StructuralDesignOptimizationofVibrationIsolatingStructures_MaxvanderKolk_Final.

StructuralDesignOptimizationofVibrationIsolatingStructures_MaxvanderKolk_Final.
.
Structural Design Optimization
of
Vibration Isolating Structures
by
Max van der Kolk
in partial fulfillment of the requirements for the degree of
Master of Science
in Mechanical Engineering
at Delft University of Technology,
to be defended publicly on Monday, 26 October 2015 at 09:45.
Student number:
Project duration:
Thesis committee:
4079035
1 December 2014 – 26 October 2015
Prof. dr. ir. F. van Keulen,
TU Delft
Prof. dr. ir. K.M.B. Jansen, TU Delft
Dr. ir. J.W. van Wingerden, TU Delft
Dr. ir. M. Langelaar,
TU Delft
Dr. ir. G.J. van der Veen,
TU Delft
Dr. ir. J. de Vreugd,
TNO
An electronic version of this thesis is available at http://repository.tudelft.nl/.
Abstract
The design of high performance instruments often involves the attenuation of poorly damped resonant
modes. These resonant modes are a limiting factor to the performance of these instruments. Current
design approaches typically start from a baseline design and introduce stiffening or damping reinforcements to tune and/or damp these modes. However, the influence on the structural damping of these
reinforcements is difficult to predict and often results in trial and error-based design approaches for
the design of damping reinforcements.
A common solution is to introduce viscoelastic material in baseline designs to increase structural
damping. These materials dissipate energy when subjected to deformation and should therefore be
located at positions which undergo large deformations during vibration. Typically, the viscoelastic
material is placed in conventional (un)constrained layer damping configurations. However, to achieve
optimized damping characteristics both the location as well as the geometry of viscoelastic material
should be optimized.
In this thesis, a multi-material topology optimization routine is presented as a systematic methodology to develop structures with optimal damping characteristics. The proposed method applies a
multi-material, parametric level set-based approach to simultaneously distribute structural and viscoelastic material within the design domain. The developed optimization routine allows for the design
of freeform, viscoelastic dampers without the limitation to conventional (un)constrained layer damping configurations and is thereby able to achieve improved damping characteristics.
The structural loss factor is applied as a performance measure to compare the damping between
different viscoelastically damped structures and as objective function during the optimization. The
viscoelastic material behavior is represented by a complex-valued material modulus, which results in
a complex-valued eigenvalue problem. The formulation of the structural loss factor is modified to
account for the complex-valued eigensolutions, resulting in accurate assessment of the structural loss
factor for designs containing viscoelastic material with high material loss factors.
The optimization routine maximizes the structural loss factor for single or multiple selected eigenmodes. An adjoint sensitivity analysis is performed to provide an exact expression for the structural
loss factor sensitivity. Based on this sensitivity information, the optimization routine is able to develop structures with optimized loss factors for the specified eigenmodes. The method is also able to
generate damping solutions for existing designs containing badly damped resonant modes.
iii
Preface
This thesis is the final result of my Master’s studies at Delft University of Technology. The project
has been a collaboration between the Structural Optimization and Mechanics group within the department of Precision and Microsystems Engineering and the Optomechatronics department at the
Netherlands Organization for Applied Scientific Research (TNO) in Delft. I am glad to have gotten
the opportunity to work on my graduation project at TNO after completing my internship here. A
lot of freedom was available to formulate this project, which allowed me to combine many different
aspects from the courses I enjoyed most. The result has been an exciting and equally challenging
graduation project.
The presented thesis would not be the same without my supervisors. I would like to thank Jan de
Vreugd for his supervision during my internship and for providing the opportunity to do my graduation work at the Optomechatronics department. Our weekly meetings have been fun and resulted in
many interesting discussions. Also, thanks are to Gijs van der Veen for the discussions we had on the
theoretical aspects of the work. Furthermore, thanks are to Matthijs Langelaar for his enthusiasm in
supervising this project and his critical questioning during our progress meetings. Finally, I would like
to thank all of them for providing the opportunity to write a scientific publication, the contributions
they made, and their efforts in revising the draft manuscripts.
These past years in Delft have been challenging and exciting, but most of all, they were a lot of fun.
I am grateful for the many great friends I got to meet and the time we spend together. I am sure
that these years would not have been the same without you! At last, I would like to thank my family,
especially my parents, for their support and kindness throughout these years and before.
Max van der Kolk
Delft, October 2015
v
Contents
Abstract
iii
Preface
v
List of Figures
ix
List of Tables
xi
Nomenclature
xiii
1 Introduction
1
2 Topology optimization
2.1 Optimization methods . . . . . . . . . . . .
2.1.1 Density-based optimization . . . . .
2.1.2 Evolutionary structural optimization
2.1.3 Level set-based optimization . . . . .
2.1.4 Black and white design . . . . . . . .
2.1.5 Multi-material topology optimization
2.2 Comparison of optimization methods . . . .
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5
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3 Parametric level set method
3.1 Level set method . . . . . . . . . .
3.2 Parametric level set method . . . .
3.2.1 Shape functions . . . . . . .
3.2.2 Multi-material formulation .
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4 Topology optimization of viscoelastically damped structures
Abstract. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Modeling of Viscoelastic Material . . . . . . . . . . . . . . . . . . . .
4.2.1 Complex material modulus. . . . . . . . . . . . . . . . . . . .
4.2.2 Structural loss factor . . . . . . . . . . . . . . . . . . . . . . .
4.3 Topology optimization of viscoelastic and structural material . . . . .
4.3.1 Multi-material boundary representation . . . . . . . . . . . . .
4.3.2 Parametric level set method . . . . . . . . . . . . . . . . . . .
4.3.3 Optimization problem for structural loss factor maximization .
4.3.4 Adjoint sensitivity analysis . . . . . . . . . . . . . . . . . . . .
4.3.5 Numerical implementation . . . . . . . . . . . . . . . . . . . .
4.4 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Case study I: cantilever beam . . . . . . . . . . . . . . . . . .
4.4.2 Design influences by complex moduli . . . . . . . . . . . . . .
4.4.3 Case study II: existing structure with limited design domain .
4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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5 Base excitation and thermal analysis
37
5.1 Base excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.2 Energy dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.3 Thermal analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
vii
viii
Contents
6 Discussion
45
7 Conclusions and recommendations
47
7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
7.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Bibliography
49
53
A Viscoelastic material model
A.1 Material model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
A.2 Energy storage and dissipation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
A.3 Loss factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
B Numerical implementation
B.1 Finite element model . . . . . . . . . . . . . .
B.1.1 Thermal model . . . . . . . . . . . . .
B.1.2 Radial basis function placement . . . .
B.2 Heaviside function: discrete and approximate.
B.3 From Level set function to topology . . . . . .
C Loss factor sensitivity analysis
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D MATLAB Code
71
D.1 Main file. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
D.2 Optimization loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
D.3 Example case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
E Commercial FEM implementation
75
List of Figures
1.1
1.2
2.1
2.2
2.3
2.4
3.1
3.2
3.3
3.4
Part of GAIA BAM: a high performance optomechatronic instrument developed at
TNO for angle measurement between stars. . . . . . . . . . . . . . . . . . . . . . . . .
An illustration of the unconstrained and constrained layer damping methods. . . . . .
1
2
Three most common optimization methods in structural design. The optimization
methods from top to bottom: (a) size, (b) shape, c topology optimization. . . . . . .
6
Two examples of designs obtained with the optimization routine developed by Robinson. 9
Examples of designs obtained by a single and multi-material BESO method. . . . . . .
9
Examples of designs obtained by a single and multi-material parametric level set method. 10
An illustration of the level set function φ(X) and the corresponding domain Ω and
boundary Γ as described by the level set φ = c. . . . . . . . . . . . . . . . . . . . . . .
Illustration of a cross section of the level set function and its parameterization. . . . .
(a): Illustration showing the cross section of the radial basis function placement. The
RBFs are aligned with the elements in the finite element discretization. The shape
functions ξi and the expansion coefficients αi are aligned with the ith element in the
design domain. (b): Illustration of a compactly-supported radial basis function as
described by Equation (3.9). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Illustration of a multi-material structure defined by two level set functions. . . . . . .
Illustration of the clamped cantilever beam containing a constrained viscoelastic layer.
The arrow indicates the location of the excitation force and position measurement
during dynamic loading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Illustration of a multi-material structure defined by two level set functions. The first
level set function φ1 determines the placement of material, while the second level set
function φ2 distinguishes between structural and viscoelastic material. . . . . . . . . .
4.3 The design domain for case study I. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 The design progression throughout the optimization routine at iteration 2, 5, 20 and
35 and the corresponding objective values. . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 The history of the objective values (a) and corresponding loss factors (b) during the
optimization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6 The history of the volume (top) and frequency (bottom) constraint values during the
optimization. The constraints are normalized with respect to the specified maximum
value. Positive values represent unsatisfied constraints . . . . . . . . . . . . . . . . . .
4.7 The deformation shapes and corresponding maximum shear strain εmax for the first
(a) and second (b) eigenmode of the optimized structure. The uncolored elements
represent void elements within the design. . . . . . . . . . . . . . . . . . . . . . . . . .
4.8 The energy dissipated for each element for the first eigenmode of the structure. The
values are normalized with respect to the maximum energy dissipated in an element.
The obtained result is directly related to the shear strain given in Figure 4.7a. The
uncolored elements represent void elements within the design. . . . . . . . . . . . . . .
4.9 A comparison between the optimized designs and their performance for the application
of multiple complex material models: (a): shear, (d): Young’s and (g): bulk modulus.
The differences in performance are highlighted by the maximum principal and maximum
shear strains within the structure for the first eigenmode. . . . . . . . . . . . . . . . .
4.10 The design and non-design domains for case study II. The non-design domain is clamped
at both sides and cannot be modified by the optimization routine. . . . . . . . . . . .
4.11 The final design of the optimization achieved after 58 iterations. The design achieved
an objective value of J = 0.3803. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
14
16
17
4.1
ix
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26
29
30
30
30
31
31
32
33
34
x
List of Figures
4.12 The history of the loss factor (a) and normalized volume and frequency constraints (b)
during the optimization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.13 The deformation shape for the first eigenmode of the optimized structure of case study
II. The dissipated energy in each element is plotted to illustrate the energy dissipation
for the first eigenmode. The uncolored elements correspond to void elements in the
design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1
5.2
5.3
5.4
5.5
Four different viscoelastic dampers: (a): single element viscoelastic layer, (b): three
elements viscoelastic layer, (c): a viscoelastic layer matching the thickness of the viscoelastic layer in the optimized design, and (d): the design generated by the optimization. The presented loss factors correspond to their first resonant mode. . . . . . . . .
Frequency response of four structures damped using viscoelastic material. The response
of the optimized structure is compared to thee other designs implementing a constrained
layer damping configuration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The figures illustrate the dissipated energy per cycle for each elements in the design. In
each illustration the values are normalized with respect to the element with maximum
energy dissipation. The distributions are given for the first and second resonant mode
of the optimized design in figures (a) and (b), and for a CLD configuration in figures
(c) and (d). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Illustrations of the heat transfer problem for a generalized structure and the situation
applied to the optimized structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Normalized temperature distribution as result of the energy dissipation due to the
deformation of the viscoelastic material. . . . . . . . . . . . . . . . . . . . . . . . . . .
A.1 Illustration of the phase difference between strain ε(t) and stress σ(t) for a harmonically
excited viscoelastic material. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.2 Illustration of the frequency and temperature dependent behavior of the shear modulus
and loss factor of a viscoelastic material. The loss factor shows a maximum for certain
operating frequency and temperatures. Image modified from Beards (1997). . . . . . .
A.3 (a): Illustration of a hysteresis loop of a viscoelastic material subjected to harmonic
excitation. The loop is found by plotting the stress σ(t) as function of the strain ε(t).
(b): The complex material modulus drawn in the complex plane to express the modulus
as function of the storage or loss moduli using the phase angle θ. . . . . . . . . . . . .
B.1 Illustration of a Q4 element applied for the discretization of the design domain. . . . .
B.2 One-dimensional illustration of the radial basis function placements. . . . . . . . . . .
B.3 Approximations of the discrete Heaviside function and its derivative. Both approximated formulations are plotted for various values of the parameter ∆. For decreasing
values of ∆ the approximations approach the discrete formulations. . . . . . . . . . . .
B.4 Illustration of the difference between discrete and approximate Heaviside functions on
the material distribution and element densities within the finite element discretization.
B.5 Visualization of both level set functions as presented by the final iteration of the optimization of case study I. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.6 The material distribution at four iterations during the optimization of case study I. . .
B.7 The material distribution at four iterations during the optimization of case study II. .
E.1 A schematic diagram of the optimization procedure using a dedicated finite element
environment to perform the FE calculations. The green blocks are performed within
the FE environment, while the orange blocks are evaluated in MATLAB or similar
programming environments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
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54
55
58
60
61
63
64
65
65
76
List of Tables
4.1
4.2
Comparison between the structural loss factors and Q-factor of a structure containing
a constrained viscoelastic layer (Figure 4.1). The loss factors γ̄ and γ are determined
using the undamped and damped eigenmodes. The relative difference is calculated
between the structural loss factors and the inverse of the Q-factor to illustrating the
improved prediction of damping behavior when using the complex-valued eigenmodes
for the loss factor calculation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A comparison between the loss factors for the optimized designs, when the performance
is evaluated with different material models than used during the optimization. . . . . .
xi
25
33
Nomenclature
Symbols
αi
Expansion coefficient corresponding to shape function ξi
αi,max
Upper bound applied to the expansion coefficients
αi,min
Lower bound applied to the expansion coefficients
∆
Parameter for approximate Heaviside formulation
δ
Kronecker delta
ε
Strain
ε0
Strain amplitude
εmax
Maximum in-plane shear strain
ε1,2
Maximum in-plane principal strains
η
Material loss factor
Γk
The boundary described by the kth level set function
γ̄r
The structural loss factor based on the rth eigenmode of an undamped system
γr
The structural loss factor based on the rth eigenmode of a damped system
γr⋆
Adjoint formulation of the structural loss factor γr
κ
Bulk modulus or thermal conductivity in context of thermal analysis
λr
Eigenfrequency of the rth eigensolution
µ1
Adjoint variables
µ2
Adjoint variable
Ωk
Domain enclosed by the kth level set function
ω
Excitation frequency
φ
Level set function
ϕr
Eigenmode r of a damped system
ψr
Eigenmode r of an undamped system
σ
Stress
σ0
Stress amplitude
ξ
Shape function
a
Parameter for approximate Heaviside formulation
b
Parameter for approximate Heaviside formulation
D
Available domain for level set functions
D
Diffusive term in Hamilton-Jacobi equation
E
Young’s modulus
xiii
xiv
Notation
E
Elasticity tensor
f
Externally applied forces
G
Shear modulus
h
Convective heat transfer coefficient
H(φ)
i
Heaviside function of the argument φ
√
The complex number: i = −1
K
Global stiffness matrix
Ke
Element stiffness matrix
KTc
Conductivity matrix
KTh
Convective matrix
M
Global mass matrix
Me
Element mass matrix
Q
Q-factor
Qcyc
Dissipated power per unit volume
ri
Radius corresponding to shape function ξi
R
Influence radius corresponding to the shape function ξ
R
Reactive term in Hamilton-Jacobi equation
Rh
Heat load introduced by convective boundary condition
Rq
Heat load introduced by dissipation in viscoelastic material
t
Time or pseudo-time variable
T∞
Environment temperature
T
Nodal temperatures
V
k
k
Vmax
V
e
Volume enclosed by the domain of the kth level set function
Volume constrained applied to the kth level set function
Volume of a single finite element
vn
Velocity field in level set-based methods
Wcyc
Dissipated energy per unit volume per cycle
X
Describes a point within the design domain D
x
Nodal displacements
Notation
′
Denotes the storage (real) component of the material modulus,
′′
Denotes the loss (imaginary) component of the material modulus
T
Transpose of a vector or matrix (superscript only)
H
Conjugate transpose of a complex vector or matrix (superscript only)
R
Real component of the vector or matrix (subscript only)
I
Imaginary component of the vector or matrix (subscript only)
Acronyms
xv
ℜ()
Real component of the argument
ℑ()
Imaginary component of the argument
∇
Gradient
div
Divergence
Acronyms
CFL
Courant-Friedrichs-Lewy
CLD
Constrained Layer Damping
CSRBF
Compactly-Supported Radial Basis Function
ERR
Element Removal Ratio
FE
Finite Element
FEM
Finite Element Method
HJ
Hamilton-Jacobi
LSF
Level Set Function
LSM
Level Set Method
MMA
Method of Moving Assymptotes
ODE
Ordinary Differential Equation
PDE
Partial Differential Equation
PLSM
Parametric Level Set Method
RBF
Radial Basis Function
TNO
Netherlands Organisation for Applied Scientific Research
TO
Topology Optimization
1
Introduction
The design of high performance instruments often involves the attenuation of badly damped resonant modes. These resonant modes are a limiting factor to the performance of these instruments.
The Optomechatronics department at the Netherlands Organization for Applied Scientific Research
(TNO) focusses on the development and design of high performance optomechatronic instruments,
such as GAIA BAM, which is shown in Figure 1.1, and MSI VNS1 . GAIA BAM measures the angle
between two stars using two telescopes and is accurate to 24 microarcsec. To put this measurement
in perspective, it is comparable to measuring the diameter of a human hair at a distance of 1000 km.
Next to the design challenges to achieve this performance, the instruments are designed to operate
in space: a highly challenging environment. The structures are subjected to strict weight constraints
to aid the transportation of the instrument and are subjected to large temperature variations during
operation. To achieve the required performance the instruments are mostly constructed from thinwalled, monolithic components to maintain dimensional stability with limited mass. The construction
of these structures results in low intrinsic damping levels (de Vreugd et al., 2014). If badly damped
resonant modes are excited during launch, transportation or operation they might introduce significant – possibly damaging – dynamic stresses in the instrument.
Figure 1.1: Part of GAIA BAM: a high performance optomechatronic instrument developed at TNO
for angle measurement between stars.
To limit the influence of external vibrations attempts are made to isolate the instruments from these
external vibrations and by attenuating badly damped resonant modes of the structure. Current design
methods typically start from a baseline design and introduce stiffening and/or damping reinforcements
1 The
reader is referred to Gielesen et al. (2013) and Tabak et al. (2013) for more information on these instruments.
1
2
1. Introduction
(a) Unconstrained.
(b) Constrained.
Figure 1.2: An illustration of (a): unconstrained and (b): constrained layer damping methods in which
the viscoelastic material (green) is combined with structural, rigid material (gray). During resonance the
viscoelastic material is subjected to extensional or shear strains for the unconstrained and constrained
configurations.
to tune and damp specific resonant modes. In most cases the stiffening reinforcements are applied
to increase the resonance frequency of the badly damped resonant modes outside the frequency spectrum of the expected external vibration. By locally introducing damping reinforcements an attempt
is made to significantly increase the damping, such that no damaging stresses are caused by the external vibration. However, the influences of these reinforcements are difficult to predict and optimal
damping characteristics are hard to achieve. This results in a trail and error based approach until the
desired performance is found. A new design methodology is required to develop a design approach for
components with specified or optimized damping performance.
Damping reinforcements are typically realized by adding viscoelastic material to the structure. The
viscoelastic material dissipates energy when subjected to deformation. Thus, to suppress the response
of a resonant mode the viscoelastic material is placed at locations which undergo large deformation
when in resonance. The viscoelastic material is cyclicly deformed and thereby dissipates energy from
the system reducing the response to the external vibrations and improving the dynamic performance of
the instrument. Multi-material structures combining structural and viscoelastic materials are known
to provide high structural damping and are widely applied in many different fields (Johnson, 1995).
Applications are for example found in automotive and aerospace industries (Rao, 2003; Rittweger
et al., 2002; Wang et al., 2008), and to increase structural damping in civil applications (Samali
and Kwok, 1995; Wang et al., 2013). The viscoelastic material is commonly applied in two different
configurations: unconstrained and Constrained Layer Damping (CLD) (Grootenhuis, 1970). Both
methods are visualized in Figure 1.2 and are defined as:
• Unconstrained layer damping: the viscoelastic material is bonded to one or both sides of a
structural, load-carrying base member. When in resonance the flexing of the structure introduces
extensional strains into the damping layers.
• Constrained layer damping: one or multiple viscoelastic layers are sandwiched and bonded
between layers of structural, load-carrying base members. During resonance the flexing of the
multilayer structure will introduce shear strains into the damping layers.
Each component within a high performance instrument should show optimal performance to satisfy
the strict requirements and specifications. Optimal performance might be defined differently for each
component. In the case of damping reinforcements the objective is to achieve maximized damping
characteristics. The challenge in designing viscoelastic dampers is to reach optimal damping performance with limited viscoelastic material. Also, the introduced viscoelastic material should not
influence the structural integrity of the design.
To approach these optimization problems either heuristic methods or mathematical optimization routines are applied. By applying experience and knowledge from previous design iterations, the heuristic
methods are capable to generate designs which perform relatively well. However, the mathematical
optimization routines provide a rigorous, systematic approach to find optimized designs, especially
3
for designs involving challenging, multi-physical problems. In structural optimization problems three
common optimization routines are applied: size, shape and topology optimization. The latter method
allows to generate complicated, freeform structures with optimized performance for many different
design objectives. This method has become an active research topic in recent years, especially with
improvements made in finite element analysis, additive manufacturing and the increased availability of
computational power. Even though advanced topology optimization packages are becoming available
for commercial use, the technology is far from fully developed and continuous efforts are made to
improve the optimization routines.
Structural optimization has also been applied for the development of viscoelastic dampers. The objective is to maximize the damping characteristics using a combination of viscoelastic and structural
material. Initial attempts applied shape optimization to unconstrained and constrained layer dampers
(Plunkett and Lee, 1970). Various parameters of the conventional damping configurations have been
used as design variable in the optimization. Besides optimizing the layout of the structural and
viscoelastic material, the optimal placement of the viscoelastic dampers have been evaluated using
optimization routines. For example to optimize the locations of viscoelastic patches to dampen the
