/dε during the phase transition.
By curve fitting, the parameters defining the material behavior can be determined.
Before the actual fitting, a conversion is applied to the experimental data. The
finite element implementation is based on the Green-Lagrange strain ε = εGL and
the Second Piola-Kirchhoff stress σ = σP K2 . The one-dimensional experimental
data is given in terms of nominal stress σnom and engineering strain εeng . The
equivalent Green-Lagrange strain and Second Piola-Kirchhoff stress can be derived from these quantities by:
1 2
ε
+ εeng ,
2 eng
σnom
=
.
1 + εeng
εGL =
(3.3)
σP K2
(3.4)
The converted experimental data is used in the least squares fitting process. The
parameter values that were subsequently obtained are listed in Table 3.1. Note,
in the final simulation, T0 = 310 K is used.
3.3.2
Three-dimensional case
The presented one-dimensional model can be generalized to three dimensions in
various ways, by defining a link between the one- and three-dimensional case.
Table 3.1: Parameter values for the proposed model found by curve fitting to experimental data.
Kε
ε0
E0
EA
2.55×10−4 K−1
5.71×10−4
20.0 GPa
68.9 GPa
KE
∆
T0
ER
619 MPa K−1
54.2×10−4
328 K
45.6 GPa
3.3
SHAPE MEMORY ALLOY CONSTITUTIVE MODELING
39
The intended application of this model in design optimization procedures has
been an important factor that has guided the choices made for the formulation
applied here. The generalization preferably conserves the history-independence
of the model in order to maintain a relatively simple sensitivity analysis. Also
a symmetric tangent operator is preferred, to keep the computational costs low
in comparison to a non-symmetric formulation, and to allow the use of standard
solvers.
A simple and efficient approach has been presented in Chapter 2. There the
link to the one-dimensional model was provided by a scalar strain definition related to strain energy. However, by this choice the fact that the R-phase transformation is pressure-insensitive and the associated transformation strain has no volumetric component (Bhattacharya and Kohn, 1996) was not accounted for. The
model presented here instead uses an effective strain based on the distortional
strain energy. Starting point for the derivation is Hooke’s law in an isotropic
three-dimensional setting. In Mandel notation, it can be expressed as:
  


σxx 
1 1 1 0 0 0




  1 1 1 0 0 0
 σyy 






 
 σ
 1 1 1 0 0 0
zz
√
 + ...


= K 
2σxy 
0 0 0 0 0 0






√
  


0 0 0 0 0 0
2σyz 




√


0 0 0 0 0 0
2σzx
(3.5)

 2
 
ε
− 13 − 13 0 0 0
xx


3


2



− 1
εyy 
− 13 0 0 0


3
 13






1
2
−


ε
−3
0 0 0 √ zz
3
3

2G 
  2εxy  ,
0
0 1 0 0
 0

 
√


 0
0
0 0 1 0 
2ε


yz


√

0
0
0 0 0 1
2εzx
which is equivalent to, in short:
σ = (KK + 2GG)ε.
(3.6)
The strain energy Πε is associated with the tensor product of stress and strain.
In vector notation, this becomes
1
1
1
Πε ∼
= σ T ε = εT (KK + 2GG)ε = KεT Kε + GεT Gε.
2
2
2
(3.7)
The expression εT Gε is associated with the distortional part of the strain energy,
εT Kε with the volumetric part. Based on this, a distortion energy related effective
strain measure is defined as
2
ε2e = εT Gε.
(3.8)
3
The factor 2/3 is included to make the energy-conjugated effective stress measure
equal to the Von Mises stress, which is convenient. The scalar effective strain εe
40
MODELING OF A SHAPE MEMORY ALLOY ACTIVE CATHETER
3.3
defined in this way is an invariant of the strain tensor, which means its value does
not depend on the coordinate system used to describe the strain components.
This is an important requirement for a meaningful effective strain definition.
In order to extend Hooke’s law to include the isochoric transformation strain
εtr , it is assumed that the transformation strain, which is a purely distortional
strain (as it does not have a volumetric component), is proportional to the distortional part of the elastic strain. This assumption is plausible as it minimizes
the total strain energy. The resulting stress-strain relation is given by:
σ = (KK + 2GG)(ε − εtr )
where
εtr = βGε.
(3.9)
The symbol β is introduced here as a scaling factor. Accounting for the transformation strain εtr in this formulation is equivalent to allowing the shear modulus
G of the material to change:
σ = (KK + 2GG)(ε − κGε) = (KK + 2ĜG)ε.
(3.10)
Here Ĝ = G(1−β), and use has been made of the fact that KG = 0 and GG = G.
In the temperature range of interest, the material is completely in the austenite phase when unloaded. Until the transformation begins, the behavior of the
material during loading can be described exactly by Hooke’s law, by just taking
Ĝ = G. As the transformation proceeds, the transformation strain is accounted
for by adjusting this “effective” shear modulus Ĝ. This formulation makes that
the modification with respect to a linear elastic model is minimal. From this point
onward this effective shear modulus Ĝ is used, and in order not to unnecessarily
clutter the notation, the caret is omitted.
To complete the formulation of the stress-strain relation, the way the effective
shear modulus depends on the effective strain has to be specified. This is done
using the previously described piecewise linear one-dimensional stress-strain relation (Equation 3.1), which provides a relation σxx = f (εxx ). In case of a tensile
test, it can be derived that the proposed three-dimensional formulation yields:
εyy = εzz =
2G − 3K
εxx ,
2G + 6K
3K
G + 3K
εxx ⇒ εxx =
εe ,
G + 3K
3K
9GK
=
εxx = 3Gεe .
G + 3K
(3.11)
εe =
(3.12)
σxx
(3.13)
The last expression can be combined with Equation 3.1, which gives an equation
from which the relation between the effective shear modulus and the effective
strain can be found:
G + 3K
σxx = f (εxx ) = f
εe = 3Gεe .
(3.14)
3K
3.3
SHAPE MEMORY ALLOY CONSTITUTIVE MODELING
41
At a given temperature, the one-dimensional model gives a linear relationship
between the uniaxial stress and strain component: its general form is σxx =
Aεxx + B. This leads to:
G + 3K
3K
B
A+
A
εe + B = 3Gεe ⇒ G =
.
(3.15)
3K
9K − A
εe
Because the parameters A and B can be derived from the formulation of the onedimensional model, this final expression relates the shear modulus directly to the
effective strain.
To use this model in a finite element implementation, next to the stress-strain
relation also the tangent operator is required. This operator can be found by
differentiating the stress given by Equation 3.10 with respect to the strain. Details
on its derivation can be found in Appendix A. The resulting tangent operator
turns out to be symmetric.
3.3.3
Plane stress case
In the current setting of modeling an active catheter, it is attractive to use shell
elements instead of solid elements. Particularly thin sections that are mainly
loaded in bending can be modeled more efficiently by shells. In order to use this
constitutive model in a thin shell element, a formulation for the plane stress case
has to be derived. Using the general 3-D formulation, it can be derived that in
plane stress, the transverse strain εzz is related to the in-plane strain components
εxx and εyy by:
εzz =
2G − 3K
(εxx + εyy ) = α(εxx + εyy ),
4G + 3K
(3.16)
where the symbol α represents (2G − 3K)/(4G + 3K), to reduce the complexity of the expressions. Note that α depends on G. Using this relation, the
transverse strain component can be eliminated from the stress-strain equation
(Equation 3.5), leading to:

  

1+α 1+α 0
 σxx 
σyy
= K 1 + α 1 + α 0 + . . .
√

0
0
0
2σxy

(3.17)

 
2−α
−(1 + α) 0
 εxx 
2G 
−(1 + α)
2−α
0 √εyy
.


3
0
0
3
2εxy
The same can be done for the effective strain definition, which becomes:
ε2e =
4
4
4 2
(α − α + 1)(ε2xx + ε2yy ) + (2α2 − 2α − 1)εxx εyy + ε2xy .
9
9
3
(3.18)
42
MODELING OF A SHAPE MEMORY ALLOY ACTIVE CATHETER
3.3
This indicates a slight complication: in the 3-D case, the effective strain only
depended on the strain components (Equation 3.8). In the plane stress case,
however, it turns out that by elimination of the transverse strain the effective
strain also has become a function of α, which in turn depends on the shear modulus
G. It turns out, that no convenient explicit expression can be found to express
G as a function of εe , as in the 3-D case. The way the relation to the onedimensional model is constructed is the same as discussed previously for the 3-D
case, so Equation 3.15 still holds. Combining Equation 3.15, Equation 3.18 and
the definition of α gives a complex equation in G, which can be solved numerically
for a given temperature and strain situation. A Newton-iteration scheme is used
to obtain the solution. Convergence is robust and usually requires only 3 to 4
iterations. Details on the equations can be found in Appendix B.
Also in the plane stress case a tangent operator is required in the finite element
implementation. There are two ways to derive it: the first is to take the tangent
operator of the three-dimensional case, and reduce it to a plane stress setting by
eliminating the transverse strain component εzz from the equations. The second
way to derive the tangent operator is to start from the stress-strain relation in the
plane stress case (Equation 3.17) and differentiate it with respect to the strain.
However, as in this case there exists no explicit relation between the shear modulus
and the (effective) strain, use has to be made of implicit differentiation, which
makes this approach slightly more cumbersome than the first one. The details of
these derivations and expressions for the resulting tangent operators can be found
in Appendix C.
Naturally, the resulting tangent operators are the same in both cases, regardless of the derivation. They are symmetric in case of a finite element formulation
based on the stress vector (σxx , σyy , σxy ) and strain vector given by (εxx , εyy , γxy ).
In fact, that is a formulation commonly used in implementations.
Both in the 3-D and the plane stress case, the stress-strain relations have been
checked against the one-dimensional case and the tangent operators have been
verified using global finite differences.
3.3.4
Discussion
The present model is certainly not the first model aimed at describing the constitutive behavior of a shape memory alloy. Neither does the proposed model cover
effects such as the asymmetry of the material behavior in tension and compression,
the dependence on a specific texture or the complex response to non-proportional
loading. Partly this is because the experimental observations presently collected
for these cases are not sufficiently conclusive. But another reason is that the
purpose of this model is not to provide an exact description of the constitutive
behavior in the finest possible detail, or to test certain theories regarding the nature of the material. Instead, the present model is formulated with the intent to
be suited for and useful in an engineering and design optimization context. This
3.3
SHAPE MEMORY ALLOY CONSTITUTIVE MODELING
500
Nominal stress, MPa
43
343 K
338 K
400
Experiment
FEA Results
333 K
328 K
300
200
100
Engineering strain, %
0
0
0.2
0.4
0.6
0.8
1
Figure 3.4: Experimental stress-strain curves at various temperatures (Tobushi et al.,
1992) together with finite element results based on the proposed SMA material model.
is achieved in this case by inclusion of the main characteristics of the R-phase
pseudo-elastic response. The present model provides good correspondence with
one-dimensional experimental data and accounts for the fact that the transformation strain is isochoric. The agreement with one-dimensional experimental data
is illustrated in Figure 3.4, which shows results of a tensile test simulation using
plate elements.
In comparison to the model presented earlier in Chapter 2, the present model
requires iterations in the plane stress case. This is a consequence of the choice to
include the isochoric transformation strain condition. It makes the present model
more accurate, but also more expensive. Having these two models in fact can
offer an advantage in an optimization setting, as it allows a trade-off between cost
and accuracy. This can be exploited in multi-fidelity optimization approaches.
Both models are history-independent, and therefore the sensitivity analysis is
significantly less involved compared to the history-dependent case (Kleiber et al.,
1997). This, together with their relative simplicity and low computational costs,
makes both these models well suited for use in design optimization of adaptive
SMA structures.
The increased complexity of the present model in the plane stress case due
to the need for internal Newton iterations can be reduced in several ways. For
instance, when a slight reduction in accuracy is acceptable, a response surface
method could be used to construct an approximation of the constitutive relations
prior to the actual finite element computation. The shear modulus and effective
44
MODELING OF A SHAPE MEMORY ALLOY ACTIVE CATHETER
3.4
strain can then be evaluated by means of interpolation instead of iterations. It
is however conceivable that such an approximate approach leads to convergence
problems. In that case the interpolated data can be used as a starting point for
the iterative process, which is likely to reduce the number of required iterations.
3.4
Finite element model
3.4.1
Electro-thermo-mechanical simulation
For simulation of active catheter concepts, in this work only steady state configurations are considered. To evaluate the capability of a design to bend, the
equilibrium configuration in full bending is of interest, therefore the dynamics of
the problem are not considered here. In the intended active catheter application,
the speed of operation is also less important compared to the range of motion.
Moreover, considering only steady-state processes keeps the simulation relatively
simple as compared to a full dynamics simulation. With regard to the feasibility of
design optimization, it is strongly preferred to not adopt unnecessary complicated
modeling approaches.
The mechanical properties of SMAs are influenced by temperature and stress
state. In the present design concepts, electrical resistive heating provides a local
heat source. Evaluation of the behavior of an active catheter design therefore
requires an electrical, thermal and mechanical simulation. The electrical simulation provides a dissipation density field which enters the thermal simulation as a
body source term. Subsequently, the thermal simulation results in a temperature
distribution. Local temperature values are used in the mechanical model where
they influence the constitutive relations.
The simulation approach described here qualifies as a sequentially coupled
multi-physics simulation. It might be argued that it is not correct to treat the
present problem as a sequentially coupled one, because the resistivity of the material in the R-phase is approximately 20% higher (Funakubo, 1987) than in the
austenite phase. As the temperature and mechanical loading state of the material
are linked to the phase fraction of the R-phase, this means that when using the
material itself via Joule heating, the electrical problem is affected by the other
problems as well. Solution of the full problem then requires a fully coupled analysis.
However, in the present situation it has been found that the effect of accounting
for this phase-dependent resistivity is not significant, and therefore sequential
treatment of the problem is justified. In order to show this, it is assumed that
the R-phase fraction γ varies from zero to one in the transition region of the
stress-strain curve. In a first approximation a linear relationship is assumed:

: γ = 0,
 εe ≤ ε1
1
ε1 < εe ≤ ε2 : γ = εε2e −ε
(3.19)
−ε1 ,

εe > ε2
: γ = 1.
3.4
FINITE ELEMENT MODEL
45
The dependence of the electrical conductivity KE to γ is expressed by
A
R
KE = (1 − γ)KE
+ γKE
,
(3.20)
A
R
where KE
and KE
represent the electrical conductivity of pure austenite and
R-phase material, respectively.
These relations have been included in the model, and fully coupled finite element simulations have been performed on the design. A staggered solution approach was adopted, consisting of repeated sequentially coupled electrical, thermal and mechanical simulations. The electrical conductivities of all elements in a
new cycle were adapted according to the results in the previous cycle. This simple
scheme converged fairly quickly, and it turned out that after 4 cycles a stable solution was reached. The difference between the obtained fully coupled solution and
the sequentially coupled one was very small. The temperatures differed less than
0.5·10−3 K, and the relative difference in the bending radius of the entire catheter
was a few parts per million. Of course, the actual relation between the electrical
conductivity and the phase fraction, temperature and mechanical loading state
might differ in details from the approximations used for this test. However, considering the tiny differences in the results, this test clearly indicates that for the
present concept it is not necessary to account for the phase-dependence of the
electrical conductivity and to adopt a fully coupled solution strategy.
The electrical and thermal problems are both described by a Poisson equation. A standard finite element implementation for this equation has been applied
here, based on a three-noded triangular element. Variations of the field variables
through the thickness of the tube wall are neglected, based on the small thickness
and relatively high conductivity. Isotropic thermal and electrical conductivity has
been used. Both problems are linear, and computationally inexpensive to solve
compared to the mechanical analysis. The heat generated by the resistive heating
enters the thermal simulation as a body source term given by Q = KE ||∇φ||2 ,
where KE is the electrical conductivity and φ is the electrical potential. The temperature field that results from the thermal simulation is used in the mechanical
simulation via the constitutive equations, as discussed in Section 3.3. For the
mechanical simulation six-noded triangular shell elements are used, described by
Van Keulen and Booij (1996). Because a nonlinear material model is used in the
mechanical simulation, numerical integration is used in this shell element, with
ten integration points over the thickness.
3.4.2
Load case, boundary conditions and symmetry
The active catheter design concept used in this chapter is a periodic structure
consisting of many repetitions of the same basic unit, as shown in Figure 3.1.
To limit the computational costs of the simulation it is attractive to exploit this
periodicity in the geometry. For this reason, also a periodic load-case is chosen to
46
MODELING OF A SHAPE MEMORY ALLOY ACTIVE CATHETER
3.4
evaluate the performance of the design. The case that will be considered is full
bending in a single plane, in this case the x, z-plane as indicated in Figure 3.5.
The electrode activation pattern used to accomplish this is shown in Figure 3.5 for
a certain voltage u. The electrodes for which no voltage is indicated remain at a
reference zero potential. With this loading, the structure will bend in the positive
rotation direction around the y-axis, using the right-hand rotation convention.
Assuming the structure can be approximated by a sequence of an infinite number
of basic segments, a symmetry condition applies between each consecutive pair
of segments. In addition, the geometry and loading conditions are completely
symmetric with respect to the x, z-plane. Therefore only a relatively small part
of the whole structure, highlighted in blue in Figure 3.5, has to be used in the
simulations.
Figure 3.5: Segmentation, applied voltage pattern and coordinate system. A full color
version is given by Figure G.3 on page 250.
In the electrical and thermal simulation the mechanical deformation is not
accounted for, since the strains are too small to affect the outcome of these simulations. The application of symmetry boundary conditions therefore is straight-
3.4
FINITE ELEMENT MODEL
47
forward. In the thermal case only convective boundary conditions are applied.
In the mechanical case, a load is applied to the structure by the spacers. These
spacers are implemented by means of truss elements under initial compression (cf.
Figure 3.1).
An aspect that also requires some discussion is the implementation of the
symmetry conditions in the mechanical case. The symmetry with respect to the
x, z-plane is not difficult to enforce, but the symmetry planes between segments
are more involved to deal with. The segment that will be analyzed is chosen
to have one plane of symmetry parallel to the x, y-plane, which is easily translated into symmetry boundary conditions. However, the other symmetry plane is
only initially parallel to the x, y-plane, but will rotate and shift as the structure
deforms.
In this case, the symmetry plane is free to rotate around the y-axis, and to
translate in x- and z-directions. Two points are sufficient to define this plane,
provided these points do not differ only in their y-coordinates. In the derivation
of the constraint equation to keep nodes on this symmetry plane, it is convenient
to select two nodes as master nodes. These master nodes are considered to be the
nodes that define the plane, and will be denoted by M 1 and M 2. A third node,
the slave node, will be denoted by S. The described situation is illustrated in
Figure 3.6. Since the y-coordinates of points are not relevant for this discussion,
locations of nodes are given by e.g. M 1 = (M 1x , M 1z ).
Figure 3.6: Definition of the symmetry plane and the distance d.
Keeping the slave node S on the plane defined by M 1 and M 2 is equivalent
to the condition that the distance of S to the plane equals zero. This distance d
is given by:
d = n · (S − M 1) = n · (S − M 2),
(3.21)
as illustrated in Figure 3.6. The normal vector of the plane n is given by:
1
M 2z − M 1z
.
(3.22)
n= p
(M 2x − M 1x )2 + (M 2z − M 1z )2 M 1x − M 2x
48
MODELING OF A SHAPE MEMORY ALLOY ACTIVE CATHETER
3.4
Evaluation of Equation 3.21 yields, in both cases:
d=
M 2z Sx − M 1z Sx + M 1z M 2x + M 1x Sz − M 2x Sz − M 1x M 2z
p
.
(M 2x − M 1x )2 + (M 2z − M 1z )2
(3.23)
The constraint equation that has to be satisfied for nodes on the symmetry
plane now simply reads h(M 1, M 2, S) = d = 0. For the implementation of this
constraint, discussed briefly in the next subsection, its Jacobian and Hessian are
needed as well. Their derivation is treated in Appendix D.
3.4.3
Solution process
The simulation aspects regarding the electro-thermo-mechanical coupling have
been addressed in Section 3.4.1. Here the mechanical analysis itself is considered.
The mechanical problem involves geometrical nonlinearities, physical nonlinearities and nonlinear constraints. Before discussing how the nonlinear constraints
are accounted for, first an overview is given of the solution process itself.
Because the intent is to use the active catheter simulations in a design optimization setting, the solution process should be robust and should not require
intervention from the user. For this reason, an incremental-iterative approach is
used, with a simple adaptive algorithm (Crisfield, 1991) to determine the best
increment size. The new increment size is determined after every increment by
the following rule:
r
ND
k+1
k
∆µ
= ∆µ
.
(3.24)
NA
Here ∆µ represents the increment size used to scale loads, initial strains and prescribed displacements. ND is the desired number of iterations per increment, and
NA is the actual number of iterations used in the previous increment. Starting
cautiously with a small increment setting in combination with this adaptive approach gives a quite robust solution strategy. In order to increase the robustness
even more, the possibility to automatically restart failed steps using half the increment size has been implemented as well. Extensive testing has shown that
generally this strategy leads to satisfactory results.
To complete this discussion on the solution process, the implementation of the
symmetry constraint has to be addressed. The symmetry constraint introduces a
number of nonlinear constraint relations between the degrees of freedom (DOFs).
Several approaches (Cheung et al., 1996, Zienkiewicz and Taylor, 2000) exist to
deal with these in a finite element setting:
1. Master-slave elimination, in which the constraint equations are used to eliminate slave DOFs out of the system equations. Master DOFs have to be
chosen such that a consistent set of equations remains. This technique is
not well suited for nonlinear cases.
3.4
FINITE ELEMENT MODEL
49
2. Penalty formulation, where the constraint is enforced to some degree by
penalizing the deviation with some penalty factor. It can be compared
to adding stiffness to the system, which resists constraint violations. Constraints are therefore not satisfied exactly. Higher penalty values give greater
accuracy, but lead to an increasing ill-conditioning of the system matrix.
3. Lagrange multiplier method. Constraint equations are included by defining
an augmented unconstrained problem with the same solution as the original
constrained problem. In this process, new variables, so-called Lagrange
multipliers, are introduced in the system. In contrast to the penalty method,
this approach can be compared to adding forces that enforce the constraints.
An exact solution can be obtained. A drawback of this approach is that the
system matrix of the augmented problem is no longer positive definite.
4. Augmented Lagrange multiplier method. This is basically a combination of
the penalty formulation and the Lagrange multiplier method. Due to the
addition of penalization terms the system matrix remains positive definite.
Penalty factors do not have to be very high, since the accuracy is obtained
by iteratively solving the Lagrange multipliers, and an exact solution can
be obtained. Several multiplier update schemes are proposed, but generally
quadratic convergence of the Newton method is lost.
After extensive testing of Methods 2 to 4, it was found that the augmented
Lagrange multiplier method generally gives the most reliable results. The solution
process is not very sensitive to the penalty factor used (in contrast to the penalty
method) and results proved accurate. The fact that the system matrix is no
longer positive definite in the Lagrange multiplier method led to problems with
the Newton process used to solve the nonlinear equations, which is explained
by the fact that convergence of Newton iterations is no longer guaranteed when
positive definiteness is lost (Silva et al., 2001).
The augmented Lagrange multiplier approach has been implemented using a
specially formulated constraint element. The element stiffness matrix ke of this
constraint element is given by:
ke = p
d2 h
dh dh T
+ (ph(U ) + λ)
,
dU dU
dU 2
(3.25)
and its contribution to the load vector f e reads:
f e = − (ph(U ) + λ)
dh
.
dU
(3.26)
In these equations h represents the constraint, U is the vector of nodal displacements, p is a penalty factor and λ is the Lagrange multiplier variable associated
with this constraint. The following conventional update rule is used:
λk+1 = λk + ph(U )
(3.27)
50
MODELING OF A SHAPE MEMORY ALLOY ACTIVE CATHETER
3.4
The multipliers are updated after every iteration. An alternative is to alternate
multiplier updates and iterations with fixed multipliers until convergence, but
generally that approach requires significantly more iterations. When starting
a new increment, the Lagrange multipliers were incremented as well, by linear
extrapolation from previous values. This proved to improve the convergence of
the entire process.
3.4.4
Results
Finite element simulations were carried out for an SMA active catheter design
model consisting of ca. 3500 elements. The input voltage was scaled to obey the
maximum temperature limit. In the mechanical analysis, an initial increment size
of 1% was used, in combination with the adaptive increment scaling algorithm.
Physical constants and values of other parameters used in the simulation are listed
in Table 3.2. Using a desired number of increments equal to five, the final solution
was obtained after 14 increments. The augmented Lagrangian formulation was
used for the symmetry plane constraint, with a penalty factor of 105 and using
continuous multiplier updates. No line searches or restarts were required during
this solution process.
Table 3.2: Physical quantity values and settings used in the finite element simulation.
Quantity
Spacer Young’s modulus
Spacer cross-sectional area
Initial spacer space
Spacer deployed length
Tube wall thickness
Tube diameter
Electrical conductivity
Applied voltage
Thermal conductivity
Convection coefficient
Ambient temperature (T0 )
Maximum operating temperature
Value
10 GPa
1.0 ·10−4 mm2
25.4 µm
50.8 µm
45 µm
1.50 mm
1.25·106 Sm-1
22 mV
21 Wm-1 K-1
2·103 Wm-2 K-1
310 K
322 K
Some interesting results of the analysis are collected in Figure 3.7. In Figure 3.7-a the temperature distribution is shown, and it can be seen that one of
the thin sections is clearly at a higher temperature than the one opposing it. This
results in a different response to the load introduced by the spacers, as is shown
in the deformed configuration in Figure 3.7-b. Without spacers, the bottom of
the depicted segment in the unloaded configuration would be at z = 0. The top
of the segment is supported in the x, y-plane and remains horizontal. But due
3.4
FINITE ELEMENT MODEL
51
to the non-uniform temperature distribution the bottom of the segment is clearly
turned in clockwise direction. The symmetry plane positioned at the bottom is
visualized in Figure 3.7-b by a continuous line. Note that the effective strains,
plotted on the deformed configuration in the same figure, are all below 1%. This
means that they are within the range of validity of the present material model.
But in addition, the highest values of the effective strain are very localized. Their
contribution to the total deformation is therefore not very significant, and overstraining at isolated locations can be considered tolerable without affecting the
validity of the overall result.
The deformation of the analyzed segment results in a total bending radius for
this active catheter design of 35.2 mm, which is also shown in the composition of
deformed segments in Figure 3.7-c. A close-up view of a section of the catheter
in bent configuration is shown in Figure 3.8. The power consumption of a single
segment is equal to 2.2 mW, and for the 90◦ turned catheter section shown in
Figure 3.7-c the total power consumption is ca. 0.2 W. To give an indication of the
effect of this power consumption, which is effectively equal to the generated heat,
the following example can be considered. Given the volume of blood present in a
vessel six times the diameter of the catheter and as long as the catheter section
Figure 3.7: Results of the finite element analysis. a) Temperature distribution, b)
Deformed geometry and effective strain, c) Composition of catheter in bent configuration.
A full color version is given by Figure G.1 on page 249.
52
MODELING OF A SHAPE MEMORY ALLOY ACTIVE CATHETER
3.4
Figure 3.8: Close-up of a catheter section in bent configuration (dimensions in mm).
required for a 90◦ bend (ca. 5.5 cm), and assuming the heat capacitance of blood
is equal to that of water at 310 K (i.e. 4.18 J/(gK)), then the time required to
heat this blood by a single degree with the heat generated in full bending equals
more than 70 seconds. In this calculation the effect of blood flow and conduction
through the vessel wall has been ignored, which even more strongly shows that
the present heat generation does not threaten the health of the patient.
The performance of the present design is promising, but at the same time it
leaves room for further improvements. The bending radius of ca. 3.5 cm might
be sufficient for most applications, but in other cases a smaller bending radius
might be preferred. The preceding discussion on peak effective strains suggests
that the present design concept still has some unused potential. And even without
locally loading the material beyond the limits of the present material model, the
deformations could possibly be increased by local shape adjustments that reduce
the peak strains. Next to that, it is conceivable that different choices for the
general shape of the segments, the tube diameter, wall thickness and the spacer
dimensions have a combined effect such that the rotation per segment increases
significantly, resulting in a smaller bending radius and therefore an even more
agile active catheter. Potentially when bending radii become sufficiently small, it
might even become possible to actuate the catheter in a kind of crawling motion,
which could be used to propel it through the vessels, instead of advancing it
by pushing from the remote insertion point. The prevention of buckling during
insertion by pushing is the main reason why conventional catheters have to be
fairly stiff. It would certainly be interesting to see whether other methods for
catheter placement are feasible.
A second aspect to consider when exploring alternative or modified designs
is reduction of the diameter of the catheter, as the 1.5 mm diameter used in the
current design might be too large in some cases. Yet another objective for design
improvement could be to reduce the sensitivity of the catheter performance to
factors that are relatively uncertain or difficult to control, such as manufacturing
inaccuracies, thermal convection coefficients or various aspects of the material
behavior.
3.5
CONCLUSIONS
53
The results presented in this section demonstrate the basic functionality of the
current design concept. They show that a respectable degree of steerability can
be achieved by an R-phase transformation shape memory alloy, and that accompanying operating temperature and heat generation do not pose a serious threat
to the patient’s health. But next to that, these results also indicate the great
potential of model-based design and the opportunities for design optimization.
The present model can be used to evaluate the performance of every conceivable
alternative active catheter design concept, and the kind of structures that can be
explored in this way are only limited by the imagination of the designer.
3.5
Conclusions
Because it serves as a challenging and typical example of a complex SMA adaptive
structure, the design of an active catheter is taken as the subject of this work. A
new active catheter concept has been proposed, aimed to reduce the number of individual parts, to improve its reliability and simplify its assembly. The limitations
imposed on the device by the in vivo operating conditions have been considered,
and a specific shape memory alloy has been selected that is able to operate in a
narrow temperature range. For this R-phase transformation shape memory alloy
a new constitutive model has been presented. This model combines a sufficiently
accurate description of the experimentally observed behavior with a good suitability for design optimization, through the fact that its history-independence allows
for efficient computation of design sensitivities. This suitability for optimization
is also the motivation for the efforts to reduce the computational cost of the finite
element model, by exploiting symmetry as much as possible. For this reason also a
special constraint element to implement a rotating symmetry plane condition has
been developed. The performance of the present active catheter design has been
evaluated by the finite element model, and the results confirmed its potential.
However, this work does not only apply to catheters. The finite element modeling concept and the constitutive model can be used to model many other shape
memory alloy adaptive structures as well. The selected material is very well
suited for in vivo biomedical applications, where the continuing trend toward less
invasive procedures stimulates the development of enhanced and miniaturized instrumentation. But also outside the medical field many (potential) shape memory
alloy applications can be identified, and also for those the modeling and design
approaches presented in this chapter can be of great use.
Returning to the active catheter design case, it is expected to continue to
serve as a fruitful example to develop and study design optimization techniques
for shape memory alloy adaptive structures. A number of directions for further
improvement of the present active catheter design have been suggested at the
discussion of the finite element results. Of course, it is possible to attempt to
achieve these improvements by a process of repeated modification and evaluation
of many designs. However, it is much more efficient to adopt a more systematic
54
MODELING OF A SHAPE MEMORY ALLOY ACTIVE CATHETER
3.5
approach, which is provided by structural design optimization techniques. The
present model has been constructed with practical use in an optimization setting
in mind, and is particularly suited for further exploration of this topic.
Based on: Langelaar, M. and van Keulen, F (2004a). Design optimization of shape
memory alloy structures. In 10th AIAA/ISSMO Multidisciplinary Analysis and
Optimization Conference, Albany, NY.
Chapter
4
Shape Optimization of an SMA
Gripper
Due to their unique properties, shape memory alloys (SMAs) are well suited for a wide
variety of (micro-)actuation applications. However, the complexity of their constitutive
behavior complicates the design process. Much effort has therefore been spent on formulating mathematical models to describe SMA phenomena and analyze SMA structures.
A few researchers have actually tried to use this knowledge in the design process by combining SMA models with optimization techniques. However, only cases with significant
limitations have been published: some are restricted to one-dimensional wire models,
others use assumed optimality criteria that might not always yield the optimal design.
This chapter presents shape optimization of arbitrarily shaped SMA structures based
on the actual material behavior itself. This allows for a clear and unambiguous formulation of the design problem, and provides a structured approach for the design of shape
memory alloy structures. A novel material model is used for this purpose. This model is
specifically tailored for transformation pseudoelasticity based on the R-phase in Ni-Ti,
a popular shape memory alloy. To illustrate the effectiveness of the proposed approach,
a shape optimization study of a miniature gripper is discussed.
4.1
Introduction
Shape memory alloys (SMAs) are materials which exhibit a diffusionless solidstate phase transformation. This transformation is associated with a certain
transformation strain that can be used for actuation. Local stress state, temperature and sometimes also magnetic field affects the transformation and therefore
55
56
SHAPE OPTIMIZATION OF AN SMA GRIPPER
4.2
these quantities can be used to control the behavior. The stresses or strains that
can be obtained from SMAs are relatively large as compared to other actuator
materials, particularly in nickel-titanium alloys, and this makes these materials
very interesting for many applications. More information on SMAs can be found
in Duerig et al. (1990) and Otsuka and Wayman (1998), among others.
An important difficulty with shape memory alloys is the fact that it is hard
to design effective devices due to the complex behavior of these materials. Computational modeling of the constitutive behavior has received a lot of attention,
as clearly the ability to analyze models of SMA structures helps to understand
and improve their design. Yet, the combination of this analysis capability with
optimization procedures leads to a much more powerful and efficient design tool.
Some researchers have already applied optimization in various SMA design problems, but to the authors’ knowledge never before has a formal design optimization
method been applied to a generic finite element (FE) model based on the actual
SMA constitutive behavior. A reason for this might be that existing constitutive
models suited for finite element analysis are often too computationally intensive or
complex for use in an optimization setting. However, recently, constitutive models
have been developed with specifically the suitability for use in design optimization
in mind (Langelaar and Van Keulen, 2004b,c) (Chapter 2 and Chapter 3 of this
thesis). Not only the agreement with experimental observations, but also computational efficiency and the possibility to efficiently compute design sensitivities
has been considered. These models therefore enable a generic design optimization
procedure of shape memory alloy structures. In this chapter, such a procedure is
explored and presented for the first time.
The structure of this chapter is as follows: first, in Section 4.2 an overview
is given of research related to design optimization of SMA structures and the
approach presented in this chapter is discussed. The constitutive model for the
shape memory alloy material used in the finite element analysis is the subject of
Section 4.3. This is followed by Section 4.4 in which the design concept, parameterization, modeling and the formulation of the design optimization showcase of
a miniature SMA gripper is presented. Results of the optimization are the topic
of Section 4.5. Finally, conclusions and future directions are given in Section 4.6.
4.2
Design optimization of SMA structures
In recent literature, there have been a few publications related to the formal design
optimization of SMA structures. Some of these limit themselves to structures
based on wires (Troisfontaine et al., 1999) or composites with embedded wires
(Birman, 1996). A thorough optimization study based on an analytical model of
a flexural actuator with corrugated core has been presented by Lu et al. (2001).
However, this analytical approach only applies to a limited set of design problems
where the geometry and loading pattern of the structure remain relatively simple.
4.2
DESIGN OPTIMIZATION OF SMA STRUCTURES
57
Research focusing on more general structures is described in the publications
by Kohl and coworkers on microvalves (Kohl et al., 2000a, Skrobanek et al., 1997),
linear actuators (Kohl and Skrobanek, 1998) and grippers (Kohl et al., 2000b).
The approach to the optimal design of SMA structures in these papers, although
it is not elaborated in great detail, is based on the assumption that the optimal
design has a homogeneous stress distribution. This is motivated by the idea that
peak stresses will lead to undesirable degradation of shape memory properties,
and that the fraction of the material participating in the transformation depends
on the stress level. A design with homogeneous stresses is therefore assumed
to offer the highest resistance to fatigue, while achieving the maximum material
participation for actuation.
Morgan and Friend show that stress concentrations can indeed accelerate the
degradation of SMA structures (Morgan and Friend, 2001), although this is certainly not the only factor in the complicated mechanism of degradation of shape
memory alloy properties. Although fatigue resistance is important, it does seem
more logical to take the actual performance of a device as the objective for the
design optimization, and take the fatigue resistance into account as a constraint,
instead of optimizing the fatigue resistance alone.
Skrobanek et al. (1997) consider a microvalve that is actuated by bending
beams laser-cut out of a thin sheet. The width of each beam varies along its
length, and the profile used has been found analytically by demanding a constant stress for a given load pattern. Kohl et al. (2000a,b), Kohl and Skrobanek
(1998) use an heuristic peak stress reduction algorithm developed by Mattheck
and Burkhardt (Mattheck and Burkhardt, 1990), which basically reduces notch
stresses by simulating the growth of trees. The same algorithm is used by Fischer
et al. (1999) in the design of a Ni-Ti flexible endoscope tip. In all these works, no
specific details are given on the optimization procedure and the material model
used, but the lack of additional information seems to indicate the use of standard
linear elasticity.
In this chapter, we propose and demonstrate an alternative approach, inspired
by methods used in the field of structural optimization. In the proposed method,
the performance of a design is analyzed by a parameterized finite element model.
This model uses a newly developed constitutive law, that describes the SMA material behavior. The model is combined with an optimization algorithm, in this
case the multi-point approximation method (MAM, Toropov et al., 1993a), in
order to iterate towards the optimal design. An advantage of this approach is
that it is very versatile, since the use of finite element modeling allows it to deal
with a great variety of designs, unlike the mentioned approaches based on analytical models. Another advantage is that the optimization problem is formulated
based on the actual performance of the structure, without the need for additional
assumptions, unlike e.g. the more heuristic methods based on assumed optimality criteria reported in literature. This makes this optimization-based approach
transparent, versatile and unambiguous. Finally, a clear advantage is that also
additional constraints can be added easily.
58
SHAPE OPTIMIZATION OF AN SMA GRIPPER
4.3
By its nature, the finite element model used to analyze the design is rather
expensive in terms of computation time, and since in this case it involves highly
nonlinear models. Moreover, the finite error remaining after the iterative solution
process used for these models can lead to numerical noise in the responses, and
this can hamper the optimization process. In this setting, the multi-point approximation method is well suited to perform the design optimization. For recent
developments in the multi-point approximation method, the interested reader is
referred to Van Keulen and Toropov (1997a) and Toropov et al. (1999a). The
basic concept of this method is to perform the optimization by solving a sequence
of sub-problems. A certain number of design points are generated and evaluated
using the analysis model, and with these points a response surface is constructed,
i.e. an analytical model is fitted to the generated points. The optimal point on
this response surface in a certain trust region is located using another optimization
algorithm: since the response surface is an analytical expression, this optimization sub-problem can be solved with little computational effort compared to the
expensive analysis model. Next to that, the conversion to a response surface also
can average out the noise in the responses. In order to check the quality of the
response surface the optimal point found on the response surface is evaluated
with the analysis model, and based on the resulting degree of correspondence the
boundaries of the trust region are adjusted. This process is repeated until convergence, which can be defined by the degree of improvement, the amount of change
in the design point or the size of the trust region.
4.3
Constitutive modeling
The focus in this chapter is on the pseudoelastic behavior in Ni-Ti due to the Rphase/austenite transformation. This is motivated by its unique small hysteresis
and high fatigue resistance, which makes this an interesting material to use in
(micro-)actuators (Otsuka and Wayman, 1998). Also the limited temperature
range in which the transformation takes place makes it feasible to use this material
in actuated devices for medical applications, which offer interesting opportunities
for SMA technology.
The constitutive model used in the finite element computations for this chapter to describe the SMA material behavior is a refined version of a hyperelastic
model by Langelaar and Van Keulen (2004c) (Chapter 2). This model is suited
for quasistatic analysis of pseudoelastic behavior with a small hysteresis. The
refinements are found in the formulation of the effective strain that now leads to
a pressure-insensitive behavior and the fact that the transformation strain is fully
isochoric, which leads to a closer agreement with the experimental observations.
A full description of this refined model is given by Langelaar and Van Keulen
(2004b) (Chapter 3). In brief, the model is based on the following effective strain
4.3
CONSTITUTIVE MODELING
59
definition related to the distortional strain energy:
ε2e =
2 T
ε Gε,
3
(4.1)
where εe is the effective strain and ε denotes the Green-Lagrange strain in vector
notation. The stress-strain relation reads:
σ = (KK + 2GG)ε.
(4.2)
Here σ is the second Piola-Kirchhoff stress in vector notation, K is the bulk
ratio and G is the effective shear ratio. The matrices G and K in the preceding
equations are constant and given by:




1 1 1 0 0 0
2 −1 −1 0 0 0
1 1 1 0 0 0
−1 2 −1 0 0 0




1 1 1 0 0 0


−1
−1
2
0
0
0
1 
 .


