EM_2014_011_-_Serphos_-_MSc_Report.

EM_2014_011_-_Serphos_-_MSc_Report.
Department of Precision and Microsystems Engineering
Incorporating AM-specific Manufacturing
Constraints into Topology Optimization
M. Reuben Serphos
Report no
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EM 2014.011
Dr.ir. M. Langelaar (TU Delft)
Dr. J. Fatemi, MSc (Dutch Space)
Prof.dr.ir. F. van Keulen (TU Delft)
Engineering Mechanics
MSc Thesis
30 May 2014
Incorporating AM-specific
Manufacturing Constraints into
Topology Optimization
Master of Science Thesis
For the degree of Master of Science in Mechanical Engineering at Delft
University of Technology
M. Reuben Serphos
May 30, 2014
Faculty of Mechanical, Maritime and Materials Engineering (3mE) · Delft University of
Technology
About the cover:
A close up of a lightweight titanium lattice ball manufactured using the Additive Manufacturing or 3D printing process. This design is a good example of AM capabilities: these hollow
balls possessing a complex external geometry could not have been manufactured in a single
part using a conventional manufacturing process. But they are incredibly light while also
stiff, opening up possibilities for future space applications.
Photo and description: ESA
The work in this thesis was supported by Dutch Space B.V. Their cooperation is hereby
gratefully acknowledged.
c Precision and Microsystems Engineering Copyright Engineering Mechanics
All rights reserved.
Abstract
Designs to be manufactured by Additive Manufacturing (AM) are subject to geometrical restrictions which are necessary to guarantee successful production. This makes it necessary to
modify the obtained topology optimized design such that it is manufacturable. By doing this
the optimality of the structure can be reduced. To achieve a manufacturable design directly
from the topology optimization requires the incorporation of such a geometrical restriction
directly into the optimization process. This research has been carried out to develop a method
to do so.
The manufacturing constraint being implemented is that a structure should be self-supporting.
For this to be the case overhanging features in the design must be at an angle of at least 45
degrees with respect to the base plate.
Three approaches have been formulated and investigated: (1)a multiple objective, (2)global
constraint and (3)a density filter. A global measure has been formulated for the multiple
objective and the global constraint and a filtering scheme for the density filter. The three
approaches have been implemented into a 2D optimization code written in MATLAB. Three
predictable reference models have been used to test the suitability of each method.
The results show that the multiple objective and the global constraint remain potential candidates but further investigation is needed to determine proper parameter values. The topologies obtained with these approaches were modified to partially meet the manufacturing constraint but remained with overhanging features. The filtering method produced topologies
that had no overhanging features. The design was a compromise between slight alterations
and the introduction of support structures. This approach however showed instabilities that
are directly linked to the proposed filtering scheme. True convergence was not achieved with
any of the approaches.
The filtering approach provides a proof of concept for the inclusion of the 45 degree overhang
restriction but further development is needed. The other two approaches are also potential
candidates for inclusion of the restriction.
Master of Science Thesis
M. Reuben Serphos
ii
M. Reuben Serphos
Master of Science Thesis
Table of Contents
Preface
xi
Acknowledgments
xiii
1 Introduction
1
1-1 The Antenna - Articulated Deployment System (A-ADS) . . . . . . . . . . . . .
1
1-2 Benefiting from AM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1-3 Topology optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2 Problem and hypothesis
7
2-1 Problem description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2-2 A-ADS hinge-bracket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2-3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
3 Method
11
3-1 Current status of commercial software . . . . . . . . . . . . . . . . . . . . . . .
11
3-1-1
Hyperworks Optistruct . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
3-1-2
Simulia ATOM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
3-2 Description 88-line MATLAB code . . . . . . . . . . . . . . . . . . . . . . . . .
14
3-3 Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
3-4 Reference models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
3-4-1
Cantilever beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
3-4-2
Tension beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
Master of Science Thesis
M. Reuben Serphos
iv
Table of Contents
4 Formulation of the overhang restriction
4-1 Using the continuous nature of TO . . . . . .
4-2 Using the discrete nature of TO . . . . . . .
4-3 Pragmatic approach . . . . . . . . . . . . . .
4-4 Restriction method . . . . . . . . . . . . . .
4-5 Local constraints . . . . . . . . . . . . . . .
4-6 Multiple objective and global constraint . . .
4-6-1 Multiple Objective . . . . . . . . . . .
4-6-2 Multiple objective function sensitivities
4-6-3 Global constraint . . . . . . . . . . .
4-7 Filter . . . . . . . . . . . . . . . . . . . . . .
4-7-1 Modification by sequential scanning .
5 Validation
5-1 Validation of the multiple objective and global
5-1-1 Parameters M, δ and pn . . . . . . .
5-1-2 Sensitivity check . . . . . . . . . . . .
5-2 Verification of filter formulation . . . . . . . .
5-2-1 Modification by sequential scanning .
6 Notes on implementation
6-1 Element selection . .
6-2 Ghost-layers . . . . .
6-3 Invalid values . . . .
6-4 Constraint in MMA .
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constraint
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49
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corner-load
mid-load . .
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corner-load
mid-load . .
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53
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8 Discussion
8-1 Modification of strut orientation . . . . . . . . . . . . . . . . . . . . . . . . . .
8-2 Stalactites in multiple objective method and global constraint method . . . . . .
8-3 Erratic objective values, instability and Ωf ac . . . . . . . . . . . . . . . . . . . .
63
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8-4 Lowered volume fraction and high compliance . . . . . . . . . . . . . . . . . . .
8-5 Additional considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
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7 Final results
7-1 Multiple Objective . . .
7-1-1 Cantilever beam
7-1-2 Cantilever beam
7-1-3 Tension beam .
7-2 Global constraint . . .
7-2-1 Cantilever beam
7-2-2 Cantilever beam
7-2-3 Tension beam .
7-3 Filter . . . . . . . . . .
M. Reuben Serphos
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Master of Science Thesis
Table of Contents
v
9 Conclusion
69
10 Recommendations
71
A A-ADS hinge-bracket
73
A-1 Design requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
A-2 Circular in-plane torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
A-3 Circular out-of-plane torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
A-4 Hinge-bracket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
B Another filter approach
79
B-1 Formulation of the filter - Modification by supportivity matrix . . . . . . . . . .
79
B-2 Validation - Modification by supportivity matrix . . . . . . . . . . . . . . . . . .
82
C Results
C-1 Multiple objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
85
C-1-1 Corner-loaded cantilever beam
C-1-2 Mid-loaded cantilever beam . .
C-1-3 Tension beam . . . . . . . . .
C-2 Global constraint . . . . . . . . . . .
C-2-1 Corner-loaded cantilever beam
C-2-2 Mid-loaded cantilever beam . .
C-2-3 Tension beam . . . . . . . . .
C-3 Filter . . . . . . . . . . . . . . . . . .
C-3-1 Corner-loaded cantilever beam
C-3-2 Mid-loaded cantilever beam . .
C-3-3 Tension beam . . . . . . . . .
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Glossary
85
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109
List of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
109
List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
109
Master of Science Thesis
M. Reuben Serphos
vi
M. Reuben Serphos
Table of Contents
Master of Science Thesis
List of Figures
1-1 A-ADS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1-2 Benefiting from AM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
2
1-3 Schematic representation of the SLM process; Source: [1] . . . . . . . . . . . . .
3
2-1 Schematic of overhanging surface . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2-2 Various levels of curl occurring as a result of residual stress in increasingly overhanging material; Source: [2] . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2-3 Sagging of material in vertically built holes; Source: [2] . . . . . . . . . . . . . .
9
2-4 Concept design produced by topology optimization . . . . . . . . . . . . . . . .
9
3-1 Response types Hyperworks Optistruct . . . . . . . . . . . . . . . . . . . . . . .
12
3-2 Manufacturing constraints Hyperworks Optistruct Member size, stress, fatigue,
draw direction, extrusion, symmetry and patterning . . . . . . . . . . . . . . . .
12
3-3 Response types Simulia ATOM . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
3-4 Manufacturing constraints Simulia ATOM Member size, draw direction, symmetry
13
3-5 Structured grid mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
3-6 (Left) Load case cantilever beam mid-load and (right) the result with: Iterations:
121; Objective value: 57.17; Volume fraction: 0.418 . . . . . . . . . . . . . . . .
16
3-7 (Left) Load case cantilever beam corner-load and (right) the result with: Iterations:
191; Objective value: 64.00; Volume fraction: 0.426 . . . . . . . . . . . . . . . .
16
3-8 (Left) Load case tension beam and (right) the result with: Iterations: 1160; Objective value: 5.49; Volume fraction: 0.474 . . . . . . . . . . . . . . . . . . . . .
17
4-1 Gradient of density field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
4-2 Local gradients in black and white topology represented by blue arrows . . . . .
21
4-3 Discontinuous aspect of topology optimization showing candidate support elements 21
4-4 Example of P-norm behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Master of Science Thesis
22
M. Reuben Serphos
viii
List of Figures
4-5 (Left) Reference model corner-loaded cantilever beam and (right) result pragmatic
approach for: mesh: 30x50; volume fraction: 0.5; filter penalization power: 3;
radius: 1.5; Obtained objective after 49 iterations: 67.57; volume fraction: 0.421
23
4-6 (Left) Reference model corner-loaded cantilever beam for: mesh: 60x100; imposed
volume fraction: ≤ 0.5; filter penalization power: 3; radius: 1.5; Obtained objective
after 270 iterations: 62.69; volume fraction: 0.404 and (right) result pragmatic
approach with: Obtained objective after 38 iterations: 67.23; volume fraction: 0.422 24
4-7 Value ρ∆ with respect to ρmax and ρi . . . . . . . . . . . . . . . . . . . . . . .
26
4-8 Plot of the logistic function with x ranging from -1 to 1
. . . . . . . . . . . . .
27
4-9 Overhang with M = 10 and δ = 0, 5 . . . . . . . . . . . . . . . . . . . . . . . .
27
4-10 Influenced elements for sensitivities . . . . . . . . . . . . . . . . . . . . . . . . .
28
5-1 Standard geometry for verification . . . . . . . . . . . . . . . . . . . . . . . . .
33
5-2 Exact OH detection in standard geometry . . . . . . . . . . . . . . . . . . . . .
34
5-3 Approximate OH detection in standard geometry, with M = 35, pn = 8 and δ = 0.2 34
5-4 Logistic function for varying values of M . . . . . . . . . . . . . . . . . . . . . .
35
5-5 Ωtot for parameter M from 10-100 . . . . . . . . . . . . . . . . . . . . . . . . .
36
5-6 Ωtot for varying parameter δ
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
5-7 Ω∗ for gradient geometry with δ = 0 giving Ωtot = 0.223 . . . . . . . . . . . . .
38
5-8 Error due to P-norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-9 Ωtot for varying P-norm power pn . . . . . . . . . . . . . . . . . . . . . . . . . .
39
40
5-10 Minimum detectable ρ difference . . . . . . . . . . . . . . . . . . . . . . . . . .
42
5-11 Density of arbitrary element (in the center of the ring) is incrementally increased
from 0 to 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-12 Comparison of the calculated and the finite difference sensitivity . . . . . . . . .
42
43
5-13 Element to be varied and element of which the sensitivity is determined . . . . .
44
5-14 Filtering the densities by sequential scanning of the elements . . . . . . . . . . .
45
5-15 Effect of P-norm power on element density directly below overhang . . . . . . .
46
5-16 Filtering the corner loaded cantilever beam by sequential scanning of the elements
46
5-17 Filtered densities for varying density of test element . . . . . . . . . . . . . . . .
47
5-18 Comparison of calculated and finite difference sensitivity . . . . . . . . . . . . .
48
6-1 Candidate support elements for an element under inspection . . . . . . . . . . .
49
6-2 Ghost-layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
7-1 Results multiple objective for cantilever corner-load obtained in 191 iterations . .
54
7-2
Ω∗
multiple objective for cantilever corner-load; Using the following parameter
values: M = 35; δ = 0.2; pn = 8 . . . . . . . . . . . . . . . . . . . . . . . . . .
55
7-3 Results multiple objective for cantilever mid-load obtained in 121 iterations . . .
56
7-4
Ω∗ global
constraint for cantilever mid-load; Using the following parameter values:
M = 35; δ = 0.2; pn = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
7-5 Results multiple objective for tension beam . . . . . . . . . . . . . . . . . . . .
57
M. Reuben Serphos
Master of Science Thesis
List of Figures
ix
7-6 Ω∗ multiple objective for tension beam; Using the following parameter values:
M = 35; δ = 0.2; pn = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
7-7 Results global constraint for cantilever corner-load obtained in 191 iterations . .
58
7-8 Ω∗ global constraint for cantilever corner-load; Using the following parameter values: M = 35; δ = 0.2; pn = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
7-9 Results cantilever mid-load obtained in 121 iterations . . . . . . . . . . . . . . .
7-10 Ω∗ global constraint for cantilever mid-load; Using the following parameter values:
M = 35; δ = 0.2; pn = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
7-11 Results tension beam obtained . . . . . . . . . . . . . . . . . . . . . . . . . . .
7-12 Ω∗ global constraint for tension beam; Using the following parameter values: M =
35; δ = 0.2; pn = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
7-13 Results filter for cantilever corner-load . . . . . . . . . . . . . . . . . . . . . . .
7-14 Results filter for cantilever mid-load . . . . . . . . . . . . . . . . . . . . . . . .
7-15 Results filter for tension beam . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
61
62
8-1 Stalactite formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8-2 Cantilever beam corner-load with parameters δ = 0.2, M = 8, pn = 8 and
Ωf ac = 103 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
8-3 Extracting a black-white topology from the filter result . . . . . . . . . . . . . .
66
A-1 Schematic representation of the hinge-bracket . . . . . . . . . . . . . . . . . . .
74
A-2 Load case circular in-plane torque (left) and result (right) . . . . . . . . . . . . .
75
A-3 Load case circular out-of-plane torque (left) and result (right) . . . . . . . . . .
75
A-4 Load case of the hinge-bracket . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
A-5 Result of the hinge-bracket . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
B-1 Involved elements in determining the supportivitiy of an element ρ0 . Numbers
denote element numbers (not corresponding to actual numbering in the MATLAB
code). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
B-2 Filtering the densities using the supportivity matrix . . . . . . . . . . . . . . . .
83
B-3 Effect of P-norm power on the density of a element at the top of the structure .
83
B-4 Filtering the corner loaded cantilever beam using the supportivity matrix . . . . .
84
Master of Science Thesis
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60
65
M. Reuben Serphos
x
M. Reuben Serphos
List of Figures
Master of Science Thesis
Preface
As long as I can remember I have had a great interest in all types of science, technology
and mechanics in general and a desire to put this to practical use. The choice to study
mechanical engineering has therefore always been obvious to me. Though besides this broad
interest a more specific passion developed after visiting the Kennedy Space Center as a child.
Discovering the vastness of what lies beyond Earth fascinated me. This sparked a passion for
space exploration and space travel, finding new worlds and expanding humanities knowledge
of the universe, which to this day has stuck with me.
This passion is what drove me to apply for a thesis topic at Dutch Space. The opportunity
to apply what I had learned during my studies in a space technology oriented environment
greatly enthused me.
In discussing possible topics, my supervisor from Dutch Space, Dr. J. Fatemi (MSc), proposed
the design of a satellite part which was to be manufactured by Additive Manufacturing (AM).
This immediately intrigued me. Taking part in the application of a new developing technology
in a space oriented environment was an ideal combination of my interests and I am very
grateful Dr. J. Fatemi (MSc) could give me this opportunity.
During the course of my literature survey it became clear that the project was going to be a bit
more fundamental than anticipated. The literature survey revealed that design methodologies
for AM were underdeveloped. Even though the use of topology optimization as a design tool
for AM is widely agreed upon as being the way to go, the approach is not complete. The tool
has not been tailored to the advantages and limitations of AM. This realization is what led
to the final version of the research goal leading to the thesis that lies before you. It is a small
step in the development of design tools specifically for AM and I hope it will lead to further
developments in its field.
I am glad I got the opportunity to do this and am thankful to those who helped make it
possible.
Master of Science Thesis
M. Reuben Serphos
xii
M. Reuben Serphos
Preface
Master of Science Thesis
Acknowledgments
I would like to take the opportunity to thank the following people for helping me with my
research and writing. First of all I would like to thank my professor Prof.dr.ir. F. van Keulen
for giving me the opportunity to do this research. I am also grateful for the assistance given
by Dr. ir. M. Langelaar whose feedback on ideas and drafts has been very valuable. A special
thanks goes to Dr. J. Fatemi (MSc) for giving me guidance and providing a critical view of
the steps made along the way. Advise given by Dr.ir. G. van der Veen has been a great help.
I would like to express my gratitude to Dutch Space B.V for providing the facilities to work
on my research at your company. I am thankful for the A-ADS team who welcomed me into
the team and were always willing to provide me with information about the A-ADS project.
The company of the interns that were part of the ’Skunk Works’ group at Dutch Space has
been much appreciated.
Finally I would like to thank my friends and family for their support and encouragement
throughout this process.
Delft, University of Technology
May 30, 2014
Master of Science Thesis
M. Reuben Serphos
M. Reuben Serphos
xiv
M. Reuben Serphos
Acknowledgments
Master of Science Thesis
Chapter 1
Introduction
Currently Additive Manufacturing (AM) is making a growth spurt in terms of technological
developments and its popularity is increasing. Due to the alternative manufacturing method,
it introduces new possibilities and allows a new approach to mechanical design.
