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A STRAIN BASED TOPOLOGY OPTIMIZATION METHOD
By
EUIHARK LEE
A dissertation submitted to the
Graduate School-New Brunswick
Rutgers, The State University of New Jersey
In partial fulfillment of the requirements
For the degree of
Doctor of Philosophy
Graduate Program in Mechanical and Aerospace Engineering
Written under the direction of
Professor Hae Chang Gea
And approved by
________________________________
________________________________
________________________________
________________________________
New Brunswick, New Jersey
October, 2011
© 2011
Euihark Lee
ALL RIGHTS RESERVED
ii
ABSTRACT OF THE DISSERTATION
A Strain Based Topology Optimization Method
by Euihark Lee
Thesis Advisor:
Dr. Hae Chang Gea
Strain energy based topology optimization method has been used since topology
optimization method was presented. Although successful examples from strain energy
based topology optimization have been presented, some of the optimal configurations of
these designs show stress concentrations or a localized large deformation which is highly
undesirable in structure design.
In this dissertation, firstly, the strain energy based topology optimization method
is reviewed to discover the cause of the problems. Moreover, strain based topology
optimization method is presented to avoid these drawbacks. Instead of minimizing strain
energy, global effective strain function is minimized. Using this function, compliant
mechanism and energy absorbing structure design problems are presented. Since these
designs require flexibility and rigidity at the same time, a multi-objective optimization
iii
problem is formulated using a physical programming method. Comparisons of design
examples from both the strain energy based topology optimization and the proposed
method are presented and discussed.
The main contributions of this dissertation are listed as follows: (1) deriving the
sensitivity of the global effective strain, (2) presenting a complaint mechanism design
scheme that can distribute the deformation within the entire mechanism to avoid localized
deformation, (3) formulating energy absorbing function by implementing seek range
class function using physical programming method.
iv
Acknowledgments
This dissertation was accomplished with kind help from lots of people. I would
like to express my deepest appreciation to my advisor Professor Hae Chang Gea for his
valuable guidance and continuous encouragement throughout my doctoral studies. His
friendly advice and high degree of motivation have been very helpful to my academic and
personal aspects of my life.
I would also like to thanks to my committee members, Professor Alberto Cuitino,
Professor Mitsunori Denda, and Professor Kang Li for their interest in this research and
for their valuable time in reviewing this dissertation.
My appreciation goes to my lab mates: Dr. Ching Jui(Ray) Chang, Dr. Bin Zheng,
Dr. Po Ting Lin, Wei Ju(Lisa) Chen, Wei Song, Xi Ke Zhao, Hui hui Qi, Xiao Bao Liu,
Zhe Qi Lin, Xiao Ling Zhang, Jian Tao Liu, Yi Ru Ren and Wang Bo. Their physical and
mental supports push me forward. I would also like to give my great appreciation to my
colleges: Sung Chul Jung, Gobong(Paul) Choi, Tushar Saraf, Pallab Barai, and all other
colleges that may help me continue my research at some points.
Most importantly, I would like to thank my lovely parents and parents in laws:
Jayoung Lee, my dad; Sanggye Choi, my mom, Choonho Jang, my father in law;
Soonyun Kim, my mother in law. Truly thank you all for your earnest encouragement and
devoted support.
v
Lastly, I want to specially devote my appreciation to my lovely wife, Hyohyun
Jang. Thank you for your sacrifice, love and encouragement. It is nearly impossible for
me to complete this research work without your accompany.
vi
Dedications
To my wife
Hyohyun Jang
and my children
Joonbum and Seoeun
vii
Table of Contents
ABSTRACT OF THE DISSERTATION ................................................................................................... ii
Acknowledgments .........................................................................................................................................iv
Dedications ....................................................................................................................................................vi
Table of Contents ....................................................................................................................................... vii
List of Figures ...............................................................................................................................................ix
List of Tables............................................................................................................................................... xii
Chapter 1. Introduction ................................................................................................................................ 1
1.1. Review of Research in Compliant Mechanism Design ........................................................................ 5
1.2. Review of Research in Energy Absorbing Design ............................................................................... 9
1.3. Review of Physical Programming ..................................................................................................... 11
1.4. Research Contribution ....................................................................................................................... 14
1.5. Outline of the Dissertation................................................................................................................. 16
Chapter 2. Background of Topology Optimization Method .................................................................... 18
2.1. Basic Formulation of Topology Optimization Method ...................................................................... 19
2.2. Microstructure Based Composite Material Models ........................................................................... 21 2.2.1. Solid Isotropic Microstructure with Penalty Model (SIMP) ...................................................... 21 2.2.2. Spherical Micro-inclusion Model .............................................................................................. 24
2.3. Topology Optimization Algorithm ..................................................................................................... 26 2.3.1. Problem Formulation for Topology Optimization ..................................................................... 27 2.3.2. Sensitivity Analysis ................................................................................................................... 29
2.4. Examples............................................................................................................................................ 31 2.4.1. Cantilever Beam: 2D ................................................................................................................. 32 2.4.2. Cantilever Beam: 3D ................................................................................................................. 33
Chapter 3. A Strain Based Topology Optimization Method ................................................................... 36
3.1. Energy Based Topology Optimization Method. ................................................................................. 36
3.2. Stress Based Topology Optimization Method. ................................................................................... 39 3.2.1. Sensitivity Analysis of Stress Based Objective Function .......................................................... 41
3.3. Strain Based Topology Optimization Method. ................................................................................... 44 3.3.1. Sensitivity Analysis of Strain Based Objective Function .......................................................... 46
3.4. Design Examples. .............................................................................................................................. 50 3.4.1. Cantilever Beam ........................................................................................................................ 50
3.5. Conclusion and Remark ..................................................................................................................... 53
Chapter 4. Compliant Mechanism Design using a Strain Based Topology Optimization Method ...... 55
viii
4.1. Energy Based Topology Optimization for Compliant Mechanism Design ........................................ 56 4.1.1. Complaint Gripper І ................................................................................................................... 58
4.2. Strain Based Topology Optimization for Compliant Mechanism Design .......................................... 61 4.2.1. Problem Formulation ................................................................................................................. 62 4.2.2. Sensitivity Analysis ................................................................................................................... 64
4.3. Numerical Examples .......................................................................................................................... 66 4.3.1. Compliant Gripper I ................................................................................................................... 66 4.3.2. Compliant Gripper II ................................................................................................................. 69 4.3.3. Displacement Inverter ................................................................................................................ 71
4.4. Conclusion ......................................................................................................................................... 73
Chapter 5. Energy Absorbing Structure Design using a Strain Based Topology Optimization Method
....................................................................................................................................................................... 74
5.1. Problem Formulation for Energy Absorption Structure Design........................................................ 76 5.1.1. Maximizing Energy Absorbing Formulation ............................................................................. 76 5.1.2. Energy Absorbing Formulation ................................................................................................. 78 5.1.3. Multi-Objective Formulation for Energy Absorbing Structure Design ..................................... 80
5.2. Numerical Example ........................................................................................................................... 81 5.2.1. A Simple Container Model ........................................................................................................ 82
5.2.1.1. The Optimal Design Using Single Objective Minimizing Strain Energy Formulation ...... 83 5.2.1.2. The Optimal using Maximizing Strain Energy Formulation with Physical Programming
Scheme ............................................................................................................................................ 85 5.2.1.3. The Optimal using Seek Range of the Strain Energy Formulation with Physical
Programming Scheme ..................................................................................................................... 88
5.3. Conclusion ......................................................................................................................................... 92
Chapter 6. Conclusion and Future Work .................................................................................................. 93
6.1. Conclusion ......................................................................................................................................... 93
6.2. Future work ....................................................................................................................................... 94
References .................................................................................................................................................... 96
Curriculum VITA ..................................................................................................................................... 101
ix
List of Figures
Figure 1.1.Three Types of Structure Design Optimization: Sizing Optimization (top), Shape
Optimization (middle), Topology Optimization (bottom) ........................................................... 4
Figure 1.2. The Comparison of Rigid-body Crimping Mechanism (top) and Compliant Crimping
Mechanism(bottom) (source: http://compliantmechanisms.byu.edu/) ......................................... 6
Figure 1.3. Ancient Long Bow (source: http://www.squidoo.com/) and Drawn Position of the Bows
(source: http://www.ftexploring.com/)......................................................................................... 7
Figure 1.4. Leonardo da Vinci's sketches of complaint catapults .................................................................... 7
Figure 1.5. Complaint Mechanism Examples in MEMS ................................................................................. 8
Figure 1.6. Examples of Energy Absorbing Structure Design: Spring, Sandwich Board, Cell Design,
Vehicle Frame ............................................................................................................................ 10
Figure 1.7. Vehicle Crash Tests (source: http://www.edaily.co.kr/, http://olpost.com/) .............................. 10
Figure 1.8. Classification of Class Function .................................................................................................. 14
Figure 2.1. Design Domain of Typical Topology Optimization Problem. .................................................... 18
Figure 2.2. The Flow of Topology Optimization Method ............................................................................. 19
Figure 2.3. The Penalty Function is SIMP Model ......................................................................................... 22
Figure 2.4. A Comparison of the SIMP Model and the Hashin-Strikhman Upper Bound for an Isotropic
material with Poisson Ratio υ=1/3 mixed with void. For the H-S Upper Bound,
Microstructures with Properties Almost Attaining the Bounds are also Shown [50]................. 23
Figure 2.5. Microstructures of Material and Void Realizing the Material Properties of the SIMP Model
with p = 3, for a Base Material with Poisson's Ratio υ =1/3. As Stiffer Material
Microstructures can be Constructed from the Given Densities, Non-structural Areas are seen
at the Cell Centers [50] .............................................................................................................. 24
Figure 2.6. Microstructure of the Spherical Micro-inclusion Material Model[51] ........................................ 25
Figure 2.7. Relative Stiffness vs. Volume Fraction for the Spherical Micro-inclusion Model and the
SIMP Model with Penalization Power p=2 ............................................................................... 26
Figure 2.8. The Design Domain and Boundary Conditions for Cantilever Beam Problem ........................... 32
Figure 2.9. The Optimal Design of Cantilever Beam Problem ..................................................................... 33