response of a frame subjected to external vibrations (Lunden, 1980). Recently, the routines using
size and shape optimization have been extended to topology optimization. These implementations
mainly investigated the distribution of viscoelastic materials within (un)constrained layer configurations. Numerous studies are available which investigate the optimal distribution of a limited volume
of viscoelastic material, such that maximized damping characteristics are obtained. Studies have investigated vibrating structures, such as simply supported beams, plates and shells (Zheng et al., 2004;
El-Sabbagh and Baz, 2013; Zheng et al., 2013). Even though these studies provide optimized results,
the designs are still limited to predefined (un)constrained layer damping configurations. Before the
optimization is initialized the designer has to predefine the dimensions and location of the viscoelastic
layer within the design domain, possibly limiting the achievable performance.
This work presents a multi-material, parametric topology optimization method for the design of viscoelastic dampers to overcome restrictions of existing optimization methods. The viscoelastic and
structural material are distributed simultaneously throughout the design domain to achieve optimized damping characteristics. There is no need to predefine a location of the viscoelastic layer, as
both materials are distributed simultaneously. Besides, the optimization routine is able to develop
arbitrary, freeform designs, since the viscoelastic material is not limited to a certain region within the
design domain. The ability to generate freeform viscoelastic dampers is expected to provide higher
levels of structural damping compared to conventional (un)constrained layer configurations.
In the next chapter the principles of topology optimization are introduced. Common methods of
topology optimization are evaluated and discussed to motivate the decision to apply a parametric
level set method for the optimization of viscoelastic dampers. The mathematical background of level
set-based methods and the extension to multi-material, parametric methods are presented in Chapter
3. The developed method is applied for the optimization of viscoelastic structures in Chapter 4. It
is to be noted that this chapter has been submitted as article to Journal of Vibration & Control
and the author apologizes for possible overlapping material discussed in the preceding chapters. The
article discusses the viscoelastic material model, the objective function and sensitivity analysis. Also,
two case studies are treated to illustrate the performance of the optimization routine. Moreover,
different material models are applied for the viscoelastic material to illustrate the importance to
select a viscoelastic material model which accurately describes its material behavior before initializing
the optimization. The thesis continuous with Chapter 5, which investigates the thermal behavior
of the viscoelastic dampers during operation. Chapter 6 discusses the results obtained using the
optimization routine. Finally, conclusions and recommendations are presented in Chapter 7.
2
Topology optimization
This chapter introduces the general principles of topology optimization and its application in structural design. The most common methods applied in structural topology optimization are introduced
and briefly discussed. A comparison between the performance of these methods for multi-material
applications is presented. Finally, a method is chosen for the optimization of viscoelastically damped
structures.
2.1. Optimization methods
In structural mechanics three types of optimization are most common: size, shape and topology optimization (Bendsøe and Sigmund, 2003). The methods are distinguished based on the design variables
used during the optimization and the freedom provided to the optimizer. For size optimization the
routine is initialized with a predefined topology. Design parameters used to parameterize the topology
are provided as design variable to the optimization, e.g. truss length or cross-sectional area. Only
these parameters are allowed to be modified during the optimization routine. The top figure of Figure
2.1 illustrates the shape optimization for a bridge-like structure. Shape optimization extends the size
optimization by introducing the shape of boundaries, holes and cross-sectional areas as design variable.
A simplified topology is presented as initial design for the optimization routine. The shape optimization has a larger design freedom compared to size optimization and is able to generate complicated
boundaries within the design to achieve optimized performance, as illustrated by the middle figure in
Figure 2.1. In topology optimization the optimization has even more freedom. The density of each
element in the finite element discretization is presented as design variable. During the optimization
the element densities are scaled to represent either solid or void material and thereby generate a new
topology to show optimized performance for the objective of interest. This method is illustrated in
the bottom image in Figure 2.1.
Topology optimization is a powerful technique to generate complicated, freeform designs with optimized performance for a specific objective. The method does not require any conceptual or base-line
design to initialize the optimization routine. Therefore, topology optimization will be applied for
the synthesis and optimization of vibration isolating structures. To generate viscoelastic dampers,
we require a multi-material topology optimization routine to distribute both materials throughout
the design domain. Many researchers have worked on the formulation and implementation of topology optimization routines and to provide a complete literature overview on the available methods is
outside the scope of this work. However, the following sections provide context regarding common
optimization formulations and motivate the decision to develop a multi-material, parametric level set
method for the structural optimization of vibration isolating structures.
2.1.1. Density-based optimization
In density-based optimization methods a density value is assigned to each element within the finite
element discretization of the design domain. These density, or pseudo-density, values are supplied
as design variables to the optimizer. The density of each element is scaled between solid and void
5
6
2. Topology optimization
Figure 2.1: Three most common optimization methods in structural design. The optimization methods
from top to bottom: a) size, b) shape, c) topology optimization. The initial structure is given on the
left with the optimized result on the right. Image source: Bendsøe and Sigmund (2003).
material. By scaling the element densities the optimization routine is able to develop topologies with
optimized performance with respect to the specified design objective. Either a discrete or continuous
scaling is applied for the element densities. The discrete method results in a raster of solid and void
elements. However, due to difficulties regarding discrete optimization, the most common methods scale
the elements continuously. The continuous scaling allows to apply gradient-based solution methods
to iteratively converge towards an optimized design. The pseudo-densities are scaled linearly between
zero and one, representing void and solid material. Similar scaling can be applied to the remaining
material properties of interest (Bendsøe, 1989):
Eijkl (X) = ρ(X)Ēijkl ,
0 ≤ ρ ≤ 1.
(2.1)
Here E describes the elasticity tensor corresponding to point X in the design domain. The elasticity
tensor is a scaled version of the constant tensor of the material Ē. The pseudo-density ρ can be
interpreted as the local density of the material for the element located at X. The intermediate material
properties might provide optimized performance for certain objective functions. However, from a
manufacturing point of view, the occurance of intermediate material in final designs is unwanted.
A penalization is applied to force the design towards solid and void elements. This is realized by
applying an exponential scaling of the pseudo-densities as:
Eijkl (X) = ρ(X)p Ēijkl ,
0 ≤ ρ ≤ 1,
p > 1.
(2.2)
This approach is referred to as Solid Isotropic Material with Penalization (SIMP) (Bendsøe, 1989;
Rozvany et al., 1992). Commonly a penalization parameter of p = 3 is applied to obtain designs with
minimal intermediate materials.
The density-based methods are often combined with gradient-based optimization routines to find the
optimal topology for the specified objective. The sensitivities with respect to the objective function
are determined on element level. During each iteration the element densities are modified using the
sensitivity information to achieve an updated topology with improved performance. Depending on
the complexity of the objective function and the number of design variables the sensitivities are either
calculated analytically, semi-analytically or numerically.
2.1.2. Evolutionary structural optimization
In many aspects the Evolutionary Structural Optimization (ESO) method is similar to the discussed
density-based approaches. The ESO method first emerged in publications by Xie and Steven (1993,
1997) and have since then been applied to various structural optimization problems. In contrast to
the density-based approaches the ESO method applies discrete element densities: each element is
either solid or void. The discrete densities result in a discrete optimization problem. To overcome
2.1. Optimization methods
7
difficulties regarding discrete optimization the ESO method applies a heuristic updating step to iteratively update the topology. The sensitivity number of each element is determined with respect to
the optimization objective. A ranking of all elements is created based on the value of their sensitivity
numbers. In each iteration a fraction of the elements with the lowest sensitivity are removed from
the design domain. The number of elements to remove are based on the Evolutionary Removal Ratio
(ERR), which specifies the ratio of elements removed with respect to all elements in the finite element
discretization. The ESO method is continued until a minimum volume fraction is achieved.
A downside of the ESO method is the ability to only remove elements from the structure, regardless if
the sensitivities would indicate otherwise. Especially during later iterations where the reintroduction
of material might increase the performance of the design. The possibility to reintroduce elements has
been introduced in the Bi-directional Evolutionary Structural Optimization (BESO) method. The
rejection ratio and inclusion ratio are used to determine the ratio of elements which are removed and
introduced during each iteration. A similar heuristic ranking is made based on the element sensitivities
to determine which elements rejected or included.
2.1.3. Level set-based optimization
The Level Set Method (LSM) applies a fundamentally different approach to describe and update the
topology. The method was originally developed by Osher and Sethian (1988) for the numerical computation of front and boundary propagation in surface motion problems. Since then, the level set method
has found various applications in different areas of research, such as fluid mechanics (Zhao et al., 1998),
image processing and segmentation (Li et al., 2011; Vese and Chan, 2002) and recently in structural
mechanics and optimization (Allaire et al., 2002; Wang et al., 2003). Rather than scaling element
densities, the level set method describes the boundary of the topology and propagates this boundary
until the design converges or a maximum number of iterations has been evaluated. The boundary
is extracted from a Level Set Function (LSF), which is defined as a three or four dimensional function. From this LSF a level set (isoline) or level surface (isocurve) is extracted. These functions allow
to define complicated boundaries without providing a parameterization for all curves of the boundary.
To update the topology the extracted boundary is propagated through the design domain to find
optimized performance. The update direction of the boundary is determined from a velocity field
obtained from a sensitivity analysis. The propagation is captured by a Hamilton-Jacobi convection
equation, which can be solved for a small time step to update the topology. This method focusses on
the movement of the boundary to update the topology. An alternative approach is to directly apply
local changes to the level set function and thereby introduce local changes to the structure’s topology.
In these Parametric Level Set Methods (PLSMs) a parameterization is applied to describe the level
function using shape functions and expansion coefficients (Wang and Wang, 2006; Luo et al., 2007).
These expansion coefficients allow local design changes of the topology and are used as the design
variables in gradient-based optimization routines. The update direction of the expansion coefficients is
determined from a sensitivity analysis linking a change of expansion coefficient to changes in topology
and performance. Both the conventional and the parametric level set methods are discussed in detail
in Chapter 3.
2.1.4. Black and white design
One of the challenges in topology optimization is to develop black and white designs in which only solid
or void elements remain. The occurrence of intermediate materials in optimized designs is often not
preferred, as the manufacturability of these intermediate material properties are highly challenging or
not possible with current manufacturing methods. Similar problems occur for mixtures of structural
and viscoelastic material and to create clearly separated material phases is an important aspect for
the development of the optimization routine. Commonly applied methods are briefly discussed. For
in depth discussion on these topics the reader is referred to Deaton and Grandhi (2013); Sigmund and
Maute (2013) and Bendsøe and Sigmund (2003).
8
2. Topology optimization
• Penalization. The goal of penalization is to make intermediate density values unattractive for
the optimization routine by introducing a non-linear relation between the pseudo-density value
and the material properties of interest. Common methods are SIMP (Bendsøe, 1989; Rozvany
et al., 1992) or RAMP (Stolpe and Svanberg, 2001) interpolation.
• Checkerboard patterns. These patterns describe areas in the mesh with alternating solid and
void elements. These regions provide artificially high stiffness due to bad numerical modeling of
these patterns (Diaz and Sigmund, 1995; Jog and Haber, 1996) and therefore are most common
in optimization problems in which this additional stiffness contributes to the objective function.
These patterns might appear when a penalization is applied to force the design to solid and void
elements.
• Filtering. Numerical filters are applied for multiple reasons. These filters smooth either the
density or sensitivity values in the mesh. This allows to obtain mesh-independent results and
remove possible checkerboard patterns from the design domain. However, the filtering might
introduce additional intermediate elements along the boundary of the design, which has to be
kept in mind.
2.1.5. Multi-material topology optimization
The problems encountered in single material optimization are also applicable for multi-material topology optimization. Besides intermediate density values, the occurrence of material mixtures are now
also a possibility, especially when the mixed elements provide ideal material properties regarding the
optimization objective. This might result in designs with large areas containing mixed, intermediate
elements, which are unfavored from a modeling and manufacturing perspective. To separate material
phases the density-based methods apply multi-material formulation of penalization methods, while the
ESO/BESO and LSMs introduce discretely separated material phases. Examples of multi-material
topology optimization are given for density-based (Sigmund, 2001b; Robinson, 2013), evolutionarybased (Huang and Xie, 2009) and level set-based methods (Allaire et al., 2014; Wang and Wang,
2004).
2.2. Comparison of optimization methods
The presented methods are all suited to be applied to multi-material topology optimization problems.
However, in this work we limit ourselves to the development of a single implementation using one of the
available methods. An numerical implementation has been developed for a multi-material optimization using density-based, evolutionary-based and level set-based approaches. By investigating these
methods, their performance, complexity and ease of numerical implementation became clear and provided motivation to choice one of these methods for the optimization of vibration isolating structures.
The following paragraphs will discuss the experience with each method. The numerical implementations are modified from available implementations in literature. For density based approaches the
well known 99-line and 88-line codes are used as starting point (Sigmund, 2001a; Andreassen et al.,
2010), which has been extended towards multi-material optimization for viscoelastic structures in the
master thesis of Robinson (2013). A BESO method is evaluated based on the methods described by
(Huang and Xie, 2009). A MATLAB implementation of the LSM is presented by (Otomori et al.,
2014). To the knowledge of the author, no available implementation of a parametric level set method
has been presented in literature. Therefore, the numerical implementation has been developed based
on publications which describe this method, i.e. Wang et al. (2014b); Luo et al. (2008b); Pingen et al.
(2009); Wang and Wang (2004).
The method proposed by Robinson (2013) has been used to evaluate the density-based approaches
for multi-material topology optimization. In the work by Robinson multi-material viscoelastically
damped structures were generated. The final designs show relatively large regions of intermediate
and mixed materials. Two examples of the final designs are shown in Figure 2.2. The routine seems
unable to generate designs with clearly separated boundaries, especially when the routine is initialized
without any predefined geometry. However, it has to be noted that a conventional SIMP penalization method has been applied to interpolate the material properties and that the application of more
specialized penalization methods might reduce the occurrence of mixed materials.
optimality - structure (c) has worse Q-factor and objective value than structure (a)
ditional allowed mass. Both structures had the same initial design, however in (a)
raint
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due=to0.4the; m
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; mvmax
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dd
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1 mass constraints.
stiffness
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Figure 5.4: Optimal structures with varying separate mass constraints
Figure 2.2: Two examples of designs obtained with the optimization routine developed by Robinson.
The optimization is subjected to constraints limiting the Q-factor of the designs to reach a certain level
of structural damping. The method shows multiple regions in which material mixtures occur. The red
shows the results
a study
on the script
top opt
multimat
separate.m.
scriptimages
uses are taken
andof
blue
colors correspond
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The shown
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ate mass constraints
applied(2013).
to each material. The intention of using separate constraints
vent the optimiser from generating large amounts of viscoelastic material by accepting
nalty on the objective function, as it was seen that in this circumstance the optimiser
Cdd = 6.0E5;
(e) pE1 = 5 ; pE2 = 5 ; Cdd = 5.9E5;
(f) pE1 = 5 ; pE2 = 7 ; Cdd = 6.4E5;
etimes
achieveIt aislow
Q-factor
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by filling
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referenceofimplementation.
This method
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designs
withhas
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However,
density-based
optimization
routines
for
multi-material
problems,
especially
when
pparent that the objective function is determined almost completely by the use ofmesh-independent
sti↵
Cdd
filtering techniques are applied. However, the heuristic updating methods are a major disadvantage
compared to the gradient-based optimization methods applied in density-based optimization. The
heuristic updating method requires large number of iterations and is relatively slow to converge. Two
examples of designs generated
59by this method are given in Figure 2.3. A single and multi-material
example are given for a typically stiffness maximization. The multi-material optimization using the
BESO method requires a relatively large number of iterations to converge. The selection of the rejec= 6.3E5; tion and
(h) p
; pE2 are
= 5difficult
; Cdd =to6.1E5;
(i) pcan
= 7 significant
; pE2 = 7 influence
; Cdd = on
6.6E5;
inclusion
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the behavior
E1 = 7 ratios
E1 have
Q
=
17.5
Q
=
11.4
1
1
and performance of the optimization routine. Also, the evolutionary methods can have difficulties
with strict constraints, such as volume or eigenfrequency constraints.
uctures with varying sti↵ness penalisations (rows have constant pE1 , columns constant pE2 )
in Figure 5.2 shows the e↵ect of changes in the two RAMP sti↵ness penalisation
it is possible also to penalise the mass, this is not physically realistic, so only
nalisations were considered. It can be seen immediately that the penalisation has
↵ect on the optimal structures’ objective and constraint values. Furthermore the
gies are very similar, though the shape is somewhat di↵erent. However, observing the
ots of the objective and constraints shows that the penalisation has a great e↵ect
(a) Single material.
(b) Multi-material.
ation behaviour. Higher penalisation
factors seem to be associated
with greater
ning a feasible design.
Figure 2.3: Examples of designs obtained by a single and multi-material BESO method. The designs
ws the e↵ects of
filter
radius
onforce
theis applied
secondat variable.
intention
arechanging
fixed on the the
left side,
while
a vertical
the middle ofThe
the right
boundary. The designs
arefiner
optimized
to maximize
theirform
stiffness.
the multi-material
optimization
two elastic materials are
estigate whether
structures
would
andInimprove
the optimal
performance,
used. The red materials has a higher stiffness than the blue material.
radius was lower.
However, it can be seen that the filter radius in fact has a very
the topology and performance. All structures are feasible to within an acceptable
A level set implementation is presented by Otomori et al. (2014) for a minimum compliance topology
achieve the same
objective value to within 5%. Looking at the structures formed,
optimization. In this method the boundary is propagated by solving the Hamilton-Jacobi convection
he same topology,
only finite
di↵erence
being
the decreased
sharpness
as the
equationwith
usingthe
a upwind
differencing
routine.
During the iterations
a regularization
is applied
reases. Structure
(a)thedisplays
additional
di↵erences
since
filter is inactive
to keep
level set slight
function
smooth and
well defined.
The the
implementation
has been limited to a
material phase,
due toofthedamping
complexitymaterial
of the mathematical
derivations
of the
shape and topologre - thereforesingle
single-element
layers
can be seen
on the
outer
ical
derivatives
required
for
the
boundary
propagation
and
hole
nucleation.
The
method
shows good
he sti↵ structural members.
performance for relatively coarse meshes. However, when decreasing the element sizes the method
58
10
2. Topology optimization
quickly becomes computational expensive, even for a single material phase.
Finally, the parametric method has been investigated. The numerical formulation is generated based
relevant publications by Wang et al. (2014b); Luo et al. (2008b); Pingen et al. (2009); Wang and Wang
(2004). The method can be seen as a mid-way between a density and level set-based formulations
(Sigmund and Maute, 2013). The parameterization allows to fully describe the level set function using
the corresponding expansion coefficients. These coefficients are then presented to a gradient-based
optimization routine, similar to the element densities in a density-based implementation. The boundaries of the material domains are extracted from the level set functions and relatively small regions
of intermediate, mixed materials are observed along the boundary between both material phases.
The computational effort is similar to a density-based implementation, since the same gradient-based
optimization routines are applicable. Two examples of designs generated by the parametric level set
method are given in Figure 2.4. The method provides multi-material designs with clearly separated
boundaries and little intermediate or mixed material regions. It has to be noted that the choice of
basis functions, their influence radius and the initial material distribution can have an influence on
the performance of the optimization method. However, the values mentioned in literature seem to
result in well-performing routines.
40
40
35
35
30
30
25
25
20
20
15
15
10
10
5
5
10
20
30
40
50
(a) Single material.
60
70
80
10
20
30
40
50
60
70
80
(b) Multi-material.
Figure 2.4: Examples of designs obtained by a single and multi-material parametric level set method.
The designs are fixed on the left side, while a vertical force is applied at the middle of the right boundary.
The designs are optimized to maximize their stiffness. The graphs illustrate the material distribution
based on a contour plot of the level set functions. In multi-material optimization two elastic materials
are used. The red material has a higher stiffness than the blue material.
The decision is made to develop a multi-material, parametric level set method for the structural
design optimization of vibration isolating structures. The close relation to both density and level
set-based optimization routines make the parametric level set method a potential improvement on
the formulations presented by Robinson. The method captures advantages of both density and level
set-based formulations. The formulation allows to apply gradient-based optimization algorithms to
update the topology, while still providing an exact boundary description of the level set function. The
mathematical framework and the parameterization of the level set-based approaches are discussed in
the following chapter.
3
Parametric level set method
This chapter presents the mathematical background of the level set-based methods. A parameterization of the level set function is applied to arrive at the parametric level set method, which is extended
to simultaneously describe multiple material domains.
3.1. Level set method
In structural mechanics the level set function is defined as a three or four dimensional function, which
allows to describe a two or three dimensional topology by extracting a level set or level surface. The
reference domain and the level set function are mathematically defined as:
Let Ω ⊆ Rd (d = 2 or 3) be an open and bounded set occupied by the region of a material
phase. Then, the shape boundary is embedded such that Γ = ∂Ω of the region as the zero
level set of the implicit function φ(X) : Rd+1 → R as
Γ = {X : φ(X) = 0}.
(3.1)
In many implementations the zero level set is chosen as reference set to extract the structure’s topology.
This results in three distinguishable domains described by the level set function:


φ(X) > 0, ∀X ∈ Ω\Γ
(material),

(3.2)
φ(X) = 0, ∀X ∈ Γ
(interface),


φ(X) < 0, ∀X ∈ D\Ω
(void).
In this notation D corresponds to the complete design domain in which the material domain Ω is
located and X specifies any point within the design domain. An illustration of a level set function
and the corresponding boundary is presented in Figures 3.1a and 3.1b. In this example the boundary
Γ is extracted from the level set c and encloses the domain Ω. The level set function allows us to
define this boundary without providing a parameterization for any curvatures of the boundary, as
this boundary is found directly from the extracted level set function. Therefore, changes applied to
the level set function will directly modify the topology. This can be illustrated by applying a vertical
translation to the level set function. During the translation a position occurs where the left and right
side of the domain are connected by a single point. When continuing to translate the level set function,
the domain Ω now describes two separated areas within the design domain, as illustrated by Figures
3.1e and 3.1f. No additional descriptions are required to describe the separation of the boundary into
separate domains, which allows us to describe complicated, evolving boundaries effortlessly.
11
12
3. Parametric level set method
(a) Level set function φ(X).
(b) Domain Ω and boundary Γ obtained
from the level set function φ(X).
(c) Level set function φ(X).
(d) Domain Ω and boundary Γ obtained
from the level set function φ(X).
(e) Level set function φ(X).
(f) Domain Ω and boundary Γ obtained
from the level set function φ(X).
Figure 3.1: An illustration of the level set function φ(X) and the corresponding domain Ω and boundary
Γ as described by the level set φ = c. Figures (a,b) give the level set function and boundary for the
initial position. The figures (c-f) illustrate the changing boundary as a result of a vertical translation of
the level set function. The translation results in a separation of the boundary in two separated domains.
3.1. Level set method
13
The level set method is an interesting approach for structural topology optimization since the method
is able to describe complicated geometries and the boundaries are directly modified by a change of the
level set function. For structural topology optimization we need to track the changes of the boundary
of the level set function to iteratively determine design updates to improve the structure’s performance. Therefore, a pseudo-time variable t is introduced to express the boundary changes through
time. The zero level set function φ(X, t) = 0 becomes a function of the spatial coordinate X and the
pseudo-time variable t.
The evolution of the zero-level set function can be described by a Hamilton-Jacobi (HJ) convection
equation and is obtained by differentiation of the level set function (Osher and Sethian, 1988). This
results in the following Partial Differential Equation (PDE):
∂φ(X, t)
dX
dφ(X, t)
=
+ ∇φ ·
= 0,
dt
∂t
dt
(3.3)
where dX/ dt is obtained from the shape sensitivity analysis and provides the design boundary velocity
field. This velocity field forces the boundary to move in the direction that improves the structure’s
performance. Any infinitesimal displacements in the tangential direction to the boundary will not
result in any boundary change. Allowing us to write dX/ dt = vn n with vn a scalar normal velocity
field and n the outward normal to the boundary (Allaire et al., 2004; Wang et al., 2003; van Dijk,
2012). The outward normal to a level set curve is defined as n = −∇φ/||φ||. Substituting these
relations in the Hamilton-Jacobi convection equation (3.3) results in:
dφ(X, t)
∂φ(X, t)
=
− ||∇φ||vn (X) = 0.
dt
∂t
(3.4)
The HJ equation is solved by numerical integration of a given pseudo-time step. The solution of this
integral is used to update the level set function φ and thereby update the topology of the structure.
Transporting the boundaries allows to describe complicated topologies and is able to merge the domain at the boundary. The design updates originate at the boundary of the level set function and the
current formulation is therefore not able to nucleate holes within already defined material domains.
Often, the optimization is initialized with many small holes to reduce the probability that the creation
of additional holes is required during the optimization.
A more mathematical approach is to introduce additional terms to the HJ equation, such that nucleation becomes possible within the solid domains. Applying the formulation as presented by van Dijk
(2012) the generalized HJ equation is given as:
∂φ(X, t)
− ||∇φ||vn (X) − D(φ) − R(φ) = 0,
∂t
(3.5)
where D is a diffusive term and R a reactive term. Since the diffusive term is mainly related to the
numerical treatment of the HJ equation it is not treated in detail. The reactive term R provides
source and sink terms within the solid domain. These reactive terms allow the optimization to nucleate holes within the solid domains and thereby overcome the nucleation problem of the original
level set method. A topological derivative is derived, which provides the sensitivity information for a
nucleation of infinitesimal holes in the design domain (Burger et al., 2004; Otomori et al., 2014). The
required derivation can become involved for complicated objective functions. Based on the sensitivity value a hole might be nucleated. The nucleation can be performed continuously throughout the
iterations or at predefined sets of iterations.
The topology is updated by integrating the HJ equation for a certain pseudo-time step. The solution
provides an new topology, which is used to update the topology of the current iteration. To solve
the HJ equation a forward finite differencing method is applied (Osher and Ronald, 2003; Zhao et al.,
1998; Otomori et al., 2014). The iterative updates are continued until the boundary converges or a
maximum number of iterations are evaluated. The implementation of the numerical integration is
non-trivial due to constraints to enforce numerical stability. The stability is enforced by the CourantFriedrichs-Lewy (CFL) condition, which limits the allowable time steps of the numerical integration.
14
3. Parametric level set method
Figure 3.2: Illustration of a cross section of the level set function and the applied parameterization by
the shape functions ξi and expansion coefficients αi . The level set φ = c is drawn to illustrate a level set
defined to extract a boundary from the LSF. Image modified from van Dijk et al. (2013).
The stability constraint is a function of the time step as well as the element size of the finite element
discretization. When reducing the mesh size the CFL condition requires a reduction of the pseudotime steps to ensure numerical stability. Thus, decreasing the mesh size will result in a significant
increase of computational effort for each iteration.
3.2. Parametric level set method
The parametric level set method is a variation on the level set-based approaches and can be seen as a
midway between density and level set-based optimization routines. Compared to the level set-based
approaches the parameterized method does not require complicated formulations for shape and topological derivatives and overcomes difficulties involved in the numerical integration of the HJ convection
equations. The parameterization describes the level set function with a combination of shape functions
and expansion coefficients. The topology is updated by changing the scaling of the different expansion coefficients. The parameterization allows to determine the expansion coefficient updates using
gradient-based optimization routines, such as the Methods of Moving Asymptotes (MMA) (Svanberg,
1987). The parametric level set method also allows for the nucleation of holes within the domain by
allowing negative values for the expansion coefficients. The remainder of this section discusses the
parametric level set method for topology optimization in detail and describes the used shape functions
and the modifications required for a multi-material implementation.
The level set function is parameterized by shape functions ξi (X) with corresponding expansion coefficients αi (t). The shape functions are only a function of the spatial coordinates X, while the expansion
coefficients are only a function of the pseudo-time variable t. The level set function is parameterized
by a summation of scaled shape functions for all N elements in the design domain:
φ(X, t) =
N
∑
ξi (X) αi (t).
(3.6)
i=1
Figure 3.2 illustrates a cross section of level set function parameterized by shape functions. The
parameterization results in a parametric problem in which we iteratively search for optimized values
of expansion coefficients to achieve an optimal topology (Wang and Wang, 2006; Luo et al., 2008b).
Two methods are available to update the topology using shape functions. The first is to substitute
the parameterization in the Hamilton-Jacobi convection equation (3.4) to obtain:
N
∑
i=1
ξi (X)
N
∑
(
)
dαi (t)
− vn
|| ∇ξi (X) αi (t)|| = 0.
dt
i=1
(3.7)
This allows us to find an expression for the velocity field for each element in the domain, given by:
3.2. Parametric level set method
15
∑N
i (t)
ξi (X) dαdt
vn = ∑N i=1(
.
)
i=1 || ∇ξi (X) αi (t)||
(3.8)
By discretizing the design domain and applying the parameterization, the set of PDEs is reduced to a
set of Ordinary Differential Equations (ODEs) Wang and Wang (2006). This set of equations can be
solved by a variety of different ODE solvers to update the expansion coefficients by integrating along
the pseudo-time variable. The optimization routine becomes equivalent to transporting the level set
function by updating the values of the expansion coefficients by solving the set of ODEs.
The second method applies a gradient-based optimization routine to update the value of the expansion
coefficients and thereby update the corresponding topology. The sensitivity of each element to the
level set function is determined analytically and presented to the gradient-based routine. During
each iteration the optimization routine updates the values of the expansion coefficients to improve
the performance of the topology. In this work the second method is applied to generate design
updates using a gradient-based routine. The formulation of the objective function is discussed in the
next chapter. The adjoint sensitivity analysis is mentioned in Section 4.3.4 with additional details
presented in Appendix C.
3.2.1. Shape functions
Various shape functions are available for the parameterization of the level set function. The choice of
shape function to apply is not straightforward, since the application of different shape functions might
influence the performance of the optimization routine. These functions influence the computational
effort involved in the parameterization, the radius of influence of each design variable and also the
sensitivity response at the level set. The general differences between shape functions are discussed in
the next sections. A detailed discussion regarding the choice of shape functions is presented by (van
Dijk et al., 2013, section 2).
Two discretization are required for the parametric level set method: one for the structural design and
the other for the placement of the shape functions. In many implementations the same discretization
is used for both applications (Wang et al., 2015; Luo et al., 2008b). This simplifies the numerical implementation, since the centers of the shape functions are aligned with the centers of the elements in
the finite element discretization. Figure 3.3a illustrates the parameterization using a single discretization for the finite element discretization and the shape function placement in which the expansion
coefficients αi are aligned with the center of the ith element. Another, more complicated approach is
two apply two different discretizations, for example by applying a finer mesh for the structural design
(Pingen et al., 2009). The finer mesh allows for more detailed analysis of the structural mechanics.
However, some form of interpolation has to be performed to relate the values of the nodes in the level
set function to the nodes in finite element discretization. In the presented implementation the same
discretization is applied for the structural design and the level set parameterization, as this simplifies
the numerical implementation of the parametric level set method.
The influence (or support) radius of the shape functions is one of the most important parameters
in the parametric level set methods. This influence radius determines the radius within the shape
function takes on a non-zero value. Depending on this radius, a single shape function might be limited
within a single element in the discretization or extend throughout the complete design domain. Three
types of level set functions are distinguishable based on their influence radius: local, mid-range and
global shape functions.
Local basis function
These shape functions are only non-zero in a small, finite part of the design domain and show minimal
overlap with neighboring elements. The influence of such a basis function is limited to the support
of the corresponding node of the finite element discretization. A change in expansion coefficient will
result in a relatively small, localized change of the level set function. These basis functions generally
require more iterations to converge towards an optimized design.
16
3. Parametric level set method
(a) Radial basis function placement.
(b) Compactly-supported radial basis function.
Figure 3.3: (a): Illustration showing the cross section of the radial basis function placement. The
RBFs are aligned with the elements in the finite element discretization. The shape functions ξi and the
expansion coefficients αi are aligned with the ith element in the design domain. (b): Illustration of a
compactly-supported radial basis function as described by Equation (3.9).
Mid-range basis function
For mid-range basis functions the influence radius is slightly larger compared to the local shape
functions. These functions show substantial overlap a number of neighboring elements. Changes of
a single expansion coefficient will therefore have a larger effect on the shape of the level set function.
Commonly radial basis functions (or similar) are applied as mid-range basis function with an influence
radius of two to four times the mesh size.
Global basis function
The influence radius of global basis functions span almost the entire design domain. Most of the basis
functions will overlap in large parts of the domain. Each expansion coefficient has a slight influence
on almost the complete level set function (de Ruiter and van Keulen, 2004; Wang and Wang, 2006).
Compared to the local and mid-range shape functions, the global functions allow to introduce large
design changes by small changes of design variables as the influence of the design variables will propagate throughout most of the design domain. The large influence radius results in high computational
effort for both the parameterization and the sensitivity calculation. Each design update has to be
determined carefully, as small changes influence the complete design domain and various combinations
of shape functions will result in the same topology.
In the presented work a radial basis function with mid-range influence radius is applied. The implementation is realized using compactly-supported RBFs. These basis functions provide a midway
between local and global basis functions. The mid-range influence radius provides overlap between
neighboring elements and extends the influence of each expansion coefficient to a slightly larger area,
while keeping the computational effort limited. Moreover, CSRBFs are shown to perform well in
structural optimization (Luo et al., 2008b; Wang et al., 2014b).
The implemented CSRBFs are described by Wendland (1995) and are defined as
{
ξi (X) =
0,
for ri (X) ≥ 1,
(
)4
(1 − ri (X) (4ri (X) + 1), for ri (X) < 1.
(3.9)
The distance ri determines the distance of a point Xi with the current position X and thereby limits
the non-zero values of the shape function to its influence radius R. The radius ri is given as
ri (X) =
||X − Xi ||
.
R
(3.10)
3.2. Parametric level set method
17
Void
Viscoelastic material
Void
Structural material
D
Figure 3.4: Illustration of a multi-material structure defined by two level set functions. The first level
set function φ1 determines the placement of material, while the second level set function φ2 distinguishes
between structural and viscoelastic material. For both level set functions the zero-level set is used to
extract the boundary.
The CSRBFs have shown good performance with an influence radius between two and four times the
mesh size. In the presented work, similar radii have provided good performance for the structural
optimization. Applying shape functions which extend along multiple parameters has similar effects as
filtering techniques in density-based optimization. Figure 3.3b presents an illustration of this radial
basis function. The material discretization and the numerical implementation of the parametric level
set method are further discussed in Appendix B.
3.2.2. Multi-material formulation
For the structural optimization of viscoelastic dampers we have to simultaneously distribute both
viscoelastic and structural material in the design domain. Therefore, the presented formulation has
to be extended to introduce a second material phase within the domain. Introducing a second level
set function allows to describe the additional material phase. The material properties of a specific
point in the domain are then determined based on the values of both level set functions. In this work
a multi-material formulation is applied, which is based on the formulation presented by Wang et al.
(2015). The formulation is slightly modified to allow for a structural and viscoelastic material to be
defined by both level set functions. In general, a multi-material formulation using k level set functions
will describe the following domains:

k
k
k


(material),
φ (X) > 0, ∀X ∈ Ω \Γ
k
k
(3.11)
φ (X) = 0, ∀X ∈ Γ
(interface),


φk (X) < 0, ∀X ∈ D\Ωk
(void).
This introduces k boundaries within the design domain. Only two level set functions are required to
define a structural and viscoelastic material phase. The first level set function is applied to distribute
any material within the domain. Then, the second level set function distinguishes between viscoelastic
and structural material inside the material domain specified by the first level set function. The material
boundaries are extracted at the zero-level set, which provides the following material domains:

1


void,
φ < 0
1
2
(3.12)
φ ≥ 0, φ < 0 structural material,


φ1 ≥ 0, φ2 ≥ 0 viscoelastic material.
Figure 3.4 illustrates a material distribution constructed from two level set functions and its ability to
describe both material regions and void within the design domain. In the next chapter the presented
parametric level set method is applied for the optimization of viscoelastically damped structures.
4
Topology optimization of
viscoelastically damped structures
In this chapter the parametric, multi-material level set is applied for the structural design optimization of vibration isolating structures. The goal of the optimization is to achieve maximum damping
characteristics by simultaneous distribution of structural and viscoelastic material using the methods presented in Chapter 3. This chapter has been submitted to Journal of Vibration & Control
and might partially overlap with material discussed in the previous chapters. The article discusses
the modeling of the viscoelastic material, the formulation of the optimization problem and presents
multiple numerical examples. Addition details regarding the applied viscoelastic material model are
presented in Appendix A, the details on the numerical implementation are given in Appendix B and
the complete derivation of the sensitivity analysis is presented in Appendix C.
19
20
4. Topology optimization of viscoelastically damped structures
Multi-Material Topology Optimization of
Viscoelastically Damped Structures using a Parametric
Level Set Method
M. van der Kolk†,‡ , G.J. van der Veen† , J. de Vreugd‡ and M. Langelaar†
†
‡
Delft University of Technology, Delft, The Netherlands
Optomechatronics, Netherlands Organisation for Applied Scientific Research, Delft, The
Netherlands
Abstract
The design of high performance instruments often involves the attenuation of poorly damped resonant modes. Current design practices typically rely on informed trial and error based modifications
to improve dynamic performance. In this article, a multi-material topology optimization approach is
presented as a systematic methodology to develop structures with optimal damping characteristics.
The proposed method applies a multi-material, parametric, level set-based topology optimization
to simultaneously distribute structural and viscoelastic material to optimize damping characteristics. The viscoelastic behavior is represented by a complex-valued material modulus resulting in a
complex-valued eigenvalue problem. The structural loss factor is used as objective function during the
optimization and is calculated using the complex-valued eigenmodes. An adjoint sensitivity analysis is
presented that provides an analytical expression for the corresponding sensitivities. Multiple numerical examples are treated to illustrate the effectiveness of the approach and the influence of different
viscoelastic material models on the optimized designs is studied. The optimization routine is able to
generate designs for a number of eigenmodes and to attenuate a resonant mode of an existing structure.
Keywords: viscoelastic damping, topology optimization, multi-material optimization, level set method,
loss factor, modal analysis, constrained layer damping
4.1. Introduction
This paper addresses the design of structures composed of both viscoelastic and structural material, to achieve optimal damping characteristics. This is implemented using a multi-material level
set-based topology optimization. The design of high-performance instruments often involves the attenuation of poorly damped resonant modes. At the Netherlands Organisation for Applied Scientific
Research (TNO) this is encountered during the design of high performance optomechatronic instruments (de Vreugd et al., 2014), such as GAIA BAM (Gielesen et al., 2013) and MSI VNS (Tabak et al.,
2013). Current design approaches typically start from a baseline design and introduce stiffening or
damping reinforcements to tune and damp these modes. However, the influence of these reinforcements
is difficult to predict, resulting in a trial and error approach. We propose an integrated multi-material
topology optimization approach as a systematic methodology for the development of these structures.
Designs involving multiple appropriately distributed materials, in specific the combination with viscoelastic materials, are known to provide high structural damping and have been applied in many
fields (Johnson, 1995), for example in automotive and aviation (Rao, 2003), aerospace (Rittweger
et al., 2002; Wang et al., 2008), and civil applications (Samali and Kwok, 1995; Wang et al., 2013).
Viscoelastic materials dissipate energy when subjected to deformation. To achieve optimal damping
the viscoelastic material has to be placed at locations which undergo large deformation during vibration. Besides, the design of the construction itself should promote deformation of the viscoelastic
material. The two most common damping geometries for viscoelastic dampers are unconstrained and
constrained layer damping (Grootenhuis, 1970). Here, the viscoelastic material is constrained on one
or both sides. During vibration the viscoelastic material is forced to deform and thereby dissipate
energy.
4.1. Introduction
21
The development of viscoelastic dampers presents a challenging optimization problem, where the goal
is to achieve optimal structural damping. Both the location as well as the geometry of the viscoelastic
layer can be determined using an optimization routine. In previous studies both analyses have been
performed separately, for example, the placement of viscoelastic patches along a vibrating frame (Lunden, 1980) and the shape optimization of individual, unconstrained and constrained viscoelastic layers
(Lumsdaine and Scott, 1998; Plunkett and Lee, 1970). The loss factor (Kerwin, 1959) is determined
for each structure to quantitatively compare the amount of structural damping for each design.
More recent studies have applied topology optimization to find optimized damping characteristics by
distribution of viscoelastic material in the design domain. The optimization method searches for the
distribution of viscoelastic material to obtain the highest loss factors for single or multiple modeshapes.
These optimization routines are implemented for unconstrained layer damping on plates (El-Sabbagh
and Baz, 2013) and shell structures under harmonic excitations (Kang et al., 2012). Here, an additional layer of viscoelastic material is optimally distributed on top of the plates to dissipate energy
during vibrations. Similar approaches are available for constrained layer treatments: for simply supported beams (Zheng et al., 2004) and vibrating plates (Ling et al., 2011; Zheng et al., 2013). In these
designs the viscoelastic layer is sandwiched by two beams or plates and its distribution within this
layer is optimized. However, these optimization routines limit the design domain of viscoelastic material to the predefined (un)constrained layer. Before the optimization, the location of the viscoelastic
layer is to be provided by the designer.
Multiple modeling approaches have been presented to model the viscoelastic behavior. The material
can be represented using models based on springs and dashpots, for example, the Maxwell or KevinVoigt models (Bert, 1973) and extensions hereof. Alternatively, the elastic and dissipative behavior
can be represented by a complex material modulus (Grootenhuis, 1970). For the modeling of harmonically excited structures, this latter method is widely applied, for example, by (Kerwin, 1959; Johnson,
1995) and the previously mentioned optimization routines. The studies apply different formulations
of the complex modulus. For example, the authors of (Lunden, 1980; Rao, 1978; El-Sabbagh and
Baz, 2013) apply a complex Young’s modulus, while the authors (Liu et al., 2013; Ling et al., 2011)
only apply a complex shear modulus. When the complex modulus is limited to the shear modulus,
only deformations with shear components dissipate energy. For the analysis of beams or plates using
constrained layer configurations, the difference between using a complex Young’s or shear modulus
might not result in significant performance changes: the obtained modeshapes show predominately
shear deformation within the viscoelastic layer. Yet, when optimizing arbitrary geometries the choice
of complex modulus might steer the optimization towards specific designs. In this paper we devote
specific attention to this point.
The implementation of the complex material modulus provides a complex-valued stiffness matrix,
resulting in a complex-valued eigenvalue problem to determine the structure’s natural modes. In the
mentioned studies, a number of approximations are made regarding the calculation of the loss factor
and its sensitivities. In some papers the loss factors are calculated using the eigenmodes obtained from
the undamped analysis (Kim, 2011; Liu et al., 2013; Ling et al., 2011; Zheng et al., 2013). However,
for low structural damping, we have observed that the eigenmodes of the undamped and damped
structure are approximately the same. At higher structural damping it will be required to move towards the complex modeshapes for the calculations. Similarly, it is observed that the sensitivities of
the loss factor are determined using approximations. For example, by neglecting the contribution of
the modeshape sensitivity to changes in the design (Ling et al., 2011), or by using an approximate
formulation (Wang et al., 2014a; Zheng et al., 2013).
Several multi-material topology optimization routines are available for various optimization methods,
such as, the density methods by (Sigmund, 2001b; Yin and Ananthasuresh, 2001), the evolutionary
implementation by (Huang and Xie, 2009) and the level set methods by (Wang and Wang, 2004;
Allaire et al., 2014). For multi-material or multi-phase optimization, the level set method has the
advantage of distinct separation between the different materials. To overcome numerical difficulties
from tracking the boundaries and solving the Hamilton-Jacobi convection equations, the level set
functions have been parameterized (Wang and Wang, 2006; Luo et al., 2007). The parametric level
22
4. Topology optimization of viscoelastically damped structures
set methods describe the level set function by a set of shape functions and expansion coefficients, while
maintaining distinct separation between material phases. A recent overview of the level set methods
and parametrization approaches is given in (van Dijk et al., 2013). The parametric level set methods
have recently been extended towards multi-material optimization (Wang et al., 2015).
In this article a multi-material topology optimization formulation is developed. Here, the viscoelastic
and structural material will be distributed simultaneously to achieve optimal damping characteristics. Within the user-specified bounds the obtained designs can have arbitrary material distributions
and geometries, possibly resulting in higher levels of structural damping. Also, there is no need to
predefine a location of the viscoelastic layer by the designer. Furthermore, the calculation of the loss
factors is performed using the complex-valued modeshapes and an exact formulation is proposed for
the loss factor sensitivities.
The article is organized as follows. Section 4.2 describes the modeling of the viscoelastic material
and investigates the obtained loss factors definitions for a constrained layer configuration. Then,
Section 4.3 provides the formulation of the parametric, multi-material level set method, the objective
and constraints, and the adjoint sensitivity analysis. In Section 4.4 the results of two case studies
are presented and discussed. The paper ends with conclusions regarding the proposed optimization
routine.
4.2. Modeling of Viscoelastic Material
In this paper, the viscoelastic material is described using a complex material modulus. The formulation
of the material model is discussed in the following sections. Furthermore, the structural loss factor
is introduced as measure to compare the structural damping between different structures. Different
methods to calculate the loss factor are shown and their performance to represent the structural
damping for viscoelastically damped structures are compared.
4.2.1. Complex material modulus
When a viscoelastic material is deformed dynamically the corresponding stresses are not in phase
with the applied strains (Tschoegl, 1989). For a harmonically excited structure the stress will lead
the applied strain by an angle θ. The strain ε and stress σ are represented as
ε(t) = ε0 sin(ωt)
(
)
σ(t) = σ0 (ω) sin ωt + θ(ω) .
(4.1)
For harmonically excited structures the observed phase angle can be represented by applying a complex
material modulus. The complex material modulus is found by dividing the stresses by the strains
(Tschoegl, 1989; Grootenhuis, 1970). This results in
E(ω) = E(ω)′ + iE(ω)′′ .
(4.2)
√
The Young’s modulus is given by a complex quantity, with i being the complex number i = −1. The
material modulus is given by the storage modulus E ′ and the loss modulus E ′′ . These components are
responsible for the elastic and dissipative behavior, respectively. Many different models are available to
describe the viscoelastic behavior. The complex material model is applied for its relative simplicity and
ease of modeling responses of viscoelastic structures to harmonic excitations. For a general viscoelastic
material the moduli depend on both temperature and excitation frequency. In the presented work
the material moduli are assumed constant to simplify the material description. The storage and loss
modulus are related through the material loss factor η and the loss angle θ:
E ′′
= tan(θ) = η.
(4.3)
E′
A similar approach can be applied to define a complex bulk (κ) or shear (G) modulus. Depending
on the applied viscoelastic material any complex modulus can be chosen to represent the dissipative
behavior:
E = E ′ (1 + iη),
κ = κ′ (1 + iη),
G = G′ (1 + iη).
(4.4)
4.2. Modeling of Viscoelastic Material
23
When a complex bulk or shear modulus is applied, the viscoelastic material only dissipates energy
when the deformation contains the corresponding bulk or shear components. However, when a complex
Young’s modulus is applied the viscoelastic material will dissipate energy for any deformation.
The behavior of the continuum body with a structural and viscoelastic material is described by the
following equilibrium condition:
{
div (Cχ ε(u)) + f − ρχ ü = 0 in D,
(4.5)
u=0
on ΓkD .
The constitutive constant is scaled using a Voigt mixing law and is based on the presence of the
material phases k: Cχ = C k χk , where χk is a function of phase Ωk . A similar scaling is applied
for the material density ρχ . The applied body forces are given by f , the displacement by u and the
strains by ε. To account for the phase lag between stresses and strains the elastic coefficients with Cχ
can take on complex values. For the remaining analysis the system is discretized in finite elements
providing the following discretized equations of motion. Due to the application of complex-valued
elastic coefficients the stiffness matrix is complex-valued:
M ẍ + (KR + iKI ) x = f ,
(4.6)
with M the global mass matrix, KR and KI respectively the real and imaginary parts of the global
stiffness matrix, x the nodal displacements and f the forces applied to the structure. The imaginary
components of the stiffness matrix model the dissipative behavior. The applied viscoelastic material
model also results in a complex eigenvalue problem:
(
)
−λ2 M + (KR + iKI ) ϕ = 0.
(4.7)
The obtained eigenfrequencies λ and eigenmodes ϕ are therefore complex-valued.
The harmonically excited, viscoelastic structures show a steady-state oscillatory deformation. Part
of the excitation energy is stored elastically, while the remainder is dissipated within the viscoelastic
material. In a stress-strain diagram, this dissipative oscillation results in a hysteresis loop. The
dissipated energy per unit volume during each cycle is determined from the contour integral along
the hysteresis loop. The hysteresis loop is constructed by plotting the stress σ(t) as function of the
strain ε(t). The dissipated energy per unit volume for a cycle Wcyc has been derived by Tschoegl
(1989). It has to be noted that this derivation can be performed for either choice of complex material
modulus. To illustrate the obtained expressions, the derivation has been performed with a complex
shear modulus, resulting in the following expressions for the dissipated energy:
∮
Wcyc = σ(t) dε(t) = πε20 G′′ .
(4.8)
By multiplication with the corresponding excitation frequency the dissipated power per unit volume
is found. Since the eigenfrequencies are given in rad/s the result is divided by 2π to obtain power:
Qdissip =
1 2 ′′
ε G .
2 0
(4.9)
With the complex material modulus, the dissipated energy can also be determined from the imaginary
part of the stiffness matrix in combination with a modeshape corresponding to the deformation. Then,
the dissipated energy per cycle is given as
Wcyc = πϕH KI ϕ.
(4.10)
The superscript ϕH indicates the conjugate transpose.
4.2.2. Structural loss factor
To quantitively compare the damping between different structures the structural loss factors are
determined. A general formulation of the structural loss factor has been proposed by Johnson and
24
4. Topology optimization of viscoelastically damped structures
Structural
Viscoelastic
Figure 4.1: Illustration of the clamped cantilever beam containing a constrained viscoelastic layer. The
arrow indicates the location of the excitation force and position measurement during dynamic loading.
Kienholz (1982). The structural loss factor describes the ratio between dissipated and stored energy.
The structural loss factor for eigenmode r is given by
γ̄r =
ψrT KI ψr
.
ψrT KR ψr
(4.11)
In the proposed formulation (4.11) the modeshapes ψr are based on the undamped system and
therefore real-valued. The used modeshapes are mass normalized and are determined by leaving out
KI from the eigenvalue problem in equation (4.7):
(
)
−λ2 M + KR ψ = 0.
(4.12)
The symbol γ̄ indicates the loss factors based on the undamped eigenmodes ψ. Applying the undamped eigenmodes introduces an approximation in the calculation of the structural loss factors. The
imaginary components introduced by the viscoelastic material are neglected. A revised formulation
has been proposed by Xu et al. (2002) to provide a better approximation of the loss factor for high
material loss factors. However, the revision is still limited to the undamped eigenmodes. Therefore,
we propose to perform the structural loss factor calculation using the complex-valued modeshapes ϕ
from the complex-valued eigenvalue problem from equation (4.7). Then, the structural loss factor is
given as:
ϕH K I ϕr
γr = Hr
.
(4.13)
ϕr KR ϕr
To compare the performance of both formulations, equations (4.11) and (4.13), the obtained structural
loss factors are compared to the Q-factor of the system. The structural loss factor and Q-factor are
related via the used equivalence:
γr = Q−1
(4.14)
r .
The Q-factor is determined based on the frequency response of the structure. The frequency response is
obtained by performing harmonic response analysis in the analyzed frequency range. The 3dB method
is applied to extract the Q-factor from the frequency response by dividing the resonant frequency by
the half-power bandwidth, as described by (Bert, 1973). A constrained viscoelastic layer is used as
example. Here, aluminum is applied as structural material with Young’s modulus 70 GPa, density
2.7 × 103 kg/m3 and Poisson ratio ν = 0.3 and for the viscoelastic material: Young’s modulus 1 GPa,
density 1 × 103 kg/m3 and Poisson ratio ν = 0.3. The analyzed beam has a length of 2.25 m, a height
of 0.15 m and unit thickness. Figure 4.1 illustrates the structure. The viscoelastic layer is centered
along the beam with a total height of 0.05 m. The structure illustrates a rather extreme scenario
in which the viscoelastic material has significant contribution to the deformation. For the first two
eigenmodes both structural loss factors and Q-factor are determined. Furthermore, the comparison is
performed for different values of the material loss factor η. The results are summarized in Table 4.1.
The relative difference between the Q-factor and the structure’s loss factor is significantly reduced
by application of the complex-valued modeshapes ϕ. This reduction is mainly applicable for designs
in which viscoelastic materials with high material loss factors are used and where large parts of the
viscoelastic material are subjected to deformation for the analyzed modeshapes.
4.3. Topology optimization of viscoelastic and structural material
The simultaneous distribution of both viscoelastic and structural material is realized using a multimaterial topology optimization routine. The optimization method is mainly based on the study
4.3. Topology optimization of viscoelastic and structural material
25
Table 4.1: Comparison between the structural loss factors and Q-factor of a structure containing a
constrained viscoelastic layer (Figure 4.1). The loss factors γ̄ and γ are determined using the undamped
and damped eigenmodes. The relative difference is calculated between the structural loss factors and
the inverse of the Q-factor to illustrating the improved prediction of damping behavior when using the
complex-valued eigenmodes for the loss factor calculation.
η
Mode
Q−1
γ̄r
Relative difference
γr
Relative difference
1.00
1
0.079
0.139
0.774
0.079
0.001
2
0.289
0.416
0.434
0.307
0.059
1
0.072
0.105
0.448
0.073
0.007
2
0.249
0.312
0.249
0.259
0.040
1
0.057
0.069
0.204
0.058
0.007
2
0.187
0.208
0.114
0.191
0.021
1
0.034
0.035
0.025
0.033
0.011
2
0.101
0.104
0.025
0.102
0.003
0.75
0.50
0.25
regarding multi-material, parametric optimization routines by (Wang et al., 2015) and briefly outlined
here. We have opted for a level-set based approach, in order to obtain clearly distinct material regions.
Trials using density-based multi-material topology optimization often resulted in designs containing
mixtures of materials that are difficult to interpret.
4.3.1. Multi-material boundary representation
The level set method was originally developed by Osher and Sethian (1988) for the numerical computation of front and boundary propagation and has been applied in the context for shape optimization
by Allaire et al. (2004); Wang et al. (2003). In level set-based optimization routines the structural
boundary is represented by the zero level set of an auxiliary scalar function, the so called Level Set
Function (LSF). Multi-material structures require multiple level set functions to describe the required
material boundaries. Each level set defines the subdomains:

k
k
k


(material)
φ (X) > 0, ∀X ∈ Ω \Γ
k
k
(4.15)
φ (X) = 0, ∀X ∈ Γ
(interface)


φk (X) < 0, ∀X ∈ D\Ωk
(void).
Here X is a point within the design domain, the index k indicates the use of k level set functions and
D corresponds to the design domain. Two level set functions are required to describe two materials
and void. The first level set function separates material from void, while the second level set function
distinguishes the domains of viscoelastic and structural material. Figure 4.2 illustrates a material
distribution using two level set functions. Using the formulation proposed earlier for elastic materials
(Wang et al., 2015), we can now define the following stiffness and mass matrices for the considered
structural and viscoelastic design problem:
{(
}
)
K(φ) = H(φ1 )
1 − H(φ2 ) Ke1 + H(φ2 )Ke2
{(
}
(4.16)
)
M (φ) = H(φ1 )
1 − H(φ2 ) Me1 + H(φ2 )Me2 .
Here H(φk ) represents the Heaviside function of the k th level set function, Kek and Mek correspond
respectively to the elementary stiffness and mass of material k. The applied Heaviside function is
26
4. Topology optimization of viscoelastically damped structures
Void
Viscoelastic material
Void
Structural material
D
Figure 4.2: Illustration of a multi-material structure defined by two level set functions. The first level
set function φ1 determines the placement of material, while the second level set function φ2 distinguishes
between structural and viscoelastic material.
both approximated and regularized, as discussed in more detail in Section 4.3.5.
4.3.2. Parametric level set method
The topology optimization is implemented by a parametric level set method. In parametric methods a
parameterization is applied to uncouple the space and time dependencies within the level set method
(Wang and Wang, 2006; Luo et al., 2007). The level set functions are constructed by the summation
of basis functions:
N
∑
φk (X, t) =
ξi (X)αik (t).
(4.17)
i=1
Here N equals the total number of shape functions included in the design domain. The shape functions
are given as ξi (X), which describe the ith shape function. The expansion coefficients αik (t) are applied
to scale the individual shape functions. The variable t represents a pseudo-time variable and indicates
the change of expansion coefficients throughout the iterations. The decoupling of the space and time
dependencies allows us to apply gradient-based optimization routines, rather than solve HamiltonJacobi convection equations to update the structure’s boundary. Besides, the parametric level set
method allows us to nucleate holes throughout the design domain by allowing negative values of
the expansion coefficients (Luo et al., 2007). Different approaches are available to parameterize the
level set function and to describe the level set functions (van Dijk et al., 2013). In this study we
apply a Compactly-Supported Radial Basis Function (CS-RBF) proposed by Wendland (1995) for the
parameterization. These shape functions also have been applied in earlier parametric level set studies
(Luo et al., 2008a; Wang et al., 2015). The shape function is given as
{
0
)4
ξi (X) = (
(1 − ri (X) (4ri (X) + 1)
if ri (X) ≥ 1
if ri (X) < 1.
(4.18)
The radius ri is given as
ri (X) =
||X − Xi ||
,
R
(4.19)
where R refers to the influence radius of the basis function. Only the neighboring elements within the
influence radius will contribute to the function value of the level set function.
4.3.3. Optimization problem for structural loss factor maximization
In the parametric level set method the shape of the level set function is completely determined by the
given expansion coefficients. By changing the values of the expansion coefficients local shape changes
4.3. Topology optimization of viscoelastic and structural material
27
are realized. Therefore, the expansion coefficients are used as design variables during the optimization.
The optimization problem is formulated as:
find αik ,
i = 1, 2, . . . , N,
k = 1, 2
∑ ϕH KI ϕr
r
max J =
γr =
HK ϕ
ϕ
R r
r
r
r
( ′
)
K + iK ′′ ϕ = λ2 M ϕ
s.t.
∑
(4.20)
ϕH
j M ϕl = δjl
λ21 ≥ λ2min
k
V k ≤ Vmax
k
k
≤ αik ≤ αi,max
.
αi,min
The optimization routine searches for the combination of expansion coefficients αik to maximize the
summation of the first r loss factors. In the presented work all loss factors have equal weight, however, it is also possible to define different weight factors for each modeshape. The optimization is
subjected to the eigenvalue problem. The δ represents the Kronecker delta, furthermore, a minimum
eigenfrequency of the structure is required to prevent development of low-frequency localized modes
in viscoelastic material regions. Additionally, constraints can be added for the volume of either structural or viscoelastic material V k . Finally, the design freedom of the optimizer is limited by specifying
k
k
a lower αi,min
and upper αi,max
bound for the design variables.
4.3.4. Adjoint sensitivity analysis
An adjoint sensitivity analysis is performed to derive an exact formulation of the loss factor sensitivities. In this analysis we assume that all modes are of multiplicity one. The adjoint sensitivity analysis
for discretized systems is discussed by Adelman and Haftka (1991) and is applied in the context of
multi-material level set methods by Allaire et al. (2014). The formulation of the adjoint problem
including a complex-valued eigenvalue problem is based on the adjoint sensitivity analysis given in
(van der Veen et al., 2014, Appendix B). A similar approach is applied to add the adjoint variables
µ1 and µ2 to the loss factor:
γr⋆ =
(
(
) )
(
)
ϕH
r K I ϕr
H
2
+
ℜ
µ
K
−
λ
M
ϕ + µ2 ϕH M ϕ − 1 .
1
r
H
ϕr K R ϕr
(4.21)
Taking the gradients with respect to the design variables αik results in
)
(
)
( H
)
∂ϕr
∂ϕr
H ∂KI
H
H ∂KR
H
2ϕ
K
+
ϕ
ϕ
−
ϕ
K
ϕ
2ϕ
K
+
ϕ
ϕ
I ∂αk
I r
R ∂αk
r
r
r ∂αk r
r
r
r ∂αk
i
i
i
i
∂γr⋆
=
(
)2
k
∂αi
ϕH
r KR ϕr


)
(
) ∂ϕ
(
2
∂λ
∂K
∂M
r
r
2
− λ2r k −
M ϕr + µH
+ ℜ µH
1 K − λr M
1
∂αik
∂αi
∂αik
∂αik
(
)
∂ϕr
H
H ∂M
+ µ2 2ϕr M k + ϕr
ϕr .
∂αi
∂αik
(
ϕH
r K R ϕr
)
(
The adjoint multipliers µ1 and µ2 are chosen such that the modeshape sensitivities
∂λ2r
∂αk
i
∂ϕr
∂αk
i
(4.22)
and eigenfre-
quency sensitivities
are not required to be calculated. The following adjoint problem has to be
solved to determine the adjoint variables:
(
    − ϕH K ϕ (2K ϕ )+ ϕH K ϕ (2K ϕ ) 
)
( r R r) I r ( r I r) R r
2
K − λ2r M
2M ϕr
µ

(ϕHr KR ϕr )

  1 = 
(4.23)
.