At Dutch Space B.V. the question has been posed as to whether this new manufacturing
technology can be benefited from. It is then the goal to apply this such that new, innovative,
superior metal products can be produced that have performance comparable to or exceeding
that of the existing products and can then compete with products made of Carbon Fiber
Reinforced Polymer (CFRP) in terms of performance and cost.
To investigate this an existing project had been selected for which one of the parts was to
be designed with AM in mind. The project selected for this was the so-called Antenna Articulated Deployment System (A-ADS).
1-1
The A-ADS
The A-ADS is a system designed to deploy large antennae dishes far away from the spacecraft
(S/C) body. The system is to be stowed in a compact volume, once in orbit it is deployed
by a deployment spring mechanism which locks into place upon full deployment. The system
must then be pointed with considerable accuracy and errors due to environmental influences
must be minimized (e.g. Thermo-elastic deformation (TED)).
The system consists of tubular CFRP booms which are connected by hinge-brackets that
house the spring- and locking mechanism and the synchro-system (a cable system that ensures
synchronous deployment of the booms). There will be several hinge-brackets on such a system.
Figure 1-1 shows a schematic representation of the A-ADS. In this figure the connection on
the far left (u-turn consisting of two 90◦ hinge-brackets) had originally been selected to serve
as a test-case for application of AM. The goal was to design a hinge-bracket that would meet
all necessary requirements with AM in mind as manufacturing method.
Master of Science Thesis
M. Reuben Serphos
2
Introduction
Figure 1-1: A-ADS
1-2
Benefiting from AM
In the literature survey related to this thesis [3] various methods of benefiting from AM were
mentioned. A summary of the three most relevant methods for industrial applications is
shown in Figure 1-2.
Figure 1-2: Benefiting from AM
Functional customization refers to the customization of designs by varying certain parameters
so that the design can be applied in slightly different situations. For example in the case of
the hinge-bracket, the same design could be used for a bracket under an angle of 45◦ instead
of 90◦ . Then without having to make a new mold, as would be the case for a CFRP part, a
3D CAD design could simply be modified and manufactured.
The application of this does not require any drastic changes to the design or design methodology. It can be implemented in the design and manufacturing process in a straightforward
manner. It is therefore not treated further in this study.
Part consolidation is the integration of multiple functions into one part. By doing this the
number of parts in a product could be reduced resulting in lower assembly costs. By doing
this the complexity of the part is increased, which for conventional methods would result in
higher manufacturing costs, but for AM this is not the case.
There is no structured or systematic way to choose which components should or shouldn’t be
integrated.
M. Reuben Serphos
Master of Science Thesis
1-3 Topology optimization
3
The last route to take is increasing the performance of a part by increasing the complexity of
the structure. Examples of this are using internal/cellular structures to create a lightweight
yet stiff structure or the design of complex compliant mechanisms to create materials with
mechanical behavior otherwise not found in monolithic materials.
Topology optimization is a method that accommodates this approach particularly well. It
exploits the design freedom available to find structures that are optimized for certain behavior (for example a maximum stiffness). Topology optimization provides a systematic way of
finding these designs.
In this study attention is focused on this last approach. It provides a measurable and structured means of design which is increasing its popularity in industry. It has been proven to be
a promising tool for design and is a natural counterpart of AM.
Various AM methods exist each with their own advantages and disadvantages. A key difference between the various AM methods is the type of materials they can handle and how the
material properties compare after manufacture to that of conventional manufacturing techniques. In [3] it was found that for metallic products requiring high mechanical performance
the best method is a powder-bed fusion method. A powder-bed fusion method selectively
melts or sinters powdered material together in a powder-bed (see Figure 1-3). Dutch Space
has access to the powder-bed method Selective Laser Melting (SLM) which fully melts the
powdered metal by laser layer-by-layer to form a product. This method has been shown to
produce metal parts with the best mechanical properties compared to other AM methods
and is therefore the most suitable for the manufacture of end-use parts in industry. For this
reason SLM is the method of choice for this research.
Figure 1-3: Schematic representation of the SLM process; Source: [1]
1-3
Topology optimization
Topology optimization is a computational method that explores various geometrical material
lay-outs in a given design space until an optimal value for a given objective and constraints has
Master of Science Thesis
M. Reuben Serphos
4
Introduction
been found. This method is free of any geometrical restrictions (apart from the discretization
resolution) unless explicitly imposed.
Two main approaches exist: the boundary evolution approach and the material distribution
approach. The first takes some initial design as starting point and evolves it by varying the
boundaries that describe it. In this case the parameters that describe the curve of the design
boundary are the design variables. To be able to introduce holes special mechanisms must
be built into the algorithm. Another boundary method that does not have this problem is
the level set method. This method defines a so-called level set function which can be manipulated, but the result is only a level set (i.e. isoline) of the function, this way the boundary
is manipulated but it is done implicitly.
The material distribution approach on the other hand starts by discretizing the design space
and then varying the amount of material in each discretized sub-domain. In this case the
variables are the amount of material in each sub-domain. In this approach the problem may
be discrete or continuous in the sense that the variable may be limited to no material or maximum material (discrete) or intermediate densities are permitted (continuous). The latter
is more common due to the awkwardness of discrete problem solving in numerical methods.
Within this category are various methods of topology optimization e.g. ground structure
method, varying sheet thickness, homogenization. The most common method used currently
is the so-called Solid Isotropic Material Penalization (SIMP) or power-law approach. In this
method the elements of the discretized design space are assigned a density which can be varied. The material properties being evaluated (in the case of compliance this would be the
modulus of elasticity) are assumed to be proportional to this density.
In practice lay-outs with either full material density or void values are preferred since intermediate densities are difficult to manufacture. For this reason a penalization factor is introduced
by raising the density to the power of this penalization factor. By doing this intermediate
densities appear less efficient to the optimizer and are thus discouraged. Every iteration the
topology is evaluated by a Finite Element Analysis (FEA). The optimization problem can
then be solved by applying an optimization algorithm like the method of moving asymptotes,
sequential quadratic programming or an optimality criterion update. Compliance minimization is one of the most common optimization problems solved with topology optimization
applied for weight saving in various industries. A way to formulate this problem (for the
SIMP method) is as follows:
min
ρ
N
X
c(ρ) = UT KU =
(ρe )p ueT ke ue
e=1
subject to KU = F
V (ρ)
=f
V0
0 < ρmin ≤ ρ ≤ 1
(1-1)
Where:
c =Compliance of the material lay-out
U =Global displacement vector
F =Force vector
K =Global stiffness matrix
ue =Element displacement vector
M. Reuben Serphos
Master of Science Thesis
1-3 Topology optimization
5
ke =Element stiffness matrix
ρ =Vector of design variables
ρmin =Vector of minimum relative densities
N =Number of elements used to discretize design domain
p =Penalization power
V (ρ) =Material volume
V0 =Design domain volume
f =Prescribed volume fraction
Loads to which the design will be subjected can be implemented in the force vector while
displacement boundary conditions can be incorporated in the displacement vector. The objective function is subject to the constraints listed below it. The first constraint describes the
equality condition that the material volume is the prescribed fraction (f ) of the total design
domain volume, i.e. the amount of volume to be distributed is predetermined. The second
constraint is the mechanical equilibrium condition and the final line bounds the density such
that it cannot become zero to avoid singularities and limits it to one, since this is of course
maximum density.
The design space is discretized into a finite number of elements, which is referred to as the
finite element mesh or mesh for short. Varying the number of these elements may have large
effects on the obtained result and computation time; mesh-dependency. Mesh-dependence
will generally result in increasingly finer structures, microstructures for example, as the mesh
is refined. This is due to the fact that in general increasing the perimeter in a design for
a fixed volume increases the efficiency (in terms of stiffness) of the material [4]. There are
various techniques to limit this, for example [4] name the following techniques:
• Perimeter control, where a constraint is added to the problem limiting the total length
of the perimeter of the design.
• Gradient methods, where for example the local density variation is constrained.
• Spatially filtering the density, which limits the density variations in the set of the admissible stiffness tensor, this in turn results in that the stiffness in a point depends on
the density of all points in the neighborhood of that point.
• Spatially filtering of the sensitivities, which modifies the design sensitivity of a specific
element based on the sensitivities in the area around that element.
Such a filter also prevents so-called checkerboarding. Checkerboarding is the occurrence of
checkerboard patterned solutions in the topology. These patterns provide, especially for high
penalization power values, a numerical work-around for intermediate densities. By checkerboarding a group of elements, the equivalent of all elements having half-density is achieved.
An artificial stiffness is the result, since (depending on element type and mesh) the elements
are only connected by their respective corners.
Master of Science Thesis
M. Reuben Serphos
6
M. Reuben Serphos
Introduction
Master of Science Thesis
Chapter 2
Problem and hypothesis
2-1
Problem description
Currently topology optimization is widely being used in the design process for products to
be manufactured with Selective Laser Melting (SLM). Generally the ‘optimal’ design found
is modified based on engineering judgment (using design rules and experience with the manufacturing process) such that it can be manufactured, i.e. complies with the manufacturing
constraints of SLM. Such modifications are for example orienting the product such that
build time is minimized while surface roughness (which is dependent on the orientation of
the surface) is acceptable, building support structures to prevent material from sagging (due
to melt pool properties, see Figure 2-3) or curling (due to residual stresses, see Figure 2-2)
and ensuring that minimum achievable dimensions are not exceeded. Such factors as surface
roughness and number of supports can have a large influence on the final cost of the product
since these factors increase post-processing.
It has been suggested by [2] that to minimize post-processing designs should be self-supporting
structures.
Some design rules for SLM are suggested by [2], a selection is presented here:
1. Minimum member size;/minimum wall thickness
2. Minimum slot size
3. Maximum member size
4. Restrict closed surfaces (to allow powder removal)
5. Overhang ≥ 45 degrees
6. Cost minimization by encouraging low build height
7. Cost minimization by minimizing the amount of support structures needed
Master of Science Thesis
M. Reuben Serphos
8
Problem and hypothesis
The choice of the selection of rules above has been based on the observation that these
rules form the basis for the other (non-selected) rules, e.g. allowable chamfer angle, adapted
radii and holes. The maximum member size (3) does not originate directly from the design
rules/criteria, but is rather based on a statement made in [5], namely that limiting the
maximum member thickness could lead to an increase in the amount of members which would
then result in less overhanging structures. Possibly, depending on the chosen parameters, this
could also lead to the formation of cellular (lattice) structures, which might prove useful for
lightweight structures.
As mentioned before (and can be found in [3]) structural elements where sagging or curling
may occur require support structures. Such structural elements are referred to as overhang
or an overhanging surface and is defined in this text as follows:
Overhang
When a surface is facing down and parallel to or inclined at an angle with respect
to the base-plate.
Figure 2-1: Schematic of overhanging surface
Up to an angle of 45◦ additional support-structures are needed. Thus in the design process overhanging structures inclined at an angle lower than 45◦ should either be avoided or
support-structures will be needed. Alternatively support-structures may be incorporated in
the design such that these do not need to be removed afterwards. Additionally the design
could be oriented differently in the build chamber to alter the relative angles.
It is rather difficult, when dealing with complex designs, to judge whether the design should
be modified (maybe resulting in lower performance) to be self-supporting, re-oriented, to introduce support-structures or some combination of this. The ability to do this well is limited
by the creativity and experience of the engineer designing the product. There exists no guide
as to when to choose one option or the other, while this restriction can have an extremely
large influence on the topology of the design.
It is therefore the goal of this study to incorporate this overhang restriction into the topology optimization, thereby eliminating the variable of engineering experience and creativity
offering a structured approach for the design for additive manufacturing (DfAM). By doing
this the design process is also further streamlined since the step of design modification for
Additive Manufacturing (AM) is eliminated.
There is even reason to believe that incorporating the restriction into the topology optimization may yield a design with superior performance to that found when not including the
restriction. This behavior was seen to occur in [6] where the incorporation of a draw direction for casting resulted in a stiffer structure than the unrestricted optimization. Including
a restriction may force the optimizer out of one local minimum into another or maybe even
into a global minimum.
M. Reuben Serphos
Master of Science Thesis
2-2 A-ADS hinge-bracket
9
Figure 2-2: Various levels of curl occurring as a result of residual stress in increasingly overhanging
material; Source: [2]
Figure 2-3: Sagging of material in vertically built holes; Source: [2]
2-2
A-ADS hinge-bracket
A simplified topology optimization problem was set-up to generate a concept for the Antenna
- Articulated Deployment System (A-ADS) hinge-bracket. The objective was to minimize
compliance. A description of the development of the problem can be found in Appendix A.
The results are shown in Figure 2-4. In this figure only elements with a density of 0.9 or
higher are shown to improve visibility. The wire frame box around the part indicates the
design domain, the right image shows a cross-section of the part.
Figure 2-4: Concept design produced by topology optimization
This result must only be viewed as a conceptual suggestion for a design and by far does not
represent a final design.
Master of Science Thesis
M. Reuben Serphos
10
Problem and hypothesis
A tubular structure can be recognized connecting the loaded area with the section of Carbon
Fiber Reinforced Polymer (CFRP) boom. Some additional internal structure is seen as well
as external reinforcements.
Even though the resulting design is not overly complex it requires some thought as to how it
would need to be oriented in the build chamber and whether modifications are necessary for
printing.
Since both circular ends of the hinge-bracket must interface with other parts of the A-ADS
system sufficient geometrical precision must be guaranteed in these areas. For the best results
the build direction would need to be along the axis of the ring. Due to the inherent 90◦ angle
of the part this could not be possible for both ends of the hinge-bracket. A compromise would
therefore need to be made. Orienting the part such that the axes of the circular ends are at
an angle of 45◦ with respect to the base plate would be a possible solution for this, however
by doing this the external reinforcements and the upper section of the tubular wall would
now need support.
On the other hand the design could be modified slightly so that the supports are not necessary,
reducing post-processing. Another alternative could be to include supporting structures which
would not need removing, for example that connect the external reinforcements to the rest of
the structure under such an angle that the result is self-supporting.
The above mentioned alternatives illustrate some of the considerations that need to be made
for a design to be manufacturable by AM. Besides this the issue of cost is introduced by the
possible post-processing needed to remove support structures.
One can imagine that for more complex design conditions these decisions become increasingly
difficult to motivate.
2-3
Summary
The problem The design that follows from the current design process is degraded from
a supposed optimal design (from topology optimization) to a likely sub-optimal design by
modification for manufacturability.
There is no way of knowing if the design has been modified in the best way and modification
is limited by the experience and creativity of the engineer.
Research goal It is the goal of this study to find a way to incorporate the 45◦ overhang
restriction into the topology optimization to provide a more structured approach for the design
of products for AM that is not limited by the experience and creativity of the engineer.
Hypothesis The overhang restriction can be incorporated into the topology optimization,
which can then be used to find designs optimized specifically for the AM method SLM possibly
even yielding designs with better performance than an unrestricted optimization.
M. Reuben Serphos
Master of Science Thesis
Chapter 3
Method
A variety of commercial topology optimization software is available on the market that can
assist engineers in concept generation of designs. It would be preferable to incorporate the
overhang restriction directly into one of these commercial software packages presenting an
industry-ready solution for the inclusion of the overhang restriction. To do this the commercial software must allow enough freedom to manipulate the software to achieve this goal. In
this section the capabilities of some commercial software packages are discussed and their
usefulness for this research is assessed.
It is found that the commercial software does not meet the needs of this research. It is not
the goal of this study to re-invent a topology optimizer and thus use is made of a publicly
available topology optimizer written in MATLAB code. The basic idea behind the code and
the key aspects of it are presented. This code is in 2D and is not sufficient to model the 3D
loads of the industrial problem from which this study emerged. This research is thus limited
to testing the concept of implementation of the overhang restriction.
Finally a section is devoted to the general framework of the research.
3-1
Current status of commercial software
Most commercial software allows the use of multiple objectives and is able to define various constraints, but the extent to which this is possible must be researched. Besides this
it must be checked whether the interface of the available software will allow custom defined
constraints/objectives and if enough freedom and access to the underlying code is available
to define these constraints/objectives.
Two software packages have been reviewed and the suitability for the implementation of the
optimization problem at hand has been determined.
The software packages that have been reviewed are Hyperworks OptiStruct and Simulia
ATOM. This software is based on the material density method but combine it with a boundary method by extracting a smoothed topology from the obtained result. After this a sizing
Master of Science Thesis
M. Reuben Serphos
12
Method
optimization is done for the final result. The material density method is the most common
method used for commercial software.
3-1-1
Hyperworks Optistruct
Hyperworks Optistruct offers, besides the material density method, also optimization by the
level-set method as a Beta functionality. This aspect has not been reviewed since it is also
not yet developed enough for standard use.
Hyperworks Optistruct allows the user to select from various response types which can be set
as either objective or constraint. Multiple responses can be chosen for both. The screenshot
in Figure 3-1 shows the response types available.
Figure 3-1: Response types Hyperworks Optistruct
Even though Hyperworks offers the possibility to perform custom mathematical operations
and analyses on various data, the input is still restricted to the predetermined responses. Custom responses cannot be designed nor is the optimization process accessible for modification.
The only option that remains is using the existing responses and constraints to formulate a
work-around that acts as the overhang restriction. Hyperworks Optistruct comes with the
possibility to apply the manufacturing constraints shown in Figure 3-2.
Figure 3-2: Manufacturing constraints Hyperworks Optistruct
Member size, stress, fatigue, draw direction, extrusion, symmetry and patterning
Though these options allow a lot of versatility in doing topology optimization, they do not
extend to the needs of this research and therefore this software has not been selected for use.