Figure 2.10. The Design Domain and Boundary Conditions for 3D Cantilever Beam Problem ................... 34
Figure 2.11. The Optimal Design of 3D Cantilever Beam Problem .............................................................. 34
Figure 2.12. The Optimal Design of 3D Cantilever Beam Problem : Top View and Front View ................. 35
Figure 3.1. Material Density Function .......................................................................................................... 38
Figure 3.2. Three Different Finite Element Analysis Systems ...................................................................... 41
Figure 3.3. Cantilever Beam with Different Material Density and Von Mises Stress Distribution: xi=0.3
(top), xi=0.7 (bottom) ................................................................................................................. 44
Figure 3.4. Three Different Finite Element Analysis Systems ...................................................................... 47
x
Figure 3.5. Cantilever Beam with Different Material Density and Von Mises Strain Distribution: xi=0.3
(top), xi=0.7 (bottom) ................................................................................................................. 49
Figure 3.6. Design Domain and Boundary Conditions for Cantilever Beam Problem .................................. 51
Figure 3.7. Optimal Design Configurations: Using Strain Based Formulation (Top) and Strain Energy
Based Formulation (Bottom) ..................................................................................................... 52
Figure 3.8. Von Mises Stress Plot for Optimal Designs: Using Strain Based Formulation (Top) and
Strain Energy Based Formulation (Bottom) .............................................................................. 52
Figure 4.1. Example of Complaint Mechanism: Crimping Mechanism [41] ................................................ 55
Figure 4.2. Design Domain and Boundary Condition for Compliant Mechanism Design Problem .............. 57
Figure 4.3. Design Domain and Boundary Condition for Complaint Gripper І ............................................ 59
Figure 4.4. The Optimal Design of Strain Energy Based Formulation for Complaint Gripper І .................. 59
Figure 4.5. The Full Compliant Gripper І Design of Strain Energy Based Formulation ............................... 60
Figure 4.6. The Strain Energy Plot of Strain Energy Based Formulation for Complaint Gripper І .............. 61
Figure 4.7. The Deformed Configuration of the Compliant Gripper І using Strain Energy Based
Formulation ................................................................................................................................ 61
Figure 4.8. Class Functions: Class 1(The Rigidity Function) and Class 2(The Flexibility Function) ........... 63
Figure 4.9. Finite Element Analysis System for Maximum Output Displacement ....................................... 65
Figure 4.10. The Optimal Design of Strain Energy Based Formulation for Complaint Gripper І ................ 67
Figure 4.11. The Full Compliant Gripper І Design of Strain Based Formulation ......................................... 67
Figure 4.12. Strain Energy Comparisons: The Strain Based Formulation (top) and The Strain Energy
Based Formulation (bottom) ...................................................................................................... 68
Figure 4.13. The Deformed Configuration of the Compliant Gripper І ........................................................ 68
Figure 4.14. Design Domain and Boundary Condition for Gripper .............................................................. 69
Figure 4.15. The Optimal Design of Strain Based Formulation for Complaint Gripper ІІ ............................ 69
Figure 4.16. The Full Compliant Gripper ІІ Design of Strain Based Formulation ........................................ 70
Figure 4.17. The Strain Energy Distribution for Compliant Gripper ІІ ......................................................... 70
Figure 4.18. The Deformed Configuration of the Compliant Gripper ІІ ....................................................... 71
Figure 4.19. Design Domain and Boundary Condition for Displacement Inverter ....................................... 71
Figure 4.20. The Optimal Design of Strain Based Formulation for Displacement Inverter .......................... 72
Figure 4.21. The Full Displacement Inverter Design of Strain Based Formulation ...................................... 72
Figure 4.22. The Strain Energy Distribution for Displacement Inverter ....................................................... 73
Figure 4.23. The Deformed Configuration of the Displacement Іnverter ..................................................... 73
Figure 5.1. Crashworthiness Test of Aircraft (resource: http://www.apg.jaxa.jp, http://www.onera.fr/) ...... 74
Figure 5.2. Design Domain and Boundary Conditions for Energy Absorbing Structure Design Problem.... 75
Figure 5.3. Vehicle Crash Test: (resource: http://autos.msn.com/) ............................................................... 78
Figure 5.4. Class Function for Energy Absorption of Structure .................................................................... 80
Figure 5.5. The Entire Design Doamin of the Simple Container Example ................................................... 82
xi
Figure 5.6. The Quarter Design Domain and Boundary Conditions of the A Simple Container Example ... 83
Figure 5.7. The Boundary Conditions for Minimizing Strain Energy Problem ............................................ 84
Figure 5.8. Optimal Design of a Simple Container Problem ......................................................................... 85
Figure 5.9. A Quarter Part of the Simple Container Model and Boundary Condition: External Loading
for Energy Absorption F1 and Self Weight Loading of Goods F2 .............................................. 86
Figure 5.10. Applied Class Functions: Maximizing Strain Energy (left) and Minimizing Global Effective
Strain (right) ............................................................................................................................... 87
Figure 5.11. Optimal Design of a Simple Container Problem ....................................................................... 88
Figure 5.12. Seek Range Class Function for Energy Absorption Formulation ............................................. 89
Figure 5.13. Optimal Design of a Simple Container Problem ....................................................................... 90
Figure 5.14. The Optimal Design Configurations using Strain Based Formulation (left) and Strain
Energy Formulation (right) ........................................................................................................ 91
Figure 5.15. The Stress Distribution Configurations using Strain Based Formulation (left) and Strain
Energy Formulation (right) ........................................................................................................ 92
xii
List of Tables
Table 1.1. Design Variables Depend on Optimization Method for Truss Optimization Problem ................... 3
Table 1.2. The Range Description of Designer’s Preference [42] ................................................................. 13
Table 3.1. The Relation of Material Density, Strain, Strain Energy .............................................................. 39
1
Chapter 1.
Introduction
Since the dawn of time, mankind has attempted to come up with various designs
in order to enhance the efficiency of an object. These efforts have contributed to the
progress of human society. For example, the circular structure of logs were used to
move heavy objects during ancient times; This concept was further developed into
wheel design, which in turn, has led to other inventions such as gears, trains, and various
other forms of transportations. In this day and age, similar types of efforts are still
practiced across many different fields of study, including engineering.
Structure design engineers always try to find ways to improve on an object design
while using minimal resources. The most common method would be to make numerous
prototype models based on the knowledge and experience of the engineers, and to then
analyze each model to determine which has the best design. However, this method not
only requires a lot of money and time, it also cannot ensure the best design of the
structure, Moreover, the quality of design is highly dependent on the engineer’s ability.
Because of these drawbacks, many engineers have conducted research to find a
methodology that could provide optimal design of the structure under specipic work
conditions. These efforts initially started over one hundred years ago. In 1869, Maxwell
[1] introduced the first theoretical work for structure optimization. Based on Maxwell’s
research, Michell [2] introduced a general theory of minimum weight structure. These
researchers provided the fundamentals of the structure optimization theory.
2
Following the World War ІІ, a great amount of research was put into the aircraft
structure design because of the boom of the aviation industry, and minimum weight
designs for aircraft structure components were thoroughly developed. Simultaneous
Failure Mode Design (SFMD) and Fully Stressed Design (FSD) method for structure
design were developed in this era. This principle was implemented to many optimization
problems in aircraft structural components designs by Shanley [3] in 1952. Prager et al.
[4][5] were presented a uniform method of treating a variety of problems of optimal
design of structures in 1968. This method consisted of two stops: The integration of an
optimality condition and the subsequent determination of the optimal distribution of
elastic stiffness from the usual differential equations of the structure. This method
established the theoretical foundation of the optimality criterion approach.
While the optimal structure design methodology was developed, many
mathematical methods were also being developed, including the Finite Element Method
(FEM). The FEM is a numerical method used to solve engineering and mathematical
problems that involve complicated geometries, loadings and material properties. FEM
was introduced in the 1940s by Hrennikoff [6] and McHenry [7], and they used one
dimensional element to solve stress problem in a continuous solid. Courant [8] introduced
pricewise continuous functions over triangular domains and used an assemblage of
triangular elements and the principle of minimum potential energy to study the St. Venant
torsion problem. In 1954 matrix structural analysis method was developed by Argyris [9].
Turner et al. [10] derived stiffness matrix for two-dimensional elements such as truss,
beam, triangular and rectangular elements in 1956. The three dimensional problems with
the tetrahedral elements was developed by Martin [11] and Gallagher et al.[12]. Based on
3
the developments of FEM theory, it was extended to enormous fields such as large
deflection, dynamic analysis, fluid flow, heat conductions, and bioengineering.
With the immense development of computer technologies, and FEM, structure
optimization began to accelerate its pace. Sizing, shape and topology optimization
problems were presented different aspects of optimal design problems. Table 1.1
describes the difference of design variables depending on the optimization method and
Figure 1.1 is illustrated the conceptual configuration of sizing, shape and topology
optimization method. In a typical sizing problem, the objective is to find out optimal
thickness of members area such as truss cross sections. By changing the thickness of each
member, the whole structure will have minimum compliance, less stress, or less
deflection. In the case of the shape optimization problem, the goal is to find out the
optimal shape of the domain to minimize a certain cost function, or objective function
while satisfying given constraints. Topology optimization is a method that optimizes
material layout within a given design space under boundary conditions. Through
topology optimization, engineers can find the best conceptual design that can satisfy the
objective design. This dissertation is a research based on topology optimization method.
Table 1.1. Design Variables Depend on Optimization Method for Truss Optimization Problem
Optimization Method Design Variable
Sizing optimization Number of members : n
Shape optimization Length of truss members: li
Topology optimization Cross section area of members: Ai
4
Figure 1.1.Three Types of Structure Design Optimization: Sizing Optimization (top), Shape
Optimization (middle), Topology Optimization (bottom)
In traditional structure design of mechanical engineering fields, stiff structure
design is usually considered as a better design than the flexible structure design.
Therefore, the minimizing strain energy or minimizing mean compliance are usually
considered as objective function since those represent stiffness of the structure in
topology optimization fields. However, flexible structure can become a better structure if
flexibility is implemented properly, and we can find many applications that implement
flexibility of the structures. A compliant mechanism and energy absorbing structure are
the two representative examples that use the advantages of flexibility. In the following
section, current researches of complaint mechanism and energy absorbing structure will
be reviewed. Moreover, multi-objective formulation will also be reviewed since both the
5
complaint mechanism design and the energy absorption structure design require the
flexibility and rigidity at the same time.
1.1. Review of Research in Compliant Mechanism Design
A mechanism is a system or structure that can transform motion, force or energy;
rigid-body mechanisms are usually considered dependent on the purpose of the
mechanisms. Traditional rigid-body mechanism is composed of links and joints, and the
mechanisms can be performed by the mobility of these joints or links. Typical rigid-body
mechanism design is illustrated in the top of Figure 1.2.
On the other hand, compliant mechanism also transfers the motion, force, or
energy, but uses flexible members instead of joints or links like that of a rigid-body
since the deformations of flexible members provide the mobility of the mechanism. The
comparison between the rigid-body mechanism and compliant mechanism is illustrated in
Figure 1.2. Since the compliant mechanisms replace hinges or joints as flexible members,
the designs are able to reduce the total number of parts that are required to accomplish a
specified task. This reduction of the parts provides numerous advantages such as easy
assembly, reduced manufacturing cost, and time. Moreover, it is not necessary to use
lubrication since there are no joints or links, and it is also possible to significantly reduce
the weight. Another advantage of compliant mechanisms is that it can be miniaturized
easily because there is no assembly required
6
Figure 1.2. The Comparison of Rigid-body Crimping Mechanism (top) and Compliant Crimping
Mechanism(bottom) (source: http://compliantmechanisms.byu.edu/)
The compliant mechanism has been used for millennia, the bow being a simple
example of this. The ancient longbow and deformation of the bow are illustrated in
Figure 1.3. The early bows were made by relatively flexible members such as wood and
animal sinew. These flexible members were able to store the strain energy and convert it
into kinetic energy to shoot the arrow.