(4.3)
,
K=
G= 3

0
0 3 0 0
0 0 0 0 0 0

0
0 0 0 0 0 0
0
0
0 0 3 0
0 0 0 0 0 0
0
0
0 0 0 3
For a given value of the shear ratio G, this stress-strain relation corresponds
to the well-known Hooke’s Law for linear elasticity. However, in this case, in
order to account for the isochoric transformation strains of the R-phase/austenite
transformation, the value of G is taken as a function of the effective strain εe .
The relation between G and εe is found by using data obtained from experimental
tests. For the current model, data presented by Tobushi et al. (1992) have been
used, in combination with a piecewise linear formulation. Results of finite element
simulations of a tensile test specimen using this model are shown in Figure 4.1,
together with the experimental results. It can be seen that they are in good
agreement. As the model is intended to represent the experimentally observed
behavior, its range of validity is limited to effective strains up to 1%. At higher
effective strain levels, a stress-induced transformation to the martensite phase
might occur, which is not described by the present model.
The tangent operator of this material model is obtained by differentiation of
the stress-strain relation in Equation 4.2, which yields:
dσ
4 dG
= KK + 2GG +
GεεT G.
dε
3εe dεe
(4.4)
It can be seen that this tangent operator is symmetric, since K, G and also the
product Gε(Gε)T are symmetric. The term dG/dεe follows from the relation
between the shear ratio G and the effective strain mentioned earlier. Further
details of these expressions, as well as the formulation of this material model
in the plane stress case, can be found in Langelaar and Van Keulen (2004b)
(Chapter 3).
60
4.4
SHAPE OPTIMIZATION OF AN SMA GRIPPER
500
Nominal stress, MPa
343 K
338 K
400
Experiment
FEA Results
333 K
328 K
300
200
100
Engineering strain, %
0
0
0.2
0.4
0.6
0.8
1
Figure 4.1: Experimental stress-strain curves at various temperatures as determined by
Tobushi et al. (1992) together with results obtained by finite element analyses using the
proposed SMA material model.
Note that the design optimization approach discussed and demonstrated in
this chapter is not restricted to the present material model; in principle any SMA
model that can be implemented in a finite element software package can be used.
However, the practical feasibility of the procedure depends on the computational
effort required for the function evaluations, relative to the available computing
resources. The present model provides robust and reasonably efficient computations, and therefore fulfills this requirement.
4.4
Gripper design case
To demonstrate the effectiveness of the proposed FE-based design optimization
approach, the design of a miniature gripper will be discussed. The conceptual
design of this gripper is shown in Figure 4.2. It consists of an identical top and
bottom arm made of folded Ni-Ti plates. In order to generate internal stresses in
the material, starting from the undeformed configuration in Figure 4.2, the end
of the top plate of the top arm is pinched towards the bottom plate, and similarly
the bottom plate of the bottom arm is pinched towards the top plate of that arm.
In this situation, the equilibrium configuration of each arm can be changed by
either heating the top or bottom plate. Heating the inner plates (i.e. the bottom
plate of the top arm and the top plate of the bottom arm) will cause the tip ends
4.4
GRIPPER DESIGN CASE
61
Figure 4.2: Gripper geometry in undeformed configuration.
to move apart, opening the gripper. Likewise, heating the outer plates will make
them move towards each other, closing the gripper. In the closing configuration,
clamping forces of 100 mN are applied in z-direction at the tips of the gripper.
Because of symmetry, only a quarter of the gripper needs to be modeled: in
this case half the top arm is used. This part together with the parameterization
of the geometry is shown in Figure 4.3. Dimensions in this chapter are given in
millimeters, unless stated otherwise. In the finite element modeling, symmetry
boundary conditions are used in the x,z-plane. The parameters chosen for this
design study are the plate thickness t, the undeformed arm height H, the actuation
plate begin and end width W1 and W2 , and the shape of the actuation plate.
This shape is described by a quadratic B-spline (see e.g. Farin, 2002), and the ycoordinates of the two middle control points are used as design variables: Y1 and
Y2 . Further geometrical details of the miniature gripper are listed in Table 4.1.
In the finite element modeling of this gripper, two cases are considered: in the
first case, it is assumed that the plates used for actuation are heated externally,
Table 4.1: Significant coordinates of B-spline control points and other points defining
the geometry of the miniature gripper.
Point
Control point 1
Control point 3
Control point 5
Tip
x [mm]
0
1
3.5
5.377
y [mm]
W1
Y1
W2
0
Point
Control point 2
Control point 4
Control point 6
Slit end
x [mm]
0.5
3
4
3.8
y [mm]
W1
Y2
W2
0
62
SHAPE OPTIMIZATION OF AN SMA GRIPPER
4.4
Figure 4.3: Gripper design parameterization. Because of symmetry, only a quarter of
the full gripper needs to be considered.
and their temperature is chosen to be 10 degrees above the reference temperature.
As the intention is to use this gripper in biomedical applications, 310 K is chosen
as a reference temperature instead of 328 K shown in Figure 4.1, as in Chapter 3. Geometrical nonlinearities are considered in the simulation, and also the
constitutive model is strongly nonlinear, which makes it necessary to use Newton iterations to find the new equilibrium configuration. An incremental-iterative
approach is used, with rather conservative settings for the increment size. The
reason for using small increments is that the analysis has to finish successfully for
all designs in the design range: next to efficiency, robustness is very important
for models used in optimization. A triangular shell element is used for the mechanical analysis (Van Keulen and Booij, 1996). For every design, a new mesh
is generated automatically based on its geometry using a preprocessor: no mesh
morphing techniques are used. Although this is one of the sources of noise in the
responses of the model, this approach is preferred for its flexibility, ease of use
and robustness.
In the second case, use is made of the Joule effect to heat the plate. This
means that heat is generated internally by sending an electrical current through
the structure by applying a voltage over the end of the plate. The design shown
in Figure 4.2 does physically not allow for individual heating of top or bottom
plates in this way, therefore a narrow slit is added in the middle of each plate in
the direction of the x-axis, as shown in Figure 4.4. The width of the slit equals
200 micrometer. Figure 4.4 also shows the way the voltage is applied, in this
case to the top plate. To simulate the performance of the gripper, an electrical,
thermal and mechanical finite element analysis is needed. Dissipated heat from
4.4
GRIPPER DESIGN CASE
63
Figure 4.4: Gripper geometry in the Joule heated case, in undeformed configuration.
The voltage difference is applied to the top plate in order to close the gripper, or to the
bottom plate in order to open it.
the electrical analysis is used as a heat source in the thermal analysis, and the
resulting temperature distribution is used in the mechanical analysis. Physical
constants used in the simulations are collected in Table 4.2. The input voltage
for every design is adjusted iteratively in order to meet a required maximum
temperature of 320 K (same as in the externally heated case), since on physical
grounds it is likely that this is the most favorable case. But of course, it is also
possible to include the input voltage as a design variable as well, and add the
temperature restriction as a constraint.
The objective for this design study is to maximize the range of motion of the
gripper tips. Therefore, the difference between the z-coordinates of the gripper tip
in open and closed configurations is taken as the objective function. The material
model is limited to a maximum effective strain of 1%, and therefore a constraint
on the effective strain is added in both the open and closed configuration. This
Table 4.2: Physical constants used in the finite element modeling.
Quantity
Electrical conductivity
Thermal conductivity
Thermal convection coefficient
Ambient temperature
Value
1.25 · 106 Sm-1
21 Wm-1 K-1
2.0 · 103 Wm-2 K-1
310 K
64
4.5
SHAPE OPTIMIZATION OF AN SMA GRIPPER
yields the following design optimization problem:
max f (x)
x
subject to:
max (εopen
) ≤ 0.01
e
max (εclosed
) ≤ 0.01
e
xlower
≤ xi ≤ xupper
,
i
i
(4.5)
i = 1...6
Here x is the vector of design variables {t, H, Y1 , Y2 , W1 , W2 } and f (x), the objective function, equals the difference between the z-coordinate of the tip of the
gripper in opened and closed position. The last constraints represent the bounds
used on the design variables. The values for the lower and upper bounds for each
design variable are listed in Table 4.3. To illustrate the effect of the effective strain
constraints on the optimal design, optimization studies have been performed with
and without these constraints.
To shorten the time consumed by the optimization process, a parallel computing approach has been used, as described by Gurav et al. (2003). In the multipoint approximation method, the evaluation of design points for the response
surface can be carried out in parallel quite easily. In this case, a heterogeneous
Linux cluster consisting of 9 dual-processor PCs has been used, yielding 18 computational nodes. To give an indication of the computational effort involved in an
analysis: the mechanical finite element model has about 10,000 degrees of freedom (order of magnitude - the exact number depends on the specific values of
the design variables), and typically 60 Newton iterations are required to perform
a single analysis. Note, both for the open and closed configuration an analysis is
required. The electrical and thermal analyses hardly contribute to the total time,
because they have much fewer degrees of freedom and do not require iterations.
4.5
Results
In total, four design optimization cases have been considered: an unconstrained
and constrained case using external heating, and an unconstrained and constrained case using Joule heating. The multi-point approximation method turned
out to be an effective tool and finished the optimization in 20 to 30 iterations.
The number of function evaluations (i.e. finite element analyses) for a complete
Table 4.3: Bounds used for the design variables.
Variable
Lower bound [mm]
Upper bound [mm]
t
0.05
0.3
H
0.3
2
Y1
0.01
1.5
Y2
0.01
1.5
W1
0.1
1.5
W2
0.1
1.5
4.5
RESULTS
65
1.2
Tip displacement range, mm
Maximum effective strain, %
0.6
1
0.8
0.4
Max. ε (open)
e
Max. εe (closed)
0.6
0.2
Unconstrained, external heating
Constrained, external heating
Unconstrained, Joule heating
Constrained, Joule heating
0
0.4
0.2
Iteration number
Iteration number
−0.2
0
10
20
30
0
0
5
10
15
20
Figure 4.5: Evolution of the objective (left) and constraint values (right) for various
cases.
optimization varied between roughly 700 and 1000, and thanks to the parallel
computing approach the whole procedure took a modest 10 to 15 hours. The
evolution of the objective function, the tip displacement range, throughout the
optimization history is shown in Figure 4.5(a), for all four cases. The evolution
of the constraint values for the Joule heated case are shown in Figure 4.5(b); for
the externally heated case the trend was similar.
The optimal designs that have been found in all four cases are listed in Table 4.4, and the corresponding values of objective and constraints in Table 4.5.
As could be expected, the addition of constraints on the maximum effective strain
Table 4.4: Design variable values at the optimum design for various cases.
t [mm]
H [mm]
Y1 [mm]
Y2 [mm]
W1 [mm]
W2 [mm]
External heating
Unconstrained Constrained
0.08792
0.09601
1.7036
1.2168
1.2965
0.8956
0.0100
0.3177
1.5000
1.4965
1.5000
1.4988
Joule heating
Unconstrained Constrained
0.09182
0.10058
1.6570
1.0941
1.3271
1.0276
1.3972
0.9789
1.4960
1.5000
1.4432
1.4423
66
4.5
SHAPE OPTIMIZATION OF AN SMA GRIPPER
Table 4.5: Objective (displacement range) and constraint values (maximum strain) at
the optimum design for various cases.
Case
Unconstrained, external heating
Constrained, external heating
Unconstrained, Joule heating
Constrained, Joule heating
Displacement
range [mm]
0.6345
0.5453
0.4765
0.3847
Max. strain
(open/closed) [%]
0.9988 / 0.9235
0.9990 / 0.9896
lowers the range of motion of the gripper tip, and also it is seen, that the range
of motion in the Joule heated case is smaller than in the externally heated case.
This can be explained by the fact that in the Joule heated case, the optimal design
somehow has to strike a balance between the shape that gives the best temperature distribution and that with the best mechanical performance in terms of
tip displacement: these two aspects are interrelated. Therefore the final design
will be a compromise. In the externally heated case, the shape of the structure
(a) Externally heated cases
(b) Joule heated cases
Figure 4.6: Optimal designs in the unconstrained (top) and constrained (bottom) case,
for both types of heating.
4.6
CONCLUSION AND OUTLOOK
67
only influences the mechanical aspects, which leaves more freedom to improve the
range of motion.
It is interesting to investigate the optimal designs not only by considering the
numerical values shown in Table 4.4, but also by looking at the actual geometries.
To this end, the optimal structures are presented in Figure 4.6(a) for the externally
heated case, and in Figure 4.6(b) for the Joule heated case. It is seen, that in
both cases the addition of the effective strain constraints leads to a design that is
less high. From the numerical data it can be seen that in addition the plates are
slightly thicker. Another interesting observation is that the shape of the plates is
quite different depending on the heating conditions: in the externally heated case,
locally the width of the plates is reduced strongly, whereas in the Joule heated
case, the width remains much more constant along the length of the plate. This
illustrates the fact that in the latter case a trade-off must be made between the
electrothermal and mechanical properties of the structure.
More insight into the way this gripping device works can be gained by looking
more closely at the analysis results. Most interesting is the Joule heated case, as
it is a genuine multi-disciplinary problem. In Figure 4.7 results of the electrical,
thermal and mechanical analysis are shown for the opened respectively closed
gripper configuration. The electrical potential shows the way the device is operated: a voltage difference is applied over the ends of either the top or bottom
plate. The resulting temperature distributions show that due to the generated
heat, the temperature of the bottom, respectively, top plate is raised. The upper
limit imposed on the temperature is respected. The Von Mises stress distribution
found in the mechanical analysis is shown on the deformed configurations. The
stress distributions differ considerably in opened and closed configurations, but
note that in the latter situation also a 100 mN clamping force is acting on the
structure.
Finally, to show even more clearly how this gripper works, the contours of the
open and closed configurations of the top gripper arm in the constrained Joule
heated case are shown together in side view in Figure 4.8. Due to the changes
in temperature distribution, the mechanical behavior of the material is changed
locally, which leads to the two different equilibrium configurations.
4.6
Conclusion and outlook
In this chapter, structured design optimization of shape memory alloy devices is
presented. Finite element modeling using a novel constitutive model for the SMA
material behavior has been used in combination with a parameterized geometric
model. A formal optimization problem was formulated and the optimization
was carried out using the multi-point approximation method. To the authors
knowledge, this is the first time a formal design optimization approach has been
applied to a general finite element based model of such a device. Its viability and
effectiveness for practical SMA structure design is demonstrated by means of an
68
SHAPE OPTIMIZATION OF AN SMA GRIPPER
(a) Analysis results for the opened
configuration.
4.6
(b) Analysis results for the closed
configuration.
Figure 4.7: Computed electrical potential (top), temperature distribution (middle) and
Von Mises stress distribution on the deformed structure (bottom) for the optimal design
in the constrained Joule heated case, in opened (left) and closed (right) gripper configurations. A full color version is given by Figure G.4 on page 251.
Figure 4.8: Gripper top arm in open (black) and closed (red) position, side view. A
full color version is given by Figure G.5 on page 251.
4.6
CONCLUSION AND OUTLOOK
69
example. It is emphasized that the presented approach is not limited to specific
shapes or material models such as those used in this study. The generality of
this approach is one of its main advantages. Another advantage is the ability to
formulate clear and unambiguous design objectives and constraints based on the
actual performance of the device.
Overall computing time was reduced by the use of parallel computing, and
decreased to a level that makes this approach suitable for practical use. Further
reductions are possible when design sensitivities are available: for example, the
multi-point approximation method would then require less function evaluations
for the construction of a response surface (Van Keulen and Vervenne, 2002). For
this aspect the material model does make a difference: the history-independent
nature of the material model used in this chapter makes that sensitivity analysis
is considerably less involved compared to other, usually history-dependent SMA
models (Kleiber et al., 1997, Langelaar and Van Keulen, 2004c). This means that
sensitivities could be computed with relatively little computational effort, which
makes using sensitivities an attractive option to increase the practical applicability
of the presented approach, particularly in larger design problems. This topic is
explored further in Chapter 7, 8 and 9 of this thesis.
70
SHAPE OPTIMIZATION OF AN SMA GRIPPER
4.6
Based on: Gurav, S.P., Langelaar, M. and van Keulen, F. Cycle-based alternating
anti-optimization combined with nested parallel computing: application to shape
memory alloy microgripper. Computers and Structures, in review.
Chapter
5
Shape Optimization under
Uncertainty
In this chapter, a new method for uncertainty-based design optimization based on an
anti-optimization approach using Bounded-But-Unknown (BBU) uncertainties is studied
on the basis of a practical application. The basic anti-optimization technique looks
at the worst case scenario by finding the worst settings of the uncertainties for each
constraint evaluation separately. This Rigorous anti-optimization technique involves
two-level optimization in which anti-optimization is nested within the main optimization,
making it computationally exhaustive. In the alternative Lombardi-Haftka approach,
anti- and main optimization are carried out alternately avoiding the nested approach,
which is quite efficient. A new Cycle-based Alternating technique based on a similar
idea is studied in this chapter. In this technique, anti-optimization is carried out at the
end of every cycle of the main optimization. The above anti-optimization techniques are
studied and compared on the basis of an illustrative elastically supported beam example.
Additionally, in the present chapter, a nested parallel computing strategy is developed
in order to make the Cycle-based Alternating technique computationally efficient when
a cluster of computers is available for function evaluation. This is particularly essential
in case of practical problems involving expensive function evaluations, e.g., using Finite
Element Analysis. The effectiveness of the Cycle-based Alternating technique combined
with nested parallel computing is demonstrated by application to the uncertainty-based
shape optimization of a shape memory alloy microgripper.
71
72
SHAPE OPTIMIZATION UNDER UNCERTAINTY
5.1
Introduction
5.1.1
Optimization involving bounded-but-unknown uncertainties
5.1
Many practical design optimization tasks involve uncertainties. In case statistical data on uncertainties is available, it can be used to construct statistical
distributions for uncertainties. If such distributions are sufficiently reliable, a
reliability-based design can be obtained by using probabilistic methods, see e.g.
the textbook by Elishakoff (1983). In general, probabilistic methods require an
abundance of experimental data, and even small inaccuracies in the statistical
data can lead to large errors in the computed probability of failure (Elishakoff,
1999). However, in case of practical applications, it often happens that there is
insufficient data available to construct reliable distributions. This particularly
occurs in early stages of a design process. In such situations, the computed reliability of a structure can exhibit large errors. This can be crucial in applications
which are required to perform without failure, or with very small probability of
failure for the entire lifespan, for example, in case of space applications. On the
other hand, the available data, in combination with engineering experience, can
be used to set tolerances or bounds on uncertainties, within which the distribution is unknown, thus identifying uncertainties as Bounded-But-Unknown (BBU)
(Ben-Haim, 1996, Ben-Haim and Elishakoff, 1990)).
The anti-optimization technique described in Elishakoff et al. (1994) tackles
the BBU uncertainties using the worst case approach. This technique involves
vertex checking of the uncertainty domain, in order to obtain the worst response
of the structure. In comparison to other approaches, this technique is computationally very efficient for problems that are monotonic with respect to the uncertainties, however its application is limited to that category of problems. A more
generalized approach, which can handle non-monotonicities, is adopted by Van
Keulen et al. (2001) and Gurav et al. (2002). Here, this generalized or Rigorous anti-optimization is applied to the uncertainty-based design optimization of
a car deck floor of a ferry. However, this technique suffers from the required large
number of expensive function evaluations, due to the underlying two-level nested
optimization. The Enhanced anti-optimization technique, in which sensitivities,
a database and parallel computing are used to improve the computationally efficiency, is studied in Gurav et al. (2005) on the basis of a practical application
related to microsystems. However, the approach still becomes computationally
expensive for an increasing number of design variables and uncertainties.
5.1.2
Cycle-based approaches
The need for and lack of a computationally efficient method to handle uncertainties in design optimization problems has motivated the search for alternative
approaches, which avoid the expensive two-level nested optimization used in the
mentioned techniques. In one such approach, proposed by Lombardi and Haftka
5.1
INTRODUCTION
73
(1998), anti- and main optimization are carried out alternately, thereby avoiding
the nested approach. This approach can converge very rapidly in case of monotonic problems and is quite efficient in terms of the required number of function
evaluations. However, in case of non-monotonic problems, for which the worst
case can fluctuate from design to design, a large number of optimization cycles
may be required to achieve convergence. Inspired by the Lombardi-Haftka technique, a slightly modified approach is adopted in the present chapter. Here, in
the proposed Cycle-based Alternating technique, anti-optimization is carried out
at the end of each cycle of the main optimization, to obtain the worst sets of
uncertainties. These worst sets of uncertainties are subsequently used during the
next cycle of the main optimization. For the initial cycle of the main optimization,
uncertainties are set to zero. In case of fluctuating uncertainties, the Cycle-based
Alternating technique is expected to converge faster than the Lombardi-Haftka
technique, since the worst case is re-evaluated after each cycle, instead of after each complete sub-optimization process. Also in comparison to the previously
proposed Enhanced anti-optimization technique, in case of fluctuating worst cases
this newly proposed approach is expected to be more efficient. Additionally, the
use of response derivatives with respect to uncertainties for estimation of worst
uncertainties during the cycle can improve the convergence of the Cycle-based
Alternating technique.
In many practical situations, the function evaluations of optimization problems involve computationally expensive finite element analysis (FEA). Moreover,
the number of such FEAs required in the uncertainty-based design optimization
can be quite high, depending on the problem at hand, resulting in impractical
computation times. For such problems, it is necessary to perform function evaluations in parallel, e.g. using a cluster of computers. In the present chapter,
a nested parallel computing approach is used in combination with the proposed
cycle-based technique, in order to optimize a shape memory alloy (SMA) microgripper, simulated by computationally expensive FEA. The parallel computing
framework used here is implemented in Python (Lutz, 2001).
The techniques for uncertainty-based optimization discussed above are embedded in a structural optimization setting using the Multipoint Approximation
Method (MAM) (Toropov et al., 1993b, 1999b, Van Keulen and Toropov, 1998).
The different techniques will first be studied and compared on the basis of an
elastically supported beam problem, considered earlier by Lombardi and Haftka
(1998). To demonstrate the ability of the proposed Cycle-based Alternating technique to solve practical problems involving expensive FEA, the uncertainty-based
shape optimization of an SMA microgripper is considered in the current chapter.
In this study, uncertainties affect relevant environmental operating conditions as
well as parameters in the SMA material model. Non-deterministic design optimization of SMA structures has thus far not been reported in the literature. By
means of the considered microgripper example, this study aims to demonstrate
the practical applicability of the proposed technique for this class of applications.
74
5.1.3
SHAPE OPTIMIZATION UNDER UNCERTAINTY
5.2
Outline
This chapter is organized as follows: first, the basic optimization problem formulation, together with a short description of the MAM optimization procedure,
is given in Section 5.2. The BBU description of uncertainties is the subject of
Section 5.3. Subsequently, in Section 5.4, various uncertainty-based design optimization techniques using BBU uncertainties are described. In Section 5.5, the
discussed anti-optimization techniques are studied on the basis of an elastically
supported beam problem. After having verified the newly proposed method on
these test problems, its application in the uncertainty-based design optimization
of an SMA microgripper is described in Section 5.6, followed by the conclusions
in Section 5.7.
5.2
Multipoint Approximation Method
5.2.1
Introduction
In the present chapter the Multipoint Approximation Method (MAM) is used as
a basis for optimization. Often practical applications involve numerical evaluation of response functions. From an optimization point of view, these types of
problems can either suffer from numerical noise or the large computational time
involved. The MAM, which is based on the sequential application of Response
Surface Methodology (Khuri and Cornell, 1996, Myers and Montgomery, 1995),
is well suited for dealing with those difficulties. The interested reader is referred
to the studies by Toropov et al. (1993b, 1999b) and Van Keulen and Toropov
(1997b, 1998). The MAM uses sequential response surface approximations to the
responses, in order to reduce the number of expensive numerical response evaluations. However, it should be noted here, that particularly in combination with
higher-order response surfaces, the method becomes increasingly expensive when
the number of design variables increases.
5.2.2
Optimization problem formulation
Designing a structure implies that a design concept has to be selected, which
subsequently has to be optimized. The latter involves the selection of design
variables, which determine, among other features, the dimensions, shapes and
materials to be used. This set of n design variables is denoted as x, with
x = (x1 . . . xn ).
(5.1)
Throughout this chapter, it is assumed that all design variables are continuous.
The behavior of the structure is described by the response functions, which are
functions of the design variables. These response functions are denoted as f with
f = (f0 . . . fm ),
(5.2)
5.2
75
MULTIPOINT APPROXIMATION METHOD
which may reflect, for example, weight, cost, buckling loads, maximum equivalent stress, or strain levels. Now the optimization problem can be formulated
mathematically as
min fo (x)
x
s.t. fi (x) ≤ 1, i = 1, . . . , m,
Aj ≤ xj ≤ Bj , j = 1, . . . , n.
(5.3)
Here, f0 is the objective function and fi are constraints. The design space is
represented by the upper and lower limits on xj , Aj and Bj , respectively.
The MAM is based on a sequential replacement of the actual optimization
problem, as described by Equation 5.3, by a series of approximate optimization
problems as schematically illustrated in Figure 5.1. The approximate optimization
problem for a cycle p can be formulated as
(p)
min f̃o (x)
x
(p)
s.t. f̃i (x) ≤ 1, i = 1, . . . , m,
(p)
(p)
Aj ≤ xj ≤ Bj , j = 1, . . . , n,
(p)
(p)
Aj ≥ Aj , Bj ≤ Bj .
(5.4)
Here, the response functions are replaced with approximate functions over the
(p)
sub-domain for a cycle. For the current approximate optimization problem, f̃i (x)
are considered as adequate approximations of fi (x) over the sub-domain (p), rep(p)
(p)
resented by the move limits Aj and Bj , see Figure 5.1. It should be noted
Sub-domain
for a cycle (p)
B2
Design space
p
B2
x2
Sub-optimum
p
A2
p
A1
A2 Plan points
A1
x
p
Optimum
B1
1
B1
Figure 5.1: Optimization using the MAM for a problem of two design variables (x1 and
x2 ).
76
5.3
SHAPE OPTIMIZATION UNDER UNCERTAINTY
(0)
(0)
here, that the move limits for the initial cycle (Aj and Bj ), can be chosen
either arbitrarily or based on engineering experience. Many times this can significantly influence the convergence. For example, if the initial domain includes the
optimum, then the optimization can converge quite rapidly.
5.3
Bounded-But-Unknown uncertainty
If the problem at hand is non-deterministic, i.e. there are uncertainties that play a
non-negligible role, the response functions also depend on the uncertainties. The
set of uncertainty variables is denoted by α , with
α = (α1 . . . αu ).
(5.5)
Consequently, the response functions depend on both design variables and uncertainties, hence f (x, α ). In the present chapter, uncertainties are modeled using
D2
D2
Nominal
D1
H
H
D1
l
Di
(a) Ellipsoidal bound for all
uncertainties together.
(b) Simple box bound with
separate bound for each uncertainty.
Di
D
u
i
(c) Simple box bounds on uncertainties: ᾱi are nominal values of uncertainties with lower (αli ) and upper (αu
i ) bounds and ε as the dimension of the space of the feasible uncertainty.
Figure 5.2: Bounds on uncertainties.
the BBU approach. Thus, several bounds can be introduced, each providing a
bound for a group of uncertainty variables or all uncertainty variables simultaneously, as illustrated in Figure 5.2. At the same time, we may want to measure
the amount of uncertainty. Thus, measures for the dimensions of the subspace
containing all possible selections of uncertainty variables are desired. This can be
cast into a mathematical framework as follows. Assuming a set with b bounds,
then a possible or feasible selection of α satisfies, see Van Keulen et al. (2001):
α, ε ) ≤ 0,
Bi (α
for i = 1, . . . , b.
(5.6)
Otherwise the selection of the uncertainty variables α is infeasible. Here, the components of ε are used to specify the dimensions of the space of feasible uncertainty
variables. We will therefore refer to these components as the levels of uncertainty.
5.4
77
UNCERTAINTY-BASED OPTIMIZATION USING ANTI-OPTIMIZATION
In the present chapter, simple box bounds, see Figure 5.2 (b) and (c), are used to
specify uncertainties as
2
(αi − ᾱi ) − ε2 ≤ 0.
(5.7)
This type of bounds generally come from a tolerance specified on a nominal value,
for example due to manufacturing inaccuracies. These bounds can be alternatively
represented in terms of lower (αil ) and upper bounds (αiu ) on uncertainties as
αil
αiu
= ᾱi − ε,
= ᾱi + ε.
(5.8)
Note that the bounds used in this study were chosen based on engineering intuition, rather than a detailed analysis, since the purpose of the present examples
is mainly to illustrate the proposed optimization technique. However, in practical situations where more detailed data is available, the same procedure can be
applied with different bounds.
5.4
Uncertainty-based optimization using Anti-optimization
5.4.1
Anti-optimization
The anti-optimization technique to tackle BBU uncertainties consists of two levels of optimization. The outer level is given by the main optimization, and the
inner level consists of the nested anti-optimization. The main optimization here
is a standard minimization problem, that searches for the best design in the design domain. Anti-optimization is performed for every constraint, in order to
obtain the worst values of constraints for each design within the main optimization. The anti-optimization problem using BBU uncertainties can be formulated
mathematically as:
min fo (x)
x
s.t. fi (x; α ∗i ) ≤ 1,
i = 1, . . . , m,
(5.9)
where α ∗i is the maximizer of
max
αi
s.t.
f∗i (x; α i )
αi , ε ) ≤ 0,
Bj (α
j = 1, . . . , b.
(5.10)
Here, f0 (x) is the objective function and fi (x, α i ) are constraints, whereas
αi , ε ) are bounds on uncertainties. The minimization as defined in EquaBj (α
tion 5.9 will be referred to as the main optimization. Notice that, in general,
the evaluation of the constraints involves, for each set of design variables, one
full anti-optimization for every individual constraint. This anti-optimization is
reflected by Equation 5.10. The anti-optimization technique in the setting of the
78
5.4
SHAPE OPTIMIZATION UNDER UNCERTAINTY
x
fi(x, D )
x2
D2
x1
Main Optimization
D1
Anti-optimization
Figure 5.3: Anti-optimization technique in the MAM setting for a problem of two
design variables (x1 and x2 ) and two uncertainties (α1 and α2 ). The big boxes indicate
the search (sub-)domains. The small open boxes indicate sets of design variables (left) or
uncertainty variables (right) for which function evaluations are carried out. The small
solid boxes indicate solutions of the approximate optimization problems.
MAM is depicted in Figure 5.3. For the applications studied in the present chapter, uncertainties bounded by simple box bounds (see Figure 5.2(b) and (c) and
Equation 5.8) are adopted. Therefore, the constrained maximization problem,
as defined by Equation 5.10, reduces to an unconstrained maximization problem
given by:
max f∗i (x; α i )
αi
(5.11)
s.t. α li ≤ α i ≤ α ui .
The above Rigorous anti-optimization technique can handle large uncertainties
safely. Moreover, it can account for discontinuities, if any. The price paid for
this flexibility is the large computing effort required for the anti-optimization
processes. In case of practical problems involving large numbers of design variables
and uncertainties, anti-optimization can become very computationally expensive.
In order to reduce the total number of expensive numerical response evaluations,
the anti-optimization technique is modified in Gurav et al. (2005) by making use
of database techniques and sensitivities.
In this Enhanced anti-optimization, derivative information, if available, is utilized to decrease the required total number of expensive function evaluations. In
many cases of computational response analysis, gradient information can often be
obtained at a fraction of the computing time as compared to the analysis itself
(De Boer and Van Keulen, 2000a, Van Keulen and De Boer, 1998a, Van Keulen
et al., 2005). This sensitivity information can be used in addition to the function
values to construct Gradient Enhanced Response Surfaces (GERS) (Van Keulen
and Vervenne, 2004, Vervenne and Van Keulen, 2002). This incorporation of sensitivities can improve the quality of the response surfaces. Alternatively, fewer
5.4
UNCERTAINTY-BASED OPTIMIZATION USING ANTI-OPTIMIZATION
79
response evaluations may be required to construct the approximations. Thus,
using derivative information may decrease the total number of expensive function
evaluations and hence may speed up the numerical optimization process. Similarly, a database technique is used to modify the anti-optimization technique in
order to reduce the number of expensive function evaluations required for each
anti-optimization. For this purpose, the worst sets of uncertainties obtained by
anti-optimizations are stored in a database. When there is enough data available
in the database, it is used to generate starting points for the anti-optimizations.
Often this can speed up the anti-optimization processes significantly. Additionally, a parallel computing strategy is combined with anti-optimization in Gurav
et al. (2005) in order to speed up the whole procedure.
5.4.2
Lombardi-Haftka Alternating anti-optimization
Although the Enhanced anti-optimization provides a quite efficient way of handling uncertainties, it becomes increasingly impractical with an increasing number of design variables and uncertainties. Therefore, it is necessary to think of
alternative approaches. In the Lombardi-Haftka approach (Lombardi and Haftka,
1998), nesting of anti-optimization within the main optimization is avoided. In
this method, the main and anti-optimization is carried out alternately as follows:
(Main optimization)
min fo (x)
x
s.t. fi (x; α ∗i ) ≤ 1,
i = 1, . . . , n.
(5.12)
for given worst set of uncertainties α ∗i . These uncertainties are kept constant for
each of the constraints until convergence of the main optimization, Equation 5.12,
has been reached. Thereafter, new settings of the uncertainties α ∗i are determined
through anti-optimizations for the optimum x∗ obtained by Equation 5.12, given
by:
max fi (x∗ ; α i )
αi
(Anti-optimization)
(5.13)
s.t. α li ≤ α i ≤ α ui .
These cycles are repeated until convergence. For the initial iteration uncertainties
α, see Figure 5.2(c). This process is considered
are chosen arbitrarily or as α ∗ = ᾱ
to have converged when:
f0 − f0prev ≤ tolerance, and
(5.14)
f0
∗
∗
αik − αikprev
≤ tolerance
i = 1, . . . , n; k = 1, . . . , u. (5.15)
∗
αik
Thus, convergence is defined based on relative change in the objective function,
Equation 5.14, and the worst set of uncertainties corresponding to each constraint,
Equation 5.15.
80
5.4.3
5.4
SHAPE OPTIMIZATION UNDER UNCERTAINTY
Cycle-based Alternating anti-optimization
The Lombardi-Haftka technique can be quite efficient in terms of number of expensive function evaluations in case of problems involving monotonicities. However,
it can suffer from bad convergence in cases for which worst uncertainties fluctuate
from design to design. Inspired by Lombardi and Haftka (1998), in the present
chapter a slightly modified approach is presented. The idea is to solve
min f(p)
o (x)
x
(p)
s.t. fi (x; α pi ) ≤ 1,
(5.16)
i = 1, . . . , n,
for given α pi . This set of uncertainties consists of the maximizers of
(p)
max fi (xp ; α i )
αi
s.t. α li ≤ α i ≤ α ui .
(5.17)
Here, Equation 5.16 represents the main optimization for the pth cycle which
is solved first to obtain corresponding sub-optimum xp . For the initial cycle, the
worst set of uncertainties α pi needed in Equation 5.16 are chosen arbitrarily or as
α, see Figure 5.2(c). Then, anti-optimization represented by Equation 5.17
α p = ᾱ
is carried out at the sub-optimum (xp ) to obtain the worst set of uncertainties
αpi ), which will be used in the next cycle of the main optimization. During every
(α
cycle of the main optimization depending on the optimization history, the size
and direction of the subdomain keeps changing until convergence is reached, as
depicted in Figure 5.4.
In the Cycle-based Alternating technique, during each cycle of the main optimization, evaluation of constraints does not involve expensive anti-optimization,
x
x2
D2
f i
x1
Main Optimization
D1
Anti-optimization
Figure 5.4: Cycle-based Alternating anti-optimization technique in the MAM setting
for a problem of two design variables (x1 and x2 ) and two uncertainties (α1 and α2 ).
Here the anti-optimization is carried out only at the end of every cycle that is for the
sub-optimal designs indicated by the solid boxes.
5.4
UNCERTAINTY-BASED OPTIMIZATION USING ANTI-OPTIMIZATION
81
which makes it computationally less expensive. Even though the technique suffers from slower convergence in case of fluctuating uncertainties, it shows better
convergence as compared to the Lombardi-Haftka technique. Moreover, with the
increase in number of iterations, the Lombardi-Haftka technique becomes computationally very expensive in terms of total function evaluations as compared
to the Cycle-based Alternating technique. Additionally, in case additional information is available, such as estimates of worst uncertainties through derivative
information, this information can be utilized during the cycle. Such additions are
expected to further improve the convergence of the proposed technique.
5.4.4
Combined Cycle-based Alternating and Asymptotic method
The Asymptotic method uses derivatives of the response functions with respect
to uncertainties to estimate worst set of uncertainties. In case of fluctuating
uncertainties, when the Asymptotic method is combined with the Cycle-based
Alternating technique, it tremendously improves the convergence of the Cyclebased Alternating technique. The Asymptotic method is fully described in Van
Keulen et al. (2001) and discussed here for the type of problems dealt with in the
present chapter.
Here, approximations for responses are constructed using Taylor series around
α = ᾱ
α for a given design x. These Taylor series read
∆fi =
1 ∂ 2 f̄i
∂ f̄i
∆αk +
∆αk ∆αl + . . . ,
∂αk
2 ∂αk ∂αl
(5.18)
with
α).
∆fi = fi (x, α ) − f̄i = fi (x, α ) − fi (x, ᾱ
(5.19)
It is important to emphasize that it is not always possible to construct the above
Taylor series. This is, for example, the case when a response function is continuous but its derivatives are discontinuous. If the above Taylor series can be
created, then first-order approximations for the response functions are obtained
by dropping all higher-order terms, giving
∆fi =
∂ f̄i
∆αk .
∂αk
(5.20)
It is important to realize that the derivatives of the response functions with respect to the uncertain variables are relatively inexpensive to calculate, provided
efficient algorithms for sensitivity analysis are available (Van Keulen et al., 2005).
Using the approximations for the response functions given by Equation 5.20, the
maximization problem given by Equation 5.17, which is used to find the worst set
of uncertainties for a given design x, is replaced by
α
max hT
i ∆α
αi
∆α
s.t.
α li
≤ αi ≤
(5.21)
α ui ,
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SHAPE OPTIMIZATION UNDER UNCERTAINTY
5.4
with
hT
i
=
∂ f̄i
∂ f̄i
,...,
∂α1
∂αu
.
(5.22)
As discussed before, in case of the Cycle-based Alternating technique, during the
cycle worst sets of uncertainties are kept constant while computing responses.
These worst sets of uncertainties are updated at the end of the optimization
cycle by anti-optimization carried out for the sub-optimum. In the combined
technique described here, during each cycle estimation of worst uncertainties by
means of the Asymptotic method (Equation 5.21) is used. However, the rigorous
check by means of full anti-optimization Equation 5.17 at the end of cycle, to
update the worst set of uncertainties, is still kept in place. Additionally, this
worst set of uncertainties obtained at the end of a cycle is used as a basis for the
Taylor series approximation for the next cycle, i.e. hT
α∗ . The Asymptotic
i |α =α
evaluation of worst uncertainties during the cycle can be solved computationally
inexpensively, whereas it can improve the convergence significantly in case of
fluctuating uncertainties. In the present chapter, the Cycle-based Alternating
technique combined with the Asymptotic method is studied using an elastically
supported beam example, in Section 5.5.
5.4.5
Parallel computing
Optimization using Parallel Computing
In many practical problems, evaluation of response functions is based on computationally expensive finite element analysis (FEA). For such problems, the time
required for design optimization involving a large number of such expensive FEAs
will easily become impractical. The duration of the optimization process can be
reduced to practical levels through the use of parallel computing. To evaluate responses involving FEAs in parallel, clusters of multiple processors can be utilized
(see e.g. Van Keulen and Toropov, 1999).
Optimization using the MAM involves various steps, such planning of experiments, response evaluation, response surface approximation, nonlinear minimization and updating of the move limits, see Section 5.2. The computing times
required for these steps for a typical optimization involving expensive FEAs are
illustrated schematically in Figure 5.5 (a). It can be clearly seen there, that evaluation of the plan points is computationally the most expensive phase, whereas
the nonlinear minimization and the move limit updates are relatively inexpensive.
When a cluster of several processors is available for computation, it can be used
to evaluate the expensive response evaluations in parallel, see Figure 5.5 (b).
It should be noted here, that a single additional response evaluation is required at the end of the cycle, to evaluate the sub-optimum of the approximate
optimization problem. In the present setting for response evaluation, splitting of
this single response evaluation is not possible, therefore this sub-optimal point can
only be evaluated on a single processor, keeping the other processors idle. This
5.4
UNCERTAINTY-BASED OPTIMIZATION USING ANTI-OPTIMIZATION
2
1
Cycle
3
4
1
5
3
2
4
5
Idle
Machines
Machines
Cycle
83
Time
Idle
Regular Evaluation
Regular Evaluation
Sub-optimum
Sub-optimum
Time
(a) Plan points evaluated sequentially.
(b) Plan points evaluated using parallel computing.
Figure 5.5: A cycle in optimization using the MAM: Steps involved are 1. Planning of
new points, 2. Evaluation of plan points, 3. Nonlinear minimization, 4. Evaluation of
sub-optimum, 5. Updating of move limits.
increases the overall idle time significantly. The parallel computing framework
used in the present research is developed in Python, see the textbook by Lutz
(2001). To start multiple threads in parallel, the Threading module from Python
is used in the current framework, which is schematically shown in in Figure 5.6.
Here, each job involves evaluation of response functions, using, for example, FEA.
The number of parallel jobs that can be started simultaneously depends on the
number of processors available for computation.
During the evaluation of an individual job, first the design parameters are sent
to the remote processor, see Figure 5.6. Then, the actual evaluation of responses
is started on the remote processor by the associated thread. When the evaluations for the job are finished, corresponding responses are received back and are
associated with the job. As soon as a processor finishes response evaluation and
becomes again available for computation, the next job in the queue is submitted
Parallel
User
Unit
Data
Responses
Response
Optimizer
Clusters
Check availability
Design
Designs
Spawn threads
Threads
Figure 5.6: A framework for parallel computing using the Threading module of Python.
84
5.4
SHAPE OPTIMIZATION UNDER UNCERTAINTY
to it. This procedure is repeated until all the jobs have been evaluated.
It should be mentioned here, that the communication between the master and
slave computers is implemented through files containing data, for example, flags
indicating whether the submitted job is finished or not. The communication using
files does increase the overhead time, however in practical cases this is negligible
compared to the time required for an individual response evaluation.
Uncertainty-based Optimization using Nested Parallel Computing
In the case of uncertainty-based design optimization using anti-optimization, the
overall number of required FEAs is quite high compared to the deterministic
optimization. Therefore, to have a practical technique that can tackle uncertainties in practical design optimization problems, the use parallel computing is
essential. In Gurav et al. (2005), parallel computing is combined with Enhanced
anti-optimization to carry out uncertainty-based design optimization of a practical microstructure analyzed using expensive FEA. However, the strategy used
there for the parallel computing is rather simple, and does not lead to the best utilization of the available computing power. In this chapter, an improved strategy
for parallel computing is combined with the Cycle-based Alternating technique to
carry out uncertainty-based design optimization efficiently.
For combining parallel computing together with the Cycle-based Alternating
technique, various strategies have been considered. The effectiveness of these
strategies for parallel computing is discussed by means of a virtual problem involving four constraints and three processors. It should be noted here, that the
different strategies differ only in the evaluation of the suboptimum at the end of
the cycle of the main optimization, see Figure 5.7(b). The evaluation of responses
Evaluation in Parallel
FEA Jobs
Nodes
FEA
FEA
FEA
Main
Opt
X ; D
X
Eval
f
PCT
f
(a) Response evaluation during the cycle of
the main optimization.
Subopt
Design
x
f i ; D i
Anti
Opt
x ; D i
FEA
fi
(b) Response evaluation at end of the cycle of
the main optimization for the suboptimum.
Figure 5.7: Response evaluation for the main optimization for the Cycle-based Alternating technique. Here worst uncertainties obtained from the previous cycle are used for
the evaluation. Response evaluation involves FEA in parallel.
5.4
UNCERTAINTY-BASED OPTIMIZATION USING ANTI-OPTIMIZATION
85
during the cycle of the main optimization involving expensive FEAs is carried out
in parallel using available nodes as shown in Figure 5.7(a) in the same way for all
strategies.
In case of Strategy I, see Figure 5.8(a), all anti-optimizations can be started
in parallel, running one anti-optimization on one node. Each anti-optimization
requires several cycles for convergence, and each iteration involves evaluation
of constraints by means of FEAs in series on the corresponding node. It can
be easily seen here, that for the current fictitious problem, the number of antioptimizations mismatch the number of available nodes. For this problem three
anti-optimizations corresponding to three constraints can be started in parallel on
three nodes. However, during the evaluation of the fourth anti-optimization, only
one node is utilized whereas all other nodes remain idle. The overall idle time
can significantly increase with the increase in the difference between the number
of constraints and number of nodes. Furthermore, increase in the computational
time for a single FEA can add up to this idle time for such problems. However,
in case the number of constraints matches the number of nodes, this strategy can
be efficient.
For Strategy II, see Figure 5.8(b), anti-optimizations are carried out in series.
However, the FEAs involved in each of these anti-optimizations are carried out
in parallel, utilizing all available nodes. At the end of every cycle of the antioptimization, a single FEA needs to be carried out. As before, this additional