3-1-2
Simulia ATOM
Simulia ATOM works in a similar fashion as Hyperworks Optistruct. Predefined response
types can be chosen for the objective function and constraints (Figure 3-3) and some manufacturing constraints are available (Figure 3-4)
M. Reuben Serphos
Master of Science Thesis
3-1 Current status of commercial software
13
Figure 3-3: Response types Simulia ATOM
Figure 3-4: Manufacturing constraints Simulia ATOM
Member size, draw direction, symmetry
Simulia ATOM also offers the possibility to use scripting with Python (an open source programming language), which seems promising at first glance. However, this option is meant
to incorporate the optimization into a larger process flow and does not grant any access to
the optimization code itself.
Simulia ATOM does not offer any advantage over Hyperworks Optistruct for this research
and is therefore also not used. It is concluded that commercial software cannot be used for
this research and use must be made of tailored code. This will allow an unlimited amount of
accessibility and freedom only limited by the programming language itself.
Master of Science Thesis
M. Reuben Serphos
14
3-2
Method
Description 88-line MATLAB code
To test the concepts developed in this research the formulations of the overhang restriction
will be implemented into a publicly available 88-line MATLAB code provided by [7]. This
code allows a relatively quick check of the results and makes understanding of the occurring
phenomena relatively easy.
In this code the optimization problem solved is a 2D problem that discretizes the design
space into square elements forming a structured grid-mesh, see Figure 3-5. New density
values are found using the Method of Moving Asymptotes (MMA). MMA is a gradient-based
optimization method, which means first-order derivatives of the objective and constraints
are used to find an optimum. The derivative of the objective function with respect to the
optimization variable is referred to as the sensitivity. The algorithm is coded to terminate
when the design change is smaller than or equal to 0.01. The design change is given by the
element with the maximum change in density.
This code provides a clear overview of the structure of the optimization, it is efficient and
easy to modify. Visual feedback is provided after every iteration, which gives insight into
the way the topology evolves. This code allows quick testing of various approaches and
most importantly it is easily available. In it’s current form this 88-line code is set-up for
compliance minimization, which is all that is needed for the simplified problem. The code
offers the possibility to choose between two mesh-independency filters; sensitivity-based and
density-based. Both filters are controlled by a parameter called the filter radius, which denotes
the radius, with respect to the center of an element, over which material is smoothed-out.
Use will be made of the density based filter because this is mathematically more consistent.
For a more detailed description of the code see [7].
Figure 3-5: Structured grid mesh
3-3
Set-up
It is not the intention to re-evaluate all the aspects of the described code but rather extend
the code by including the discussed restrictions.
Using the 88-line code some reference problems are set-up which will act as a control for the
implementation of the overhang restriction. These reference models and motivation of choice
are explained further in the next section. Three main approaches are presented for the implementation of the 45◦ overhang restriction. Before implementation the effect of the approaches
is validated by applying them to some standard geometry in 2D. The required sensitivities
are validated using the finite difference method. The three approaches are implemented in
the 88-line code and optimization is run for the 2D reference models. The results and behavior are analyzed based on the evolution of the topology and objective and constraint values.
M. Reuben Serphos
Master of Science Thesis
3-4 Reference models
15
Based on this a conclusion can be made about the suitability of each approach for further
development.
3-4
Reference models
A number of reference models have been set-up to assess the effect of the overhang restriction.
In this section these reference models will be introduced and further explained as will the
reason for their respective choice.
For the 2D models ran with the 88-line code an elasticity modulus of one is used, for these
models the interest is purely conceptual and this value is not of great importance since
(in linear analysis) this only has a scaling effect on the compliance. For the same reason
applied loads are unit loads. By using these unit values the objective value obtained from
the optimization is dimensionless. It does not give the actual compliance of the structure but
rather a measure of compliance which can be minimized and that allows comparison between
different designs. The Poisson ratio used is 0.3.
All reference models were produced with the following parameters:
Mesh:
Penalization power:
Volume fraction:
Filter used:
Filter radius:
3-4-1
30x50 elements
3
≤ 0.5
Density filter
1.5
Cantilever beams
A classic example used for the illustration of compliance minimization with topology optimization is the so-called cantilever beam. For a 2D rectangular design space the boundary
condition imposed is that one side of the design space is fully clamped while on the opposite
side a force is applied parallel to the clamped side. The result for this particular problem has
been well documented and is therefore easily verified.
Figure 3-6 (left) shows the rectangular design space, clamped on the left side (all degrees of
freedom of the nodes here are set to zero), while the red arrow shows the applied load on
a single node. In this case the load is applied at mid-height. The result is shown in Figure 3-6 (right). The degree of darkness represents the density of each element in the figure,
i.e. white is void, black is fully dense and gray represents intermediate densities. The results
were obtained in 121 iterations.
Master of Science Thesis
M. Reuben Serphos
16
Method
Figure 3-6: (Left) Load case cantilever beam mid-load and (right) the result with:
Iterations: 121; Objective value: 57.17; Volume fraction: 0.418
For the same reasons as for the mid-loaded cantilever beam another version of the cantilever
beam is used where the load is applied in the bottom right corner. The schematic representation of the load case and the result can be seen in Figure 3-7. The result was obtained in
191 iterations.
Figure 3-7: (Left) Load case cantilever beam corner-load and (right) the result with:
Iterations: 191; Objective value: 64.00; Volume fraction: 0.426
3-4-2
Tension beam
For a final reference model the load is applied, once again, on the opposite side of the clamped
condition at mid-height, except now the force is directed perpendicular to the face. This
problem is referred to in this text as the tension beam. The load case and result obtained in
1160 iterations are shown in Figure 3-8.
This model has been chosen for two reasons. The first being that the result is easy to predict
when no additional restrictions are placed on it. The second is that the solution exists almost
exclusively out of overhang which makes for an interesting case once the overhang restriction
M. Reuben Serphos
Master of Science Thesis
3-4 Reference models
17
is applied. This problem is presumed to exploit any weaknesses of the implementation of the
restriction in the simplest way.
Figure 3-8: (Left) Load case tension beam and (right) the result with:
Iterations: 1160; Objective value: 5.49; Volume fraction: 0.474
Master of Science Thesis
M. Reuben Serphos
18
M. Reuben Serphos
Method
Master of Science Thesis
Chapter 4
Formulation of the overhang
restriction
In order to implement a restriction on the geometry of the output of the topology optimization, a method must be found to identify the geometrical property. In this case the geometrical
property in question is the presence of supporting material under an angle of at least 45 degrees. The geometrical properties of every (intermediate) solution of the topology optimizer
can be addressed in two ways, that is, by assessing the discrete nature of the model or the
continuous nature. The discrete nature of the model refers to the finite element discretization
of the design space and the continuous nature of the model refers to the density field produced
by the various density values of each element. To find out whether material is supported or
not a transition from high density material to low density material must be identified, this
indicates a border, the angle of this border will have to meet the imposed requirement for the
restriction to be implemented successfully.
4-1
Using the continuous nature of TO
Taking a look at the continuous nature of the model it is straightforward that the gradient
of the density can be used to find these transition areas, i.e. a high gradient indicates a
sharp transition from high to low density. The direction of this gradient vector could then be
compared to some reference vector to quantify the angle under which supporting material is
present/missing. A restriction could be imposed directly on this angle to control the material
distribution such that material is always properly supported. For this approach a value must
be defined for which the gradient is deemed to represent a border. This is necessary since
the topology optimization (using the SIMP method) always encounters intermediate densities
during the optimization process. Figure 4-1 demonstrates this in a section of a result of a 2D
topology optimization using the 88-line MATLAB code. In the middle of the image moving
toward the top right corner (A) it can be seen that a member can already be identified even
Master of Science Thesis
M. Reuben Serphos
20
Formulation of the overhang restriction
though there is no direct black to white transition, however it becomes increasingly blurred.
The bottom right corner (B) also shows such a situation. The bottom left corner (C) shows a
situation where a clear distinction is present. A value must thus be defined where the gradient
can be regarded as a border and where this is not applicable.
Figure 4-1: Gradient of density field
By choosing a high value for the gradient it is ensured that only true borders are identified,
however a consequence of this is that the optimizer may then avoid these gradients in cases
where it is very beneficial to have a member under an unallowable angle by using densities
just below the specified gradient. Since intermediate densities are undesired this is potentially
a big drawback of this method. In other words this restriction competes with the penalization
of element densities (introduced to make ‘black and white’ solutions more attractive for the
optimizer). A potential solution could be to implement a continuation method, i.e. define
the restriction in such a way that it is not active in the initial part of the optimization and
as the process progresses so does the restriction become more active. By doing this members
will already be partially formed when the restriction becomes active.
Another difficulty that arises in the determining of the boundaries, even in cases where clear
black-white structures are generated, is that due to the discrete nature of the mesh the
gradient along a boundary is not continuous. Taking the grid mesh as an example, see
Figure 4-2, it can be seen that along an inclined boundary a staircase pattern is present,
this gives gradients in completely different directions making it difficult to identify the angle
of the boundary locally. A solution for this may be found by applying an interpolation
over the elements from which the gradient is then determined. This introduces the need
for interpretation of the topology, thus introducing an additional step in the optimization
process. This step must allow mathematical differentiation in order to determine sensitivities
for proper use of the Method of Moving Asymptotes (MMA) optimizer. This additional step
is expected to make the new optimization problem overly complex.
4-2
Using the discrete nature of TO
From the point of view of the discrete nature of the optimization process it can be deduced that
for an element to be supported in an allowable fashion there must be an adjacent element that
is (at least) equally dense and located within the allowable range of a supporting structure.
Figure 4-3 shows a section of a grid mesh, the filled element represents an element with some
density. The hatched region in the image indicates where support material could be placed
to be able to place material in the element in question. For this particular mesh it can be
M. Reuben Serphos
Master of Science Thesis
4-2 Using the discrete nature of TO
21
Figure 4-2: Local gradients in black and white topology represented by blue arrows
seen that there are three candidate elements of which at least one would have to have at least
the same density of the to-be-supported element. The overhang restriction can be enforced
by imposing a relation between the density of elements that fall within the hatched region
and the element in question. From the image it can be seen that for the grid type mesh and
a 45 degree overhang restriction, there are three main elements (outlined) that can serve as
supports. Of these three elements the one with the highest density will determine whether
enough support is provided. Hence a maximum value must be determined. To be able to
Figure 4-3: Discontinuous aspect of topology optimization showing candidate support elements
make use of this property in the optimization algorithm (in order to determine sensitivities)
a mathematical differentiable and continuous description must be formulated to find the
maximum value. A way to specify this relation mathematically is to define the P-norm as a
function of the densities in the selected elements Eq. (4-1). Taking a finite value of pn , the
output of the P-norm will approximate the highest density value of the selected elements.
A condition can then be specified for this value. Imposing this condition on each element
should then result in a design that complies with the design rule. The P-norm for k elements
numbered from 1 to k is defined as follows:
ρmax = lim
pn →∞
v
u k
uX
pt
n
ρpn
(4-1)
a
a=1
Where:
ρmax = the maximum value of the included density values
pn = a parameter that influences how well the true maximum value is approximated
a = the element number for a = 1...k
k = the number of elements being evaluated
ρa = the density value of element a
Master of Science Thesis
M. Reuben Serphos
22
Formulation of the overhang restriction
An example of what such an approximation looks like can be seen in Figure 4-4. In this
figure two functions have been plotted, one for an inverted parabola (function 1) and one for
a parabola (function 2). The P-norm is then plotted in cyan with a (high) value for pn of 50.
Additionally, as a reference, the actual maximum value of each point is plotted (function 3)
using a ‘max’ command.
Figure 4-4: Example of P-norm behavior
A drawback of this method is that the selection of elements is highly dependent on the meshtype used and will thereby influence the structure being created, though this issue might be
negligible for elements of small enough size. Furthermore, if this approach is applied to a
grid mesh it might promote checkerboarding, i.e. a checkerboard member/structure (of one
element thickness) is a sufficient means of support. As an initial indication this is fine, but
for a manufacturable and numerically accurate model this would need to be modified by for
example a density filter or some other means that imposes a minimum member thickness.
It is presumed that the density filter will suffice. This approach is expected to be the most
straightforward for implementation and have the best chance of successful implementation,
therefore this has been selected as the approach to take for this study.
4-3
Pragmatic approach
As a very first attempt to generate a result that complies with the overhang restriction a
pragmatic approach has been examined.
Using the 88-line MATLAB code an extra section of code is added after every optimization
iteration.
Each element is evaluated sequentially. Per element the maximum density of the candidate
support elements is determined (for this approach the ‘max’ function in MATLAB is used and
M. Reuben Serphos
Master of Science Thesis
4-3 Pragmatic approach
23
not the P-norm). If this density is equal to or higher than that of the element being evaluated
then no action is undertaken. If the opposite is the case, the density of the candidate support
element is set equal to that of the element being evaluated. The process in pseudocode is as
follows:
1.
2.
3.
4.
Determine candidate support elements for element ‘i’ with density ρi
Determine maximum density value of candidate support elements ρmax
If ρmax ≥ ρi then do nothing
If ρmax < ρi then replace ρmax with ρi
The algorithm has been executed for the corner-loaded cantilever beam. The result is shown
in Figure 4-5.
Figure 4-5: (Left) Reference model corner-loaded cantilever beam and (right) result pragmatic
approach for: mesh: 30x50; volume fraction: 0.5; filter penalization power: 3; radius: 1.5;
Obtained objective after 49 iterations: 67.57; volume fraction: 0.421
In this algorithm the volume fraction constraint is not taken into account. Furthermore
the optimizer does not have any way of taking the new behavior of the model into account
when generating new values. Despite this, the result in Figure 4-5 shows an altered design
with structures that support overhang that can reach an objective value near that of the
reference model and only a slight difference in the occupied volume fraction. These results
look promising, but when the algorithm is executed for the same problem with a finer mesh a
very different result emerges. This approach is extremely mesh-dependent and uncontrollable
in terms of optimization.
The supports in Figure 4-6 look quite illogical in certain places (for example on the far left).
The stiffness of the support is not taken into account when generating the structure, so the
supports do not necessarily improve the stiffness of the structure. Material is simply placed
wherever an element is overhanging, the candidate with the highest density is taken as the bias
as to where to add this material. Especially in void areas, where the values of the elements
are near zero and the densities might even be equal for all candidate support elements, this
bias becomes meaningless and the generated support structures look arbitrary in shape.
Master of Science Thesis
M. Reuben Serphos
24
Formulation of the overhang restriction
Figure 4-6: (Left) Reference model corner-loaded cantilever beam for: mesh: 60x100; imposed
volume fraction: ≤ 0.5; filter penalization power: 3; radius: 1.5;
Obtained objective after 270 iterations: 62.69; volume fraction: 0.404
and (right) result pragmatic approach with:
Obtained objective after 38 iterations: 67.23; volume fraction: 0.422
4-4
Restriction method
Once a formulation for the quantification of overhang is defined a next step can be made,
that is a method for imposing the restriction on the optimization problem can be formulated.
Three main approaches have been identified for imposing the overhang restriction: add a
constraint to the optimization problem, include an additional term in the objective function
that accounts for the restriction (penalty function) or apply a filter over the design that adapts
the design so that it complies with the desired restriction.
4-5
Local constraints
With the help of Eq. (4-1) a constraint can be defined for overhanging elements. That is, for
each element to be supported the difference between the density of the element under inspection and the candidate support element with the highest density must be zero or negative.
In other words the supporting element has a density equal or higher than that of the element
under inspection. With this the optimization problem, Eq. (1-1), can now be rewritten as:
min
ρ
N
X
c(ρ) = UT KU =
(ρe )p ueT ke ue
e=1
subject to KU = F
V (ρ)
=f
V0
ρ∆ ≤ 0
(4-2)
0 < ρmin ≤ ρ ≤ 1
with
ρ∆ = ρi − ρmax
M. Reuben Serphos
(4-3)
Master of Science Thesis
4-6 Multiple objective and global constraint
25
which is the difference, as described earlier, for each element in the design space.
In this description a constraint is assigned to each element in the design space. It is in
general desirable to have as little constraints as possible due to the computational cost. A
large number of constraints also makes it more difficult for the optimizer to find a solution.
By imposing the restriction in this manner as many constraints as elements in the design
space are added. In general this is a lot (too many for a well-posed optimization problem).
Therefore an alternative approach should be considered.