7
Figure 1.3. Ancient Long Bow (source: http://www.squidoo.com/) and Drawn Position of the Bows
(source: http://www.ftexploring.com/)
Compliant catapults are another example of the usage of compliant mechanism in
early ages. A catapult is a device used to throw or hurl a projectile a great distance
without the ad of explosive device. The early catapults are made by wood, and it stored
the strain energy by deflection of wood, and shoots the object when strain energy is
released. Figure 1.4 shows several conceptive sketches of compliant catapults which were
drawn by Leonardo Da Vinci.
Figure 1.4. Leonardo da Vinci's sketches of complaint catapults
8
(a) (b)
Figure 1.5. Complaint Mechanism Examples in MEMS
(a) Compliant Member in a Micro-engine (b) Micro-engine that Uses Several Compliant Members
(Courtesy of Sandia National Laboratories, www.sandia.gov.)
In recent times, compliant mechanisms are used in various fields such as grippers,
actuators and suspension systems. They also can be miniaturized and used in Micro-
Electro-Mechanical-System (MEMS) or embedded structures [1]-[16], and Figure 1.5
shows the application of compliant mechanism in MEMS.
In the topology optimization method, the energy based formulation [17]-[20] is
used for the compliant mechanism design problem in general. Various applications for
compliant mechanism designs based on energy formulation have been reported in
literature [21]-[25].
Since compliant mechanism works as a mechanism as well as a structure, two
contradictory functionalities, flexibility and rigidity, must be considered simultaneously.
For the flexibility design objective, maximizing a selected nodal displacement (mutual
potential energy) is employed; for the rigidity design objective, minimizing the mean
compliance (strain energy) is commonly used. To accommodate both design objectives,
the solution requires a multi-objective optimization formulation. Various formulations
9
which combine these two design objectives have been proposed and many successful
examples have been presented [26]-[30]. However, configurations of many optimized
compliant mechanisms are very similar to the rigid link mechanisms which replace joins
connector with compliant members. It is obvious that these compliant members will
endure large deformations under applied force in order to perform the specified motions
and large deformation will produce high stress which is very undesirable in compliant
mechanism design. Moreover, although various forms have been used in combining both
flexibility and rigidity such as weighted sum and ratios, there have always problems on
the solution convergence. In this dissertation, these problems are discussed in detail and
new formulation is introduced to solve the problems.
1.2. Review of Research in Energy Absorbing Design
Another application that utilizes the advantages of flexibility as a design criterion
is energy absorbing structures design which can protect the structural integrity under
excessive external force. A spring is one of the representative examples for energy
absorbing structure, and there are many examples such as sandwich box, cell design and a
vehicle frame as shown in Figure 1.6. Energy absorbing structure design is an important
consideration in packaging industry field, and a sandwich box and cellular design are the
typical design for packaging. This energy absorbing characteristic can preserve the goods
from external forces that might occur during transportation, or storage.
10
Figure 1.6. Examples of Energy Absorbing Structure Design: Spring, Sandwich Board, Cell Design,
Vehicle Frame
Another example for energy absorption design is a car frame design. When
engineers design the vehicle frame, the capacity of energy absorption should be
considered to ensure the safety of the passengers. The ability of a structure to protect is
called crashworthiness, and it is commonly tested when investigating the safety of
vehicles. Figure 1.7 shows the crash tests of vehicles.
Figure 1.7. Vehicle Crash Tests (source: http://www.edaily.co.kr/, http://olpost.com/)
11
Since the energy absorption is an important criterion, many areas in topology
optimization fields have been studied. Mayer et al.[31] implements topology optimization
method to the crashworthiness design by maximizing crash energy absorption for a given
volume constrain. Soto and Diaz [32] introduced a simplified model for crashworthiness
design using topology optimization method. Soto [33] applied topology optimization
method and heuristic rules on the design of energy-absorbing structure. Pederson [34]-
[38] presented conceptual design for crashworthiness using 2 dimensional frame model.
Gea [39] developed the topology optimization of regional strain energy and Jung et al.[40]
proposed designing a method of an energy absorbing structure using a multimaterial
model.
One challenging issue of energy absorbing structure design with topology
optimization problem is how both flexibility and rigidity can be handled at the same time.
Detailed discussion and new formulation for energy absorbing structure design is
proposed in Chapter 5.
1.3. Review of Physical Programming
In many cases, it is required for optimizer to process two or more objective
functions at the same time, and there is many research that has proposed for multi
objective optimization problems last few decades such as weighted sum, fraction or
physical programming. However, when dealing with two conflicting objective functions
using weighted sum or fraction, many solution convergence problems will come with it.
For example, in the weighted sum method, specific weights must be selected beforehand
however the choice of weights may easily bias the solution towards one way or another,
12
even arrive at the undesirable solutions. On the other hand, if the minimization of a
fraction form is used, the solution can arrive at the extreme cases by minimizing the
strain energy only. Although the objective function becomes very good, the optimized
solution is completely useless for compliant mechanisms.
Physical programming is one of the representative examples for multi objective
optimization problems. Physical programming was introduced by Messac in 1996 [42]
and it involves designer’s preference into optimization problem in terms of physically
meaningful terms and parameters. Physical programming formulation is presented as:
1
min:sN
i ix
i
P r x
(1.1)
where x is a design variable of optimization problem and Ns represents number of design
goals. Pi represents class function which involved designer’s decisions, and ri is an
original objective function of minimize or maximize optimization problem.
To implement the designer’s preference into physical programming formulation,
the range of ri(x) should be predetermined and it can be defined in Table 1.2. The
designer specifies the ranges of different levels of preference for each objective.
Therefore, it is not necessary to define meaningless weights for each objective function.
13
Table 1.2. The Range Description of Designer’s Preference [42]
Name Range Description
Highly Desirable ri ≤ ri1
An acceptable range over while the improvement that results
from further reduction of the preference metric is desired,
but is of minimal additional value
Desirable ri1 ≤ ri ≤ ri2 An acceptable range that is desirable
Tolerable ri2 ≤ ri ≤ ri3 An acceptable, tolerable range
Undesirable ri3 ≤ ri ≤ ri4 A range that, while acceptable, is undesirable
Highly Undesirable ri4 ≤ ri ≤ ri5 A range that, while still acceptable, is highly undesirable
Unacceptable ri5 ≤ ri The range of value that the objective function may not take
In physical programming, the designer classifies objective functions into the
following four different categories in Figure 1.8. Class-1 can be used for minimization
problem because class function will consider smaller function value r as a better solution.
Maximization problem can be implemented in Class-2 whose larger r value will be an
optimal solution. Class-3 and Class-4 can be used when optimizer is trying to find a
certain value or range.
In this dissertation, both compliant mechanism and energy absorbing structure
design are required for multi-objective optimization schemes because flexibility and
rigidity should be formulated into the objective function. For that, physical programming
technique is applied because the engineer can involve the designer’s preference into the
formulation which is highly desirable for both compliant mechanism design and energy
absorbing structure design. Detail formulation using physical programming scheme is
described in each topic.
14
(a) (b)
(c) (d)
Figure 1.8. Classification of Class Function
(a) Class 1: Minimize (b) Class 2: Maximize (c) Class 3: Seek Value (d) Class 4: Seek Range
1.4. Research Contribution
A strain based topology optimization method is presented and solved in this
dissertation. Several achievements of my research are listed as follows:
Sensitivity analysis of strain formulation
When the effective strain is considered as an objective function, the sensitivity
of effective strain can also be derived. However, because the effective strain
15
function is not directly differentiable with design variable; the strain based
topology optimization method cannot be implemented. In Chapter 3, the
analytical solution of sensitivity calculation is derived using three different
finite element analysis systems.
A compliant mechanism design scheme with strain based topology
optimization method.
In general, compliant mechanism design is formulated using energy based
function. However, the optimal design using energy based formulation shows
localized deformation or stress concentration. To avoid that, strain based
formulation is presented, and the optimal designs of strain based formulation
are deformed evenly using entire structure. Therefore, the optimal designs of
the strain based formulation will have less possibility of the malfunction, or
mechanism failures than the energy based optimal design.
A energy absorption structure design scheme
One challenge for energy absorbing structure design is that the design should
have both energy absorption capacity and strength. In this dissertation, we
changed the function format as an function of energy absorption capacity and
By changing the function, the design can be converged that can have high
energy absorption capacity and enough strength to protect inside of the
structure
16
1.5. Outline of the Dissertation
This dissertation consists of five chapters covering strain based topology
optimization method. Especially, the flexible structure optimization problem is mainly
considered such as compliant mechanism and energy absorbing structure. All the details
are introduced and discussed in the rest of the dissertation. In chapter 2, the background
of topology optimization method is described to make the reader of this dissertation
understand my research more easily. After this, my research will be presented more in
deep in three chapters.
In chapter 3, three different types of the objective function will be studied. Firstly,
strain based formulation and the drawback of that is presented. Since stress is an
important consideration of structure design, the stress based formulation and sensitivity
of stress objective function are described. From the sensitivity derivation, we find that
some particular cases, the stress based formulation cannot process the optimization
algorithm. To avoid these drawbacks from both energy and stress formulation, strain
based topology optimization formulation is presented with sensitivity analysis. The
numerical example also shown to compare the optimal designs between energy based and
strain based formulation.
In chapter 4, the compliant mechanism design scheme is presented using strain
based formulation. The objective in this chapter is to design compliant mechanism
without high localized deformation which can easily be found from energy based optimal
design. To solve the problem, global effective strain is used for structure integrity.
Physical programming scheme is also implemented to handle flexibility and rigidity at
17
the same time. Moreover, three different types of compliant mechanism example are
presented
In chapter 5, the methodology of the energy absorption structure design is studied.
For that, two loading cases are considered as a problem setting. The first force is for
energy absorption when external force is applied and the second force is to support
weight of the interior of the object since energy absorption structures are usually used to
protect goods or persons. For energy absorption consideration, new function is presented
instead of using maximizing or minimizing function. To support goods global effective
strain function is implemented since it can avoid the drawbacks of the energy based
formulation. The optimal design configurations using different objective function is also
shown at the end of the chapter.
Finally in chapter 6, the main achievements of this research are briefly
summarized and further extensions of the work are proposed.
18
Chapter 2.
Background of Topology Optimization Method
Topology optimization method is a method to find out optimal material
distribution within predefined design domain. Through that it can give the best
conceptual design that can satisfy all design requirements. Topology optimization
problem includes objective function, design domain and design constraints. Objective
function represents the goal of the optimization method which is defined as minimization
or maximization. A scheme of design domain is shown in Figure 2.1, where Ft is the
external force, Ω is the design domain, Ωs denotes a solid domain and Ωv represents a
sub-domain without material. Topology optimization methods are based on FEM and
sensitivity analysis. These methods utilize each finite element in the mesh for FEM, and
each finite element is assigned as a design variable which is the material density of the
element. By updating material density of each element, structure design can be improved
to optimal design. The computational flow is illustrated in Figure 2.2.
Figure 2.1. Design Domain of Typical Topology Optimization Problem.
19
Figure 2.2. The Flow of Topology Optimization Method
2.1. Basic Formulation of Topology Optimization Method
In order to formulate topology optimization problem, objective function and
constrains should be implemented. Therefore, the problem can be presented as:
Minimize(or Maximize)
0 1,
f
subject to dv V
v or v
(2.1)
20
where f(ρ) is the objective function, V is the upper bound of solid volume, and ρ is a
density distribution in domain.