FEA can increase the overall idle time substantially. Additionally, during the cycle of each anti-optimization, if the number of FEAs are a multiple of the number
of available nodes, then these nodes can be utilized efficiently. However, if there
is a mismatch, then this will further increase the overall idle time. Moreover,
the increase in computational time for the individual FEA adds up to the overall
Evaluation in Parallel
Evaluation in Parallel
x
f 1;D 1
Subopt
Design
Anti
Opt
x
Nodes
x; D 1
FEA Jobs
FEA
Nodes
FEA
f1
f 2 ; D 2
FEA
f 3 ; D 3
FEA
PCT
f
(a) Parallel computing Strategy I:
anti-optimizations are carried out in parallel,
one on each node.
Subopt
Design
x
f i ; D i
Anti
Opt
x ;D i
PCT
fi
(b) Parallel computing Strategy II: FEAs for
each single anti-optimization are carried out
in parallel.
Figure 5.8: Various strategies in combining parallel computing with the Cycle-based
Alternating technique.
86
5.5
SHAPE OPTIMIZATION UNDER UNCERTAINTY
D
Nodes Jobs
Wait
FEA
x
Anti
Opt
f 1
f1
x
Subopt
Design f i
f 2
f 3
f 4
Scheduler
PCT
Figure 5.9: Evaluation of responses at the end of the cycle of the main optimization, i.e.
at the suboptimum. Here anti-optimization for each constraint is carried out, in order to
obtain the worst values. Corresponding worst uncertainties are used for the next cycle.
Anti-optimizations are carried out in parallel using nested parallel computing.
idle time, making the strategy computationally less efficient for such problems.
In the present chapter, Strategy III, referred here as nested parallel computing
due to the involved two levels in parallel computing, is combined with the Cyclebased Alternating technique, and applied for the SMA microgripper optimization
involving uncertainties. This third strategy overcomes the disadvantages of the
preceding two approaches, and is schematically depicted in Figure 5.9. In the
nested parallel computing approach as proposed here, at the end of each cycle of
the main optimization, the anti-optimizations are first started in parallel. Next,
a program called “Scheduler” is started in the background. This Scheduler collects all the designs to be evaluated within each cycle of every anti-optimization,
thus synchronizing all anti-optimizations. All designs collected by the Scheduler
are subsequently evaluated in parallel. This nested parallel computing approach
involves two levels. In the outer level, anti-optimization processes are started in
parallel, whereas in the inner level, the actual evaluation of designs within all
anti-optimizations is carried out in parallel as well. In this nested approach, efficiency is not affected seriously when the numbers of anti-optimizations and nodes
do not match. Moreover, making a common list of all FEAs and then evaluating
them in parallel utilizes the available nodes more efficiently. A disadvantage of
this approach is that the implementation is significantly more complex, which is
the price for the better utilization of the available computing power.
5.5
Test case
An elastically supported beam, see Figure 5.10, has been used by Lombardi and
Haftka (1998) to test the anti-optimization technique in the presence of nonlinearities. One of the features of this problem is the strong dependence of the worst
5.5
87
TEST CASE
0.014
P
x0
0.012
0.01
0.008
EI
w(x)
0.006
0.004
k
0.002
0
1
s
0.8
x0
L
0.6
0.4
0.2
0
(a) Problem setting:
P = 1000 N, k
EI = 1000 Nm2 .
here L = 1 m,
= 500000 N/m,
0
0.2
0.4
s
0.6
0.8
(b) Plot of the integrated displacement as a function of
x0 and s.
Figure 5.10: Elastically supported beam example.
case uncertainty values on the design variable. The different anti-optimization
techniques discussed in Section 5.4 are studied on the basis of this example. For
comparative study, for this test example similar notation as in Lombardi and
Haftka (1998) is used.
In this problem, a beam loaded by a concentrated force P is supported elastically by a support with stiffness k, to limit its vertical displacement w, as shown
in Figure 5.10 (a). The goal of the optimization here is to optimally place the
elastic support, in order to minimize the integral of the displacement over the
length of the beam. The location of the concentrated force P is uncertain, and
can vary over the whole length of the beam. Assuming the nominal location of
the concentrated force at the center of the beam (x̄0 = 0.5L), with ε = 0.5L, the
lower and upper bounds on uncertainty (x0 ) can be given as
xl0
xu0
= x̄0 − ε = 0,
= x̄0 + ε = L.
(5.23)
Here, the nonlinear displacement function w(x, s, x0 ) is the objective of the antioptimization. It can be determined analytically by integration of the fourth-order
differential equation governing the deformation of the beam:
EI
1
∂4w
= P δ(x − x0 ) − kw(x)δ(x − s).
∂x4
(5.24)
The main and anti-optimization problems are formulated as follows: solve the
main optimization to obtain minimizers s for given x0
Z L
min
w(x, s, x0 ) dx s.t. 0 ≤ s ≤ L,
(5.25)
s
0
88
5.5
SHAPE OPTIMIZATION UNDER UNCERTAINTY
and solve the anti-optimization to obtain maximizers x0 for given s as
Z
max
x0
L
w(x, s, x0 ) dx s.t.
0 ≤ x0 ≤ L.
(5.26)
0
The actual integrated displacement is plotted in Figure 5.10 (b). It can be clearly
seen here, that the displacement function is highly nonlinear, and displays a
saddle-like response formed due to the discontinuity depending upon the location of the load (x0 ) with respect to the location of the support (s). As a result,
the worst location of the load (x0 ) strongly depends on the location of the elastic
support (s) and can fluctuate from point to point. In a first study, for selected
designs (s) at regular intervals, anti-optimization is carried out to find
R the worst
location of uncertainty (x0 ) and the corresponding worst response ( w), see Figure 5.11. The worst uncertainty (x0 ) can be clearly seen to vary with respect to
the design (s), and in addition a discontinuity can be identified at s = 0.5L. Here,
the optimum location of the support is at 0.5m, whereas the worst location of the
load is either at 0.35m or at 0.65m.
The uncertainty-based optimization problem as described by Equation 5.25
and Equation 5.26 is carried out here using three different techniques. Results
are compared in terms of convergence and total number of function evaluations.
A comparison of the optimization history of the different techniques is shown in
Figure 5.12, and the obtained optimum design and the total number of function
evaluations are compared in Table 5.1. It is clear, that the alternating techniques
require more steps until convergence compared to the Rigorous approach, as shown
in Figure 5.12 and Figure 5.13. In the Lombardi-Haftka case, due to the fluctuating uncertainties the process did not converge. The convergence of the Rigorous
technique requires only a small number of steps, however, the total number of
−1
−3
x 10
0.8
−2
0.7
−3
0.6
∫w
−4
0.5
x0
−5
−6
0.4
−7
0.3
−8
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
s
R
Plot for the worst response ( w) obtained
by anti-optimization.
0.9
0.2
0.1
0.2
0.3
0.4
0.5
s
0.6
0.7
0.8
0.9
Plot for the worst uncertainty (x0 ) obtained
by anti-optimization.
Figure 5.11: Anti-optimization results as a function of the design s for the elastically
supported beam test case.
5.5
89
TEST CASE
−3
4
x 10
Rigorous
Cyclebased
Cyclebased + Asymptotic
Objective
3
2
1
0
0
2
4
6
8
Steps
10
12
14
Figure 5.12: Optimization history for the elastically supported beam problem: comparison of uncertainty-based optimization using different techniques.
∫δ
0.014
1
0.012
0.9
0.01
0.8
0.008
0.7
0.006
0.6
0.004
x0
0.5
0.002
0.4
0
0.3
1
0.2
0.5
0.1
x0
0
0
0.1
0.2
0.4
0.3
0.5
0.6
0.7
0.8
0.9
s
(a) Plot of the integrated displacement
function together with the optimization
history for the Lombardi-Haftka
Method.
1
0
0
0.1
0.2
0.3
0.4
0.5
s
0.6
0.7
0.8
0.9
1
(b) Contour plot for the integrated
displacement function together with the
optimization history for the
Lombardi-Haftka Method.
Figure 5.13: Optimization history using the Lombardi-Haftka method for the elastically
supported beam problem.
90
5.6
SHAPE OPTIMIZATION UNDER UNCERTAINTY
Table 5.1: Comparison between uncertainty-based optimization results using different
methods for the elastically supported beam test problem.
Method
Rigorous
Cyclebased
Cyclebased + Asymptotic
Design
variable s
0.4987
0.4985
0.4997
Objective
R
w
1.1993e-03
1.2048e-03
1.1997e-03
Function
evaluations
720
676
100
function evaluations is the highest for this approach. Moreover, this number can
rapidly increase further when the number of design variables, uncertainties and
optimization cycles increases. Table 5.1 clearly shows, that in spite of requiring a
larger number of steps, the cycle-based techniques result in fewer function evaluations. Moreover, if the number of steps required for the convergence is reduced,
the gain in the number of function evaluations can be substantial. In case of the
combined cycle-based and asymptotic technique, a further spectacular reduction
in the number of function evaluations is realized. This is due to the incorporation
of derivative information for predicting the worst uncertainties during each cycle.
It should be noted here, that the required derivatives are obtained analytically for
this problem. For practical problems, sensitivity analysis should be carried out,
which should be available at low computational cost in order to keep this method
efficient.
5.6
SMA Microgripper optimization under uncertainty
5.6.1
Introduction
The problem considered in this section is the shape optimization of a shape memory alloy microgripper. Shape memory alloys (SMAs) are materials in which a
solid-state phase transformation can occur under the influence of a change in temperature or stress state. Internally, the lattice structure of the alloy changes from
one configuration to another. The transformation is accompanied by a transformation strain, that can be used for actuation. Compared to other actuator
materials, SMAs are capable of generating relatively large strains and stresses.
This makes these materials very interesting for many applications. For further
information about SMAs, see e.g. Otsuka and Wayman (1998) or Duerig et al.
(1990). The focus of the present study is on the so-called R-phase transformation
in nickel-rich NiTi alloys. The thermomechanical behavior of this material has
been studied experimentally by Tobushi et al. (1992), and stress-strain curves
at various temperatures are shown in Figure 5.14. Unlike most SMAs, this Rphase transformation is characterized by a small hysteresis and a relatively narrow
5.6
SMA MICROGRIPPER OPTIMIZATION UNDER UNCERTAINTY
91
thermal operating range. These properties are attractive for actuator applications
(Kohl et al., 2004). The fact that in this case relatively small temperature changes
can still induce significant SMA effects, makes that also in vivo medical applications might be possible using this alloy. The present microgripper design study is
situated in that context.
600
σ [MPa]
343 K
338 K
333 K
328 K
400
200
ε [%]
0
0.2
0.4
0.6
0.8
1.0
Figure 5.14: Experimental stress-strain data at different temperatures for a NiTi alloy
(Tobushi et al., 1992). Thick and thin lines represent loading and unloading curves,
respectively (left). The right diagram shows a schematic stress-strain diagram illustrating
the piecewise linear approximation used in the SMA model.
The constitutive model used to describe this SMA behavior focuses on the
temperature range of 328–343 K, where the hysteresis is sufficiently small to be
neglected. It is based on a piecewise linear approximation, which is fitted to the
stress-strain curves, as schematically illustrated in Figure 5.14. A more detailed
description of this model and its generalization to a three dimensional setting
can be found elsewhere (Langelaar and Van Keulen, 2004b,c) (Chapter 2 and
Chapter 3). Note that the selected temperature range is not directly suited for in
vivo applications, but because the transformation temperatures can be influenced
by heat treatments and alloy composition (Sawada et al., 1993), lowering this
range to compatible temperatures is possible.
5.6.2
Microgripper model
Before discussing the formulation of the optimization problem, the microgripper
design concept is presented here. The conceptual design of this gripper is shown in
Figure 5.15. It consists of an identical top and bottom arm made of folded Ni-Ti
plates. An initial deformation is applied in order to generate internal stresses in
the material, which are required to make use of the shape memory effect. Starting
from the undeformed configuration in Figure 5.15, the ends of the outer plates are
pinched toward the inner plates. In this situation, the equilibrium configuration
of each arm can be changed by changing the temperature of either the inner or
92
SHAPE OPTIMIZATION UNDER UNCERTAINTY
5.6
outer plates. Resistive heating is used for this purpose, and to guide the electrical
current through individual plates a slit is present along the length of each plate.
Heating the inner plates will cause the tip ends to move apart, opening the gripper.
Similarly, heating the outer plates will make them move toward each other, closing
the gripper. In the closing configuration, clamping forces of 100 mN are applied
in z-direction at the tips of the gripper, acting against the closing forces. A
related microgripper design problem has been studied before by Langelaar and
Van Keulen (2004a) (Chapter 4). However, in that case, uncertainties were not
considered in the shape optimization. In the present problem, uncertainties in
both the operating conditions and the SMA material properties are taken into
account.
Figure 5.15: Conceptual gripper geometry in the undeformed configuration.
Because of symmetry, only a quarter of the gripper needs to be modeled: in
this case half the top arm is used. This part, together with the parameterization of
the geometry, is shown in Figure 5.16. The design variables chosen for this design
problem are the plate thickness t, the undeformed arm height H, the actuation
plate front width W2 , and the shape of the actuation plate. This shape is described
by a quadratic B-spline (Farin, 2002), and the y-coordinates of the two middle
control points are used as design variables: Y1 and Y2 . Previous design studies
have shown that the plate end width W1 always remains at its upper bound,
therefore it is excluded from the present design problem and set to 15 mm, based
on previous studies. Further geometrical details of the miniature gripper can be
found in Table 5.2.
The gripper is simulated by finite element analysis of the parameterized design
shown in Figure 5.16. For both the opened and closed case, a quasi-static electrical, thermal and mechanical analysis is performed, to simulate the SMA behavior
5.6
93
SMA MICROGRIPPER OPTIMIZATION UNDER UNCERTAINTY
Table 5.2: Significant coordinates of B-spline control points and other points defining
the geometry of the miniature gripper.
Point
Control point 1
Control point 3
Control point 5
Tip
x [mm]
0
1
3.5
5.377
y [mm]
15
Y1
W2
0
Point
Control point 2
Control point 4
Control point 6
Slit end
x [mm]
0.5
3
4
3.8
y [mm]
15
Y2
W2
0
Figure 5.16: Design parameterization of the gripper. Because of symmetry, only a
quarter is considered.
under the influence of Joule heating. Dissipated heat from the electrical analysis
is used as a heat source in the thermal analysis, and the resulting temperature
distribution is used in the mechanical analysis. Physical constants used in the
simulations are listed in Table 5.3. Particularly the mechanical analysis is computationally intensive, because of the nonlinear SMA material model as well as
the consideration of geometrical nonlinearities. An adaptive incremental-iterative
scheme is used to ensure robust convergence. A triangular shell element is used
for the mechanical analysis (Van Keulen and Booij, 1996).
5.6.3
Optimization problem formulation
The objective chosen for this design study is to maximize the range of motion
of the gripper tips, i.e. the stroke of the gripper. Therefore, the difference between the z-coordinates of the gripper tip displacement utip
in open and closed
z
configurations is taken as the objective function:
open
closed
max utip,
(x) − utip,
(x)
z
z
x
(5.27)
94
5.6
SHAPE OPTIMIZATION UNDER UNCERTAINTY
Table 5.3: Physical constants used in the finite element modeling.
Quantity
Electrical conductivity
Thermal conductivity
Thermal convection coefficient
Ambient temperature
Value
1.25 · 106 Sm-1
21 Wm-1 K-1
2.0 · 103 Wm-2 K-1
328 K
Here (x) represents the vector of design variables. The design variables for the
current problem, see Figure 5.16, are listed in Table 5.4.
Table 5.4: Design variables (x) for the design optimization of the SMA gripper.
Design variable
Plate thickness
Gripper arm height
Plate shape control point 1
Plate shape control point 2
Plate front width
Applied voltage
Symbol
t
H
Y1
Y2
W2
V
Lower bound
0.05
0.3
0.01
0.01
0.1
0.001
Upper bound
0.3
2
1.5
1.5
1.5
0.5
Unit
mm
mm
mm
mm
mm
V
The considered gripper optimization problem also involves a number of constraints. The validity of the material model is limited to a certain strain range,
therefore a constraint on the effective strain εe is added in both the open and
closed configuration. In addition, motivated by the possibility to use this SMA
material for in vivo active devices, the thermal operating range is limited to 10 K.
This means that per element in the finite element mesh, the following constraints
are added to the optimization problem:
(i)
gε(i) =
(i)
gT
εe
≤1
(max)
εe
T (i) − Tmin
=
≤1
Tmax − Tmin
(max)
(5.28)
(5.29)
The maximum effective strain εe
is set to 1%, and the minimum and maximum
temperature values used are 328 and 338 K, respectively. A minimum value is
included in the formulation in order to scale the temperature constraint properly.
To reduce the number of individual constraints and to make the anti-optimization
approach feasible, a Kreisselmeier-Steinhauser (Kreisselmeier and Steinhauser,
1983) constraint aggregation function is used. This aggregation function in its
5.6
SMA MICROGRIPPER OPTIMIZATION UNDER UNCERTAINTY
95
standard form is given by:
N
X
1
KS(g) = ln
eρgi
ρ
i=1
!
(5.30)
where N is the number of individual constraints gi (e.g. the number of elements)
and ρ is a parameter that determines the bias of the aggregation. A higher value
of ρ puts larger weight on higher constraint values, tending toward a maximumoperator, whereas a lower weight has the opposite effect, tending towards an
averaging operator. In the present study, a modified Kreisselmeier-Steinhauser
function is used. Here, in order to reduce the contribution of local violations, the
individual element constraint values are weighted by the associated element area
Ai as
!
N
X
1
1
ρgi
KSA (g) = ln PN
·
Ai e
(5.31)
ρ
i=1 Ai i=1
Note that the use of this Kreisselmeier-Steinhauser function can not prevent a
small number of isolated individual violations of the aggregated constraints. However, for a suitable choice of the parameter ρ the violations remain very small and
limited in number. A value of 40 has been used here, as this turned out to give a
satisfactory behavior.
The resulting optimization problem is now given by:
(p)
open
closed
(x) − utip,
(x)
max utip,
z
z
x
Subject to:
(p)
(p)
KSA (gεopen (x; α 1 )) ≤ 1
(p)
(p) closed
(x; α 2 )) ≤ 1
KSA (gε
(p)
(p)
KSA (gTopen (x; α 3 )) ≤ 1
(p) closed
(p)
KSA (gT
(x; α 4 )) ≤ 1
l
u
x ≤x≤x
(5.32)
The lower and upper bounds of the design variables are represented by xl and xu ,
(p)
respectively, and αi denotes the worst settings for the uncertainties corresponding to each of the constraints in optimization cycle p, obtained as the maximizers
of the following anti-optimization problems:
(p)
s.t. α l ≤ α 1 ≤ α u
(p)
s.t. α l ≤ α 2 ≤ α u
(p)
s.t. α l ≤ α 3 ≤ α u
(p)
s.t. α l ≤ α 4 ≤ α u
max KSA (gεopen (x(p) ; α 1 ))
α1
max KSA (gεclosed (x(p) ; α 2 ))
α2
max KSA (gTopen (x(p) ; α 3 ))
α3
max KSA (gTclosed (x(p) ; α 4 ))
α4
(5.33)
Here x(p) represents the suboptimal design obtained in the main optimization
problem Equation 5.32 and anti-optimizations are carried out for this design for
96
5.6
SHAPE OPTIMIZATION UNDER UNCERTAINTY
each constraint, as specified in Equation 5.33. For evaluation of the objective, a
complete electro-thermo-mechanical analysis is required for both the opened and
closed configuration. However, for the constraint values, which are the objectives
in the anti-optimization problems, no full gripper simulation is required. For instance, for KSA (gεopen ), only an electro-thermo-mechanical analysis is required for
the opened configuration. Likewise for KSA (gTopen ), an electro-thermal analysis
for the opened configuration is sufficient. In relation to the computational effort
required for the nonlinear mechanical analysis involving an incremental-iterative
solution process, the computational effort required for this electro-thermal analysis is virtually zero. Therefore, significant computational savings were possible
by exploiting these observations in the practical implementation, leading in this
case to a reduction of the computational effort by approximately a factor 4.
The uncertainty variables contained in the set α that are selected for the
present design problem are listed in Table 5.5, together with their nominal values
and their bounds. The ambient temperature is considered uncertain because it
is hard to control. The convection coefficient is difficult to determine unless the
environmental conditions are well known and stable, which is not likely to be the
case. The other uncertainty variables are parameters of the SMA constitutive
model, and these account for any inaccuracy that might be present in the measurements, as well as unknown aspects of the SMA behavior that have not been
included in the modeling. One could think of, for instance, the minor hysteresis
that has been neglected in the formulation of the SMA model. The range for
these parameters has been chosen such, that it covers a substantial deviation of
the modeled material behavior. This is illustrated by the stress-strain diagram
shown in Figure 5.17, which visualizes the effect of the uncertainties in EA , ER ,
Table 5.5: Uncertainty variables α for the gripper problem, their deterministic or nominal values together with their upper and lower bounds.
Uncertainty variable
Symbol
h
Nominal
value
2.0
Lower
bound
1.8
Upper
bound
2.2
Face convection
coefficient [kWm-2 K-1 ]
Ambient temperature [K]
Austenite Young’s
modulus [GPa]
R-phase apparent
Young’s modulus [GPa]
Initial apparent transition
Young’s modulus [GPa]
Poisson ratio
Ta
EA
328
68.939
327.6
63.0
328.4
75.0
ER
45.612
43
48
E0
20.006
19.5
20.5
ν
0.3333
0.3
0.36
5.6
SMA MICROGRIPPER OPTIMIZATION UNDER UNCERTAINTY
97
E0 and Ta on the one-dimensional material model. The variation of the ambient
temperature is assumed to directly affect the temperature of the SMA material,
which is the worst case situation.
500
σ
xx
[MPa]
400
T = 338 K
300
200
100
T = 328 K
εxx [%]
0
0
0.2
0.4
0.6
0.8
1.0
Figure 5.17: One-dimensional stress-strain curves according to the deterministic model
(thick lines) and the range covered by the uncertain material parameters (gray), at different temperatures.
In order to visualize the effect of the uncertainties on the material model in
the plane stress setting used in the gripper model itself, and to include the effect
of the uncertainty in the Poisson ratio ν, two new quantities are defined. A first
measure to illustrate the effect is the difference between the largest and smallest
Von Mises stress value ∆σV M at a certain strain state, for any combination of
uncertainty values in the defined ranges:
∆σV M (ε1 , ε2 ) = max σV M (ε1 , ε2 ; α) − min σV M (ε1 , ε2 ; α)
α
α
(5.34)
where ε1 and ε2 are the principal strains, and σV M is the Von Mises stress. In
order to be able to judge the relative magnitude of this difference, also a quantity
Ψ is introduced where ∆σV M is normalized by the nominal value of the Von Mises
stress at the considered strain state:
Ψ(ε1 , ε2 ) =
∆σV M (ε1 , ε2 )
σV M (ε1 , ε2 )
(5.35)
These two quantities are visualized in Figure 5.18 (a) and (b), for various values
of the nominal temperature. Again the variation of the ambient temperature is
98
SHAPE OPTIMIZATION UNDER UNCERTAINTY
(a) Maximum absolute differences in Von Mises
stress.
5.6
(b) Maximum differences in Von Mises stress relative to the nominal values.
Figure 5.18: Maximum absolute (left) and relative (right) differences in Von Mises
stress at different strain states due to the effect of the uncertainties, at different nominal
temperatures. A full color version is given by Figure G.6 on page 252.
5.6
SMA MICROGRIPPER OPTIMIZATION UNDER UNCERTAINTY
99
assumed to directly affect the temperature of the material. Note that the relative
effect of the uncertainties is quite large, roughly 20–25% on average, and that
the uncertainties affect different strain states differently. Because of this, the
equilibrium configuration of the gripper will therefore most likely be affected by
the uncertainties. Hence it is hard to make a statement about which combination
of uncertainty variables will result in the worst value for strain constraints.
In contrast to the strain constraint, the temperature constraint is affected only
by the uncertainties in the ambient temperature and the convection coefficient.
In fact, on physical grounds it is clear that the largest value for the temperature
constraint is obtained when the ambient temperature uncertainty is at its upper
bound and the convection coefficient is at its lower bound. But in this study, no
use is made of this knowledge, and the temperature constraints are treated in the
most general way.
Actually, it might be argued that the uncertainties affecting the temperature distribution are not very relevant, since in practice a temperature-controlled
voltage could be applied, that ensures that the temperature limit is respected.
However, this example is mainly meant to illustrate the versatility and effectiveness of the proposed technique in dealing with uncertain factors. Eliminating or
circumventing some of the uncertainties might in fact be possible for this design
problem, but in many other problems this might not be the case at all.
5.6.4
Results
Using the Cycle-based Alternating anti-optimization technique proposed in this
chapter, design optimization of the SMA microgripper has been performed. The
even more efficient combined Cycle-based Alternating and Asymptotic method
could not be employed because sensitivity information was not yet available. Linear approximations have been used for response surfaces, both in main and antioptimization. The Python-based nested parallel computing framework reduced
the total time of the optimization process, and a cluster of 14 CPUs (1 GHz
Pentium) was used. Both the deterministic and uncertainty-based optimization
converged after ca. 20 iterations, and the evolution of the objective and constraint
values are depicted in Figure 5.19 and Figure 5.20, respectively.
The objective history of the deterministic case shows a sharp spike at the
fourth step, where the stroke of the gripper even becomes negative. This is clearly
undesirable, and possibly is caused by the inability of that design to generate the
required clamping force. However, the optimizer recovers in the subsequent step.
The activity of the constraints shown in Figure 5.20 shows that all constraints
are relevant to the design problem. This is confirmed by the fact that the final
constraint values shown in Table 5.6 are all very close to 1.
The design variable values and the responses of the final designs are listed in
Table 5.6. The stroke in case of the uncertainty-based design is ca. 15% smaller
100
5.6
SHAPE OPTIMIZATION UNDER UNCERTAINTY
0.6
0.4
Objective
0.2
0
−0.2
Deterministic Opt
Cycle−based Alternating Anti−opt
−0.4
−0.6
0
5
10
15
20
25
Steps
Figure 5.19: Optimization history: objective function.
1.8
1.4
Deterministic Opt
Cycle−based Alternating Anti−opt
1.6
Deterministic Opt
Cycle−based Alternating Anti−opt
1.2
1
1.2
Constraint 2
Constraint 1
1.4
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0
5
10
Steps
15
20
(a) Evolution of KSA (gεopen ).
2
0.2
0
25
5
10
15
20
25
Steps
(b) Evolution of KSA (gεclosed ).
2
Deterministic Opt
Cycle−based Alternating Anti−opt
Deterministic Opt
Cycle−based Alternating Anti−opt
1.8
1.5
Constraint 4
Constraint 3
1.6
1.4
1.2
1
1
0.8
0
5
10
15
Steps
open
(c) Evolution of KSA (gT
).
20
25
0.5
0
5
10
15
Steps
closed
(d) Evolution of KSA (gT
).
Figure 5.20: Optimization history: constraints.
20
25
5.6
SMA MICROGRIPPER OPTIMIZATION UNDER UNCERTAINTY
101
than that of the deterministic design. Accounting for uncertainties essentially requires the design to move further away from the deterministic constraints, resulting in a reduction of the objective. Note that in the present study, only the effect
of uncertainties on constraint functions is considered due to the present setting
of the optimizer. In contrast, the effect of uncertainties on the objective function
value is considered separately at the end of both optimizations. This is done by
means of a separate anti-optimization for the optimal design, in order to anticipate
the influence of uncertainties on the objective function. This anti-optimization is
carried out separately for both the deterministic and uncertainty-based optimal
design. In case of the deterministic optimization, the stroke reduces from 0.4341
mm (Table 5.6) to 0.3259 mm, whereas in the uncertainty-based case it reduces
from 0.3684 mm (Table 5.6) to 0.2728 mm. Note that the presented technique can
in principle be extended to include this influence of uncertainties on the objective
in a similar way as this has been done for constraints.
The total number of FEAs and the wall clock time required for both deterministic and uncertainty-based optimization are also listed in Table 5.6. The
uncertainty-based case requires 30 times as many evaluations as the deterministic
case, which clearly shows the need of sophisticated techniques such as Nested Par-
Table 5.6: Comparison between deterministic and uncertainty-based optimization results
using Cycle-based Alternating anti-optimization for the SMA microgripper problem.
Optimal Response
f [mm]
KSA (gεopen )
KSA (gεclosed )
KSA (gTopen )
KSA (gTclosed )
Optimal Design
t [mm]
H [mm]
Y1 [mm]
Y2 [mm]
W2 [mm]
V [V]
Number of FEA
Number of FEA (relative)
Wall clock time [hours:min]
Wall clock time (relative)
Deterministic
Uncertainty-based
0.4341
0.9992
0.9999
0.9999
0.9979
0.3684
0.9776
0.9983
0.9998
0.9982
0.0974
1.1703
1.2485
1.1125
1.4001
0.0603
221
1
8:17
1
0.1022
1.0463
1.3047
1.0208
1.2703
0.0553
6645
30
62:12
7.5
102
SHAPE OPTIMIZATION UNDER UNCERTAINTY
5.6
allel Computing in order to make the uncertainty-based optimization practical.
Furthermore, the fact that the anti-optimizations involve only parts of the total
analysis, combined with the high utilization of the parallel computing resources
by the use of Nested Parallel Computing, makes that the uncertainty-based optimization procedure is 4 times more efficient, relative to the deterministic case.
This is illustrated by the fact that the duration of the uncertainty-based case
equals merely 7.5 times the duration of the deterministic optimization, while 30
times as many function evaluations are involved.
The geometries of the SMA gripper corresponding to the optimal designs obtained in the deterministic and uncertainty-based optimization are depicted in
Figure 5.21. From the top and side views, it can clearly be seen that the deterministic design is wider at the front and also higher. A geometrical difference that
cannot be seen in Figure 5.21 is that the plates are 5% thicker in the uncertaintybased design. The operation of the gripper is demonstrated in Figure 5.22, which
shows a side view of the optimal uncertainty-based gripper in the open and closed
configuration.
Figure 5.21: Undeformed geometries of the optimal designs obtained by deterministic optimization (left) and optimization considering bounded-but-unknown uncertainties
(right).
5.6
SMA MICROGRIPPER OPTIMIZATION UNDER UNCERTAINTY
103
Figure 5.22: Side view of the optimized gripper in opened (light gray) and closed (dark
gray) configuration.
In the case of uncertainty-based optimization, the values of the uncertainty
variables that yield the worst (i.e. largest) constraint values for the final design
are listed in Table 5.7 for all four constraints. For the temperature constraints,
only the uncertainties in the thermal quantities are meaningful, and it turns out
that indeed their worst case values are as predicted earlier. Inspection shows that
all of the uncertainty variables are at either their lower or upper bound for the
final design, except the face convection coefficient h at the strain constraint for
the opened case KSA (gεopen ). Evaluation of this constraint with h at its lower
and upper bound confirmed that indeed the worst value is obtained for h at an
interior value.
When considering the evolution of this uncertainty variable h during the optimization process, as depicted in Figure 5.23, it turns out that in many other steps
also interior worst-case values are found, in both strain constraints. Also, for the
Table 5.7: Worst set of uncertainties obtained for the uncertainty-based optimum for
each constraint for the SMA microgripper problem.
Normalized
uncertainty
h/h̄
Ta /T¯a
EA /E¯A
ER /E¯R
E0 /E¯0
ν/ν̄
KSA (gεopen )
KSA (gεclosed )
KSA (gTopen )
KSA (gTclosed )
1.08485
1.00122
1.08792
0.94273
0.97470
1.08
0.9
1.00122
1.08792
0.94273
0.97470
1.08
0.9
1.00122
-
0.9
1.00122
-
104
5.7
SHAPE OPTIMIZATION UNDER UNCERTAINTY
1.4
Uncertainty No. 1
Constraint No. 1
1.3
Uncertainty No. 1
Constraint No. 2
1.3
1.2
1.2
1.1
1.1
1
0.9
0
1
5
10
15
20
25
Plot for the normalized worst value of the
uncertainty in face convection coefficient h/h̄
and the worst value of constraint
KSA (gεopen ) obtained by anti-optimization
corresponding to suboptimal design at each
step.
0.9
0
5
10
15
20
25
Plot for the normalized worst value of the
uncertainty in face convection coefficient h/h̄
and the worst value of constraint
KSA (gεclosed ) obtained by anti-optimization
corresponding to suboptimal design at each
step.
Figure 5.23: Anti-optimization results in terms of worst set of uncertainties and constraint corresponding to suboptimal design at each step for the SMA microgripper problem.
strain constraint in the opened case KSA (gεopen ), the uncertainty variable h initially stays at the lower bound, but changes to values equal or close to the upper
bound. From this observation, it can be concluded that the present uncertaintybased SMA gripper optimization problem indeed exhibits fluctuating uncertainties. The worst case for uncertainty variable h clearly is design-dependent, and
also takes interior values. The complexity and nonlinearity of the model leads
to non-monotonicity in the uncertainty-based design optimization. In general,
for complex models, it is hard to predict which combination of uncertainties will
result in the worst constraint values, and whether even interior worst cases are
possible. Therefore, the proposed anti-optimization technique is a good choice for
such problems, even though it is computationally more involved than approaches
that rely on, e.g., vertex checking.
5.7
Discussion and conclusions
The Cycle-based Alternating anti-optimization technique combined with nested
parallel computing has successfully been applied to the design optimization of
an SMA microgripper involving bounded-but-unknown uncertainties. This allows
for practical optimization-based design of SMA devices in situations where, for
example, material properties and operating conditions are not exactly known,
but where bounds on these unknown quantities can be specified. The worst set of
uncertainties turned out be design-dependent, and the Cycle-based Alternating
technique proved to be able to handle this situation effectively.
5.7
DISCUSSION AND CONCLUSIONS
105
The optimization problem involving bounded-but-unknown uncertainties is
solved using an anti-optimization technique. To demonstrate and compare various anti-optimization approaches, the elastically supported beam problem from
literature has been used. For this test problem, the worst set of uncertainties is
also strongly dependent on the design, and fluctuates considerably from design to
design. This situation is a challenging case for efficient anti-optimization, however,
many problems of realistic complexity, such as the SMA microgripper problem,
share this characteristic. For such problems exhibiting fluctuating worst uncertainties, it is shown that the Cycle-based Alternating technique is more efficient
than the existing Lombardi-Haftka Alternating technique in terms of convergence
and required computational effort. Rigorous anti-optimization proves to be better than the cycle-based techniques in terms of convergence properties. However,
since the number of function evaluations required for the rigorous approach can
quickly become prohibitive as the problem size increases, in terms of the number
of design variables, uncertainties and constraints, the rigorous approach quickly
becomes impractical for problems involving computationally expensive function
evaluations.
Further improvement of the effectiveness of the Cycle-based Alternating technique is possible with the use of derivative information. It is demonstrated by
means of the elastically supported beam problem that combining the Cycle-based
Alternating technique with the Asymptotic method, which uses derivative information for prediction of worst cases, is significantly more efficient than all other
examined techniques, in terms of both the convergence and number of function
evaluations. In the present study however, this combined Cycle-based Alternating and Asymptotic method is not applied in the SMA microgripper optimization,
because of the lack of the required derivative information.
For the problems involving computationally expensive response evaluations,
use of parallel computing is essential to make the above optimization techniques
computationally feasible. For the SMA microgripper problem, which involves
computationally expensive finite element simulations, the Cycle-based Alternating technique combined with nested parallel computing enabled the practical application of anti-optimization. For this problem, the total number of FEAs required for uncertainty-based optimization is 30 times that of the deterministic
optimization. However, a high utilization of the computing cluster due to Nested
Parallel Computing strategy in combination with a smart treatment of function
evaluations in the anti-optimization procedures reduced the total wall clock time
for uncertainty-based optimization to merely 7.5 times that of deterministic optimization. Hence, for this modest computational cost, the proposed techniques
enable designers to account for bounded-but-unknown uncertainties in the design
optimization process, and it has been shown that this approach is indeed feasible
even for relatively complex and large problems as the considered SMA structure.
It is therefore expected, that by making it feasible to account for uncertainties in
the design process in a systematic way, the proposed methodology contributes to
the development of better, safer and more reliable SMA structures and devices.
106
SHAPE OPTIMIZATION UNDER UNCERTAINTY
5.7
Based on: Langelaar, M., Yoon, G.H., Kim, Y.Y. and van Keulen, F. Topology
optimization of shape memory alloy thermal actuators using element connectivity
parameterization. Computer Methods in Applied Mechanics and Engineering, in review.
Chapter
6
Topology Optimization of SMA
Actuators
This chapter presents the first application of topology optimization to the design of
shape memory alloy actuators. Shape memory alloys (SMAs) exhibit strongly nonlinear,
temperature-dependent material behavior. The complexity in the constitutive behavior
makes it difficult to apply the conventional element density-based topology optimization
to the topology design of SMA structures. Therefore, in the present study, the recently
developed element connectivity parameterization (ECP) formulation is applied, which
offers important advantages for complex nonlinear topology optimization problems. A
history-independent constitutive model of SMAs is employed which allows efficient adjoint sensitivity analysis. The effectiveness of the proposed technique is illustrated by
several numerical examples.
6.1
Introduction
6.1.1
Shape memory alloys
This chapter presents a procedure to perform topology optimization of shape
memory alloy (SMA) structures. In SMA materials, a solid state phase transformation can take place under local temperature change and/or stress state variation. This transformation is associated with a transformation strain that can
be utilized for actuation. SMA actuators are applicable in situations requiring
relatively large deflections, combined with substantial mechanical loading (Duerig
et al., 1990, Otsuka and Wayman, 1998). SMAs are used in a wide variety of ap107
108
TOPOLOGY OPTIMIZATION OF SMA ACTUATORS
6.1
plications, and present developments particularly focus on emerging applications
in medicine and microsystems (Büttgenbach et al., 2001, Duerig et al., 1999, Kohl,
2004, Van Humbeeck, 1999).
It is expected that systematic, computer-assisted design techniques such as
topology optimization will play an important role in further development of future
SMA actuators, as their complexity increases. Many presently existing SMA
actuators rely on a relatively simple layouts, e.g. straight wires or helical springs.
Clearly, for such layouts, understanding the behavior of a one-dimensional SMA
wire or spring is sufficient for effective design (see e.g. Liang and Rogers, 1992a).
This changes radically as soon as SMA actuators of more complex geometries
have to be designed. As a consequence, it is necessary to understand and model
the material behavior in a 2D or 3D setting. The complex material behavior and
the fact that often several physical domains (mechanical, thermal, electrical) are
involved make it very difficult to design effective SMA actuators by intuition.
Therefore, utilization of powerful systematic design techniques such as topology
optimization is crucial for the realization of these structures.
6.1.2
Topology optimization
Topology optimization aims to find the optimal material distribution in a given
domain to fulfill a certain objective. The technique, initially developed for the
design of linear elastic structures (Bendsoe and Kikuchi, 1988), has been applied to a wide variety of engineering problems (see e.g. Bendsoe and Sigmund,
2003). In comparison to the alternative shape and sizing design optimization
approaches, the unique strength of topology optimization is that it can generate
arbitrary topologies. The parameterization used in shape and sizing optimization
is typically much more restrictive. On the other hand, in topology optimization
the analysis is generally less accurate, and it is harder to include e.g. stress or
buckling constraints. These properties make that topology optimization is most
applicable in the initial design stage. However, since decisions made in that stage
have a strong effect on the final performance of a design, the merit of topology
optimization is evident.
The conventional approach basically consists of controlling the material parameters of elements in a finite element model, e.g. the Young’s moduli, using
density design variables. In this way, in e.g. structural problems, the presence of
void regions in the structure can be approximated by very low stiffness elements.
The majority of the reported applications is restricted to linear analysis models. However, next to the expansion of topology optimization to other physical
domains, in recent years, researchers have started to explore its application to
nonlinear problems. The focus is mainly on structural problems involving geometrical nonlinearities and/or elastoplasticity (Buhl et al., 2000, Jung and Gea,
2004, Maute et al., 1998). These nonlinear problems pose new challenges to the
conventional topology optimization approach, for instance due to the excessive
distortion of highly compliant elements (Yoon and Kim, 2005). In addition, in
6.1
INTRODUCTION
109
case of complex nonlinear materials such as SMAs, the conventional approach of
relating material properties to design variables easily leads to complications and
ambiguity (Pedersen, 2002).
The Element Connectivity Parameterization (ECP) approach proposed by
Yoon and Kim (2004, 2005) presents a new paradigm that avoids these difficulties. In this ECP scheme, not the properties of elements, but the connectivity
between the elements is controlled by design variables. This is accomplished by
introducing zero-length links between elements. By changing the stiffness of these
links, different topologies can be realized. A drawback of the originally proposed
“external” ECP method (Yoon and Kim, 2004, 2005) is, that the number of degrees of freedom increases considerably. However, a recently developed “internal”
ECP version (Yoon et al., 2005) is employed here, in which the introduction of additional nodes, a modified arrangement of elements and links, and the application
of condensation completely eliminates this disadvantage.
6.1.3
SMA material modeling
The application of topology optimization to the design of SMA structures has
not been reported before. Next to the outlined difficulties involved in topology
optimization of nonlinear problems in general, this is most likely due to the lack
of sufficiently simple, though practical SMA material models that are suited for
optimization, in spite of the large number of models that have already been developed (Birman, 1997). In this chapter, a new SMA model is presented, which
is well suited for (topology) optimization. The shape memory effect considered is
superelasticity due to the R-phase/austenite transformation in NiTi. Generally,
SMAs exhibit hysteresis in the stress-strain-temperature behavior. However, in
case of this transformation, the hysteresis is particularly small, and the cyclic
stability of the transformation is excellent. These are attractive properties for
actuator applications (Kohl, 2004, Otsuka and Wayman, 1998). However, the reversible transformation strain of the R-phase transformation is considerably less
(ca. 0.8%) than that of the martensite transformation in NiTi (up to ca. 7%).
Therefore, it is crucial to optimize designs to utilize the relatively small transformation strain in the most effective way.
6.1.4
Outline
This chapter is organized as follows: in Section 6.2, the new SMA material model
is presented, together with a comparison to experimental data. A one-dimensional
constitutive model is generalized to a three-dimensional setting, and subsequently
also the formulation in a plane stress situation is discussed. Detailed derivations
can be found in Appendices A till D. In addition, in Section 6.2.6, the strategies
used to ensure robust convergence of the SMA finite element model used in the
topology optimization procedure are discussed.
110
TOPOLOGY OPTIMIZATION OF SMA ACTUATORS
6.2
Section 6.3 is devoted to the modified Element Connectivity Parameterization.
After discussion of the difficulties encountered in the conventional density-based
topology optimization approach in nonlinear problems, the ECP concept is presented as well as its detailed formulation. An important advantage of the ECP
method is that, in contrast to density-based approaches using design-dependent
material properties, the sensitivity analysis does not require differentiation of the
material model, as shown in Section 6.3.4. This aspect is particularly important
in case of complex nonlinear material models, where differentiation can quickly
become quite involved. Section 6.3 is concluded by a description of a new interpolation function used in the proposed ECP formulation, that allows for the use
of penalization similar to the popular SIMP model (Solid Isotropic Material with
Penalization, Bendsoe and Sigmund, 2003).
In Section 6.4, regularization techniques are studied in order to avoid certain
numerical artifacts of the topology optimization process. It is shown, that limiting
the design space by the use of nodal design variables results in the desired behavior. Subsequently, Section 6.5 presents the results of the application of topology
optimization to the design of SMA thermal actuator structures. Various results
of a study on the characteristics of the SMA topology optimization problem are
presented, including the effect of the load magnitude and direction, the mesh
density, and the initial design. Finally, based on the observations made in this
study, conclusions are drawn with regard to the potential and effectiveness of the
developed techniques for SMA topology optimization in Section 6.6.
6.2
SMA modeling
6.2.1
Design optimization considerations
In order to assist the design process of SMA structures, and to obtain a deeper
understanding of the underlying phenomena that lead to the peculiar behavior
of SMA materials, much research has already been done on the computational
modeling of SMA constitutive behavior (Birman, 1997). In contrast to existing
models, the SMA constitutive model presented here is developed with an emphasis
on its suitability for design optimization, as well as on its ability to adequately
represent the observed material behavior. Design optimization generally is an
iterative process that requires many evaluations of the analysis model. Thus,
to make design optimization practically feasible, it is important to consider the
efficiency of the analysis model. In the present work, as a first simplification,
the dynamic response of the actuator is neglected, and a quasistatic simulation
is performed. This is sufficient in many applications where the response speed is
not critical.
A second aspect that is of great importance in design optimization – and
particularly topology optimization – is the computational complexity of the sensitivity analysis. In history-dependent models, sensitivity analysis is much more
6.2
SMA MODELING
111
involved than in the history-independent case, and adjoint sensitivity analysis is
impractical (Kleiber et al., 1997). Internal variables, such as phase fractions used
in most of the existing SMA models, render those models history-dependent. By
neglecting the (rather small) hysteresis of the R-phase transformation, it turns
out to be possible to eliminate internal state variables, resulting in a historyindependent model. In that case, (adjoint) sensitivity analysis can be carried out
at the end of the analysis, without the need to account for every increment during
the evolution of the model over time.
6.2.2
One-dimensional R-phase transformation modeling
Figure 6.1 shows experimental data obtained by Tobushi et al. (1992) for the
stress-strain-temperature behavior of a Ti-55.3wt%Ni alloy exhibiting the R-phase
transformation. The modeling procedure described in the present chapter focuses
on this particular alloy, but it can similarly be applied to other alloys in which
the R-phase transformation occurs. The hysteresis between loading and unloading
behavior is practically negligible at temperatures above 328 K. The focus of the
present model is mainly on applications with a limited thermal operating range,
such as (in vivo) actuator applications, and the range 328–343 K has been selected
as the temperature range of interest. Given this small range, strains due to
thermal expansion are ignored in this formulation. The temperatures are still high
for in vivo use, but the range can be shifted by changes in the alloy composition
or thermal processing (Gyobu et al., 1996, Sawada et al., 1993).
Several one-dimensional models for the stress-strain-temperature behavior for
the R-phase transformation in NiTi have been published, and a review is given by
Langelaar and Van Keulen (2004c) (Chapter 2). By focusing on a temperature
range where the hysteresis is negligible, simplifications can be applied. The onedimensional model used in the present research is similar to the model proposed
earlier (Langelaar and Van Keulen, 2004c) (Chapter 3), and is summarized here.
Based on the shape of the experimental curves, the stress-strain relations are
approximated by a three-part piecewise linear function, as depicted in Figure 6.1:

: EA εxx ,
 εxx ≤ ε1
ε1 < εxx ≤ ε2 : ET (εxx − ε1 ) + EA ε1 ,
σxx =
(6.1)

εxx > ε2
: ER (εxx − ε2 ) + ET (ε2 − ε1 ) + EA ε1 ,
where ER is a constant parameter. In order to represent the experimental data
well, ε1 , ε2 , ET and EA are chosen to be linearly dependent on the temperature
T , in the following way:
ε1 (T ) = Kε (T − T0 ) + ε0 ,
ε2 (T ) = ε1 (T ) + ∆,
ET (T ) = KE (T − T0 ) + E0 ,
0
EA (T ) = KA (T − T0 ) + EA
.
(6.2)
112
6.2
TOPOLOGY OPTIMIZATION OF SMA ACTUATORS
600
σ [MPa]
xx
343 K
338 K
333 K
328 K
400
ER
ET
200
EA
1
ε [%]
0
0.2
0.4
0.6
0.8
2
xx
1.0
Figure 6.1: Experimentally obtained stress-strain curves at various temperatures (Tobushi et al., 1992). Thin lines are unloading curves (left). Schematic representation of
piecewise linear one-dimensional SMA stress-strain relation (right).
ε1 and ε2 are the transition strains that mark the points where the piecewise
linear function changes its slope. In this approximation, these points correspond
to the strains at which the R-phase transformation starts and finishes. ET is the
apparent Young’s modulus dσxx /dεxx during the phase transition. EA can be
interpreted as the Young’s modulus of the austenite phase, and was taken as a
constant in the previous model (Langelaar and Van Keulen, 2004c) (Chapter 3).
However, for the topology optimization problems considered in the present chapter, it has been found that making EA depend on temperature as well provides
more useful sensitivity information in low-stressed areas of the design. This idea
has been implemented as follows:
0
EA (T ) = KA (T − T0 ) + EA
(6.3)
0
Here EA
, KA and T0 are parameters of the material model, and T represents
the temperature. This modification improves the convergence of the optimization
procedure. Next to this, it also provides a slightly better approximation of the
experimental data.
The proposed model is calibrated to experimental data by determining the
material parameters using curve fitting. Before the actual fitting, a conversion
is applied to the experimental data. The finite element implementation is based
on the Green-Lagrange strain εGL and the Second Piola-Kirchhoff stress σP K2 .
The one-dimensional experimental data is given in terms of nominal stress σnom
and engineering strain εeng . The equivalent Green-Lagrange strain and Second
Piola-Kirchhoff stress can be obtained from these quantities by:
1 2
ε
+ εeng ,
2 eng
σnom
.
=
1 + εeng
εGL =
(6.4)
σP K2
(6.5)
6.2
SMA MODELING
113
Table 6.1: Parameter values for the SMA model.
Kε
∆
0
EA
2.47×10−4 K−1
52.7×10−4
66.3 GPa
KE
E0
ER
678 MPa K−1
18.8 GPa
45.6 GPa
ε0
T0
KA
5.89×10−4
328 K
188 MPa K−1
The converted experimental data is used in the least squares fitting process, and
the parameter values that were subsequently obtained are listed in Table 6.1.
This one-dimensional model will be referred to by σxx (εxx , T ). For subsequent
derivations, it is useful to present an alternative description of the piecewise linear
stress-strain equations in the following generalized form:
σxx = Ai (T )εxx + Bi (T ).
(6.6)
The terms Ai (T ) and Bi (T ) follow from the original formulation given in Equation 6.1, and the indices refer to the three segments of the piecewise linear relation:

 εxx ≤ ε1
ε1 < εxx ≤ ε2
Ai (T ) =

εxx > ε2

 εxx ≤ ε1
ε1 < εxx ≤ ε2
Bi (T ) =

εxx > ε2
6.2.3
: A1 = EA ,
: A2 = ET ,
: A3 = ER ,
(6.7)
: B1 = 0,
: B2 = (EA − ET )ε1 .
(6.8)
: B3 = (EA − ET )ε1 + (ET − ER )ε2 .
Three-dimensional modeling
The transformation pseudoelasticity due to the R-phase transformation is described by a path-independent nonlinear thermo-elastic formulation. Use is made
of the experimentally observed characteristics of the R-phase transformation,
namely that it is associated with an isochoric transformation strain, and that the
transformation behavior is insensitive to hydrostatic pressure (Kakeshita et al.,
1992, Miyazaki and Wayman, 1988). The starting point of this formulation is
Hooke’s Law for linear elasticity, split into volumetric and deviatoric parts:
1
σ = K(tr ε)I + 2G ε − (tr ε)I ,
3
where K and G represent the shear and bulk moduli, respectively.
(6.9)
114
6.2
TOPOLOGY OPTIMIZATION OF SMA ACTUATORS
In Mandel notation, this can also be written as:

  

σxx 
1 1 1 0 0 0




 σyy 
  1 1 1 0 0 0






 σ
 
 1 1 1 0 0 0
zz
√
 + ...


= K 
2σxy 
0 0 0 0 0 0






√

  

0 0 0 0 0 0
2σyz 




√


0 0 0 0 0 0
2σzx
 2
− 13 − 13 0 0
3
2
− 1
− 13 0 0
3
 31
1
2
−
−3
0 0
3
3
2G 
 0
0
0
1 0

 0
0
0 0 1
0
0
0 0 0

 
εxx 
0




  εyy 

0



 




0 √εzz
,

0
√2εxy 

 



0 

√2εyz 



1
2εzx
(6.10)
which is equivalent to, in short:
σ = (KK + 2GG)ε.
(6.11)
Upon transformation, the isochoric phase transformation strain must be accounted
for. In this nonlinear elastic model, it is found that the distortional strain energy
is minimized when the transformation strain εtr is aligned with the distortional
strain component Gε. This can equivalently be represented by a variable shear
modulus G̃:
σ
σ
= (KK + 2GG)(ε − εtr ) where εtr = βGε
= (KK + 2G̃G)ε with G̃ = G(1 − β).
⇒
(6.12)
(6.13)
Here use has been made of KG = [0] and GG = G. To relate the value of the
effective shear modulus G̃ to the strain state, an scalar equivalent strain measure
is used based on the deviatoric strain energy, given by:
ε2e =
2 T
ε Gε.
3
(6.14)
This scalar distortion measure is invariant, and equal to 4/3(I2 + I12 /3) = 4/3J2 ,
where I1 and I2 are the first and second invariant of the Green-Lagrange strain
tensor, and J2 is the second invariant of its distortional part. The factor 2/3 is
included to make this strain measure energy-conjugated to the Von Mises stress,
which covers the pressure-insensitive property of the transformation. The constitutive stress-strain relation for R-phase transformation pseudoelasticity now
simply takes the following form:
σ = (KK + 2G(εe )G)ε.
(6.15)
Note that here the shear modulus can vary depending on the value of the effective
strain, and the tilde used before in Equation 6.13 to indicate this variability
6.2
SMA MODELING
115
will be omitted from this point onwards. The relation between the value of the
shear modulus and the effective strain G(εe ) is found by considering a tensile
test. In that case, the stress-strain behavior is described by the one-dimensional
model σxx (εxx , T ) presented in Equation 6.1. In addition, the proposed threedimensional model given by Equation 6.10 in combination with Equation 6.14
yields:
εyy = εzz =
2G − 3K
εxx
2G + 6K
⇒
3K
G + 3K
εxx , εxx =
εe
G + 3K
3K
9GK
=
εxx = 3Gεe
G + 3K
(6.16)
εe =
(6.17)
σxx
(6.18)
Combining these expressions with the one-dimensional model results in a single
equation that can be used to determine G for a given value of εe :
G + 3K
σxx (εxx , T ) = σxx
εe , T = 3Gεe .
(6.19)
3K
Using the generalized notation introduced in Equation 6.6 subsequently gives:
G + 3K
εe + Bi (T ) = 3Gεe ⇒
3K
3K
Bi (T )
G=
Ai (T ) +
.
9K − Ai (T )
εe
Ai (T )
(6.20)
(6.21)
This final expression relates the shear modulus directly to the effective strain.
Finally, the tangent operator of this constitutive model is found by differentiation
of Equation 6.15 to the strain, which yields:
[K]T = KK + 2GG +
4 dG
GεεT G.
3εe dεe
(6.22)
The detailed derivation is given in Appendix A.
6.2.4
Plane stress case
Using the general 3-D formulation, it can be derived that in plane stress, the
transverse strain εzz is related to the in-plane strain components εxx and εyy by:
εzz =
2G − 3K
(εxx + εyy ) = α(εxx + εyy ),
4G + 3K
(6.23)
where the symbol α is introduced to represent the term (2G − 3K)/(4G + 3K), to
reduce the complexity of the expressions. Note that α depends on G. Using this
116
TOPOLOGY OPTIMIZATION OF SMA ACTUATORS
6.2
relation, the transverse strain component can be eliminated from the stress-strain
equation (Equation 6.10), leading to:
  


1+α 1+α 0
 σxx 
σyy
= K 1 + α 1 + α 0 + . . .

√
0
0
0
2σxy

(6.24)
 

2−α
−(1 + α) 0
 εxx 
2G 
−(1 + α)
2−α
0 √εyy
.


3
0
0
3
2εxy
The same can be done for the effective strain definition, which becomes:
ε2e =
4 2
4
4
(α − α + 1)(ε2xx + ε2yy ) + (2α2 − 2α − 1)εxx εyy + ε2xy .
9
9
3
(6.25)
This indicates a slight complication: in the 3-D case, the effective strain only
depends on the strain components (Equation 6.14). In the plane stress case, however, by elimination of the transverse strain, the effective strain also has become
a function of α, which in turn depends on the shear modulus G. It turns out, no
convenient explicit expression can be found to express G as a function of εe , as
in the 3-D case. Combining Equation 6.21, Equation 6.25 and the definition of
α gives a nonlinear equation in G, which can be solved numerically for a given
temperature and strain situation. Newton-Raphson iterations are used to obtain
the solution. Convergence is robust and usually requires only 3 to 4 iterations.
Details on the equations can be found in Appendix B.
Also in the plane stress case a tangent operator is required in the finite element
implementation. There are two ways to derive it: the first is to take the tangent
operator of the three-dimensional case, and reduce it to a plane stress setting
by eliminating the transverse strain component εzz from the equations. This
is worked out in more detail in Appendix C.1. The second way to derive the
tangent operator is to start from the stress-strain relation in the plane stress
case (Equation 6.24) and to differentiate it with respect to the strain. However,
as in this case there exists no explicit relation between the shear modulus and
the (effective) strain, use has to be made of implicit differentiation, which makes
this a rather complicated derivation. The details are given in Appendix C.2.
Naturally, the resulting tangent operator is the same in both cases, regardless
of the derivation. It is symmetric in case of a finite element formulation based
on the stress vector (σxx , σyy , σxy ) and strain vector given by (εxx , εyy , γxy ), as
commonly used in implementations.
6.2.5
Verification and discussion
The presented SMA constitutive model in the plane stress formulation has been
implemented in a standard four-noded quadrilateral element (see e.g. Bathe,
6.2
SMA MODELING
117
1995). This element is most popular in topology optimization, and is also used
in this research. The implementation of the material model has been verified by
simulating a tensile test. Figure 6.2 shows the good agreement between experimental stress-strain data from Tobushi et al. (1992) and results obtained using
the finite element analysis.
More particular constitutive aspects such as the influence of texture, multiaxial non-proportional loading effects and tension-compression asymmetry are not
considered in the present model, for several reasons. On the one hand these effects
are considered to be refinements of the principal material behavior, and of lesser
importance for the present application. The present model is mainly aimed at
an efficient formulation that is able to capture the first-order characteristics, with
a sufficient degree of accuracy for design optimization in an early design stage.
Also, firm and conclusive experimental evidence that allows incorporation of these
effects in the formulation is currently still lacking.
On the other hand, including more detailed constitutive effects will almost
certainly require the use of additional internal variables in the model, which will
render the adjoint sensitivity computation essential for topology optimization intractable. A simpler and even more computationally efficient model has been
formulated before (Langelaar and Van Keulen, 2004c) (Chapter 2), however that
model did not account for the isochoric transformation strain, and the transformation was not pressure-independent. Therefore the present model is a practical
compromise between representing the physical characteristics of the material as
good as possible and maintaining suitability for (topology) optimization.
500
Nominal stress, MPa
343 K
338 K
400
Experiment
FEA Results
333 K
328 K
300
200
100
Engineering strain, %
0
0
0.2
0.4
0.6
0.8
1
Figure 6.2: Simulated and experimentally determined stress-strain curves at various
temperatures.
118
TOPOLOGY OPTIMIZATION OF SMA ACTUATORS
6.2.6
Robust analysis techniques for optimization
6.2
A vital property of the simulation model used for the topology optimization procedure is robustness. The SMA constitutive model is strongly nonlinear, and
also geometrical nonlinearities are considered in the finite element analysis. To
evaluate the responses necessary for the next step in the optimization process,
the analysis is performed using an incremental approach combined with Newton
iterations. Newton iterations have a tendency to overshoot the solution, which
can in turn lead to excessive corrections and subsequent divergence. Closer to
the equilibrium solution, the Newton process is stable and converges quadratically. By applying the load in a number of smaller increments, the deviation from
the equilibrium path is reduced and the Newton iterations in each increment are
more likely to converge. However, the total computational effort required for the
analysis generally increases with the number of increments. Since topology optimization itself is already a time-consuming iterative process, it is important to
strive towards using the minimum number of increments the analysis requires.
It has also been observed in case of the present SMA finite element model,
when the configuration of the system deviates too far from the equilibrium path,
the tangent operator in some cases can lose its positive definiteness during the
Newton iterations. This subsequently lead to divergence (when using a generalpurpose solver for the non-positive definite system). We believe this behavior is
the result of a combination of the strong and sharp nonlinearities in the SMA
model, the inclusion of geometrical nonlinearities and the overshoot and overcorrection tendency of the Newton iterations. The present material model involves
significant changes in effective stiffness due to the phase transformation, which
lead to numerical convergence problems. Conventional remedies, such as using
relaxed or damped Newton variants worked to some degree, but the parameters involved turned out to be strongly case-dependent. Adding a line search to
the Newton iteration was found to work robustly, but was prohibitively expensive.
Therefore, to prevent this type of numerical problem, sufficiently small increments
turned out to be the most practical solution.
To combine the two objectives, robustness and efficiency, an adaptive solution
strategy is applied to the SMA analysis used for topology optimization, schematically depicted in Figure 6.3. Since in topology optimization, changes in the design
are generally gradual, the changes in the response are generally small as well, in
consecutive optimization steps. Starting the Newton iterations from the solution
obtained for the preceding design therefore can lead to very fast convergence.
This procedure is referred to in Figure 6.3 as “Continued analysis”. To increase
the chances of success, it is only attempted under certain conditions, as described
in Figure 6.3. If in spite of these precautions this continued analysis fails, a
full analysis is performed using N increments. If this full analysis also does not
converge, the setting for the number of increments N is increased and another
attempt is made. This continues until a certain maximum, so that the process
can not continue indefinitely.
6.3
ELEMENT CONNECTIVITY PARAMETERIZATION METHOD
119
Start
> 10 steps since
beginning or last
failed restart?
No
Yes
Average absolute
change in density
< 0.025?
> 15 steps since
last increase of N?
Yes
N = max(N - 1, 1)
No
No
N=N+1
Yes
Continued analysis
from previous solution
No
Fail
N = number of
increments used in
previous full analysis
Full analysis
with N increments
Fail
Success.
N = 14?
Yes
Failure.
Figure 6.3: Solution strategy used in the topology optimization analysis.
This strategy proved to work very well in practice. Particularly the continued
analysis, possible because a history-independent model is used, resulted in swift
convergence. No detailed study was made to obtain the optimal settings for
this strategy, but using the constants shown in Figure 6.3 proved to result in
satisfactory behavior. Note that although this strategy dramatically improved
the robustness of the analysis, it cannot guarantee convergence in all cases. In
fact, in the Results section, a case will be described where the analysis eventually
failed. But in the majority of practical cases the proposed strategy resulted in a
robust as well as efficient solution process.
6.3
Element Connectivity Parameterization method
6.3.1
Difficulties in conventional method (density approach)
Three difficulties arise when the conventional density-based topology optimization
method is applied to nonlinear problems. First, an interpolation needs to be
defined between the material properties of elements and their densities. The
behavior of the optimization strongly depends on this interpolation, and in case
of complex nonlinear material models, finding a proper interpolation can be very
difficult (Pedersen, 2002). The second problem is that because material properties
are functions of the design variables, the sensitivity analysis can require a full
120
6.3
TOPOLOGY OPTIMIZATION OF SMA ACTUATORS
differentiation of the material model, which can become very involved for complex
material models. Finally, in the density-based approach, elements that should not
be part of the design are represented by very compliant material. As noted before
by Yoon and Kim (2005), among others, in nonlinear finite element models these
compliant elements can suffer from excessive distortion, due to which their tangent
stiffness matrices can lose positive definiteness. This seriously affects stability and
convergence of the entire model.
6.3.2
Basic idea and mesh layout
Because of the observed difficulties of the conventional density-based method,
the recently developed element connectivity parameterization (ECP) formulation
(Yoon and Kim, 2004, 2005, Yoon et al., 2006b) is used here. Unlike the density
based approach, the layout is not described by varying the densities and modifying
the material properties of elements, but by varying the connectivity between elements. Computationally, this concept is implemented using zero-length links that
connect the nodes of neighboring elements with additional nodes, see Figure 6.4.
The internal element can involve geometrical and material nonlinearities, however
the connecting links behave linearly. The stiffnesses of the links connected to an
individual element are controlled by a single design variable.
Note that parameterization of element connectivity requires separation of elements, which increases the number of degrees of freedom of the problem significantly. This has been identified as a disadvantage of the original external ECP
approach (Yoon and Kim, 2004, 2005). The present arrangement with additional
nodes is chosen because it allows for condensation of the element-link patch (see
Figure 6.4), which reduces the number of degrees of freedom to that of the original
mesh with directly connected elements. In the condensed patch, the element connectivity is dealt with internally, hence this approach is denoted as internal ECP.
Additional nodes
Zero­length links
uyj
uOUTER
uINNER
uxj
kL(e)
uyi
uxi
y
Patch used for condensation
x
Figure 6.4: Mesh and cell layout used in the internal element connectivity parameterization approach.
6.3
ELEMENT CONNECTIVITY PARAMETERIZATION METHOD
121
External forces and boundary conditions are applied to the additional nodes.
Because element connectivity instead of element properties is parameterized,
clearly there is no need to define a material model interpolation. In addition,
sensitivity analysis for ECP is straightforward, because only the zero-length links
are affected by the design variables. Finally, since elements remain stiff, and only
the stiffness of the one-dimensional links between the elements is modified, numerical problems due to excessive distortion of compliant elements are prevented
(Yoon and Kim, 2005). For these reasons, the ECP approach is very suitable for
topology optimization of structures involving complex nonlinear materials, such
as the SMA actuators studied in this thesis.
6.3.3
Governing equations
Without loss of generality, the focus of the present research is on two-dimensional
plane stress problems, to limit the required computational effort. A zero-length
link element in the considered 2-D setting, connecting nodes i and j is governed
by:
T i
T
(e) = Fx Fyi Fxj Fyj ,
(6.26)
KL uix uiy ujx ujy
where the nodal displacement components uβα , where α = x or y, β = i or j,
are indicated in Figure 6.4. Fαβ represent the components of the associated force
(e)
vector, and the link stiffness matrix KL is given by:


1
0 −1 0
0
1
0 −1
I2 −I2
(e)
(e)
(e) 