4-6
4-6-1
Multiple objective and global constraint
Multiple Objective
The overhang can also be viewed as a value which is to be minimized along with the compliance
in the objective function of the optimization problem. The mathematical formulation of this,
i.e. a multi-objective optimization problem, takes the following form:
min
ρ
UT KU + Ωf ac Ωtot =
N
X
(xe )p uTe ke ue + Ωf ac Ωtot
e=1
subject to KU = F
V (ρ)
=f
V0
0 < ρmin ≤ ρ ≤ 1
(4-4)
Where:
Ωtot = A function that defines a value for the level of overhang
Ωf ac = A weight factor for the additional overhang term
For the function that represents the total overhang of the geometry (Ωtot ) it can be said,
based on previous arguments, that the level overhang can be given by the difference between
the density of the element being evaluated (ρi ) and the maximum value of the candidate
supporting elements (ρmax ). This is given by (ρ∆ ). Setting the overhang of an element, Ω,
equal to ρ∆ , however, will result in negative values when ρmax is greater than ρi . This would
mean that the total objective value would be improved (lowered in value) for these cases,
which does not represent reality. In fact the level of overhang for these cases should simply be
zero. In Figure 4-7 a 3D plot can be seen of ρ∆ for values of ρmax and ρi ranging from zero to
one. Furthermore, overhang is present for any positive value of ρ∆ and should thus return a
positive value for these cases. Once again examining Figure 4-7 it is seen that for cases where
ρmax is kept constant (e.g. ρmax = 0) and ρi is increased, ρ∆ increases linearly. This would
imply that overhang is more acceptable when ρi has intermediate densities, which is not the
case. This would promote intermediate densities because for these values of ρi overhang would
be perceived as more acceptable. This is not desired. The function for Ωtot should thus not
be linear, rather it should give a zero value for any negative ρ∆ and a maximum value for any
positive ρ∆ . Taking the maximum value as one, overhang can then be defined as follows:
Master of Science Thesis
M. Reuben Serphos
26
Formulation of the overhang restriction
For an element that is not properly supported, i.e. candidate support elements
have a value lower than that of the element under inspection, the overhang has a
value of one and when the element is supported it has a value zero.
(
Ω=
1
0
if
if
ρ∆ > 0
ρ∆ ≤ 0
(4-5)
Figure 4-7: Value ρ∆ with respect to ρmax and ρi
This response could be achieved by multiplying ρ∆ with a step-function. This approach,
however, does not yield a differentiable function. To achieve the desired response for Ωtot use
is made of a differentiable approximation of a step-function, the so-called logistic function
(see Figure 4-8):
f=
1
1 + e−M ·x
(4-6)
Where:
M = A parameter that controls the steepness of the curve (i.e. higher values result in a
steeper curve)
x = The independent variable in the function
Replacing the independent variable (x) in Eq. (4-6) by ρ∆ the desired response is obtained.
It can be seen in Figure 4-8 that for x = 0 the function value is 0.5 which is not in agreement
with Eq. (4-5), and that the function is still positive for negative values of x. The error of the
function for x < 0 can be minimized by setting the parameter M higher, however extremely
high values of M will result in numerical difficulties due to high gradients in the curve. The
error at x = 0 will, however, always remain at 0.5 for finite values of M . This error can be
reduced by shifting the entire curve to the right, but this comes at a cost. By doing this the
value of ρ∆ for which overhang is registered is increased. An extra parameter, δ, is introduced
to be able to tweak this property by replacing x = x − δ, see Figure 4-9.
Up till now the function is only applicable to each individual element. A global measure for
the amount of overhang in a certain geometry can be formulated using this function, referred
to as the overhang coefficient (Ωtot ). It is defined by the sum of the overhang of all elements
normalized by the number of elements, N , in the design domain.
M. Reuben Serphos
Master of Science Thesis
4-6 Multiple objective and global constraint
27
Figure 4-8: Plot of the logistic function with x ranging from -1 to 1
Figure 4-9: Overhang with M = 10 and δ = 0, 5
Ωtot =
N
X
Ωi
i=1
now with:
Ωi =
1
N
(4-7)
1
1+
e−M (ρ∆ −δ)
(4-8)
Alternatively this can be formulated as:
Ωtot =
N
X
Ω∗i
(4-9)
i=1
with:
Ω∗i =
1
1+
e−M (ρ∆ −δ)
1
N
(4-10)
Where Ωi and Ω∗i are the overhang and normalized overhang of an individual element i respectively.
Now a global measure of overhang for any geometry is available with a value between zero
and one which can now be minimized.
Master of Science Thesis
M. Reuben Serphos
28
4-6-2
Formulation of the overhang restriction
Multiple objective function sensitivities
Since the new objective function is the sum of the original (compliance) term and the new
df
overhang term (Ωf ac Ωtot ) the total sensitivities ( dρ
) will also be a sum of the derivatives of
each term:
∂c
∂Ωtot
df
=
+ Ωf ac
dρ
∂ρ
∂ρ
(4-11)
Figure 4-10: Influenced elements for sensitivities
∂c
The derivatives of the original term, ∂ρ
, remain unchanged. An array is generated with the
derivative of Ωtot with respect to each element and ordered accordingly. To calculate the
sensitivity of Ωtot with respect to an arbitrary element, ρi , the effect on the overhang of
surrounding elements should also be taken into account. Figure 4-10 shows an element ρi
(black) and its surrounding elements. Varying the density of ρi will affect the overhang of
the element itself and elements ρ1 , ρ2 and ρ3 as is indicated in the figure. The elements ρ1 ,
ρ2 and ρ3 are referred to as the influenced elements. Elements ρA , ρB and ρC denote the
candidate support elements. The overhang of the rest of the elements in the design space will
not be affected by this density change and thus the contributions of these elements to the
tot
total derivative is zero. With this the sensitivity dΩ
dρi becomes:
N
dΩtot
d X
=
Ω∗
dρi
dρi i=1 i
=
∂Ω∗i
∂Ω∗1 ∂Ω∗2 ∂Ω∗3
+
+
+
∂ρi
∂ρi
∂ρi
∂ρi
(4-12)
(4-13)
with
Ω∗i =
1
1+
ρ∆ = ρi −
e−M (ρ∆ −δ)
1
N
(4-14)
q
pn
ρpAn + ρpBn + ρpCn
To evaluate the partial derivatives in Eq. (4-13) the chain rule of differentiation is applied:
M. Reuben Serphos
Master of Science Thesis
4-6 Multiple objective and global constraint
29
dΩ∗i
∂Ω∗i ∂ρ∆
=
dρi
∂ρ∆ ∂ρi
(4-15)
dΩ∗i
∂Ω∗i ∂ρ1∆
=
dρ1
∂ρ1∆ ∂ρ1
(4-16)
dΩ∗i
∂Ω∗i ∂ρ2∆
=
dρ2
∂ρ2∆ ∂ρ2
(4-17)
dΩ∗i
∂Ω∗i ∂ρ3∆
=
dρ3
∂ρ3∆ ∂ρ3
(4-18)
Working this out then gives the following terms:
dΩ∗i
1
=
dρi
N
(−M )e−M (ρ∆ −δ)
−
(1 + e−M (ρ∆ −δ) )2
!

(4-19)
1
p −1)
n
n pn
n
n
) · ρ1C
+ ρp1C
+ ρp1B
dΩ∗i
1  e−M (ρ1∆ −δ) · (ρp1A
=
−
dρ1C
N
(1 + e−M (ρ1∆ −δ) )2

1
p −1)
n
n pn
n
n
) · ρ2B
+ ρp2C
+ ρp2B
dΩ∗i
1  e−M (ρ2∆ −δ) · (ρp2A
=
−
dρ2B
N
(1 + e−M (ρ2∆ −δ) )2
dΩ∗i
dρ3A

=
1 
−
N
e−M (ρ3∆ −δ)
1
n pn
n
+ ρp3B
+ ρp3C
)
e−M (ρ3∆ −δ) )2
n
(ρp3A
·
(1 +
·


(4-20)


(4-21)

pn −1)
ρ3A

(4-22)
Where
ρ1∆ = ρ1 −
q
n
n
n
ρp1A
+ ρp1B
+ ρp1C
(4-23)
q
n
n
n
ρp2A
+ ρp2B
+ ρp2C
(4-24)
q
n
n
n
ρp3A
+ ρp3B
+ ρp3C
(4-25)
pn
ρ2∆ = ρ2 −
pn
ρ3∆ = ρ3 −
pn
ρ1B = ρ2A
(4-26)
ρ2C = ρ3B
(4-27)
ρ1C = ρ2B = ρ3A = ρi
(4-28)
Substituting these terms back into Eq. (4-13) yields the sensitivity of Ωtot with respect to
element ρi .
Master of Science Thesis
M. Reuben Serphos
30
Formulation of the overhang restriction
Once all Ωtot derivatives have been calculated the two arrays of derivatives (compliance and
overhang multiplied by the weight factor) can be summed elementwise to obtain an array
with the total derivatives of each element; the sensitivities.
4-6-3
Global constraint
The formulation derived in the previous section can also directly be used for a global constraint. In this case an equality constraint is introduced that requires that Ωf ac Ωtot = 0.
With this the new optimization problem becomes:
min
ρ
N
X
T
c(ρ) = U KU =
(ρe )p uTe ke ue
e=1
subject to KU = F
V (ρ)
=f
V0
Ωf ac Ωtot = 0
(4-29)
0 < ρmin ≤ ρ ≤ 1
4-7
Filter
An alternative method to impose a geometrical restriction to the design is by applying a
filter. A (density-based) filter modifies the densities of the elements to meet the imposed
requirements after which Finite Element Analysis (FEA) and calculation of the sensitivities
are performed with the new, modified, densities. This is done after the optimizer update.
That is the density matrix ρ becomes ρ̃ by:
ρ̃ = F ∗ ρ
(4-30)
Where F is a convolution matrix representing the filter.
The sensitivities of the elements must then be calculated by the chain rule:
dc
∂c ∂ ρ̃
=
dρ
∂ ρ̃ ∂ρ
(4-31)
Using a filtering approach can be thought of as using a virtual fabrication process where ρ
represents the fabrication plan and ρ̃ the result. For the specific problem at hand, restricting
overhang, two approaches have been identified for the design of the filter. Overhang can be
eliminated by either removing material from an unsupported element or by adding material to
(at least one of) the candidate support elements to achieve a supported element. The latter,
however, may present an ambiguity problem in the case where all candidate support elements
have an equal value. A choice must be made by the filter to which of the elements material
will be added. During construction of the filter there is no information available to motivate
which element should then be selected, thus making this choice ambiguous (this would be a
M. Reuben Serphos
Master of Science Thesis
4-7 Filter
31
lot like the pragmatic approach presented earlier). For this reason the approach that removes
material is chosen to be developed further.
4-7-1
Modification by sequential scanning
For this approach the maximum value of the candidate supporting elements is again checked
using the P-norm. The maximum density value is then compared with the density of the
element being analyzed, if the maximum density is larger than or equal to the analyzed
element (no overhang) no action is undertaken and the density remains unchanged. If the
maximum density of the candidate support elements is lower (overhang is present) the density
of the element under inspection is set equal to that of the maximum density of the candidate
support element, thereby eliminating overhang. The mathematical formulation of this for an
element ρi can be stated as follows:
(
ρi
ρmax
ρ̃i =
if
if
ρi ≤ ρmax
ρi > ρmax
(4-32)
or equivalently:
ρ̃i = min(ρi , ρmax ) = 1 − max(1 − ρi , 1 − ρmax )
(4-33)
Now, once again, the maximum value can be determined by using the P-norm. This yields
the following differentiable equation for the modified densities (ρ̃):
ρ̃i = 1 −
q
pn
(1 − ρi )pn + (1 − ρmax )pn
(4-34)
Applying this sequentially to each element from the bottom to the top of the design space will
eliminate any overhanging elements. Since the bottom row of elements is always supported
these densities do not need to be modified. Therefore the scanning of the elements is started
one row higher.
Filter sensitivities
Recalling Eq. (4-31) the sensitivity with respect to an element ρj is then found to be:
N
X
dc
∂c ∂ ρ̃i
=
dρj
∂ ρ̃i ∂ρj
i=1
(4-35)
Using the chain rule of differentiation and Eq. (4-34) gives:
1
∂ ρ̃i
∂ρmax
∂ρi
−1
= ((1 − ρi )pn + (1 − ρmax )pn ) pn · (1 − ρmax )pn −1
+ (1 − ρi )pn −1
∂ρj
∂ρj
∂ρj
Master of Science Thesis
!
(4-36)
M. Reuben Serphos
32
Formulation of the overhang restriction
with
∂ρmax
∂ρj
!
∂ρmax
∂ρj
!
=
(ρ̃pAn
+
ρ̃pBn
+
1
−1
ρ̃pCn ) pn
i6=j
·
∂ ρ̃A
ρ̃pAn
∂ρj
+
∂ ρ̃B
ρ̃pBn
∂ρj
+
∂ ρ̃C
ρ̃pCn
∂ρj
!
(4-37)
=0
i=j
and
∂ρi
= δij
∂ρj
(4-38)
ρ̃B
ρ̃C
Where δij is the Kronecker delta. In Eq. (4-37) ∂∂ρρ̃Aj , ∂∂ρ
and ∂∂ρ
should have been previously
j
j
determined by calculating the sensitivities of each element from the bottom of the design space
to the top. For the bottom row of elements the sensitivities are:
∂ ρ̃i
∂ρj
!
= δij
(4-39)
Bottom row
These elements have the same value after filtering as before filtering, a change in the unfiltered
density will translate to the same change in the filtered density. Density values of all other
elements have no influence.
M. Reuben Serphos
Master of Science Thesis
Chapter 5
Validation
In order to verify the expressions formulated for the overhang in Chapter 4 some standard
test geometry has been set-up in MATLAB, i.e. a portal, a ring and a gradient. This has
been done by defining an array with dimensions that correspond to that of a desired mesh, the
elements in the array represent material elements that may assume a value between zero and
one. The value represents the density of the element. Initially all values in this array have a
value of zero. A value of one is assigned to selected elements such that when displayed in a
colormap the desired geometry is obtained. To generate a gradient the value of the element
is made dependent on the index values. The resulting (grayscale) colormaps are shown in
Figure 5-1. This method corresponds to the way the problem is set-up in the 88-line code
from [7]. The test geometry used has been created in a 50x50 element mesh.
(a) Portal
(b) Ring
(c) Gradient
Figure 5-1: Standard geometry for verification
5-1
Validation of the multiple objective and global constraint
Using a function set-up in MATLAB to calculate the overhang according to the formulation
described in section 4-6, a matrix is generated with the values of Ω∗ for each element and Ωtot .
Plotting this matrix in a colormap reveals which elements have been registered as overhanging
as can be seen in Figure 5-3.
Master of Science Thesis
M. Reuben Serphos
34
Validation
For reference a function is defined in MATLAB in which no use is made of the formulation
described in Section 4-6. Rather use is made of the ‘max’ command and an if statement to
determine the maximum value of the candidate support elements and register the overhang
of an element (as 0 or 1) respectively. Ω∗ is then N1 for an overhanging element and 0 for a
supported element. A matrix is constructed in which all these values are stored. Summing
Ω∗ of all elements will then give the exact value for Ωtot . Plotting the constructed matrix in
a colormap will then show exactly which elements are overhanging, see Figure 5-2 where the
overhanging elements are depicted in red.
Comparing the results of the exact determination and the formulated approximation of the
overhang and Ωtot , it is seen that for the portal and ring a satisfactory result is obtained.
The gradient, however, is not approximated well at all. In the following section it will be
explained that this is due to the choice of the parameters.
(a) Portal - Exact values of Ω∗
with Ωtot = 0.011
(b) Ring - Exact values of Ω∗
with Ωtot = 0.010
(c) Gradient - Exact values of
Ω∗ with Ωtot = 0.980
Figure 5-2: Exact OH detection in standard geometry
(a) Portal - Approximate values
of Ω∗ with Ωtot = 0.011
(b) Ring - Approximate values
of Ω∗ with Ωtot = 0.011
(c) Gradient - Approximate values of Ω∗ with Ωtot = 0.000
Figure 5-3: Approximate OH detection in standard geometry, with M = 35, pn = 8 and δ = 0.2
5-1-1
Parameters M, δ and pn
Next the effects of the parameters M , pn and δ are investigated. Each parameter is varied
for each of the three test geometries.
Recalling the logistic function used to approximate a step function, the effect of the parameters
on the overhang detection is deduced. The parameter M controls the steepness of the function,
for higher values of M a better approximation of the step function is obtained. This can be
M. Reuben Serphos
Master of Science Thesis
5-1 Validation of the multiple objective and global constraint
35
seen in Figure 5-4. Choosing M too high, however, makes it more difficult for the optimizer
to find a solution due to the abrupt transitions in the curve. For low values of M , it can
be seen in Figure 5-4, that Ωi will take on higher positive values for negative values of ρ∆
while all these values should actually return a Ωi of zero. This will result in an Ωtot that is
higher than it should be. In Figure 5-5 the effect on Ωtot of varying M is plotted for the three
geometries. M is varied from 10 to 100 keeping pn = 8 and δ = 0.2. In these figures it can be
seen that the Ωtot converges at about M = 50. For the portal and the ring the value to which
Ωtot converges is in accordance with the exact values obtained in the previous section. The
gradient geometry converges to zero, which is far from the exact value, this has to do with
the δ parameter and the error introduced by the P-norm approximation. This is discussed in
the following sections.
Returning to the logistic function in Figure 5-4 it is shown that the δ parameter shifts the
center of the curve along the horizontal. This has a similar effect as M in terms of Ωi at
ρ∆ = 0. Increasing δ will lower Ωi , in return the value for which Ωi reaches one is also shifted.
In effect this means that overhang is detected for higher values of ρ∆ . This parameter should
be kept as close as possible to zero and negative values should be avoided. For negative values
of δ, negative values of ρ∆ are identified as overhang, which in general is undesirable. The
lower the value of M the more important it is to increase δ.
For the gradient in Figure 5-1c ρ is increased from 0 to 1 over 50 elements (50x50 mesh),
and thus the difference in ρ between each row is only 0.02. This means that for any δ value
larger than 0.02 overhang will not or barely be detected in this case. Setting δ = 0 results
in a higher Ωtot for the gradient geometry, but the value is still not in correspondence with
the calculated exact value, a plot of Ω∗ for this case is shown in Figure 5-7. Figure 5-6 shows
the effect of varying δ on Ωtot for the three respective test geometry cases. Figure 5-6c shows
that the correct value for Ωtot is found when δ is negative, implying that the detected ρ∆ is
negative. This is due to the error introduced by the P-norm approximation.