However, it is very difficult to solve problems using equation (2.1) directly, and
numerical calculation method can be used to have approximated solution. To do that,
firstly, topology optimization method should be discretized from the design domain Ω
into N finite elements and each element is assigned a design variable ρi. A design variable
vector ρ is used to represent the material distribution in the design domain. Therefore, the
formulation can be changed as:
1
Minimize(or Maximize)
0 1, 1, ,
N
i i
i
f
subject to v V
v or i N
(2.2)
where vi is the volume of the ith element.
However, the binary topology optimization problem using equation (2.2) shows
the lack of solutions [45]. The numerical solutions are shown numerical instabilities such
as the checkerboard effect and mesh dependence problems. To overcome these problems,
several methods are proposed to both restrictions method and relaxation methods.
Restriction methods decrease the design set by implementing extra constraints to avoid
oscillation. For example, Beckers [46], Haber et. Al [47] solved problems using an extra
bound of the perimeter of the boundary between solid and void subdomain. Relaxation is
the method that increase the design set to accomplish existence of solutions. A
homogenization method was applied to topology optimization [48] by using a perforated
21
microstructure before computing the effective material properties which makes the
material density varies from 0 to 1.
In this study, relaxation methods are used to solve problems of topology
optimization and the material model is presented in the following section.
2.2. Microstructure Based Composite Material Models
Topology optimization method allows the design variable, the element density, to
vary from 0 to 1. The reason is that the instabilities of the binary topology optimization
problem are very difficult to solve. Since the element material density is between 0 and 1,
each element can be considered as a new composite material which has a different micro
structure in it. This composite material should be determined and applied during finite
element analysis. There are many material models are applied to topology optimization
method. Bendsøe and Kikuchi [48] introduced the Hole-in-cell microstructure model in
1988. Solid Isotropic Microstructure with Penalty Model was present by Bendsøe [49] in
1989, and Gea [51] presented the Spherical Micro-inclusion model in 1996. In topology
optimization field, SIMP model is used commonly because of its simplicity. Detailed
description is discussed in the following section.
2.2.1. Solid Isotropic Microstructure with Penalty Model (SIMP)
Bendsøe introduced a material model, Solid Isotropic Microstructure with Penalty
(SIMP), in 1989[49]. SIMP material model can simplify topology optimization and finite
element analysis procedures. In the SIMP model the elastic tensor Eijkl is defined as:
22
0
, 1p
ijkl ijklE x x E P
V x d
(2.3)
where x is the design variable of each element which is the volume density, 0 ≤ x ≤1, and
Ω is the design domain, p represents a penalty coefficient to material, which is greater
than 1 to provide penalty on stiffness of the material. V is the volume of material in the
design domain. The effect of different penalty values p to the relative stiffness (E / E0) is
illustrated in Figure 2.3. As shown in Figure 2.3, the curve has a tendency to become a
step function when the penalty power p increases. Therefore topology optimization
method can be more efficient to converge material density (x) as a void (x = 0) or solid (x
= 1) in SIMP model. Hence, the result of topology optimization will tend to generate an
optimal design with mostly solid and void phase.
Figure 2.3. The Penalty Function is SIMP Model
23
SIMP model has been called artificial or a fictitious material model even though it
provides expected condensed topology results. The reason is that it does not have any
physical interpolation to the region with intermediate density value. In 1999, Bendsøe
and Sigmund [50] analyzed and compared material models such as the Hashin-
Strikhman upper bound, SIMP model. A comparison between the Hashin-Strikhman
upper bound and SIMP model is illustrated in Figure 2.4. The Microstructures of material
and void realizing the material properties of the SIMP model can be seen in Figure 2.5 As
the microstructures are developed from material density and penalty values, density
variable is quite natural for the SIMP material model.
Figure 2.4. A Comparison of the SIMP Model and the Hashin-Strikhman Upper Bound for an
Isotropic material with Poisson Ratio υ=1/3 mixed with void. For the H-S Upper Bound,
Microstructures with Properties Almost Attaining the Bounds are also Shown [50]
24
Figure 2.5. Microstructures of Material and Void Realizing the Material Properties of the SIMP
Model with p = 3, for a Base Material with Poisson's Ratio υ =1/3. As Stiffer Material
Microstructures can be Constructed from the Given Densities, Non-structural Areas are seen at the
Cell Centers [50]
2.2.2. Spherical Micro-inclusion Model
Homogenization method formulation was developed based on the relation
between the elastic tensor and the volume density of periodic microstructure. By using
SIMP model, topology optimization problem can be converted as a relatively simple
problem. However, it is required that we need to select an appropriate penalty value p for
volume density function. Gea [51] introduced a new microstructure-based design domain
method using the spherical micro-inclusion model. In Figure 2.6, the micro-inclusion
material model is illustrated. Infinitely many isotropic spherical voids are embedded in an
isotropic material. For this composite model, the material property was derived by means
of Mori-Tanaka’s mean field theory in conjunction with Eshelby’s equivalence principle
and his solution of an ellipsoidal inclusion.
25
Figure 2.6. Microstructure of the Spherical Micro-inclusion Material Model[51]
Assuming Poisson’s ration of the base material υ0=1/3, the effective Young’s
modulus and shear modulus are calculated by:
0
0 0
0
, 0 12
cE E c
c
(2.4)
and
0
0 0
0
8, 0 1
15 7
cc
c
(2.5)
where c0 is the volume faction of the base material in the element. The relation between
the relative stiffness (E/E0) of the sphere micro-inclusion model and volume fraction c0 is
shown as the continuous curve in Figure 2.7 as solid curve line. The SIMP model with
p=2 is also shown as the dashed curve line to compare between two different models.
26
Figure 2.7. Relative Stiffness vs. Volume Fraction for the Spherical Micro-inclusion Model and the
SIMP Model with Penalization Power p=2
As we can see, these two curves are fairly close to each other. Therefore, the sphere
micro-inclusion model could be used for the multi material model, and it gives not only
rigorous formulation, but also easy application into topology optimization problem.
2.3. Topology Optimization Algorithm
In this section, practical formulation of topology optimization method is described.
First of all, the objective function and constraints are introduced in terms of finite
element analysis form. Furthermore, sensitivity analysis is explained to obtain optimal
configuration of the problem.
27
2.3.1. Problem Formulation for Topology Optimization
In the previous section, the formulation of topology optimization problem was
explained in equation(2.2). Considering the objective of topology optimization problem
as minimizing the total strain energy of the structure, the formulation can be expressed as:
0
1
1 2
1min
2
. .
, , , ,
T
x
N
i i
i
N i
X U KU
s t V X x v V
KU F
X x x x x x x
(2.6)
where U is the displacement vector, K is the stiffness matrix. KU=F is the equilibrium
equation where F is the loading vector that applies to the structure. xi is the design
variable of ith element and vi is the volume of an element. X is the design variable vector
and N is the number of design variable.
If we assume that the current design point is located at point Xk, the objective
function can be converted as a linear approximation form using the first order of Taylor
expansion
0
1
0
1
1 2
min
. .
, , , ,
Nki
i ix
i i
N
i i
i
N i
X x xx
s t V X x v V
KU F
X x x x x x x
(2.7)
28
If we consider the offsetting of variable xi as yi, i iy x x , the optimization
problem can be expressed as variable set of yi and equation (2.7) can be rewritten as
follows:
1
1
1
1
1 2
min
. .
, , , , 0
Ni
ix
i i
N
i i
i
N i
Y yy
s t V Y y v V
KU F
Y y y y y y
(2.8)
Moreover, we also can consider vi as a scale factor and then variable yi can be converted
as zi = viyi. The derivative dzi = vidyi and using those relations, equation (2.8) can be
further written as:
1
1
1
1
1 2
min
. .
, , , , 0
Ni
ix
i i
N
i
i
N i
Z zz
s t V Z z V
KU F
Z z z z z z
(2.9)
This formulation can eliminate the distortion effect during the optimization process when
element size is not uniform since z variable are including element volume information.
The relation between different variables are shown in equation (2.10)
29
i i i i iz v y y x x (2.10)
2.3.2. Sensitivity Analysis
In order to process the topology optimization algorithm, we need to know which
element should be increased in material density. In mathematics, gradient of the function
represents the direction of the greatest rate of increase. Therefore, by calculating the
gradient of the objective function, we can define which element should increase the
material density. This can be expressed as:
i i i
i i i
d dy
dz y z
(2.11)
From equation(2.10), we know dyi=dxi and dzi=vidyi, so equation (2.11) can be converted
to:
1 1 1
2
Ti i i i
i i i i i
d dyU KU
dz x z x v x v
(2.12)
Since both stiffness matrix K and displacement vector V depend on the design
variable, equation (2.11) can be extended using chain rule and it follows as:
1 1
22 2
T T T
i i i
U KU KU U K U U
x v v x x
(2.13)
30
The derivative of equilibrium equation, KU=F, can be derived as:
0i i
i i
K UU K
x x
K UU K
x x
(2.14)
and equation (2.13) can in turn be written as:
1 1
2 2
T T
i i
KU KU U U
x v v x
(2.15)
Since global stiffness matrix is assembled using element stiffness matrix, ke, and each
element stiffness matrix is a function of material density of each element, equation (2.15)
can be expressed as element level, and it is rewritten as:
1 1
2 2
T T e
i i
kU KU u u
x v v x
(2.16)
where u is a element displacement vector. Assuming the material stiffness is a function of
the design variable, element stiffness matrix can be defined as follows:
0i
e i ek f x k (2.17)
31
where 0
ek is the stiffness matrix with full material. Using equation(2.17), element stiffness
matrix can be derived as:
0 ' '
0i
i e i iei e e
i i i i
f x k f x f xkf x k k
x x f x f x
(2.18)
Substituting the derivative of element stiffness term back into equation(2.16), the
sensitivity of total strain energy in respect to design variable can be derived as:
'1
2 2
iT T
e e e
i i
f xU KU u k u
x v vf x
(2.19)
Using equation (2.19), the graidient of each element can be calculated and base on
that material density can be updated.
2.4. Examples
In this section, numerical examples of topology optimization are shown using
minimizing strain energy formulation. Each example illustrates the boundary condition
and objective function. The material properties are used as Poisson’s ratio ν = 0.3 and
Young’s modulus E0 = 1e6.
32
2.4.1. Cantilever Beam: 2D
The Cantilever Beam problem is one of the typical topology optimization
problems. The design objective is to minimize strain energy while satisfying constraints
such as volume. The design domain and boundary condition is illustrated in Figure 2.8.
Left edge is fixed to support structure and force is applied at the middle point of right
edge in the downward direction. In this example, the volume constraint is set at 40
percent of the total design domain.
Figure 2.8. The Design Domain and Boundary Conditions for Cantilever Beam Problem
This is a very well known problem, and optimal designs have been presented in
many articles. The optimal configuration is illustrated in Figure 2.9.
33
Figure 2.9. The Optimal Design of Cantilever Beam Problem
2.4.2. Cantilever Beam: 3D
In this section, the three dimensional cantilever beam problem is described.