H
2ϕr M
0
µ2
0
28
4. Topology optimization of viscoelastically damped structures
When the adjoint variables are known, the loss factor sensitivities are determined from the remaining
terms in Equation (4.22):
(
)
(
)
( H
)
( H
)
H ∂KR
H ∂KI
ϕr KR ϕr ϕr ∂αk ϕr − ϕr KI ϕr ϕr ∂αk ϕr
i
i
∂γr⋆
=
(
)2
k
∂αi
ϕH
r KR ϕr
(4.24)

) 
(
∂K
∂M
∂M
+ ℜ µH
− λ2r k ϕr  + µ2 ϕH
ϕr .
1
r
∂αik
∂αi
∂αik
To determine the loss factor sensitivity with respect to all design variables αik requires to solve the
adjoint problem (4.23) once per iteration for each considered mode. Then, equation (4.24) is solved
for each design variable. The required sensitivities with respect to the stiffness and mass matrices can
be derived from equations (4.16) and (4.17).
4.3.5. Numerical implementation
For the numerical implementation of the presented method, the design domain is discretized in fournode quadrilateral (Q4), square plane stress finite elements. The ersatz material model is applied
to scale the material properties around the level set boundaries. This formulation is applied for its
simplicity, but note that mixed materials can still appear in the domain near boundaries between
material regions. This local effect can be reduced by mesh refinement by reducing the minimal size
of the transition region between material regions. The Heaviside function and its derivative are
implemented with the approximations (Wang et al., 2003):


φk ≤ −∆

a,
(
)
(φk )3
3(1−a) φk
k
1+a
H(φ ) =
+ 2 , −∆ < φk < ∆
4
∆ − 3∆3


1,
φk ≥ ∆
(4.25)

(
)
( k )2

 3(1−a) 1 − φ
, |φk | ≤ ∆
∂H
4∆
∆
=
∂φk 
b,
|φk | > ∆.
This implementation of the smooth Heaviside can introduce some mixed regions within the domain.
However, it has been observed that the problem has a tendency to result in clearly separated material
regions. In the following numerical examples, the parameters are implemented as: a = 0.001, b =
k
k
0.0005, ∆ = 1, αi,min
= −2, αi,max
= 2. The parameter b is kept to a non-zero value to allow possible
design changes after a point in the design has exceeded ∆. Otherwise, the sensitivity information
equals zero and the design cannot be changed once the absolute value of the LSF has exceeded ∆.
For each element in the discretization a shape function is introduced. Hence, the centers of the shape
functions are aligned with the centers of the elements in the mesh. The influence radius R is kept
between two and four times the mesh size. The optimization problem is solved using the Method of
Moving Asymptotes (MMA) as given in (Svanberg, 1987). This method iteratively solves nonlinear
programming problems and is suited for general and structural optimization problems. In each step
of the iterative process a strictly convex approximating subproblem is generated and solved. The
resulting topologies are visualized based on the level set functions.
4.4. Numerical examples
This section presents a number of case studies to study the optimal design of damped structures for
maximum dissipation, and to illustrate the performance of the proposed optimization routine. First,
a cantilever beam is optimized. Secondly, a comparison is made between the application of different
complex material moduli and their influence on the final designs. Finally, the optimization routine is
applied to an existing structure with a limited design space.
4.4.1. Case study I: cantilever beam
The design domain of the cantilever beam is given in Figure 4.3. The complete cantilever beam is
provided as design domain for the optimization. The cantilever is fixed on the left side, the remaining
4.4. Numerical examples
29
Design domain
H
L
Figure 4.3: The design domain for case study I.
boundaries are unconstrained. The domain is discretized with L = 70 and H = 20 elements. The
first level set function describes the structural material. For the structural material steel is applied
with Young’s modulus E1 = 200 GPa, Poisson ratio ν1 = 0.3 and density ρ1 = 7.85 × 103 kg/m3 . Any
damping introduced by the structural material is neglected, as this is much smaller than the damping
provided by the viscoelastic material. The second level set function models the viscoelastic material
and general material properties are applied to model the viscoelastic behavior with Young’s modulus
E2 = 1 GPa, Poisson ratio ν2 = 0.3 and density ρ2 = 1.0 × 103 kg/m3 . A complex shear modulus with
a material loss factor of ηshear = 1.0 is applied to model the dissipative properties. The value of the
shear modulus is derived from the Young’s modulus and Poisson ratio as
G=
E
.
2(1 + ν)
(4.26)
The optimization is initialized with two constant level set functions. The initial values are chosen
slightly above the zero level set, such that the design is initially filled with material which is a mixture of structural material with a small amount of viscoelastic material available. Since the level set
functions are initialized as constant values above the zero level set, the design domain is completely
filled by elements with scald material properties with respect to the values of both initial LSFs. The
objective of the optimization is to maximize the average of the structural loss factors corresponding to
the first two eigenmodes. This represents a simplified form of optimizing the dissipation within a certain frequency band which contains both eigenfrequencies. The optimization is constrained such that
only 40% of the design domain may be occupied by viscoelastic material. Furthermore, a frequency
constraint is applied to keep the first eigenfrequency within 50% of the eigenfrequency of the initial
design. We mainly apply the frequency constraint to prevent the creation of low-frequency internal
modes in the viscoelastic material regions with high structural loss factors. Therefore, the chosen
percentage can be chosen to match any practical frequency requirements.
The optimization is terminated when all constraints are satisfied and when the final objective value
is within 1% of the previous three objective values. Figure 4.4 shows the progression of the design
throughout the iterations, with the final design given in Figure 4.4d. The general layout of the viscoelastic material resembles a constrained-layer damping configuration, as the viscoelastic material is
mainly distributed along the centerline of the cantilever. After 35 iterations the final design achieved
an objective value of J = 0.5075. The result can be compared with a cantilever with the same aspect
ratio with a conventional constrained layer damping configuration, which provides a structural loss
factor between roughly 0.26 and 0.39 for a centered, viscoelastic layer between respectively 3 and
7 elements. The optimized design is able to achieve higher structural loss factors compared to the
conventional configurations.
The history of the objective values and the corresponding loss factors of the two modeshapes are given
in Figure 4.5. Similarly, Figure 4.6 presents the progression of the volume and frequency constraints.
The constraint values are normalized with respect to the maximum value of the constraint. For this
formulation positive values represent unsatisfied constraints, while negative values correspond to satisfied constraints. All constraints remain satisfied at the end of the optimization.
30
4. Topology optimization of viscoelastically damped structures
(a) Iteration 2, J = 0.1651.
(b) Iteration 5, J = 0.3761.
Structural
(c) Iteration 20, J = 0.4553.
Viscoelastic
(d) Iteration 35, J = 0.5075.
Figure 4.4: The design progression throughout the optimization routine at iteration 2, 5, 20 and 35
and the corresponding objective values.
1
0.7
0.9
Mode 1
Mode 2
0.6
0.8
0.5
Loss factor (−)
Objective (−)
0.7
0.6
0.5
0.4
0.3
0.4
0.3
0.2
0.2
0.1
0.1
0
0
5
10
15
20
Iteration
25
30
0
0
35
5
10
(a) Objective.
15
20
Iteration
25
30
35
(b) Loss factors.
Frequency constraint (−)
Volume constraint (−)
Figure 4.5: The history of the objective values (a) and corresponding loss factors (b) during the
optimization.
1.5
Structural material
Viscoelastic material
1
0.5
0
−0.5
0
5
10
15
20
25
30
35
5
10
15
20
Iteration
25
30
35
0
−0.5
−1
−1.5
−2
−2.5
0
Figure 4.6: The history of the volume (top) and frequency (bottom) constraint values during the
optimization. The constraints are normalized with respect to the specified maximum value. Positive
values represent unsatisfied constraints
4.4. Numerical examples
(a) Shear strain εmax mode 1.
31
(b) Shear strain εmax mode 2.
Figure 4.7: The deformation shapes and corresponding maximum shear strain εmax for the first (a)
and second (b) eigenmode of the optimized structure. The uncolored elements represent void elements
within the design.
The application of the complex shear modulus results only in energy dissipation when the viscoelastic
material is subjected to shear deformation. Therefore, the performance of the obtained designs can
be visualized by plotting the shear deformation of each element. The maximum in-plane shear strain
is determined as:
√(
)2
εx − εy
εmax =
+ ε2xy .
(4.27)
2
To compare the deformation between both modeshapes a base excitation is applied. The eigenmodes
of the structure only provide information on the deformation shape and not on the actual amplitude
of the deformation. Therefore, the first and second eigenmodes are excited with the same harmonic
acceleration profile at the clamped side of the cantilever. Figure 4.7 shows the resulting deformation
shapes and the maximum shear strain. The shear strain is mostly visible in the viscoelastic material.
In both eigenmodes the viscoelastic material is effectively used, as almost all viscoelastic elements are
subjected to shear strain. In the second mode, slightly higher strain values are observed, corresponding
to the slightly higher structural loss factor of the second modeshape. Figure 4.8 illustrates a contour
plot of the energy dissipated per element for the first modeshape of the structure. The obtained
energy is directly related to the shear deformation observed for this modeshape, as illustrated in
Figure 4.7a. The dissipated energy is calculated by Equation (4.10) and is normalized with respect to
the element with highest dissipation. The direct relation between shear deformation and dissipated
energy is observed when comparing the figures of shear strain and dissipated energy (Figures 4.7a and
4.8).
4.4.2. Design influences by complex moduli
In the previous case study the viscoelastic material has been described with a complex shear modulus.
However, as discussed in Section 4.2 the viscoelastic material can also be represented with either a
complex Young’s or complex bulk modulus. To illustrate the influence of the different modeling
Figure 4.8: The energy dissipated for each element for the first eigenmode of the structure. The values
are normalized with respect to the maximum energy dissipated in an element. The obtained result is
directly related to the shear strain given in Figure 4.7a. The uncolored elements represent void elements
within the design.
32
4. Topology optimization of viscoelastically damped structures
(a) Complex shear modulus,
JG = 0.6002.
(b) Max principal strain.
(c) Max shear strain.
(d) Complex Young’s modulus,
JE = 0.8823.
(e) Max principal strain.
(f) Max shear strain.
(h) Max principal strain.
(i) Max shear strain.
Structural
Viscoelastic
(g) Complex bulk modulus,
Jκ = 0.6940.
Figure 4.9: A comparison between the optimized designs and their performance for the application of
multiple complex material models: (a): shear, (d): Young’s and (g): bulk modulus. The differences in
performance are highlighted by the maximum principal and maximum shear strains within the structure
for the first eigenmode.
approaches, the optimization of case I will be evaluated separately for a complex shear, Young’s and
bulk modulus. For each complex modulus a corresponding material loss factor of η = 1 is applied. The
remaining material parameters and the design space are kept the same. However, in this optimization
only the structural loss factor corresponding to the first eigenmode is considered as the objective. The
optimizations were terminated after the objective value converged and all constraints were satisfied.
The optimization with a shear, Young’s and bulk modulus provide the objective values: JG = 0.6002,
JE = 0.8823, Jκ = 0.6940. The obtained designs are given in Figure 4.9. The material layout of
the three designs show significant differences. For the complex shear modulus the material is mainly
located in the center of the design, similar to conventional constrained layer configurations. For the
complex bulk modulus, the material is mainly located at the outer edge of the domain, where the
normal strains are the highest. In the design using a complex Young’s modulus, the viscoelastic
material is mainly located at the right side of the cantilever, where it is subjected to both normal and
shear strains. The performance of the designs are illustrated by plotting the in-plane maximum shear
strains, equation (4.27), as well as the maximum in-plane principal strains within the structure.
ε1,2
εx + εy
=
±
2
√(
εx − εy
2
)2
+ ε2xy .
(4.28)
The obtained figure are presented in Figure 4.9. The design based on a complex shear modulus shows
almost identical principal strains as maximum shear strains, illustrating that most of the viscoelastic
material is subjected to shear deformation. On the other hand, the design using the complex bulk
modulus shows higher maximum principal strains, illustrating that the viscoelastic material is mainly
subjected to normal strains. The complex Young’s modulus contains a combination of both shear and
normal deformations. The higher loss factor achieved by the complex Young’s modulus is understood,
since any deformation of the viscoelastic material will result in energy dissipation for the complex
Young’s modulus, while in the complex shear and bulk modulus only the shear or normal strain
components dissipate energy.
4.4. Numerical examples
33
Table 4.2: A comparison between the loss factors for the optimized designs, when the performance is
evaluated with different material models than used during the optimization.
Modulus for performance check
Modulus during
optimization
G
E
κ
G
0.6002
0.6298
0.0341
E
0.1838
0.8823
0.4834
κ
0.0206
0.7168
0.6940
Performance cross-check
To illustrate that the obtained designs only show optimized performance for the material model used
during the optimization, an additional verification is performed. The structural loss factor for each
optimized design is evaluated with the other two complex material models. This is to show the impact
of the used material model on the optimization process. This is done for all combinations of material
moduli and results are given in Table 4.2. Since the complex Young’s modulus dissipates energy for
any deformation, it is observed that all designs evaluated with this modulus provide reasonable or
even improved performance. However, a significant performance difference is observed between the
complex shear and bulk modulus.
4.4.3. Case study II: existing structure with limited design domain
The second case study illustrates the ability to generate a design to damp a particular mode of an
existing structure. In this case study a clamped beam is studied as existing structure. The beam
is clamped at both sides and cannot be modified by the optimization. A limited design domain is
defined on top of the cantilever. The optimization routine is only allowed to develop designs within
this domain. An illustration of the design and non-design domains are given in Figure 4.10. The objective of the optimization is to achieve maximum loss factor for the first eigenmode of the structure.
The material properties are the same as applied in Section 4.4.1 and a complex shear modulus with
a material loss factor of ηshear = 1 is applied for the viscoelastic material.
Design domain
Non design domain
Figure 4.10: The design and non-design domains for case study II. The non-design domain is clamped
at both sides and cannot be modified by the optimization routine.
The optimization is subject to two constraints: the final design should utilize more than half of the
total design domain and the volume fraction of the viscoelastic material is limited to half of the total
design domain. The frequency constraint is omitted in this optimization routine, since the existing
structure prevents the creation of very low frequency resonant modes.
The final design achieved after 58 iterations is given in Figure 4.11. The optimization results in
J = 0.3803. Figure 4.12a presents the history of the loss factor and Figure 4.12b shows the history
of the volume and frequency constraints. All applied constraints remain satisfied after 58 iterations.
The optimization has developed a design which resulted in a loss factor of 0.3803 for the first eigen-
34
4. Topology optimization of viscoelastically damped structures
Structural
Viscoelastic
Figure 4.11: The final design of the optimization achieved after 58 iterations. The design achieved an
objective value of J = 0.3803.
0.4
0
Structural material
Viscoelastic material
−0.1
0.35
−0.2
Volume constraint (−)
Loss factor (−)
0.3
0.25
0.2
0.15
−0.3
−0.4
−0.5
−0.6
−0.7
0.1
−0.8
0.05
0
0
−0.9
10
20
30
Iteration
40
(a) Loss factor.
50
−1
0
10
20
30
Iteration
40
50
(b) Volume and frequency constraints.
Figure 4.12: The history of the loss factor (a) and normalized volume and frequency constraints (b)
during the optimization.
mode. The performance of the design is illustrated by plotting the dissipated energy per element
during oscillation in Figure 4.13. During the first eigenmode the structure within the design domain
resonates in phase with the beam. The relative displacement of the top part is larger compared to the
displacement of the beam, straining the viscoelastic material and dissipating energy during oscillation.
The behavior of the generated design is similar to tuned mass damper systems and attenuates the
first resonant mode of the structure.
Figure 4.13: The deformation shape for the first eigenmode of the optimized structure of case study
II. The dissipated energy in each element is plotted to illustrate the energy dissipation for the first
eigenmode. The uncolored elements correspond to void elements in the design.
4.5. Discussion
A systematic design approach has been presented for the development of damped structures using a
combination of viscoelastic and structural materials. The optimization routine has shown to be capable to generate structures with high loss factors and improve the damping of a given structure. While
the presented method provides promising results, a number of difficulties and assumptions remain.
The current method optimizes the structures for a specified eigenmode. However, by the introduction
4.6. Conclusions
35
of structural and viscoelastic material the possibility exists to introduce new modeshapes at lower
frequencies. Also, the change of the design might switch the order of modeshapes, especially for designs where eigenfrequencies are close together. A mode tracking procedure could be added to the
proposed method to handle such events. Moreover, when multiplicity of eigenvalues is encountered
additional measures are required to determine the eigenmode sensitivities, since the eigenmodes can
change discontinuously during a crossing of eigenvalues.
Moreover, the applied viscoelastic material model has a significant influence on the performance of
the optimization routine. As illustrated in Section 4.4.2, the application of different complex moduli
results in large differences between the optimized designs. Therefore, an accurate material model
is required to describe the viscoelastic behavior. Of particular importance are its dissipative properties, as the optimization routine will exploit any dissipative behavior included in the material model.
Finally, the implemented viscoelastic material model contains two assumptions: both temperature
and excitation frequency dependencies are neglected. This considerably simplifies the behavior of
the viscoelastic material. However, depending on the applied viscoelastic material, both parameters
can have significant influence on the dissipative properties of the material. To extend the model
to capture the thermal behavior requires a separate thermal analysis to determine the steady state
temperature during oscillation. If temperature peaks occur within the viscoelastic material, its dissipative properties are reduced, resulting in lower structural loss factors than predicted. This results
in a fully coupled thermomechanical problem that must be solved iteratively. A workaround is to
introduce additional thermal constraints within the optimization routine to keep the temperature of
the viscoelastic material between specified boundaries. The coupling with thermal behavior is left as
future work.
4.6. Conclusions
A well-performing topology optimization approach has been presented that can generate multi-material
designs with optimized damping properties. The employed level-set based formulation yields a clear
separation between various material phases. Structural loss factors using complex-valued eigenmodes
have been found to provide an accurate assessment of damping characteristics, even for designs with
high volume fractions of viscoelastic material combined with high material loss factors.
In various examples, we have demonstrated that designs were obtained achieving higher structural
loss factors than conventional constrained layer configurations. Also, the optimization method is
successfully applied to attenuate a resonant mode of an existing structure. The generated design
shows behavior similar to tuned mass damper systems and adds significant damping to the resonant
mode of interest. The method has the potential to be applied to complicated structures and can be
extended towards three-dimensional designs. The influence of different viscoelastic material models
is investigated and significantly different results are found. To achieve suitable designs, the chosen
viscoelastic material has to adequately reflect the behavior of the actual material.
5
Base excitation and thermal
analysis
In this chapter the viscoelastic structures are excited by a base excitation to investigate their response for a range of excitation frequencies. The performance of the optimized structure is compared
with conventional damping solutions to illustrate the achieved performance improvement. Besides,
a steady-state thermal analysis is performed to investigate the thermal response of the viscoelastic
damper when subjected to an external vibration.
5.1. Base excitation
The designs generated by the optimization have optimized loss factors for specific eigenmodes. In
practice the structures are most likely to be excited by a broad frequency spectrum. These vibrations
are introduced by varying external forces or by displacements imposed by their supporting structure.
The excitation by support motion is chosen to investigate the structure behavior to external vibration. The displacement of the support motion is imposed on the fixed DOFs of the optimized designs.
This represents various scenarios in which the vibration is imposed at a structure’s support, such as
vibration carried by spacecraft, vehicles or the response to ambient vibrations.
The DOFs x of the discretized system are partitioned in two sets to model the imposed support
motion. The external displacements are imposed on the support DOFs x2 . The remaining DOFs x1
are left unconstrained. The equations of motion of the structure
M ẍ + Kx = f ,
with K = KR + iKI , is partitioned accordingly:

  
ẍ
M11 M12
K

  1  +  11
M21 M22
ẍ2
K21
   
x
0
  1 =   .
K22
x2
r
K12
(5.1)
(5.2)
The forces r in the right hand side represent the reaction forces between the moving structure and the
moving support. The partitioning allows us to find an expression for the response of the free degrees
of freedom from the first equation:
M11 ẍ1 + M12 ẍ2 + K11 x1 + K12 x2 = 0.
(5.3)
A harmonic displacement is imposed at the support and therefore both the displacements x2 as well
as the corresponding accelerations ẍ2 are known. In steady-state the structure is assumed to show
a harmonic response at the same frequency. The amplitude of the response is found by rearranging
Equation (5.3):
(
)−1 (
)
x1 = −ω 2 M11 + K11
ω 2 M12 − K12 x2 ,
(5.4)
37
38
5. Base excitation and thermal analysis
(a) Single element γ = 0.2280.
(b) Three elements γ = 0.3142.
(c) Thick layer γ = 0.4108.
(d) Optimized, γ = 0.4858.
Figure 5.1: Four different viscoelastic dampers: (a): single element viscoelastic layer, (b): three
elements viscoelastic layer, (c): a viscoelastic layer matching the thickness of the viscoelastic layer in
the optimized design, and (d): the design generated by the optimization. The presented loss factors
correspond to their first resonant mode.
with ω being the frequency of the external vibration. The viscoelastic material results in complexvalued response for the displacements x1 , which are converted towards real-valued displacements by
taking in account the phase differences of each DOF. When the response of all degrees of freedom
are known, the reaction forces as result of the imposed motion can be obtained by solving the second
equation from Equation (5.2):
r = M21 ẍ1 + M22 ẍ2 + K21 x1 + K22 x2 .
(5.5)
In practice the structures are mostly excited by a wide frequency spectrum. A support excitation is
applied to the optimized design to investigate its frequency response around its first eigenmode. This
also allows us to compare the performance of the optimized design with conventional constrained layer
damping configurations. The optimized design is compared with three other designs. These designs
contain a constrained viscoelastic layer of various thicknesses, ranging from a very think, single layer
of viscoelastic material to a thick layer, that matches the width observed in the optimized design.
Figure 5.1 illustrates the different viscoelastic dampers. All structures have the same dimensions and
only the internal distribution of viscoelastic material differs. The loss factors of the three designs are:
0.2280, 0.3142 and 0.4108, which are all below the loss factor of the optimized design: 0.4858.
The structures are subjected to a harmonic support motion in the vertical direction, which is applied
at the left side of the design. The response are measured at the top right corner of the design domain and the resulting frequency response functions are given in Figure 5.2. A unit displacement is
applied for illustrative purposes. Therefore, the frequency response function will show unit amplitude
at low frequencies, as the tip will follow the applied displacement. All designs show a magnification of
the applied excitation near their first resonant frequencies. All three conventional constrained layer
damping configurations show larger amplitudes compared to the optimized design. Even the CLD
configuration with the thickets viscoelastic layer, shown in Figure 5.1c, shows larger amplitudes. The
frequency response clearly illustrates the performance increase realized by the material distribution
generated by the optimization routine.
The optimized design shows a lower resonance frequency compared to CLD damping of a single and
three layers thick viscoelastic layer. However, the relatively thick layer shows and even lower first
resonance frequency. The eigenfrequency of the optimized design is roughly 25% lower compared to
the design using a single layer of viscoelastic material. When we are interested to achieve the highest
damping for a broad frequency range, the deviation of resonance frequency might not be of specific
importance. For a case in which a design should be optimized to damp an excitation with a specific
frequency, the applied frequency constraint should be enforced more strictly. This would make sure
5.2. Energy dissipation
39
that the frequency of the obtained design corresponds with the specific excitation. However, in most
practical cases the external vibration contains a relatively wide frequency band and the resonance
frequency of the viscoelastic damper is not of specific interest.
1
10
Amplitude (−)
Single element
Three elements
Thick layer
Optimized
0
10
2
3
10
10
Frequency (Hz)
0
Phase (deg)
−50
−100
−150
−200
−250
2
3
10
10
Frequency (Hz)
Figure 5.2: Frequency response of four structures damped using viscoelastic material. The corresponding designs are shown in Figure 5.1. The designs are subjected to a support motion with unit amplitude
at the left side of the designs and the response of the structure is measured at the top right corner
of the cantilever. The response of the optimized structure (dark blue) is compared with three structures containing a conventional constrained layer damping configuration. These designs contain different
thicknesses of viscoelastic material: a single layer (red), three layers (green) and a layer matching the
thickness observed in the optimized design (cyan). The optimized design shows an amplitude reduction
in comparison to the designs with conventional damping treatments.
5.2. Energy dissipation
The viscoelastic material dissipates energy when subjected to deformation. The imposed support
motion will result in deformation of the structure and especially when the structure is excited in its
eigenfrequency. The dissipated energy is determined using the deformation shape of the structure
when subjected to a steady, harmonic excitation. For this analysis the deformation shape obtained in
Equation (5.4) is used. The dissipated energy is given by the following two expressions:
Wcyc = πxH ℑ (K) x,
′′
Wcyc = πεT
xy G εxy V,
(5.6)
with V being the volume of viscoelastic material, since the expression given in Equation (4.9) provides the dissipated energy per unit volume. The second expression can be evaluated based on the
chosen material modulus to describe the dissipative behavior. Similar expression can be written for
the bulk and Young’s modulus. Equation (5.6) provides the total energy dissipated for a single cycle
for the investigated deformation mode. It has to be noted that these expressions are based on a single
harmonic response with constant amplitude. If resonances with decreasing amplitudes or multiple
40
5. Base excitation and thermal analysis
(a) Optimized design excited at its first eigenfrequency.
(b) Optimized design excited at its second eigenfrequency.
(c) CLD configuration excited at its first eigenfrequency.
(d) CLD configuration excited at its second
eigenfrequency.
Figure 5.3: The figures illustrate the dissipated energy per cycle for each elements in the design. In
each illustration the values are normalized with respect to the element with maximum energy dissipation.
The distributions are given for the first and second resonant mode of the optimized design in figures (a)
and (b), and for a CLD configuration in figures (c) and (d).
harmonics are studied, the complete integral along stress and strain needs to be evaluated to determine the dissipated energy.
To illustrate the energy dissipation the structures are subjected to a steady, harmonic oscillation. The
resulting deformation modes are used to determine the dissipated energy per cycle. The equations are
evaluated at element level to investigate the dissipation of each individual element and their relative
contributions to the total energy dissipation. The element displacements or strains are determined
and multiplied with either the element matrix or material modulus. The material parameters are
scaled based on the level set function values at the investigate locations. The expressions to evaluate
the dissipation on element level become:
{(
}
(
))
(
)
e
H
1
2
1
2
2
Wcyc = πx(X) H(φ (X))
1 − H φ (X)
Ke + H φ (X) Ke x(X),
{(
e
Wcyc
1
= πεxy (X)H(φ (X))
))
(
)
1
2
1 − H φ (X)
G′′e + H φ2 (X) G′′e
(
2
}
(5.7)
εxy (X).
The dissipated energy per element provides the same result for both methods. The element dissipation is illustrated for the optimized structure when a base excitation is applied in its first and second
eigenfrequency. The structures are subjected to a unit input and the obtained dissipations are normalized with respect to the maximum dissipation for a single element. Figures 5.3a and 5.3b show
the obtained results. The highlighted elements are all located within the viscoelastic elements and
most of the elements contribute to the total energy dissipation in both the first and second resonant
modes. The obtained results clearly correspond to the shear deformation as illustrated in Figure 4.7.
The difference in the observed distribution between the shear strain and the element dissipations are
explained by the squared relation of the element strains within the expression of the dissipated energy,
as given in Equation (5.6).
5.3. Thermal analysis
41
To investigate the improvements the same analysis is performed for a structure with a CLD configuration. The dissipated energy is presented in Figures 5.3c and 5.3d. In the CLD configuration the
dissipation is limited to the thickness of the viscoelastic layer. The optimization has improved its
damping characteristics by locally extending the viscoelastic layer to locate more viscoelastic material
on locations which are subjected to shear deformation. Besides, the optimization routine has slightly
modified the modeshapes of the structure to allow larger shear deformation within the viscoelastic
material. The maximum energy dissipation has thereby been moved towards the center of the design,
rather than at the tip of the cantilever as observed for the CLD configuration.
5.3. Thermal analysis
The obtained distribution of dissipation energy provides a steppingstone towards a thermal analysis
of the design. If we make the assumption that all dissipated energy is converted into heat, the energy
dissipated by each element now represents a thermal load for each element within the domain. This
thermal load is applied to the structure in a thermal analysis to determine the steady-state temperature distribution of the designs. From this analysis the temperature of the viscoelastic material
is mainly of interest, since the dissipative properties can deteriorate by local temperature changes.
Furthermore, the viscoelastic material is often concentrated within the designs and limits the conduction of heat by its low thermal conductivity. This results in both localized heat loads and localized
temperature increases. The remainder of this chapter presents a thermal analysis to investigate the
steady-state temperatures of the designs as a result of a harmonic base excitation.
The thermal analysis is simplified by assuming isotropic thermal behavior of both structural and
viscoelastic materials and the thermal coefficients are assumed to be constant. The thermal behavior
of the viscoelastic material is not yet included within the model, as this allows to perform a single linear
analysis to obtain the temperature distributions. However, if the temperature dependent behavior
of the viscoelastic material is included, a fully coupled thermomechanical analysis has to be solved
iteratively to determine the steady-state temperatures. The thermal behavior of the continuum body
with structural and viscoelastic material is described by the following equilibrium and boundary
conditions



in D,
−div(κχ ∇T ) + Q − cχ ρχ Ṫ = 0
(5.8)
q = hχ (T − T∞ )
on Γkconv ,


k
T = T∞
on Γfix ,
which are illustrated in Figure 5.4a. The first line describes the thermal conduction within the continuum. Note that in the context of this thermal analysis κ refers to the thermal conductivity in
stead of the bulk modulus of the material. A similar notation is used as in Equation (4.5) to scale
the conductivity κχ , convective heat transfer coefficient hχ , specific heat cχ and material density ρχ .
This scaling is based on the presence of the material phases k: κχ = κk χk , where χk is a function of
phase Ωk . The equilibrium is subjected to two boundary conditions: a convective boundary condition
on Γkconv and an imposed temperature at Γkfix . It is chosen to keep the temperature at the support of
the structure equal to the environment temperature T∞ .
The problem is discretized using finite elements to find a solution for the temperature distribution.
A detailed description of the formulation of the thermal problem is presented in Appendix B, Section
B.1.1. The resulting problem to solve becomes:
(KTc + KTh ) T = RQ + Rh ,
(5.9)
with KTc the conductivity matrix, KTh the convective matrix, RQ the heat load introduced by viscoelastic material and Rh the convective boundary condition. A similar partitioning is applied as in
the base excitation analysis to partition the free and imposed DOFs. In the analysis the heat conduction of the viscoelastic material is significantly lower than the structural material: κ1 = 10κ2 . For
the convective boundary condition natural convection is assumed with a similar convective boundary
coefficient for both material phases. The temperature of the fixed boundary is assumed to remain
42
5. Base excitation and thermal analysis
(a) General heat transfer problem.
(b) Boundary conditions for optimized geometry.
Figure 5.4: Illustrations of the heat transfer problem. Figure (a) provides a sketch of the domain
and boundary definitions. The domain Ωk contains both two material phases Ω1 and Ω2 . Its boundary
is separated in two parts: Γkfix and Γkconv which respectively impose a fixed temperature and apply a
convective boundary condition to the corresponding boundary. In figure (b) the boundary conditions
are applied for the optimized structure.
constant and equal to the environment temperature T∞ . All remaining boundaries are subjected to a
convective boundary condition with an assumed convective heat transfer coefficient for free convection
in air equal to hχ = 5 W/(m2 K). Figure 5.4b provides an illustration of the boundary conditions
applied to the optimized structure.
The energy dissipation obtained in the previous section is applied as heat load to determine the
steady-state temperature distribution corresponding to the applied excitation. The temperature distribution as result of the excitation at the first resonance frequency of the structure is given in Figure
5.5a. It is chosen to visualize the temperature distribution using a normalized temperature difference
with respect to the environment temperature T∞ . We are mostly interested to investigate the temperature distribution of the structure, since the actual obtained temperature distribution is highly
dependent on the applied excitation and its amplitude. Furthermore, the temperature problems scales
linearly with the heat load RQ , which is expected to change most significantly in this problem. Even
though the convective problem does not scale equally to RQ , its effect is limited for free convection
in air, especially for smaller structures. Therefore, the investigation of the normalized temperature
distribution initially provides enough information regarding the observed temperature distribution
and possible localized temperature increases within the optimized design.
The obtained results provide a temperature distribution as expected. The boundary Γkfix remains
equal to the chosen environment temperature an the most pronounced temperature increases are located within the center of the viscoelastic layer. This is in correspondence with the distribution of
dissipated energy in the designs, as shown in Figure 5.3a. The convective boundary condition results
in small reduction of the temperature near the top, right and bottom edges of the domain. The
low thermal conductivity of the viscoelastic material prevents the generated heat from distributing
throughout the design and peak temperature values are observed in relatively confined areas within
the viscoelastic material. The second resonant mode provides similar results, as illustrated in Figure
5.5b. The applied distribution of dissipated energy shows relatively more energy dissipation on the
right half of the domain compared to the energy distribution in the first resonant mode, as illustrated
in Figure 5.5b. These differences are observed in the resulting steady-state temperature distribution
in Figure 5.3b. Again, peak temperature values are achieved within the center of the viscoelastic
material.
The presented thermal analysis provides brief insights into the expected steady-state temperature
distributions of the optimized design for a harmonic excitation in its first and second resonant mode.
The analysis confirmed that the temperature increases are mostly to be expected within the viscoelastic
5.3. Thermal analysis
43
material and that temperature peaks occur at relatively small areas of the viscoelastic material.
Depending on the frequency and magnitude of the excitation, the type of viscoelastic material, and
the boundary conditions of the structure, these temperature peaks could deteriorate the damping
properties of the viscoelastic material. As this is very case dependent, a careful analysis has to
performed to identify the temperature distribution within the viscoelastic material to determine if
any possible deterioration of its damping characteristics might occur.
(a) Mode 1.
(b) Mode 2.
Figure 5.5: Normalized temperature distribution as result of the energy dissipation due to deformation
of the viscoelastic material. The temperature distribution is normalized with respect to the highest
temperature difference in the structure. The structure is subjected to an imposed temperature at the
left edge, while the remaining edges are subjected to a convective boundary condition. The results are
shown for the first (a) and second (b) resonant mode of the optimized structure.
6
Discussion
A systematic design approach has been presented for the development of viscoelastically damped
structures. The optimization routine is able to develop structures with high structural loss factors
and increases the structural damping for a given structure. Wile the proposed optimization routine
provides promising results, a number of difficulties and assumptions remain.
The chosen material model to represent the viscoelastic material behavior has significant influence on
the performance of the optimization routine. As illustrated in Section 4.4.2, the application of different complex material moduli results in large differences between the optimized designs. An accurate
description of the viscoelastic behavior is therefore required to achieve realistic, well-performing designs. Of particular importance are the dissipative properties included in the material model, since
the optimization routine will exploit any dissipative behavior to achieve higher structural loss factors.
Moreover, the current implementation has neglected the temperature and frequency dependencies of
the viscoelastic material. These assumptions simplify the behavior of the viscoelastic material. To
gain accurate descriptions of the structural loss factor it is important to include both temperature
and frequency dependencies within the material model, since both parameters can deteriorate the
dissipative properties of the viscoelastic material. The thermal analysis mainly illustrates localized
temperature increases within the viscoelastic material, which might locally deteriorate the dissipative
capabilities. By including the temperature and frequency dependencies a fully coupled thermomechanical problem is obtained that has to be solved iteratively to determine the steady-state temperatures
and the corresponding structural loss factor.
In addition, the material description by the level set functions neglects any interaction forces between
both materials. It assumes that the displacements of the viscoelastic material are imposed by the
structural material and that both displacements are therefore identical. However, in practice both
materials are connected by an adhesive interface, which possibly reduces the obtainable structural loss
factor of the design. More advanced finite element modeling allows to study these interface problems
in more detail. Methods such as (interface enriched) generalized finite element methods provide means
to accurately study interface effects, while maintaining a relatively simple, structured mesh (Soghrati
et al., 2010).
45
7
Conclusions and recommendations
7.1. Conclusions
The goal of this research has been to develop a topology optimization routine for the structural design
of vibration isolating structures. In this routine the optimization is able to distribute both structural
and viscoelastic material simultaneously throughout the design domain to achieve optimized damping
characteristics. Previous research by TNO and TUDelft presented a density-based topology optimization routine that is able design viscoelastic dampers using both materials. Even though promising
results were obtained, large regions of the design were occupied by material mixtures. Therefore,
the research has been continued to investigate alternative methods for the optimization of vibration
isolating structures using viscoelastic materials.
The performance of three widely applied topology optimization methods have been compared for
multi-material optimization problems. The material separation, complexity and ease of numerical
implementation were evaluated. The choice was made to adopt a variation on the level set-based approaches and apply a parametric level set method for this multi-material optimization problem. The
parametric method performs as a midway between density-based and level set-based formulations
and shows advantages of both methods. Gradient-based optimization is possible, which reduces the
involvement required for shape and topological derivatives in classical level set methods. Besides, the
parametric method still provides an exact description of the boundary between both material domains.
The parametric level set method is applied for the multi-material optimization of viscoelastic dampers.
This allows to simultaneously distribute structural and viscoelastic material without any requirements
of the designer to specify the location and dimensions of a viscoelastic layer. The implementation
shows a well-performing topology optimization routine that generates multi-material designs with
optimized damping characteristics. The optimization routine maximizes the structural loss factor of
the design for resonant modes of interest. In this formulation a modified structural loss factor is implemented by including the imaginary components of the stiffness matrix in the eigenvalue analysis.
The full complex modeshapes are obtained and used for the calculation of the structural loss factor.
It has been shown that this method agrees much better to the damping characteristics compared to
previously proposed measures.
The optimization routine is able to generate designs with optimized damping characteristics for a
specific number of eigenmodes. The optimization of a cantilever beam clearly shows improved performance compared to conventional constrained layer configurations. The freeform distribution of the
viscoelastic material allows to achieve higher structural loss factors compared to conventional design
approaches. Moreover, the optimization is able to improve the damping characteristics of existing
designs. The design of a tuned-mass-damper like system is presented, in which the optimization develops structures to improve the damping of a specific resonant mode of a given system.
47
48
7. Conclusions and recommendations
The improvements of the design are visualized by a frequency response analysis of the structure. This
analysis confirmed the improved damping characteristics as predicted by the increased structural loss
factors. The optimized design shows a reduced amplitude compared to the conventional constrained
layer damping configurations for the same excitation. The frequency response analysis also allowed
to determine the local energy dissipation by the viscoelastic material and thereby determine the corresponding steady-state temperature distribution within the designs. The thermal analysis indicated
localized temperature increases, which potentially result in local deterioration of the viscoelastic material and thereby reduce its damping characteristics.
7.2. Recommendations
The proposed optimization routine has provided promising results for the optimization of viscoelastic
dampers. The method inspires further research regarding the development of the parametric level set
approach in topology optimization, the extension towards advanced finite element modeling and the
verification and validation of the presented designs. The following list proposes some topics for future
research and developments.
• The verification and validation of the proposed optimization method and the applied material
models to represent the viscoelastic behavior. Experimental work could verify the predicted
behavior of the optimization routine and validate its effectiveness for the design of viscoelastic
dampers. Besides, the improvements by the modified structural loss factor can be verified by
experimental frequency response analysis.
• The current implementation only specifies a minimum frequency constraint and limits the available material volumes for both materials. In the presented investigation these constraints have
presented the design of structures completely formed by viscoelastic material, by requiring a certain stiffness enforced by the frequency constraint. However, for industrial applications it might
be beneficial to include additional strength, stiffness or stress constraints within the optimization
routine.
• Besides, the effectiveness of the optimization routine can be illustrated by applying this method
for a high performance, mechatronic system to show the applicability of the optimization routine
in industry. This would require extension of the current optimization towards three-dimensional
structures to fully cope with industrial design challenges. The three-dimensional optimization
could be realized by applying commercial finite element packages (e.g. ANSYS or Comsol) to
perform the finite element analysis, while the optimization routine is performed within Matlab
or similar environments Some thoughts on this approach are given in Appendix E.
• Furthermore, the current implementation should be extended to handle temperature and frequency dependencies of the viscoelastic material. The challenge of this research is to include
a fully coupled thermomechanical analysis within the optimization routine. Iterative solution
methods are required to find the steady-state temperature and structural loss factor and the
temperature and frequency dependencies have to be included within the sensitivity analysis
regarding the structural loss factor and possible thermal constraints to limit the temperature
within the viscoelastic material.
• Finally, the optimization routine itself might be extended towards advanced finite element modeling. The parametric level set method allows to introduce a separate discretization for the
structural problem. This provides the opportunity to apply methods such as interface enriched
generalized finite element modeling to provide accurate analysis by including the boundaries
of the multi-material structures in the finite element analysis while still using relatively simple meshes. Moreover, these methods would allow application of boundary conditions on the
interfaces between materials even though the interfaces are updated in each iteration.
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A
Viscoelastic material model
This appendix provides a detailed discussion on the material model, which represents the elastic and
dissipative properties of the viscoelastic material. Besides, additional material and derivations are
presented to supplement the implemented theory in Section 4.2.
A.1. Material model
If a viscoelastic material is subjected to deformation the corresponding stresses are not in phase with
the applied strains (Tschoegl, 1989). This is in contrast to purely elastic or viscous materials, in which
there is either no phase difference between the stresses and strains or a phase difference of exactly 90
degrees. In the case of a harmonically excited structure the stress will lead the applied strain by a
phase angle θ. The strain ε and stress σ are represented as
ε(t) = ε0 sin(ωt),
σ(t) = σ0 (t) sin(ωt + θ(ω, T )).
(A.1)
An illustration of this relation is show in Figure A.1. The behavior of the harmonically excited
structure and the observed phase difference can be represented by introducing a complex material
modulus. This modulus is found by dividing the stresses by the strains (Tschoegl, 1989; Grootenhuis,
1970), resulting in:
E(ω, T ) = E(ω, T )′ + iE(ω, T )′′ .
(A.2)
The storage E ′ and loss modulus E ′′ are related to the material loss factor η and the loss angle θ:
E(ω, T )′′
= tan(θ(ω, T )) = η(ω, T ).
E(ω, T )′
(A.3)
1
Strain
Stress
0.5
0
−0.5
−1
0
0.1
0.2
0.3
0.4
0.5
time (s)
0.6
0.7
0.8
0.9
1
Figure A.1: Illustration of the phase difference between strain ε(t) and stress σ(t) for a harmonically
excited viscoelastic material.
53
54
A. Viscoelastic material model
Figure A.2: Illustration of the frequency and temperature dependent behavior of the shear modulus and
loss factor of a viscoelastic material. The loss factor shows a maximum for certain operating frequency
and temperatures. Image modified from Beards (1997).
Figure A.2 illustrates the loss factor as function of both temperature and frequency. The loss factor
significantly decreases near the extremities of both frequency and temperature. This behavior has to
be kept in mind when selecting viscoelastic materials which show maximal damping characteristics
for the required operating frequency and temperature. The proposed method introduced in Section
4.2 simplifies the material model by neglecting the temperature and frequency dependencies of the
viscoelastic material and assuming the viscoelastic material remains close to its intended operating
frequency and temperature.
Single DOF model
To illustrate the implementation of the hysteretic model a single degree of freedom, mass spring
damper system is compared for viscous and viscoelastic damping. Each structure is subjected to a
harmonic excitation, which results in the following equations of motion:
mẍ + cẋ + kx = F0 eiωt ,
mẍ + k(1 + iη)x = F0 eiωt .
(A.4)
The complex stiffness introduces a damping term to the equation of motion, similar to the viscous
damping coefficient. It becomes even more clearly visible when comparing both transfer functions:
Hc =
Hv =
1
(k −
ω 2 m)
+ iωc
,
1
.
(k − ω 2 m) + ikη
(A.5)
The viscoelastic damping is not scaled by the excitation frequency and shows a phase difference, which
is not necessarily aligned with the velocity, as in the viscous model. The viscoelastic model will also
provide damping for very low frequencies, which is not an accurate representation of the material
model. However, when the viscoelastic material is subjected to an excitation, the complex material
model provides a relatively straightforward method to model both elastic and dissipative behavior.
A.2. Energy storage and dissipation
Viscoelastic material that is subjected to deformation will dissipate energy. For a harmonically excited
structure, the dissipated energy is determined from a hysteresis loop. Figure A.3a illustrates the
elliptical hysteresis loop when plotting the stresses σ(t) as function of the strains ε(t). The area
A.2. Energy storage and dissipation
55
(a) Hysteresis loop.
(b) Material modulus in complex plane.
Figure A.3: (a): Illustration of a hysteresis loop of a viscoelastic material subjected to harmonic
excitation. The loop is found by plotting the stress σ(t) as function of the strain ε(t). (b): The complex
material modulus drawn in the complex plane to express the modulus as function of the storage or loss
moduli using the phase angle θ.
captured by the hysteresis loop describes the dissipated energy in the viscoelastic material during
each cycle. The dissipated energy is found by solving the following contour integral:
∮
Wcyc = σ(t) dε(t).
(A.6)
The strain and stress definitions from Equation A.1 are applied and the integral is solved for one
cycle:
∮
(
)
Wcyc (ω) = σ0 (ω) sin ωt + θ(ω) ε0 ω cos(ωt) dt
(A.7)
(
)
= πε0 σ0 (ω) sin θ(ω) .
The relation between the stress and strain is applied to simplify the obtained result:
σ0 =
E
.
ε0
(A.8)
Also, the material modulus E is expressed in its corresponding storage or loss modulus by the relations
obtained from the complex plane, as illustrated in Figure A.3b. The equation is rewritten to obtain:
Wcyc (ω) = πε20 E ′′ (ω).
(A.9)
The power dissipated by the viscoelastic material is captured by multiplying the dissipated energy per
cycle with the corresponding excitation frequency. Note that the excitation frequency is converted to
Hertz by dividing by 2π:
1
Qdissip (ω) = ε20 E ′′ (ω)λ2 .
(A.10)
2
The dissipated power corresponds to a constant value for a single oscillation. This allows to perform
steady-state calculations on the achieved temperatures within the viscoelastic material. However, if
a thermal analysis performed on time scales smaller than a single period additional calculations are
required to determine the dissipations as function of time during a single period.
The viscoelastic material also stores energy elastically during each cycle. Applying similar calculations
the elastically stored energy per cycle is found. The stored energy is a function of the strain amplitude
and the storage modulus.
1
(A.11)
Wstored (ω) = ε20 E ′ (ω).
2
56
A. Viscoelastic material model
The expressions for the dissipated and stored energy are expressed as function of the Young’s modulus.
However, it should be noted that these expressions can also be given as function of the bulk or shear
modulus of the material. These formulations can be applied when its assumed the viscoelastic material
only shows energy dissipation during specific types of deformation, e.g. shear or elongation.
A.3. Loss factor
The material loss factor, Equation (A.3) is commonly applied to describe the dissipative behavior of
the viscoelastic material. To describe a structure containing both structural (elastic) and viscoelastic
material a formulation is required to quantify a structure’s loss factor. The structural loss factor has
been introduced by Johnson and Kienholz (1982) and is expressed as a fraction of dissipated energy
by stored energy during each cycle Carfagni et al. (1998), which is similar to the inverse definition of
the Q-factor:
γ=
Energy dissipated per cycle
Stored energy per cycle
(A.12)
Since the loss factor is mostly determined at a certain resonance frequency of the structure it is
often expressed as a function of the corresponding modeshape and the global stiffness matrix. These
expressions are similar to Equations (A.7) and (A.11) derived in the previous section. This provides
the following expression for the structural loss factor:
γr =
ϕH
r K I ϕr
.
H
ϕr KR ϕr
(A.13)
Notice that the scaling of the modeshapes can be chosen arbitrarily, since the modeshape only provide
information on the shape of the deformation and no information regarding the actual displacements.
In the formulation of the loss factor the modeshapes appear in both numerator and denominator,
which makes the expression of the loss factor independent of the chosen scaling of the modeshapes.
B
Numerical implementation
This appendix provides information on the numerical implementation of the presented multi-material,
parametric level set method. The development of the finite element model is discussed and viscoelastic
material behavior is introduced in the element stiffness matrices. The placement of the shape functions
within the finite element discretization is illustrated. Moreover, the consequences of applying an
approximated Heaviside function are discussed to illustrate the possible introduction of intermediate
materials.
B.1. Finite element model
The design domain is discretized using two-dimensional, plane-stress, four-node quadrilateral finite
elements (Q4). Each element contains four nodes with each two degrees of freedom. The nodes are
located at the corners of the element. A single element is illustrated in Figure B.1. The numerical
implementation only applies square elements with a = b. Linear shape functions are applied:
1
(a − x)(b − y),
4ab
1
N2 =
(a + x)(b − y),
4ab
1
N3 =
(a + x)(b + y),
4ab
1
N4 =
(a − x)(b + y).
4ab
The deformation matrix B contains the derivatives of the shape functions:


N1,x
0
N2,x
0
N3,x
0
N4,x
0




B= 0
N1,y
0
N2,y
0
N3,y
0
N4,y  ,


N1,y N1,x N2,y N2,x N3,y N3,x N4,y N4,x
N1 =
(B.1)
(B.2)
where the subscript ,x or ,y illustrates the derivative with respect to x or y. For the structural,
isotropic material the plane-stress constitutive matrix D is used:


1 ν
0

E 


D=
(B.3)
ν 1
0 .
2
1−ν 

0 0 1−ν
2
For the viscoelastic material the constitutive matrix is given as function of the bulk κ and shear G
modulus. The Young’s modulus is related to the bulk and shear modulus (Tschoegl, 1989, Chapter
1):
E = 3κ(1 − 2ν),
E = 2G(1 + ν).
(B.4)
57
58
B. Numerical implementation
Figure B.1: Illustration of a Q4 element applied for the finite element discretization of the design
domain. The method only applies square elements with a = b.
This allows to define different material responses as result of bulk or shear deformations by specifying a
separate material loss factor for the bulk and shear modulus. A complex Young’s modulus is realized
by specifying the same material loss factor for both moduli. The relations are substituted in the
constitutive matrix D:


3κ(1 − 2ν)ν
3κ(1 − 2ν)
0
 (1 − ν 2 )
(1 − ν 2 )





 3κ(1 − 2ν)ν
3κ(1 − 2ν)
D=
(B.5)
.
0

 (1 − ν 2 )
2
(1 − ν )




0
0
G
The element stiffness matrix is found by solving the integral:
∫
Ke =
B T DB dVe ,
(B.6)
Ve
with Ve the volume of the element. In general these integrals are solved numerically. However, this
work applies the same elements throughout the complete design domain. Therefore, the integration
is only required to be performed once and has been evaluated numerically. The material properties
of the elements are scaled according to the element densities obtained from the optimization routine.
Similarly, the element mass matrix is found by evaluating:
∫
Me =
ρN T N dVe ,
(B.7)
Ve
with N being a vector containing the shape functions Ni and ρ being the density of the material.
In the multi-material implementation an element stiffness and mass matrix are calculated for each
material phase. The kth element matrix is illustrated by the notation Kek , Mek . An assembly routine
is applied to position and sum the components of the element matrices within the global matrices. The
elements are placed such that connections between neighboring elements are realized. The material
properties of each element in the design domain are based on the value of the LSF. A Heaviside
function is evaluated to determine which material phase is represented by the element and performs
the required scaling of the material properties. The global matrices are given as function the element
stiffness matrices and both level set functions:
{(
}
)
k
1
2
1
2
2
K(φ ) = H(φ )
1 − H(φ ) Ke + H(φ )Ke ,
{(
}
(B.8)
)
k
1
2
1
2
2
M (φ ) = H(φ )
1 − H(φ ) Ke + H(φ )Ke .
B.1. Finite element model
59
The volume of each material phase is determined using the Heaviside functions and by integrating
over the design domain:
∫
V1 =
H(φ1 ) dΩ,
∫D
(B.9)
2
V =
H(φ1 )H(φ2 ) dΩ.
D
To investigate the damping characteristics of the structure it is possible to investigate the strains or
stresses available within the viscoelastic material. The optimized structures are expected to show most
strain within the viscoelastic material, as this will directly dissipate energy. The strain components
are obtained from the nodal displacements u and the deformation B as:
[
ε = εx
]T
εy
= Bu,
(B.10)
= Dε = DBu.
(B.11)
εxy
with the stress distribution given as:
[
σ = σx
]T
σy
σxy
A strain transformation is applied to arrive at coordinate independent measures for the maximum
in-plane principal strains ε1,2 and the maximum in-plane shear strain εmax :
ε1,2
εmax
√(
)2
εx + εy
εx − εy
=
±
+ ε2xy ,
2
2
√(
)2
εx − εy
=
+ ε2xy .
2
(B.12)
B.1.1. Thermal model
The strains calculated during a steady harmonic oscillation allow to calculate the energy dissipated
during a singel cycle of the viscoelastic damper. The strains within each finite element describe
the local energy dissipated by each element. To analyse the resulting steady-state temperature, the
described finite element is extended to include element and nodal temperatures. Each node in Figure
B.1 is given a additional DOF to describe its temperature. The temperature at the center of the
element is then given by the average of the four surrounding nodal temperatures. The discretized
equation to describe the thermal problem is given by:
CTT Ṫ + KTT T = R.
(B.13)
Since we are only interested in the steady-state temperature the time derivatives of the temperatures
Ṫ become zero and the thermal capacity matrix C drop out of the equation. To find the steady-state
temperature distribution the following problem is solved:
KTT T = R.
(B.14)
Here KTT being the thermal conduction matrix (also referred to as the thermal stiffness matrix) and
R the heat load imposed on the structure. In this analysis we limit ourself to convective boundary
conditions and imposed boundary temperatures. To include the convective boundary conditions it is
required to modify the thermal stiffness matrix and the boundary loads accordingly. When including
the convective boundary conditions the steady-state problem is given by
(KTc + KTh ) T = RQ + Rh .
(B.15)
The thermal conduction matrix is here given by KTc , the convective matrix KTh , the internal heat
sources (or sinks) RQ and the convective boundary conditions Rh . The global matrices are assembled
from the corresponding element matrices, which are found by integration of the shape functions for a
single element. The matrices are found by evaluating the following integrals. Note that the convective
60
B. Numerical implementation
boundary conditions only influence the nodes located on the boundary. The shape functions regarding
the convective boundary conditions have been modified accordingly.
∫
∫
KTc =
B T κB dV,
RQ =
N T Q dV,
V
V
∫
∫
(B.16)
T
T
KTh =
N N h dS,
Rh =
N hT∞ dS.
S
S
B.1.2. Radial basis function placement
Various methods are available to place the RBFs within the discretized design domain. In this implementation it is chosen to align the centers (knots) of the RBFs with the nodes of the finite element
discretization. This means that for each node in the finite element model a RBF is placed and scaled
with the corresponding expansion coefficient αi . To align the RBF knots with the nodes greatly simplifies the numerical implementation of the parametric level set method. The alignment of the knots with
the nodes allow to directly extract the function value after the level set function has been constructed
by summation. The values of the LSF are known directly above the nodes in the discretization. For
implementations with different placement of the RBF knots, an interpolation has to be performed to
extract the level set function value in between the location of the expansion coefficients.
Figure B.2: One-dimensional illustration of the radial basis function placement. The RBFs are aligned
with the centers of the elements in the finite element discretization. The shape functions ξi and the
expansion coefficients αi are aligned with the centers of the ith element in the design domain.
The placement of the RBFs are illustrated for a one-dimensional situation in Figure B.2. The expansion
coefficients α are directly above the center of the ith element in the discretization. The level set
function is constructed by summing the contributions of all scaled shape functions. Since this only
results in a vertical shift of the LSF, the values at the center of the elements are directly extracted.
The material properties of each element are related to the level set function values at the center of
each element. The simplicity of using a single discretization for both the level set function and the
structural design domain results in possible interplay between the level set function and the structural
model (van Dijk et al., 2013). However, these effects have not been observed in the case studies
analyzed within this thesis.
B.2. Heaviside function: discrete and approximate
A Heaviside function is applied to the LSFs to determine the material properties of the elements in
the material discretization as stated in Equation (B.8). In general the Heaviside function represents
a discontinuous function whose values equal zero for negative arguments and equal one for positive
arguments. The Heaviside function, or unit step function, is represented by the discrete form as:
{
0,
φk < 0,
k
H[φ ] =
(B.17)
1,
φk > 0.
B.2. Heaviside function: discrete and approximate
61
3
1
Derivative of Heaviside function
0.9
Heaviside function
0.8
0.7
0.6
0.5
0.4
∆ =1.25
∆ =1
∆ =0.75
∆ =0.5
∆ =0.25
0.3
0.2
0.1
0
−1.5
−1
−0.5
0
0.5
Level set function value
1
(a) Heaviside function H(φ).
1.5
2.5
2
1.5
1
0.5
0
−1.5
−1
−0.5
0
0.5
Level set function value
1
(b) Derivative of Heaviside function
1.5
dH(φ)
.
dφ
Figure B.3: Approximations of the discrete Heaviside function and its derivative. Both approximated
formulations are plotted for various values of the parameter ∆. For decreasing values of ∆ the approximations approach the discrete formulations.
The implementation of a discrete Heaviside function results in a discrete optimization problem, where
the design variables are either void or solid material. To overcome difficulties related to discrete
optimization problems an approximated Heaviside function is implemented to continuously scale the
material density between void and solid. The material description of a parametric level set method
using an approximated Heaviside function becomes similar to density-based optimization methods
with continuous pseudo-density descriptions (Sigmund and Maute, 2013). The Heaviside function is
described using the analytical approximation as (Wang et al., 2003):


φk ≤ −∆,

a,
( k
)
k 3
(φ
)
3(1−a)
φ
k
H(φk ) =
+ 1+a
(B.18)
4
∆ − 3∆3
2 , −∆ < φ < ∆,


1,
k
φ ≥ ∆.
The parameter a is chosen as a small number (≤ 1 × 10−3 ) to represent void material. The parameter
∆ determines the range (width) and slope of the approximated Heaviside function. Decreasing the
value of ∆ makes the approximation approach the discrete Heaviside function. Figure B.3a illustrates
the Heaviside function for different values of ∆. The derivative of the Heaviside function is used during
the sensitivity analysis for the optimization. In the direct formulation the derivative corresponds to
the Dirac delta function and the approximate formulation is found by taking the derivative of the
approximate Heaviside function:

(
( k )2 )

3(1−a)