Figure 5-4: Logistic function for varying values of M
Master of Science Thesis
M. Reuben Serphos
36
Validation
(a) Portal
(b) Ring
(c) Gradient
Figure 5-5: Ωtot for parameter M from 10-100
M. Reuben Serphos
Master of Science Thesis
5-1 Validation of the multiple objective and global constraint
37
(a) Portal
(b) Ring
(c) Gradient
Figure 5-6: Ωtot for varying parameter δ
Master of Science Thesis
M. Reuben Serphos
38
Validation
Figure 5-7: Ω∗ for gradient geometry with δ = 0 giving Ωtot = 0.223
The error introduced by the P-norm can be shown to increase linearly with the density of the
candidate support elements. Recalling the P-norm and taking ρ to be the actual maximum
value of the candidate support elements, the approximated maximum value ρmax can be
written as:
ρmax =
q
pn
=ρ
ρpn + (αρ)pn + (βρ)pn
p
p
n
1 + αpn + β pn
(5-1)
(5-2)
with:
0≤α≤1
(5-3)
0≤β≤1
(5-4)
The error is then found to be:
p
err = ρmax − ρ = ρ( pn 1 + αpn + β pn − 1)
(5-5)
showing that the error is linearly dependent on ρ. This explains why, in Figure 5-7, a gradient
is seen for the detected overhang. A minimum error is found for α = β = 0 while a maximum
error is found for α = β = 1, which is the case for the gradient.
Figure 5-8 illustrates how the calculated maximum value ρmax deviates from the actual maximum value ρ increasingly for higher values of ρ with α = β = 1.
This error can be minimized by increasing pn since:
lim err = 0
pn →∞
M. Reuben Serphos
(5-6)
Master of Science Thesis
5-1 Validation of the multiple objective and global constraint
39
Figure 5-8: Error due to P-norm
Now it is proven that for the gradient geometry a negative value of ρ∆ is detected as a result
of the P-norm error. With α = β = 1, for an arbitrary element ρi , the actual difference in
density between each row 0.02 and pn = 8 the following is found:
with
√
8
√
8
ρmax = ρ 3
(5-7)
ρi = ρ + 0.02
(5-8)
ρ∆ = ρ + 0.02 − 1.15ρ
(5-9)
3 ≈ 1.15 and
then for ρ = 1:
= −ρ · 0.15 + 0.02 = −0.13
(5-10)
Thus the detected ρ∆ is negative at the top of the gradient geometry. In Figure 5-9 the effect
of an increasing P-norm power on Ωtot is shown for the three test geometries.
Master of Science Thesis
M. Reuben Serphos
40
Validation
(a) Portal - varying pn from 1 to 1000 with δ = 0.2
and M = 35
(b) Ring - varying pn from 1 to 1000 with δ = 0.2
and M = 35
(c) Gradient - varying pn from 1 to 19000 with δ = 0
and M = 35
Figure 5-9: Ωtot for varying P-norm power pn
M. Reuben Serphos
Master of Science Thesis
5-1 Validation of the multiple objective and global constraint
41
As expected the values converge for extremely high pn , though the portal and the ring already
are very close to the limit value for very low pn . This is due to the black-white nature of the
geometry. The gradient needs extremely high values of pn which is due to the small difference
in ρ between each row. It can be calculated that for pn = 8, δ = 0.2 and M = 35 and the
assumption that overhang is registered for Ω = 0.8 the minimum difference in density to be
registered as overhanging is:
1
1+
= 0.8
(5-11)
⇒ ρ∆ ≈ 0.24
(5-12)
e−35(ρ∆ −0.2)
Taking into account the maximum error introduced by the P-norm with ρ = 1 the difference
for which overhang starts to register is:
0.24 + ρ(1 −
√
8
3) = 0.39
(5-13)
This is assumed to be sufficient for further investigations.
Figure 5-10 shows a density plot with the exact overhang and the approximated overhang
calculations. The density plot shows a small section of gradient that increases the density in
the following manner: ρ = 0 > ρ = 0.2 > ρ = 0.6 > ρ = 1 from bottom to top. The transition
from ρ = 0 to ρ = 0.2 is not detected as well as the others since this is below the previously
calculated minimum difference. This accounts for the deviation in the approximate and exact
values of Ωtot , 0.05 and 0.06 respectively.
5-1-2
Sensitivity check
To check the sensitivities of the overhang term in the multiple objective use is made of the
finite difference method. To do this an arbitrary element is selected in the test geometries and
the density of this element is incrementally increased to eventually produce a dot. Figure 5-11
shows this for the ring, with the dot in the center of the ring. For every value of this dot
Ωtot and the sensitivity of Ωtot with respect to the density of the dot is calculated using the
equation for the sensitivity derived in Subsection 4-6-2. These calculated sensitivities are
plotted in Figure 5-12a. Using the Ωtot value calculated for every density of the dot and
the increment of density change (here set to 0.01) the finite difference approximation can be
computed, this result is shown in Figure 5-12b. As can be seen in these plots the calculated
values and the finite difference values agree, indicating that the sensitivities are correct. This
was also seen for the other test geometries.
Master of Science Thesis
M. Reuben Serphos
42
Validation
(a) Geometry with small section of gradient with ρ = 0; ρ = 0.2; ρ = 0.6;
ρ = 1 from bottom to top, all other elements have ρ = 1
(b) Exact Ω∗ with Ωtot = 0.060
(c) Approximate Ω∗ with Ωtot = 0.050
Figure 5-10: Minimum detectable ρ difference
Figure 5-11: Density of arbitrary element (in the center of the ring) is incrementally increased
from 0 to 1
M. Reuben Serphos
Master of Science Thesis
5-1 Validation of the multiple objective and global constraint
43
(a) Calculated sensitivity
(b) Finite difference sensitivity
(c) Ωtot for the density of dot
Figure 5-12: Comparison of the calculated and the finite difference sensitivity
Master of Science Thesis
M. Reuben Serphos
44
5-2
Validation
Verification of filter formulation
In this section the filter and its sensitivities are checked. The filter is applied to the portal
geometry. Ωtot is determined for the portal before and after filtering using the exact method
and the density distribution is visually inspected. To more easily identify the density gradients the colormaps have been set to vary from dark red indicating low density to dark blue
indicating high density.
To verify the derived sensitivities for the filter a coarse version of the portal geometry is used
due to the long computation time. The portal is now constructed in a 15x15 element mesh.
The sensitivity of a single element with respect to another arbitrarily selected element is de∂ ρ̃i
termined, i.e. ∂ρ
. The element with respect to which the sensitivity is being determined, ρj ,
j
is varied om zero to one. For each value of ρj the value ρ̃i is determined and stored, using this
information the finite difference approximation can be determined over the entire domain of
ρj from one to zero. This is then compared to the calculated values. The elements chosen to
be inspected are shown in Figure 5-13.
Figure 5-13: Element to be varied and element of which the sensitivity is determined
5-2-1
Modification by sequential scanning
The results for the sequential filtering method are shown in Figure 5-14. The first image
shows the original geometry while the two others show the filtered geometry using pn = 8 and
pn = 1000 respectively. The overall result is as desired; the mid-section of the overhanging
part has been removed under a 45◦ angle. On the other hand some unexpected gradients are
seen, but for the very high value of 1000 for pn this disappears. Closer inspection shows that
this result also shows the same alternating pattern in densities except the differences between
density in each row are so small that they are not visible.
Looking at the calculated (exact) Ωtot it is found that the overhang coefficient has increased
after filtering. For the result with pn = 1000 the overhang too is not zero. This is directly
linked to the alternating density pattern. Every other row contributes positively to the
overhang coefficient since these elements are not supported properly.
These deviations from the desired result can be explained again by the error introduced by
the P-norm:
The elements in the bottom row have the correct values (ones and zeros), the values in the
M. Reuben Serphos
Master of Science Thesis
5-2 Verification of filter formulation
45
second to bottom row are slightly lower than they should be. Taking the middle element of
one of the pillars as a representative point, it is clear that ρi and ρmax in Eq. (4-34) in reality
are both one resulting in ρ̃ = 1. Nevertheless in calculating ρmax an error is introduced as
described in Subsection 5-1-1.√Since all candidate support elements are of density one the
error is at its maximum, i.e. pn 3 which for pn = 8 is about 1.15. Plugging this in to Eq. (434) gives ρ̃ = 0.85, which corresponds to the value produced by MATLAB. Note that since
pn is an even number the term (1 − ρmax )pn is positive, if an uneven power is chosen complex
values will be registered in MATLAB.
Moving on to the void section on the same row, taking the same approach ρi = 0 and
ρmax = 0. Unlike for the situation with the solid elements, for elements √
with a value of zero
ρmax returns zero, with no error. This time Eq. (4-34) becomes ρ̃ = 1 − pn 2 which for pn = 8
gives ρ̃ = −0.09. Doing this experiment for subsequent rows confirms the alternating pattern
seen in Figure 5-14b.
It is clear that these errors are of the same type but originate from a different source. Needless
to say increasing pn reduces these errors, but they are not completely eliminated which has
already been demonstrated by setting pn = 1000. For this value for the P-norm power an Ωtot
is found near that of the unfiltered geometry. This is because the error produces a distinct
border of elements with a value slightly above zero where the overhanging structure used to
be under which elements of negative value are present (how this value varies with respect to
pn is shown in Figure 5-15). The (exact) overhang calculator registers this as overhang (ρ∆ is
smaller but still present) explaining the near unfiltered value of Ωtot . This border is not seen
in Figure 5-14c but it is present albeit for very low values of ρ∆ .
Applying the filter to the corner loaded cantilever beam gives an indication of the effect the
filter will have, see Figure 5-16.
(a) Original
Ωtot = 0.011
densities
with
(b) Filtered densities with pn =
8 and Ωtot = 0.398
(c) Filtered densities with pn =
1000 and Ωtot = 0.010
Figure 5-14: Filtering the densities by sequential scanning of the elements
Master of Science Thesis
M. Reuben Serphos
46
Validation
Figure 5-15: Effect of P-norm power on element density directly below overhang
(a) Original topology
(b) Filtered topology
Figure 5-16: Filtering the corner loaded cantilever beam by sequential scanning of the elements
Sensitivity check
Varying the density of an element ρj in the portal produced the results shown in Figure 5-18
for the value and sensitivity of an element ρ̃i . The element being varied, ρj (annotated in
Figure 5-13) provides indirect support to element ρ̃i . For ρj = 0 element ρ̃i is unsupported
and thus the density is zero after filtering. As the density of ρj increases elements along the
diagonal path leading to ρ̃i , which were previously unsupported, become supported. Hence
ρ̃i is increasingly supported allowing the density of this element to remain after filtering in
an increasing quantity. Figure 5-17 shows the unfiltered and filtered densities of the portal
for ρj equal to zero and one. Comparing Figure 5-17b and Figure 5-17d the additional strip
of elements can be identified.
From this it can be deduced that the value of ρ̃i should be approximately proportional to ρj .
As is seen in Figure 5-18a the value of ρ̃i is indeed roughly proportional to ρj . Comparing
the calculated and finite difference sensitivities in Figure 5-18b and Figure 5-18c respectively,
it is seen that the sensitivity has been properly determined. Doing the same experiment for
other element pairs also produced agreeing results.
M. Reuben Serphos
Master of Science Thesis
5-2 Verification of filter formulation
47
(a) Reference portal with ρj = 0
(b) Filtered densities with ρj = 0
(c) Reference portal with ρj = 1
(d) Filtered densities with ρj = 1
Figure 5-17: Filtered densities for varying density of test element
Master of Science Thesis
M. Reuben Serphos
48
Validation
(a) Value of ρ̃i vs. ρj
(b) Calculated sensitivity
(c) Finite difference sensitivity
Figure 5-18: Comparison of calculated and finite difference sensitivity
M. Reuben Serphos
Master of Science Thesis
Chapter 6
Notes on implementation
In this chapter some issues encountered in the implementation of the previously described
restriction methods into the 88-line topology optimization MATLAB code are discussed.
6-1
Element selection
All three restriction methods require the identification of candidate support elements. The
multiple objective and global constraint approach also require the additional influenced elements to be identified for the calculation of the sensitivities. For this study a mask has
been used to properly select the correct elements. That is, indices of the candidate support
elements and influenced elements were predetermined with respect to the indices of the element under inspection. In the mesh used element rows are numbered from top to bottom and
columns from left to right. Recalling Figure 6-1, for an element under inspection with row
index i and column index j the candidate support elements (in the case of 45◦ overhang) are
located one row lower, i.e. with row index i + 1. The column index of the candidate support
elements are then j − 1, j and j + 1 from left to right.
A next step would be to introduce an element-selector which automatically selects the correct
candidate support elements and influenced elements depending on the desired orientation and
build angle. This would allow a more general formulation that could be applied to unstructured meshes and also allow inclusion of the build orientation into the optimization problem.
Figure 6-1: Candidate support elements for an element under inspection
Master of Science Thesis
M. Reuben Serphos
50
6-2
Notes on implementation
Ghost-layers
In determining the overhang with the previously described mask, an issue arises when doing
so for the elements along the boundaries of the design space. At these locations the mask
will require elements that fall outside the design space, which do not exist. To account for
these missing elements an extra layer of ‘ghost’ elements is added around the design space.
This is done by embedding the density field into a larger matrix. The surrounding elements
are given a value of zero, which corresponds to the inability to support any other elements,
see Figure 6-2. The bottom layer of elements must be perceived as supported, but instead
of setting the ghost layer of elements at this boundary to one, the bottom layer elements
are simply not included in the operation. That is element scanning starts one layer higher.
In determining the overhang coefficient as well as when filtering the densities, skipping the
bottom row of elements results in no contribution to the overhang constraint and no changes
due to filtering respectively. This corresponds to the elements being supported.
To include the influenced elements needed when determining the sensitivities another ghost
layer is needed. Thus two ghost layers have been added; GL=2 elements.
Figure 6-2: Ghost-layer
6-3
Invalid values
In calculating the sensitivities cases occur where density values of zero are raised to a negative
1
−1
power (e.g. 0 pn ), for this MATLAB will return ‘Inf’ (infinity) as a solution. In all the cases
where this happens the value is also multiplied by zero resulting in an undefined numerical
result, in MATLAB this gives a ‘NaN’ value which stands for Not-a-Number. These values
have been replaced by zero for the calculations to make sense.
6-4
Constraint in MMA
The Method of Moving Asymptotes (MMA) function used for the optimization (for a detailed
description see [8]) is designed for optimization problems in which constraint functions are
M. Reuben Serphos
Master of Science Thesis
6-4 Constraint in MMA
51
of the form: fconstraint ≤ 0. This means that the equality constraint presented in Subsection
4-6-3 cannot be implemented as such. Instead this becomes Ωf ac Ωtot ≤ 0, which still gives
the optimizer the incentive to set the overhang to zero even though negative values are not
possible.
Master of Science Thesis
M. Reuben Serphos
52
Notes on implementation
M. Reuben Serphos
Master of Science Thesis
Chapter 7
Final results
In this chapter the results of the implementation of the formulated restriction methods, i.e.
multiple objective, global constraint and filter, applied to the load cases described in Chapter
3 are summarized. The obtained results are compared to the reference models and differences
are identified. The complete set of results can be found in Appendix C.
7-1
Multiple Objective
Various runs were executed for all three load cases for several values of Ωf ac . The chosen
Ωf ac values differ by an order of magnitude.
In all three load cases the topology changes for a sufficiently high Ωf ac . In the cases of
the corner-loaded and mid-loaded cantilever beam struts in the design are reoriented. All
topologies obtained with the multiple objective remained with overhanging sections where
stalactite-like formations appeared. This is particularly visible for the tension beam. For
increasing Ωf ac the volume fraction decreases. The overhang coefficient shows erratic behavior
for Ωf ac values of 102 and up for the corner-loaded and mid-loaded cantilever beams. For the
tension beam this is already the case for Ωf ac = 10. The objective and compliance value also
start to show erratic behavior for Ωf ac values of 103 and up.
None of the results converge nicely. Taking the results obtained for the same amount of
iterations as was necessary to obtain the reference model still allows comparison of designs.
In the following subsections these results are presented for each load case along with a more
detailed description of the overall results.
7-1-1
Cantilever beam corner-load
Using a weight factor of 10 for the corner-loaded cantilever beam has no noticeable effect on
the topology. The obtained compliance is almost identical to that of the reference model.
Slight changes in topology start to appear when the weight factor is increased to 102 . The
compliance is increased and the overhang coefficient has been decreased significantly. For
Master of Science Thesis
M. Reuben Serphos
54
Final results
a weight factor of 103 much more drastic changes can be seen in the design. The volume
fraction is pushed down further and the compliance is increased to about 159. The optimizer
ends up cycling around this point after about 500 iterations. Design changes throughout the
process are small though, the optimization can be terminated earlier while keeping the same
general topology. The trend in the objective value during the optimization process is erratic
showing many sharp peaks. This behavior is encountered in both terms that make up the
objective, the compliance and overhang term. The topology obtained after 191 iterations
Figure 7-1c shows the elimination of many of the struts encountered in the reference design.
Instead a triangular structure is formed with two struts under 45 degree angles with another
overhanging strut connecting the tip of the triangle to the constrained side of the design space.