Design domain and boundary conditions are shown in Figure 2.10. The four corners of
the left side are fixed and a vertical load is applied at the middle of the lower right edge.
The design objective is also to find the stiffest structure with a volume constraint of 10%.
The optimal design of 3 dimensional cantilever beam is illustrated in Figure 2.11 and
Figure 2.12
34
Figure 2.10. The Design Domain and Boundary Conditions for 3D Cantilever Beam Problem
Figure 2.11. The Optimal Design of 3D Cantilever Beam Problem
35
Figure 2.12. The Optimal Design of 3D Cantilever Beam Problem : Top View and Front View
36
Chapter 3.
A Strain Based Topology Optimization Method
Since topology optimization method was introduced, most of the objective
function is formulated based on energy form, and strain energy formulation is one of the
general formulation for topology optimization method. The formulation and numerical
examples of strain energy based formulation are already explained in the previous chapter.
In this chapter, strain energy formulation is studied in more detail, and the
drawbacks of strain energy formulation are described. After that, the other types of
formulation are described as stress based formulation and strain based formulation.
3.1. Energy Based Topology Optimization Method.
Strain energy is the potential energy stored in a structure by elastic deformation,
and it is equal to work that must be done to produce this deformation. The mathematical
formulation of strain energy is presented as:
. TS E d (3.1)
where σ denotes stress and ε is strain. Since FEM is a numerical calculation method,
equation (3.1) can be written as numerical formulation, and it is followed as:
37
1 1 1
.2 2 2
T T T
e e eS E U F U KU u k u (3.2)
Each element stiffness matrix, ke, includes material properties information such as
Young’s modulus and Poisson’s ratio. Especially, Young’s modulus can be described in
terms of material density since topology optimization method uses SIMP model.
Therefore, element strain energy can be written as:
0 0 0
T T
i e e if x E u Cu f x E D (3.3)
where C is a positive definite dimensionless matrix and D0 is a strain stress relation
matrix without Young’s modulus. The strain stress relation matrix D0 can be described as:
0
1 0 0 01 1
1 0 0 01 1
1 0 0 01 11
1 21 1 2 0 0 0 0 0
2 1
1 20 0 0 0 0
2 1
1 20 0 0 0 0
2 1
D
(3.4)
38
Figure 3.1. Material Density Function
From equation(3.3), element strain energy can be expressed in terms of material density
function f(xi), Young’s modulus of base material E0, and εTD0ε. Since E0 is the constant,
material density function f(xi) works as penalty function of the strain as shown in
Figure 3.1. It means that strain energy is not only the function of the element deformation,
but also the function of the material density. Because of that, there is a possibility of the
distortion of the element strain. For example, when element has large strain, material
should be added to decrease the element strain. However, when element is low density
value, high strain value can be converted as low strain energy, because of the material
density function f(xi). From Figure 3.1 blue circle area shows that when material density
is less than 0.6, material density function converts the strain value to strain energy as less
than 1/5 of strain value. Therefore, optimizer will remove the material instead of adding
the material even though strain is high when material density is low. In this case, element
strain will be increased and optimal configuration will have problems such as high stress
concentration, or localized high deformation. The relation of the material density, strain,
39
Table 3.1. The Relation of Material Density, Strain, Strain Energy
xi εTD0 ε f (xi)E0ε
TD0 ε xi
High High High Increased
Low High
High Increased
Low Decreased or Increased
High Low Low Decreased
Low Low Low Decreased
and strain energy is shown in Table 3.1. Moreover, energy based topology optimization
does not include any criteria for stress even if it is one of the important criteria of
structure design. Therefore, section, stress based topology optimization formulation is
presented in the following section. After that, strain based formulation is introduced.
3.2. Stress Based Topology Optimization Method.
Since stress is one of the important criteria for structure analysis, stress based
topology optimization method is studied in this section. To formulate stress as an
objective function, we need a scalar measure of stress state, and effective stress could be
used as an objective function. The formulation of effective stress is described as:
2 1
0
T D (3.5)
40
where denote effective stress and σ is a stress, and using stress and strain relation
effective stress can be expressed in terms of material properties and strain:
2 2 2 2 2 2
0 0 0
T
i if x E D f x E (3.6)
Equation (3.6) can also be decomposed of displacement vector and stiffness matrix terms
and it is followed as:
2 2 2 2
0
2 2
0
2 2
0
* * 2 2
0
( )
( )
( )
i
T
i e e
T
e i e
T
e e e e i
f x E
f x E u Cu
u f x E Cu
u k u k f x E C
(3.7)
In equation(3.7), new element stiffness matrix ke* is defined, and stress based topology
optimization formulation can be described as:
*
0
1
1 2
min
. .
, , , ,
T
N
i i
i
N i
U K U
s t V X x v V
KU F
X x x x x x x
(3.8)
41
where UTK*U represents the summation of ueTke
*ue. In the following section, sensitivity
of stress based objective function is derived to determine which element should be added
the materials.
3.2.1. Sensitivity Analysis of Stress Based Objective Function
After we have defined problem formulation, we also need to derive the sensitivity
of new objective function which is stress based formulation. To derive sensitivity, three
different finite elements analysis systems are introduced in Figure 3.2. System A is a
general finite element analysis system with SIMP model, and system B is a system to
calculate new force vector F* using displacement vector U from system A and new
stiffness matrix K*. System C is a systme to calculate another displacement vector V
using force vector F* from system B and stiffness matrix K from system A.
Figure 3.2. Three Different Finite Element Analysis Systems
(a) System A (b) System B (c) System C
Applying the chain rule, derivative of UK*U can be expanded as:
42
*
* *2T
T T
i i i
d dU dKU K U K U U U
dx dx dx (3.9)
Using the relation between system B and system C, K*U can be replaced as KV
and because of KU΄= -K΄U, equation (3.9) can be converted as:
*
* 2T T T
i i i
d dK dKU K U U V U U
dx dx dx (3.10)
As we discuss the derivative of stiffness matrix in previous chapter, equation (3.10) can
simply be converted as:
'2
* *
2
'
*
'2
'2 2
iiT T T
e e e e e e
i i i
i iT T
e e e e e e
i i
f xf xdU K U u k v u k u
dx f x f x
f x f xu k v u k u
f x f x
(3.11)
and it can be simplified as:
* *
'2
iT T
e e e e e
i i
f xdU K U u k u k v
dx f x (3.12)
From equation(3.12), it is clear that sensitivity of all elements will be a zero when
ke*ue= keve, and we know it is true when material is distributed evenly into entire design
43
domain with constant input force conditions. In general, evenly distributed material is
considered as an initial design of the topology optimization method. However,
optimization algorithm cannot update material density to minimize stress because all
sensitivity of elements are zero.
To verify that, Cantilever beam problem is presented in Figure 3.3. Left edge is
fixed and Force is applied at the right bottom corner. Base Young’s modulus E0 and
Poisson’s ration are equally applied. The only difference between the top and bottom
design domain is material density. In case of top configuration, material density xi=0.3 is
distributed evenly and for bottom configuration, material density xi=0.7 is distributed
evenly. The Von Mises Stress distribution is plotted on the bottom of the each design
domain. The figure clearly shows that even though the material densities are different,
stress distribution are identity when all elements have equal material density value.
Therefore, when material density is changed evenly, it does not affect the stress of each
element because sensitivity is zero.
Even if evenly distributed material density is one of the particular case, effective
stress objective function cannot be directly implemented into topology optimization
method. In the following section, strain based topology optimization formulation is
presented to avoid the drawbacks both strain based formulation and stress based
formulation.
44
Figure 3.3. Cantilever Beam with Different Material Density and Von Mises Stress Distribution:
xi=0.3 (top), xi=0.7 (bottom)
3.3. Strain Based Topology Optimization Method.
In case of strain energy based formulation, we discussed that the material density
function f (xi) will distort the strain information if the material density is relatively small
in the element. In order to eliminate this distortion, the material density function f (xi)
must be eliminated and it can be simply achieved by dividing the material density
function from strain energy formulation. Since E0 is constant, it really does not matter
45
whether dividing E0 or multiplying E0 to the objective function. Therefore we can also
divide strain energy function by base Young’s modulus E0. Then we can define new
formulation in terms of effective strain as followins as:
02
00 0( )
T TT Te e e e e ee e
u k u u k uu Cu D
f x E E (3.13)
This effective strain formulation not only removes the strain distortion effect but
can also represent the stress bound of the structure. In general, stress is considered as one
of the constraints for minimum stress problem in topology optimization method, and that
is expressed as a constraint of the effective stress as:
i allowf x (3.14)
where σallow is a allowable stress or upper bound of stress. If we divide both side of
equation (3.14) by material density function f(xi), left term can be expressed as effective
strain and E0 as:
0 allow
i
Ef x
(3.15)
From this equation, minimizing effective strain can be represented as minimizing stress
bound since E0 is a constant.
46
From this derivation, effective strain formulation gives us two big advantages
1. Effective strain formulation can avoid strain information distortion
during optimization algorithm
2. Minimizing effective strain problem also represent minimizing stress
bound problem.
Therefore, the new formulation for strain based topology optimization problem
can be formulated as:
2
1
0
1
1 2
min
. .
, , , ,
N
i
N
i i
i
N i
s t V X x v V
KU F
X x x x x x x
(3.16)
3.3.1. Sensitivity Analysis of Strain Based Objective Function
Since we proposed strain based topology optimization formulation, one of the
most important tasks is to evaluate the sensitivity of the functions with respect to design
variables, and the mathematical calculation form is discribed as:
2
1 1
n nT
i e e
i ii i
d du Cu
dx dx
(3.17)
47
Figure 3.4. Three Different Finite Element Analysis Systems
(a) System A (b) System B (c) System C
Since displacement of element nodes, ue is not differentiable and gradient of C is
zero, sensitivity of effective strain cannot be derived directly. Three different finite
element analysis systems are introduced to derive the senstivity of effective strain in
Figure 3.4. If we consider the design domain to be filled with base materials, and to
impose nodal displacement fields as ue, the summation term can be viewed as the total
strain energy of the new system which is system B. Let us define the global stiffness
matrix as K1, and then, the sensitivity of the summation of the global effective strain
becomes:
1
1 0
11
0 0
2 1
E E
TnT
e e
ii i
TT
i i
d d U K Uu Cu
dx dx E
dU dKK U U U
dx dx
(3.18)
dK1/dxi is zero because K1 is independent with material density; therefore
1
0
1
E
T
i
dKU U
dx
term can be ignored. K1U is actually the loading vector generated by the imposed
displacement field in the new system which is system C. If we apply this loading vector
48
as the applied force to the original system, we will have a system governing equation as
KV= K1U. Substituting the left side of the system governing equation into equation (3.18)
and applying the derivative results of the original system K´U+KU´=0, we arrive at the
following expression:
1
0
0 0
2 2n
ii
TT T
e e
i i
du
dx
dU dKCu K U U V
E dx E dx
(3.19)
Derivative of stiffness matrix respect to meterial density is a well know problem and it is
explained in previous chapter. Therefore, sensitivity of global effective strain can be:
1 0
'2
E
n
Ti
e e e
ii i
T
e e
f xdu u k v
dx f xCu
(3.20)
The sensitivity of global effective strain is derived in equation (3.20). However,
the sensitivity of that can be zero when preset displacement input is applied instead of
preset input force. It can be verified by simple cantileber beam examples as shwon in
Figure 3.5. The preset displacement input is applied at the botton right corner to the
downward direction and the strain distributions from two different material density
structures are ploted in the Figure 3.5. From the figure it shows that even though the
material density is different, strains distributions are indentity when preset displacement
input is applied. The reason is that material density will not affect the strain of each
element when material density is chagned evenly.