= kL
KL = kL 
,
(6.27)
−1 0
1
0
−I2 I2
0 −1 0
1
(e)
where I2 represents the 2-by-2 unity matrix, and kL is the link stiffness associated
to the particular e th element this link belongs to. Subsequently, the governing
equation of the patch of the e th element and its associated links is given by:
"
# (
) (e)
(e)
(e)
ext (e)
kL I8
−kL I8
uOUTER
FOUTER
=
,
(6.28)
(e)
(e)
(e) ·
(e)
0
−kL I8 kL I8 + KE
uINNER
(e)
(e)
where KE is the stiffness matrix of the internal element, and ext FOUTER represents the force applied to the outer nodes of this patch. The displacement
components of the patch are partitioned in components associated with the outer
and inner nodes. In the following, the dimension subscript of the unity matrix I
will be omitted. Equation 6.28 is valid in the linear setting. To analyze the geometrically and materially nonlinear SMA actuators, the Total Lagrangian (TL)
formulation is used. A full description of the TL formulation can be found in
reference works (e.g. Bathe, 1995). The final system equations are given by:
R(U ) = ext F − int F (U ) = 0,
(6.29)
122
TOPOLOGY OPTIMIZATION OF SMA ACTUATORS
6.3
where R is the residual force vector, given by the difference between the external
and internal force vectors. Cases with displacement-dependent external forces are
not considered in this chapter. Newton iterations are used to solve this nonlinear
system of equations, resulting in:
K(U k )∆U k+1 = R(U k ),
U k+1 = U k + ∆U k+1 ,
(6.30)
where the tangent stiffness matrix K is given by the negative Jacobian −∂R/∂U ,
and the superscript denotes the iteration number. The contribution of an individual patch to this system is given by:
"
# (
)
(e)
(e)
(e)
kL I
−kL I
∆uOUTER
·
= ...
(e)
(e)
(e)
(e)
(e)
−kL I kL I + KE (uINNER )
∆uINNER
(
)
(6.31)
(e)
ext (e)
FOUTER − int FOUTER
.
(e)
0 − int FINNER
The right-hand side vector contains the residual contributions of the outer and
inner nodes, where:
(e)
(e)
(e)
int (e)
FOUTER = kL uOUTER − uINNER ,
(6.32)
(e)
(e)
(e)
(e)
int (e)
FINNER = int FE − kL uOUTER − uINNER ,
(6.33)
(e)
where int FE is the internal force vector of the e th internal element.
In order to reduce the number of degrees of freedom at system level, condensation is applied to each patch, and only the degrees of freedom of the outer
(e)
nodes uOUTER are used at system level. The condensed stiffness matrix KC and
(e)
residual RC for the eth patch are found to be:
2 h
i−1
(e)
(e)
(e)
(e)
(e)
KC = kL I − kL
kL I + KE
,
(6.34)
h
i−1
(e)
(e)
(e)
(e)
(e)
(e)
int (e)
RC = ext FOUTER − int FOUTER − kL kL I + KE
FINNER . (6.35)
After solving the condensed system, the inner displacement increments are computed for each patch from the outer increments by:
h
i−1 n
o
(e)
(e)
(e)
(e)
(e)
(e)
∆uINNER = kL I + KE
kL ∆uOUTER − int FINNER .
(6.36)
The condensed system has exactly the same number of degrees of freedom as the
original mesh before additional nodes and zero-length links were added. In the
implementation of the ECP method, the inclusion of link elements and additional
nodes is therefore accounted for by modifications to the element stiffness and
internal force vector routines, instead of actually constructing a modified mesh.
6.3
6.3.4
ELEMENT CONNECTIVITY PARAMETERIZATION METHOD
123
Sensitivity analysis
In the topology optimization problems considered here, the response of interest f
can generally be written as:
f = LT U .
(6.37)
Here L is a constant vector that is used to select the displacement components
of interest, and U is the total displacement vector. The distinction between inner and outer displacements, and the effect of condensation will be considered
later. The sensitivity of this response f is determined by the adjoint method.
Adjoint sensitivity analysis has large computational advantages over direct sensitivity analysis, when dealing with large numbers of design variables and relatively
few response quantities, as in topology optimization. The derivation involves including the effect of the state equations given in Equation 6.29 by the formulation
of the following Lagrangian:
L = LT U + λT R(U (γ), γ).
(6.38)
In the equilibrium configuration, L is equivalent to the response f , since in that
case R equals zero. The design variables in the ECP method, given by γ, deter(e)
mine the link stiffness kL of the links in each patch. Since different expressions
can be used to relate the link stiffnesses to the design variables, the sensitivity
of the response to the link stiffness is considered first. Differentiation of L with
(e)
respect to kL yields:
!
∂R ∂U
∂R
∂L
T ∂U
T
=L
+λ
+
.
(6.39)
(e)
(e)
(e)
∂U ∂k (e)
∂k
∂k
∂k
L
L
L
L
This equation holds for any linear or nonlinear history-independent model, such
as the presented SMA model. Rearranging yields:
∂L
(e)
∂kL
= LT − λT K
∂U
(e)
∂kL
∂R
+ λT
(e)
,
(6.40)
∂kL
where the definition of the system tangent stiffness matrix K introduced in Equation 6.30 has been used. In the adjoint sensitivity analysis technique, the vector
of multipliers λ is now chosen such that the term with the sensitivities of the
(e)
displacements ∂U /∂kL in Equation 6.40 vanishes:
LT − λT K = 0 ⇒ Kλ = L.
(6.41)
The symmetry of K has been used here. When using a direct solver, K is available
in decomposed form after the analysis, and the multipliers λ can be obtained at
a fraction of the computational effort required for the nonlinear analysis. Sensitivities are obtained by subsequent substitution of λ into Equation 6.40:
∂L
(e)
∂kL
= λT
∂R
(e)
∂kL
.
(6.42)
124
TOPOLOGY OPTIMIZATION OF SMA ACTUATORS
6.3
In the implementation, condensation is applied to the adjoint sensitivity equations
analogously as to the equilibrium equations, in order to re-use the decomposed
system matrix based on the outer displacements, denoted by KOUTER . This gives:
KOUTER λOUTER = LOUTER .
(6.43)
Given the outer multipliers, the inner multipliers can subsequently be found by
applying the equivalent of Equation 6.36 to each patch:
h
i−1 n
o
(e)
(e)
(e)
(e) (e)
(e)
λINNER = kL I + KE
kL λOUTER − LINNER .
(6.44)
From Equation 6.31, it can be derived that the contribution of a particular patch
to the partial derivatives of the residual used in Equation 6.42 is given by:
 
(e)
− u(e)

∂R(e)
OUTER − uINNER
=
.
(6.45)
(e)
(e)
 u(e)

−u
∂k
L
OUTER
INNER
Combining Equation 6.45 and Equation 6.42 subsequently yields:
∂f
∂L
(e)
(e)
(e)
(e)
=
λ
−
λ
u
−
u
=
.
INNER
OUTER
OUTER
INNER
(e)
(e)
∂kL
∂kL
(6.46)
Finally, for a given relation between design variables γ and link stiffnesses kL ,
the sensitivity with respect to the design variables can be evaluated by applying
the chain rule:
∂f
∂f T ∂kL
=
.
(6.47)
∂γi
∂kL ∂γi
(e)
In the implementation, the sparsity of ∂kL /∂γi can be exploited. The presented
sensitivity analysis has been verified by finite differences. Note that nowhere in
the computation of the adjoint sensitivities, differentiation of the material model
is required. This is a clear advantage of the ECP approach.
6.3.5
Interpolation function
In order to understand the effect of the link stiffness on the stiffness of the patch
as a whole, it is insightful to rewrite the condensed stiffness matrix given in
Equation 6.34 using the Woodbury formula (see e.g. Golub and Loan, 1996). In
its most general form, the Woodbury formula states that:
h
i−1
h
i−1
A + UVT
= A−1 − A−1 U I + VT A−1 U
VT A−1 .
(6.48)
(e)
(e)
Using U = KE , V = I and A = kL I, it follows that:
!2
"
#−1
h
i−1
1
1
1
(e)
(e)
(e)
(e)
KE I + (e) KE
.
kL I + KE
= (e) I −
(e)
kL
kL
kL
(6.49)
6.3
ELEMENT CONNECTIVITY PARAMETERIZATION METHOD
125
Substitution in Equation 6.34 yields, while dropping the (e) superscripts for readability:
−1
1
KC = KE I +
KE
.
(6.50)
kL
From this equation, it is clearly seen how the link stiffness kL influences the total
condensed stiffness of the patch, KC : when the link stiffness approaches zero,
KC will tend toward a very compliant stiffness matrix (kL I). On the other hand,
when kL becomes large, the condensed stiffness matrix tends toward the original
element stiffness KE . Equation 6.50 can be reformulated into:
1
−1
−1
−1
K
K−1
=
K
.
(6.51)
I
+
E = KE + [kL I]
C
E
kL
This last relation shows that the condensed stiffness of the patch essentially is
the result of Reuss (series) homogenization of the original element and a material
with a stiffness matrix kL I. A more elaborate investigation of the relation between
homogenization techniques and the present ECP approach is presented by Yoon
et al. (2006b). To determine the way the link stiffness affects the condensed
stiffness of the patch, a matrix norm can be used to turn this relation into a
scalar expression. In this case, the largest eigenvalue norm is used, but similar
results are obtained using other matrix norms (Yoon et al., 2006b). It follows
that:
1
1
1
||KC ||
1
.
(6.52)
=
+
⇒
=
||KE ||
||KC ||
||KE || kL
||KE ||
1+
kL
The quantity ||KC ||/||KE || expresses the relative stiffness of the (condensed)
patch with respect to the stiffness of the original element, in terms of the chosen
matrix norm. It is referred to as the stiffness ratio in this chapter, and Figure 6.5(a) depicts the way this quantity varies with the normalized link stiffness.
Figure 6.5(a) appears to show a smooth transition from a very low stiffness to
a stiffness ratio equal to 1, when the link stiffness varies from low to high. At
the point where the link stiffness equals the norm of the element stiffness matrix
||KE ||, the stiffness ratio equals 0.5. Note, however, that the normalized link
stiffness is shown on a logarithmic scale. On a linear scale, the transition from
low to high stiffness is very abrupt and nearly step-wise.
If the link stiffness would directly be used as a design variable in the topology
optimization problem, the strong non-smoothness would make convergence of the
process very difficult. Clearly, a more gradual relation between the stiffness ratio
and the design variable is desirable. This can be accomplished by introducing
an interpolation function based on the inverse of the stiffness ratio/link stiffness
relation of Equation 6.52, and use that to define a relation between design variable
γ and link stiffness kL :
kL =
||KE ||
1
γ −1
0 ≤ γ ≤ 1.
(6.53)
126
6.3
TOPOLOGY OPTIMIZATION OF SMA ACTUATORS
1
KC
0.8
KE
0.6
0.4
0.2
kL
K
0
−4
−2
10
0
10
2
10
(a)
10
10
E
4
3
10
kL
KE
2
10
1
10
n=1
0
10
n=3
−1
10
−2
10
γ
−3
10
0
0.2
0.4
0.6
0.8
1
(b)
1
KC
0.8
KE
0.6
n=1
0.4
n=3
0.2
γ
0
0
0.2
0.4
0.6
0.8
1
(c)
Figure 6.5: Graphs showing the stiffness ratio versus the normalized link stiffness (a),
the normalized link stiffness interpolation function (b), and the stiffness ratio versus
design variable γ (c). Dashed lines in (b) and (c) illustrate the effect of penalization
with different exponents n.
6.4
ELEMENT CONNECTIVITY PARAMETERIZATION METHOD
127
Ideally, to disconnect elements, the link stiffness should be zero, and to fully
connect elements, the link stiffness should be infinite, as is the case with the
relation given by Equation 6.53. In practice, however, this approach would result
in various numerical problems, such as divisions by zero and ill-conditioning or
non-positive definiteness of the system matrix. For this reason, the link stiffness
relation needs to be modified and a suitable lower and upper limit for kL has to
be determined. Given the monotonic influence of the link stiffness on the stiffness
ratio shown in Figure 6.5(a), suitable limits can be found by allowing deviation
by a certain tolerance ε from the ideal situation:
||KC || (min) k
= ε,
||KE || L
||KC || (max) k
= 1 − ε.
||KE || L
(6.54)
Using Equation 6.52, it follows that:
(min)
kL
=
||KE ||
,
1
ε −1
(max)
kL
=
||KE ||
.
1
1−ε − 1
(6.55)
In this chapter, ε = 0.001 has been used. This results in normalized link stiffnesses in the range of 10-3 to 103 , as indicated in Figure 6.5(a). Inspired by
Equation 6.55, the design variable/link stiffness interpolation function proposed
in Equation 6.53 can be altered as:
kL =
||KE ||
1
ε+γ(1−2ε)
−1
0 ≤ γ ≤ 1.
(6.56)
The advantage of this formulation is that the domain of the design variable γ is still
exactly [0, 1]. This relation is shown in Figure 6.5(b), and the resulting relation
between the stiffness ratio and the design variable γ is depicted in Figure 6.5(c).
Clearly a linear interpolation is achieved. Subsequently, penalization can be added
to this interpolation, which helps designs to converge toward black-white designs
by making intermediate design variable values less efficient. In this chapter, an
exponential penalization function is used, which finally results in the following
interpolation function:
kL =
||KE ||
1
ε+γ n (1−2ε)
−1
0 ≤ γ ≤ 1.
(6.57)
The element stiffness matrix norm ||KE || is evaluated once at the start of the optimization process. In the case studies presented in this chapter, using a constant
penalization exponent n = 3 or n = 4 turned out to improve the convergence
of the optimization process. Validation of the outlined ECP topology optimization approach for various two- and three-dimensional linear and geometrically
nonlinear compliance minimization problems is presented elsewhere (Yoon et al.,
2006a,b).
128
TOPOLOGY OPTIMIZATION OF SMA ACTUATORS
6.4
Problem formulation and regularization
6.4.1
Problems considered
6.4
The aim of this research is to develop a topology optimization procedure for
the design of SMA actuators. In this case, SMA structures are considered that
can generate a displacement under a constant load. The actuation is triggered
by a change in the temperature of the structure. This study is limited to twodimensional plane stress structures, that are subjected to uniform changes in
temperature. The geometry, applied load and boundary conditions of the two
types of design problems considered in this chapter are depicted in Figure 6.6.
Both problems involve a constant load F acting on the SMA structure from the
top of the design domain, and supports are defined at either the whole bottom or
only part of it. The structures are of unit thickness, and symmetry is enforced. In
the analysis, only the left half of the structures is considered. Unless mentioned
otherwise, the topology optimization process is started with a uniform 50% density distribution. The optimization is performed using the Method of Moving
Asymptotes by Svanberg (1987), which is often claimed to be the most effective
optimizer for topology optimization problems.
F
F
U
Fully supported
bottom
16 mm
16 mm
U
20 mm
Partly supported
bottom
7.2 mm 5.6 mm
7.2 mm
Figure 6.6: Geometry, applied load and boundary conditions of problems considered for
SMA topology optimization.
The objective of the design problem is to maximize the stroke of the actuator.
The stroke is evaluated by taking the difference in displacement of the output
point at the lowest and highest temperature:
Objective:
max |UT =TA − UT =TB |.
γ
(6.58)
In all cases studied here, TA = 328K and TB = 343K has been used. In the
6.4
PROBLEM FORMULATION AND REGULARIZATION
129
presented examples, the point where the load is applied is selected as the output
point. In general, other choices are possible. In many topology optimization problems, a mass constraint is commonly added to the problem formulation. However,
it has turned out that for this SMA actuator design problem, the mass constraint
is not necessary. Unlike in e.g. compliance minimization, using more mass does
not by definition result in a better objective value. However, using no material at
all obviously also does not result in an effective actuator. Somewhere inbetween
those two extremes, an optimal amount of mass exists, that can be found by
the optimization process. It is not necessary to constrain the mass using an explicit constraint, in order to create a meaningful and well-behaving optimization
problem.
6.4.2
Numerical artifacts in topology optimization
It was found that in case of SMA actuator design, possibly because of the objective to increase the displacement difference, the optimization tends to exploit
numerical artifacts such as one-node hinges and associated mechanism deformation modes. A clear case of such a mechanism mode is shown in Figure 6.7, which
was encountered during the topology optimization process. From an optimization
point of view, for the given discretized model, exploiting mechanism modes indeed
can lead to very good objective values in certain cases. However, in the actual
design problem, hinges are usually not desired, as particularly for micro-actuators
they simply cannot be manufactured.
Another well-known artifact often encountered in topology optimization is the
100 N
λ=1
λ = 0.5
λ=0
Figure 6.7: Mechanism deformation mode occurring during the design process. The
details on the right show the deformation at a sequence of load factors λ.
130
TOPOLOGY OPTIMIZATION OF SMA ACTUATORS
6.4
so-called checkerboard pattern (see e.g. Bendsoe and Sigmund, 2003). This is
an alternating pattern of elements with maximum and minimum density, that
in the finite element discretization results in a favorable stiffness-to-weight ratio.
Checkerboard pattern problems were encountered less frequently in trials than
mechanism modes, possibly because in contrast to the classical compliance minimization problem, the stiffness-to-weight ratio is not an important factor in the
design process. Regularization of the topology optimization problem is required
to prevent the optimizer to pursue and find unphysical and meaningless solutions.
6.4.3
Effect of filtering
To regularize the topology optimization process and to avoid these problems, filtering is often applied. This is a heuristic technique that consists of replacing the
sensitivity of each element ∂f /∂ρi by a weighted spatial average of the sensitivities in its neighborhood. Filtering has been introduced by Sigmund (1997), and
usually a weighting function Hi is used that decays linearly with the distance to
the center of the element:
˜
1
∂f
=
∂γp
γp
PN
∂f
q=1 Hq γq ∂γq
PN
q=1 Hq
,
(6.59)
with
(
R − dist(p, q) dist(p, q) ≤ R,
Hi =
0
dist(p, q) > R.
(6.60)
Here dist(p, q) is an operator that produces the distance between the center of elements p and q, and R represents the radius of the domain of influence of the filter.
However, the addition of a filtering step in the optimization procedure basically
means that the design description and the sensitivities are no longer consistent.
This can result in rather erratic changes in the objective, systematic worsening
or even a total failure. For the SMA topology optimization, these effects were
frequently encountered, as shown in Figure 6.8 and Figure 6.9. Moreover, the
Sigmund filtering turned out to be unable to prevent the one-node connections
and mechanism modes. A reason why filtering appears to work well for e.g. compliance minimization problems, but fails to regularize the SMA actuator topology
optimization problem, could be that in compliance minimization all sensitivities
have the same sign. Adding material reduces the compliance, or at least does not
increase it. In the presently studied SMA actuator design problems however, the
objective is to maximize the stroke of the actuator. It was observed that both
positive and negative sensitivities of this objective can occur. We speculate that
this fact might be related to the reason why the heuristic filtering approach is not
effective in this class of problems.
6.4
131
PROBLEM FORMULATION AND REGULARIZATION
150 N
0.2
Stroke [mm]
0.15
0.1
0.05
Iteration
0
0
200
400
600
800
Figure 6.8: Obtained design (left) and evolution of the objective function during the
optimization process (right), for a case using a sensitivity filter technique. Note the fact
that the stroke actually decreases as the optimization progresses.
150 N
0.5
Stroke [mm]
0.4
0.3
0.2
0.1
0
0
Iteration
10
20
30
40
Figure 6.9: Obtained design (left) and rather erratic evolution of the objective function
during the optimization process (right), for a case using a sensitivity filter technique. In
this case, the design gradually dissolved and the analysis failed.
132
TOPOLOGY OPTIMIZATION OF SMA ACTUATORS
6.4.4
Nodal design variables
6.4
The use of nodal design variables was found to be a simple and effective method to
regularize the SMA actuator topology optimization problem. Unlike the conventional element-based approach, where design variables are assigned to elements,
the design variables are assigned to the nodes in the finite element mesh, as illustrated in Figure 6.10. The design variable values for each element are then
determined by the average of the nodal design variables of its nodes:
n
γi =
1X
ηj .
n j=1
(6.61)
Here γi represents the density associated with element i, n equals the number of
nodes of the element and ηj are the design variables associated with these nodes.
Figure 6.10: Nodal design variables and corresponding element values for a given layout
of nodal design variables.
In this way, the design space is restricted, and undesirable numerical artifact
patterns, such as checkerboards and one-node hinges, can no longer be represented
in this formulation. Thus, they are banned from the set of possible solutions of
the optimization problem, and physically more meaningful solutions can be generated. For this reason, nodal variable approaches have recently gained popularity
(Bendsoe and Sigmund, 2003, Guest et al., 2004, Matsui and Terada, 2004, see
e.g.). Several alternative regularization methods have been proposed (see e.g.
Poulsen, 2003, Sigmund and Petersson, 1998, Yoon et al., 2004, among others),
but in general these methods tend to be rather involved. In contrast, the nodal
variable approach is simple and straightforward to implement. And unlike in
the case of filtering approaches, the design description and the sensitivities are
formulated in a fully consistent way:
m
m
k=1
k=1
X ∂f ∂γk
∂f
1 X ∂f
=
=
.
∂ηj
∂γk ∂ηj
n
∂γk
(6.62)
Here m is the number of elements node j is associated with, here conceptually
represented by the set k = 1 . . . m.
6.5
SMA TOPOLOGY OPTIMIZATION RESULTS
133
The continuity of the density field, that is inherent to the nodal variable approach, results in the advantage that it prohibits undesired density patterns such
as checkerboards and one-node connections. On the other hand, a disadvantage
of this approach is that it unavoidably implies that edges of the design will consist
of elements with intermediate density values, as shown for an example layout in
Figure 6.10. This makes that optimization results appear to be slightly blurred.
However, by using a suitable interpolation function, the effect of the elements with
intermediate densities on the response of the structure can be controlled. Experience has shown that adequate penalization, such that elements with intermediate
density have a less-than-proportional stiffness, generally improves convergence
and leads to sufficiently clear results.
6.5
SMA topology optimization results
As this is the first research where topology optimization is applied to SMA thermal
actuator design, several studies have been conducted to investigate the nature and
characteristics of this class of problems. In this section, the results of the following
studies are reported:
Effect of the load magnitude
Effect of mesh refinement
Effect of the load direction
Effect of the initial design
Although this limited study does not intend to give a full characterization of the
SMA topology optimization problem, several interesting aspects are revealed by
these investigations, that illustrate the complex nature of this strongly nonlinear
problem. All results are obtained using the nodal design variable formulation
presented in the previous section, and for the problem formulations given in Section 6.4.1.
6.5.1
Effect of load magnitude
Unlike in linear problems, the SMA topology optimization problem shows a clear
effect of the load magnitude on the resulting design. Both the fully and partly
supported design problems, introduced before in Figure 6.6, have been evaluated
for loads of 50, 75, 100 and 150 N. Results for the fully supported bottom case
are shown in Figure 6.11, and for the partly supported case in Figure 6.12. Note
that the designs obtained in the various cases are considerably different, clearly
showing the influence of the load magnitude on the optimization process.
There appears to be a tendency to use more material in case of larger loads.
This can be understood from the fact that the SMA material is most effective
134
TOPOLOGY OPTIMIZATION OF SMA ACTUATORS
50 N
75 N
100 N
150 N
6.5
Figure 6.11: Topologies found by topology optimization in the fully supported case, for
load magnitudes of 50, 75, 100 and 150 N.
6.5
SMA TOPOLOGY OPTIMIZATION RESULTS
50 N
75 N
100 N
150 N
135
Figure 6.12: Topologies found by topology optimization in the partly supported case, for
load magnitudes of 50, 75, 100 and 150 N.
136
6.5
TOPOLOGY OPTIMIZATION OF SMA ACTUATORS
when in the low temperature situation, it is in the transformed, R-phase state
while in the high temperature situation, it should be preferably in the austenite
phase. That situation leads to the largest difference in mechanical properties
of the structure, which will result in the largest stroke. Inspection shows that in
optimized structures, the strain distribution in large parts of the structures indeed
is as described here. This is shown in Figure 6.13, where the volume fraction of
the R-phase is shown together with the deformed structures, in both temperature
cases for the final design in the 75 N fully supported case. The volume fraction
of the R-phase is used here as a comparison measure, and is defined as:


0
εe ≤ εe(1)
(6.63)
φR = εe(1) < εe ≤ εe(2) (εe − εe(1) )/(εe(2) − εe(1) )


εe > ε2
1
The transition strains εe(1) and εe(2) are the strain levels at which the piecewise
linear stress-strain relation switches from one branch to another, as derived in
Appendix B. For the definition of this comparative phase fraction measure φR , it
is assumed that a linear relation exists between the effective strain and the R-phase
volume fraction, which is not necessarily correct. However, it is a useful measure
to compare to what extent the material is in a state where the transformation
from austenite to R-phase is ongoing or has completed, or in a state where it is still
about to begin. As can be seen in Figure 6.13, in the high temperature situation
hardly any material is in the transformed state, whereas in the low temperature
situation practically the complete structure has transformed into the R-phase.
T = 328 K
T = 338 K
T = 328 K
T = 338 K
R−phase [%]
100
90
80
70
60
50
40
30
20
10
0
Figure 6.13: Deformed structures in the low and high temperature situation for the fully
supported 75 N case (left), and volume fractions of R-phase in both situations (right).
Displacements have been scaled by a factor 5.
The physical explanation for the designs that are obtained by the topology optimization process is that structures are generated for which the effective strains
are such, that in the low temperature phase nearly all material transforms into
6.5
SMA TOPOLOGY OPTIMIZATION RESULTS
137
the R-phase, whereas in the high temperature phase nearly all material remains in
the austenite phase. This apparently results in the most effective structure. This
interpretation also explains why generally with higher load levels designs with
thicker structural members are obtained: in order to keep the strain in this “most
effective range”, the structure has to become stiffer when the load is increased.
Next to that, it is observed that often winding or meandering structures are generated, and configurations are obtained that show a considerable amount of bending
deformation. Strains in the right magnitude help to increase the difference between the low and high temperature configuration, whereas bending deformations
in many cases can lead to larger displacements at the point of interest. Note that
particularly the winding structures, such as, for example, those seen in the 150 N
partly supported case, are rather uncommon in topology optimization results.
In the partly supported 50 N case (Figure 6.12), some excess material remains
on the top left and right side, that is not connected to the final structure. This is
possible because no mass constraint is used, so there is no motivation for the optimizer to remove parts that do not contribute to the objective. Because the void
elements are not exactly void numerically, but have some finite stiffness, generally
material that is not actively contributing but obstructing the actuator by adding
stiffness, is removed during the optimization. In this 50 N case, apparently, the
disconnected parts remaining from earlier design configurations in the optimization process do not adversely contribute to the resulting stroke. Likewise, in the
100 N partly supported case, two inward-pointing “spikes” remain at the bottom
of the structure, apparently because of the same reason.
A final aspect that deserves attention is a problem that occurred in the 150 N
fully supported case, but a tendency towards this problem was also observed in
other cases. The original result for this case is depicted in Figure 6.14, which shows
a structure in which the part where the load is applied and the rest of the structure
(the “arms”) are nearly disconnected. No one-node hinges are formed, since that
is impossible in the nodal design variable setting. However, effectively still hingelike structures are created, by areas of very low stiffness, as is clearly shown in the
deformed configuration in Figure 6.14. This is an undesired effect, because the
aim is to obtain a clear topology with distinct solid and void regions, that actually
is manufacturable. Possibly, the very compliant regions acting as hinges could be
realized by using a secondary material or by using porous microstructures, but
this is not the objective of the present design study. In this case, fortunately,
the tendency of this problem to form such hinges could be suppressed by slightly
extending the non-design domain. The result obtained with that modification is
the one shown in Figure 6.12. Strikingly, the performance of the modified case
actually turned out to be better than that of the structure with the hinges (a
stroke of 0.3758 versus 0.3401 mm).
As a comparative study, it is interesting to investigate the performance of
all obtained designs at various load levels for different load magnitudes than
the design load. For this purpose, scans have been made at load levels ranging
from 0 to 200 N, with increments of 25 N, and the strokes of all actuators under
138
TOPOLOGY OPTIMIZATION OF SMA ACTUATORS
6.5
150 N
150 N
Figure 6.14: Optimal design obtained in the 150 N fully supported case with the original
non-design domain. Note the fact that the structure is nearly disconnected.
6.5
SMA TOPOLOGY OPTIMIZATION RESULTS
139
these loads have been plotted in Figure 6.15. It can be seen that for both the
fully and partly supported case, the 50 N designs are clearly local optima, since
all structures designed for larger loads outperform these designs. This could be
related to the strongly nonlinear nature of the material model, which possibly at
lower load levels results in a problem that has different characteristics than at
higher load levels. For example, the fact whether or not the phase transformation
is induced by the loading condition significantly affects the local apparent stiffness.
This can definitely influence the sensitivities, and, accordingly, affect the direction
the structural optimization process takes in the design space.
0.4
Stroke [mm]
0.3
0.2
50 N
75 N
100 N
150 N
0.1
Applied load [N]
0
0.6
0.4
50
75
100
50
75
100
150
200
Stroke [mm]
50 N
75 N
100 N
150 N
0.2
Applied load [N]
0
150
200
Figure 6.15: Actuator stroke as a function of the applied load, for the designs found
by topology optimization in case of a fully supported (top) and partly supported (bottom)
bottom edge.
140
6.5
TOPOLOGY OPTIMIZATION OF SMA ACTUATORS
For larger loads, some other local optima are revealed, and particularly the
partly supported 100 N case is clearly inferior to the 150 N design loaded at 100
N. The nonlinearity of the SMA actuator design problem is likely to result in a
nonconvex topology optimization problem, where it cannot be guaranteed that a
global optimum is obtained. However, performing the design at different loads
and making load-scans, as shown in Figure 6.15, increases the chance of finding
the optimal design layout.
6.5.2
Effect of mesh refinement
In case of the previously shown results, a mesh of 25 by 40 elements has been
used, which is the default mesh for all problems studied in this chapter. Only
the left half of the structure is simulated, as symmetry is assumed and enforced,
without loss of generality. In this subsection, the same design problems have
been evaluated using a refined mesh of 50 by 80 elements. Results for loads of
50, 75 and 100 N for the fully and partly supported bottom case are shown in
Figure 6.16. The strokes for coarse and fine cases are listed in Table 6.2.
Table 6.2: Strokes for optimal designs for various load cases, on coarse and fine meshes.
Bottom boundary
condition
Fully supported
Fully supported
Fully supported
Fully supported
Partly supported
Partly supported
Partly supported
Partly supported
Load magnitude [N]
50
75
100
150
50
75
100
150
Stroke [mm]
(25×40 mesh)
0.0449
0.1835
0.3237
0.3758
0.0826
0.2617
0.2704
0.5010
Stroke [mm]
(50×80 mesh)
0.0800
0.3423
0.3269
(Analysis failed)
0.1189
0.4515
0.5507
(Analysis failed)
Clearly the finer mesh allows for much more detail in the final design, since
no mesh-independence measures or enforced length scales have been used in these
studies, in order to reveal the effect of mesh refinement. For the 50 N cases, the
topologies show some resemblance to the coarse mesh solutions, although some
more detail and thinner structures have developed. The refinement has resulted
in all cases in a clear increase in performance, as shown by the strokes listed in
Table 6.2. Unlike in the 50 N cases, the other cases show no correspondence in
optimal topologies between the coarse and refined meshes. Interesting winding
structures are again obtained, particularly in the 100 N cases. Generally, the
design evolution during the optimization process is more complex and sometimes
exhibits drastic changes during optimization. This was also observed for the
coarse mesh case, but is even more apparent in the refined case. Possibly the
6.5
SMA TOPOLOGY OPTIMIZATION RESULTS
50 N
50 N
75 N
75 N
100 N
100 N
141
Figure 6.16: Topologies found by topology optimization in the fully supported (left column) and partly supported (right column) cases, using a refined mesh, for load magnitudes of 50, 75, 100 N.
142
6.5
TOPOLOGY OPTIMIZATION OF SMA ACTUATORS
fact that the design space is greatly enlarged contributes to this effect. As an
illustrative example of the interesting design evolution that occurs during the
optimization process, the objective history of the 75 N fully supported case is
depicted in Figure 6.17, together with some intermediate designs. The objective
curve, although it varies smoothly with the iteration number (due to the use of
consistent sensitivities), shows several “bumps” at points where drastic topological
changes occurred.
Note that the number of iterations is rather large in this case, i.e. over 800.
Most coarse mesh design problems converged in 200-300 iterations, but in the
refined mesh cases, the number of design variables is larger. This apparently
makes that the optimizer needs more iterations to determine the optimal topology.
In addition, optimizer settings were chosen such that the process would not stop
prematurely. At roughly iteration 400 the main features of the final design were
already clear in the present case.
0.4
Stroke [mm]
(e)
0.3
(d)
(c)
(e)
(d)
0.2
(b)
0.1
(c)
(a)
(b)
Iteration
(a)
0
0
200
400
600
800
Figure 6.17: Evolution of the objective and the design for the 75 N fully supported case
on a refined mesh. Note the rather drastic changes in the topology during the optimization
process.
No results are shown for the 150 N cases on the refined meshes, because convergence problems were encountered during the optimization process. It turns out,
that these cases experience buckling in the low temperature case, which is subsequently exploited by the optimizer. The process is driven towards a very unstable
structure which is close to a bifurcation point. This eventually results in a failure
to converge within the maximum number of increments. An unconverged result,
6.5
SMA TOPOLOGY OPTIMIZATION RESULTS
143
obtained after 335 iterations in the fully supported case, is shown in Figure 6.18,
together with the evolution of the objective function during the optimization. At
roughly iteration 150, the buckling phenomenon starts to develop in the low temperature structure. Very high objective values are obtained, compared to those
found in comparable cases: at its peak value in iteration 287, the stroke equals
an unparalleled 8.5 mm. Deformed structures for the low and high temperature
situation are shown in Figure 6.18 as well. In spite of the fact that nodal design
variables are used, a structure that effectively acts as a hinge has formed by an aggregate of elements with intermediate density. It should be noted, however, that
this concerns an unconverged design. Clearly, the low temperature case shows a
very large deformation, due too the buckling that has occurred. Note, by the way,
that the void elements seem to be excessively distorted, but this does not lead
to numerical problems because of the ECP approach. The sudden drop in objective in the next step is caused by the fact that the structure no longer buckles,
due to the change in design suggested by the optimization algorithm. Although
in subsequent iterations the objective starts to improve again, at iteration 335
convergence can not be reached and the process is halted.
In these 150 N cases, a fundamental difficulty of nonlinear problems is encountered: the possibility of numerical complexities such as multiple solutions,
instability, buckling and bifurcation points. There are two approaches to deal
with this situation: one possibility is to try to avoid these phenomena by, for
example, including a buckling constraint, or a constraint on the displacement.
Another option, however, is to embrace these features of the problem, since a
careful exploitation of buckling phenomena could be very interesting for the design of high-performance SMA binary actuators. The objective values found in
the present case clearly illustrate the potential. Care should be taken to make
sure that, when making use of buckling, the structure indeed can be cycled between the two configurations that are found by the analysis used here; possibly
another simulation approach is required in that case. However, this subject is not
pursued further in this thesis.
6.5.3
Effect of load direction
To study the effect of the load direction, the fully and partly supported cases with
a 150 N load were also evaluated for upward loads. This resulted in completely
different topologies, as shown in Figure 6.19. Possibly, geometrical nonlinearities,
in combination with the complex nonlinear SMA material model, are the cause
of these differences.
It is interesting to subsequently evaluate the designs obtained with a downward
load using an upward load, and vice versa, to check whether any local optima can
be identified. The results of these cross-examination checks are listed in Table 6.3.
It can be seen that the solution obtained for the fully supported downward case
is a local optimum, since the structure designed with an upward load, loaded with
the downward load, performs 4% better. In all other cases, the design load clearly
144
TOPOLOGY OPTIMIZATION OF SMA ACTUATORS
6.5
150 N
10
Stroke [mm]
Iteration 287
8
6
4
2
0
−2
0
T = 328 K
Iteration
100
200
300
T = 338 K
Figure 6.18: Unconverged result of topology optimization on a refined mesh in case of a
150 N load in the fully supported case (left), together with the evolution of the objective
(middle) and the deformed structures at iteration 287.
6.5
SMA TOPOLOGY OPTIMIZATION RESULTS
150 N
150 N
150 N
150 N
145
Figure 6.19: Obtained designs for a 150 N downward and upward loads, for cases with
fully (left) and partly supported bottom (right).
Table 6.3: Stroke of SMA actuator structures evaluated for different loading directions.
Design case
Fully supported
Fully supported
Fully supported
Fully supported
Partly supported
Partly supported
Partly supported
Partly supported
Design load
150 N down
150 N up
150 N up
150 N down
150 N down
150 N up
150 N up
150 N down
Applied load
150 N down
150 N down
150 N up
150 N up
150 N down
150 N down
150 N up
150 N up
Stroke [mm]
0.3758
0.3919
0.3117
0.1867
0.5010
0.4329
0.3635
0.3418
146
6.5
TOPOLOGY OPTIMIZATION OF SMA ACTUATORS
gave the best performance. It is also interesting to note that there are striking
differences in the performance of designs under upward or downward loading. This
also illustrates the nonlinearity of this design problem. Geometrically nonlinear
effects can lead to differences in the deformation behavior with regard to the
loading direction. For instance, the design obtained for the 150 N downward load
in the fully supported situation is found to have a stroke of 0.3758 mm in case of
downward loading, but only yields 0.1867 mm (less than half) when loaded in the
upward direction.
6.5.4
Design improvement from a baseline design
The effect of the initial design used in the topology optimization process is studied
by a practical problem concerning the improvement of an existing design, referred
to as the baseline design. The baseline design used in this study is depicted
in Figure 6.20, and is a classical meander-like structure as conventionally used
for planar SMA actuators. It is attempted to improve this design using the
developed topology optimization procedure. In order to make a fair comparison,
the performance of the baseline design is optimized first, by selecting the load
magnitude for which it delivers its maximum stroke. For the present design, this
optimal load turns out to be 200 N (see Figure 6.20).
200 N
0.15
Stroke [mm]
0.1
0.05
Applied load [N]
0
100
200
300
400
Figure 6.20: Baseline actuator design and its performance versus various applied loads.
The maximum stroke is obtained for a load of 200 N.
Instead of starting with a uniform density distribution of 50%, the topology
optimization procedure was started with the baseline design. Generally, for nonconvex problems such as the present SMA topology optimization problem, the
starting configuration can have a large effect on the final solution. Many local
optima might exist, and which one the process will converge to, depends on the
chosen initial design. To avoid any bias, normally uniform density distributions
6.5
147
SMA TOPOLOGY OPTIMIZATION RESULTS
200 N
0.4
Stroke [mm]
Optimized design
0.3
0.2
Baseline design
0.1
Applied load [N]
0
100
200
300
400
Figure 6.21: Actuator design obtained by topology optimization starting from the baseline design, and its performance versus various applied loads, together with the performance curve of the baseline design. The optimized design outperforms the baseline
designs for the entire range of loads considered.
are used in topology optimization. In this case, however, the process is deliberately started from the baseline design in order to investigate the ability of the
topology optimization procedure to improve the performance of an existing design.
The resulting design is shown in Figure 6.21. The topology of the structure
has changed, the two sides have merged in the center. In addition, the structure has become less slender, and some slight winding can be recognized. But
most interesting is the striking difference in performance: at the design load, the
optimized design shows a 134% better performance compared to the baseline design. Moreover, although this is indeed the peak value of the optimized design,
also at other load levels the structure obtained by topology optimization clearly
outperforms the baseline design, as shown by the load-stroke data in Figure 6.21.
It is interesting to examine whether such well-performing designs can also be
found when starting from a uniform density distribution, in the present case.
Using boundary conditions similar to those used in the baseline design, as well
as for the fully and partly supported bottom cases used in the previous studies,
topology optimization has been performed as well for this 200 N upward load,
and resulting topologies are depicted in Figure 6.22.
The strokes of all cases relevant to this problem are listed in Table 6.4. All
designs obtained by topology optimization show a large improvement with respect
to the baseline design, of the order of a factor 2 or better. The design obtained
with the partly supported bottom boundary condition has the smallest stroke of
the optimized designs, and clearly is a local optimum. The structure obtained
in the fully supported case could also be realized in the partly supported case,
148
TOPOLOGY OPTIMIZATION OF SMA ACTUATORS
6.5
200 N
200 N
200 N
Figure 6.22: Optimal designs obtained for a 200 N upward load, for different boundary
conditions. Optimization is started from a uniform density distribution.
6.6
SMA TOPOLOGY OPTIMIZATION RESULTS
149
Table 6.4: Stroke of SMA actuator structures.
Design case
Baseline design
Optimized design starting from baseline
Optimized design starting from uniform density distribution
200 N upward load, fully supported bottom
200 N upward load, partly supported bottom
Stroke [mm]
0.1401
0.3273
0.3297
0.3321
0.2780
and shows a better objective (+19%). The design obtained in the case that used
the baseline boundary conditions and that was started from a uniform density
distribution (Figure 6.22, top figure) has a very interesting topology, of which the
upper part actually resembles the shape obtained by the optimization starting
from the baseline design. Compared to the latter, it shows an even slightly better
performance, but it is debatable whether this difference is significant. It can be
argued that the less than perfect approximation of the boundary shape of the
structure due to the coarseness of the mesh and the presence of elements with
intermediate density values make that the accuracy of the analysis is not optimal.
However, this comment is not unique to the present SMA optimization problem,
but applies to most topology optimization procedures in general.
It is conceivable, that SMA topology optimization is more sensitive to analysis
inaccuracies than topology optimization problems involving linear materials, since
the material behavior varies significantly with the strain state and magnitude, and
the performance of the active structures is closely linked to this SMA material
behavior. However, this difference in sensitivity is difficult to quantify, since there
is no clear way to compare such different problems. Still, it would be of interest to
study the effect of using different analysis and design meshes, and to investigate
how the results of procedures using more refined analysis meshes compare to
the results obtained in the present study. Moreover, given the finite accuracy of
topology optimization design models in general, also for the present SMA case
it is recommended to extract the shape of the design obtained by the topology
optimization process, and to perform a subsequent shape optimization using a
high-fidelity analysis model, as demonstrated before in e.g. Chapter 4. Such
subsequent detailed design, as well as the suggested refinements, are however not
the focus of the present chapter, and can be seen as extensions of the procedures
introduced here. The redesign example discussed in this subsection convincingly
illustrates the merit of the proposed topology optimization procedure, by more
than doubling the performance, and in addition the results in Figure 6.21 and
Figure 6.22 show the influence of initial design and the boundary conditions on
the topology optimization results.
150
6.6
TOPOLOGY OPTIMIZATION OF SMA ACTUATORS
6.6
Conclusions
The successful application of a topology optimization method to the design of
shape memory alloy thermal actuators has been presented for the first time. The
results presented in this chapter illustrate that the proposed procedure is capable
of generating a wide variety of innovative, non-trivial, well-performing designs,
and that in addition it is able to improve existing structures significantly. The
effectiveness of the procedure relies on two aspects: the efficient and historyindependent SMA material model presented in Section 6.2 (see also Chapter 3),
and the new efficient and robust ECP topology optimization paradigm. The presented SMA model exploits the small hysteresis of the pseudoelastic behavior of
the R-phase/austenite transformation. The resulting formulation combines an
adequate representation of experimental observations with excellent suitability
for optimization. The history-independent nature of the model enables adjoint
sensitivity analysis, which is essential for performing topology optimization. The
strong nonlinearity of the model nonetheless requires an elaborate adaptive solution strategy in order to ensure the robustness of the finite element simulations.
Also, it should be noted that the SMA model does not cover intricate phenomena
such as the effect of crystallographic texture or peculiarities of the material behavior under non-proportional multi-dimensional loading. However, as explained,
such detailed modeling would be neither justified nor practical, given the present
state of experimental findings and the intended use of the model for (topology)
optimization.
Difficulties that have seriously complicated the application of topology optimization to nonlinear problems have been shown to be alleviated by the introduced Element Connectivity Parameterization method. Due to the radically
different concept of parameterizing the connectivity between elements instead
of modifying the properties of elements themselves, the problem of the conventional density-based method with e.g. excessive distortion of compliant elements
is avoided. In addition, in the ECP method there is no need to arbitrarily define
a relation between the design variables and the material properties, and the associated complexities in sensitivity analysis and convergence of the optimization
process are consequently prevented. These aspects are of particular importance
for problems involving complex nonlinear material models, as in the present case
of shape memory alloys. Since the ECP method presented in this chapter does
not have the disadvantage of increasing the number of degrees of freedom, and
given its effectiveness and advantages, it is expected to be an attractive option
for many nonlinear topology optimization problems.
It was found that restriction of the design space by the use of nodal design
variables was effective in avoiding undesired numerical artifacts, such as unrealistic designs containing one-node hinge patterns. In contrast, the heuristic filtering technique, that has been applied successfully in compliance minimization
problems, proved to be ineffective and unable to regularize the topology optimization problem for SMA structures. The strong nonlinearity of the SMA material
6.7
ACKNOWLEDGMENTS
151
combined with the geometrically nonlinear formulation result in a nonconvex optimization problem, which poses a challenge for the optimization algorithm. In
several cases, clearly local optima were found. A study of the characteristics of
the SMA topology optimization problem has revealed that it shows a strong dependence on the magnitude and direction of the applied load. Interestingly, a
volume constraint turned out to be unnecessary, as there apparently exists an
optimal amount of material for a given problem setting.
The displacements and strokes of the actuators remained relatively modest in
most of the design cases. This is directly related to the fact that the transformation strain in case of the R-phase transformation is rather small. In addition, the
considered problems all involved planar structures. In configurations where also
transverse bending is possible, as in shell structures, much larger displacements
can be realized. In one case, the optimization process was halted because the
analysis failed to converge. The convergence difficulties were related to a buckling phenomenon that was observed in the structure. A methodology how to avoid
or rather exploit this behavior is yet to be realized. Also, many existing SMA
actuators are actuated by a nonuniform temperature field, generated by resistive
heating. The problems considered in the present chapter were limited to a uniform
temperature, but following the approach of Sigmund (2001) an electrothermomechanical formulation is also possible. These and other extensions of the presented
topology optimization technique will be the subject of future investigations.
6.7
Acknowledgments
The authors would like to thank Krister Svanberg for giving permission to use his
MMA code for this research.
152
TOPOLOGY OPTIMIZATION OF SMA ACTUATORS
6.7
Based on: Langelaar, M. and van Keulen, F (2006). Sensitivity analysis of shape
memory alloy shells. Computers and Structures, in review.
Chapter
7
SMA Sensitivity Analysis
This chapter presents procedures to perform efficient design sensitivity analysis for the
shape memory alloy (SMA) actuators considered in this thesis. Design sensitivities
are derivatives of response quantities with respect to the design variables. Availability of sensitivity information at low computational cost can dramatically improve the
efficiency of the optimization process, as it enables use of efficient gradient-based optimization algorithms. The formulation and computation of design sensitivities of SMA
shell structures using the direct differentiation method is considered, in a steady state
electro-thermo-mechanical finite element context. Finite difference, semi-analytical and
refined semi-analytical sensitivity analysis approaches are discussed and compared in
terms of efficiency, accuracy and implementation effort, based on a representative finite
element model of a miniature SMA gripper.
7.1
Introduction
Shape memory alloys (SMAs) are active materials with a high power density,
capable of producing comparatively large actuation strains and stresses (Otsuka
and Wayman, 1998). Their actuation properties originate from a solid-state phase
transformation, which is affected by changes in temperature or stress, and strains
associated with this transformation can be used for actuation. SMA actuators
are widely used in wire or spring configurations, but upcoming applications in,
e.g., medical instrumentation or microsystems also demand more complex shapes.
However, designing effective multi-dimensional SMA actuators is a challenging
task, due to the complex behavior of the material and the fact that often electrical, thermal and mechanical aspects have to be considered simultaneously. For
153
154
SMA SENSITIVITY ANALYSIS
7.2
this reason, interest in the application of systematic computational design approaches, such as design optimization techniques, to the design of SMA structures
is increasing.
Design optimization has been applied to SMA wire-based configurations (Birman et al., 1996, Troisfontaine et al., 1999) and to SMA structures modeled by
analytical models (Lu et al., 2001). However, the models used in these studies
cannot be extended to more general SMA structures. In addition, others have applied peak stress reduction algorithms (Mattheck and Burkhardt, 1990) to more
complex SMA designs (Kohl et al., 1999, Skrobanek et al., 1997). However, their
approach is less versatile than the more general and systematic design optimization techniques developed in the structural optimization community, based on a
formal mathematical problem formulation combined with optimization algorithms
(see e.g. Haftka and Gürdal (1992) for an overview).
To realize efficient SMA design optimization suited for a wide range of problems of realistic complexity, theavailability of sensitivity information is crucial.
Various approaches exist to perform sensitivity analysis, and the available techniques and their characteristics are discussed extensively in dedicated books and
review papers (Choi and Kim, 2005a,b, Haftka and Gürdal, 1992, Kleiber et al.,
1997, Van Keulen et al., 2005). An essential aspect is that with the appropriate
techniques, design sensitivities can often be obtained at low computational cost,
compared to the response evaluation itself. This advantage is particularly evident
in the case of history-independent nonlinear models (Kleiber et al., 1997). In that
case the analysis itself is quite expensive, since the nonlinearity usually requires
an incremental-iterative solution strategy. In comparison to this significant computational effort, the sensitivity analysis for path-independent models is far less
demanding.
The sensitivity analysis presented in this chapter is based on a simple constitutive model for SMA behavior based on the R-phase transformation in NiTi
(Langelaar and Van Keulen, 2004b) (Chapter 3). In contrast to the majority of
existing SMA models, this model is history-independent and therefore well suited
for use in sensitivity analysis and design optimization. This chapter starts with
an overview of various sensitivity analysis approaches in Section 7.2. The present
work is aimed particularly at SMA shell structures, as these can generate large
actuator displacements through bending deformation. The most general case of
actuation by means of resistive heating is considered, which requires a sequentially coupled electrical, thermal and mechanical finite element analysis. Simpler
situations, e.g. a given temperature distribution, are also covered by this general formulation. Section 7.3 discusses the derivation and computation of design
sensitivities for SMA shell structures in this setting. Numerical results based
on finite difference, semi-analytical and refined semi-analytical sensitivity analysis approaches are subsequently presented and discussed in Section 7.4, using a
representative case study of a miniature SMA gripper, followed by conclusions.
7.2
SENSITIVITY ANALYSIS APPROACHES
7.2
155
Sensitivity analysis approaches
Several approaches exist to perform sensitivity analysis, and the available techniques and their characteristics are briefly reviewed here. For more detailed information, the interested reader is referred to dedicated books by Haftka and Gürdal
(1992), Kleiber et al. (1997), Choi and Kim (2005a,b) and also the recently published review by Van Keulen et al. (2005).
In the following subsections, the system response of interest will be denoted
by f and the state variables by u. For simplicity, only a single design variable
s is considered, without loss of generality. The response is considered to be a
function of both u and s, where the state variables also implicitly depend on
the design variable, i.e. f = f (u(s), s). Adjoint formulations are not considered,
since for the intended shape optimization problems they are not expected to offer
significant advantages over the direct differentiation method.
7.2.1
Variational approaches
Figure 7.1 shows a schematic overview that illustrates how the various approaches
to perform sensitivity analysis are related to the governing equations. In so-called
continuum or variational approaches, first the continuum governing equations are
differentiated with respect to the design variables, and the resulting sensitivity
equations are subsequently discretized. In the discrete approaches, on the other
hand, the design differentiation is applied to the discretized governing equations.
Continuum governing equations
Design differentiation
Discretization
Continuum sensitivity equations
Discrete governing equations
Discretization
Finite
differences
(Semi)­analytical
design differentiation
Sensitivity code generation
Design sensitivities
Figure 7.1: Overview of design sensitivity analysis techniques.
In the variational approach, the sensitivity equations are not linked to the
discretization used for the analysis. This means that theoretically, a different
discretization can be used, which makes this approach more flexible than the
other methods. However, in practice, in order to make the method efficient and
156
7.2
SMA SENSITIVITY ANALYSIS
to limit inconsistencies, it is usually beneficial to use the same discretization.
Another advantage is that in case internal routines and quantities of an analysis
program can not be accessed, the variational approach could be considered for
sensitivity analysis (Choi and Duan, 2000). For the present SMA simulation,
the variational approach will not be considered, since other methods are at least
equally efficient, and require less implementation effort.
7.2.2
Discrete approaches: finite differences
Discrete design sensitivity approaches are based on the discretized system equations. In the following subsections, the system response of interest will be denoted
by f and the state variables by u. For simplicity, only a single design variable s
is considered, without any loss of generality. The response is considered to be a
function of both u and s, where the state variables also implicitly depend on the
design variable, i.e. f = f (u(s), s).
A popular sensitivity analysis approach is the use of finite differences to approximate the design derivatives. One or more additional simulations are required
to evaluate the responses in perturbed design configurations f (s ± ∆s), after completing the nominal analysis f (s), e.g.:
Forward finite difference:
Central finite difference:
f (s + ∆s) − f (s)
df
≈
ds
∆s
f (s + ∆s) − f (s − ∆s)
df
≈
ds
2∆s
(7.1)
(7.2)
Critical for the accuracy of the obtained derivatives is the selection of the
proper design perturbation ∆s. A larger perturbation worsens truncation errors
due to the truncation of the Taylor series, a smaller perturbation can lead to
round-off or condition errors. In practical situations, it can be difficult to select
a suitable perturbation, particularly in case of “noisy” numerical models. With
noise is meant the erratic variation of the response due to numerical reasons,
for example due to adaptive mesh refinement or finite convergence tolerances for
iterative procedures. When due to this noisy character the response is not a
smooth function of the design variable, the finite difference approach can produce
highly inaccurate results.
Next to these weaknesses, the fact that the finite difference method requires
one or more additional function evaluations for every design variable makes it
rather costly. To reduce the computational effort, use can be made of efficient reanalysis techniques, that can evaluate or approximate the perturbed responses at
a fraction of the computational cost of a full analysis. Particularly in case of pathindependent nonlinear problems, the efficiency of the finite difference approach
can be improved significantly, when after the nominal analysis the perturbed
analysis is started from the nominal solution, instead of starting from the initial
configuration. In this setting, care must be taken to correct for the finite residual that remains after convergence of the solution procedure, to ensure accurate
7.2
SENSITIVITY ANALYSIS APPROACHES
157
sensitivity results when using reanalysis. An effective remedy has been suggested
by Haftka (1985), who has shown that subtracting the nominal residual from the
applied load in the perturbed problem alleviates this problem.
An additional improvement of the basic finite difference method has initially
been suggested by Lyness (1967), Lyness and Moler (1967), and recently described
by Squire and Trapp (1998). By using complex arithmetic, the finite difference
approach can be made insensitive to the numerical round-off error, which allows
the use of very small perturbations. The complex forward difference equation for
the design sensitivity with respect to s is given by:
df
= Im(f (s + i∆s))
ds
(7.3)
Drawbacks of this approach are that the entire analysis has to be performed using
complex operations, which is up to four times as expensive as a normal analysis using only real numbers, and that twice the amount of memory is required.
Naturally, full access to the source code is also necessary for this approach. The
implementation effort, on the other hand, is reported to be relatively modest
(Anderson and Nielsen, 2001).
7.2.3
Discrete approaches: semi-analytical design sensitivities
An apparently straightforward way to compute sensitivity information based on
the discretized system equations is to differentiate the discrete equations analytically with respect to the design variables. However, it turns out that for most
models of realistic complexity, this differentiation is tedious, error-prone and requires a large implementation effort. Therefore, the popular semi-analytical methods use analytical design differentiation only at the global level of the discretized
equations, and the lower level derivatives are approximated by finite difference
techniques. To illustrate this method, a linear finite element setting is considered, governed by:
Ku = f .
(7.4)
This is a common equation in a solid mechanics context, where K represents
the tangent stiffness matrix, u the displacement vector and f the load vector.
The load vector and the stiffness matrix are considered to be independent of the
displacements. In a general design optimization setting, K and f can depend on
design variable s, hence also the state variable vector u implicitly depends on s.
The sensitivity of a response f (u(s), s) is now given by:
df
∂f
∂f T du
=
+
.
ds
∂s
∂u ds
(7.5)
It is seen, that in order to evaluate this response sensitivity, the sensitivity of
the state variables du/ds is required. An equation to evaluate this quantity is
158
7.2
SMA SENSITIVITY ANALYSIS
obtained by design differentiation of the governing equations (Equation 7.4):
dK
du
df
du
df
dK
u+K
=
⇒ K
=
−
u .
(7.6)
ds
ds
ds
ds
ds
ds
This equation for the design sensitivity has the same structure as the original governing equation (Equation 7.4). The right-hand side is referred to as the pseudoload, since “loading” the structure with this “load” results in the sensitivities of
the displacements. In the semi-analytical method, the design derivatives df /ds
and dK/ds needed to compute this pseudo-load are not evaluated analytically,
but by finite differences. Usually forward finite differences are used:
dK
K(s + ∆s) − K
≈
,
ds
∆s
df
f (s + ∆s) − f
≈
.
ds
∆s
(7.7)
When a direct solver is used, the system matrix K has already been decomposed, and du/ds can be computed by backsubstitution, at a fraction of the
computational cost of the analysis itself. When an iterative solver is used, the
preconditioning effort can be reused.
Slender structures where elements undergo significant rotations have shown
reduced accuracy of semi-analytical sensitivities. This problem is attributed to
increased errors in the finite difference components used for the pseudo-load calculation. Various remedies have been suggested, including the use of higher order
finite difference formulae, an “exact” formulation (Olhoff et al., 1993) and the
so-called “refined” semi-analytical formulation, which is based on exact differentiation of the rigid body modes of elements (De Boer and Van Keulen, 2000b, Van
Keulen and De Boer, 1998b). Element rigid body modes are displacement fields
that do not generate internal forces, i.e. on element level the following holds:
Kur = 0.
(7.8)
Design differentiation of this condition gives
dK
dur
dK
dur
ur + K
=0 ⇒
ur = −K
.
ds
ds
ds
ds
(7.9)
The sensitivity of rigid body modes ur is easy to compute analytically, and the
part of the pseudo-load vector term associated with these modes (dK/ds)ur can
therefore be evaluated exactly, using the relation above. An alternative to avoid
the finite difference inaccuracies might also be to use the complex variable approach discussed in the previous section.
7.2
7.2.4
SENSITIVITY ANALYSIS APPROACHES
159
Discrete approaches: sensitivity code generation
A third discrete approach is to consider not even the discrete governing equations, but operate directly on the source code by which they are implemented.
Programs have been developed that parse this source code and generate derived
code that, when compiled, is able to compute the design derivatives directly. Some
well known programs are for example ADIFOR (Bischof et al., 1996) and ADOLC (Griewank et al., 1996), among others. This technique could be regarded as
automated analytical differentiation, and therefore the resulting sensitivities are
traditionally referred to as “automated derivatives” or “computational derivatives”.
Sensitivity code generation techniques cannot exploit the structure of the governing equations, such as, for example, the reuse of the decomposed tangent stiffness matrix in semi-analytical approaches. Also, fully analytical partial design
derivatives not seldomly require considerably more computational effort compared to the finite difference approach used in semi-analytical methods. Still,
potentially this approach allows for the implementation of design sensitivities
with minimal human implementation effort. However, generally for larger programs, the application of sensitivity code generators is not straightforward, and
human intervention is often needed to obtain reasonably efficient derivative code
(Bartholomew-Biggs et al., 2000, Coleman and Jonsson, 1999, Sherman et al.,
1994). Using sensitivity code generation selectively for parts of a code, possibly
in combination with a semi-analytical formulation, seems a better approach than
applying it to complete, complex programs at once.
Typically, a design sensitivity computation for a case of realistic complexity takes the time of several function evaluations, in the case of straightforward
application of these methods (Bartholomew-Biggs et al., 2000, Coleman and Jonsson, 1999). For the present SMA case, these code generation techniques have
not been used, because manual differentiation provides more insight and control,
and because no sensitivity code generator is currently available for the Pascal
programming language used for the SMA finite element analysis implementation.
7.2.5
Discussion
Having briefly reviewed the options for design sensitivity analysis, it can be concluded that the most appropriate methods to implement for the present SMA
case are the direct semi-analytical method, as well as the finite difference method
combined with a fast reanalysis technique. The latter could be particularly useful
in case of the active catheter model discussed in Chapter 3, because the symmetry constraint in that model complicates the semi-analytical design sensitivity
analysis. Basically this constraint makes that a directional derivative should be
used. When the constraint would be enforced with a penalty method, there is
no problem, and the standard semi-analytical formulation can be used. However,
with the employed augmented Lagrangian method, with its iteratively updated
160
SMA SENSITIVITY ANALYSIS
7.3
Lagrange multipliers, the situation is more complicated. A possible solution is
to revert to the standard Lagrangian formulation after convergence, but using
finite differences in combination with fast reanalysis is expected to require far less
implementation effort. This problem is addressed in more detail in Chapter 9.
Concerning the SMA model, it is observed that the constitutive equations are
based on a piecewise linear stress-strain relation (see e.g. Chapter 2 and Chapter 3). This leads to the situation that non-differentiable states exists in the
simulated structures when the material is exactly at one of the transition points,
which could yield complications in the sensitivity analysis. In these points, formally only the existence of directional derivatives is well-defined. Similar problems
occur in e.g. elastoplastic models, but investigations have shown that, although
theoretically this non-differentiability presents a problem, in practical numerical
cases, it is highly unlikely that material states in the integration points exactly
correspond to these transition points, and therefore the practical effect on the
accuracy of the sensitivity is negligible (Cho and Choi, 2000, Schwarz and Ramm,
2001, Vidal and Haber, 1993). For this reason, similar to the elasto-plastic case,
no special treatment is applied to the transition points in the sensitivity analysis
of the present SMA model.
7.3
Sensitivity analysis of SMA finite element model
This section treats the formulation of semi-analytical design sensitivities for electrothermally actuated SMA structures, as the comparison of approaches given in
Section 7.2 showed this to be an attractive approach for such problems. The
other attractive method, finite differences in combination with fast reanalysis, is
applied as described by Haftka (1985), and requires no further explanation. This
section will also briefly outline essential details of the coupled finite element simulation of the SMA structures, to the extent that is required to adequately specify
the formulation of the sensitivity equations.
For the electrical and thermal simulation, the field equation formulation and
the associated three-noded triangular element as given by Fagan (1992) is used.
The mechanical analysis is performed using triangular shell elements developed by
Van Keulen and Booij (1996), in combination with the SMA constitutive model
described in Chapter 3. Details of the used elements are not given here, but can
be found in the mentioned references. Topologically equivalent meshes are used
for the electrical, thermal and mechanical cases, thus no mapping of quantities
between disciplines is required, which simplifies the coupling in the multidisciplinary analysis. However, the approach outlined here basically also applies to
cases using different elements and meshes. The Einstein summation convention
is used in this section, and partial derivatives are abbreviated by ∂f /∂xi = f,i .
7.3
7.3.1
SENSITIVITY ANALYSIS OF SMA FINITE ELEMENT MODEL
161
Electrical and thermal case
The isotropic electrical and thermal steady-state problems are governed by the
same partial differential equation, known as the Poisson equation:
Kφ,ii + Q = 0.
(7.10)
Here φ represents the potential field, which corresponds to the electrical potential
in the electrical case, and the temperature in the thermal case. K is the isotropic
conductivity, a material property, and Q is a local source term. The boundary
conditions (BCs) for Equation 7.10 can be the following:
Dirichlet BCs: φ = φ̂,
Neumann BCs: Kφ,i ni = q̂,
Robin BCs:
Kφ,i ni = h(φ − φ∞ ).
(7.11)
The Dirichlet and Neumann BCs prescribe either the potential φ̂ at or the flux q̂
through the boundary, given by the outward boundary normal n. The Robin BC
only applies to the thermal case, where it describes the convection at the surface.
Here h is the convection coefficient and φ∞ represents the ambient temperature.
Discretization of Equation 7.10 using the finite element method results in a
system of linear equations. To distinguish between the electrical and thermal
cases, subscripts e and t are used:
Electrical case:
Thermal case:
Ke (s)ue (s) = fe (s),
Kt (s)ut (s) = ft (ue (s)).
(7.12)
Note, in the thermal case, the load vector is a function of the electrical variables,
since resistive heating is used to generate internal heat sources in the structure.
In the finite element setting, the locally dissipated energy q can be evaluated by:
q = uTe Ke ue .
(7.13)
This equation is to be evaluated on element level, and each contribution is added
to the thermal load vector ft in the thermal analysis.
The sensitivities for the electrical case can be obtained directly using the direct
semi-analytical approach given in Equation 7.6:
due
dKe
dfe
−1
= Ke
−
ue ,
(7.14)
ds
ds
ds
Forward finite differences are used to approximate the derivatives of Ke and fe .
Similarly, the thermal case yields:
dut
dft
dKt
−1
= Kt
−
ut .
(7.15)
ds
ds
ds
162
SMA SENSITIVITY ANALYSIS
7.3
However, the thermal load vector ft is a function of the electrical variables, and
its design derivative dft /ds is used in this expression. It is possible to evaluate
this term by means of finite differences, but this would require an additional
perturbed electrical analysis for every design variable. A more efficient approach
is to compute the analytic design derivative of q (Equation 7.13):
due
dKe
dq
= 2uTe Ke
+ uTe
ue .
ds
ds
ds
(7.16)
The design derivative of the electrical tangent matrix Ke in this expression can
again be approximated by finite differences. This term is also used in the sensitivity analysis for the electrical state variables, given in Equation 7.14. Therefore,
in the present implementation, the design sensitivity of the dissipated heat given
in Equation 7.16 and the electrical state variable sensitivity are evaluated simultaneously, to avoid repeated calculations. The procedure has been verified by
comparing the results to finite difference computations.
7.3.2
Mechanical case
In the modeling of the mechanical case, geometrical and physical nonlinearities are
included. A Total Lagrangian approach is adopted, and the SMA material behavior is represented by the constitutive model described in Chapter 3. The coupling
between the thermal and mechanical analysis originates from the temperaturedependence of the SMA model. Finite element discretization yields the following
discrete nonlinear governing equations:
R(u(s), ut (s), s) = f (s) − fint (u, ut (s), s) = 0
(7.17)
Here, the residual R is a function of the nodal displacement vector u and the
temperatures ut , and is given by the difference between the external force vector
f and the internal force vector fint . Configuration- or temperature-dependent
external forces are not accounted for in the considered problem, hence only fint
is a function of u and ut . This set of nonlinear equations is solved using an
incremental-iterative method. Design differentiation of Equation 7.17 yields:
dR
∂R ∂R du
∂R dut
=
+
+
.
ds
∂s
∂u ds
∂ut ds
(7.18)
The quantity −∂R/∂u is referred to as the tangent stiffness matrix K. Rearranging and using this definition of K gives:
du
∂R
∂R dut
−1
=K
+
.
(7.19)
ds
∂s
∂ut ds
The design sensitivities of the temperature field dut /ds have already been computed after the thermal analysis, and the explicit partial derivative of the residual
7.3
SENSITIVITY ANALYSIS OF SMA FINITE ELEMENT MODEL
163
with respect to the design variable ∂R/∂s can be computed conveniently using
finite differences, following the semi-analytical approach. The remaining term
in the pseudo-load vector contains the expression ∂R/∂ut , which represents the
coupling between the thermal and mechanical problem. This expression can be
interpreted as a matrix with nm rows and nt columns, where nm is the number
of mechanical degrees of freedom and nt the number of thermal degrees of freedom. Fortunately, this matrix is very sparse, and in practice it is never formed
explicitly. Moreover, from Equation 7.17, it follows that when considering the
influence of the temperature distribution ut on the residual R, only the internal
load component of the residual has to be considered:
∂fint
∂R
=−
.
∂ut
∂ut
(7.20)
Note that this term needs to be computed only once, regardless of the number
of responses or design variables. Two approaches have been explored and implemented to compute this coupling term: one using finite differences and another
based on a fully analytical derivation, which both are discussed hereafter.
Finite difference approach
It is important to reduce the quantity used in the finite difference calculation as
much as possible, to avoid unnecessary computations. Straightforward application
of finite differences to the complete internal force vector would require as many
internal force evaluations as there are thermal degrees of freedom. Given the
complexity of the material model and its non-negligible evaluation time, this
would be a rather inefficient procedure. Therefore, before discussing the finite
difference approach to evaluate the thermo-mechanical coupling term, the internal
force contribution of an individual element is considered more closely.
The internal force contribution of an individual finite element is given by:
fint = DT σ
with Dij =
∂εi
.
∂uj
(7.21)
Here σ and ε are vectors of generalized stresses and strains, respectively. All
quantities in this subsection are element-based quantities, unless stated otherwise. From Equation 7.21, and given the fact that D does not depend on the
temperature, it follows that:
∂σ
∂R
∂fint
=−
= −DT
.
∂ut
∂ut
∂ut
(7.22)
Hence, by evaluating the term ∂σ/∂ut , the coupling term can be obtained. Note
that the element stress will only depend on the local temperature. The shell
element used in this study uses only a single integration point, and its temperature
164
SMA SENSITIVITY ANALYSIS
7.3
is given by the average value Tavg of the nodal temperatures of its three corner
(i)
nodes ut :
3
1 X (i)
Tavg =
u .
(7.23)
3 i=1 t
From this, it follows that the design sensitivity of the average element temperature
equals:
3
(i)
1 X dut
dTavg
=
.
(7.24)
ds
3 i=1 ds
Subsequently, the coupling term contribution for a given element becomes:
∂R dut
∂σ
dTavg
T
=D
−
.
(7.25)
∂ut ds
∂Tavg
ds
The total thermo-mechanical sensitivity coupling term in the system-level pseudoload vector given in Equation 7.19 is obtained by assembling these element contributions. The derivative of the element stress vector σ with respect to the average
element temperature Tavg is evaluated using forward finite differences:
∂σ
σ(Tavg + ∆Tavg ) − σ(Tavg )
≈
.
∂Tavg
∆Tavg
(7.26)
Note that the cost of this procedure is comparable to a single stress update as
performed in every Newton iteration, and that these terms can be reused for
multiple design variables.
Analytical approach
As shown in the previous section, the derivative of the element stress vector σ
with respect to the average element temperature Tavg is the only nonstandard
term that needs to be computed in order to evaluate the thermo-mechanical coupling sensitivity related term of the pseudo-load vector (Equation 7.19). Instead
of using finite differences to compute the this term, as in Equation 7.26, it is also
possible to use a fully analytical approach. The advantage is that possible accuracy problems of the finite difference approach are avoided, which is known to
suffer from sensitivity to the perturbation size, and poor performance with noisy
numerical models. The SMA material model internally uses an iterative solution
process, which is terminated once a certain tolerance criterion is satisfied. This
is a typical source of numerical noise, that could spoil the accuracy of the finite
difference approach.
To derive the term ∂σ/∂Tavg analytically, it is necessary to consider the SMA
material model in more detail. In-depth information on this model can be found in
Chapter 3 and Chapter 6. The focus here is on the derivatives of the components
7.3
SENSITIVITY ANALYSIS OF SMA FINITE ELEMENT MODEL
165
of the stress tensor, since the derivatives of the generalized stresses easily follow
once these stress tensor component derivatives have been obtained.
In the current plane stress setting, the SMA constitutive equations are formulated by:
  