Some vague stalactite formation can be recognized on this overhanging strut. Looking at the
plot of Ω∗ in Figure 7-2c it is indeed seen that this overhanging strut is the major contributor
to overhang in this design. For a weight factor of 104 a similar design as for a weight factor
of 103 is found. The volume fraction continues to decrease as the compliance of the structure
increases. Stalactite formations are present around the same area. The same general behavior
is found for the objective value, namely an erratic distribution of values and cycling which
occurs after about 650 iterations. The overhang coefficient has been reduced and is lower
than any of the other results, but so is the volume fraction. Increasing the weight factor to
105 results in incoherent patches of material appearing and disappearing in the design space
with no structural characteristics. The optimizer fails to find a topology that can provide a
reasonable value of compliance, values are of the order 106 . Cycling is observed after about 70
iterations. The volume fraction has reduced to a value around a mere 0.047 and the overhang
coefficient has become meaningless.
The results for the multiple objective approach for the corner-loaded cantilever beam are
shown in Figure 7-1. The reference model is shown along with the result for a weight factor
Ωf ac of 102 and 103 respectively.
The Ω∗ values along with their respective overhang coefficients for these results are shown in
Figure 7-2. An improvement in Ωtot can be seen, which corresponds with a less overhanging
elements.
(a) Reference model Objective
value: 57.17; Volume fraction:
0.418
(b) Ωf ac : 102 Objective value:
69.91; Volume fraction: 0.406;
Compliance: 68.94; Ωf ac Ωtot :
0.97
(c) Ωf ac : 103 ; Objective value:
181.26; Volume fraction: 0.220;
Compliance: 168.25; Ωf ac Ωtot :
13.01
Figure 7-1: Results multiple objective for cantilever corner-load obtained in 191 iterations
M. Reuben Serphos
Master of Science Thesis
7-1 Multiple Objective
(a) Ω∗ for the reference model
with Ωtot =0.0274
55
(b) Ω∗ for Ωf ac =102 with
Ωtot =0.0114
(c) Ω∗ for Ωf ac =103 with
Ωtot =0.0165
Figure 7-2: Ω∗ multiple objective for cantilever corner-load; Using the following parameter values:
M = 35; δ = 0.2; pn = 8
7-1-2
Cantilever beam mid-load
As with the corner-loaded cantilever beam a weight factor of 10 has no noticeable effect on
the design of the mid-loaded cantilever beam. For a weight factor of 102 struts start to
rearrange, but the overhang coefficient actually becomes higher. The onset of some stalactite
formations can be recognized. The bottom section of the design starts to raise resulting in an
overhanging strut. The compliance increases and the volume fraction decreases. The trend
in overhang coefficient is not smooth. Setting the weight factor to 103 results in a design in
which stalactite formations are more evident and the struts have been drastically rearranged.
The overhang coefficient has increased further and the volume fraction has become lower.
The compliance and objective value have increased and their behavior is erratic. Further
increasing the weight factor to 104 , the volume fraction is reduced till about 30 iterations at
which point the volume fraction becomes too low to form a coherent structure of reasonable
compliance.
The results for the mid-loaded cantilever beam for Ωf ac equal to 102 and 103 obtained with
a multiple objective are shown in Figure 7-3. The results shown here were obtained in 121
iterations.
The topology for the multiple objective becomes asymmetric and Figure 7-4 reveals that, even
though one of the struts has been angled at 45◦ , a large portion of the structure (front/lower
strut) has been angled at a smaller angle. The upper overhanging strut is extended introducing
more overhanging elements.
7-1-3
Tension beam
For a weight factor of 10 stalactite formations are seen along the overhanging underside of the
structure. A decrease in volume fraction is seen and the objective function and its compliance
term shows an increase. The overhang coefficient has been reduced. A strongly decreased
volume fraction is seen for a weight factor of 102 and stalactite formation has become more
evident. The compliance and objective function have increased. The overhang coefficient has
also increased slightly with respect to the result for a weight factor of 10. For a weight factor
of 103 , the volume fraction is further reduced. The remaining topology shows many stalactite formations and a reduced overhang coefficient. The objective value and compliance have
increased further. Setting the weight factor to 104 results in incoherent patches of material
Master of Science Thesis
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Final results
(a) Reference model Objective
value: 64.00; Volume fraction:
0.426
(b) Ωf ac : 102 ; Objective value:
67.57; Volume fraction: 0.387;
Compliance: 64.93; Ωf ac Ωtot :
2.64
(c) Ωf ac : 103 ; Objective value:
256.6159;
Volume fraction:
0.201; Compliance: 218.034;
Ωf ac Ωtot : 38.582
Figure 7-3: Results multiple objective for cantilever mid-load obtained in 121 iterations
(a) Ω∗ for the reference model
with Ωtot =0.0352
(b) Ω∗ for Ωf ac =102 with
Ωtot =0.0294
(c) Ω∗ for Ωf ac =103 with
Ωtot =0.0452
Figure 7-4: Ω∗ global constraint for cantilever mid-load; Using the following parameter values:
M = 35; δ = 0.2; pn = 8
M. Reuben Serphos
Master of Science Thesis
7-2 Global constraint
57
appearing and disappearing throughout the design space.
In the reference model the tension beam load case needed 1160 iterations to meet the original termination criteria set in the 88-line MATLAB code. When running the optimization
algorithm for the multiple objective the output showed cycling behavior around 80 iterations.
Therefore this was set as the new termination criteria.
The results for the multiple objective problem are shown in Figure 7-5 and Figure 7-6. These
figures show that the volume fraction occupied is drastically reduced for an increasing weight
factor which increases the compliance. The overhang coefficient is slightly improved.
(a) Reference model Objective
value: 5.49; Volume fraction:
0.474; Number of iterations:
1160
(b) Ωf ac : 10; Objective value:
7.35; Volume fraction: 0.354;
Number of iterations: 80; Compliance: 7.01; Ωf ac Ωtot : 0.34
(c) Ωf ac : 102 ; Objective value:
18.78; Volume fraction: 0.166;
Number of iterations: 80; Compliance: 15.48; Ωf ac Ωtot : 3.30
Figure 7-5: Results multiple objective for tension beam
(a) Ω∗ for the reference model
with Ωtot =0.0486
(b) Ω∗ for Ωf ac =10 with
Ωtot =0.0387
(c) Ω∗ for Ωf ac =102 with
Ωtot =0.0393
Figure 7-6: Ω∗ multiple objective for tension beam; Using the following parameter values:
M = 35; δ = 0.2; pn = 8
7-2
Global constraint
In general the designs obtained by the global constraint approach have comparable features
to those obtained by the multiple objective approach. Reorientation of struts and stalactite
formation can be found in all the designs. The lowering of volume fraction is also a recurring
trait. Various values of Ωf ac do not affect the results very much. For the cantilever beams
a value of 105 does lead to instability in the form of light patches of material appearing and
disappearing in the design space. Since Ωf ac has little influence on the result comparison will
only be made with results obtained with Ωf ac = 10.
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Final results
The erratic behavior described for the multiple objective is also present with the global constraint. In this case the erratic behavior is already present for Ωf ac = 10.
7-2-1
Cantilever beam corner-load
Figure 7-7 shows the topology results results for the corner-loaded cantilever beam terminated
after 191 iterations.
(a) Reference model Objective
value: 57.17; Volume fraction:
0.418
(b) Ωf ac : 10; Objective value:
87.52; Volume fraction: 0.34
Figure 7-7: Results global constraint for cantilever corner-load obtained in 191 iterations
The Ω∗ values along with their respective overhang coefficients for these results are shown
in Figure 7-8. A significant improvement in Ωtot is seen. The objective value is significantly
increased for the added constraint and the volume fraction has decreased. As with the multiple
objective, struts have been angled at 45◦ and more pronounced stalactite formations appear
at the top of the design.
(a) Ω∗ for the reference model
with Ωtot =0.0274
(b) Ω∗ for Ωf ac =10 with
Ωtot =0.0115
Figure 7-8: Ω∗ global constraint for cantilever corner-load; Using the following parameter values:
M = 35; δ = 0.2; pn = 8
7-2-2
Cantilever beam mid-load
Running the optimization for the mid-loaded cantilever beam gives the results shown in
Figure 7-9. Just as with the corner-loaded cantilever beam the effects are now much more
pronounced than for the multiple objective. Formation of stalactites and a very low volume
M. Reuben Serphos
Master of Science Thesis
7-2 Global constraint
59
fraction are observed. An increase of about 147% in objective value is seen accompanied by
a slight decrease in overhang coefficient. The structure has become asymmetric, which is an
unexpected result. Other struts have been altered such that these are under a smaller angle
than the prescribed constraint this can be deduced from Figure 7-10.
(a) Reference model Objective
value: 64.00; Volume fraction:
0.426
(b) Ωf ac : 10; Objective value:
158.31; Volume fraction: 0.25
Figure 7-9: Results cantilever mid-load obtained in 121 iterations
(a) Ω∗ for the reference model
with Ωtot =0.0352
(b) Ω∗ for Ωf ac =10 with
Ωtot =0.0333
Figure 7-10: Ω∗ global constraint for cantilever mid-load; Using the following parameter values:
M = 35; δ = 0.2; pn = 8
7-2-3
Tension beam
Just as with the multiple objective the tension beam showed cycling, therefore the comparison
model is terminated for a fewer amount of iterations than the reference model. In this case
250 iterations.
As seen in Figure 7-11 and Figure 7-12 the results for the tension beam show the same
trends but again much more pronounced. An extremely low volume fraction results in higher
compliance and stalactite formation is present. The overhang coefficient is generally much
lower.
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Final results
(a) Reference model Objective
value: 5.49; Volume fraction:
0.474; Number of iterations:
1160
(b) Ωf ac : 10; Objective value:
81.36; Volume fraction: 0.07;
Number of iterations: 250
Figure 7-11: Results tension beam obtained
(a) Ω∗ for the reference model
with Ωtot =0.0486
(b) Ω∗ for Ωf ac =10 with
Ωtot =0.0234
Figure 7-12: Ω∗ global constraint for tension beam; Using the following parameter values:
M = 35; δ = 0.2; pn = 8
7-3
Filter
As was the case in validating the sensitivities, computation time for the topology optimization with filter is extremely long. For reasonable computing times the mesh sizes for the
test models have been reduced, maintaining the aspect ratio of the design space, to 25x15
elements.
None of the optimization runs with the filtering method converge. As the optimization progresses the topology becomes more pronounced but also lowered in volume fraction. This
continues till at some point errors occur in the Method of Moving Asymptotes (MMA) optimizer. The objective and volume fraction values leading to this point do not show a smooth
distribution, though there are not as many peaks and sharp transitions as in the results obtained for the multiple objective and global constraint approach.
The topologies shown in this section have been obtained by terminating the algorithm near
error introduction, but as pronounced as possible. In these topologies elements were encountered with negative valued densities.
Overhang of these results has not been calculated due to the alternating density nature of the
topology for this filter (also seen in section 5-2). This makes the calculated overhang values
meaningless.
M. Reuben Serphos
Master of Science Thesis
7-3 Filter
61
Cantilever beam corner-load
The topology found for the corner-loaded cantilever beam by the filtering method has a very
low volume fraction. The objective/compliance is very high. Nevertheless by visual inspection
it can be deduced that this design has no overhang whatsoever.
The structure has been modified in a similar fashion as with the multiple objective and global
constraint, except in this case internal supporting structures have been added to eliminate
the remaining overhang.
(a) Reference model Objective
value: 63.60; Volume fraction:
0.46
(b) Objective value: 583.93;
Volume fraction: 0.24 ; Number
of iterations: 35
Figure 7-13: Results filter for cantilever corner-load
Cantilever beam mid-load
Similar to the corner-loaded cantilever beam, the mid-loaded beam has a low volume fraction
a high objective/compliance value and no overhang.The design has again been slightly altered
in general shape and augmented by internal supporting structures..
(a) Reference model Objective
value: 63.41; Volume fraction:
0.45; Number of iterations: 75
(b) Objective value: 474.55;
Volume fraction: 0.24; Number
of iterations: 98
Figure 7-14: Results filter for cantilever mid-load
Tension beam
Once again the obtained structure is low in volume fraction and high in objective/compliance
but without overhanging elements.
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Final results
(a) Reference model Objective
value: 5.36; Volume fraction:
0.48; Number of iterations: 304
(b) Objective value: 115.11;
Volume fraction: 0.15; Number
of iterations: 35
Figure 7-15: Results filter for tension beam
M. Reuben Serphos
Master of Science Thesis
Chapter 8
Discussion
The results of the multiple objective and the global constraint approach show three main
behaviors in the topology when introducing the overhang restriction. These are: slight modifications in strut orientation, stalactite formation, and the lowering of the volume fraction.
In general the behavior is seen to be the same for both methods but more exaggerated for the
global constraint. Which could be expected since the global constraint is a strict condition
for which the optimizer must find a solution, while in the case of a multiple objective formulation the solution is allowed to be a compromise between the two objective terms. The fact
that overhang is still present in the global constraint approach is due to the inner workings
of the optimizer itself, Method of Moving Asymptotes (MMA). In cases where no solution
can be found that complies with the imposed constraint the optimizer presents a non-feasible
solution instead.
Additionally both these restriction methods show erratic behavior of the objective, compliance and Ωtot value at some point.
The filter succeeded in creating structures that were fully supported by a combination of
reorienting struts and introducing support structures.
In the following sections the observed behaviors will be discussed in further detail.
8-1
Modification of strut orientation
The fact that all three methods show reorientation of struts to meet the 45 degree overhang
restriction indicates potential for all methods. The multiple objective and global constraint
also altered struts such that these introduced more overhang. This may be due to the large
contribution of these elements to the compliance of the structure. Another explanation for
this can be that due to the lowered volume fraction and other alterations in the design the
optimizer cannot find any other solution that provides a lower compliance value.
The filtering method demonstrated the ability to create support structures in the cases in the
cases where elements had a large contribution to the compliance. The multiple objective and
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64
Discussion
global constraint resort to the formation of stalactites. The tension beam illustrates this very
well since the only way to comply with the added restriction is to add support structures and
the results for this load case showed a lot of stalactite formation.
8-2
Stalactites in multiple objective method and global constraint
method
Formation of stalactite-like structures appear where elements need support. This phenomenon
can be explained by taking a closer look at what the implications are of such a surface for the
additional overhang term. A representation of these stalactites is shown in Figure 8-1. The
contribution to the overhang term (marked in red) is decreased by reducing the number of
unsupported elements to a single element. This way instead of having multiple unsupported
elements, only one unsupported element remains. Such a structure does indeed present a
good solution to the problem of reducing overhang, however it is impractical and not of any
use in the fabrication process of Selective Laser Melting (SLM). For this it is necessary that
the structure is fully supported and not simply reduced to a single point.
Since these stalactites form where radical changes in topology are necessary it can also be
interpreted as a suggestion for the location of placement of support structures. The question
remains why these stalactites are not extended to form fully developed support structures.
This must be attributed to the nature of the way the overhang is formulated. The overhang
behaves like a step function and as mentioned before the M-parameter determines how steep
the curve is. If the value is too high the gradients in the function become extremely high and
optimization is difficult. It seems that the chosen value for M of 35 is still too high and that
this obstructs the optimizer from finding solutions with support structures. Setting this value
to M = 8 gives a result for the corner-loaded cantilever beam in which support structures do
form, see Figure 8-2.
Figure 8-1: Stalactite formation
8-3
Erratic objective values, instability and Ωf ac
The effect of the steep step function also explains the erratic behavior observed for the objective, compliance and overhang values. The steep transition in overhang coefficient causes
these values to vary drastically for every iteration. For lower values of M this behavior will
be smoother.
Since the filter does not make use of the overhang coefficient formulation the sharp transitions here originate from a different source. In this case the design may still experience drastic
M. Reuben Serphos
Master of Science Thesis
8-4 Lowered volume fraction and high compliance
65
Figure 8-2: Cantilever beam corner-load with parameters δ = 0.2, M = 8, pn = 8 and Ωf ac =
103
changes due to lack of support, but these changes are the result of the sensitivity behavior.
One may recall that the filtering method is also based on the approximation of a discrete
valued concept. Namely that an element under inspection takes on the value of a candidate
support element if support is not sufficient, otherwise nothing happens. The sharpness of
this transition is determined by the value of the P-norm. A lower P-norm value will result
in a worse approximation of the value to be assigned to the element under inspection which
will diminish the quality of the filter. In Section 5-2 it was shown that the P-norm power
is also responsible for negative values in the filtered design. The optimizer does not succeed
in avoiding these negative values which at some point leads to instability in the algorithm.
The Finite Element Analysis (FEA) in the algorithm cannot handle these negative values and
‘NaN’ values are produced. Consequently these ‘NaN’ values become the input for the MMA
algorithm which results in an error. This is an important aspect of the filter that must be
dealt with in further development. Another filtering method has also been formulated in the
process of this research but not further developed due to time constraints. This method can
be found in Appendix B. In this method densities are never negative but do take on values
far greater than one, which is the drawback of this method.
The multiple objective and global constraint also show some instability. This occurs when
Ωf ac is chosen to be very high. When this is done the volume fraction is lowered to such a
point that not enough material is available to produce a coherent structure. Elements take
on strongly varying density values but the optimizer is unable to find a solution anymore.