49
Figure 3.5. Cantilever Beam with Different Material Density and Von Mises Strain Distribution:
xi=0.3 (top), xi=0.7 (bottom)
In previous section and this section, the stress and strain based topology
optimization methods are discussed. From these studies, it was found out that the proper
formulation is needed to be applied according to the input boundary condition. For the
preset displacement input problem, the stress based formulation can be applied instead of
the strain based formulation since the sensitivity of strain is zero. However, the stress
formulation is not working well for structure integrity. The reason is that the rigid
structure has higher stress than the soft structure when constant input displacement is
50
applied. Therefore, minimizing stress formulation will lead to the low density structure
design which is infeasible design from manufacturer point of view. Instead of applying
the minimizing formulation, maximizing stress formulation is also studied, but integrity
of the structure is not successful either. To resolve this issue, further investigation is
needed in the future. In contradiction to preset displacement input problem, the preset
force input problem can be solved by the strain based formulation but not be solved by
the stress formulation because of sensitivity calculation. In this thesis, the research is
focusing the strain based formulation which is applied preset force input as a boundary
condition.
3.4. Design Examples.
In this section, design examples of the strain based topology optimization are
presented, and it is compared with energy based topology optimization design. cantilever
beam problem is studied in this section. The material properties of all problems are used
as Young’s modulus E0=1e6 Pa, Poisson’s ratio ν = 0.3.
3.4.1. Cantilever Beam
In this example, cantilever beam design problem is studied. Total design domain
is a rectangular shape which has width of 120 and height of 40. Mesh size of each
element is defined as 2×2 so the total elements number of design domain is 1200. Left
51
edge of design domain is fixed and force is applied at the right bottom corner in the
downward direction as shown in Figure 3.6. Volume constrain of the problem is set as 40
percent of the total volume.
The optimal design configuration of strain based topology optimization method is
illustrated the top of the Figure 3.7, and the optimal design using strain energy
formulation is shown on the bottom for comparison purpose. In Figure 3.8 Von Mises
Stress is plotted for both optimal designs, and the optimal design of a strain based
formulation that can avoid the high stress concentration.
Figure 3.6. Design Domain and Boundary Conditions for Cantilever Beam Problem
52
Figure 3.7. Optimal Design Configurations: Using Strain Based Formulation (Top) and Strain
Energy Based Formulation (Bottom)
Figure 3.8. Von Mises Stress Plot for Optimal Designs: Using Strain Based Formulation (Top) and
Strain Energy Based Formulation (Bottom)
53
3.5. Conclusion and Remark
In this chapter, the different formulations for the topology optimization method
are presented and discussed. First of all, strain energy based formulation is studied. In
case of the strain energy based formulation, we found out that strain information can be
distorted by material density function. Moreover, strain energy based formulation does
not have any consideration for the stress criteria. Next, stress based formulation is
introduced since stress is the one of the important criteria. However, the sensitivity of
stress formulation can be zero when material is evenly distributed. Therefore, stress
formulation cannot be used for topology optimization method.
To avoid these drawbacks, strain based topology optimization method is
presented. Strain formulation cannot only avoid the strain distortion but also represent
minimizing stress bound. Moreover, the sensitivity of the global effective strain is
derived by introducing three different analysis systems. By comparing the optimal
designs of both strain energy based optimization and proposed method, it is clear that
strain based optimization method can avoid stress concentration of the structure.
However, even though strain based optimal design does not have high stress
concentration, overall configurations for both cases are similar to each other. The reason
is that for minimizing strain energy is convex single objective problem so from the initial
design, it will be converged well without confliction. This means that there is less
possibility to have low material with high strain. However, if the objective function is
formulated with two contrary objectives, we can expect that the optimal design between
strain based formulation and strain energy based formulation will have different
54
configuration. Therefore, in following chapter, a multi-objective optimization problem is
presented.
55
Chapter 4.
Compliant Mechanism Design using a Strain Based
Topology Optimization Method
Energy based topology optimization method has been used in the design of
compliant mechanisms for many years. Although many successful examples from the
energy based topology optimization have been presented, optimized configurations of
these designs are often very similar to their rigid linkage counterparts except that, it uses
compliant joints in place of rigid links. The compliant mechanism design in Figure 4.1
shows several compliant joints circled in red. It is obvious that these complaint joints
will endure large deformations under the applied forces in order to perform the specified
motions. These large deformations will produce high stress which is very undesirable in
compliant mechanism design.
Figure 4.1. Example of Complaint Mechanism: Crimping Mechanism [41]
56
In this chapter, a strain based topology optimization method is proposed to
produce optimal compliant mechanism. Instead of minimizing the strain energy for
structural rigidity, a global effective strain functional is minimized in order to distribute
the deformation within the entire mechanism while maximizing the structural rigidity.
Furthermore, the physical programming method [42]-[44] is adopted to accommodate
both flexibility and rigidity design objectives. The remainder of this chapter is organized
as follows: First, the energy based topology optimization for compliant mechanisms is
reviewed and the problems associated with this formulation are discussed. Next, the
proposed strain based topology optimization method is presented and the implementation
of physical programming in the multi-objective optimization problem is described. Then,
solution procedures including sensitivity analysis is derived. Finally, comparisons of
design examples from both the energy based topology optimization and the strain based
method are presented and discussed.
4.1. Energy Based Topology Optimization for Compliant
Mechanism Design
Consider a topology optimization problem setting for a compliant mechanism
design, the applied force F in to the design domain Ω will drive the output node
producing a desirable displacement in the direction of Uout as shown in Figure 4.2, where.
Ω represents entire design domain, Ωs denotes solid areas and Ωv is areas without material.
57
Figure 4.2. Design Domain and Boundary Condition for Compliant Mechanism Design Problem
To consider the flexibility of the compliant mechanism, the displacement at the
output point can be formulated as the mutual potential energy (MPE):
TMPE U (4.1)
where U is the displacement vector and Φ is a unit vector that can be viewed as a dummy
load. At the same time, the rigidity of the design is evaluated by the strain energy (SE)
which can be expressed as
1
2
TSE U KU (4.2)
58
where K is a global stiffness matrix of the design
Since both flexibility and rigidity should be accommodated for complaint
mechanism design, two objective functions, maximizing mutual strain energy and
minimizing strain energy, are needed. However, as we discussed in previous chapters,
strain energy formulation can misrepresent the strain information which is highly
associated with deformation. Therefore, compliant mechanism may also have localized
large deformation at compliant joints if the traditional minimizing strain energy
formulation is used. To perform the desired function as a mechanism, these compliant
joints will suffer from localized large deformations and consequently high stresses when
external forces are applied. In the following section, numerical example of strain energy
based compliant mechanism design is presented. Then, our proposed solutions are
presented in section 4.2.
4.1.1. Complaint Gripper І
Compliant gripper design problem is studied in this example. Because of
symmetry, only half of the design domain is considered. The device is fixed at the bottom
and external force is applied downward at the upper left corner as shown in Figure 4.8.
The material properties are Young’s modulus E0=100Pa, Poisson’s ration ν=0.3, and the
15 percent of the total design domain volume is applied as the upper bound of the volume
constraint.
59
Figure 4.3. Design Domain and Boundary Condition for Complaint Gripper І
The design objective is to generate a compliant gripper that produces a downward
motion at the lower right corner. The optimal design of the strain based formulation is
presented in Figure 4.4. Since this design domain is only half of the entire design domain,
the fully gripper design is also illustrated in Figure 4.5
Figure 4.4. The Optimal Design of Strain Energy Based Formulation for Complaint Gripper І
60
Figure 4.5. The Full Compliant Gripper І Design of Strain Energy Based Formulation
To verify the localized high deformation, strain energy distribution of optimal
design is plotted in Figure 4.6. One can easily find that some localized high strain regions
are found in the straight member. The higher strain energy will cause higher stress and
these results matches with our derivations very well. Therefore, we need to find out the
formulation that can avoid stress concentration when mechanism is performed. In the
next section, strain based topology optimization method for complaint mechanism design
is presented. The deformed shape is also illustrated to verify the motion of the design.
61
Figure 4.6. The Strain Energy Plot of Strain Energy Based Formulation for Complaint Gripper І
Figure 4.7. The Deformed Configuration of the Compliant Gripper І using Strain Energy Based
Formulation
4.2. Strain Based Topology Optimization for Compliant
Mechanism Design
To overcome the drawbacks of the energy based topology optimization
formulation for compliant mechanism design, a strain based topology optimization
method is presented in this section. The problem formulation of the proposed method
will be discussed first and the sensitivity will be derived. Finally, the implementation of
62
the physical programming method for multi-objective optimization formulation for
compliant mechanisms is presented.
4.2.1. Problem Formulation
As we discussed in the previous section, the material density function in the SIMP
model will distort the solution by discounting the localized large deformation if the
artificial material density is relatively small in the same element. In order to accurately
account for the local displacement, the effect from the penalty must be removed from the
formulation. As we explained in Chapter 3, effective strain 2 can be considered as a
new formulation instead of the strain energy function since it can be derived by dividing
the material density function and Young’s modulus of the base material, which is a
constant, from the element strain energy. Since the effective strain is a direct indicator of
deformation, and deformation is the direct indicator of rigidity, minimizing the effective
strain can be used in place of the rigidity design objective of the compliant mechanism.
Considering both flexibility and rigidity simultaneously requires multi-objective
optimization formulations, and both of the weighted-sum, and the fraction formulations
experiences numerical and convergence problems. A physical programming framework is
implemented in this study. One of the advantages of physical programming is that it can
incorporate designers’ preference by specifying desirability ranges. This is particularly
important in the compliant mechanism design because the flexibility function and the
rigidity function have completely different numerical orders and different sensitivity with
respect to design variables. The physical programming method can remove problems by
defining proper class functions with suitable desirability.
63
Based on the effective strain formulation and the physical programming method,
the strain based topology optimization problem can be defined as
2
1 2
1
0
1
1 2
Minimize
Subject to
( )
, , , ,
nT
i
i
N
i i
i
N i
P P U
KU F
V X x v V
X x x x x x x
(4.3)
where P1 and P2 are two class functions in the physical programming method. In our
study, P1 is a preference function representing the smaller value of global effective strain
the better design, whereas P2, representing the larger value of output displacement the
better design. Both class functions are shown in Figure 4.8. The choices of ranges
between r 0 and r* on both class functions are pre-defined by the designer. In this way,
optimization algorithm can process both the flexibility and rigidity at the equal level and
provide satisfactory solutions.