1+α 1+α 0
 σxx 
σyy
= K 1 + α 1 + α 0 + . . .

√
0
0
0
2σxy

(7.27)
 

εxx 
2−α
−(1 + α) 0

2G 
−(1 + α)
2−α
0 √εyy
,


3
0
0
3
2εxy
with α = (2G − 3K)/(4G + 3K), where K is the bulk modulus. The temperaturedependent quantity here is the shear modulus G, and since α is a function of G,
it depends on T as well. From the definition of α, it follows that:
∂α
∂
∂G
2G − 3K
18K
=
.
(7.28)
=
2
∂T
∂T 4G + 3K
(4G + 3K) ∂T
With this, differentiation of Equation 7.27 with respect to the temperature T
yields:





∂σxx

 ∂T 
 ∂G 6K(3K − 2G) 1 1 0
∂σyy

1 1 0 + . . .
=
öT

(4G + 3K)2
 ∂ 2σxy 
 ∂T
0 0 0
∂T
(7.29)

 

εxx 
2−α
−(1 + α) 0

2 
−(1 + α)
2−α
0 √εyy
.


3
0
0
3
2εxy
The remaining unknown in this equation is ∂G/∂T . This term is not easily
evaluated, because no explicit equation is available for G(T ). Therefore, implicit
differentiation has to be used. The equation that is solved iteratively in order to
obtain G and the effective strain εe is given in Chapter 6 by:
Z = Ci (T ) +
Di (T )
− G = 0,
εe
where the effective strain is given by
r
4 2
4
4
εe =
(α − α + 1)(ε2xx + ε2yy ) + (2α2 − 2α − 1)εxx εyy + ε2xy .
9
9
3
(7.30)
(7.31)
Ci and Di are quantities that depend on the temperature and the material parameters. Their exact form as well as their derivatives are given in Appendix E.
166
7.3
SMA SENSITIVITY ANALYSIS
Note also the presence of α in the effective strain definition, which makes Z a
nonlinear function of G. Implicit differentiation of Equation 7.30 yields:
Z(T, G(T )) = 0 ⇒
∂Z
∂Z
∂Z ∂G
∂G
∂T
.
+
=0⇒
= − ∂Z
∂T
∂G ∂T
∂T
∂G
(7.32)
Using this relation, the temperature derivative of G is found to be given by:
∂G
=
∂T
dCi
1 dDi
dT + εe dT
4Di K(εxx +εyy )2 (2α−1)
(4G+3K)2 ε3e
.
(7.33)
+1
With this expression, the temperature sensitivity of the stress components given
in Equation 7.29 can be evaluated analytically. No iterations are required, in
contrast to the normal stress update procedure. No special measures are taken
at the transition points, where strictly speaking only directional derivatives are
defined. However, as discussed previously, it is unlikely that this will result in a
severe loss of accuracy. In spite of the expectation that the finite difference appraoch discussed before might suffer from a possible noisy behavior of the stress
function, no indications of such problems were found in practical tests. A comparison between this analytical approach and the finite difference option revealed
no significant differences in accuracy or evaluation time. In the implementation of
the semi-analytical sensitivity analysis, the analytical approach as discussed here
has been used.
7.3.3
Effective strain sensitivity
An important response quantity used in optimization problems involving the
present SMA model is the effective strain. To make sure the model is not used
outside its range of applicability, constraints are included that limit the maximum
effective strain to εmax
= 1%:
e
max εe − εmax
≤ 0.
e
z
(7.34)
Here z is the transverse coordinate of the shell element. Because the maximum
effective strain within an element occurs at the outside layer (see Appendix F), in
practice only the case |z| = t/2, where t represents the shell thickness, needs to be
considered. Since including this maximum effective strain condition for every individual element leads to a large number of constraints, and since in practice a small
number of isolated and limited constraint violations can be tolerated, the following
Kreisselmeier-Steinhauser (Kreisselmeier and Steinhauser, 1983) constraint aggre(i)
gation function can be used to combine all element constraints gi = εe − εmax
e
into a single constraint, given by:
!
N
X
1
1
ρgi
KSA (g) = ln PN
·
Ai e
.
(7.35)
ρ
i=1 Ai i=1
7.3
SENSITIVITY ANALYSIS OF SMA FINITE ELEMENT MODEL
167
Here the element contributions are weighted by the element area Ai , to reduce
the significance of small local violations. The parameter ρ controls the tolerance
of the aggregated constraint and can be chosen depending on the demands of the
application. This approach has also been used in the SMA gripper optimization
described in Chapter 5. A larger ρ-value allows smaller isolated violations. The
effective strain constraint is now given by:
KSA (g) − 1 ≤ 0
(7.36)
Since the Kreisselmeier-Steinhauser aggregation of effective strain constraints
is an important response quantity, and because obtaining the design sensitivity
of this quantity is nontrivial, detailed expressions for this sensitivity are derived
here. The sensitivity of KSA (g) is given by:
dKSA (g)
=
ds
P
i
i
ρAi eρgi dg
ds +
i
eρgi dA
ds −
P
ρ i Ai eρgi
P
i
P
Pi
i
dAi
ds
Ai
·
P
i
Ai eρgi
.
(7.37)
Note that not only the sensitivity of the constraint dgi /ds is required, but also
the sensitivity of the element area dAi /ds plays a role. This term is relevant when
dealing with shape design variables. In the implementation, it is evaluated using
forward finite differences.
Remains the sensitivity of the maximum effective strain constraint gi itself,
which is equal to the sensitivity of the effective strain at a given position z. Since
the maximum occurs at the outside layer of the shell (see Appendix E), only the
effective strains at both outside layers are evaluated, and also only the sensitivities
of these quantities are relevant. Analytical or semi-analytical evaluation of the
effective strain sensitivity is preferred over finite differences, because the latter
would require solving the perturbed displacement vector for every design variable.
These analytical sensitivities are discussed here, as several nontrivial steps are
involved.
In the present shell element setting, the effective strain is given by
s
4
2
2
9 A1 ((εxx + zκxx ) + (εyy + zκyy ) ) + . . .
(7.38)
εe =
4
4
2 ,
9 A2 (εxx + zκxx )(εyy + zκyy ) + 3 (εxy + zκxy )
where
α=
2G − 3K
,
4G + 3K
A1 = α2 − α + 1,
A2 = 2α2 − 2α − 1.
(7.39)
The εij and κij quantities are the membrane strain and curvature components at
the midplane of the shell. Design differentiation of εe yields:
dεe
∂εe T dε ∂εe dz
∂εe ∂α
∂α dG
=
+
+
+
.
(7.40)
ds
∂ε ds
∂z ds
∂α ∂s
∂G ds
168
7.4
SMA SENSITIVITY ANALYSIS
Here the term
∂εe T dε
∂εe dεxx
∂εe dκxy
=
+ ... +
(7.41)
∂ε ds
∂εxx ds
∂κxy ds
stands for the complete summation of products of partial effective strain sensitivities with respect to the strain components and their respective total sensitivities.
These strain component sensitivities are derived from the sensitivities of the generalized strains ε, computed using
dD
du
dε
=
u+D .
(7.42)
ds
ds
ds
The midplane membrane strains εij correspond directly to the generalized membrane strains, but the generalized curvature components kij are related to the
midplane curvatures κij by kij = κij A, where A is the element area. It follows
that the sensitivity of the curvature components is given by:
dκij
1 dkij
kij dA
=
− 2
.
(7.43)
ds
A ds
A ds
Terms dD/ds and dA/ds are evaluated using forward finite difference. Full expressions for the partial derivatives of εe with respect to the strain components
and α required in Equation 7.40 are not all shown here, but follow directly from
Equation 7.38, for example:
∂εe
2
=
(2A1 (εxx + zκxx ) + A2 (εyy + zκyy )) , etc.
∂εxx
9εe
(7.44)
The sensitivity of the shear modulus dG/ds, also needed in Equation 7.40, is
computed similarly to dG/dT in the previous section, using implicit differentiation
of Equation 7.30:
∂Z
∂Z dG
dZ
=
+
=0
ds
∂s
∂G ds
where
⇒
∂Z
dG
∂s
= − ∂Z
,
ds
∂G
∂Z
Di ∂εe ∂α
=− 2
− 1,
∂G
εe ∂α ∂G
(7.45)
(7.46)
and
∂Z ∂Ci dT
∂Ci
=
+
+
∂s
∂T ds
∂s
Di
ε2e
∂Di dT
∂Di
+
∂T ds
∂s
1
− ...
εe
!
∂εe T dε ∂εe dz
∂εe ∂α
+
+
.
∂ε ds
∂z ds
∂α ∂s
(7.47)
Here T again denotes the temperature in the integration point, which for the
present shell element equals the averaged element temperature. The terms ∂Ci /∂s,
∂Di /∂s, dz/ds and ∂α/∂s are all evaluated by forward finite difference. The effective strain is non-differentiable at εe = 0, and for this situation the average of
the directional sensitivities is used, which results in a value of zero.
7.4
EVALUATION
7.4
Evaluation
7.4.1
Numerical results
169
The various sensitivity analysis procedures discussed in this chapter for the electrothermo-mechanical SMA finite element model are applied to the SMA gripper
structure used in Chapter 5. This model represents the most general and complete
case that can be considered with the modeling approach presented in this research,
and therefore serves as a representative problem. The configuration considered
is the deterministic optimal design, and sensitivities computed using global finite
difference (GFD), direct semi-analytical (SA) and direct refined semi-analytical
(RSA, (De Boer and Van Keulen, 2000b, Van Keulen and De Boer, 1998b)) methods are compared. In the mechanical finite difference case, the perturbed analysis
is started from the final nominal configuration, which raises the efficiency of this
method significantly. The correction for finite difference sensitivities of iteratively
solved problems suggested by Haftka (1985) is used to obtain accurate results.
The response quantities and design variables that are considered are listed in Table 7.1 and Table 7.2, respectively. The first 6 design variables control the gripper
shape and the applied voltage, the latter 6 in fact are uncertainty variables that
are not directly controlled by the designer. Still, in an optimization procedure,
sensitivities of both types of variables are needed, and in this section no further
distinction is made. Nominal values are used for the uncertainty variables.
Design sensitivities have been evaluated using a number of relative design
perturbations, ranging from 10-14 to 10-2 , to compare the stability and accuracy
of the results. This is more informative and relevant than the numerical values
of the sensitivities by themselves. A large quantity of sensitivity data has been
generated, therefore only a selection of the most interesting results is discussed
here. These results of interest are shown in Figure 7.2, Figure 7.3 and Figure 7.4.
Table 7.1: Response quantities considered in the numerical study of SMA design sensitivity analysis.
Response quantity name
Tip displacement [mm]
Aggregated maximum effective strain
constraint, open configuration
Aggregated maximum effective strain
constraint, closed configuration
Aggregated maximum temperature
constraint, open configuration
Aggregated maximum temperature
constraint, closed configuration
Symbol
uz
KSA (gεopen )
Nominal value
0.4341
0.9992
KSA (gεclosed )
0.9999
KSA (gTopen )
0.9999
KSA (gTclosed )
0.9979
170
7.4
SMA SENSITIVITY ANALYSIS
L z
0.13
SA
RSA
GFD
d u /dY
−0.05
2
SA
RSA
GFD
d u /dW
L z
2
0.12
−0.055
−0.06
0.11
−0.065
0.1
−0.07
0.09
−15
10
−10
10
−5
10
0
10
10
−15
−10
10
−5
10
0
10
Figure 7.2: Logarithmic sensitivities dL uz /dY2 (left) and dL uz /dW2 (right), computed
using various methods, as a function of the relative design perturbation.
1.4
SA
RSA
GFD
dLKSA(gclosed
)/dY1
ε
0.4
0.38
SA
RSA
GFD
dLKSA(gclosed
)/dH
ε
1.3
0.36
1.2
0.34
0.32
1.1
0.3
1
0.28
−15
10
−10
10
−5
10
0
10
10
−15
−10
10
−5
10
0
10
Figure
7.3:
Logarithmic
sensitivities
dL KSA (gεclosed )/dY1
(left)
and
closed
dL KSA (gε
)/dH (right), computed using various methods, as a function of
the relative design perturbation.
SA
RSA
GFD
dLKSA(gopen
)/dE0
ε
−0.014
SA
GFD
dLKSA(gopen
)/dW2
T
0.12
−0.016
0.11
−0.018
0.1
0.09
−0.02
−15
10
10
−10
−5
10
0
10
10
−15
−10
10
−5
10
0
10
Figure
7.4:
Logarithmic
sensitivities
dL KSA (gεopen )/dE0
(left)
and
open
dL KSA (gT )/dW2 (right), computed using various methods, as a function of
the relative design perturbation.
7.4
EVALUATION
171
Table 7.2: Design (and uncertainty) variables considered in the numerical study of SMA
design sensitivity analysis.
Variable name
Plate thickness
Gripper arm height
Gripper shape control point 1
Gripper shape control point 2
Plate front width
Applied voltage
Face convection coefficient
Ambient temperature
Austenite Young’s modulus
R-phase apparent Young’s modulus
Initial apparent transition Young’s modulus
Poisson ratio
7.4.2
Symbol
t
H
Y1
Y2
W2
V
h
Ta
EA
ER
E0
ν
Nominal value
0.0974 mm
1.1703 mm
1.2485 mm
1.1125 mm
1.4001 mm
0.0603 V
2 kWm-2 K-1
328 K
68.939 GPa
45.612 GPa
20.006 GPa
0.333333
Discussion
In all results, the influence of the relative design perturbation can clearly be seen.
For both small and large values of the perturbation, the accuracy of the sensitivities deteriorates. This is in agreement with the expectations discussed earlier:
reducing the perturbation increases the round-off error, while increasing the perturbation also increases the error due to the truncation of the approximating
Taylor series used in the finite difference formulation.
Regarding the semi-analytical approaches, the range of perturbations that
yield stable results varies depending on the response quantity and variable considered. Generally, it is observed that this range is smaller for sensitivities of
mechanical responses (uZ , εe -constraints) with respect to shape design variables.
Typical examples are shown in Figure 7.2 and Figure 7.3, and typically accurate
sensitivities are only obtained for relative perturbations in the order of 10-9 up
to 10-7 . As sensitivities become smaller, this range even reduces to 10-9 –10-8
(Figure 7.2). In case of either non-shape variables or non-mechanical responses,
the relative perturbation range for stable sensitivities is considerably larger, approximately 10-9 up to 10-4 . A possible explanation for this observation is that
the nonlinearity of the relationship between the pseudo-load and the shape design variables is considerably stronger in comparison to the other cases. This is
indicated by the fact that the truncation error already becomes evident at relatively small perturbations of approximately 10-7 . No indications have been found
that the (electro-)thermo-mechanical coupling adds to the inaccuracy, as voltage
sensitivity of mechanical quantities turned out to be fairly stable, comparable to
results shown in Figure 7.4.
172
SMA SENSITIVITY ANALYSIS
7.4
Computation of accurate shape design sensitivities, particularly in case of
slender structures, is known to be challenging, as discussed in Section 7.2.3.
The refined semi-analytical (RSA) method has been proposed to reduce errors
in pseudo-load vectors, which increased with increasing rigid body motions of elements. In the present numerical example, it is not clearly observed that this RSA
method improves the accuracy of the sensitivity computation. In one case, concerning dL uz /dW2 in Figure 7.2, the RSA shows clearly a larger range of stable
perturbations. However, in several other cases there is no improvement over the
conventional semi-analytical (SA) method. In fact, some cases (e.g. dL uz /dY2 in
Figure 7.2) show a slight decrease in accuracy when applying the RSA method.
This could be attributed to the increased number of numerical operations that
have to be performed, compared to the SA case. Apparently, the plausible source
of the inaccuracy in the present case, the strong nonlinearity of the problem and
hence its susceptibility to truncation errors, is not affected by the RSA method.
Possibly for cases in which elements are subjected to larger rigid body motions,
results might be different.
The finite difference approach, due to the use of the nominal configuration
as a starting point for the perturbed analyses in the mechanical case, proves to
be a competitive approach for the present problem. In most cases only one or
two iterations are necessary to converge to the perturbed solution. Given the
fact that the mechanical analysis itself requires roughly 100 iterations, this is a
relatively low computational cost, even when it has to be repeated for every of
the 12 considered design variables. For the linear electrical and thermal analysis,
the GFD approach in contrast is relatively costly, since for each design variable
a complete additional analysis is required. However, compared to the mechanical problem, the computational effort spent on the linear electrical and thermal
problem is hardly significant (<1%). Therefore, considering the SMA analysis as
a whole, the computational cost of GFD-based sensitivity analysis for the present
number of variables is still relatively modest.
The range of relative design perturbations that yield accurate results for the
GFD method is generally comparable to that of both semi-analytical cases, although differences can be found in either direction. For example, for the sensitivities in Figure 7.2, the GFD method shows the most stable behavior, whereas for
dL KSA (gεclosed )/dH in Figure 7.3, the opposite is true. The smallest stable perturbation is approximately equal to that of the (R)SA cases or slightly larger. The
latter might be due to the finite accuracy of the analysis, as a tolerance is used to
terminate the Newton iterations. Apart from this, no indication was found that
the accuracy of the GFD approach suffered from numerical noise of the model,
although it cannot be ruled out that the present case just represents a favorable
design point. The largest stable perturbation is in this case influenced by the
truncation error in the Taylor approximation of the response quantity, instead of
that of the pseudo-load terms. This explains the observed differences, but the fact
remains that a priori selection of an appropriate relative perturbation is difficult.
7.5
CONCLUSIONS
173
Table 7.3: Time consumption of sensitivity analysis of an SMA miniature gripper shell
finite element model, with 12 design variables.
Method
Total time [s]
Index w.r.t. SA [%]
Sensitivity analysis time [s]
Time per variable [s]
No sensitivity analysis
226
-
SA
476
100
250
20.8
RSA
521
109
295
24.6
GFD
552
116
326
27.2
Finally, discussion of the computational cost of sensitivity analysis is also of
interest. Table 7.3 shows the time consumption of the various sensitivity analysis methods considered in this chapter. It turns out, that the SA method is
computationally the most efficient, followed by the RSA method. The least efficient method is the restarted GFD approach, but to put this in perspective, it
turns out that for the considered case with twelve design variables, this method
is only 16% slower than the best-performing SA method. Given the considerable
difference in implementation effort, this makes that for many practical cases the
GFD approach would be preferred. Note also that per design variable, the cost
of sensitivity analysis roughly corresponds to 10% of the cost of a single analysis.
7.5
Conclusions
In this chapter, the sensitivity analysis of SMA structures actuated by Joule
heating has been considered. The behavior of these structures is described by
a nonlinear path-independent model, which allows for sensitivity analysis procedures that require significantly less computational effort than the analysis itself. A
restarted global finite difference (GFD) approach as well as direct semi-analytical
(SA) and refined semi-analytical (RSA) procedures have been studied in this context. As the number of design variables in the intended SMA shape optimization
applications is generally modest, an adjoint formulation has not been considered
in this chapter.
Numerical testing on an SMA miniature gripper model has revealed that particularly for shape design sensitivities of mechanical response quantities, the selection of a proper relative design perturbation is critical. By testing a range
of perturbations, it was found that in certain cases only a small interval exists
for which accurate sensitivities are obtained. The semi-analytical approaches appeared to be slightly more sensitive to the perturbation, compared to the GFD
method. The reported improved accuracy of the RSA method over the SA formulation was not observed in all cases, probably because the dominant errors in
the present case are not related to large rigid body motions, but rather to the
nonlinearity of the SMA model.
174
SMA SENSITIVITY ANALYSIS
7.5
Generally, one or two iterations proved sufficient to obtain a perturbed solution in the GFD case, starting from the nominal final configuration. For the linear
electrical and thermal analyses, the superior efficiency of the semi-analytical approach is undisputed. However, considering the entire SMA analysis as a whole,
the GFD method turns out to be quite competitive for the considered class of
problems, due to the relatively high cost of the incremental-iterative scheme used
to solve the nonlinear mechanical problem. For the representative miniature gripper example, the GFD approach proved to be only 16% slower than the SA case,
for 12 design variables. This makes it an attractive option for sensitivity analysis,
particularly when its comparatively straightforward and generic implementation
is taken into account.
In contrast, the complexity of the computation of the pseudo-load in the semianalytical approaches was found to increase significantly due to the fact that the
problem involves a coupled electrical, thermal and mechanical analysis. Particularly the term associated with the thermo-mechanical coupling required additional
implementation effort. A fully analytical as well as a finite-difference-based formulation has been implemented, and it was found that both procedures perform
well. In spite of the fact that an iterative process is used inside the material
model, inaccuracies due to numerical noise have not been detected in case of the
finite-difference approach, and results obtained with both methods were in good
agreement.
An additional consideration is the sensitivity analysis of derived quantities. An
important response quantity in the used SMA model is the maximum effective
strain. In case of the finite difference method, evaluation of its sensitivity is
straightforward, as the finite difference formula can simply be applied, using the
computed nominal and perturbed responses. In contrast, because of the complex
SMA material model, in the semi-analytical cases a rather lengthy and complex
procedure is required to compute the sensitivity of the effective strain from the
state variable sensitivities.
In conclusion, it can be stated that for the present class of SMA problems,
given a suitable ratio of design variables versus response quantities, the restarted
finite difference approach is a viable option for sensitivity analysis, particularly
considering the significantly smaller implementation effort. However, for larger
problems with considerably more degrees of freedom, the advantage of semianalytical methods will increase, as the cost of system matrix decomposition will
increase sharply. Also, when the number of design variables increases, the GFD
approach will become increasingly unattractive, as the computational cost per
design variable is higher than in the (R)SA methods. For large numbers of design
variables, an adjoint semi-analytical approach is likely to be the most efficient.
The developed sensitivity analysis methods are expected to be of great use in
further work on design optimization of SMA structures.
Based on: Langelaar, M. and van Keulen, F. (2006). Sensitivity analysis and
optimization of a shape memory alloy gripper. Computers and Structures, in review.
Chapter
8
Gradient-based Shape Optimization
of an SMA Gripper
Optimization techniques can be used to improve the design process of shape memory
alloy (SMA) structures. This chapter presents the shape optimization of a miniature
SMA gripper using two gradient-based optimization algorithms, Sequential Quadratic
Programming (SQP) and the Method of Moving Asymptotes (MMA). The use of gradient information enables faster optimization or allows more design variables, in comparison to direct methods. To obtain gradient information, the sensitivity analysis of the
sequentially coupled electro-thermo-mechanical SMA gripper problem is investigated.
Finite difference and semi-analytical sensitivity analysis techniques for this problem are
compared on accuracy, stability, efficiency and implementation aspects. Furthermore,
using the SMA gripper as a representative example, the effectiveness and computational
efficiency of the gradient-based optimization methods is compared to a direct responsesurface-based approach.
8.1
Introduction
Shape memory alloys (SMAs) are interesting materials for many (micro-)actuation
applications, because of their unusually high energy density (Otsuka and Wayman, 1998). SMAs can generate relatively large actuation strains and stresses,
controlled by modest changes in temperature. Often SMAs are applied in the form
of wires or springs, but in many emerging applications in, e.g., microsystems or
medical instruments, more complex structures are considered. However, designing
effective planar or three-dimensional SMA actuators is no easy task, due to the
175
176
GRADIENT-BASED SHAPE OPTIMIZATION OF AN SMA GRIPPER
8.1
complexity of the thermo-mechanical behavior. Moreover, often the temperature
of SMA actuators is controlled by direct resistive heating of the material itself,
which makes that its geometry as well as its electrical, thermal and mechanical
properties are all interacting in a complicated way. For this reason, efforts are
made to develop systematic model-based design procedures for SMA structures.
Particularly design optimization techniques are promising, as these methods offer
an approach to handle design problems in a structured and formal way (Haftka
and Gürdal, 1992).
Several examples of the application of design optimization concepts to SMA
design problems can be found in recent literature. Skrobanek et al. (1997) have
reported the use of a heuristic peak stress reduction algorithm in the design of
SMA microvalves, which can be viewed as a kind of shape optimization. The same
procedure has been used by Fischer et al. (1999) to reduce peak stresses in the
design of a flexible SMA endoscope tip. Finite element modeling is used, so this
method can be applied to a wide variety of structures. However, this approach
does not involve a formal optimization problem formulation. This significantly
limits its applicability, as constraints and objectives regarding other aspects than
peak stress reduction cannot be included.
Lu et al. (2001) applied a combination of graphical optimization and sequential quadratic programming (SQP) to an SMA design problem, using an elaborate
analytical model of an SMA actuator with a corrugated core. Unfortunately, analytical modeling is not practical for more general SMA structures. Recently,
Dumont and Kuhl (2005) reported the optimization of an SMA spring actuator
using a genetic algorithm. Genetic algorithms are stochastic optimization procedures that are typically quite robust, but also very costly in terms of the number
of required evaluations. Therefore, this approach is clearly limited in terms of
the complexity of the problem and the number of design variables that can be
considered.
In contrast, this chapter presents gradient-based design optimization of a shape
memory alloy (SMA) structure, a miniature gripper. In an earlier article, we
have reported the shape optimization of a similar SMA gripper using a direct response surface method (Langelaar and Van Keulen, 2004a) (Chapter 4). However,
gradient-based methods are potentially superior to direct approaches in terms of
efficiency, in case the design sensitivities of the problem can be computed at relatively low cost. This implies that solutions can be obtained faster and with less
computational effort, or that more design variables can be included. The presented approach is not limited to design of grippers, since finite element modeling
is employed, which can be used to model virtually any kind of SMA structure. To
allow for efficient sensitivity analysis, the finite element model uses a dedicated
history-independent SMA constitutive model.
This article is organized as follows: first, the SMA gripper design concept
and modeling is presented in Section 8.2, together with the formulation of the
design optimization problem. Subsequently, Section 8.3 discusses the sensitivity
analysis of this gripper, using finite difference and semi-analytical methods. The
8.2
SMA MINIATURE GRIPPER
177
shape optimization results for the SMA gripper are presented in Section 8.4.
The performance of the previously used response surface method (Multi-point
Approximation Method (Toropov et al., 1993a)) is compared to two gradientbased methods, sequential quadratic programming (SQP) and Svanberg’s Method
of Moving Asymptotes (Svanberg, 1987), followed by conclusions.
8.2
SMA miniature gripper
8.2.1
Concept and modeling
The design concept for the miniature SMA gripper used in this chapter is shown
in Figure 8.1. The gripper consists of an identical top and bottom arm made of
folded Ni-Ti plates. The gripper is intended to exploit the shape memory behavior
due to the R-phase transformation in Ni-Ti, as described in an experimental
study by Tobushi et al. (1992). This transformation has a very small hysteresis,
and allows this gripper to operate in a relatively narrow temperature window.
An initial deformation is applied in order to generate internal stresses in the
material, which are required to make use of the shape memory effect. Starting
from the undeformed configuration in Figure 8.1, the ends of the outer plates are
pinched toward the inner plates during assembly. In this situation, the equilibrium
configuration of each arm can be changed by changing the temperature of either
the inner or outer plates. Resistive heating is used for this purpose, and to guide
the electrical current through individual plates a slit is present along the length
of each plate. Heating the inner plates will cause the tip ends to move apart,
opening the gripper. Similarly, heating the outer plates will make them move
towards each other, closing the gripper. In the closed configuration, clamping
forces of 100 mN are applied in z-direction at the tips of the gripper.
Because of symmetry, only a quarter of the gripper needs to be modeled: in
this case half the top arm is used. This part together with the parameterization
of the geometry is shown in Figure 8.2. Seven design variables have been selected
for this design problem: the plate thickness t, the undeformed arm height H, the
actuation plate front and end width W1 and W2 , and the shape of the actuation
plate. This shape is described by a quadratic B-spline (see Farin, 2002), and
the y-coordinates of the two middle control points are used as design variables:
Y1 and Y2 . Finally, the voltage V to operate the gripper is used as the seventh
design variable. Further geometrical details of the miniature gripper are listed in
Table 8.1.
The gripper is simulated by finite element analysis of the parameterized design shown in Figure 8.2. For both the opened and closed case, a quasi-static
sequentially coupled electrical, thermal and mechanical analysis is performed, to
simulate the SMA behavior under the influence of Joule heating. Dissipated heat
from the electrical analysis is used as a heat source in the thermal analysis, and
the resulting temperature distribution is used in the mechanical analysis. Physical
178
GRADIENT-BASED SHAPE OPTIMIZATION OF AN SMA GRIPPER
8.2
Figure 8.1: Conceptual gripper geometry in the undeformed configuration.
Figure 8.2: Design parameterization of the gripper. Because of symmetry, only a
quarter is considered.
8.2
SMA MINIATURE GRIPPER
179
Table 8.1: Significant coordinates of B-spline control points and other points defining
the geometry of the miniature gripper.
Point
Control point 1
Control point 3
Control point 5
Tip
x [mm]
0
1
3.5
5.377
y [mm]
W1
Y1
W2
0
Point
Control point 2
Control point 4
Control point 6
Slit end
x [mm]
0.5
3
4
3.8
y [mm]
W1
Y2
W2
0
Table 8.2: Physical constants used in the finite element modeling.
Quantity
Electrical conductivity
Thermal conductivity
Thermal convection coefficient
Ambient temperature
Value
1.25 · 106 Sm-1
21 Wm-1 K-1
2.0 · 103 Wm-2 K-1
328 K
constants used in the simulations are listed in Table 8.2.
Particularly the mechanical analysis is computationally intensive, because of
the nonlinear SMA material model as well as the consideration of geometrical
nonlinearities. A dedicated SMA material model is employed (Langelaar and Van
Keulen, 2004b) (Chapter 3), aimed at the R-phase transformation in Ni-Ti, which
is used in this gripper. In the considered temperature range (328–343 K), hysteresis is practically absent, which allows simplifications in the constitutive modeling.
Neglecting hysteresis furthermore renders the model history-independent, which
simplifies the sensitivity analysis considerably (Kleiber et al., 1997). Triangular
shell elements are used for the mechanical analysis (Van Keulen and Booij, 1996),
and the nonlinear SMA constitutive model is included using numerical integration
over the thickness of the shells. To ensure robust convergence for all combinations
of design variables, an adaptive incremental-iterative scheme is used.
8.2.2
Optimization problem
The aim of the design problem is to maximize the displacement range of the tips of
this miniature gripper, so it can be used for gripping and clamping a wide range
of objects. This translates into maximizing the difference between the gripper
tip z-positions in the opened and closed configurations. Furthermore, constraints
are added that restrict both the strain and the temperature of the structure.
Strain constraints are necessary to remain within the range of applicability of the
material model, which has been based on experimental data for strains up to 1%.
Temperature constraints are added to limit the thermal operating range of the
180
GRADIENT-BASED SHAPE OPTIMIZATION OF AN SMA GRIPPER
8.2
gripper, which is an important requirement in, e.g., biomedical applications. In
this way, this gripper design problem serves as a realistic example of SMA shape
optimization.
It is not practical to include these strain and temperature constraints for every
single finite element in the mesh, as the number of constraints would become prohibitively large. Furthermore, in a practical setting, limited and small constraint
violations are tolerable, and strictly constraining every element individually would
be overly stringent. Therefore, in this chapter, the individual element constraints
are combined into global constraints using the Kreisselmeier-Steinhauser constraint aggregation function (Kreisselmeier and Steinhauser, 1983). A slightly
adapted version is used here, that also accounts for the area A of each element.
In this way, small local violations are tolerated to some extent, but still the constraint restricts the design from becoming unacceptable in a global sense. The
design optimization problem can now formally be written as:
open
closed
max f (x) = 2 utip,
(x) − utip,
(x)
z
z
x
Subject to:
g1 (x) = KSA (gεopen (x)) − 1 ≤ 0,
g2 (x) = KSA (gεclosed (x)) − 1 ≤ 0,
g3 (x) = KSA (gTopen (x)) − 1 ≤ 0,
g4 (x) = KSA (gTclosed (x)) − 1 ≤ 0,
x ≤ x ≤ x.
(8.1)
The vector x represents the design variables, and x and x contain their lower and
upper bounds, respectively. The individual variables and their bounds are listed in
Table 8.3. The aggregated strain constraints are given by g1 and g2 , one for each
configuration. Likewise g3 and g4 represent the temperature constraints of the
open and closed gripper. The individual element constraints and the aggregation
function are given by:
(i)
gε(i) =
(i)
gT
εe
≤ 1,
(max)
εe
T (i) − Tmin
=
≤ 1,
Tmax − Tmin
1
KSA (g) = ln
ρ
1
PN
i=1
A(i)
(8.2)
(8.3)
·
N
X
!
(i) ρg (i)
A e
.
(8.4)
i=1
For this gripper, Tmin = 328 K and Tmax = 338 K have been used. The maximum
(max)
effective strain εe
is set to 0.01. The definition of the effective strain εe is
given in Chapter 3.
To get a good impression of the various optimization methods studied in this
chapter, each method will be applied to this problem starting from three different
initial designs, as listed in Table 8.4. All these designs are infeasible starting
8.3
SENSITIVITY ANALYSIS
181
Table 8.3: Design variables (x) for the design optimization of the SMA gripper.
Design variable
Plate thickness
Gripper arm height
Plate shape control point 1
Plate shape control point 2
Plate front width
Plate end width
Applied voltage
Symbol
t
H
Y1
Y2
W1
W2
V
Lower bound
0.05
0.3
0.01
0.01
0.1
0.1
0.01
Upper bound
0.3
2
1.5
1.5
1.5
1.5
0.5
Unit
mm
mm
mm
mm
mm
mm
V
Table 8.4: Starting points used for the design optimization of the SMA gripper.
Case
A
B
C
t [mm]
0.08464
0.09645
0.20712
H [mm]
0.452
1.415
0.459
Y1 [mm]
1.018
0.816
1.307
Y2 [mm]
0.965
0.912
0.661
W1 [mm]
1.390
1.151
0.714
W2 [mm]
1.379
0.590
1.212
V [V]
0.06411
0.05032
0.06810
points, and they have a relatively poor objective function value. In fact, these
points were found as sub-optimal points in the first iteration of three different
runs of the multi-point approximation method, and they have subsequently been
used as starting points for the SQP and MMA studies.
Before proceeding with the discussion of the results of the design optimization,
first the sensitivity analysis of this problem is treated in the following section.
8.3
Sensitivity analysis
8.3.1
Methods
In order to be able to use the efficient gradient-based optimization methods, it
is necessary to compute the derivatives of the responses (objective, constraints)
with respect to the design variables, i.e. the design sensitivities. For the present
sequentially coupled gripper model involving a complex nonlinear SMA material
model, the sensitivity analysis is not trivial, and therefore some selected aspects
are highlighted in this section. An in-depth discussion of the sensitivity analysis
for this type of problems is given in Chapter 7.
Various methods exist for evaluating design sensitivities, and extensive discussions can be found in, e.g., books by Haftka and Gürdal (1992), Kleiber et al.
(1997) and the recent review by Van Keulen et al. (2005). Considering the fact
that the SMA gripper is modeled with a history-independent nonlinear model,
182
GRADIENT-BASED SHAPE OPTIMIZATION OF AN SMA GRIPPER
8.3
and that the number of design variables is not far greater than the number of
responses, appropriate methods for sensitivity analysis are either the restarted
Global Finite Difference (GFD) approach or the direct Semi-Analytical approach
(SA). As the model involves shell elements, for which the SA method is known to
be unreliable for large rotations, an improved SA formulation such as the Refined
Semi-Analytical (RSA) method (Van Keulen and De Boer, 1998b) should also be
considered.
The GFD method used here is based on forward finite difference, i.e. the design
sensitivities of a response f with respect to a design variable s are approximated
by:
f (s + ∆s) − f (s)
df
≈
,
(8.5)
ds
∆s
where ∆s is a small design perturbation. Since the present problem is historyindependent, the evaluation of the mechanical problem can be carried out by
restarting iterations from the final unperturbed solution, thereby saving a costly
incremental-iterative solution process. This reduces the computational cost significantly, and in combination with its relatively straightforward implementation,
GFD sensitivity analysis becomes an attractive option. The residual correction
proposed by Haftka (1985) has been applied to improve the accuracy of the computed sensitivities.
The SA and RSA approaches, in contrast, utilize analytical differentiation of
the discretized equations. For example, for a typical linear finite element problem
given by KU = F, design differentiation results in:
dK
dU
dF
dU
dF dK
−1
U+K
=
⇒
=K
−
U .
(8.6)
ds
ds
ds
ds
ds
ds
At this point however, the quantities dF/ds and dK/ds are evaluated using finite
difference, similar to Equation 8.5. Note that the expression for dU/ds involves
the inverse of the system matrix K−1 , which can be reused when a direct solver
is used. This makes that sensitivities can be obtained at a fraction of the cost of
a full analysis. In case of RSA, inaccuracies in the finite difference approximation
of dK/ds that occur in mechanical problems with large rotations are reduced by
exact differentiation of rigid body modes (Van Keulen and De Boer, 1998b). The
following subsections focus particularly on the non-standard aspects of the (R)SA
sensitivity analysis for the present problem.
8.3.2
Handling of interdisciplinary coupling
The analysis of the SMA gripper involves a sequentially coupled electrical, thermal
and mechanical problem. This implies that in the sensitivity analysis, as in the
analysis itself, the coupling between the disciplinary models has to be accounted
for. The first coupling is found between the electrical and thermal problem, where
the heat generated by the electrical current serves as a body source term for the
8.3
SENSITIVITY ANALYSIS
183
thermal simulation. In the SA setting, this results in an electrical contribution
to the load design derivative dF/ds in Equation 8.6, that accounts for the effect
of a changing voltage on the thermal load. This contribution could be evaluated
by finite differences, but this would require an additional perturbed electrical
simulation for every design variable. In the SA context, the aim is to avoid
such expensive solutions, and therefore an analytical approach is considered. The
dissipated heat q on element level is given by:
q = uTe Ke ue .
(8.7)
Here ue represents the electrical state variables (i.e. the electrical potential) and
Ke the element conductivity matrix. Design differentiation of q subsequently
results in:
due
dKe
dq
= 2uTe Ke
+ uTe
ue .
(8.8)
ds
ds
ds
The design derivative of the electrical tangent matrix Ke can again be approximated by finite differences. This term is also used in the SA sensitivity analysis for
the electrical case, and can therefore be reused. The procedure has been verified
by comparing the results to finite difference computations.
The second coupling in the present SMA actuator problem exists between
the thermal and mechanical problem, where the local temperature distribution
affects the mechanical material behavior. The discrete governing equations for
the nonlinear mechanical problem are given by:
R = fext (s) − fint (um (s), ut (s), s) = 0.
(8.9)
Here R is the global unbalance or residual load vector, which is iteratively reduced
to zero in the solution process. The terms fext and fint represent the external and
internal load vectors, respectively, and um and ut contain the system displacements and temperatures. The SA approach now yields the following equation for
the displacement sensitivity:
dum
∂R dut
∂R
−1
= Km
+
.
(8.10)
ds
∂s
∂ut ds
Here use is made of the fact that the mechanical tangent matrix Km is given by
−∂R/∂um . The term ∂R/∂s is evaluated using finite differences, and the temperature sensitivity dut /ds has already been calculated at this point. To evaluate
Equation 8.10, still the term ∂R/∂ut needs to be computed, which expresses the
interdisciplinary coupling between the mechanical and thermal problem. A finite
difference approach is possible, however this requires as many perturbations as
the number of thermal degrees of freedom. It is more attractive to first further
specify this term analytically. From the fact that the external load is independent
of temperature, it is found that
∂R
∂fint
=−
.
∂ut
∂ut
(8.11)
184
GRADIENT-BASED SHAPE OPTIMIZATION OF AN SMA GRIPPER
8.3
Furthermore, since on element level the internal force is given by:
fint = DT σ
with Dij =
∂εi
,
∂uj
(8.12)
for each individual element, with σ and ε as the element generalized stress and
strain vectors, Equation 8.11 eventually becomes:
∂R
∂σ
= −DT
.
∂ut
∂ut
(8.13)
Hence, by evaluating the term ∂σ/∂ut , the coupling term can be obtained. As
only local temperatures affect element stress values, using this element-level formulation the cost of computing this term using finite differences is comparable to
merely a single stress update. However, a fully analytical treatment is possible
as well, by differentiation of the SMA constitutive model. A specific challenge in
this approach is considered in the following subsection.
8.3.3
Handling of the implicit SMA material model
When considering differentiation of the SMA constitutive relations used in the
present model, a challenge is encountered in the fact that the SMA material
model itself is not formulated in a fully explicit way. This differentiation is necessary in for example the discussed thermo-mechanical sensitivity coupling term, or
the sensitivities of derived quantities such as stresses or effective strains. Without
addressing the full details of its formulation (these can be found elsewhere (Langelaar and Van Keulen, 2004b, 2006) (Chapter 3 and Chapter 6)), by discussing
the structure of the equations this subsection aims to make clear how this problem
is resolved.
The SMA stress-strain relation used in the present shell elements takes the
following form:
σ = C(G(T, ε))ε.
(8.14)
Here C represents the constitutive tangent matrix, that relates the generalized
stresses σ to the generalized strains ε. In this SMA model, C is not constant,
but changes with the value of the effective shear modulus G. This shear modulus
in turn is a function of the temperature T and the current strains ε. However,
the way G depends on T and ε is not formulated explicitly, but is given by an
implicit equation of the following structure:
G = C(T ) +
D(T )
.
f (G, ε)
(8.15)
Here C and D are governed by material properties and vary with temperature,
and f is a complex function of G itself as well as the strain vector ε. During
8.3
SENSITIVITY ANALYSIS
185
finite element calculations, in order to evaluate the material response using Equation 8.14, the shear modulus G is determined from Equation 8.15 by solving the
following nonlinear equation using Newton iterations:
Z(G) = C(T ) +
D(T )
− G = 0.
f (G, ε)
(8.16)
Note that T and ε are given at this point, and only G is still to be determined.
The fact that G is not formulated as an explicit function leads to a slight complication in the analytical sensitivity analysis involving this material model. For
example, in order to analytically evaluate the term ∂σ/∂ut used in the thermomechanical sensitivity coupling (Equation 8.13), differentiation of Equation 8.14
to T yields:
∂C ∂G
∂σ =
ε.
(8.17)
∂T um =constant
∂G ∂T
Since G is not explicitly formulated, the term ∂G/∂T can not be evaluated directly. To deal with this situation, implicit differentiation must be applied to
Equation 8.16:
Z(T, G(T )) = 0 ⇒
∂Z
∂Z ∂G
∂G
∂Z
+
=0 ⇒
=−
∂T
∂G ∂T
∂T
∂T
∂Z
∂G
−1
.
(8.18)
In a similar way, analytical expressions can be derived for sensitivities of quantities
derived from the state vector, the displacement vector um . Further detailed
expressions can be found in Chapter 7.
8.3.4
Comparative evaluation of sensitivity analysis techniques
The discussed finite difference, semi-analytical and refined semi-analytical sensitivity analysis methods have been implemented to perform the sensitivity analysis
of the considered SMA gripper. In terms of implementation, GFD requires the
least effort, followed by SA and finally RSA. Particularly the analytical treatment of the interdisciplinary coupling terms and the SMA material model make
the implementation of (R)SA rather involved in this case. However, next to implementation effort it is important to evaluate how these three sensitivity analysis
methods perform in terms of accuracy and efficiency.
All methods utilize finite difference approximations at some point, and this
involves a relative design perturbation. Accuracy of finite difference derivatives
is known to depend on the choice of this relative perturbation, and therefore
sensitivities have been computed for relative perturbations ranging from 10−14 to
10−2 . The design point used for this sensitivity analysis was the optimal design
found in a previous study (Chapter 5), given by x = (t, H, Y1 , Y2 , W1 , W2 , V )
=(0.0974, 1.1703, 1.2485, 1.1125, 1.5000, 1.4001, 0.0603). The results of two
typical cases are shown in Figure 8.3, which clearly shows that a perturbation
186
8.3
GRADIENT-BASED SHAPE OPTIMIZATION OF AN SMA GRIPPER
range exists where stable values are obtained, and that both too large and too
small perturbations lead to increasing deviations. In general it was found that for
mechanical responses the stable range was smaller than for thermal ones, and that
shape design variables were more sensitive to the perturbation used. Considering
all results, no method was found to consistently outperform the others in terms
of accuracy. RSA did not result in clear improvements over SA for the present
problem.
1.4
SA
RSA
GFD
dL(g2)/dH
0.4
SA
RSA
GFD
dL(g2)/dY1
1.3
0.35
1.2
1.1
0.3
1
−14
10
−12
10
−10
10
−8
10
−6
10
10
−4
10
−2
−14
10
10
−12
−10
10
−8
10
10
−6
−4
10
10
−2
Figure 8.3: Logarithmic sensitivities dL g2 /dH (left) and dL g2 /dY1 (right), computed
using various methods, as a function of the relative design perturbation.
The three sensitivity analysis methods have also been compared in terms of
their computational efficiency. For this comparison, the same design was used as
for the perturbation study, and a machine equipped with a 2 GHz AMD Athlon
processor was used for the evaluation. Table 8.5 lists the resulting time consumption of each method for an analysis of the SMA gripper in both the open and
closed case, combined with a full sensitivity analysis of the five responses of the
optimization problem given in Equation 8.1. It can be seen that the SA method
is the most efficient for the present problem, followed by RSA and GFD. Note
however also that in terms of total evaluation time, GFD only requires 12% more
time than SA. When also considering the fact that the implementation of the
GFD method is far less involved than the SA method for the present problem,
the GFD method in fact is an attractive option.
Still, based on its superior efficiency for the present problem, and since it had
been implemented already, the semi-analytical method was chosen for use in the
gradient-based design optimization of the SMA gripper discussed in Section 8.4.
Furthermore, considering the results of the perturbation variation study, a relative
perturbation of 10-7 was selected.
8.3.5
Impact of numerical noise on design sensitivities
The extensive numerical studies of the sensitivities of this SMA gripper problem
computed by the GFD and (R)SA methods have shown that in most cases, the
8.3
187
SENSITIVITY ANALYSIS
results are in good agreement. However, due to the numerical noise that can
be observed for certain responses, in rare cases the GFD approach sometimes
produces inaccurate results. Two interesting examples have been encountered at
the (feasible) design point x∗ =(0.1546, 0.8774, 0.8977, 0.5614, 1.2687, 1.1719,
0.036085), for the strain constraint in the open case as a function of variations in
H and Y2 , respectively. The behavior of this response is illustrated in Figure 8.4.
0.25
−0.06
g1
g1
−0.08
0
−0.1
−0.25
H [mm]
−0.5
0.6
0.7
0.8
0.9
1
1.1
−0.12
Y2 [mm]
0.3
1.2
0.4
0.5
0.6
0.7
0.8
Figure 8.4: Response data for the strain constraint in the open case (g1 ), against
variations in H (left) and Y2 (right, with closeup) at a certain design point x∗ (o).
The responses in Figure 8.4 show nonsmooth variations, linked to numerical
noise in the response evaluation. This noise could be caused by remeshing effects,
or by the fact that an iterative solution process is used for solving the nonlinear
mechanical problem, which is terminated at a certain finite error. Still, clearly
a trendline can be identified from the graphs. Ideally, the gradients used in
the optimization process should follow this trendline, but in reality sensitivity
information only has a local validity. When a model suffers from noise, it can
be debated whether it makes sense to use derivative information and gradientbased optimization methods. Response surface based methods (such as the Multipoint Approximation Method used in this chapter) might be better suited for
optimization of such noisy functions, as the approximations tend to dampen the
effect of noise.
Table 8.5: Time consumption of sensitivity analysis of the SMA miniature gripper finite
element model, with 7 design variables.
Method
Total time [s]
Index w.r.t. SA
Sensitivity analysis time [s]
SA
372
100
146
RSA
398
107
172
GFD
416
112
190
Analysis only
226
188
GRADIENT-BASED SHAPE OPTIMIZATION OF AN SMA GRIPPER
8.4
Table 8.6: Sensitivity values of g1 at x∗ obtained by various methods, with respect to
H and Y2 .
Sensitivity
From trendline (approx.)
GFD
SA
∂g1 /∂H [mm-1 ]
1.211
-29.318
1.213
∂g1 /∂Y2 [mm-1 ]
79.514×10−3
-4.520×10−3
-4.523×10−3
Table 8.6 illustrates the effect numerical noise can have on the sensitivities.
It lists sensitivity values based on the trendlines taken from Figure 8.4, together
with results obtained by SA and GFD approaches. In the case of ∂g1 /∂H, the
SA result matches the trendline well, but the GFD result is clearly inaccurate,
both in sign and in magnitude. Apparently the perturbation used in the GFD
computation has been such, that noise has affected the accuracy. The authors
want to emphasize that this is the only case of this type of inaccuracy of GFD
results that has been encountered so far, in spite of extensive testing. Still, it
illustrates the potential susceptibility of GFD sensitivities to large inaccuracy
due to numerical noise.
In case of ∂g1 /∂Y2 , SA and GFD results are in good agreement, but fail to
capture the slope of the trendline that appears to be present in Figure 8.4, which
again clearly differs both in sign and in magnitude. As shown in the inset in
Figure 8.4, the response g1 (Y2 ) is locally rather noisy, showing fast variations.
Apparently, the SA and GFD results are indeed accurate locally, but they fail to
represent the response behavior on a slightly larger scale. This phenomenon could
clearly reduce the effectiveness of gradient-based optimization methods. Again,
the authors want to stress that observations of this phenomenon have been very
rare for the present problem, but still the presented case illustrates why caution
should be taken when using gradient-based algorithms in combination with noisy
response functions.
8.4
Gripper design optimization
8.4.1
Optimization methods
The optimization of the design problem defined in Equation 8.1 has been performed by three different methods, one direct and two gradient-based methods.
Each of the methods has been tested using the three different starting points given
in Table 8.4. The direct method used is the Multi-point Approximation Method
(MAM) (Toropov et al., 1993a), which essentially is a combination of design of experiments techniques, response surface building and a move limit strategy. After
evaluating various designs at different sampling points, a response surface is con-
8.4
GRIPPER DESIGN OPTIMIZATION
189
structed and optimization is performed within the current trust region using this
approximation. The point that is obtained is called the sub-optimal point, and
this design is subsequently evaluated using the model, in order to judge the quality of the approximation. Depending on the accuracy and several other heuristic
indicators, the trust region is adapted and new sampling points are planned for
the next iteration.
The strengths of the MAM are its robustness in case of noisy function evaluations, through the use of approximations. Further advantages are its versatility
and ease of application, as it allows treating the model as a “black box”. However, generally considerable numbers of sampling points and function evaluations
are needed, and for nontrivial problems parallel computing is generally required
to obtain results within a reasonable timeframe. Furthermore, both as advantage and disadvantage, the MAM involves a large number of optimizer parameter
settings that can be tuned to the specific problem. This includes the order and
shape of the response surface, parameters affecting the move limit strategy, and
parameters related to the design of experiments. In this study, linear response
surfaces were used to limit the number of required function evaluations, and a few
trial runs have been performed in order to find practical settings for other parameters. Note that more extensive tuning might further improve the performance of
this method, but as this requires a significant additional effort, such tuning was
not considered fair in comparison to the other methods that hardly require any
tuning.
The gradient-based methods that are considered are the Method of Moving
Asymptotes (MMA) by Svanberg (1987) and the Sequential Quadratic Programming method (SQP), (see e.g. Fletcher, 1980). MMA approximates the responses
by smooth monotonic approximations with asymptotes, which leads to convex approximations. MMA has proven its effectiveness for problems with large numbers
of design variables, such as topology optimization problems, and it is interesting
to test its performance in the present SMA shape optimization problem. SQP is
generally accepted as the best general-purpose gradient-based optimization algorithm, and the specific implementation used here is the one implemented in the
Matlab Optimization Toolbox (Version 2.1).
8.4.2
Optimization results
A complete overview of the performance of the three methods and the designs
obtained is given in Table 8.7. It can be seen that the designs obtained with the
MAM method are consistently inferior to those found with the gradient-based
methods. In part, this is due to the convergence settings used. If convergence
tolerances would have been tightened, it is possible that better objectives could be
obtained with the MAM approach, however at the cost of additional iterations and
function evaluations. Furthermore, more extensive and elaborate tuning of the
parameters and settings of the MAM method might improve the optimal designs
as well, but to allow a fair comparison between the methods in a practical setting,
190
GRADIENT-BASED SHAPE OPTIMIZATION OF AN SMA GRIPPER
8.4
Table 8.7: Results overview of the optimization study of the SMA miniature gripper.
Method
Case
Objective
f [mm]
Constraints
g1
g2
g3
g4
Variables
t
H
Y1
Y2
W1
W2
V
Cost
Iterations
Evaluations
Time (rel.)
Averaged
time (rel.)
A
MAM
B
A
MMA
B
0.8916
0.7975
0.8541
0.9164
0.9166
-0.005
0.003
0.000
-0.003
-0.014
-0.002
-0.000
-0.002
-0.010
0.001
0.000
-0.003
-0.001
-0.041
-0.000
-0.001
0.0957
1.2130
1.4970
1.1580
1.5000
1.4960
0.0605
0.1030
1.0910
1.1090
0.8910
1.3660
1.0860
0.0585
0.0858
1.2550
1.3530
1.2360
1.4960
1.3200
0.0647
24
348
264
28
310
309
34
569
373
208
C
A
SQP
B
0.9166
0.9149
0.9137
0.9159
-0.007
-0.050
-0.000
-0.016
-0.000
-0.040
0.000
-0.002
-0.004
-0.044
-0.000
-0.002
-0.004
-0.050
0.001
-0.001
0.000
-0.042
-0.000
-0.002
0.0959
1.3023
1.0132
1.0935
1.5000
1.5000
0.0600
0.0959
1.3010
1.0180
1.0970
1.5000
1.5000
0.0600
0.0960
1.3020
1.0130
1.0970
1.5000
1.5000
0.0599
0.0958
1.3030
1.0070
1.0860
1.5000
1.5000
0.0599
0.0962
1.3250
0.9701
1.0190
1.5000
1.5000
0.0595
0.0960
1.3120
1.0060
1.0460
1.5000
1.5000
0.0597
22
22
200
20
20
182
29
29
264
13
17
155
6
11
100
16
22
200
142
C
C
100
we have limited the tuning effort for this study. Moreover, linear response surfaces
have been used in the MAM studies, to limit the number of sampling points
required in every iteration. When quadratic response surfaces would have been
used, it might be that the higher quality of the approximation would lead to faster
convergence. However, the number of sampling points required for using quadratic
approximations depends quadratically on the number of design variables, which
makes this option unattractive for larger design problems.
The designs obtained by the gradient-based methods are in good agreement,
regardless of the starting point used. This could indicate that the present problem
has convex properties, although this cannot be concluded with certainty based on
this modest number of trials. In comparison, it turns out that MMA is more
consistent in finding the same optimum than SQP, but it also uses more iterations. Variables W1 and W2 are at their upper bounds, indicating that when the
application would allow wider gripper plates, further performance improvement
could be possible. Active constraints that can be identified in various cases are
either the strain or temperature constraints of the opened case, indicating that
this case is apparently the most critical for this design problem.
There is some variation in the number of iterations between the different cases
and methods, and generally the gradient-based methods converge slightly faster.
SQP shows the least iterations, but in some iterations a line search is used, which
8.4
GRIPPER DESIGN OPTIMIZATION
191
makes that the number of function evaluations is close to that used by MMA.
Furthermore, the MAM method clearly requires much more function evaluations,
as it is based on response surface building on sets of sampling points. However,
next to comparing this number of evaluations, in practical situations it can also
be relevant to consider the time required for the optimization process. Assuming
sufficient computing facilities and software licenses are available to evaluate all
sampling points of every MAM iteration simultaneously, and given the fact that
evaluation of the responses requires less time than evaluating the responses and
their gradients, it is found that in terms of time, the difference between the MAM
approach and the gradient-based methods is much smaller than just a comparison
of the number of function evaluations indicates. These time comparisons have
been normalized with respect to the fastest case (set to 100). Note that every
MAM iteration in this view requires the time of two function evaluations (sampling
and sub-optimal point evaluation), which for the present problem both require
60% of the time of a simulation with sensitivity analysis (see also Table 8.5). On
average, in terms of time between starting the process and obtaining an answer,
the gradient-based approaches beat the MAM method by only a factor 2 or less.
Considering the fact that setting up a sensitivity analysis for a given problem can
be a very complex and time-consuming task, this shows that for practical purposes
response-surface-based methods are in fact quite competitive, for problems with
a modest number of design variables. On the other hand, purely based on the
computational effort and the quality of the results, the gradient-based methods
clearly turn out to be superior in this case, showing a reduction of 94 to 96% of
the average number of function evaluations in comparison to MAM.
Note that comparing optimization algorithms is not as exact as it may seem.
Much depends on the settings for certain parameters of the optimizer, the starting point used, and the termination criteria. For the gradient-based cases, the
optimization process was terminated when the norm of relative variable change
became smaller than 10-3 , or when the absolute objective change dropped below 10-4 . Constraint violations smaller than 10-3 were tolerated. For the MAM,
similar but slightly less strict convergence criteria were used.
Figure 8.5 and Figure 8.6 show the history of the objective function and the
thickness design variable t during the optimization process, for all three algorithms, for the cases using Starting point C. The objective initially worsens, as the
starting point is infeasible, and the optimizers initially try to achieve feasibility. It
can also be seen that the MAM objective does not improve significantly anymore
after 25 iterations, and its final objective is lower than that of the gradient-based
methods. This could be caused by the fact that linear approximations are used,
while close to convergence quadratic approximations would probably perform better. An interesting extension of the MAM method as used here might therefore be,
to include an algorithm that allows adaptive switching between approximations,
based on the convergence characteristics.
The evolution of the plate thickness illustrates key differences between the
three methods. The MAM method makes use of an adaptive move limit strategy,
192
GRADIENT-BASED SHAPE OPTIMIZATION OF AN SMA GRIPPER
1
8.4
Objective [mm]
MMA
MAM
SQP
0.5
0
Iterations
−0.5
0
5
10
15
20
25
30
35
Figure 8.5: History of the objective value for the three different optimizers used, in case
of Starting point C.
0.7 t [mm]
MMA
MAM
SQP
0.6
0.5
0.4
0.3
0.2
0.1
Iterations
0
0
5
10
15
20
25
30
35
Figure 8.6: History of the thickness design variable for the three different optimizers
used, in case of Starting point C.
8.5
GRIPPER DESIGN OPTIMIZATION
193
which makes that initially rather large design changes are possible, but gradually
the move limits are tightened as the process zooms in toward the optimum. The
MMA method intrinsically limits the changes of design variables, and no large
steps are observed here. The thickness gradually evolves to its final optimal
value. Finally, in case of SQP, no explicit limits are imposed on the changes of
design variables, which is reflected in the fairly large thickness changes observed
both at the beginning and at a later stage of the process. A full discussion of all
the convergence histories for all responses, variables, methods and starting points
is outside the scope of this chapter, but the two examples shown in Figure 8.5
and Figure 8.6 give a good representation of the typical observations also found
in other cases.
Figure 8.7: Temperature distribution [K] on the optimal miniature SMA gripper in the
open and closed configuration. A full color version is given by Figure G.7 on page 253.
Finally, the geometry of the optimal SMA gripper design is shown in Figure 8.7, in both the open and closed configuration. The design obtained by MMA
for Starting point C has been used here. Figure 8.7 also shows the temperature
distributions in both configurations, and it can clearly be seen that either the
inner or outer plates are activated by the applied voltage. It can also be seen
that the maximum temperature exceeds the value used in the individual element
constraints (338 K), since constraint aggregation was used. The maximum temperature in the optimal design was found to be 338.76 K. However, the excessive
temperatures only are present in a small area of the complete gripper. It depends
on the demands of a specific application whether the given degree of constraint
violation is acceptable. If not, the constraint violation can be reduced easily by
e.g. increasing the ρ-parameter in the aggregation function (Equation 8.4) or
by reducing the limits used in the individual constraints. In a trial run, it was
found that increasing the ρ-parameter from 40 to 80 resulted in a 40 to 50% decrease of the maximum violations. However, the tightening of the constraints also
decreased the objective by 6%.
194
8.5
GRADIENT-BASED SHAPE OPTIMIZATION OF AN SMA GRIPPER
8.5
Conclusions
As recognized by other authors, the application of design optimization techniques
shows great promise for the design of SMA structures, as the complexity of the
problem often defies trial-and-error or intuition-based approaches. The research
presented in this chapter extends the available techniques for SMA design optimization in several ways. Firstly, use is made of a versatile multi-disciplinary
finite element model, that considers electrical, thermal and mechanical aspects in
order to evaluate the performance of a design. Next to this, the design problem is
cast into a formal optimization problem, which allows the consideration of clear
performance-based objectives and constraints. Furthermore, an efficient semianalytical sensitivity analysis procedure has been developed for the finite element
model. This finally allows the use of gradient-based optimization algorithms, that
are clearly more effective than methods that only utilize the responses. A key enabler of this approach is the history-independent SMA constitutive model used in
this work, which allows efficient sensitivity analysis.
The comparison of sensitivity analysis approaches revealed that a restarted finite difference approach only required slightly more computation time compared
to the semi-analytical approach, for the miniature SMA gripper case. Since implementation of the finite difference method is much less involved, this method turns
out to be an attractive option. Some caution must however be taken with the use
of this method in combination with noisy responses, as it was found that in rare
cases it can produce inaccurate sensitivity results. Furthermore, the presented
method is restricted to history-independent models.
Comparison of the SQP, MMA and MAM optimization approaches clearly
show that the response-surface-based MAM method required far more function
evaluations, and resulted in an optimum design of lesser quality. Here it should
be noted that no extensive tuning of the optimizer settings was performed beyond what could be considered reasonable, in order to allow a fair comparison.
Furthermore, if the algorithm would be able to adaptively switch between the
type of approximations that are used (e.g. linear or quadratic), improvements in
convergence and efficiency are certainly possible.
The gradient-based SQP and MMA methods yielded more or less comparable
results. MMA obtained slightly more consistent results, whereas SQP converged
faster. Based on the results of this study, the most efficient approach in terms
of performance is to use semi-analytical sensitivity analysis in combination with
the SQP optimizer. From another point of view however, when merely the time
needed to obtain a design optimization result is of interest and sufficient parallel
computing power is available, the MAM method only requires double the time
of the fastest gradient-based method, while not requiring a sensitivity analysis
implementation. Furthermore, the MAM approach can also be applied to historydependent problems. As long as the number of design variables remains modest,
it should certainly be considered.
8.6
ACKNOWLEDGMENTS
195
In conclusion, the presented results clearly demonstrate the potential of gradientbased design optimization for SMA structures. Extension of this approach to
models involving history-dependent material behavior, such as the majority of
existing SMA constitutive models, remains as a challenge for the future.
8.6
Acknowledgments
The authors would like to thank Krister Svanberg for giving permission to use his
MMA code for this research.
196
GRADIENT-BASED SHAPE OPTIMIZATION OF AN SMA GRIPPER
8.6
Based on: Langelaar, M. and van Keulen, F. (2007). Gradient-based Design
Optimization of Shape Memory Alloy Active Catheters. 15th AIAA/ASME/AHS
Adaptive Structures Conference, Waikiki, Hawaii, April 23–26, 2007, accepted.
Chapter
9
Gradient-based Optimization of
SMA Active Catheters
The design of an active catheter is an example of a challenging design problem of a
Shape Memory Alloy (SMA) adaptive structure. The objective is to find a geometry
that combines the electrical, thermal and mechanical properties of the structure in such a
way that optimal bending performance is achieved. This chapter introduces the application of an efficient gradient-based design optimization procedure to this design problem.
The specific model used focuses on the R-phase transformation in NiTi, and involves
multi-point constraints to implement symmetry conditions. The nonlinear mechanical
analysis is carried out using an incremental-iterative approach in combination with an
augmented Lagrangian technique to account for the nonlinear constraints. Sensitivity
analysis is performed using finite differences in combination with fast reanalysis, where
a new correction term is applied to the multi-point constraints that significantly improves the accuracy. The proposed gradient-based optimization approach is compared
to an alternative direct method, and a clear advantage in terms of the number of required function evaluations is achieved. The application of design optimization yields
active catheter designs that clearly outperform previous versions. It is expected that
the presented method will prove useful for the design of other SMA adaptive structures
as well.
9.1
Introduction
The improvement of the steerability and controllability of catheters by integrating miniaturized distributed actuators has recently attracted considerable interest
197
198
GRADIENT-BASED OPTIMIZATION OF SMA ACTIVE CATHETERS
9.1
(see e.g. Dario et al., 1991, Kaneko et al., 1996, Lim et al., 1995, Mineta et al.,
2001, Park and Esashi, 1999b), and also Chapter 3. By the integration of actuators, catheters can be turned from passive devices into adaptive instruments,
whose shape can be modified in a controllable way. This capability is very helpful
in accurately navigating to remote treatment locations of the body in a noninvasive manner (Tendick et al., 1998).
Most of the active catheter prototypes that have been developed make use
of shape memory alloys (SMAs) for actuation. SMA actuators are an attractive choice in comparison to other options, because they offer the highest work
density, relatively large actuation strains and stresses, and good biocompatibility
(Humbeeck, 2001). Moreover, because SMAs can be used for direct actuation,
the designs can remain relatively simple. However, since the geometry of a design
is directly linked to its electrical, thermal and mechanical performance, which in
turn all affect the overall functionality of SMA devices, designing effective structures is a difficult task, particularly in a three-dimensional setting. This chapter
therefore introduces the use of gradient-based design optimization applied to the
design of an SMA active catheter, and shows that this offers a systematic and
effective procedure to design such SMA adaptive structures.
Several examples of the application of design optimization concepts to SMA
design problems can be found in recent literature (e.g Dumont and Kuhl, 2005,
Fischer et al., 1999, Langelaar and Van Keulen, 2004a, Lu et al., 2001, Skrobanek
et al., 1997). However, none of these studies used efficient gradient-based optimization algorithms. This limits their applicability to relatively simple problems
with few design variables. The aim of the present chapter is to develop and
demonstrate an approach that is more scalable, and better suited for design optimization of SMA structures of higher complexity and with many design variables.
The case of an active catheter design problem is used as a carrier, as it involves
all the aspects relevant to the more complex type of design problems we intend
to solve.
First, Section 9.2 discusses briefly the active catheter concept and the designs
that will be considered in the present study, as well as the computational modeling and analysis procedure. Note that the proposed optimization-based design
approach is not limited to the specific SMA active catheters used in this chapter,
but that it can be applied to other design problems involving SMA structures
as well. In order to enable the use of efficient gradient-based optimization algorithms, design sensitivities need to be computed, and Section 9.3 treats the
sensitivity analysis used for the present problem by a finite difference approach
involving reanalysis. As the finite element model of the present problem involves
multi-point constraints that are handled by an augmented Lagrangian approach,
a novel correction is used to ensure the accuracy of the sensitivities when using
reanalysis techniques, inspired by the correction for nonlinear elastostatic problems previously proposed by Haftka (1985). After validation of the sensitivity
analysis, Section 9.4 presents the results of various design optimization studies,
and a comparison is made between a direct method, the Multipoint Approxima-
9.2
PROBLEM FORMULATION
199
tion Method (MAM, Toropov et al., 1993a) and the gradient-based Sequential
Quadratic Programming (SQP) algorithm, followed by the conclusions.
9.2
Problem formulation
9.2.1
Active catheter design concept
Most active catheter prototypes reported in the literature consist of an assembly
of many individual parts, which increases the complexity and cost of the manufacturing process and reduces the overall reliability and robustness of the devices.
Therefore, in a previous paper, we have proposed an active catheter design aimed
at maximum integration of electrical, thermal and structural functions (see Chapter 3). In the present chapter, this concept is further extended and optimized for
various diameters. The active catheter concept and its proposed fabrication process are shown in Figure 9.1. A pattern is laser-cut or etched in a small diameter
SMA tube, and subsequently initial strains are introduced in the resulting structure by inserting spacers. Straining the structure is essential in order to make use
of the SMA effect for actuation. The pattern is chosen such, that the spacers can
be inserted easily, while the structure as a whole is stretched axially.
Figure 9.1: Proposed fabrication of the new active catheter concept using laser cutting
from a small diameter SMA tube (a). After laser cutting (b–c), the structure is stretched
(d) and spacers are inserted (e), in order to generate internal stresses. A full color
version is given by Figure G.2 on page 250.
The final structure of the schematic concept shown in Figure 9.1 contains
orthogonal bending sections, and by combining these, the catheter can bend in
any direction. The actuation takes place by sending a current through parts
of the structure, which locally heats the SMA structure because of its inherent
resistivity. Figure 9.2 illustrates a voltage pattern that could be applied to this
structure in order to induce bending. A more elaborate description of this active
200
GRADIENT-BASED OPTIMIZATION OF SMA ACTIVE CATHETERS
9.2
catheter concept can be found in Chapter 3. Note that although the design of an
active catheter typically also involves other aspects as well, such as a control and
sensory system, this chapter focuses on the design of its mechanical structure,
since that is the part that is essential for achieving the desired functionality.
Figure 9.2: Segmentation, applied voltage pattern and coordinate system. A full color
version is given by Figure G.3 on page 250.
Given the limitations imposed by the in vivo working environment of devices
such as active catheters, the temperature range in which the actuation units are
operated is limited. For human applications, the ambient temperature is given by
the body temperature, i.e. 37◦ C. Based on various clinical models on tolerable
temperatures, a conservative estimation for the upper limit was determined to
be 49◦ C (see Chapter 3). This small temperature window does not allow for
an effective use of conventional SMAs. However, it was found that the R-phase
transformation, which occurs in certain NiTi alloys, can be applied in this setting.
In addition, the hysteresis typical for SMAs is particularly small in case of the
R-phase transformation, which is an attractive property for actuator applications
(Otsuka and Wayman, 1998, Tobushi et al., 1996). The inherent drawback of using
the R-phase transformation is that the actuation strain is considerably smaller
9.2
PROBLEM FORMULATION
201
than that of the generally used martensite transformation of NiTi, offering only
0.7% recoverable strain versus 5–7% for the martensite case. This limited amount
of actuation strain makes it important to optimize the design as much as possible,
to achieve maximum functionality and the largest possible amount of bending.
9.2.2
Active catheter modeling
The active catheter is controlled by applying voltage to electrodes on the SMA
structure, which results in local temperature changes that influence the material
behavior, resulting in a change in the equilibrium configuration of the structure.
In this way, by choosing a certain voltage pattern, the catheter can be bent in
any direction, as illustrated in Figure 9.2 for positive bending in the x, z-plane.
Therefore, to simulate the bending performance of SMA active catheter designs, a