Increasing the value of Ωf ac allows the overhang term to become more important in finding
the optimal solution. As Ωf ac is increased the overhang coefficient is seen to decrease as would
be expected. Along with this the volume fraction of the topology is lowered which leads to
the unstable behavior. The global constraint is less dependent on the weight factor. This is
because for the MMA optimizer the value of the constraint is aimed to be set to zero or lower
regardless of its magnitude. Setting Ωf ac sufficiently high though will still cause instability
for the same reasons as the multiple objective. This is related to the way the optimization
problem is solved in the MMA algorithm.
8-4
Lowered volume fraction and high compliance
The lowering volume fraction is presumed to be the result of behavior of the sensitivities.
The sensitivities of the three methods (see Figure 5-12 and Figure 5-18) show that a zero
value is obtained for low density values. In other words a minimum is found for a density of
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Discussion
zero. In minimizing the overhang coefficient this is, numerically, an acceptable method. This
manifests as a lowering of the volume fraction of the topology.
Naturally, decreasing the volume fraction will result in increasing compliance values. The high
compliance of the designs should not be a reason to render the obtained results worthless. The
produced topology still shows useful structures. Taking the result obtained by the filter for
the mid-loaded cantilever beam as an example the potential can be shown. Elements with a
value lower than 0.3 have been set to zero and all other element densities have been increased
to one. The resulting black-white topology shown in Figure 8-3 has a volume fraction of 0.5
and no overhanging elements. An FEA has been performed on this structure and a compliance
of 53.88 is the result. This value is lower than that of the reference model (which was 63.41).
Though the reference model has not been post-processed to be fully black and white this
result indicates the potential of the topology.
(a) Result from filtering approach for mid loaded cantilever
beam with compliance value:
474.55; Volume fraction: 0.24;
Number of iterations: 98
(b) Threshold applied
elements above 0.3
for
(c) Filtered black-white structure with compliance: 53.88;
Volume fraction: 0.5
Figure 8-3: Extracting a black-white topology from the filter result
8-5
Additional considerations
Up till now the filter has shown the best results in terms of topology but the computation
time is a major drawback. The main factor for this is the calculation of the sensitivities.
For each element the sensitivity must be calculated with respect to all other elements in the
design domain. This data is then stored for calculation of the sensitivities of the subsequent
elements. A total of N · N calculations must be done (N is the number of elements in the
design domain). For the implementation for this research memory pre-allocation has already
been applied but a solution might be found in vectorization of operations. The nature of the
filter might make this difficult since for every calculation the results of previous elements are
needed (the sequential nature of the filter). The number of elements in the design domain for
a 2D problem is
N = nx · ny
(8-1)
where nx and ny are the number of elements along the horizontal (x-axis) and vertical (yaxis) of the design domain respectively. Making the number of calculations needed n2x · n2y .
Extending this to 3D would mean the number of calculations needed would increase by a
factor n2z where nz is the number of elements along the z-axis. Unless the method can be
M. Reuben Serphos
Master of Science Thesis
8-5 Additional considerations
67
implemented in a more efficient way practical application of this method may be ill-fated.
Besides this extension to 3D also introduces a further increase in error when calculating ρmax .
In this case there are nine candidate support elements which need to be taken√into account.
Analogous to Eq. (5-5) the maximum introduced error is then found to be pn 9 − 1. This
applies to all approaches
One must not lose sight of the big picture in developing these restriction methods. Eventually it is the goal to also take part orientation into account. It could very well be that by
arranging the part to be built in a different orientation, i.e. changing the build direction,
that modifications are not necessary. The tension beam is an example of this. If the entire
beam were to be oriented to be built at a 45 degree angle the optimal design could be built
without modification. It is therefore necessary to extend these restriction methods so this can
be taken into account. In Section 6-1 element selection was discussed. If successfully applied
the automated element selector can serve as a way to indicate the build direction. By defining
this as a variable the optimization algorithm can be extended to include part orientation. A
nested optimization could be a possible approach.
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M. Reuben Serphos
Discussion
Master of Science Thesis
Chapter 9
Conclusion
The goal of this research has been to include the 45◦ overhang manufacturing constraint for
Selective Laser Melting (SLM) into a topology optimization algorithm. Three methods to do
this have been proposed, these are:
• By replacing the objective function by a multiple objective
• By adding a global constraint
• By applying a density filter
For the multiple objective and the global constraint a global measure for overhang in a design
has been formulated. A filtering scheme has been formulated for the density filter which
modifies the densities such that no overhanging elements remain.
The three methods have been implemented in a 2D topology optimization algorithm written
in MATLAB. Tests have been carried out on three predictable reference models. All three
methods produced results that showed promising behavior, but only the results obtained by
the filtering method fully met the manufacturing constraint. The filter method generated
results that were modified versions of the reference models augmented by supporting structures. Even the most extreme case where the reference design consisted of purely overhanging
elements, a topology has been found where support structures have formed.
The other two methods should not be immediately discarded though. It has been found that
a detailed investigation of parameter choice for these methods may yield parameters that lead
to useful results.
The results revealed that the filter also needs further development due to instabilities in the
optimization process. The instabilities have been presumed to be the repercussion of densities
taking on negative values. Moreover the implementation of the filter as used in this research
proved to be slow and convergence has not been achieved. Nonetheless the concept of a filtering method has been demonstrated to be a candidate for inclusion of the 45◦ overhang
manufacturing constraint. The multiple objective and global constraint still show potential
but have not been proven to generate the desired result.
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Conclusion
The formulated restriction methods form a first step into obtaining a manufacturable design
for SLM directly from topology optimization. The methods proposed form a basis for further
research and provide a proof of concept.
M. Reuben Serphos
Master of Science Thesis
Chapter 10
Recommendations
The approaches presented and investigated in this research have shown promising results,
but it is clear that further development is needed. There are various aspects of the proposed
approaches which can be developed. A list of some topics for future work and ideas for further
investigation follows.
• The influence of the parameters associated with the overhang coefficient on the results
of the topology optimization must be further investigated. Due to time constraint this
could not be properly done in this research, though an indication of its potential was
given in Chapter 8.
• The behaviour of the three methods may be influenced by other parameters in the
optimization as well. These have not been investigated. The settings for the Method of
Moving Asymptotes (MMA) algorithm are some of these parameters. Other parameters
are the volume fraction, penalization power and filter radius of the mesh-independency
filter.
• The filtering scheme presented in this thesis results in instabilities. This needs to be
resolved. The filter concept presented in the appendix may provide a solution for this.
• The filtering method should be further improved so that computation time is reduced.
A possible direction could be the vectorization of certain operations.
• Various implementation methods might also be possible. Applying a continuation
method to the multiple objective or global constraint may be beneficial. Some suggestions could be:
– Make the weight factor dependent on the iteration
– Reset the optimization to start from an initial design which has already been
optimized for compliance only
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Recommendations
– Allowing a varying amount of overhang in the constrained approach instead of the
strict rule that overhang must be at most zero.
• To further advance in the direction of an SLM-ready topology optimized result, inclusion
of orientation is necessary. This can be done by writing an element selecting algorithm.
By including build direction in this selector the optimization problem can be extended
to treat this as a variable.
• Other methods of including the restriction could also be considered. For example a
damage approach where overhanging elements are present but do not contribute to the
stiffness of the structure.
• In this research the choice has been made to focus on the discrete nature of the topology
optimization process. Making use of the continuous nature of topology optimization
remains another path which can be taken to incorporate restrictions.
• It might also be of interest to consider other topology optimization methods than the
Solid Isotropic Material Penalization (SIMP) method. The level-set method may prove
to be a favorable alternative due to the well defined boundaries.
M. Reuben Serphos
Master of Science Thesis
Appendix A
A-ADS hinge-bracket
In this chapter a description is given of how the topology optimization has been done for the
A-ADS hinge-bracket.
First the design driving requirements are determined. Next two sub problems are set up to
prepare the topology optimization. Finally the set-up for the topology optimization of the
hinge-bracket is presented.
A-1
Design requirements
A list of requirements has been made that drive the design. These requirements follow from
the operational requirements. The system is a precision pointing system and therefore stiffness
(and eigenfrequency) play a large role. Based on the allowable eigenfrequency, dimensions and
material properties of the booms a required rotational stiffness of 30kN m /rad for the hingebracket around the y- and z-axis has been determined, see Figure A-1. In general, however,
stiffer is better.
Besides the stiffness requirement it is desirable to minimize the mass of the hinge-bracket since
this translates to cost in the sense of rocket fuel needed to transport the system into space.
Additionally this also reduces material cost when manufacturing with Additive Manufacturing
(AM).
The hinge-bracket will be of metallic material but will have to interface with the Carbon Fiber
Reinforced Polymer (CFRP) booms. The lay-up of the booms will be designed such that the
Thermo-elastic deformation (TED) is minimized to minimize disturbances in pointing. This
will result in a mismatch of coefficient of thermal expansion (CTE) at the interface between
boom and hinge-bracket. The final design should be able to cope with this. The TED of
the total hinge assembly (hinge-bracket including synchro pulley-system, spring and lock
mechanism) is also restricted and must be smaller than 0.005◦ in the operational temperature
range, which is −150◦ C to 90◦ C.
These main design requirements for the hinge-bracket are summarized below.
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A-ADS hinge-bracket
• Rotational stiffness of at least 30kN m /rad around y- and z-axis (derived from allowable
system eigenfrequency)
• Mass of hinge assembly (incl. brackets, hinge unit, synchro-pulleys) < 2.6 kg
• Interface to CFRP boom with inner diameter 100 mm ±0.5 mm and CTE=−0.2micron /m·K
• Design for minimum mass, compatible with strength, stiffness and TED requirement
• TED of the hinge assembly < 0.005◦ in operational temp. range
• Operational temp. −150◦ C to 90◦ C
Figure A-1: Schematic representation of the hinge-bracket
For this thesis the problem has been greatly simplified by excluding the thermal mismatch
of the assembly. For a functional design it is crucial that this is taken into account. Hence
the following results are purely demonstrative for the topology optimization process. This
leaves the main design requirements of stiffness and mass. This has been interpreted as a
compliance minimization problem of which the expected result is a conceptual design which
gives insight in how material can be best distributed.
The problem can be split up into two main load cases; loading around the z-axis and loading
around the y-axis. Loading around the z-axis is an in-plane torque, while loading around the
y-axis is an out-of-plane torque. These load cases have first been set-up independently in two
sub problems.
An experimental topology optimization code provided by the TU Delft has been used to
obtain the results in this chapter. The finite elements in this algorithm are solid isotropic and
do not support rotations. The mesh generated is a 3D grid mesh. The optimizer is based on
Method of Moving Asymptotes (MMA). The results presented in this chapter were obtained
with the following parameters:
Penalization power: 3
Filter radius:
1.8
Volume fraction:
0.5
M. Reuben Serphos
Master of Science Thesis
A-2 Circular in-plane torque
A-2
75
Circular in-plane torque
The first 3D model, presented here, is that with a circular load case of in-plane torque. The
design space is cuboid with dimensions 50x50x100 elements. One of the square faces is fully
clamped and the torque is applied over a ring of nodes. Such a load case is a common one
in practice and the results can be compared with the many practical applications and can be
expected to be tubular.
Figure A-2 shows the schematic representation of the problem. The red arrows indicate the
clamped degrees of freedom applied to all nodes on the respective side, while the green arrow
shows the applied load on the group of nodes that fall within the green box. The clamped
condition is applied to the pink face and the green ring indicates the volume over which the
torque is applied. The finite elements of the optimizer do not allow simple application of a
torque. Instead, to emulate a torque, the ring has been divided into subsections on which a
force is applied tangent to the ring. Doing this over the entire ring will effectively result in
torque. The saw-tooth appearance of the ring is due to all the force vectors on the subsections.
Figure A-2: Load case circular in-plane torque (left) and result (right)
A-3
Circular out-of-plane torque
The second load case is the out-of-plane torque applied over a ring. As in the in-plane torque,
one face is fully clamped and a ring divided into subsections is used to impose a torque. In
this case the force vectors are applied perpendicular to the ring and the magnitude of the
force varies sinusoidally with its angular location on the ring. The design space dimensions
are kept the same as for the circular in-plane torque case.
Figure A-3: Load case circular out-of-plane torque (left) and result (right)
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A-4
A-ADS hinge-bracket
Hinge-bracket
For the hinge-bracket the design space is a 150x150x250 element cuboid in which one of the
square faces is fully clamped. Attached to this face is a tube that represents the CFRP tube to
which the hinge-bracket will connect. This plays an important role in a later stage of design
in the modeling of the thermal mismatch between the tube and the bracket. This section
is also of importance to be able to take the effect of the adhesive connection into account.
Due to the limitations of the code being used the CFRP tube can only be modeled as an
isotropic material, therefore some replacement values must be determined to get the proper
response. Since the thermal mismatch is the main reason for the inclusion of the tube, the
critical stiffness for this situation is chosen for the bulk value of the tube. The most important
stiffness in this case is that tangential to the tube, the tube will be manufactured such that
this will be 89 GPa.
The inner diameter of the tube is 100mm and the wall thickness will be 2mm. The length
of the modeled tube is set at 60mm. To make sure the optimizer sees the tube as potential
connection surface and not the clamped face, a rectangular void is imposed against the entire
(clamped) face. A remainder of tube is left protruding the void space, this protrusion will
serve as adhesive surface. The length of this surface has been chosen to be 25mm since this
is the maximum length commonly used for adhesive surfaces. A cylindrical void is imposed
where a spring and latching system will be inserted, the diameter of the cylinder is 47mm
and the height is set to 64.2mm. Forces will be applied at the (base) connection point of
this spring and latching system. The connection is ring-shaped and therefore so will be the
forces. The two load cases presented earlier (circular in-plane torque and circular out-of-plane
torque) will be applied. The torques will both be of unit magnitude.
Figure A-4 shows a schematic representation of the boundary conditions and loads, pink
represents the fully clamped surface (with the arrows indicating the clamped DOFs), yellow
indicates void space and the forces are shown in green.
The results for this configuration are shown in Figure A-5.
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A-4 Hinge-bracket
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Figure A-4: Load case of the hinge-bracket
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A-ADS hinge-bracket
Figure A-5: Result of the hinge-bracket
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Master of Science Thesis
Appendix B
Another filter approach
B-1
Formulation of the filter - Modification by supportivity matrix
In this chapter another filtering approach is presented. The idea behind this filter is to modify
the element densities by scaling these with a measure that accounts for the amount of support
available.
To achieve this, first a new definition is introduced, the so-called supportivity of an element.
The supportivity is a value between zero and one assigned to an element that quantifies to
what extent an element is supported, i.e. a maximally supported element has a supportivity
of one while an absolutely unsupported element has a value of zero. One must note however
that the supportivity of one element is dependent on the supportivity of the elements below
it. An element that is not supported, in turn, cannot support any other element, thereby
influencing the supportivity of an element above it.
For ease of implementation the supportivity of each element is stored in a matrix, the supportivity matrix, so that these values can be called for the calculation of supportivity of elements
above. By executing an element-wise multiplication (denoted by ◦) with the supportivity
matrix and density matrix, a modified density matrix is obtained in which the densities of all
unsupported elements have been scaled such that they are supported and all other densities
are modified accordingly. By this the desired effect of the filter has been achieved. The key
difference in this filter is that the densities are scaled with a value between zero and one and
can thus never take on negative values.
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Another filter approach
The supportivity has been defined as:
Si = Smaxi · ρmaxi
(B-1)
Where:
Si
= The supportivity of element i
Smaxi = The maximum value of the supportivity of the candidate support elements for
element i
ρmaxi = The maximum density of the candidate support elements for element i
In Eq. (B-1) Smaxi accounts for the dependence on all elements below element i. All these
elements must be supported as should these elements be supported by the elements below
them and so on. This is thus a global term. Smaxi is approximated by again using the P-norm.
The supportivity is then scaled with the maximum density value of the candidate support
elements by multiplying with ρmaxi , since an element can only support as much material as
it is supported by. ρmaxi is a local term.
A matrix is constructed containing the supportivity of each element, the supportivity matrix.
In this matrix all the values of the bottom row will always have a value of one. This accounts
for the support of the base-plate on which the design would be built.
The new element densities now become:
ρ̃ = Smax ◦ ρmax ◦ ρ
(B-2)
ρ̃i = Smaxi · ρmaxi · ρi
(B-3)
or for an arbitrary element ρi :
Per example the calculation of the modified density is demonstrated for an element ρ0 as
displayed in Figure B-1. The numbering denotes the element numbers, note that this numbering does not correspond to the numbering of the elements in the code. The element under
inspection is at the top, ρ0 , elements 1-3 are the first line of support, these are supported by
elements 4-8 and these are then in turn supported by elements 9-15. For the modified density
of ρ0 we have:
ρmax0 =
Smax0 =
M. Reuben Serphos
q
ρp1n + ρp2n + ρp3n
(B-4)
S1pn + S2pn + S3pn
(B-5)
pn
q
pn
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B-1 Formulation of the filter - Modification by supportivity matrix
81
Figure B-1: Involved elements in determining the supportivitiy of an element ρ0 .
Numbers denote element numbers (not corresponding to actual numbering in the MATLAB code).