Figure 4.8. Class Functions: Class 1(The Rigidity Function) and Class 2(The Flexibility Function)
64
4.2.2. Sensitivity Analysis
One of the most important tasks in topology optimization is to evaluate the
sensitivity of functions with respect to design variables. Two class functions are used to
consider both rigidity and flexibility in the strain based topology optimization.
The first class function is the global effective strain energy function. By some
simple derivations of the chain rules, sensitivity of the class function P1 can be further
reduced to the sensitivity analysis of the summation of the global effective strain as
1 11 1
1i i
dP rdP dr
dx dr dx (4.4)
21
1
n
i
ii i
dp d
dx dx
(4.5)
The sensitivity of global effective strain is already derived in previous chapter, and
sensitivity can be expressed as:
0
2
1
'2
E
Ti
e e e
i
n
i
ii
f xu k v
f x
d
dx
(4.6)
The second class function is maximizing mutual strain energy function and other
finite element analysis systems are introduced to derive sensitivity of mutual strain
energy in Figure 4.9. System A is the analysis system with SIMP model to calculate
65
Figure 4.9. Finite Element Analysis System for Maximum Output Displacement
(a) System A (b) System D
displacement vector U and System B is also used SIMP model but external force is used
as a unit dummy load vector that represents the desired motion of complaint mechanism.
The derivative of mutual strain energy can be expressed as:
'
iT T T
e e e
i i i i
f xd d dKU U K W U W u k w
dx dx dx f x
(4.7)
where ve1 is displacement vector from system D.
Using equation (4.6) and(4.7), the sensitivity of objective function can be
expressed as
'21 2
1 1 0 2
'2
E
p i iT
e e e e e e
n i i i
dP f x f xdP dPu k v u k w
dx dr f x dr f x
(4.8)
66
Since sensitivity is derived, topology optimization algorithm can be performed to
find out optimal design of the compliant mechanism. In the following section, nemerical
examles are presented using the strain based topology optimization method.
4.3. Numerical Examples
In this section, design examples of the strain based topology optimization for
complaint mechanism are presented. First, the optimal designs of the strain energy based
formulation and the strain based formulation are compared and discussed using the
compliant gripper І example. Then, two additional examples are presented to
demonstrate the proposed method. In all examples, the material properties are Young’s
modulus E0=100Pa, Poisson’s ratio υ=0.3.
4.3.1. Compliant Gripper I
As it was presented before in section 4.1.1, compliant gripper І is studied again.
The design domain and boundary conditions are equally defined as Figure 4.3. The
optimal design using strain based formulation is illustrated in Figure 4.10 and the full
model of the complaint gripper І is shown in Figure 4.11. From the optimal design, we
can see that the optimal configuration is different from the optimal configuration using
strain energy formulation in Figure 4.4. Especially, the straight member is removed and
the curved member is generated to reduce strain energy concentration.
To examine the impact of these two different designs, strain energy distributions
of both solutions are plotted in Figure 4.12 and the figure clearly shows that the strain
67
energy is greatly reduced for the curved member which is generated by strain based
formulation.
Figure 4.10. The Optimal Design of Strain Energy Based Formulation for Complaint Gripper І
Figure 4.11. The Full Compliant Gripper І Design of Strain Based Formulation
68
Figure 4.12. Strain Energy Comparisons: The Strain Based Formulation (top) and The Strain
Energy Based Formulation (bottom)
In addition to the lower strain energy, the compliant mechanism should give a
proper motion as to how we define it. To verify, the motion of the optimal design
deformed shape is illustrated in Figure 4.13. From the finite element analysis results of
the optimal designs, the output displacement of the solution from the strain based method
is 22% larger than that from the strain energy based method. It is because the solution
from the strain based method has larger leverage on the right side of the structure.
Figure 4.13. The Deformed Configuration of the Compliant Gripper І
69
4.3.2. Compliant Gripper II
The second example is another gripper design. The design domain is very similar
to that of the first gripper, but the boundary conditions are very different as shown in
Figure 4.14. The upper left corner is fixed and the bottom is only constrained in the y-
direction due to the symmetric condition. The force is applied at the lower left corner
toward the right and the desired output displacement is a downward motion at the lower
right corner.
Figure 4.14. Design Domain and Boundary Condition for Gripper
The optimal topology configurations of the gripper and the full model of the
gripper is illustrated in Figure 4.15 and Figure 4.16
Figure 4.15. The Optimal Design of Strain Based Formulation for Complaint Gripper ІІ
70
Figure 4.16. The Full Compliant Gripper ІІ Design of Strain Based Formulation
The strain energy of each element for optimal design are plotted to demonstrate
that the strain energy is well distributed in Figure 4.17 and the deformed shape is also
illustrated to verify the motion of the optimal design in Figure 4.18
Figure 4.17. The Strain Energy Distribution for Compliant Gripper ІІ
71
Figure 4.18. The Deformed Configuration of the Compliant Gripper ІІ
4.3.3. Displacement Inverter
In the third example, a displacement inverter is studied. Because of the symmetric
condition, only half of the design domain is modeled. The force is applied at the upper
right corner in the downward direction and the displacement is expected to be produced
at the lower right corner towards left direction as shown in Figure 4.19.
Figure 4.19. Design Domain and Boundary Condition for Displacement Inverter
72
The optimized design of the problem shown in Figure 4.20 and the full inverter is
also shown in Figure 4.21. Once again, the deformation is well distributed as we expected
from Figure 4.22 . From the Figure 4.23, it is clear that the optimal design deformed
properly as what we specified.
Figure 4.20. The Optimal Design of Strain Based Formulation for Displacement Inverter
Figure 4.21. The Full Displacement Inverter Design of Strain Based Formulation
73
Figure 4.22. The Strain Energy Distribution for Displacement Inverter
Figure 4.23. The Deformed Configuration of the Displacement Іnverter
4.4. Conclusion
In this chapter, the strain based topology optimization method for compliant
mechanism was presented. Instead of the strain energy used in common topology
optimization formulation, a global effective strain functional is implemented for
maximizing the rigidity of compliant mechanism design. This new formulation can
reduce localized high deformation in compliant joints generated by the strain energy
based formulation. Drawbacks from the weighted sum and the fraction methods are
eliminated by the implementations of the physical programming method. Numerical
examples further demonstrated the advantages of the proposed method for compliant
mechanism design.
74
Chapter 5.
Energy Absorbing Structure Design using a Strain
Based Topology Optimization Method
Energy absorbing structure is also one of the representative applications using
flexibility of the structure. Since flexible members have more deformation than rigid
members, by using flexibility of the structure, we can absorb more energy than rigid
structure. In this chapter, we are focusing on developing the formulation for topology
optimization method to find the optimal configuration for energy absorbing structure
design. In general, energy absorbing structures are used widely with the purpose of safety.
In the packaging field, energy absorption concept is applied to protect goods such as
sandwich board boxes or cell designs. The energy absorption design is also implemented
for safety of humans especially in transportation field. For example, the vehicle frame
should have energy absorbing ability to ensure safety of the passenger in an accident,
while aircraft structures should also be considered for its crashworthiness in case of the
hard landing as shown in Figure 5.1
Figure 5.1. Crashworthiness Test of Aircraft (resource: http://www.apg.jaxa.jp, http://www.onera.fr/)
75
Figure 5.2. Design Domain and Boundary Conditions for Energy Absorbing Structure Design
Problem
When we consider energy absorption, we apply excessive load to the structure.
However, we also need to consider the stiffness of the structure, which can support the
goods that is contained in the structure. So we also need to apply another force that can
represent the weight of the goods, Because of that, 2 loading cases are studied in this
chapter. These two forces are presented as F1 and F2 in Figure 5.2. F1 denotes the
external force to simulate energy absorption of the structure and F2 denotes the weight of
the goods for it have enough stiffness to support the goods.
In this chapter, a new formulation is proposed for energy absorbing structure
design problem of topology optimization method. First of all, the formulation for
protection from the first loading cases is studied. After that, physical programming
formulation is implemented to handle these two loading cases. Next, the comparison of
energy based and strain based formulation is also explained and numerical examples are
presented and discussed before concluding remarks are made.
76
5.1. Problem Formulation for Energy Absorption Structure
Design
The problem formulation is studied to implement the energy absorption structure
design concept into topology optimization method. In the first section, maximizing
energy absorbing formulation is studied and the drawback of that is explained. Next, new
formulation is presented to avoid the drawback of maximizing formulation. As it was
discussed before, we are considering two loading cases in this chapter. Therefore, multi-
objective formulation is presented to handle both energy absorbing and weight supporting
at the same time.
5.1.1. Maximizing Energy Absorbing Formulation
To enhance the energy absorption capacity of the structure, maximizing objective
function is usually applied in optimization problem. Since first loading F1 represents the
external loading for energy absorption, the maximizing strain energy formulation can be
expressed as:
1 1
1max
2
TU F (5.1)
U1 denotes the displacement field when the first loading is applied to the design
domain. In optimization method maximizing objective function should be converted as a
minimizing function, and by multiplying negative sign to objective function, maximizing
77
formulation can simply be converted as minimizing function, therefore the equation (5.1)
can be converted as:
1 1
1min
2
TU KU (5.2)
Physically, soft material can absorb more energy than stiff material, so the
structure made with soft material will have more energy absorption capacity than a
structure made with stiff material. In topology optimization method, optimal design can
be accomplished by adding or removing material density, and since soft material can
absorb more energy, material density of each element could be decreased to the lower
bound of material density. Then the optimal design of the energy absorption structure can
be empty space. This design may have infinite energy absorption capacity that can be an
ideal optimal design from the energy absorption point of view, but it is an infeasible
design for manufacturing purposes.
This problem also can be proved by deriving sensitivity of the equation(5.2). The
sensitivity of strain energy is already derived in Chapter 2, and since we have negative
sign in front of strain energy, the sensitivity of equation (5.2) can be derived as:
'
1 1 1 1
1
2 2
iT T
e e e
i i
f xU KU u k u
x f x
(5.3)
78
u1eTkeu1e is the element strain energy and we know energy is always a positive
value and
'
i
i
f x
f x is also positive because 0 ≤ xi ≤ 1. Therefore, sensitivity value of all
elements should be positive, and it means that material should be removed to accomplish
the objective function. Therefore, the optimal design will shows as zero material density.
5.1.2. Energy Absorbing Formulation
In a practical sense, energy absorbing structure requires that they not only absorb
energy ability but also provide strength for protection. For example, the front of a car
frame should absorb as much energy as it can, but it should also have the strength to
protect the passengers inside the vehicle as shown in Figure 5.3. If the vehicle frame was
designed only from an energy absorption point of view, impact would cause serious
injury to the passengers. Therefore, the strength of the frame must also be considered.
Figure 5.3. Vehicle Crash Test: (resource: http://autos.msn.com/)
79
From the strength point of view, minimizing strain energy can be considered as a
new objective function instead of maximizing formulation. However, it will give the
stiffest design that has lower energy absorption capacity. Therefore, the main challenge
of the energy absorbing structure design is to find out how the formulation can
accomplish the energy absorbing ability and strength using strain energy terms.