sequentially coupled electrical, thermal and mechanical analysis is required. Finite
element modeling is used to construct a computational model. The electrical and
thermal problems are linear, whereas the mechanical problem involves geometrical and physical nonlinearities. The latter is solved by an incremental-iterative
solution strategy, and involves shell elements (Van Keulen and Booij, 1996), truss
elements with initial strains to model the spacers, and special constraint elements
to enforce a symmetry constraint discussed below (see also Chapter 3).
For the present active catheter application, a suitable alloy has been selected,
which exhibits the R-phase transformation. This Ti-55.3wt%Ni alloy has been
experimentally characterized by Tobushi et al. (1992), and based on the experimental data and known properties of the R-phase transformation, a constitutive
model has been developed as described in Chapter 3. The details of this model
are not repeated here, as the focus of the present chapter is on the sensitivity
analysis and gradient-based design optimization of the SMA structure. However,
it is important to note that in contrast to many conventional constitutive models
for SMA behavior, the used model is history-independent, since the hysteresis of
the R-phase transformation is sufficiently small to be neglected. In this way, the
model essentially classifies as a temperature-dependent nonlinear elastic model.
The critical advantage of the history-independence of the model is that it significantly simplifies the sensitivity analysis (see e.g. Kleiber et al., 1997).
In an optimization process, typically many function evaluations of different
designs are required, therefore it is important to exploit methods to limit the
computational costs. For the present active catheter model, use is made of the
symmetry and periodicity of the structure and its loading, by considering only
the segment highlighted in Figure 9.2. The sides of this segment must remain in
the x, z symmetry plane, and the top and bottom also act as symmetry planes
with respect to the connecting segments. However, the top and bottom symmetry
planes, both initially parallel to the x, y-plane, must be free to rotate around the
y-axis, and to translate in the x, z plane, as the catheter bends. This condition has
been implemented by multi-point constraint equations, which enforce that a set of
nodes remains in a single plane, which is free to rotate and translate as described.
202
GRADIENT-BASED OPTIMIZATION OF SMA ACTIVE CATHETERS
9.2
A full discussion of the formulation of these constraints can again be found in
Chapter 3 and is not repeated here. An augmented Lagrangian approach is used
to account for the constraints in the solution process, as this method turned out
to offer the highest robustness and efficiency. However, this has implications for
the sensitivity analysis, as is discussed in more detail in Section 9.3.
9.2.3
Design optimization formulation
In order to find the active catheter design that offers maximum bending, design
optimization is applied. The objective of the present optimization problem is
to maximize the bending, which is equivalent with minimization of the bending
radius. Two constraints are included in the optimization problem: firstly, the
SMA material is restricted to a certain strain range, as the range of validity of
the used material model is limited to 1% effective strain. Secondly, the operating
temperatures are limited to the range of 37–49◦ C as discussed before. Including
strain and temperature constraints for every individual element would lead to a
prohibitively large number of constraints, therefore a Kreisselmeier-Steinhauser
(KS) constraint aggregation procedure is applied (Kreisselmeier and Steinhauser,
1983). This leads to the following formulation of the design optimization problem:
min f (x) = R(x)
x
Subject to:
g1 (x) = KSA (gε (x)) − 1 ≤ 0,
g2 (x) = KSA (gT (x)) − 1 ≤ 0,
x ≤ x ≤ x.
(9.1)
Here R represents the bending radius measured from the centerline of the tube,
εe is the effective strain, a scalar measure for the strain used in the SMA constitutive model (see e.g. Chapter 3), and T is the element temperature. The
vector x represents the design variables, and x and x contain their lower and upper bounds. The aggregated strain and temperature constraints are given by g1
and g2 , respectively. The individual element constraints and the used aggregation
function are given by:
(i)
gε(i) =
(i)
gT
εe
≤ 1,
(max)
εe
T (i) − Tmin
=
≤ 1,
Tmax − Tmin
KSA (g) =
1
ln
ρ
1
PN
i=1
A(i)
(9.2)
(9.3)
·
N
X
!
A(i) eρg
(i)
.
(9.4)
i=1
Here A is the element area, N is the number of elements and ρ is a parameter
used in the KS function, which is set to 40 in this study. Furthermore, the strain
9.3
SENSITIVITY ANALYSIS
203
Table 9.1: Design variables and their bounds.
Variable
Wall thickness ratio t/D
Shape parameter H/Href
Shape parameter W/Wref
Spacing factor ratio ∆/∆ref
Applied voltage V [V]
Lower bound
0.03
0.8
0.1
0.5
0.005
Upper bound
0.2
2
2
3
0.05
and temperature limits are 1% and 37–49◦ C, respectively. The KSA aggregation
function is a slightly modified version of the standard KS function, which reduces
the importance of small localized constraint violations by including the element
areas. The degree of violation can be controlled by the parameter ρ as well as by
adjusting the constraint limits, depending on the demands of the application.
For the present optimization study, three related but different design concepts
are considered, as illustrated in Figure 9.3. The geometries in Figure 9.3 show
only a quarter of the analysis segment highlighted in Figure 9.2, since the full
segment is constructed by connecting four of the basic units shown in Figure 9.3.
The concepts differ mainly in the design of the flexible section, which contains
one, two or three full loops, hence the designs will be referred to as the 1-loop,
2-loop or 3-loop design. The diameter D of the tube is considered a parameter,
since certain medical applications will require a certain diameter depending on
vessel size and required working channel. The design variables selected for the
present optimization study are the wall thickness t, the two shape variables H
and W as indicated in Figure 9.3, the spacing-factor ∆ that indicates with what
factor the spacers will increase the associated gaps in the structure, and finally
the applied voltage V . The wall thickness is normalized by the tube diameter,
and the other geometrical variables are normalized with respect to their initial
value in the reference designs shown in Figure 9.3, after those designs have been
scaled according to the required design diameter. The bounds used for these
design variables are listed in Table 9.1.
9.3
Sensitivity analysis
9.3.1
Method
In order to enable the use of efficient gradient-based optimization algorithms, it
is necessary to compute design sensitivities for the present active catheter model.
Various approaches exist to perform this sensitivity analysis, which are reviewed
by e.g. Haftka and Gürdal (1992), Kleiber et al. (1997), Van Keulen et al. (2005).
The choice of the most efficient method depends on the nature of the problem.
204
GRADIENT-BASED OPTIMIZATION OF SMA ACTIVE CATHETERS
9.3
0.15
0.1
H
0.05
0
W
0
0.05
0.1
0.15
0.2
0.25
0.3
0.25
0.3
0.25
0.3
0.2
0.1
H
W
0
0
0.05
0.1
0.15
0.2
0.25
0.2
0.15
0.1
H
W
0.05
0
0
0.05
0.1
0.15
0.2
Figure 9.3: Default designs for the 1-loop, 2-loop and 3-loop case (dimensions in mm).
9.3
SENSITIVITY ANALYSIS
205
In the present case, the model is a path-independent nonlinear model involving
nonlinear constraints. However, the solution technique that is used in the simulation, the augmented Lagrangian approach, in fact changes the nature of the
problem. Auxiliary variables called Lagrange multipliers are introduced as internal variables, and these are updated in each iteration by an additional update
rule, given by:
λk+1 = λk + ph(uk )
(9.5)
Here λk represents the multiplier value at iteration k, p is the penalty factor
used and h is the constraint that needs to be satisfied, which is a function of the
displacement field u. When internal variables are present in a model, the model
has to be treated as a path-dependent, transient model (Kleiber et al., 1997). As
the solution is determined by both the displacements and the multiplier values at
the final stage, the design derivatives of the multipliers need to be computed as
well in order to evaluate the total design sensitivities of the system. In order to do
this, in fact after each iteration a sensitivity analysis has to be carried out, since
Equation 9.5 links new multiplier values to previous multiplier and displacement
values.
Based on these observations, it appears that a costly transient sensitivity analysis procedure has to be applied. However, in fact the underlying problem that
is being solved still is a path-independent nonlinear model with nonlinear constraints. Because of the solution approach that was chosen, it has been turned
into a transient type of problem. In case a penalty or Lagrange multiplier method
would be applied, the problem would remain path-independent and sensitivities
could be computed efficiently. Therefore, we observe that the choice of the solution method influences the required sensitivity analysis. However, when the final
solution of the original problem is the same, it is justified to use any method that
is applicable to either approach, since the underlying problem that has been solved
remains unchanged. The penalty method results in approximate solutions, since
the constraints are not satisfied exactly when a finite penalty value is used. The
Lagrange multiplier method in contrast yields the same solution as the adopted
augmented Lagrange multiplier technique. In this method, the auxiliary multipliers are not used as internal variables but are treated as degrees of freedom,
in the same way as the displacements are treated. Hence, the problem remains
path-independent and efficient sensitivity analysis techniques suited for this type
problems can be used.
A previous study of options for design sensitivity analysis of similar SMA
design optimization problems without constraints has shown that attractive options for these nonlinear path-independent problems are either the semi-analytical
method or finite differences in combination with reanalysis (see Chapter 7). The
semi-analytical method was found to be slightly more efficient, but this was offset
by the fact that its implementation is considerably more costly. For the present
case, one option might be to transform the problem into the Lagrange multiplier formulation, after obtaining the solution using the augmented Lagrangian
206
GRADIENT-BASED OPTIMIZATION OF SMA ACTIVE CATHETERS
9.3
approach, and to subsequently use the semi-analytical approach to compute the
sensitivities. Using the Lagrange multiplier method for the analysis itself is not
feasible, as it proved to suffer from convergence problems due to the loss of positive definiteness of the tangent system matrix (see Chapter 3). However, implementation of the proposed transformation and the subsequent semi-analytical
sensitivity analysis procedure will require a substantial effort, and in addition the
Lagrange multiplier formulation requires a solver that is able to handle the nonpositive definite matrix. Furthermore, as the problem is transformed, the original
decomposed system matrix can not be reused, because both the formulation and
the number of degrees of freedom are modified. Therefore, this approach would
reduce the efficiency of the semi-analytical approach considerably.
A more attractive option is to make use of the finite difference method in
combination with reanalysis. Finite differences involve a nominal and a perturbed
solution, and by starting the perturbed analysis from the nominal solution, the
cost of this method approaches that of the semi-analytical method (see Chapter 7).
And in fact, there is no restriction on using the same augmented Lagrangian
solution approach for the perturbed case as well. Implementation of this finite
difference approach is not as complex as the described semi-analytical option,
and moreover this method is far more efficient than adopting the alternative total
transient or global finite difference techniques. Note again that in the present
problem, the transient nature is only a result of the adopted solution strategy,
and not an inherent property of the problem itself. The presented reasoning
therefore does not hold for cases where the problem itself is path-dependent.
9.3.2
Implementation and validation
The proposed sensitivity analysis procedure consisting of forward finite differences
in combination with reanalysis has been implemented in an in-house dedicated
finite element code. The reanalysis only applies to the mechanical part of the simulation, the electrical and thermal problems are linear and a full analysis is used
to solve the associated perturbed cases. A correction is applied to the residual
of the perturbed mechanical problem, in order to ensure accurate sensitivities, as
proposed by Haftka (1985). The finite residual remaining from the nominal case
is subtracted from the residual of the perturbed case. This improves the accuracy,
particularly in case of small perturbations, since it isolates the influence of the
design variable perturbation from further iterative improvement of the solution.
But the residual is not the only finite error that exists in the nominal solution.
Since also the constraint equations h = 0 are solved by an iterative process in
the augmented Lagrangian approach, a small but finite error remains and the
constraints are not exactly satisfied when the iterative process is terminated. In
case of the constraints in the present SMA active catheter model, the error is expressed by the distances of nodes to the defined symmetry plane. These generally
finite distances result in nonzero equality constraint values ∆h at the final solution of the nominal case. By subtracting these ∆h values from the corresponding
9.3
SENSITIVITY ANALYSIS
207
constraints in the perturbed case, again the effect of the design perturbation is
isolated from further improvement of the nominal solution, in the same way the
residual is corrected as proposed by Haftka (1985). This correction is expected
to result in more accurate sensitivities, particularly for smaller perturbations.
To the best knowledge of the authors, improving the accuracy of sensitivities by
this constraint correction for problems with multi-point constraints has not been
reported before.
In order to investigate and validate the proposed procedure and the effect
of the constraint correction, first a small test problem will be considered. This
simplified test problem has the same components as the full catheter problem,
i.e. a sequentially coupled electrical, thermal and mechanical simulation, where
the mechanical problem involves the same shell, truss and constraint elements as
the full model. This test problem is depicted in Figure 9.4, which shows four
shell elements using the SMA material model also used for the catheter (in the
rectangle ACDF), one linear elastic bar element (CG) and two constraint elements
(ABG and ACG). These constraint elements restrict the motion of nodes B and C
to the x, z-plane. The edge AF is clamped, and a load is applied to the side CD as
indicated. Edge DEF is constrained in y- and z-direction, and also the position of
F
E
0
A
D
−1
z
−2
−3
B
0
2
C
4
4
6
10
x
2
G
8
0
y
Figure 9.4: Test problem used to test the sensitivity analysis procedure. Dashed lines
show the deformed configuration.
208
GRADIENT-BASED OPTIMIZATION OF SMA ACTIVE CATHETERS
9.3
node G is fixed. Without constraints, the loading would result in a contraction in
y-direction, causing nodes B and C to move away from the x, z-plane. Note that in
this test problem it would have been much simpler to constrain the y-displacement
of these nodes instead of using the constraint elements, but this is not relevant
for the purpose of this test model. The aim is to verify the sensitivity analysis
procedure using an equivalent but simpler model. The electrical and thermal
analysis only involve the rectangle ACDF, where they affect the SMA material.
The electrical boundary conditions are given by prescribed voltages on edges AF
and CD, and the thermal case involves only surface convection.
In order to validate the proposed sensitivity analysis procedure and the correction applied to the constraints, three cases are considered:
1. Finite differences using a full analysis for the perturbed case (no constraint
correction necessary!), providing reference values
2. Finite differences using a restarted perturbed analysis, with constraint correction
3. Finite differences using a restarted perturbed analysis, without constraint
correction
In both cases using reanalysis, the proven residual correction proposed by Haftka
(1985) has been applied. Tests have been done with various settings of the convergence tolerance, to also examine their effect on the accuracy of the resulting finite
difference sensitivities. The nominal convergence tolerance equalled 10-5 . This
tolerance relates to the sum of the norm of the unbalanced forces with respect to
the norm of the total applied forces and the norm of the displacement increment
normalized by the norm of the total displacement. The response considered is the
aggregated strain constraint value KSA (gε ) as also used in Equation 9.1. Various
design variables have been used, and sensitivities have been computed for a range
of relative design perturbations, in order to evaluate the accuracy and stability of
the different approaches.
A typical result is shown in Figure 9.5 for the case of the shell thickness design
variable t. It is seen that when using tolerance settings similar to the nominal
case, the effect of the constraint correction in the restarted case is minor. In that
situation, both corrected and uncorrected restarted cases clearly lack accuracy as
the relative design perturbation is reduced, in comparison to the full finite difference approach. This reduced range of stable and accurate sensitivity results for
the restarted case is presumably due to the fact that for increasingly smaller perturbations, the restarted iterative solution process terminates more quickly, since
the unbalance terms generated by the design perturbation also reduce. When the
same tolerance level is used in the perturbed case as is used in the nominal analysis, the perturbed result might not fully include the (subtle) effect of the change in
the design variable, which leads to inaccurate sensitivity results. This explanation
is supported by the fact that the accuracy improves drastically when additional
9.3
SENSITIVITY ANALYSIS
209
−32
−34
−36
−38
−40
dLKS(gε)/dt (full)
dLKS(gε)/dt (restarted, full Newton iterations, tolerance 10−5)
dLKS(gε)/dt (restarted, modified Newton iterations, tolerance 10−10)
−42
−14
dLKS(gε)/dt (restarted, modified Newton iterations, tolerance 10
)
dLKS(gε)/dt (restarted, uncorrected, 10 additional iterations, full Newton)
−44
∆t/t
−14
10
10
−12
−10
−8
10
−6
10
−4
10
−2
10
0
10
10
Figure 9.5: Logarithmic sensitivities of the aggregated effective strain constraint for the
test problem, as a function of the relative design perturbation in the thickness t.
−0.1
dLR/dt (full)
dLR/dt (restarted, tolerance 10−5)
−10
dLR/dt (restarted, tolerance 10
)
−0.15
−0.2
−0.25
∆t/t
−0.3 −12
10
10
−10
10
−8
10
−6
10
−4
10
−2
0
10
Figure 9.6: Logarithmic sensitivities of the bending radius R for the 1-loop catheter
case, as a function of the relative design perturbation in the thickness t.
210
GRADIENT-BASED OPTIMIZATION OF SMA ACTIVE CATHETERS
9.4
iterations are performed after the restarted perturbed analysis has converged to
the nominal tolerance level, or equivalently when tighter tolerance settings are
applied. This is illustrated in Figure 9.5 by the results of the restarted cases
with constraint correction with reduced tolerance levels, which approximate the
full finite difference results very closely. Modified Newton iterations were used
in these cases, since these turned out to be as effective as full Newton iterations.
Particularly for large problems, this will considerably reduce the required computational effort. To allow a fair judgement of the effect of leaving out the constaint
correction, 10 additional full Newton iterations were performed in that case. This
however did not improve the accuracy, as is seen in Figure 9.5, and it can be concluded that this is due to the fundamental error of not accounting for the finite
accuracy of the constraint conditions for the nominal case.
Sensitivity analysis results for the active catheter case are depicted in Figure 9.6, and here the constraint correction as it was verified by the test problem
study has been included. The results shown are for the sensitivity of the bending radius R with respect to the catheter wall thickness variable t for the 1-loop
design concept, but for other concepts, design variables and responses, similar
results have been obtained. Also here the beneficial effect of tightening the tolerance settings of the perturbed analysis can be observed. The results of the
restarted case with a tolerance of 10-10 approximate the full finite difference sensitivity results very closely, while when using the nominal tolerance setting of 10-5 ,
acceptable, but clearly less accurate results are found. Due to the use of modified Newton iterations, the cost of the sensitivity analysis is still relatively low.
This is illustrated by the fact that in this active catheter case, the unavoidable
mesh genereration for the perturbed design takes twice the time required for the
modified Newton process to obtain the perturbed solution. In comparison with
the full finite difference approach, the computational time for a single evaluation
including sensitivities is reduced by a factor 2.5, for the present case with 5 design variables. When the problem is scaled up to larger models with more design
variables, this factor will increase even further. Based on the findings obtained in
this sensitivity analysis study, the restarted finite difference approach using modified Newton iterations and a tolerance level of 10-10 is selected to evaluate the
sensitivities in the optimization problems, using a relative perturbation of 10-6 .
9.4
Optimization results
The design optimization of the proposed SMA active catheter concepts is performed using two different approaches, the Multi-point Approximation Method
originally proposed by Toropov et al. (1993a) and Sequential Quadratic Programming (see e.g. Fletcher, 1980). The MAM is a direct method which relies on
building a series of approximate subproblems by response surface building, in
combination with a move limit strategy. In every iteration, optimization of the
subproblem results in a sub-optimal point, which is subsequently evaluated using
9.4
OPTIMIZATION RESULTS
211
the full model in order to determine the accuracy of the response surface. The
sample points required for the response surfaces are evaluated in parallel, in order
to reduce the duration of the optimization process. Note that evaluations for optimization using MAM do not involve sensitivity analysis. The second algorithm,
SQP, is generally regarded as the best general-purpose gradient-based optimization algorithm, and the specific implementation used here is the one found in
the Matlab Optimization Toolbox (version 3.0.3). The gradients are computed
using the finite difference approach in combination with reanalysis, as discussed
in Section 9.3.
The used SQP implementation sometimes activates a line search procedure
within an iteration after obtaining a search direction from the solution of the QP
subproblem. This is why the number of iterations in general does not match the
number of function evaluations. The SQP runs are started using the first suboptimal point found in the corresponding MAM run as initial point. The MAM
involves various heuristic parameters and settings that affect the generation of
sampling points, the response surface fitting process and the move limit strategy.
A moderate effort to tune these parameters has been made, and it was found
that using linear response surface based on 10 sampling points yielded robust
and steady convergence. Possibly further experimentation with the parameters
might further reduce the total number of required function evaluations, but at
the expense of further computational investments.
A full listing of the results of the design optimization for the 1-loop, 2-loop and
3-loop concepts using these two algorithms is given in Table 9.2. The obtained
optimal bending radii are in close agreement in most cases, only in the 2-loop case
with a 0.5 mm diameter tube the MAM converges to a local optimum. Typical
convergence histories for both SQP and MAM are shown in Figure 9.7, which concerns the 2-loop 1.5 mm case. It is seen that the MAM requires many iterations
particularly when it approaches the optimum, and possibly changing the move
limit strategy parameters or the approximation settings may improve its convergence rate further. Note also that the obtained bending radius for the 1-loop 1.5
mm case of 29.9 mm is a clear improvement over the 35 mm obtained by a previous design (see Chapter 3). However, more impressive improvements are made
when the design concept itself is changed as well instead of merely optimizing it,
as illustrated by the 2-loop 1.5 mm case, which manages to produce a bending
radius of 20.5 mm, an improvement of over 40%.
In general, Table 9.2 clearly shows that SQP requires less iterations and less
function evaluations than the MAM, but its disadvantage is that sensitivity analysis is required, which increases the cost of the evaluations. Accounting for the fact
that evaluations including sensitivity analysis for the considered cases are roughly
twice as costly as a simulation alone, the advantage in total computational time of
SQP over MAM is a factor 4 on average. When considering instead the duration
of the optimization procedure in a parallel computing environment, the number
of MAM iterations is relevant. Note that each iteration of MAM takes the time
of two simulations because of the evaluation of the sub-optimal point, and that
212
GRADIENT-BASED OPTIMIZATION OF SMA ACTIVE CATHETERS
9.4
these MAM iterations should be compared to the evaluations used in the SQP
case, which cannot exploit the parallel infrastructure (unless the simulation itself
is parallellized, but this is not the case here). A direct comparison can be made,
since the SQP evaluations including sensitivity also take the time of roughly two
simulations. When comparing the average total duration, it turns out that the
MAM is only 7% slower than SQP, assuming the evaluation of sampling points
can be fully parallellized. When instead looking at the average ratio over all cases,
the difference between MAM and SQP is still only 18%, in favor of the latter.
When considering the actual optimal designs that were obtained, as listed in
Table 9.3, it turns out that, surprisingly, there are considerable differences in the
values of the design variables obtained by MAM and SQP for the same cases,
even though the objective and constraint values listed in Table 9.2 are in close
agreement. The largest variation is found in the shape variables H/Href and
W/Wref . Investigation of the logarithmic sensitivities of the final designs reveals
that indeed these variables have only a weak relation to the response values.
Apparently the shape of these designs is less important than the used diameter,
tube wall thickness, spacer length and applied voltage, at least in case of the
current parameterization. Possibly a finer parameterization allowing more shape
variations might change this situation, but this question is not explored further
in the present study. What is clear however, is that considerable improvements
MAM
SQP
Bending radius [mm]
50
40
30
20
Iterations
0
5
10
15
20
25
30
Figure 9.7: Successful convergence of the objective value for the 1.5 mm 2-loop MAM
and SQP cases.
9.4
OPTIMIZATION RESULTS
213
have been made with respect to an earlier design presented in Chapter 3, and
that various design concepts can be explored and compared in a systematic way,
using the proposed design optimization procedure.
A final aspect to note is that depending on the selected catheter diameter,
a different design concept turns out to yield the best performance in terms of
bending ratio. Table 9.4 lists the bending radii of all the considered cases, normalized by the catheter diameter. For a diameter of 0.5 mm, the 3-loop design
results in the best performance. For the 1.0 mm case, the 2-loop design performs
best, although the margin with the 3-loop case is small. For the 1.5 mm diameter catheter, again the 2-loop case comes first, with this time a clearer difference
from the 3-loop design. These observations can be explained by the fact that the
physical aspects that determine the performance of these active catheter structures are not linearly dependent on their dimensions. An optimal design at a
Table 9.2: Optimal responses for various design concepts and cases, as well as the number of iterations and function evaluations, for the direct MAM method and the gradientbased SQP method.
Case
1-loop
Diam.
Diam.
Diam.
Diam.
Diam.
Diam.
2-loop
Diam.
Diam.
Diam.
Diam.
Diam.
Diam.
3-loop
Diam.
Diam.
Diam.
Diam.
Diam.
Diam.
design
0.5 mm,
0.5 mm,
1.0 mm,
1.0 mm,
1.5 mm,
1.5 mm,
design
0.5 mm,
0.5 mm,
1.0 mm,
1.0 mm,
1.5 mm,
1.5 mm,
design
0.5 mm,
0.5 mm,
1.0 mm,
1.0 mm,
1.5 mm,
1.5 mm,
Bending
radius
[mm]
KSA (gε )
KSA (gT )
Iterations
Evaluations
MAM
SQP
MAM
SQP
MAM
SQP
11.43
11.43
20.16
20.00
29.94
29.92
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
0.9998
1.0000
28
11
27
23
17
12
206
19
199
34
129
18
MAM
SQP
MAM
SQP
MAM
SQP
10.47
7.921
14.08
14.07
20.58
20.51
0.9999
1.0000
0.9997
1.0000
1.0000
1.0000
0.7650
0.8551
0.9999
1.0000
1.0000
1.0000
19
9
14
11
27
12
143
15
108
17
199
23
MAM
SQP
MAM
SQP
MAM
SQP
7.201
7.199
14.17
14.17
22.29
22.28
1.0000
1.0000
1.0000
1.0000
0.9313
0.9241
0.9998
1.0000
0.9999
1.0000
0.9999
1.0000
19
11
22
9
28
18
143
12
164
12
206
38
214
GRADIENT-BASED OPTIMIZATION OF SMA ACTIVE CATHETERS
9.4
given scale can not simply be scaled by a factor in order to obtain an optimal
design at another scale. However, as shown here, with the help of optimization
well-performing designs can be generated for any scale, in a systematic manner.
In order to illustrate and analyze the way the SMA active catheter designs
function, the temperature distributions of the 1.0 mm diameter 1-, 2- and 3-loop
optimal designs (SQP) are plotted in Figure 9.8. Clear differences can be seen
Table 9.3: Optimal values of the design variables for various design concepts and cases,
as obtained by the direct MAM method and the gradient-based SQP method.
Case
1-loop
Diam.
Diam.
Diam.
Diam.
Diam.
Diam.
2-loop
Diam.
Diam.
Diam.
Diam.
Diam.
Diam.
3-loop
Diam.
Diam.
Diam.
Diam.
Diam.
Diam.
design
0.5 mm,
0.5 mm,
1.0 mm,
1.0 mm,
1.5 mm,
1.5 mm,
design
0.5 mm,
0.5 mm,
1.0 mm,
1.0 mm,
1.5 mm,
1.5 mm,
design
0.5 mm,
0.5 mm,
1.0 mm,
1.0 mm,
1.5 mm,
1.5 mm,
t/D
H/Href
W/Wref
∆/∆ref
V [V]
MAM
SQP
MAM
SQP
MAM
SQP
0.0362
0.0351
0.0500
0.0431
0.0502
0.0483
1.3722
1.2378
1.4282
0.8552
0.8732
1.5418
1.5140
1.1401
1.1868
1.9255
1.1871
0.8434
2.7287
2.8051
2.2421
2.4549
2.2550
2.3039
0.0175
0.0176
0.0208
0.0215
0.0242
0.0244
MAM
SQP
MAM
SQP
MAM
SQP
0.1149
0.0350
0.0414
0.0399
0.0500
0.0441
1.0979
1.4491
1.3605
1.4852
1.5050
1.3614
0.6036
0.9224
0.4790
1.2347
0.1000
1.2267
1.3908
1.9342
1.8190
1.8490
1.7473
1.8160
0.0164
0.0230
0.0312
0.0315
0.0348
0.0362
MAM
SQP
MAM
SQP
MAM
SQP
0.0347
0.0356
0.0374
0.0376
0.0389
0.0377
1.2096
1.2534
1.6041
1.3356
0.8000
1.6335
0.9962
1.0450
0.7755
0.6253
0.6661
2.0000
2.8024
2.7617
2.8235
2.8147
2.6511
2.6695
0.0273
0.0271
0.0351
0.0350
0.0410
0.0414
Table 9.4: Optimal bending radii normalized by the catheter diameter, for all considered
SQP cases.
Case
Diam. 0.5 mm, SQP
Diam. 1.0 mm, SQP
Diam. 1.5 mm, SQP
1-loop
22.8600
20.0000
19.9467
2-loop
15.8420
14.0700
13.6733
3-loop
14.3974
14.1700
14.8533
9.4
OPTIMIZATION RESULTS
215
Figure 9.8: Temperature distributions on parts of the deformed catheter structure for
the best designs of all three concepts, for a 1.0 mm diameter. A full color version is
given by Figure G.8 on page 254.
216
GRADIENT-BASED OPTIMIZATION OF SMA ACTIVE CATHETERS
9.4
between the different concepts. In the 1-loop case, the highest temperatures are
found in the activated winding section. In the 2-loop case, the winding section has
been increased by adding a second loop, and as a result its resistivity has increased
as well. Consequently, in comparison to the 1-loop case, a larger part of the current
is not flowing through the winding section, but through the support structure itself
to a nearby electrode. This is reflected in the temperature distribution, which now
shows considerable heating of the support structure as well. Note that this does
not render this design less effective: in fact, the bending performance of this 2loop design exceeds that of the 1-loop design by 30%. Finally, in the 3-loop design
the addition of yet another loop results in a further change of the temperature
distribution. In this case, the resistivity of the winding section has increased up to
the point where apparently most of the heat is generated in the support structure
instead of the part intended for actuation. Still the activated winding section
shows higher temperatures than the opposing inactive side, but the difference is
smaller than the operating limits would allow. Maximum temperatures instead
occur in the support structure. In spite of this, the performance is still very close
to that of the 2-loop case.
These observations provide hints for further improvement of these designs.
By modifying the electrode arrangement, e.g. by adding an electrode in the
middle of the winding section and letting the current flow from there, the heating
could be localized more. Potentially this will further improve the performance of
these active catheters, and new optimal designs could be found using the same
optimization procedure. Note also that the temperature range shown in Figure 9.8
exceeds the maximum temperature of 322 K mentioned earlier. Due to the type of
aggregated constraint that has been used in the formulated optimization problem,
limited local violations can occur. In this case, the maximum violation is found
in the 1-loop design, of close to 1.4 K in a spot on the winding section. The
violations in the other cases are smaller, up to a maximum of around 0.7 K.
Overall, typically only a small fraction of the total structure exceeds the 322 K
temperature limit. If this is unacceptable for the application, adjusted designs can
be generated by performing the optimization with tightened constraints, e.g. by
increasing the Kreisselmeier-Steinhauser ρ parameter or modifying the constraint
limits.
Finally, Figure 9.9 illustrates the improvements the best designs at each diameter have achieved over the previous design given in Chapter 3. The obtained
bending radii are visualized in Figure 9.9 by plotting each optimized catheter in its
deformed configuration. Since the design obtained in the previous study already
appeared potentially useful for certain applications, these improved designs show
even greater promise. Clearly also reducing the diameter improves the bending
radius of the instruments, however at some point manufacturing limitations will
prevent further miniaturization. Also, for catheter applications, a certain inner
diameter is required for the working channel. In guidewire applications, no working channel is required, and as far as manufacturing techniques allow, further
diameter reduction and improved bending behavior are certainly possible.
9.5
DISCUSSION AND CONCLUSIONS
217
Figure 9.9: Deformed optimal active catheters for various diameters, together with the
deformed design presented earlier in Chapter 3.
9.5
Discussion and conclusions
An efficient optimization-based design procedure for SMA adaptive structures has
been developed and demonstrated in this chapter. The design of an active catheter
has been chosen as a carrier application, as it presents a complex and nontrivial
SMA design problem. A sequentially coupled electro-thermo-mechanical finite
element model has been used to evaluate the performance of the catheter designs.
The use of finite element analysis yields a versatile design approach that can be
applied to a large variety of SMA design problems. Three different active catheter
design concepts have been considered in the present study.
The thermal operating range of active catheters is limited, and hence use is
made of the R-phase transformation. A simplified constitutive model is used,
that renders the analysis path-independent, which significantly simplifies sensitivity analysis. The computation of sensitivities is necessary to use efficient and
scalable gradient-based optimization algorithms. Specifically in the present design problem, multi-point constraints were included in the mechanical model, in
218
GRADIENT-BASED OPTIMIZATION OF SMA ACTIVE CATHETERS
9.5
order to reduce its complexity by exploiting symmetries. A novel sensitivity analysis procedure has been proposed and verified, which relies on finite differences
in combination with fast reanalysis of the perturbed design, using an augmented
Lagrangian approach to handle the constraint equations. A tighter convergence
tolerance was in the perturbed cases than for the nominal analysis, to improve the
accuracy of the design sensitivities. In combination with a constraint correction
for the finite accuracy involved in solving the nominal nonlinear constraint equations, this was demonstrated to yield an accurate and efficient sensitivity analysis
procedure.
The proposed gradient-based design optimization procedure has been shown
to be very effective and efficient, requiring far less function evaluations and computational time than an alternative direct method. However, when sufficient computational resources are available for parallel computation, the used Multi-point
Approximation Method turned out to be a close match to the gradient-based SQP
algorithm in terms of total duration of the optimization procedure, even without
extensive tuning of its settings. Particularly for cases where implementation of
the sensitivity analysis is difficult or costly, or for cases involving different SMA
models that cannot exploit the path-independent properties used in the present
case, this type of direct optimization procedures form a practical alternative. It
has been found that when considering active catheters with different diameters,
simple scaling of designs results in suboptimal performance. Instead, with the
help of the proposed design optimization procedure, designs can easily and systematically be tailored to specific applications, yielding optimal performance.
The designs obtained from the optimization studies at various diameters show
large improvements in bending capacity in comparison to previously published
results. An analysis of the obtained optimal designs however also pointed out,
that a different arrangement of the electrodes used to activate the SMA actuation sections may lead to further improvements. Also a more fine-grained shape
parameterization of the design concepts could lead to higher performance, as this
would enlarge the design space. The studied cases involving only two shape variables turned out to be rather insensitive to shape changes. However, an efficient
optimization method such as the presented gradient-based optimization procedure is essential when dealing with design problems with many design variables.
Given its effectiveness and versatility, it is expected that the application of design
optimization to complex problems such as the design of adaptive SMA structures
will increase further, and the presented approach could form a powerful tool for
the design of many future SMA devices.
Chapter
10
Conclusions and Future Directions
10.1
Conclusions
The conclusions discussed in this final chapter have been organized according to
the four major topics treated in this thesis: modeling of shape memory alloy
(SMA) behavior, shape optimization without using sensitivities, topology optimization and finally gradient-based shape optimization. These topics all relate
to the central question of this research, which is how and to what extent design
optimization techniques can be utilized for designing SMA structures.
10.1.1
SMA modeling
Consideration of SMA modeling is essential, since the complexity of a model is
inversely proportional to its suitability for design optimization. Since the target
applications for this research are miniaturized actuators for medical instruments
and microsystems, the focus has been on the SMA behavior due to the R-phase
transformation in NiTi. The R-phase transformation is well suited for actuation
applications due to its small hysteresis, and the modest required thermal operating
range is compatible with in vivo use.
Chapter 2 and Chapter 3 both propose new constitutive models to describe
the behavior of interest. By selecting a specific operating range, and due to the
negligible hysteresis of the material in this range, it was possible to formulate
models that are considerably less complex than conventional SMA models. It
should be noted, however, that conventional models are generally more widely
applicable, and many of the recently proposed models also attempt to include
the influence of crystallographic texture, non-proportional loading and tension219
220
CONCLUSIONS AND FUTURE DIRECTIONS
10.1
compression asymmetry. These aspects have been neglected in the presented
models, which are formulated with the intent to balance between accuracy and
suitability for design optimization. The models proposed in Chapter 2 and Chapter 3 represent different compromises between these two aspects. Both models
result in a correct representation of known one-dimensional experimental data,
and the main difference is found in the way the one-dimensional formulation is
generalized to a three-dimensional setting. In comparison, the approach taken in
Chapter 2 is computationally simpler, it results in an explicit formulation, but
the isochoric nature of the R-phase transformation is not strictly accounted for.
Chapter 3 extends the approach followed in the formulation of this model, and
includes the known aspects of the transformation. This however leads to a more
complex model, which in the plane stress case involves an implicit equation, that
needs to be solved iteratively. However, considering its less approximate nature,
this model has subsequently been used for optimization in the rest of this thesis.
A key property of both models is their history-independence, which is accomplished through neglecting the small hysteresis of the R-phase transformation. This no longer makes it necessary to use internal variables to keep track
of the state of the material. Essentially, in this way both models approximate
the considered material as a temperature-dependent nonlinear elastic solid. The
advantage of this history-independence is that it significantly simplifies sensitivity
analysis. For history-dependent materials, the sensitivity analysis must account
for the entire loading path, hence time integration of sensitivities is necessary.
However, in the present history-independent case, sensitivities can be computed
in one step, after completing the full analysis. This advantage has been essential for the topology optimization in Chapter 6 and the sensitivity analysis and
gradient-based shape optimization of Chapters 7, 8 and 9.
Therefore, in summary, it can be concluded that the specific SMA
models introduced in this thesis are very well suited for design optimization of the class of SMA structures of interest, and their suitability
for design optimization forms a key element in enabling the realized
optimization techniques.
Furthermore, Chapter 3 introduced a novel design for an active catheter, which
by means of integration is expected to be more reliable and easier to assemble
than the designs used in recent prototypes. The design was simulated using the
formulated SMA material model, and showed promising performance. Although
the transformation strain offered by the R-phase transformation is limited, it has
been demonstrated that due to the structural design of the active catheter, still
a respectable degree of bending could be achieved. Nonetheless, further improvement proved to be possible when shape optimization was applied, as shown in
Chapter 9. This case clearly illustrates the potential for design optimization for
this type of SMA applications involving complex structures.
10.1
CONCLUSIONS
10.1.2
221
Direct shape optimization
Chapter 4 demonstrates the viability and effectiveness of shape optimization
of an SMA gripper, using the response-surface-based multi-point approximation
method (MAM, Toropov et al., 1993a). This is a direct method, which implies
that it does not require design sensitivity information. This is generally not the
most efficient approach, but it has the advantages that it is robust, requires no
sensitivity analysis and has the property to smoothen noisy responses. Parallel
computing has been used to reduce the duration of the optimization procedure
to a practical level. The shape optimization of the gripper was based on a clearly
formulated optimization problem, and finite element modeling was used to evaluate the performance of designs. This versatile and generic approach applies to
many devices, and all kinds of objectives and constraints can be accounted for,
in contrast to more limited analytic or heuristic methods previously reported in
literature.
Chapter 5 extends the gripper optimization considered in Chapter 4 with the
consideration of bounded-but-unknown uncertainties. These are uncertainties in
e.g. loading conditions or material properties, characterized by their bounds. For
innovative products or in an early design stage, precise statistical data on uncertainties is generally not available. Probabilistic approaches therefore cannot be
used. However, often it is possible to define bounds on the values of the variable
parameters, based on expert opinions. Considering the effect of uncertainties is
important in order to generate robust and reliable designs. Parallel computing
has been used to make the optimization feasible for practical applications. Instead of rigorous anti-optimization, an alternating technique was used that yielded
substantial efficiency improvements. It was also shown, that the availability of
sensitivity information potentially could improve the efficiency even more, but
this has not yet been demonstrated for an SMA problem.
In order for the gripper design to remain feasible, a decrease in performance
of 15% was observed compared to the deterministic design. In general, when
accounting for uncertainties, the performance of designs will decrease. However,
in return the optimized design will perform reliably even under varying conditions.
This approach to optimization is believed to be of great practical value, as in many
realistic design situations uncertainty and variability cannot be ignored.
It can be concluded that deterministic and non-deterministic shape
optimization of SMA structures of realistic complexity is feasible, even
without the use of gradient information. However, in order to reduce
the duration of the optimization process to a practical level, parallel
computing facilities and optimization algorithms that can exploit these
are indispensable.
222
10.1.3
CONCLUSIONS AND FUTURE DIRECTIONS
10.1
Topology optimization
A procedure for topology optimization of SMA structures has been described in
Chapter 6. In fact, this is the first application of topology optimization techniques to SMA structural design. Given the facts that topology optimization
relies on adjoint sensitivity analysis, and that this adjoint sensitivity analysis is
very cumbersome for history-dependent models (Kleiber et al., 1997), the models described in Chapters 2 and 3 turn out to be very well suited for use in SMA
topology optimization. To realize SMA topology optimization, use has been made
of a novel approach to parameterize the design, the so-called Element Connectivity Parameterization (ECP, Yoon and Kim, 2005). Although the ECP approach
used in Chapter 6 does not require the sensitivities of the material model itself,
its history-independence still is essential. The proposed topology optimization
procedure has been demonstrated on problems involving planar SMA structures
under constant loading, and actuated by uniform temperature changes, but the
proposed method applies to more general problems as well.
The ECP approach differs from conventional approaches to topology optimization in the aspect that not the material properties of elements, but the connections
between elements are controlled by the design variables. This approach offers various advantages, such as improved stability of geometrically nonlinear calculations
and absence of arbitrariness in material property interpolation functions. Furthermore, as already noted, sensitivity analysis is straightforward and does not
require differentiation of the underlying material model. Unlike earlier versions
of the element connectivity parameterization concept (e.g. Yoon and Kim, 2004),
the improved ECP approach used for SMA topology optimization does not increase the number of degrees of freedom of the system. Hence the solution time
and complexity remains comparable to that of a standard finite element model of
similar dimensions.
A heuristic algorithm was developed to improve the robustness of the SMA
finite element analyses during topology optimization. The drastic design changes
possible in the topology optimization procedure combined with the strongly nonlinear behavior of the SMA structure require an adaptive incremental-iterative
solution strategy. Except for one case where the topology optimization resulted
in an instable structure, the solution strategy proved to be effective and reliable.
It was found in that particular case, that exploiting instability and buckling phenomena resulted in a large improvement of performance, and this is in fact an
interesting direction for further research.
Nodal design variables were used to counteract the tendency of the optimization problem to generate artificial solutions containing checkerboard patterns or
one-node hinges. In contrast, conventional heuristic filtering approaches to achieve
regularization of the design problem turned out to be ineffective in case of SMA
structures. The reason for this is the fact that sensitivities in the latter case can
have positive or negative values, while in case of compliance minimization the
sensitivities are always positive. The use of node-based instead of element-based
10.1
CONCLUSIONS
223
design variables resulted in a slight increase in the total number of design variables, however the used MMA optimizer was able to deal with this without any
problems.
It was found that mesh density and load intensity strongly affected the topology optimization results. The latter is a result of the nonlinearity of the SMA
material behavior, which results in different optimal designs for different load levels. The fact that increasing mesh densities lead to designs with finer details is
common to topology optimization, and several remedies have been proposed to
impose a minimum lengthscale (Guest et al., 2004, Poulsen, 2003, Zhou et al.,
2001). Furthermore, results were found to depend on the initial design used,
which is an indication that the optimization problem is nonconvex, and has multiple local optima. Nonlinearity of SMA model and the associated nonconvexity
of the SMA design problem are a likely cause for the existence of local minima.
Hence, achieving global optimality remains a fundamental challenge in topology
optimization of nonlinear problems. However, this does not mean that the proposed optimization technique has no practical value. The technique is able to
generate original and effective design concepts with very little input from the designer, apart from specifying the problem. Furthermore, as has been shown by an
example, the proposed method is able to significantly improve the performance
of existing designs.
In conclusion, the combination of a dedicated SMA model, adaptive nonlinear solution techniques, Element Connectivity Parameterization, nodal design variables and the Method of Moving Asymptotes
has enabled topology optimization of SMA structures. Obtaining globally optimal topologies however remains a challenge.
10.1.4
Gradient-based shape optimization
The use of sensitivity information can drastically improve the efficiency of optimization procedures. However, this improvement can only be realized when
the sensitivities can be computed with little additional computational effort. For
this reason, Chapter 7 reviewed the options for sensitivity analysis for the SMA
model introduced in Chapter 3, and described the formulation of restarted finite
difference and semi-analytical sensitivities in detail, for a general electro-thermomechanical setting. The history-independent nature of the constitutive model allowed for efficient sensitivity analysis. The implementation of the semi-analytical
approaches turned out to be considerably more involved than that of the finite
difference alternative, because of the coupling between the electrical, thermal and
mechanical analysis and the implicit nature of the SMA material model. In order to evaluate the required coupling terms, both analytical and finite difference
approaches were studied. No clear difference in performance or accuracy was
observed, however the analytical approach required considerably more implementation effort.
224
CONCLUSIONS AND FUTURE DIRECTIONS
10.1
The same observation was made for the calculation of sensitivities of quantities
derived from the state variables, such as e.g. equivalent strains. After evaluation
of the state vector sensitivity, in the semi-analytical approach a long analytical
calculation is required to evaluate the equivalent strain sensitivity, which in addition needs to be implemented separately for every derived response quantity of
interest. Hence, the rather complex and implicit nature of the material model
makes that the derivation and implementation of sensitivities of these derived
quantities requires a significant effort. In contrast, in the restarted finite difference case, the perturbed solution is available. Therefore, evaluating sensitivities
of derived quantities is straightforward, since the same finite difference approach
can be applied as is used for the state variables. Of course, the accuracy of finite
difference sensitivities also depends on the design perturbation used, but this also
holds for semi-analytical formulations.
For a studied SMA gripper case, it was found that stable sensitivity values were
obtained for relative design perturbations of 10-7 . Furthermore, the more elaborate refined semi-analytical approach showed no clear accuracy improvement, and
the range for which stable sensitivity values were obtained was more or less similar
for all considered methods. For the considered problem, the computational cost of
a complete evaluation including restarted finite difference sensitivity analysis was
only slightly higher (16%) than an evaluation involving semi-analytical sensitivity analysis. When also considering its relatively modest implementation effort, it
was concluded that the restarted finite difference approach is an attractive option
for the present electro-thermo-mechanical SMA model.
Chapters 8 and 9 subsequently demonstrate the use of design sensitivities in
representative SMA design optimization studies, using gradient-based optimization methods. Chapter 8 reports on shape optimization of an SMA miniature
gripper, and convincingly demonstrates the advantage of gradient-based methods
over direct approaches. An approximately twenty-fold reduction in the number
of function evaluations was realized, through the use of gradient-based optimizers (MMA and SQP), in comparison to optimization using the response-surface
based MAM method. Next to this, the consistency and accuracy of the results
obtained using the gradient-based methods clearly was superior. In all fairness,
it should be noted that the quality of the MAM results can possibly be improved
by spending further effort on finding better settings for the optimizer parameters,
for this specific problem. It should also be noted that implementation of sensitivity analysis requires additional effort, and that for practical cases with only
few design variables, this effort may not be justified. Particularly, when sufficient
parallel computing facilities are available, the versatility and ease of use of direct approaches that can benefit from parallellization, such as the MAM method,
could be preferred over the efficiency offered by gradient-based methods.
Chapter 9 finally demonstrated the use of a gradient-based optimization method
on the SMA active catheter design problem, introduced earlier in Chapter 3. This
chapter illustrated, that with the help of efficient design optimization techniques,
different design concepts can be evaluated and compared in a systematic way.
10.1
CONCLUSIONS
225
Considerable improvements in bending performance were realized, in comparison
to the baseline design studied in Chapter 3. The sensitivity analysis was performed using finite differences, in combination with fast reanalysis. By restarting
perturbed analyses from the nominal solution and using modified Newton iterations, the computational cost of the sensitivity analysis was minimized. A new
correction was applied to the multi-point constraints present in the model, which
was shown to improve the accuracy of the sensitivities. The gradient-based SQP
optimizer was used to perform the optimization, and compared to the direct MAM
optimizer. Although the latter was able to complete the design optimization process in nearly the same timespan, given sufficient parallel computing resources,
clearly also in this case gradient-based design optimization proved to be the most
efficient way to optimize designs of SMA structures.
Gradient-based optimization techniques can most efficiently optimize designs of SMA structures, and the required cost-effective sensitivity analysis is enabled by the SMA models proposed in this thesis.
Sensitivity analysis by means of finite differences in combination with
fast reanalysis offers a good compromise between efficiency and implementation effort.
10.1.5
Overall conclusion
The research presented in this thesis clearly illustrates the possibilities that design optimization techniques offer for improving and enhancing design procedures
of SMA structures. Improved design capabilities are essential to further exploit
the potential of SMAs in applications such as instruments for minimally invasive
procedures or micro-actuators. Shape optimization was demonstrated with and
without the use of sensitivity information, and with and without accounting for
uncertainties in loading conditions and/or material properties. Furthermore, in
order to generate innovative shapes and design concepts, a topology optimization
procedure has also been developed for the design of SMA structures. This technique can be applied in the early stages of a design process, where design decisions
have the largest impact. Being able to utilize optimization already at that stage
is certainly expected to have a significant impact on the quality and performance
of the final designs. Substantial performance improvements were demonstrated
in various design problems, obtained using the structured and systematic methods of design optimization. It is therefore expected that designers of future SMA
devices will increasingly make use of the demonstrated numerical modeling and
optimization techniques, in order to improve the effectiveness and reliability of
their designs.
However, it should be mentioned that a key factor in enabling these design optimization procedures has been the history-independent SMA model used in this
study, targeted specifically at the R-phase transformation in NiTi. This makes
that parts of the presented work can not be directly extended to design cases of
226
CONCLUSIONS AND FUTURE DIRECTIONS
10.2
SMA structures involving different material behavior and significant hysteresis.
Particularly, the sensitivity analysis and topology optimization procedures relied
on the history-independence of the problem, and generalizing these to historydependent cases is still a challenge. However, the direct methods shown for deterministic and uncertainty-based shape optimization can directly be applied to
history-dependent problems, and sensitivity analysis for history-dependent problems, although more involved, is certainly possible. Chapter 8 and 9 clearly show
the benefits of exploiting gradient information, for problems within the scope of
this thesis. Hence, possibly also for history-dependent problems, gradient-based
design optimization could be of interest, although the increased cost of the sensitivity analysis will reduce its computational advantages to some extent. The
following section will elaborate further on these and other future directions.
10.2
Future directions
Although this thesis aimed to cover a wide range of aspects related to design
optimization of SMA structures, there are still various topics that could be explored further. Also, many of the techniques considered in this research showed
possibilities for further development. An overview of several interesting directions
for future work is given here, divided between activities aimed at validation and
realization, and opportunities and suggestions to further extend the presented
approaches.
10.2.1
Validation and realization
The following suggestions can be made with regard to further validation or practical realization of aspects of this research:
The material models presented in Chapter 2 and Chapter 3 are based on
one-dimensional experimental data, in combination (in case of Chapter 3)
with knowledge of the underlying physical characteristics of the R-phase
transformation. Further experimental validation and assessment of the accuracy of these models is recommended.
In relation to the previous point, there is a clear opportunity for more
extensive experimental investigation of R-phase-transformation-based material behavior. Little is know of, e.g., the effect of crystallographic texture,
nonproportional loading, complex multi-dimensional loading patterns, complex thermomechanical loading, and the extent and implications of tensioncompression asymmetry. Findings could be used to further refine the material model, as long as this does not unnecessarily complicate the design
optimization.
Strong indications exist that shifting of transformation temperatures of the
material studied in this thesis is possible, which allows its application for in
10.2
FUTURE DIRECTIONS
227
vivo medical applications. However, this has not been experimentally verified. Furthermore, it is likely that changing alloy composition to shift transformation temperatures also affects the shape of the stress-strain curves.
Further experimental studies are required to explore this aspect in more
detail.
The active catheter design presented in Chapter 3, and the optimized designs
found in Chapter 9, could be worked out in more detail and realized in
practice, using e.g. laser cutting or chemical etching to manufacture the
structure from an SMA tube. Once fabricated, its performance could be
evaluated and the possibility to steer and even propel the catheter by means
of crawling motions could be assessed.
Although a careful study has been made in Chapter 3 on what temperature and heat output is tolerable inside arteries, based on information that
currently is available in literature, it is advisable to in addition carry out
a clinical study to collect more experimental data and knowledge on how
living tissues react to thermal loading. Furthermore, for an effective design of an SMA active catheter, it is important to determine key properties
such as the thermal convection coefficient h between the catheter and the
blood more accurately, through in vivo measurements. Tangwongsan et al.
(2004) already demonstrated that this is possible, by performing such measurements in living pigs. Compared to their results, the values used in this
research were adequate, but slightly conservative. More detailed and precise
information for the human case would certainly be useful.
Furthermore, regarding all optimal designs that have been described in the
chapters focusing on shape or topology optimization, the generated designs
could actually be realized and their performance could be evaluated and
compared to the predictions made by the models.
10.2.2
Extensions
The scope of the research presented in this thesis has been clearly defined in
Chapter 1. It has been necessary to narrow down the subject of this work, in
order to achieve a sufficient focus and depth. However, here in this final chapter
there is an opportunity to look beyond the outlined research area, and consider
the possibilities and future extensions that are indicated by the obtained results.
In this research, dynamic effects have been ignored, since in the intended
applications the dynamic performance of devices was not considered to be
a critical aspect. However, there may be other applications where in fact
the dynamic response of devices is very relevant. In that case, the dynamic
and transient behavior of SMA actuators can no longer be ignored, and has
228
CONCLUSIONS AND FUTURE DIRECTIONS
10.2
to be described by appropriate models. When combined with the demonstrated optimization techniques, it will become possible not only to optimize
actuators for stroke or force, but also for e.g. their response or cycle time.
Note, however, that sensitivity analysis for transient problems is more involved than for the case considered in this work, and that proper modeling
of thermal and mechanical dynamics of SMA materials is a challenging task
in itself.
The optimization approaches presented in this thesis could in principle also
be applied to SMA design problems involving more general SMA behavior,
such as the martensite transformation in NiTi, which offers a much larger
actuation strain (up to 7%), but also involves considerable hysteresis. Note
here, that since the hysteresis is no longer negligible, a history-independent
formulation as used for the R-phase transformation is no longer adequate.
Hence, the associated advantages of using a history-independent model with
respect to sensitivity analysis will also vanish. Still, it is expected that also
for this case the use of design optimization techniques offers considerable
benefits.
Regarding the optimization under uncertainty discussed in Chapter 5, the
combination of the Asymptotic approach with the Cycle-based Alternating technique for optimization under Bounded-But-Unknown uncertainties
showed great promise in a small analytical beam example. It is interesting
to further study the performance of this approach on a problem of realistic complexity, such as the presented SMA miniature gripper. Sensitivities
have been derived and implemented, as described in Chapter 7, therefore
basically the key ingredients are available.
Although Chapter 6 already showed quite a number of topology optimization
results, from another perspective the topic has only been touched upon in
this research, and several interesting challenges remain:
– To avoid the mesh-dependency of results, measures to control the minimum member size in the generated structures could be included.
– The topology optimization formulation could be extended to the more
general electro-thermo-mechanical case, to deal with Joule-heated actuators with nonuniform temperature distributions.
– The topology optimization procedure could be extended to shell structures and three-dimensional solid structures. This would allow application of topology optimization to the active catheter design problem
studied in Chapter 3 and Chapter 9, and possibly totally different design concepts may emerge from this.
– In one case, the topology optimization failed because of buckling instabilities, therefore measures to avoid these instabilities are of interest.
10.2
FUTURE DIRECTIONS
–
–
–
–
229
This aspect has already been the subject of a recent study by Kemmler et al. (2005). Furthermore, once the stability problems have been
solved, it could be attractive to try to exploit buckling and postbuckling
effects, as a means to increase the displacement range of the generated
SMA actuators. Also the generation of bistable structures is of interest.
The accuracy of the analysis used in topology optimization could be
improved, by using a different discretization and geometry description
technique. The jagged edges, combined with the fact that SMA actuator topology optimization often tends to form very thin, detailed and
curved structures makes that the accuracy of the analysis is suboptimal. An improvement could be found in the use of recently proposed
level-set techniques (see e.g. De Ruiter, 2005, Kwak and Cho, 2005, Liu
et al., 2005, Wang and Wang, 2006, Wang et al., 2004), which allow
the separation of geometrical description and the analysis mesh, and
which are able to generate structures with smooth boundaries.
Furthermore, to ensure that designs remain within the range of applicability of the constitutive model, strain constraints could be included
in the topology optimization problem formulation. However, such constraints require special measures to handle the optimization, as they
are known to lead to hard-to-find singular optima and spoil the convergence characteristics of topology optimization problems (Duysinx and
Bendsoe, 1998, Guo et al., 2001, Stolpe and Svanberg, 2003). Possibly, the constraint aggregation approach used for the microgripper case
(Chapter 5, Chapter 8) could be a viable solution, particularly when
combined with recently proposed adaptive approaches for the penalty
parameter (see Martins and Poon, 2005).
Note, however, that on the other hand, the designs resulting from
topology optimization appeared to avoid excessive strains. It seems
that it is more advantageous to use the material in a moderate strain
range, in order to improve the performance of the designs. Therefore,
the question whether there is a real need to include strain constraints
in topology optimization of SMA actuators should also be addressed
critically.
Nonconvexity of topology optimization problems involving nonlinear
models leads to possible convergence to local minima. Measures to
reduce this nonconvexity, or to somehow escape local optima could be
studied, in order to decrease the chance that optimization processes
converge prematurely to inferior designs. Possibly also the interpolation function used provides a sort of design-dependent scaling of the
problem, and this may affect its convexity properties.
Considering the mentioned extension of this work to more general
history-dependent SMA models, a fundamental challenge is to apply
topology optimization to problems involving such models. Topology
230
CONCLUSIONS AND FUTURE DIRECTIONS
10.2
optimization, with usually thousands of design variables, relies heavily
on adjoint sensitivity analysis. However, for history-dependent problems, such sensitivity analysis is much more involved than for the
history-independent case considered in this thesis. Therefore, application of topology optimization to history-dependent problems still
presents a challenge.
Regarding the sensitivity analysis presented in Chapter 7, it could be considered to also formulate and implement an adjoint sensitivity analysis procedure for the electro-thermo-mechanical case. This would be of interest for
problems where the number of design variables is significantly larger than
the number of responses.
Slightly outside the scope of the present thesis, the recommendation could
be made to include more adaptive algorithms in the Multi-point Approximation Method used in Chapter 4, 5, 7 and 8, or to include a procedure
that is able to determine important properties of a problem prior to the
actual optimization process, which would subsequently be able to adjust
the optimizer settings accordingly. This would relieve the user from the
task of finding proper settings for the many parameters of this optimizer,
and it would also allow the algorithm to change its behavior during the
optimization process, and consequently to converge faster.
Finally, based on the findings and the analysis results of the optimized active
catheter designs given in Chapter 9, it is of interest to create and optimize
modified design concepts. It is expected that this will lead to further performance improvements. The modeling might also be further refined, for
example by using solid instead of shell elements, for designs with relatively
large tube wall thicknesses. Note, however, that for a more realistic active catheter design study, the formulation of the design problem should
be based on the requirements of the actual clinical procedure for which the
instrument is designed. Next to its dimensions and active bending performance, also aspects such as its bending stiffness and buckling resistance
might be of interest. However, all these design aspects can be included in
the formulation of the design optimization problem.
The length of these lists of suggestions for future work indicates that still
enough challenges remain for further research on this topic. Hopefully, the design
optimization results from this thesis provide inspiration to confront these challenges, and to make further contributions to improve the procedures for designing
SMA structures.
Appendix
A
Derivation of 3-D Tangent Operator
The tangent operator dσ/dε for the three-dimensional constitutive model can be
found by differentiating the stress-strain relation with respect to the strain. This
three-dimensional constitutive relation is given by:
σ = (KK + 2GG)(ε − κGε) = (KK + 2ĜG)ε.
(A.1)
Using G instead of Ĝ to simplify the notation, it follows that:
dσ
dG dεe
= KK + 2GG + 2Gε
.
dε
dεe dε
(A.2)
The derivative of the effective strain εe to the strain itself is found from its definition:
r
2 T
2 T
dεe
2 1 T
2
εe = ε Gε ⇒ εe =
ε Gε ⇒
=
ε G.
(A.3)
3
3
dε
3 εe
This leads to the final expression of the tangent operator in the three-dimensional
case:
dσ
4 dG
= KK + 2GG +
GεεT G.
(A.4)
dε
3εe dεe
The term dG/dεe is easily found from the expression for G:
3K
B
dG
3KB 1
G=
A+
⇒
=−
.
(A.5)
9K − A
εe
dεe
9K − A ε2e
The tangent operator is symmetric, since K, G and also the product Gε(Gε)T
are symmetric.
231
232
APPENDIX A
Appendix
B
Shear Modulus/Effective Strain
Relation in Plane Stress Case
The starting point for defining the equations to solve G and εe numerically is
given by:
the relation between G and εe obtained from the one-dimensional behavior:
3K
Bi (T )
Di (T )
G=
Ai (T ) +
= Ci (T ) +
,
(B.1)
9K − Ai (T )
εe
εe
where Ci (T ) and Di (T ) are introduced for convenience, and the index i =
1, 2, 3 refers to the corresponding part of the piecewise linear approximation
to the one-dimensional stress-train curve used in the material model,
the expression for the effective strain in the plane stress case:
r
4 2
4
4
εe =
(α − α + 1)(ε2xx + ε2yy ) + (2α2 − 2α − 1)εxx εyy + ε2xy , (B.2)
9
9
3
and the definition of α:
α=
2G − 3K
.
4G + 3K
(B.3)
In order to solve this set of equations, Equation B.1 is rewritten to define a
function Z:
Di (T )
Z = Ci (T ) +
− G = 0.
(B.4)
εe
233
234
APPENDIX B
The effective strain is eliminated by substituting Equation B.2 in Equation B.4,
resulting in a nonlinear equation Z(G) = 0. To solve this equation with Newton
iterations, the derivative of Z with respect to G is required. Differentiation gives:
dZ
4(G − Ci (T ))K(εxx + εyy )2 (2α − 1)
− 1.
=−
dG
(4G + 3K)2 ε2e
(B.5)
Note, α is to be replaced by Equation B.3, but is maintained in this expression
for readability. Convenient starting points for the iterations are the transition
points of the piecewise linear model. These transition points differ from the
transition strains ε1 , ε2 used in the one-dimensional formulation, since in general
εxx 6= εe . Using the relations between εxx and εe derived before, it is found that
the transition points in terms of the effective strain εe are given by:
εe(1) (T )
εe(2) (T )
9K − EA (T )
ε1 (T ),
9K
9K − ET (T )
∆(T ).
= εe(1) (T ) +
9K
=
(B.6)
(B.7)
These transition points subdivide the effective strain range in three intervals. In
the austenite interval, i.e. εe ≤ εe(1) , no iterations are required since D1 = 0
and G = C1 . Here only εe needs to be evaluated (Equation B.2), and if indeed
εe ≤ εe(1) the procedure is finished. In the second and third interval, iterations
are necessary, and likewise the subsequent solution is checked against the relevant
transition strains.
Appendix
C
Plane Stress Constitutive Tangent
Operator
Two approaches are described in this appendix for the derivation of the constitutive tangent operator in the plane stress case. The first is based on reduction of
the 3-D tangent operator to the plane stress case, the second on differentiation of
the stress-strain relation in the plane stress setting. The results are identical.
C.1
Method 1: based on the 3-D tangent operator
The tangent operator in the three-dimensional case, which is taken as a starting
point for this derivation of the tangent operator in plane stress, is given by:
dG dεe
4 dG
dσ
= KK + 2GG + 2Gε
= KK + 2GG +
GεεT G.
dε
dεe dε
3εe dεe
(C.1)
Combining this with the the relation between G and εe , obtained from the onedimensional behavior and given by
3K
B
D
G=
A+
=C+ ,
(C.2)
9K − A
εe
εe
yields
4D
4(G − C)3
dσ
= KK + 2GG − 3 GεεT G = KK + 2GG −
GεεT G = T. (C.3)
dε
3εe
3D2
235
236
C.1
APPENDIX C
In Equation C.2, C and D are introduced for convenience. These are constants
in this setting, which depend on the temperature. In the case of vector notation,
as used here, the tangent operator takes the form of a square matrix. In this
appendix, this tangent matrix will be denoted by T, and its components by T[i,j] .
These expressions were derived for stress and strain vectors given by
√
√
√
σ = (σxx , σyy , σzz , 2σxy , 2σyz , 2σzx )T
and
√
√
√
ε = (εxx , εyy , εzz , 2εxy , 2εyz , 2εzx )T .
(C.4)
(C.5)
√
In the plane stress case,
vectors can be used: σ = (σxx , σyy , 0, 2σxy )T
√ reduced
T
and ε = (εxx , εyy , εzz , 2εxy ) . However, the vectors commonly used in finite element implementations for plane stress and shell elements are σ = (σxx , σyy , σxy )T
and ε = (εxx , εyy , γxy )T . This means it is necessary to eliminate the row and column associated with εzz from the matrix, and to apply other transformations to
convert the expressions based on the original stress-strain vectors to the situation
using the new vectors. First, starting from the reduced version of Equation C.1,
the transverse strain component εzz is eliminated, which results in the expressions
given in Equation C.6 till Equation C.9. Next, the expression in Equation C.9
is transformed to a new set of stress and strain vectors as used in finite element
formulations, which yields the final symmetric plane stress tangent operator given
in Equation C.11.
It can be seen from this last expression that when the original tangent operator
matrix T, that was used as a starting point, was symmetric, this reduced tangent
operator for the plane stress case will also be symmetric, when using the specified
stress and strain vectors are used. For the present material model, this is the
case, since the tangent operator given by Equation C.1 indeed is symmetric.
[4,3] T[3,3]
[4,2]