Where:
S1 = Smax1 · ρmax1
(B-6)
with:
(B-7)
Smax1 =
ρmax1 =
q
pn
q
pn
S4pn + S5pn + S6pn
(B-8)
ρp4n + ρp5n + ρp6n
(B-9)
and
(B-10)
S4 = Smax4 · ρmax4
(B-11)
S5 = Smax5 · ρmax5
(B-12)
S6 = Smax6 · ρmax6
(B-13)
with:
(B-14)
Smax4 =
ρmax4 =
Smax5 =
ρmax5 =
Smax6 =
ρmax6 =
q
pn
q
pn
pn
pn
+ S11
S9pn + S10
(B-15)
ρp9n + ρp10n + ρp11n
(B-16)
q
pn
q
pn
pn
pn
pn
+ S12
+ S11
S10
(B-17)
ρp10n + ρp11n + ρp12n
(B-18)
q
pn
q
pn
pn
pn
pn
+ S13
S11
+ S12
(B-19)
ρp11n + ρp12n + ρp13n
(B-20)
(B-21)
Since the supportivity of all the elements on the bottom row, elements
√ 9-15, are one (supported
by the base-plate), Smax4 , Smax5 and Smax6 are each equal to pn 3, this is also the case for
Smax7 and Smax8 . This will approximate one for large values of pn , the deviation from one is
an error that is introduced by the use of the P-norm. With this we now get:
√
S4 =
pn
S5 =
pn
S6 =
pn
√
√
3 · ρmax4
(B-22)
3 · ρmax5
(B-23)
3 · ρmax6
(B-24)
(B-25)
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Another filter approach
substituting and rewriting then gives:
S1 =
√
pn
q
3 pn ρp9n + ρp10n + ρp11n + ρp10n + ρp11n + ρp12n + ρp11n + ρp12n + ρp13n ·
q
pn
ρp4n + ρp5n + ρp6n
(B-26)
v
u 13
6
uX p X
√
pn
pt
=
3· n
ρi n ·
ρpi n
i=9
(B-27)
i=4
Analogously S2 and S3 can be found to be:
S2 = Smax2 · ρmax2
(B-28)
v
u 14
7
uX p X
√
pn
pt
=
3· n
ρi n ·
ρpi n
i=10
(B-29)
i=5
S3 = Smax3 · ρmax3
(B-30)
v
u 15
8
uX p X
√
pn
pt
ρpi n
ρi n ·
=
3· n
i=11
(B-31)
i=6
By substituting all this into Eq. (B-3) the modified density is found to be:

ρP hys0 = 
√
pn
v

u 13
8
15
7
14
6
u
X
X
X
X
X
X
n
ρpi n  ·
ρpi n ·
ρpi n +
ρpi n ·
ρpi n +
3 · pt
ρpi n ·
i=9
i=4
i=10
i=5
i=11
i=6
v
u 3
uX p
pt
n
ρi n · ρ0
i=1
(B-32)
It can be seen from this example that all elements that could possibly contribute to the
support of element ρ0 all the way down to the base of the design space are involved in this
method.
B-2
Validation - Modification by supportivity matrix
For verification the filter is applied to the portal geometry, Ωtot is determined for the portal
before and after filtering using the exact method and the density distribution is visually
inspected. To more easily identify the density gradients the colormaps have been set to vary
from dark red indicating low density to dark blue indicating high density.
Figure B-2 shows the results for the filter using the supportivity matrix. In this case all
elements that were originally void (ρ = 0) are still void after filtering, i.e. no negative values.
For values above zero, as with the sequential filter, the bottom row of elements all have the
correct values (ones or zeros), but every sequential row has a value that deviates from what
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Master of Science Thesis
B-2 Validation - Modification by supportivity matrix
83
it should be. In this case a relatively smooth gradient is seen (as opposed to the alternating
pattern in the sequential filter). The element densities increase as they near the top of the
geometry. This divergence from the correct value can again be attributed to the P-norm
approximation. In this case the error is introduced in two ways, first of all by the ρmaxi
term and secondly by the Smaxi term in Eq. (B-3). The error introduced by the Smaxi term
increases for every row since it includes the ρmaxi and Smaxi values of the elements below
it. Thus the error accumulates further towards the top of the geometry. For pn = 8 the
top element of the geometry has a value of 30620, which is an absolutely unacceptable value
considering ρ may only vary from one to zero. Increasing pn to 1000 this value reduces to
1,086 (this behavior is shown in Figure B-3).
Besides the introduced error the general behavior of the filter is as desired. The overhanging
mid-section of the portal has been properly removed, leaving 45 degree angles at the edges.
The overhang coefficient found for the filtered geometry is not zero, this is due to the gradient
in the geometry which registers as overhang.
(a) Original
Ωtot = 0.011
densities
with
(b) Filtered densities with pn =
8 and Ωtot = 0.193
(c) Filtered densities with pn =
1000 and Ωtot = 0.193
Figure B-2: Filtering the densities using the supportivity matrix
Figure B-3: Effect of P-norm power on the density of a element at the top of the structure
Applying the filter to the corner loaded cantilever beam gives an indication of the effect the
filter will have, see Figure B-4.
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Another filter approach
(a) Original topology
(b) Filtered topology
Figure B-4: Filtering the corner loaded cantilever beam using the supportivity matrix
The need for extremely high values of P-norm power is a large drawback of this filter and
will most-likely cause difficulties in implementation in the topology optimization. The accummulating error makes the quality of the filter mesh-dependent, the error is increasingly
large for larger meshes. Extending this approach to 3D will increase the accumulative error
even further making implementation even more challenging. Nevertheless the concept may
provide a more numerically stable alternative to the sequential filter introduced in Section
4-7, though much development is needed. Due to time constraints this approach has not been
developed any further.
M. Reuben Serphos
Master of Science Thesis
Appendix C
Results
In this chapter the results for the three overhang restriction methods are presented. Each
method has been applied to the corner-loaded and mid-loaded cantilever beam and the tension beam. Graphs of the objective, volume fraction and where applicable the overhang
coefficient and compliance values for each iteration are given. The obtained topology and
where applicable Ω∗ are given for the final iteration unless stated otherwise in the caption.
C-1
C-1-1
Multiple objective
Corner-loaded cantilever beam
(a) Ωf ac = 10
(b) Ωf ac = 102
(c) Ωf ac = 103
Figure C-1: Topology
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Results
(a) Ωf ac = 10
(b) Ωf ac = 102
(c) Ωf ac = 103
Figure C-2: Ω∗
(a) Ωf ac = 10
(b) Ωf ac = 102
(c) Ωf ac = 103
Figure C-3: Objective value
(a) Ωf ac = 10
(b) Ωf ac = 102
(c) Ωf ac = 103
Figure C-4: Compliance value
(a) Ωf ac = 10
(b) Ωf ac = 102
(c) Ωf ac = 103
Figure C-5: Overhang coefficient value
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C-1 Multiple objective
87
(b) Ωf ac = 102
(a) Ωf ac = 10
(c) Ωf ac = 103
Figure C-6: Volume fraction
(a) Ωf ac = 104
(b) Ωf ac = 105
Figure C-7: Topology
(a) Ωf ac = 104
(b) Ωf ac = 105
Figure C-8: Ω∗
(a) Ωf ac = 104
(b) Ωf ac = 105
Figure C-9: Objective value
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Results
(a) Ωf ac = 104
(b) Ωf ac = 105
Figure C-10: Compliance value
(a) Ωf ac = 104
(b) Ωf ac = 105
Figure C-11: Overhang coefficient value
(a) Ωf ac = 104
(b) Ωf ac = 105
Figure C-12: Volume fraction
M. Reuben Serphos
Master of Science Thesis
C-1 Multiple objective
C-1-2
89
Mid-loaded cantilever beam
(a) Ωf ac = 10
(b) Ωf ac = 102
(c) Ωf ac = 103
Figure C-13: Topology
(a) Ωf ac = 10
(b) Ωf ac = 102
(c) Ωf ac = 103
Figure C-14: Ω∗
(a) Ωf ac = 10
(b) Ωf ac = 102
(c) Ωf ac = 103
Figure C-15: Objective value
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Results
(a) Ωf ac = 10
(b) Ωf ac = 102
(c) Ωf ac = 103
Figure C-16: Compliance value
(a) Ωf ac = 10
(b) Ωf ac = 102
(c) Ωf ac = 103
Figure C-17: Overhang coefficient value
(a) Ωf ac = 10
(b) Ωf ac = 102
(c) Ωf ac = 103
Figure C-18: Volume fraction
(a) Ωf ac = 104
Figure C-19: Topology
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C-1 Multiple objective
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(a) Ωf ac = 104
Figure C-20: Ω∗
(a) Ωf ac = 104
Figure C-21: Objective value
(a) Ωf ac = 104
Figure C-22: Compliance value
(a) Ωf ac = 104
Figure C-23: Overhang coefficient value
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Results
(a) Ωf ac = 104
Figure C-24: Volume fraction
C-1-3
Tension beam
(a) Ωf ac = 10
(b) Ωf ac = 102
(c) Ωf ac = 103
Figure C-25: Topology
(a) Ωf ac = 10
(b) Ωf ac = 102
(c) Ωf ac = 103
Figure C-26: Ω∗
M. Reuben Serphos
Master of Science Thesis
C-1 Multiple objective
(a) Ωf ac = 10
93
(b) Ωf ac = 102
(c) Ωf ac = 103
Figure C-27: Objective value
(a) Ωf ac = 10
(b) Ωf ac = 102
(c) Ωf ac = 103
Figure C-28: Compliance value
(a) Ωf ac = 10
(b) Ωf ac = 102
(c) Ωf ac = 103
Figure C-29: Overhang coefficient value
(a) Ωf ac = 10
(b) Ωf ac = 102
(c) Ωf ac = 103
Figure C-30: Volume fraction
Master of Science Thesis
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Results
(a) Ωf ac = 104
Figure C-31: Topology
(a) Ωf ac = 104
Figure C-32: Ω∗
(a) Ωf ac = 104
Figure C-33: Objective value
(a) Ωf ac = 104
M. Reuben Serphos
Master of Science Thesis
C-1 Multiple objective
95
(b) Ωf ac = 104
(c) Ωf ac = 104
Figure C-34: Volume fraction
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C-2
C-2-1
Results
Global constraint
Corner-loaded cantilever beam
(a) Ωf ac = 10
(b) Ωf ac = 102
(c) Ωf ac = 103
Figure C-35: Topology
(a) Ωf ac = 10
(b) Ωf ac = 102
(c) Ωf ac = 103
Figure C-36: Ω∗
(a) Ωf ac = 10
(b) Ωf ac = 102
(c) Ωf ac = 103
Figure C-37: Objective value
M. Reuben Serphos
Master of Science Thesis
C-2 Global constraint
97
(b) Ωf ac = 102
(a) Ωf ac = 10
(c) Ωf ac = 103
Figure C-38: Overhang coefficient value
(b) Ωf ac = 102
(a) Ωf ac = 10
(c) Ωf ac = 103
Figure C-39: Volume fraction
(a) Ωf ac = 104
(b) Ωf ac = 105
Figure C-40: Topology
(a) Ωf ac = 104
(b) Ωf ac = 105
Figure C-41: Ω∗
Master of Science Thesis
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Results
(a) Ωf ac = 104
(b) Ωf ac = 105
Figure C-42: Objective value
(a) Ωf ac = 104
(b) Ωf ac = 105
Figure C-43: Overhang coefficient value
(a) Ωf ac = 104
(b) Ωf ac = 105
Figure C-44: Volume fraction
M. Reuben Serphos
Master of Science Thesis
C-2 Global constraint
C-2-2
99
Mid-loaded cantilever beam
(a) Ωf ac = 10
(b) Ωf ac = 102
(c) Ωf ac = 103
Figure C-45: Topology
(a) Ωf ac = 10
(b) Ωf ac = 102
(c) Ωf ac = 103
Figure C-46: Ω∗
(a) Ωf ac = 10
(b) Ωf ac = 102
(c) Ωf ac = 103
Figure C-47: Objective value
Master of Science Thesis
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Results
(b) Ωf ac = 102
(a) Ωf ac = 10
(c) Ωf ac = 103
Figure C-48: Overhang coefficient value
(b) Ωf ac = 102
(a) Ωf ac = 10
(c) Ωf ac = 103
Figure C-49: Volume fraction
(a) Ωf ac = 104
(b) Ωf ac = 105
Figure C-50: Topology
(a) Ωf ac = 104
(b) Ωf ac = 105
Figure C-51: Ω∗
M. Reuben Serphos
Master of Science Thesis
C-2 Global constraint
101
(a) Ωf ac = 104
(b) Ωf ac = 105
Figure C-52: Objective value
(a) Ωf ac = 104
(b) Ωf ac = 105
Figure C-53: Overhang coefficient value
(a) Ωf ac = 104
(b) Ωf ac = 105
Figure C-54: Volume fraction
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C-2-3
Results
Tension beam
(a) Ωf ac = 10
(b) Ωf ac = 102
(c) Ωf ac = 103
Figure C-55: Topology
(a) Ωf ac = 10
(b) Ωf ac = 102
(c) Ωf ac = 103
Figure C-56: Ω∗
(a) Ωf ac = 10
(b) Ωf ac = 102
(c) Ωf ac = 103
Figure C-57: Objective value
M. Reuben Serphos
Master of Science Thesis
C-2 Global constraint
103
(b) Ωf ac = 102
(a) Ωf ac = 10
(c) Ωf ac = 103
Figure C-58: Overhang coefficient value
(b) Ωf ac = 102
(a) Ωf ac = 10
(c) Ωf ac = 103
Figure C-59: Volume fraction
(a) Ωf ac = 104
(b) Ωf ac = 105
Figure C-60: Topology
(a) Ωf ac = 104
(b) Ωf ac = 105
Figure C-61: Ω∗
Master of Science Thesis
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Results
(a) Ωf ac = 104
(b) Ωf ac = 105
Figure C-62: Objective value
(a) Ωf ac = 104
(b) Ωf ac = 105
Figure C-63: Overhang coefficient value
(a) Ωf ac = 104
(b) Ωf ac = 105
Figure C-64: Volume fraction
M. Reuben Serphos
Master of Science Thesis
C-3 Filter
C-3
C-3-1
105
Filter
Corner-loaded cantilever beam
(a) Topology after 35 iterations
(b) Objective value
(c) Volume fraction
Figure C-65: Corner-loaded cantilever beam
C-3-2
Mid-loaded cantilever beam
(a) Topology after 98 iterations
(b) Objective value
(c) Volume fraction
Figure C-66: Mid-loaded cantilever beam
C-3-3
Tension beam
(a) Topology after 35 iterations
(b) Objective value
(c) Volume fraction
Figure C-67: Tension beam
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Results
Master of Science Thesis
Bibliography
[1] W. Meiners, C. Over, K. Wissenbach, and R. Poprawe, “Direct generation of metal parts
and tools by selective laser powder remelting (slpr),” Proceedings of SFF, Austin, Texas,
1999.
[2] D. Thomas, The development of design rules for selective laser melting. PhD thesis,
University of Wales, 2010.
[3] M. R. Serphos, “Design of a metallic industrial part for selective laser melting.”
[4] M. P. Bendsoe and O. Sigmund, Topology optimization: theory, methods and applications.
Springer, 2003.
[5] D. Brackett, I. Ashcroft, and R. Hague, “Topology optimization for additive manufacturing,” in 22nd Annual International Solid Freeform Fabrication Symposium, pp. 348–362,
2011.
[6] M. Zhou, R. Fleury, Y.-K. Shyy, H. Thomas, and J. Brennan, “Progress in topology
optimization with manufacturing constraints,” in Proceedings of the 9th AIAA MDO conference AIAA-2002-4901, 2002.
[7] E. Andreassen, A. Clausen, M. Schevenels, B. S. Lazarov, and O. Sigmund, “Efficient
topology optimization in matlab using 88 lines of code,” Structural and Multidisciplinary
Optimization, vol. 43, no. 1, pp. 1–16, 2011.
[8] K. Svanberg, “The method of moving asymptotes-a new method for structural optimization,” International journal for numerical methods in engineering, vol. 24, no. 2, pp. 359–
373, 1987.
Master of Science Thesis
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M. Reuben Serphos
Bibliography
Master of Science Thesis
Glossary
List of Acronyms
AM
Additive Manufacturing
SLM
Selective Laser Melting
FEA
Finite Element Analysis
CFRP
Carbon Fiber Reinforced Polymer
A-ADS
Antenna - Articulated Deployment System
S/C
spacecraft
TED
Thermo-elastic deformation
CTE
coefficient of thermal expansion
DfAM
design for additive manufacturing
SIMP
Solid Isotropic Material Penalization
MMA
Method of Moving Asymptotes
List of Symbols
c
Compliance of the material lay-out
U
Global displacement vector
F
Force vector
K
Global stiffness matrix
ue
Element displacement vector
Master of Science Thesis
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110
Glossary
ke
Element stiffness matrix
ρ
Vector of design variables
ρmin
Vector of minimum relative densities
N
Number of elements used to discretize design domain
p
Penalization power
V (ρ)
Material volume
V0
Design domain volume
f
Prescribed volume fraction
ρmax
The maximum value of the included density values
pn
A parameter that influences how well the true maximum value is approximated
a
The element number for a = 1...k
k
The number of elements being evaluated
ρi
The density value of element i
Ωtot
A function that defines a value for the level of overhang
Ωf ac
A weight factor for the additional overhang term
M
A parameter that controls the steepness of the logistic function curve
x
The independent variable in the logistic function
F
A convolution matrix representing a density filter
ρ̃
Vector of filtered densities
err
Error introduced by P-norm
GL
Ghost-layer width
nx
Number of elements along the x-axis of the design domain
ny
Number of elements along the y-axis of the design domain
nz
Number of elements along the z-axis of the design domain (in 3D)
Si
The supportivity of element i
Smaxi
The maximum value of the supportivity of the candidate support elements for
element i
ρmaxi
The maximum density of the candidate support elements for element i
M. Reuben Serphos
Master of Science Thesis
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