1 − φ∆
,
|φk | ≤ ∆,
dH
4∆
=
(B.19)

dφk
b,
|φk | > ∆.
The parameter b is again chosen as small number to represent the zero derivative below and above the
continuous region of the Heaviside function. In Figure B.3b the derivative of the Heaviside is plotted for various values of ∆. For decreasing values the derivative will approach the Dirac delta function.
The approximation of the Heaviside function has consequences for the material distribution in the
finite element discretization. If a discrete Heaviside function is applied the material corresponds to either solid or void elements, since the discrete function allows no intermediate density values. However,
the introduction of the approximated Heaviside function does allow intermediate density values to appear near the boundary between material phases. The ersatz material model is applied to scale the
material properties of elements with intermediate pseudo-density values with respect to the function
value of the approximated Heaviside function. Intermediate densities might provide favorable material
properties for the objective of the optimization, resulting in large regions of intermediate material in
62
B. Numerical implementation
the final design. In multi-material implementations, these intermediate material regions might overlap and thereby introduce elements with mixed material properties. Intermediate and mixed elements
are unacceptable for the final, optimized designs. These mixtures often provide unrealistic material
behavior and should be avoided in the optimized topology. If these mixed elements have a strong
presence in the final designs additional penalization methods can be implemented. For the current
implementation the final design have shown little mixed elements and provide crisp material boundaries, as highlighted in Section 4.4 and B.3. Therefore no additional penalization has been applied.
Figure B.4 illustrates the differences between the finite element material discretization for a discrete
and approximated Heaviside function. The figure shows the difference for domains containing single
and multiple materials. A discrete Heaviside function provides a reasonable approximation of the
curves in the boundary, as illustrated by Figure B.4a. The representation of the curved boundaries
could be improved with different approaches, for example by applying smaller element sizes, by application of finite elements with different shapes or by applying improved meshing algorithms to generate
a conforming mesh. The topology is shown for an approximated Heaviside function in Figure B.4b.
The crisp boundary between solid and void elements has disappeared and gray, intermediate elements
are introduced near material boundary. Similar behavior is observed for multi-material structures.
A discrete Heaviside function provides distinct separation between all material phases, as given in
Figure B.4c. The approximated Heaviside again loses the crisp boundaries, as illustrated in Figure
B.4d. Elements with mixed material properties are obtained due to the overlap of both intermediate
materials. The mixed domains are mostly obtained near the boundary between the structural and
viscoelastic phase and are illustrated by the purple hue surrounding the boundary in Figure B.4d.
B.2. Heaviside function: discrete and approximate
(a) Single material, discrete Heaviside function.
Black and white elements correspond to solid and
void elements.
(c) Multi-material, discrete Heaviside function.
Blue and red elements represent two different
material domains.
63
(b) Single material, approximate Heaviside function. Gray elements illustrate elements of intermediate density.
(d) Multi-material, approximate Heaviside function. Lightly colored elements represent elements
of intermediate density, while mixed colors (purple) illustrate a mixture of material properties
from each material domain.
Figure B.4: Illustration of the difference between discrete and approximate Heaviside functions on
the material distribution and element densities within the finite element discretization. The cyan lines
illustrate the boundaries obtained from the LSF. The ersatz material model is applied to scale the
material properties of each element with the corresponding LSF value. For a single material (a,b)
the approximation introduces intermediate material in the neighborhood of the boundary. Moreover,
mixed material properties are observed near the boundary between two materials for the multi-material
implementation (c,d).
64
B. Numerical implementation
(a) LSF φ1 .
(b) LSF φ2 .
Figure B.5: Visualization of both level set functions as presented by the final iteration of the optimization of case study I. The first LSF φ1 represents the placement of material. The second LSF φ2
distinguishes the viscoelastic material within the material domain as specified by the first LSF. The black
contour line highlights the zero-level set.
B.3. From Level set function to topology
The conversion from level set function towards topology optimization is illustrated based on the results generated in case study, as described in Section 4.4.1. The multi-material topology optimization
is extracted from both level set functions. The shape of both level set functions at the final iteration
are presented in Figure B.5. The steep transition between maximum and minimum values result in
crisp transitions between the available material phases. The steep boundaries of the level set function
are clearly visible in the second level set function in Figure B.5b.
Any mixed material regions are more pronounced during the initial iterations of the optimization. For
roughly the first 10 iterations mixed elements are observed near the zero-level set. Another region
of mixed materials are barely visible on the right side of the structure where some material phases
overlap during these iterations. The material distribution at four iterations is visualized in Figure
B.6. The mixed materials along the boundary are visible in the initial iterations, while these have
disappeared almost completely in the final design. Similarly, the mixed material properties on the
right side of the structure are not observed in the final design. The plots also illustrate that the
optimization hardly encounters mixed material phases for the used formulation and description of the
objective function. Therefore, no additional penalization method is required to force the optimization
towards clearly separated material domains.
Figure B.7 illustrates a set of similar figures created for case study II, as treated in Section 4.4.3.
These results again show mixed material domains during the initial iterations and hardly any mixed
elements in the final design.
B.3. From Level set function to topology
65
(a) Iteration 2.
(b) Iteration 4.
(c) Iteration 6.
(d) Iteration 35.
Figure B.6: The material distribution at four iterations during the optimization of case study I. Red and
blue elements correspond respectively to structural and viscoelastic material. The cyan line highlights
the zero-level set extracted from the level set functions. The mixture of these colors illustrates elements
with mixed material. The mixed materials are mostly encountered along the zero-level set, as well as on
the right side of the domain.
(a) Iteration 2.
(b) Iteration 3.
(c) Iteration 30.
(d) Iteration 58.
Figure B.7: The material distribution at four iterations during the optimization of case study II. Red
and blue elements correspond respectively to structural and viscoelastic material. The cyan line highlights
the zero-level set extracted from the level set functions. The mixture of these colors illustrates elements
with mixed material. The mixed materials are mostly encountered along the zero-level set, as well as on
the right side of the domain.
C
Loss factor sensitivity analysis
This appendix discusses the adjoint sensitivity analysis used in the optimization. The structural loss
factor is used as objective function during the optimization. The loss factor is given as function of the
complex-valued modeshapes ϕ as obtained from the eigenvalue analysis. The structural loss factor
for mode r is given as
ϕH K I ϕr
,
(C.1)
γr = Hr
ϕr K R ϕr
with KI = ℑ (K) and KR = ℜ (K). The optimization applies the expansion coefficients αik used in
the parameterization of the level set function as design variables. Therefore, we are interested in the
sensitivity of the structural loss factor with respect to infinitesimal change of the expansion coefficients.
The presented sensitivity analysis assumes a multiplicity of 1 for all modeshapes in the structure. If
higher multiplicity is encountered different measures are required to determine the sensitivity of the
structure. In this implementation no difficulties have been encountered during the optimization as
result of eigenmodes with a multiplicity larger then one.
The sensitivity is given by the derivative of the γr with respect to the expansion coefficients αik :
(
)
(
)
( H
)
( H
)
∂ϕr
∂ϕr
H ∂KI
H
H ∂KR
ϕr KR ϕ 2ϕH
K
+
ϕ
ϕ
−
ϕ
K
ϕ
2ϕ
K
+
ϕ
ϕ
I ∂αk
I r
R ∂αk
r
r
r ∂αk r
r
r
r ∂αk
i
i
i
i
∂γr
=
. (C.2)
(
)
2
∂αik
ϕH
r KR ϕr
The loss factor sensitivity contains various terms that have to be evaluated: the imaginary and
R
I
real components of the stiffness matrix sensitivities: ∂K
and ∂K
and the mode sensitivity of the
∂αk
∂αk
i
i
r
eigenmode of interest: ∂ϕ
. The eigenmode sensitivity requires to solve many equations to obtain the
∂αk
i
change of DOF within the eigenmode by changing a single design variable. This provides matrix of
RN ×n with N number of DOFs in the finite element discretization and n number of design variables.
We would prefer a method to find the sensitivities of the structural loss factor without evaluation
all eigenmode sensitivities. Therefore, an adjoint sensitivity analysis is applied, which provides an
efficient method to calculate the sensitivities of low number of response functions to a large amount
of design variables. The method of adjoint sensitivities are discussed by Adelman and Haftka (1991).
In the adjoint method additional terms are introduced to the objective function, which are multiplied
by the adjoint variables. The expressions we add to the objective value are the constraints applied to
the structural loss factor: the eigenvalue problem and the mass normalization of the modeshapes
γr =
ϕH
r KI ϕr
,
ϕH
r KR ϕr
s.t.
(K − λr M ) ϕr = 0,
ϕH M ϕ = 1.
(C.3)
These terms are added to the structural loss factor using the adjoint variables µ1 and µ2 . Notice that
µ1 is a vector containing equal number of DOFs as the eigenmodes ϕ. The adjoint formulation then
becomes:
(
(
) )
(
)
ϕH
r KI ϕr
⋆
H
2
γr = H
+ ℜ µ1 K − λr M ϕr + µ2 ϕH
(C.4)
r M ϕr − 1
ϕr KR ϕr
67
68
C. Loss factor sensitivity analysis
We only require the real part of the second term to equal zero, which has been illustrated by (van der
Veen et al., 2014, Appendix B). By taking the derivative of the adjoint formulation with respect to
the design variables results gives:
(
)
(
)
( H
)
( H
)
∂ϕr
∂ϕr
H
H ∂KI
H
H ∂KR
2ϕ
K
ϕ
K
ϕ
+
ϕ
ϕ
−
ϕ
K
ϕ
2ϕ
K
+
ϕ
ϕ
R
r
I
r
I
r
R
r
k
k
k
k
r
r
r ∂α
r
r
r ∂α
∂αi
∂αi
i
i
∂γr⋆
=
(
)2
k
H
∂αi
ϕr KR ϕr


(
)
(
) ∂ϕ
2
∂K
∂M
∂λ
(C.5)
r
r
2

+ ℜ µH
− λ2r k −
M ϕr + µH
1
1 K − λr M
∂αik
∂αi
∂αik
∂αik
(
)
∂ϕr
H
H ∂M
+ µ2 2ϕr M k + ϕr
ϕr .
∂αi
∂αik
∂λ2
r
In these sensitivities both the modeshape sensitivities ∂ϕ
and the eigenfrequency sensitivities ∂αrk
∂αk
i
i
appear. Both of these sensitivities are multiplied by the adjoint variables. Therefore, by correctly
choosing the values for µ1 and µ2 , we are able to obtain an expression for the structural loss factor
in which the calculation of the modeshape and eigenfrequency sensitivities are not required. To find
the values for the adjoint variables the following statements must hold:

(
)
)( H
)( H ) ( H
(
(
))
(
) ∂ϕ
K
K
ϕ
2ϕ
K
−
ϕ
K
ϕ
2ϕ
ϕH
R
I r
I
r
r
r
r
2

 r R r
+ ℜ µH
+ µ2 2ϕH
= 0,
(
)2
1 K −λ M
rM
k
∂α
ϕH
K
ϕ
i
R r
r
∂λ2r
M ϕr = 0.
∂αik
(C.6)
A trivial solution for this equation is to have the modeshape and eigenfrequency sensitivities equal
zero. However, this is not the solution we are looking for, since we attempt to find a value for the
adjoint variables which satisfies these equations. Therefore, we have to find a solution for the following
relations:
µH
1
(
ϕH
r K R ϕr
)(
)
)( H
) ( H
(
)
(
))
(
2ϕr KR
2ϕH
r KI − ϕr KI ϕr
H
2
+ ℜ µ1 K − λ M
+ µ2 2ϕH
(
)2
r M = 0,
ϕH
r KR ϕr
(C.7)
and
µ1
∂λ2r
M ϕr = 0.
∂αik
(C.8)
The adjoint variables are found by solving these two statements simultaneously. After rearrangeing,
the adjoint problem becomes:
    − ϕH K ϕ (2K ϕ )+ ϕH K ϕ (2K ϕ ) 
(
)
( r R r) I r ( r I r) R r
2
2
µ
K − λr M
2M ϕr

(ϕHr KR ϕr )
  1 = 


.
µ
2ϕH
M
0
2
0
r
(C.9)
Notice that in the second equation the eigenfrequency sensitivity is left out. Since this is only a scalar
multiplication its value will make no difference when the equation has to equal zero. Besides, the
additional multiplication by 2 is introduced to obtain a symmetric matrix on the left hand side. The
symmetry of this matrix allows to apply efficient solvers to numerically solve the system. By applying
the values found by this adjoint problem, the statements (C.7) and (C.8) equal zero and will drop from
the sensitivity expression (C.5). Only the remaining terms from this expression need to evaluated to
determine the loss factor sensitivities:
69
(
(
)
)
( H
)
H ∂KI
H ∂KR
ϕ
K
ϕ
ϕ
−
ϕ
ϕ
ϕ
I r
r
r ∂αk r
r
r ∂αk
i
i
∂γr⋆
=
)2
(
k
∂αi
ϕH
r KR ϕr

(
) 
∂K
∂M
∂M
+ ℜ µH
− λ2r k ϕr  + µ2 ϕH
ϕr .
1
r
k
∂αi
∂αi
∂αik
(
ϕH
r K R ϕr
)
(C.10)
To solve this expression we need to find the values of the adjoint variables and the sensitivity of the
mass and stiffness matrices. These can be obtained by differentiating Equation (B.8) with respect to
the design variables αik . We only need to solve the adjoint variable problem once for each investigate
modeshape during each iteration. Using these values of the adjoint variables allows us to efficiently
compute the structural loss factor sensitivities with respect to all design variables αik . If we would not
have applied the adjoint sensitivity analysis, the calculation of the structural loss factor sensitivities
would require to solve both the eigenfrequency and eigenmode sensitivities for the sensitivity with
respect to all design variables. For this sensitivity analysis the adjoint method provides an exact and
efficient method to evaluate the structural loss factor sensitivities.
D
MATLAB Code
This appendix describes the developed MATLAB implementation for the presented topology optimization routine. In this appendix the working principles of the code are explained. The most important
components are highlighted here and the implementation is related to formulations and equations as
discussed earlier in the thesis. Moreover, a minimal working example is provided for the reader to
evaluate the code and experiment with simple case studies.
The m-code contains two major parts: the main file init_VEM.m and the optimization loop opt_VEM_1.m.
Important parameters and variables are organized within structures to conveniently pass information
between the different functions and store the obtained results in an organized manner. The following
structures are used throughout the implementation:
• O.: mesh information, such as element dimensions, numbering and global coordinates.
• fix.: information in boundary conditions, such as fixed DOFs and imposed temperatures.
• prop_k.: material properties for the kth material.
• rbf.: stores information on the radial basis functions and its dimensions.
• opt.: parameters, variable and results involved or obtained from the optimizer.
• MMA.: all information required for the MMA optimizer, such as objective values, sensitivities and
results from previous iterations.
The following sections describes the function of the MATLAB code and discussed the various lines of
the code.
D.1. Main file
The main file init_VEM.m applies the settings for the optimization routine and initializes the iterations.
After the iterations are finished or the solution is converged, the code is resumed to perform the postprocessing.
Lines 27-52
In these lines the number of elements and dimensions are defined. These elements construct the finite
element discretization using during the structural analysis and for the placement of the RBFs. The
element numbering and the corresponding nodal coordinates are generated by elem_def.m. The square
elements allow for a simple mesh generation by iteration through all x and y elements. Finally, the
required boundary conditions are specified and the subjected nodes are extracted from the mesh.
71
72
D. MATLAB Code
Lines 53-75
The material properties for both material are specified. The bulk and shear modulus of each material
are derived from the specified Young’s modulus. The amount of viscoelastic damping is introduced
by specifying the material loss factor for the bulk or shear modulus using prop_k.eta_bulk and/or
prop_k.eta_shear. A complex Young’s modulus is realized by specifying the same loss factor for the
bulk and shear modulus.
Lines 76-102
These lines define the RBFs used for the parameterization of the level set functions. In the implementation the RBF knots are aligned with the nodes of the finite element analysis. For simplicity
the knots are aligned with FE nodes. The function csrbf.m determines the value of a shape function
given the current location and its distance from the knot of the shape function. A simple for-loop
is evaluated to build a single matrix that contains the values of the shape functions within a matrix
surrounding the knot of the basis function.
The obtained matrix rbf.shape will be used to construct the complete LSF by summation of all these
matrices multiplied by their corresponding expansion coefficients αi . It has to be noted that small
changes to the initial values of αi might result in different performance during the optimization.
Lines 103-140
These lines define optimization parameters, such as the investigated number of eigenmodes, the applied
volume constraints and maximum number of iterations to consider. Furthermore, the set of variables
required for the MMA optimizer are initialized. Line 136 calls the optimization routine opt_VEM_1.m.
Lines > 140
All lines after 140 are evaluated once the optimization is converged or the maximum number of
iterations have been evaluated. In these lines some post-processing is performed and can be extended
as necessary.
D.2. Optimization loop
The optimization loop is initialized by calling opt.VEM_1.m. This function provides all the analysis
required to build the level set functions, to evaluate the current objective and constraint function,
to calculate the sensitivities and to evaluate the MMA optimizer. The iterations are continued until
the convergence criteria are achieved or until the maximum number of iterations are evaluated. The
individual steps are briefly presented in the following paragraphs.
Lines 3-25
These lines preallocate some variables to allow for easy storage during the optimization routine.
Besides, the element stiffness and mass matrices are requested from elem_matrices.m. This function
can also be used to determine the element constitutive matrices as well as the element thermal matrices
for conduction and convective heat problems. The iterations are started after these lines are evaluated.
Lines 27 - 55
Information of the previous iteration is stored, which is provided to the optimizer for the current
iteration. This only takes place after iteration 2. Then, the level set function is constructed by
summation of the shape functions, which are scaled with the corresponding value of α. Notice that
these shape functions are stored in a slightly larger domain. This allows to easily coupe with the
overlap of the shape functions by only extracting the domain of interest.
Lines 59 - 62
These lines convert both level set functions to the corresponding Heaviside function and the derivative
of the Heaviside function by evaluating Equations (B.18) and (B.19) for each RBF knot.
D.3. Example case
73
Lines 65 - 82
The first lines assemble the global stiffness and mass matrices using the assemble.m function. This
function loops through all elements of the design domain, determines their degrees of freedom and
places the scaled element stiffness or mass matrices at the corresponding locations within the global
matrices. Faster assembly methods are possibly by evading these for-loops and using smart functions
to construct the sparse matrices. The material properties are scaled using Equation (B.8). These
global matrices do not include any boundary conditions yet. Therefore, lines 68 and 69 remove any
rows and columns from the global matrices that correspond to fixed elements. Using these matrices
the eigenvalue problem is solved. The eigensolutions are sorted based on smallest magnitude to obtain
the first n complex-valued eigensolutions. The obtained eigenmodes are also stored in vector which
correspond to the length of the original global matrices in lines 76 to 82. This is to allow for easier
calculations during the calculation of the loss factor sensitivity and also for animation purposes.
Lines 85 - 94
This loop determines the loss factor, the adjoint variable problem and the loss factor sensitivities
for the eigenmodes of interest. The function LossFactor.m evaluates Equation (4.13) to determine
the loss factor corresponding to this modeshape. Then, the functionAdjointProblem.m solves the
adjoint variable problem as given in Equation (4.23). Finally, LossFactor_Sensitivity.m determines
the loss factor and volume constraint sensitivities using the previously calculated values of the adjoint
variables.
Lines 96 - 167
If multiple eigenmodes are considered for the optimization, their loss factors are combined to determine
the objective value of the current design.
Lines 187 - 199
The current objective, constraint and sensitivity values are reshaped to satisfy the requirements of the
MMA optimizer. When all values are stored, the MMA optimization is called and the design variables
α are updated with their new values.
Lines > 200
The convergence statements are evaluated, some results are outputted towards the command window
and result_plot is called to visualize the changes made to the level set functions and the material
distribution.
D.3. Example case
The provided code performs an example optimization of a cantilever beam. The optimization aims
to maximize the structural loss factor of the first two resonant modes of the structure. The reader is
encouraged to modify various parameters and investigate the many different designs presented by the
optimization routine.
E
Commercial FEM implementation
This appendix presents some brief thoughts on how the presented parametric level set method can be
implemented with commercial finite element software. In this example MATLAB and ANSYS illustrate the respective programming and finite element environments. However, the discussed method
can be applied with any programming environment and commercial finite element software as long as
the possibility exist to communicate the required numerical information between both environments.
The implementation of commercial software will slightly simplify the steps towards a three dimensional analysis by providing assembly, mesh generation and robust solvers from their toolbox.
The diagram in Figure E.1 shows a schematic diagram of the optimization procedure in which two
different computational environments are used. A dedicated finite element solver is applied to generate
and update the mesh of the domain. A separate programming environment is responsible for the
sensitivity calculation and optimization procedure. This part can be written in MATLAB, python, C
or similar languages, as long as the language allows efficient use of sparse matrix, vector calculations
and is able to interface with the native code of the finite element environment. Some additional
thoughts on this method are given in the following list:
• The parametric level set method allows for the implementation of two discretization one for
the structural analysis and the other for the construction of the level set functions. Combining
a dedicated FE routine with a separate Matlab evaluation for the optimization would be an
interesting scenario to apply these two different discretizations. The three-dimensional mesh
can be created as fine as required, or even have non-uniform element sizes throughout the
design. From the Matlab point of view a very careful analysis is required for the construction of
both level set functions and the distribution of shape functions. Applying the knots of each RBF
to match the FE discretization might not be required and a separate, more coarse discretization
of the RBFs could be applied.
• Care has to be taken during the sensitivity analysis of the system. In the case multiple discretizations are applied, the FE nodes do not necessarily overlap with the knots in the LSFs.
Therefore, an interpolation step is required to determine the LSF value to match the location
of the FE node. Similar, the sensitivity analysis might become complicated when the nodes are
not aligned with the RBF knots. Therefore, all the nodes captures by a single RBF wil have a
certain contribution to the sensitivity of changes in the height of that basis function.
• Attention is required for the communication and interfacing between both methods. This is
especially of importance to develop a robust method, capable of handling different types of
finite elements. Also, the interface between both tools requires clear communication on the
construction of the elements and the number of the degrees of freedom. Without information on
the construction of the elements and the relations between the degrees of freedom, the elementwise calculation of the sensitivities becomes quite difficult.
75
76
E. Commercial FEM implementation
Start
Initialize shape
functions and
optimizer settings
Finite element
environment
Contruct, import
design domain
MATLAB
environment
Apply boundary
conditions
Generate mesh
Perform analysis
Construct LSF
Extract global matrices,
eigensolutions, element
numbering, list of DOF
Calculate: objective,
constraint functions and
sensitivities
Call optimizer to find
design update
Update mesh
Update LSFs with new
design variables
Modify structure to
match updated LSF
No
Design converged?
Yes
End
Figure E.1: A schematic diagram of the optimization procedure using a dedicated finite element environment to perform the FE calculations. The green blocks are performed within the FE environment,
while the orange blocks are evaluated in MATLAB or similar programming environments.
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