It is very difficult to formulate using strain energy function directly. However, if
we consider the strain energy value as a variable of the certain function, by changing
function type we can simply control the problem converged to certain range that can
accomplish both energy absorption capacity and strength. Therefore, we can convert
strain energy objective function into the function of strain energy, and it can be described
as:
1 1
1min
2
TP U F
(5.4)
By using class functions of physical programming method, designer can simply change
the problem as minimization, maximization, seek value or seek range. Therefore, if we
can define target range of the strain energy, topology optimization algorithm can give the
design that have high energy absorption ability with enough energy using class function
as shown in Figure 5.4.
80
Figure 5.4. Class Function for Energy Absorption of Structure
5.1.3. Multi-Objective Formulation for Energy Absorbing Structure
Design
As we discussed in the previous section, we also need to consider the second
loading case F2 to support the self weight of goods and the minimizing strain energy
formulation is applied in general. However, as we discussed in Chapter 2, energy based
formulation shows the distortion of the strain information. Therefore strain based
formulation is used for rigidity of the structure, as follows as:
2min (5.5)
Since physical programming method is applied for energy absorbing formulation,
equation (5.5) can also be converted to a form of physical programming using class 1
function in Figure 1.8. Therefore, the problem formulation for energy absorbing design
could be described as:
81
2
1 1 1 2
1
1 1
2 2
0
1
1 2
Minimize
Subject to
( )
, , , ,
nT
i
i
N
i i
i
N i
P U F P
KU F
KU F
V X x v V
X x x x x x x
(5.6)
There are two different loading and analysis is implemented as KU1=F1 and KU2=F2. The
sensitivity of objective function can be derived since we already derived the sensitivity of
both strain energy and global effective strain in previous chapters.
21 2
1 1 2 2
1 1 0 2 0
' '1 1
E E
i iT Tne e e e e e
n i i i
f x f xdP dP dPu k u u k v
dx dr f x dr f x
(5.7)
5.2. Numerical Example
In this section, numerical examples of the energy absorbing structure design are
described. As discussed in previous section, we presented the formulation that can have
high energy absorption capacity and enough strength using physical programming
scheme, and different types of the class functions are studied to verify the correlation of
the class functions and optimal design configuration.
82
5.2.1. A Simple Container Model
A simple container model is presented in this section for an energy absorption
structure design and full design domain is illustrated in Figure 5.5. To reduce the
calculation cost, entire design domain can be simplified to only one quarter part of the
whole design domain by using symmetric conditions as shown in Figure 5.6. Each top
end surface is fixed to represent holding area of the container and symmetric boundary
conditions are applied on the cut surfaces. Figure 5.9 shows the quarter design domain
and boundary conditions. The volume constrain is set up to 40 percent of the total volume
and lower bound of material density xi is selected as 0.2 since container should cover all
surface.
Figure 5.5. The Entire Design Doamin of the Simple Container Example
83
Figure 5.6. The Quarter Design Domain and Boundary Conditions of the A Simple Container
Example
5.2.1.1. The Optimal Design Using Single Objective Minimizing
Strain Energy Formulation
Since the optimal design of traditional structure design is usually considered the
stiffest structure as an optimal design, minimizing strain energy formulation is studied in
this section. For that, external force F1, and weight of goods F2, are considered as one
finite analysis system as shown in Figure 5.7 and objective function can be formulated as:
1 2min U F F (5.8)
84
Figure 5.7. The Boundary Conditions for Minimizing Strain Energy Problem
The optimal design configuration is illustrated in Figure 5.8. The top left figure is
the optimal design of the quarter domain and the bottom left figure is full model of the
simple container. Top and bottom right figures are the optimal configuration with
material density less than 0.2 removed. The absorbing energy when external force is
applied at the corner is 6.6J, and based on this value, the ability of energy absorption for
other cases is compared.
85
Figure 5.8. Optimal Design of a Simple Container Problem
5.2.1.2. The Optimal using Maximizing Strain Energy
Formulation with Physical Programming Scheme
As we discussed in previous section, we expected that maximizing strain energy
formulation to increase energy absorption capacity will give the empty space or lower
bound of material density structure which is an infeasible design to manufacture. In this
section, the optimal design using maximum formulation is illustrated to verify that. The
86
Figure 5.9. A Quarter Part of the Simple Container Model and Boundary Condition: External
Loading for Energy Absorption F1 and Self Weight Loading of Goods F2
problem is considered as two loading cases as shown in Figure 5.9: external force for
energy absorption F1 and self weight of goods F2 are applied separately.
The physical programming scheme is used to formulate these two objectives as an
objective function, and it can be described as:
2
1 1 1 2
1
1min
2
na T
i
i
P U F P
(5.9)
Two different class functions P1a and P2 is defined as shown for maximizing
strain energy and minimizing global effective strain as shown in Figure 5.10
87
Figure 5.10. Applied Class Functions: Maximizing Strain Energy (left) and Minimizing Global
Effective Strain (right)
The optimal design is illustrated in Figure 5.11. As we expected before, there is
no structure for energy absorption of external force at the corner, especially within the
red circle area. This design may be good for energy absorption from the mathematical
point of view, but it is not the optimal solution in practical terms because there is not
enough strength for protection of the goods. Therefore, instead of applying maximizing
class function, seeking range class function is implemented in following section.
88
Figure 5.11. Optimal Design of a Simple Container Problem
5.2.1.3. The Optimal using Seek Range of the Strain Energy
Formulation with Physical Programming Scheme
As it was discussed in previous sections, the structure design for energy
absorption requires not only the high energy absorption ability, but also the sufficient
89
stiffness. To satisfy these two conflicting objective, seek range class function can be
implemented. Therefore the objective function can be formulated as:
2
1 1 1 2
1
1min
2
nb T
i
i
P U F P
(5.10)
where P1b represent seek range class function, and it is shown in Figure 5.12
The optimal design using equation (5.10) is illustrated in Figure 5.13. The total strain
energy of the optimal design is 15.1J, and it is dramatically increased in comparison to
the optimal design using equation(5.8). Moreover it shows the structure configuration
unlike the optimal design using(5.9), and that structure provides strength when external
force is applied at the corner. From the optimal design, it is clear that the seek range class
function can provide the high energy absorption structure with feasible design in
manufacturer point of view.
Figure 5.12. Seek Range Class Function for Energy Absorption Formulation
90
Figure 5.13. Optimal Design of a Simple Container Problem
91
To support the goods that are contained within the container, global strain
formulation is implemented as shown in equation(5.10). To verify the advantages of
strain formulation, the optimal design of both strain based and strain energy based
formulation is compared in Figure 5.14. Threshold is applied to both optimal designs to
compare the stress distribution when external force F1 is applied, and Von Mises stress
distribution configuration is illustrated in Figure 5.15. The maximum stress of the strain
energy based optimal design is 2270Pa, and the maximum stress of the strain based
optimal design is 1307Pa. As we expected, the strain based formulation can avoid stress
concentration, and this is another example that proves the prove advantages of strain
based formulation.
Figure 5.14. The Optimal Design Configurations using Strain Based Formulation (left) and Strain
Energy Formulation (right)
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Figure 5.15. The Stress Distribution Configurations using Strain Based Formulation (left) and Strain
Energy Formulation (right)
5.3. Conclusion
In this chapter, energy absorbing structure design scheme with two loadings is
presented. By changing maximum strain energy formulation as the function of the strain
energy, the problem can be easily converted to maximum, minimum, seek range
problems. Because the formulation should be considered in the area of both the energy
absorption capacity and the strength of structure, seek range problem can be used. From
the numerical example, we verify that by using seek range class function, we can have
the optimal design with energy absorption ability and strength of stress. Also using strain
based formulation; we can avoid stress concentrations that can easily be shown from
strain energy based formulation.
93
Chapter 6.
Conclusion and Future Work
Strain based topology optimization method is proposed and some results on
different types of problems such as compliant mechanism and energy absorption structure
are shown in this dissertation. The achievement of this study is concluded in this chapter
and further research can that be expended based on this research is also discussed as
follow.
6.1. Conclusion
In general, topology optimization method is formulated by energy function such
as strain energy or mean compliance. However, energy function distorts strain
information because it includes the material density function from SIMP model.
Therefore, the material cannot be distributed properly when element has high strain with
low material density. To correct this problem, strain formulation is introduced and
developed, and the optimization method based on strain formulation is developed as well.
The strain based topology optimization method can provide the optimal design
configurations that can avoid stress concentration or high localized deformation. Since
stress or deformation concentration causes the problems such as structure failure, fatigue,
or malfunction of the mechanisms, the strain based topology optimization can present
safer designs in comparison to the energy based topology optimization method could.
The numerical examples in chapter 3, chapter4, and chapter 5 verified that by comparing
two different optimal designs of both strain based and energy based formulation.
94
Multi objective function is also formulated for both compliant mechanism and
energy absorbing structure design since these problem should be considered the
flexibility and rigidity of the structure at the same time. To formulate objective function,
physical programming scheme is applied since it can involve the designer’s preference
into the formulation. Especially, for energy absorption structure design, both the energy
absorption capacity and the strength of protect purpose can be formulated by using seek
range class function.
The study in this dissertation about strain based topology optimization method
extensively extends the scope of applications of topology optimization. Furthermore, this
research also provides not only more safe design but also more efficient design in
comparison to the energy based topology optimization method.
6.2. Future work
Although this research suggests a topology optimization method using strain
formulation, not all of the structure models were studied. Futher studies can be performed
on optimizing other applications of topology optimization. Some possibilities are pointed
out as follows:
Implementation of Dynamic Problems
The study in this dissertation is based on static situations. However, in
practical application, most of the structure faces the dynamic situation. For
example, in case of the simple cantilever beam, it will have the vibrations
caused by wind. Also, some of the compliant mechanism used for vehicle or
95
airplane can also give vibrations to the entire structure. Therefore, dynamic
criteria are also required for topology optimization method.
Nonlinear analysis for both the compliant mechanism and the energy
absorbing structure design
The strain based topology optimization is formulated based on linear problem.
However, the compliant devices or energy absorption structure are usually
undergone by large deformation which is structure. Moreover, to enhance the
efficiency of the complaint mechanism or the energy absorbing capacity,
nonlinear material can be used. Therefore, nonlinear analysis can be studied
more for the both compliant mechanism design and the energy absorption
structure design.
Uncertainty for Energy Absorbing Structure Design
In chapter 5, we presented the methodology of energy absorbing structure
design and we predefined the external force locations. However, we cannot
exactly define where and what direction the external force will be applied.
Therefore, further study regarding the uncertainty of the external force should
be conducted.
96
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101
Curriculum VITA
Euihark Lee
1996~2000 Bachelor of Science
Mechanical Engineering
Kyunggi University, Suwon, Republic of Korea
2000~2006 Air force officer
Air Force, Seoul, Republic of Korea
2004~2006 Master of Science
Mechanical Engineering
Kyunggi University, Suwon, Republic of Korea
2006~2011 Teaching Assistant and Grader
Mechanical and Aerospace Engineering
Rutgers University, New Brunswick, NJ
Publications Xiaobao Liu, Euihark Lee, Hae Chang Gea, Ping An Du,
“Compliant Mechanism Design using Strain Based Topology
Optimization Method,” ASME IDETC/CIE, 2011