T


T[1,1] − T[1,3] T[3,1]
[3,3]
σxx 

T

σyy
=  T[2,1] − T[2,3] T[3,1]
[3,3]


 σxy
T[3,1]
√1
T
−
T
[4,1]
[4,3]
T[3,3]
2


σxx 
σyy
=


σxy

T
T[1,1] − T[1,3] T[3,1]
[3,3]

 T[2,1] − T[2,3] TT[3,1]
[3,3]
 T[3,1]
√1
T
−
T
[4,1]
[4,3] T[3,3]
2
[4,1]
[4,4]
[3,2]
T[2,2] − T[2,3] T[3,3]
T[3,2]
√1
T
−
T
[4,2]
[4,3]
T[3,3]
2
T
T
[3,2]
T[1,2] − T[1,3] T[3,3]




T
 
√1
T
− T[1,3] T[3,4]
2 [1,4]
[3,3]  ε
xx 
T[3,4] 
√1
ε
T
−
T
.

yy
[2,3] T[3,3]
2 [2,4]
 γ 
T
xy
[3,4]
1
2 T[4,4] − T[4,3] T[3,3]
  εxx 
T

T[2,4] − T[2,3] T[3,4]
ε
[3,3]
 √ yy 
T[3,4]
2εxy
1
√
T[4,4] − T[4,3] T[3,3]
2
T
T[1,4] − T[1,3] T[3,4]
[3,3]
[4,3] T[3,3]
T[3,2]
T[1,2] − T[1,3] T[3,3]
T[3,2]
T[2,2] − T[2,3] T[3,3]
T[3,2]
√1
T
−
T
[4,2]
[4,3] T[3,3]
2
[4,3] T[3,3]





σxx 
εxx 
T[1,1] T[1,2] T[1,3] T[1,4] 







T[2,1] T[2,2] T[2,3] T[2,4]   εyy 
σyy


= 
⇒
0 
εzz 
T[3,1] T[3,2] T[3,3] T[3,4]  





√


√
T[4,1] T[4,2] T[4,3] T[4,4]
2σxy
2εxy





T
T
T
T
[1,1]
[1,2]
[1,3]
[1,4]
σxx 
εxx 








 T[2,1]
T[2,2] T[2,3] T[2,4]   εyy 
σyy


=  T[3,1]
T
T[3,4]
  εzz  ⇒
0 
−1 − T[3,3]
− T[3,3] − T[3,2]





[3,3]
√

√

2σxy
2εxy
T[4,1]
T[4,2] T[4,3] T[4,4]


T[3,1]
T
T




T[1,2] − T[1,3] T[3,2]
0 T[1,4] − T[1,3] T[3,4]
T[1,1] − T[1,3] T[3,3]
εxx 
σ
[3,3]
[3,3]



xx 





T


  εyy 
T[2,1] − T[2,3] T[3,1] T[2,2] − T[2,3] T[3,2]
σyy
0 T[2,4] − T[2,3] T[3,4]
T
T


[3,3]
[3,3]
[3,3]
= 
  εzz  ⇒
0 

0
0
−1
0





√
√


T
T[3,2]
T[3,4]
2σxy
2εxy
T
−
T
0
T
−
T
T[4,1] − T[4,3] T[3,1]
[4,2]
[4,3] T[3,3]
[4,4]
[4,3] T[3,3]
[3,3]


T[3,1]
T
T[3,4] 



T[1,1] − T[1,3] T[3,3]
T[1,2] − T[1,3] T[3,2]
T
−
T
[1,4]
[1,3]
T
σ
[3,3]
[3,3]
 xx 
 εxx 


T[3,1]
T[3,2]
T[3,4] 
σyy
= 
T[2,1] − T[2,3] T[3,3] T[2,2] − T[2,3] T[3,3] T[2,4] − T[2,3] T[3,3]  √εyy 
√

T[3,1]
T[3,2]
T[3,4]
2σxy
2εxy
T
−T
T
−T
T
−T
⇒
(C.11)
(C.10)
(C.9)
(C.8)
(C.7)
(C.6)
C.1
METHOD 1: BASED ON THE 3-D TANGENT OPERATOR
237
238
APPENDIX C
C.2
Method 2: based on the plane stress equations
C.2
Starting point for the derivation of the tangent operator are the following equations:
The stress-strain relation in the plane stress setting:

  

1+α 1+α 0
 σxx 
σyy
= K 1 + α 1 + α 0 + . . .
√

0
0
0
2σxy

 

εxx 
2−α
−(1 + α) 0

2G 
−(1 + α)
2−α
0 √εyy
,


3
0
0
3
2εxy
(C.12)
where α = (2G − 3K)/(4G + 3K).
The definition of the effective strain:
ε2e =
4 2
4
4
(α − α + 1)(ε2xx + ε2yy ) + (2α2 − 2α − 1)εxx εyy + ε2xy . (C.13)
9
9
3
The relation between G and εe obtained from the one-dimensional behavior:
3K
B
D
G=
A+
=C+ .
(C.14)
9K − A
εe
εe
where C and D are introduced for convenience. These are constants in this
setting, which depend on the temperature.
The tangent operator basically is the derivative of the stress with respect to the
strain. However, the stress as given in Equation C.12 is not only a direct function
of the strain, but also of α (which depends on G) and G, which in turn depends
on εe , where εe is a function of the strain components (Equation C.13). So, in
vector index notation, the tangent operator is basically given by:
∂σj
∂σj dα
∂σj ∂G
dσj
=
+
+
.
(C.15)
dεi
∂εi
∂α dG
∂G ∂εi
The terms ∂G/∂εi are not obvious to obtain, as an explicit relation is not present.
This difficulty can be resolved as follows: the shear modulus G is linked to the
strains through the effective strain, in the equation that relates G to the onedimensional stress-strain data (Equation C.14), given by the general expression
G = C + D/εe . This function is implicit, because also the expression for εe
involves G, as is clear from Equation C.13. To determine the terms ∂G/∂εi , a
function Z is introduced defined as Z = ε2e (G−C)2 −D2 = 0. Using this function,
C.2
METHOD 2: BASED ON THE PLANE STRESS EQUATIONS
239
by means of implicit differentiation the ∂G/∂εi terms can be determined. The
basic idea is:
∂Z
∂εe
∂εe ∂G
∂G
(G − C)2 + 2ε2e (G − C)
= 0 = 2εe
+
⇒ (C.16)
∂εi
∂εi
∂G ∂εi
∂εi
2
e
−2εe ∂ε
∂G
∂εi (G − C)
=
=
e
2
2
∂εi
2εe ∂ε
∂G (G − C) + 2εe (G − C)
e
− ∂ε
∂εi (G − C)
∂εe
∂G (G
− C) + εe
.
(C.17)
The derivatives needed to evaluate this expression are easily found:
∂εe
1 ∂(ε2e )
2
2(α2 − α + 1)εxx + (2α2 − 2α − 1)εyy , (C.18)
=
=
∂εxx
2εe ∂εxx
9εe
2
1 ∂(εe )
2
∂εe
=
=
2(α2 − α + 1)εyy + (2α2 − 2α − 1)εxx , (C.19)
∂εyy
2εe ∂εyy
9εe
∂εe
1 ∂(ε2e )
4
2
=
=
εxy =
6εxy ,
∂εxy
2εe ∂εxy
3εe
9εe
(C.20)
and
1 ∂(ε2e )
1 ∂(ε2e ) ∂α
2
∂εe
∂α
=
=
=
(2α − 1)(εxx + εyy )2
∂G
2εe ∂G
2εe ∂α ∂G
9εe
∂G
2
18K
=
.
(2α − 1)(εxx + εyy )2
9εe
(4G + 3K)2
(C.21)
(C.22)
This leads to the following dG/dεi derivatives:
(G − C) 2(α2 − α + 1)εxx + (2α2 − 2α − 1)εyy
∂G
=−
,
18K
9 2
∂εxx
(2α − 1)(εxx + εyy )2 (4G+3K)
2 (G − C) + 2 εe
(G − C) 2(α2 − α + 1)εyy + (2α2 − 2α − 1)εxx
∂G
=−
,
9 2
18K
∂εyy
(2α − 1)(εxx + εyy )2 (4G+3K)
2 (G − C) + 2 εe
(G − C)
∂G
=−
6εxy .
9 2
18K
∂εxy
(2α − 1)(εxx + εyy )2 (4G+3K)
2 (G − C) + 2 εe
The tangent operator for the plane stress


1+α 1+α 0
dσ
=K 1 + α 1 + α 0 + . . .
dε
0
0
0

2−α
−(1 + α)
2G 
−(1 + α)
2−α
3
0
0
(C.23)
(C.24)
(C.25)
case is now given by:
  ∂G T
 
0
 Qxx  
 ∂εxx 

∂G

0 + Qyy · ∂εyy
,

 
 ∂G 

3
2εxy
∂ε
xy
(C.26)
240
C.2
APPENDIX C
where
Q1
=
=
=
Q2
=
=
and
(
dα
K dG
(εxx + εyy ) + 23 (2 − α)εxx − . . .
2
2
dα
3 (1 + α)εyy − 3 G(εxx + εyy ) dG
2
dα
2
2
K − G (εxx + εyy )
+ (2 − α)εxx − (1 + α)εyy
3
dG 3
3
dα
2
2
K− G
− (1 + α) (εxx + εyy ) + 2εxx ,
3
dG 3
(
dα
K dG
(εxx + εyy ) + 23 (2 − α)εyy − . . .
2
2
dα
3 (1 + α)εxx − 3 G(εxx + εyy ) dG
dα
2
2
K− G
− (1 + α) (εxx + εyy ) + 2εyy ,
3
dG 3
dα
18K
=
.
dG
(4G + 3K)2
(C.27)
(C.28)
(C.29)
(C.30)
(C.31)
(C.32)
The expression in Equation C.26 is the correct equation for the tangent operator based on the stress-strain vectors (σxx , σyy , σxy ) and (εxx , εyy , εxy ). It is
generally nonsymmetric. To obtain the symmetric tangent operator related to the
conventional stress-strain vectors (σxx , σyy , σxy ) and (εxx , εyy , γxy ), which mostly
used in finite element implementations, a modification is required: the third column needs to be divided by two. This finally leads to the following expression:


1+α 1+α 0
dσ
=K 1 + α 1 + α 0 + . . .
dε (γ)
0
0
0
  ∂G T (C.33)

 
2−α
−(1 + α)
0
Qxx  

 ∂εxx 

2G 
∂G
−(1 + α)
2−α
0  + Qyy ·
.
∂εyy

 
3

 1 ∂G 
0
0
3/2
2εxy
2 ∂εxy
It turns out that this tangent operator is, again, symmetric, and equal to Equation C.11.
Appendix
D
Jacobian and Hessian of Symmetry
Plane Constraint
To solve a finite element problem containing multi-point constraints using an
Lagrangian or augmented Lagrangian approach, the Jacobian and Hessian of the
constraints are required. This appendix presents the derivation of these quantities
for the following symmetry plane constraint h, introduced in Chapter 3:
h=
M 2z Sx − M 1z Sx + M 1z M 2x + M 1x Sz − M 2x Sz − M 1x M 2z
p
= 0. (D.1)
(M 2x − M 1x )2 + (M 2z − M 1z )2
Here M 1, M 2 and S represent the location of two master nodes and a slave
node, respectively. The constraint equation restricts the slave node to a plane
parallel to the y-axis, defined by the two master nodes. In order to clarify the
notation, the original coordinates of the nodes are collected in a single vector X,
and the associated displacements are collected in the displacement vector U :
X = (M 1x , M 1z , M 2x , M 2z , Sx , Sz )T ,
U = (M 1u , M 1w , M 2u , M 2w , Su , Sw )T ,
x = X + U = (a, b, c, d, e, f )T .
(D.2)
(D.3)
(D.4)
Subscript u and w denote a displacement component in x- and z-direction, respectively. The components of the actual nodal location vector x are represented
by simple symbols to clarify the notation. For example, a represents M 1x + M 1u .
The constraint equation in this notation reads
h=
R
e(d − b) + f (a − c) + bc − ad
p
=√ ,
Q
(c − a)2 + (d − b)2
241
(D.5)
242
APPENDIX D
where Q and R are introduced to simplify the following expressions. It can be
seen that the constraint is a nonlinear function of the nodal displacements. Differentiating the constraint h to the displacement vector U yields the Jacobian
dh/dU :
dR
Q − 1 R dQ
d
R
dh
√
=
= dU √2 dU .
(D.6)
dU
dU
Q
Q Q
Also, the Hessian d2 h/dU 2 can be expressed as
d
d2 h
=
2
dU
dU
dh
dU
=
d2 R 2
dU 2 Q
−
1 dR dQ T
2 dU dU Q
√
Q2 Q
−
1 dQ dR T
2 dU dU Q
d2 Q
3 dQ dQ T
− 21 R dU
2Q
4 R dU dU
√
2
Q Q
+ ...
.
(D.7)
In these equations, the derivatives of R and Q are given by:
dR
= (f − d, c − e, b − f, e − a, d − b, a − c)T ,
dU
dQ
= 2(a − c, b − d, c − a, d − b, 0, 0)T ,
dU


0 0 0 0 0 0
0 0 0 0 0 0


0 0 0 0 0 0
d2 R

,
=

dU 2
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0


2
0 −2 0 0 0
0
2
0 −2 0 0


−2 0
d2 Q
2
0 0 0


=
2 0 0
dU 2
 0 −2 0

0
0
0
0 0 0
0
0
0
0 0 0
(D.8)
(D.9)
(D.10)
(D.11)
The Jacobian and Hessian are used in the implementation of this constraint condition.
Appendix
E
Derivatives of Material Parameters
Ci and Di
As also discussed in Chapter 6, the piecewise linear stress-strain equations describing the one-dimensional SMA constitutive behavior can be cast in the following
generalized form:
σxx = Ai (T )εxx + Bi (T ).
(E.1)
The terms Ai (T ) and Bi (T ) follow from the original formulation, and the indices
refer to the three segments of the piecewise linear relation:

: A1 = EA ,
 εxx ≤ ε1
ε1 < εxx ≤ ε2 : A2 = ET ,
(E.2)
Ai (T ) =

εxx > ε2
: A3 = ER .

: B1 = 0,
 εxx ≤ ε1
ε1 < εxx ≤ ε2 : B2 = (EA − ET )ε1 ,
Bi (T ) =
(E.3)

εxx > ε2
: B3 = (EA − ET )ε1 + (ET − ER )ε2 .
The generalization to a three-dimensional setting yields the following relation
between the effective strain εe and the shear modulus G:
3K
Bi (T )
Di (T )
Ai (T ) +
G=
= Ci (T ) +
.
(E.4)
9K − Ai (T )
εe
εe
The quantities Ci (T ) and Di (T ) are defined here for convenience. Their derivatives are used in the computation of ∂G/∂T , and are given by:
i
27K 2 dA
dCi
dT
=
,
dT
(9K − Ai )2
dAi
i
3K(9K − Ai ) dB
dDi
dT + 3KBi dT
=
,
dT
(9K − Ai )2
243
(E.5)
244
APPENDIX E
where
dAi
dT
dBi
dT

 εe ≤ εe(1)
=
ε
< εe ≤ εe(2)
 e(1)
εe > εe(2)

 εe ≤ εe(1)
=
ε
< εe ≤ εe(2)
 e(1)
εe > εe(2)
:
:
:
:
:
:
dA1
dT
dA2
dT
dA3
dT
dB1
dT
dB2
dT
dB3
dT
= 0,
= KE ,
= 0.
(E.6)
= 0,
= −KE ε1 + (EA − ET )Kε , (E.7)
= KE ∆ + (EA − ER )Kε .
Appendix
F
Maximum Effective Strain Values
Occur at Outer Layers
In shell elements, the effective strain used in the SMA material model described in
Chapter 3 can vary throughout the thickness of the element. Since the maximum
effective strain is relevant to optimization problems, it is of interest to investigate
whether the maximum always occurs at the outer layers of the shell. This would
reduce the number of quantities that need to be considered. In this appendix, it
is shown that this is indeed the case.
The strain components in every layer ε̃ij are a function of the transverse
coordinate z (with z = 0 at the midplane) and the midplane membrane strains
εij and curvatures κij :
 
 


ε̃xx 
εxx 
κxx 
ε̃yy
= εyy
+ z κyy
.
(F.1)
 
 


ε̃xy Layer
εxy Midplane
κxy Midplane
Note that the distinction between layer and midplane components is only made
in this appendix. Since the strain components can vary from layer to layer, the
effective strain εe can vary as well. It is defined in terms of layer strain components
as:
r
4 2
4
4
(α − α + 1)(ε̃2xx + ε̃2yy ) + (2α2 − 2α − 1)ε̃xx ε̃yy + ε̃2xy ,
(F.2)
εe =
9
9
3
where
α=
2G − 3K
.
4G + 3K
245
(F.3)
246
APPENDIX F
By substitution of Equation F.1, εe can be expressed in midplane quantities and
z. The square root complicates the further derivations, hence the square of the
effective strain will be considered:
4 2
(α − α + 1)((εxx + zκxx )2 + (εyy + zκyy )2 ) + . . .
9
4
4
(2α2 − 2α − 1)(εxx + zκxx )(εyy + zκyy ) + (εxy + zκxy )2 .
9
3
ε2e =g(z) =
(F.4)
Note that since the effective strain is nonnegative, the square operator preserves
the monotonicity. In the remainder of this appendix, ε2e is therefore considered
instead of εe itself, and is referred to as g(z), which is a quadratic function of z.
To proof that this nonnegative function reaches its extreme values for the outer
layers, it has to be verified that the second order derivative is nonnegative for
all possible strains and curvatures, such that the quadratic function g(z) cannot
have an internal maximum in an interval. This second order derivative is given
by:
d2 g
8
8
8
= (α2 − α + 1)(κ2xx + κ2yy ) + (2α2 − 2α − 1)κxx κyy + κ2xy ,
dz 2
9
9
3
(F.5)
which simplifies to
d2 g
8
8
8
= (α2 − α + 1)(κxx + κyy )2 + κ2xy − κxx κyy .
dz 2
9
3
3
Subsequent substitution of the definition of α gives:
8
d2 g
2
2
=
Ψ(κ
+
κ
)
+
κ
−
κ
κ
xx
yy
xx yy
xy
dz 2
3 |
{z
} | {z }
≥0
with
Ψ=
(F.6)
(F.7)
≤0
4G2 + 6GK + 9K 2
> 0.
(4G + 3K)2
(F.8)
In order to determine the sign of d2 g/dz 2 , the magnitudes of the terms larger
than and smaller than zero have to be compared. The term Ψ involving G and
K plays a crucial role in this comparison. This Ψ turns out decrease when the
ratio G/K increases, as shown in Figure F.1. Because the shear modulus has its
largest value before the onset of the phase transformation, the critical situation
to consider is the austenite state. In that state, the moduli G and K are related
to the Young’s modulus and Poisson ratio in the following way:
G=
E
,
2(1 + ν)
K=
E
G
3(1 − 2ν)
⇒
=
.
(3(1 − 2ν)
K
2(1 + ν)
(F.9)
Combining this relation with the definition of Ψ given in Equation F.8, Ψ can
be expressed as a function of the Poisson ratio ν. As shown in Figure F.1, Ψ
increases with an increase in ν.
247
MAXIMUM EFFECTIVE STRAIN VALUES OCCUR AT OUTER LAYERS
1
1
Ψ
Ψ
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
ν
G/K
0
0
0.5
1
1.5
0
0
0.1
0.2
0.3
0.4
0.5
Figure F.1: The term Ψ as defined in Equation F.8, as a function of G/K (left) and
of ν (right).
As an (unrealistic) worst case, the case ν = 0 is considered, which gives:
8 d2 g
3 dz 2
=
=
1
(κxx + κyy )2 + κ2xy − κxx κyy
3
1 2
κxx − κxx κyy + κ2yy + κ2xy .
3
(F.10)
(F.11)
Since
κ2xx − κxx κyy + κ2yy ≥ (κ2xx − 2κxx κyy + κ2yy ) = (κxx + κyy ) ≥ 0,
(F.12)
it follows that even in this worst case, d2 g/dz 2 cannot become negative. Therefore
it can be concluded that the maximum value of the effective strain εe must be
located at the outer layer of the shell.
248
APPENDIX F
Appendix
G
Full Color Illustrations
Figure G.1: (Figure 3.7) Results of the finite element analysis. a) Temperature distribution, b) Deformed geometry and effective strain, c) Composition of catheter in bent
configuration.
249
250
APPENDIX G
Figure G.2: (Figure 3.2) Proposed fabrication of the new active catheter concept using
laser cutting from a small diameter SMA tube (a). After laser cutting (b–c), the structure
is stretched (d) and spacers are applied (e), in order to generate internal stresses.
Figure G.3: (Figure 3.5) Segmentation, applied voltage pattern and coordinate system.
FULL COLOR ILLUSTRATIONS
(a) Analysis results for the opened
configuration.
251
(b) Analysis results for the closed
configuration.
Figure G.4: (Figure 4.7) Computed electrical potential (top), temperature distribution
(middle) and Von Mises stress distribution on the deformed structure (bottom) for the
optimal design in the constrained Joule heated case, in opened (left) and closed (right)
gripper configurations.
Figure G.5: (Figure 4.8) Gripper top arm in open (black) and closed (red) position,
side view.
252
APPENDIX G
(a) Maximum absolute differences in Von Mises
stress.
(b) Maximum differences in Von Mises stress relative to the nominal values.
Figure G.6: (Figure 5.18) Maximum absolute (left) and relative (right) differences
in Von Mises stress at different strain states due to the effect of the uncertainties, at
different nominal temperatures.
FULL COLOR ILLUSTRATIONS
253
Figure G.7: (Figure 8.7) Temperature distribution [K] on the optimal miniature SMA
gripper in the open and closed configuration.
254
APPENDIX G
Figure G.8: Temperature distributions on parts of the deformed catheter structure for
the best designs of all three concepts, for a 1.0 mm diameter.
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Summary
Design Optimization of Shape Memory Alloy Structures
This thesis presents various methods to apply design optimization principles to
the design of shape memory alloy (SMA) structures. SMAs are materials that
exhibit a solid state phase transformation, influenced by changes in temperature
and stress, which can be used for actuation. In comparison to other active materials, SMAs can generate relatively large strains and stresses, and are therefore
of interest for a variety of (micro-)actuation applications. Designing an effective
SMA structure, however, presents a challenge, because of the complexity of the
SMA material behavior and the fact that often the electrical, thermal and mechanical properties of a structure have to be considered simultaneously. For this
reason, this research aims to explore how and to what extent design optimization
techniques can contribute to the solution of SMA design problems. Little previous
work has been reported on this topic.
The approach taken to investigate the applicability and effectiveness of various
design optimization approaches for SMA structures is to apply these to representative practical cases. In this thesis, the focus is on applications emerging from
the medical field, where the popularity of minimally invasive therapy leads to a demand for miniaturized instruments with enhanced functionality. Because of their
mentioned properties, SMAs are promising candidates for use as active materials
in these instruments. The shape memory effect due to the so-called austenite/Rphase transformation in nickel-rich NiTi alloys is considered in particular, because
it is characterized by a small hysteresis and excellent cyclic stability, making it
well suited for the targeted applications. Specifically, SMA miniature grippers
and steerable catheters are used in the case studies.
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276
SUMMARY
In order to develop a generic and versatile design methodology, finite element modeling is used to evaluate the functionality of SMA structures. In this
thesis, a nonlinear quasi-static finite element model involving a sequentially coupled electrical, thermal and mechanical analysis is applied. First, a novel threedimensional model is developed to describe the SMA material behavior due to the
austenite/R-phase transformation. An empirical approach is used, combined with
the fundamental properties of the underlying phase transformation, to capture the
essence of the material behavior. The limited thermal operating range required
for the applications allows substantial simplifications, which result in a historyindependent model. This history-independent nature reduces the computational
cost and complexity of design sensitivity analysis in comparison to conventional
SMA models, which allows the use of efficient gradient-based optimization procedures. The model is illustrated by the analysis of a novel concept for a steerable
catheter, equipped with integrated SMA actuation sections.
Subsequently, shape optimization studies of a miniature SMA gripper are presented. A direct optimization approach is first explored, which does not require
the computation of design sensitivities. The stroke of gripper jaws is maximized,
subject to constraints on the operating temperature and the maximum strain,
in order to remain within the range of validity of the constitutive model. The
Multi-point Approximation Method is used to perform the optimization, which
relies on response surface fitting in combination with a move limit strategy. In
combination with parallel computing to reduce the duration of the procedure,
this method is demonstrated to be an effective way to generate optimized gripper
designs.
The same miniature gripper is also used in an optimization study in which
the influence of uncertainties is considered as well. Since usually in the initial
design stage reliable probabilistic data to describe the nature of uncertainties is
not available, the uncertainties considered in this study are characterized only
by the bounds of the domain in which the uncertain parameters may vary. The
design optimization problem is formulated such, that the optimization aims to
maximize the expected performance, while the worst combination of uncertainties
still yields a feasible design. By means of an adapted anti-optimization approach,
which avoids the use of an expensive nested optimization procedure by alternating
between main- and anti-optimization cycles, and in combination with parallel
computing, also this non-deterministic SMA shape optimization procedure was
performed successfully against strongly reduced computational cost. Again the
Multi-point Approximation Method was used as the optimizer.
In order to efficiently handle optimization problems involving more design
variables and expensive evaluations, gradient-based methods are preferred over
direct approaches. Therefore, various options for sensitivity analysis of the SMA
finite element models used in this thesis are investigated. A comparison is made
between full finite differences, finite differences in combination with fast reanalysis, semi-analytical and refined semi-analytical direct differentiation methods. The
implicit nature of the SMA material model and the coupling between the electrical,
SUMMARY
277
thermal and mechanical problems require special attention in the semi-analytical
cases. The study reveals that all methods yield sensitivities of comparable accuracy, and that the semi-analytical approaches have only a slight advantage in
terms of computational effort over finite differences with fast reanalysis, while the
implementation of the latter is considerably less involved. The relative efficiency
of the sensitivity analysis for the considered class of SMA models is to a large
extent due to the history-independent nature of the used SMA material model.
The availability of design sensitivities enables shape optimization of SMA
structures using gradient-based algorithms. The effectiveness of this approach
is demonstrated using optimization case studies involving the SMA miniature
gripper, as well as the SMA steerable catheter model introduced earlier. The latter contains nonlinear multi-point constraints that are used to enforce symmetry
and periodicity conditions, which are handled in the model using an augmented
Lagrangian approach. For this situation, finite differences in combination with
fast reanalysis is selected as the preferred method for sensitivity analysis. A
correction is introduced to account for the finite error in the constraint satisfaction in the nominal case. In combination with stricter convergence settings for
the perturbed cases, this correction is shown to clearly improve the accuracy of
the results. Modified Newton iterations are used in the reanalysis, which further
reduces the required computational effort. The algorithms considered are Sequential Quadratic Programming and the Method of Moving Moving Asymptotes. All
gradient-based optimization procedures turn out to require far less function evaluations in comparison to the previously used direct method, and the optimum
design is generally located more precisely.
Finally, also topology optimization of SMA thermal actuators is considered in
this thesis. In topology optimization, no prior assumptions are made regarding
the geometrical layout of the design, and only a design domain and boundary
conditions are defined. In this work, a novel approach to parameterize the layout
design problem is applied, based on modifying the connectivity between finite
elements instead of changing their material properties. This Element Connectivity Parameterization concept avoids difficulties that conventional approaches
encounter in this setting, such as the ambiguity of defining an interpolation of
material properties, loss of numerical stability due to excessive distortion of elements with low stiffness, and complex sensitivity analysis. With this technique,
in combination with the Method of Moving Asymptotes, topology optimization of
SMA structures could be performed successfully. Next to the capacity to generate effective SMA actuator structures from a neutral initial configuration, it was
also found that this topology optimization procedure was able to achieve large
performance improvements by adapting a given design.
The combined modeling and design approaches reported in this thesis constitute a set of versatile and effective design procedures for SMA structures, that
shows great potential for application in a wide variety of specific applications.
Particularly in the development of future SMA devices with complex geometries,
such as, for example, enhanced miniature medical instruments, it is expected
278
SUMMARY
that designers can benefit considerably from systematic design optimization procedures as developed in this research. Furthermore, the element connectivity
parameterization approach used in the topology optimization of SMA thermal
actuators also offers advantages over conventional approaches in other topology
optimization problems involving nonlinear material behavior. Finally, the novel
active catheter designs proposed in this research showed good performance after
optimization, and could very well serve as a basis for future prototypes.
Samenvatting
Ontwerpoptimalisatie van Geheugenmetaalconstructies
Dit proefschrift presenteert verscheidene methoden om ontwerpoptimalisatie toe
te passen op geheugenmetaalconstructies. Geheugenmetalen zijn metaallegeringen die een fasetransformatie kunnen ondergaan in de vaste toestand, onder invloed van veranderingen in temperatuur en mechanische spanning. De rekken die
met deze transformatie gepaard gaan kunnen gebruikt worden voor actuatie.
In vergelijking met andere actieve materialen zijn geheugenmetalen in staat
relatief grote rekken en spanningen te genereren. Er bestaat hierom een grote interesse voor het gebruik van geheugenmetalen in een scala van (micro-)actuatietoepassingen. Echter, het ontwerpen van een goed functionerende geheugenmetaalconstructie wordt bemoeilijkt door twee factoren. Ten eerste is het materiaalgedrag van geheugenmetaal niet eenvoudig te voorspellen, en ten tweede dienen bij
het ontwerpen veelal de electrische, thermische en mechanische eigenschappen van
een constructie tegelijkertijd beschouwd te worden. Dit onderzoek heeft daarom
tot doel te onderzoeken op welke wijze en in hoeverre ontwerpoptimalisatiemethoden het ontwerpen met geheugenmetaal kunnen vergemakkelijken.
De toepasbaarheid en effectivititeit van verscheidene ontwerpoptimalisatiemethoden voor geheugenmetaalconstructies is onderzocht door deze gebruiken in
representatieve toepassingen. Dit proefschrift richt zich hierbij met name op opkomende toepassingen in de gezondheidszorg, waar de populariteit van minimaal
invasieve behandelingen een grote vraag naar geminiaturiseerde instrumenten met
uitgebreide functionaliteit tot gevolg heeft. Geheugenmetaal is een aantrekkelijke optie voor het realiseren van de actuatie in dergelijke instrumenten. De
nadruk in dit onderzoek ligt op het geheugenmetaalgedrag ten gevolge van de
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SAMENVATTING
zogenaamde austeniet/R-fase transformatie in NiTi legeringen. Deze transformatie vindt plaats binnen een relatief klein temperatuurbereik, heeft een geringe
hysterese en een uitstekende cyclische stabiliteit. Dit zijn zeer geschikte kenmerken voor actuatiedoeleinden in het beoogde toepassingsgebied. Praktische
geheugenmetaaltoepassingen die in dit proefschrift met name beschouwd worden
zijn geminiaturiseerde grijpinstrumenten en bestuurbare catheters.
Om een breed toepasbare en veelzijdige ontwerpmethodologie te ontwikkelen,
worden eindige-elementenmodellen gebruikt om het functioneren van geheugenmetaalconstructies te evalueren. In dit proefschrift wordt een niet-lineair quasistatisch model gebruikt, dat bestaat uit een sequentieel gekoppelde electrische,
thermische en mechanische analyse. Een drie-dimensionaal materiaalmodel is
ontwikkeld dat het gedrag van NiTi beschrijft ten gevolge van de austeniet/Rfase transformatie. Een empirische benadering is toegepast, in combinatie met
de fundamentele eigenschappen van de onderliggende fasetransformatie, om zo de
essentie van het materiaalgedrag in een model te vatten. Het beperkte temperatuurbereik dat de toepassingen vereisen en de geringe hysterese maken belangrijke
vereenvoudigingen mogelijk. Dit leidt tot een pad-onafhankelijk model, dat bij
uitstek geschikt is voor ontwerpoptimalisatie. Het ontwikkelde materiaalmodel is
toegepast in de analyse van een nieuw concept van een bestuurbare catheter met
geı̈ntegreerde actuatiegedeeltes van geheugenmetaal.
Vervolgens worden vormoptimalisatiestudies van een geminiaturiseerd grijpinstrument van geheugenmetaal behandeld. Als eerste wordt een directe optimalisatie beschouwd, waarin geen ontwerpafgeleiden worden gebruikt. De uitslag
van de bekken van de grijper wordt gemaximaliseerd, met inachtneming van restricties betreffende de werktemperaturen en de maximale rek. Een MeerpuntsBenaderings-Methode wordt gebruikt voor het uitvoeren van de optimalisatie.
Deze methode maakt gebruik van het construeren van analytische benaderingen
van het modelgedrag, hetgeen de rekenkosten reduceert. In combinatie met parallelle berekeningen om de totale duur van de procedure te verminderen, wordt
aangetoond dat deze methode een effectieve manier biedt om geoptimaliseerde
grijperontwerpen te genereren.
Hetzelfde miniatuur grijpinstrument wordt tevens gebruikt in een optimalisatiestudie waarin ook de invloed van onzekerheden wordt beschouwd. Doorgaans
zijn in de beginfase van het ontwerpproces onvoldoende betrouwbare statistische
gegevens beschikbaar om onzekerheden te kunnen karakteriseren. Hierom worden
in deze studie onzekerheden slechts gedefinieerd door de grenzen waarbinnen de
waarden van de onzekere parameters kunnen variëren. Het ontwerpoptimalisatieprobleem is zodanig opgesteld dat het tracht de verwachte prestaties te maximaliseren, terwijl de meest ongunstigste combinatie van onzekerheden nog steeds
een toelaatbaar ontwerp geeft. Als optimalisatieprocedure is een aangepaste
anti-optimalisatie benadering gebruikt, gebaseerd op een Meerpunts-BenaderingsMethode. Hierbij wordt afgewisseld tussen hoofd- en anti-optimalisatiecycli, wat
de totale rekentijd sterk reduceert. Tevens zijn ook hier parallelle rekenmethoden
toegepast, waarmee deze niet-deterministische vormoptimalisatie van een geheu-
SAMENVATTING
281
genmetaalconstructie succesvol afgerond kon worden binnen een beperkte tijdsduur.
Naarmate optimalisatieproblemen meer ontwerpvariabelen bevatten en modellen meer rekentijd vergen, wordt het aantrekkelijker om, in plaats van directe,
gradiënt-gebaseerde methoden te gebruiken. De benodigde gradiëntinformatie
wordt berekend middels gevoeligheidsanalyse met het gebruikte model. Beschouwd
zijn globale eindige differenties, globale eindige differenties in combinatie met
snelle heranalyse, semi-analytische en verfijnde semi-analytische directe differentiatiemethoden. De semi-analytische formulering vereist speciale aandacht voor
het impliciete karakter van het materiaalmodel en de koppeling tussen het electrische, thermische en mechanische probleem. Deze studie laat zien dat alle
beschouwde methoden resulteren in ontwerpafgeleiden van vergelijkbare nauwkeurigheid. Verder blijken semi-analytische benaderingen efficiënter dan de eindige
differentie methode in combinatie met snelle heranalyse. Echter, de implementatie van de laatstgenoemde methode is aanzienlijk eenvoudiger. De relatief lage
rekenkosten van de gevoeligheidsanalyse voor de beschouwde klasse van geheugenmetaalmodellen zijn overigens voor een groot deel te danken aan het padonafhankelijke karakter van het gebruikte materiaalmodel.
De beschikbaarheid van ontwerpafgeleiden maakt gradiëntgebaseerde vormoptimalisatie van geheugenmetaalconstructies mogelijk. De effectiviteit van deze benadering wordt aangetoond middels optimalisatiestudies van een geminiaturiseerd
grijpinstrument van geheugenmetaal, alsmede van de eerder beschreven bestuurbare catheter. Een bijzonderheid van het model van deze catheter is, dat het
niet-lineaire beperkende voorwaarden tussen de vrijheidsgraden van meerdere
punten bevat. Deze worden gebruikt om de symmetrie- en periodiciteitscondities op te leggen. In de gebruikte oplossingsmethode worden deze voorwaarden
behandeld volgens de zogenaamde augmented Lagrangian benadering. In deze
situatie hebben globale eindige differenties in combinatie met snelle heranalyse
de voorkeur voor de gevoeligheidsanalyse. Een correctie is geı̈ntroduceerd om de
invloed van een restfout in de vervulling van de nevenvoorwaarden, die resteert na
het oplossen van het nominale probleem, in rekening te brengen. In combinatie
met strengere convergentievoorwaarden in de geperturbeerde gevallen, leidt deze
correctie tot een duidelijke verbetering van de nauwkeurigheid van de ontwerpafgeleiden. Gemodificeerde Newton-iteraties worden gebruikt in de heranalyse,
op basis van de gedecomponeerde nominale systeemmatrix, hetgeen de benodigde
rekentijd verder reduceert. Sequentieel Kwardratisch Programmeren en de Methode van Verschuivende Asymptoten zijn toegepast als optimalisatiealgorithmes.
Deze gradiëntgebaseerde optimalisatieprocedures blijken in alle gevallen veel minder functie-evaluaties te vereisen dan de eerder gebruikte directe methode, en zijn
tevens in staat het optimale ontwerp preciezer te bepalen.
Tot slot wordt ook topologie-optimalisatie van thermische actuatoren van
geheugenmetaal beschouwd in dit proefschrift. In topologie-optimalisatie wordt
de geometrie van het uiteindelijke ontwerp volledig vrijgelaten, en slechts het ontwerpdomein en de randvoorwaarden worden gedefiniëerd. In deze studie wordt
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SAMENVATTING
een nieuwe topologische parameterisatie toegepast, die is gebaseerd op het veranderen van de connectiviteit tussen eindige-elementen. Gebruikelijke topologische parameterisaties zijn daarentegen gebaseerd op het veranderen van materiaaleigenschappen van elementen. Deze Element Connectiviteits Parameterisatie
gaat drie problemen die gebruikelijke benaderingen ondervinden uit de weg. Ten
eerste is het niet nodig een arbitraire keuze te maken voor de interpolatie van materiaaleigenschappen. Daarnaast is deze aanpak ook ongevoelig voor numerieke
instabiliteiten door buitensporige vervorming van elementen met lage stijfheid.
Ten derde speelt het materiaalmodel geen rol in de gevoeligheidsanalyse, hetgeen
de gevoeligheidsanalyse aanzienlijk vereenvoudigd. Met behulp van de Methode
van Verschuivende Asymptoten konden innovatieve geheugenmetaalconstructies
worden gegenereerd voor willekeurige belastingsgevallen. Tevens bleek de ontwikkelde topologie-optimalisatieprocedure in staat om grote prestatieverbeteringen
te realiseren bij het aanpassen van een reeds gegeven ontwerp.
De modellerings- en ontwerpbenaderingen die in dit proefschrift worden beschreven vormen tezamen een verzameling van veelzijdige en effectieve ontwerpprocedures voor geheugenmetaalconstructies. Het is de verwachting dat ontwerpers voor een breed scala van specifieke geheugenmetaaltoepassingen baat zullen
hebben bij de ontwikkelde systematische procedures. Dit geldt in het bijzonder voor de ontwikkeling van toekomstige geheugenmetaalconstructies met complexe geometrieën, zoals bijvoorbeeld geavanceerde en geminiaturiseerde medische instrumenten. Daarbij biedt de Element Connectiviteits Parameterisatie,
zoals die toegepast is bij de topologie-optimalisatie van thermische actuatoren
van geheugenmetaal, ook perspectieven voor topologie-optimalisatie waarin ander niet-lineair materiaalgedrag een rol speelt. Tenslotte behaalde het nieuwe
ontwerp voor een bestuurbare catheter, zoals dat in dit onderzoek is voorgesteld,
goede resultaten na ontwerpoptimalisatie. Het zou hierom uitstekend kunnen
dienen als een basis voor toekomstige prototypes.
List of Publications
Publications in refereed journals and conference proceedings, related to this dissertation:
Langelaar, M. and Van Keulen, F. (2003). A simple R-phase transformation
model for engineering purposes. In Proceedings of the European Symposium
On Martensitic Transformations and Shape-Memory, ESOMAT 2003, August 17–22, Cirencester, England.
Gurav, S., Langelaar, M., Goosen, J., and Van Keulen, F. (2003). Different
approaches to deal with Bounded-But-Unknown uncertainties: Application
to MEMS. In Proceedings of the 5th World Conference on Structural and
Multidisciplinary Optimization, 19–23 May, Venice, Italy.
Gurav, S., Langelaar, M., Goosen, J., and Van Keulen, F. (2003). Boundedbut-unknown uncertainty optimization of micro-electro-mechanical-systems.
In Proceedings of the Second M.I.T. Conference on Computational Fluid and
Solid Mechanics, June 17–20, M.I.T. Cambridge, MA, U.S.A.
Langelaar, M. and Van Keulen, F. (2004). A simple R-phase transformation model for engineering purposes. Materials Science and Engineering: A
- Structural Materials: Properties, Microstructure and Processing, 378(1–
2):507–512.
Langelaar, M. and Van Keulen, F. (2004). Modeling of a shape memory alloy
active catheter. In Proceedings of the 12th AIAA/ASME/AHS Adaptive
Structures Conference, Palm Springs, CA, U.S.A.
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LIST OF PUBLICATIONS
Langelaar, M. and Van Keulen, F. (2004). Design optimization of shape
memory alloy structures. In Proceedings of the 10th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, August 30–September
1, Albany, NY, U.S.A.
Langelaar, M., Yoon, G.H., Kim, Y.Y. and Van Keulen, F. (2005). Topology
Optimization of Shape Memory Alloy Actuators using Element Connectivity
Parameterization. In Proceedings of the 6th World Congress on Structural
and Multidisciplinary Optimization, May 30–June 3, Rio de Janeiro, Brazil.
Langelaar, M., Yoon, G.H., Gurav, S., Kim, Y.Y. and Van Keulen, F. (2005).
Modeling and Design of Shape Memory Alloy Actuators. In Proceedings of
the 6th IEEE EuroSimE Conference, April 18–20, Berlin, Germany, invited
keynote contribution.
Langelaar, M., Yoon, G.H., Gurav, S., Kim, Y.Y. and Van Keulen, F.
(2005). Analysis and design techniques for shape memory alloy microactuators for space applications. In Proceedings of the 5th ESA Round Table on Micro/Nano Technologies for Space, 3–5 October, Noordwijk, The
Netherlands.
Langelaar, M. (2005). Designing Memory Metal Devices. Design optimization of shape memory alloy structures – targeted at medical and microsystem
applications. In Dingeldein, N., editor, DIMES White Papers 2005, pages
41–52, DIMES, Delft.
Gurav, S., Langelaar, M., Goosen, J., and Van Keulen, F. (2005). Uncertaintybased design optimization of shape memory alloy microgripper using combined cycle-based alternating anti-optimization and nested parallel computing. In Proceedings of the 6th World Congress on Structural and Multidisciplinary Optimization, May 30–June 3, Rio de Janeiro, Brazil.
Zeoli, M., Van Keulen, F. and Langelaar, M. (2005). Fast Reanalysis of Geometrically Nonlinear Problems After Shape Modifications. In Proceedings
of the 6th World Congress on Structural and Multidisciplinary Optimization,
May 30–June 3, Rio de Janeiro, Brazil.
Langelaar, M. and Van Keulen, F. (2006). Sensitivity analysis of shape
memory alloy shells. In Proceedings of the 3rd European Conference on
Computational Mechanics, 5–8 June, Lisbon, Portugal.
Langelaar, M. and Van Keulen, F. (2006). Sensitivity analysis and optimization of a shape memory alloy gripper. In Proceedings of the 8th International
Conference on Computational Structures Technology, 12–15 September, Las
Palmas de Gran Canaria, Spain.
LIST OF PUBLICATIONS
285
Langelaar, M. and Van Keulen, F. Modeling of shape memory alloy shells
for design optimization. Computers and Structures, in review.
Langelaar, M. and Van Keulen, F. Sensitivity analysis of shape memory
alloy shells. Computers and Structures, in review.
Langelaar, M. and Van Keulen, F. Sensitivity analysis and optimization of
a shape memory alloy gripper. Computers and Structures, in review.
Langelaar, M., Yoon, G.H., Kim, Y.Y. and Van Keulen, F. Topology optimization of shape memory alloy thermal actuators using element connectivity parameterization. Computer Methods in Applied Mechanics and Engineering, in review.
Gurav, S.P., Langelaar, M. and Van Keulen, F. (2006). Cycle-based alternating anti-optimization combined with nested parallel computing: application to shape memory alloy microgripper. Computers and Structures, in
review.
Yoon, G.H., Kim, Y.Y., Langelaar, M. and Van Keulen, F. Theoretical
aspect of the internal element connectivity parameterization approach for
topology optimization. In preparation, to be submitted to International
Journal of Numerical Methods in Engineering.
Langelaar, M. and Van Keulen, F. (2007). Gradient-based design optimization of shape memory alloy active catheters. In Proceedings of the 15th
AIAA/ASME/AHS Adaptive Structures Conference, April 23–26, Waikiki,
Hawaii, U.S.A., accepted.
Langelaar, M. and Van Keulen, F. (2007). Design optimization of shape
memory alloy active structures using the R-phase transformation. In Proceedings of the 14th International SPIE Symposium on Smart Structures and
Materials, 18–22 March, San Diego, CA, U.S.A., accepted.
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LIST OF PUBLICATIONS
Acknowledgements
First of all, I thank José, for all the support, encouragement and inspiration she
gave me over the past few years, and for helping me balance my professional and
personal life. I also thank my parents, family and friends for all their interest
and support. Furthermore, I would like to thank my supervisor Fred van Keulen,
for his commitment, enthusiasm and advice, and for giving me the freedom to
develop my own ideas. Also each time when the final date of my contract was
approaching, he managed to find a source of funding that allowed me to continue
my research, for which I am very grateful. I also thank all my collegues from
the Structural Optimization and Computational Mechanics group in Delft: Jan,
Hans, Marianne, Javad, Jacqueline, Marten Jan, Koen, Andriy, Gerard, Teun,
Sham, Peterjan, Chiara, Bert, Gih-Keong and Caspar. They all together created
an enjoyable and stimulating working environment.
A special thanks goes to Sham Gurav, with whom I collaborated on including
uncertainties in shape memory alloy design optimization and on the development
of parallel computing scripts, which have been of great use for many of the results in this thesis. I also especially would like to mention Jan Booij, for solving
numerous computer problems, providing a well-maintained working environment,
tolerating my occasional computer pranks and not even complaining when due to
my use of an obsolete operating system the entire network of the group was hacked.
Furthermore, I thank Niels Bakker and Paul Breedveld from the Biomedical Engineering department, for sharing their expertise on minimally invasive surgery, and
Vincent Henneken and Warner Venstra for sharing their insight and experiences
regarding microsystem design and development.
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ACKNOWLEDGEMENTS
∗
Furthermore, I would like to say
to Professor Yoon Young Kim
of Seoul National University, for his great hospitality and for creating the ideal
conditions for a fruitful collaboration. Also his guidance and advice during the
goes out to
regular progress meetings was invaluable. An equally strong
Gil Ho Yoon, who taught me all the secrets of topology optimization and element
connectivity parameterization. With his knowledge and ideas, he contributed
significantly to the realization of the topology optimization procedure discussed
in this thesis. Next to this, I also feel indebted to Professor Yoon Young Kim
and Gil Ho Yoon for introducing me to a wide range of Korean customs, food and
drinks, and in this regard I also should mention Yongkeun Park and Young-Soo
Joung for being excellent company.
I furthermore thank Dr. Andrzej Ziolkowski of the Polish Institute of Fundamental Technological Research, for his discussions on modeling of shape memory
alloys, for sharing his knowledge on the R-phase transformation with me, and
for sending me the relevant literature. I also want to acknowledge Karsten Svanberg for the use of his Method of Moving Asymptotes, Sivan Toledo for the use
of his fast direct sparse linear solver library TAUCS, and Leo Breebaart for the
use of his LATEX thesis template. Finally, I also would like to acknowledge the
Delft Center of Mechatronics and Microsystems, which has financially supported
a considerable part of this research.
Matthijs Langelaar
Delft, October 2006
∗ “Kamsa
hamnida”, meaning “Thank you”, in Korean.
About the author
February 12, 1976
Born in Apeldoorn, the Netherlands.
1988 – 1994
High school, CLA Gymnasium in Apeldoorn. Graduated cum laude.
1994 – 1999
BSc. and MSc. study in Mechanical Engineering at
University of Twente, the Netherlands. Graduated
cum laude, on a graduation project carried out at
the Product Application Center of Corus, IJmuiden,
the Netherlands, supervised by Prof.dr.ir. J. Huétink
and Dr.ir. H. Vegter. Title of MSc-thesis: Verification of the Vegter Yield Criterion. This thesis
was awarded the Dutch Central Institute of Industrial
Development (Centraal Instituut Voor Industrie Ontwikkeling) prize in November 2000, by the Royal Holland Society of Sciences and Humanities (Koninklijke
Hollandsche Maatschappij der Wetenschappen).
2000 – 2001
Researcher at the Control Design Engineering group
of the Institute of Robotics and Mechatronics. This
institute is a division of DLR, the German Aerospace
Center, located in Oberpfaffenhofen, Germany. Participant in European Union funded project “REALSIM”,
aimed at real-time simulation for the design of multiphysics systems.
289
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ABOUT THE AUTHOR
2001 – present
Ph.D. candidate at the Structural Optimization and
Computational Mechanics group, which is part of the
Precision and Microsystem Engineering Department of
the Faculty of Mechanical, Materials and Maritime Engineering of Delft University of Technology, Delft, the
Netherlands.
November 2004 –
January 2005
Visiting scholar at the Integrated Design and Analysis
of Structures Laboratory of Prof. Y.Y. Kim, at Seoul
National University, Seoul, Korea. Collaborated with
Dr. G.H. Yoon on the development of topology optimization of shape memory alloy structures.
From January 2007, Matthijs Langelaar is appointed as a post-doctoral researcher at the Precision and Microsystem Engineering Department, and will
work on a project aimed at topology optimization of microcomponents including
reliability aspects, which is part of the national MicroNed research program.
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