M.Tech_Thesis _surendra_singh

Topology Optimization of Structural Stiffness and Structural
Frequency
A thesis submitted in partial fulfillment of the requirements for
the award of the degree of
M.Tech
in
Machine Design
By
Surendra Singh
DEPARTMENT OF MECHANICAL ENGINEERING
INDIAN INSTITUTE OF TECHNOLOGY
GUWAHATI-781039
MAY 2014

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Bonafide Certificate
This is to certify that the project titled Topology Optimization of Structural Stiffness and
Structural Frequency is a bonafide record of the work done by
Surendra Singh (124103021)
in partial fulfillment of the requirements for the award of the degree of Master of Technology in
Machine Design of the INDIAN INSTITUTE OF TECHNOLOGY, GUWAHATI, during the year
2013-2014.
Dr. A.N. Reddy Dr. Deepak Sharma Prof. P. Mahanta
Advisor Advisor Head of the Department
Project Viva-voce held on: 12/05/2014.
Prof. Rajiv Tiwari Dr. Sangamesh Deepak R Prof. Arbind K. Singh
Internal Examiner Internal Examiner External Examiner

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Acknowledgements
First and foremost, I would like to thank my advisor, Dr. Aneem Narayanan Reddy for being an
incredible role model, teacher and mentor. The time, technical knowledge and advice he shared on a
daily basis were invaluable and so greatly appreciated.
I would also like to express my warm appreciation to my second advisor, Dr. Deepak Sharma for his
encouragement, beneficial comments, contribution and helpful advice. Much of my achievement
would not have been possible without them and I am deeply grateful for their dedication to my success.
Secondly, I would like to thank examination committee members Prof. Rajiv Tiwari and Dr.
Sangamesh Deepak R for the sharing of their time and knowledge help me in completion of this
project.
Finally, I would like to thank my family for their unwavering love and support, both financially and
emotionally. I would not be where I am today, or the person I am today, without them. I would also
like to thank all of my friends that I have met along the way for always keeping my spirits high and
their kind words of encouragement throughout the journey.

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Abstract
In this thesis, two problems are addressed using topology optimization. The first problem deals with
the design of stiff-structure under given loading conditions whereas the second problem with the
design of a structure for the optimal first natural frequency. The optimal frequency problem is solved
for both maximization and minimization of frequencies.
The stiff-structure problem is useful in automobile industry and also in civil engineering structural
design. On the other hand the frequency maximization problem is useful in design of mechanical
components and earthquake resistant structures. Furthermore, the frequency minimization problem has
application in energy harvesting.
The stiff-structure problem is implemented based on the uniform stress criteria and it is compared with
the conventional mean compliance minimization. The frequency maximization problem has been
solved and compared the results that were presented in literature. In addition, the numerical artifact i.e.,
mode localization in frequency maximization has been discussed with an effective example.
The frequency minimization problem has been solved and found that the algorithm is putting the mass
away from the support. Also a flexible structure has been connected between support and the mass.
Taking the results of topology optimization as inspiration, we also developed few realistic intuitive
designs that have low frequency.

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Table of Contents
Bonafide Certificate…………………………………………………………………………...……. ii
Acknowledgements…………………………………………………………..................................... iii
Abstract……………………………………………………………………………………………... iv
Table of Contents…………………………………………………………………………....……..... v
Table of Figures…………………………………………………………………………………….. vii
1 Introduction
1.1 Background and Motivation of Stiff Structure Optimiation..………………………………… 1
1.2 Background and Motivation of Structural Frequency Optimization….……………………… 1
1.3 Introduction to Topology Optimization………………………...……………………………. 2
1.4 Objective of the Thesis……………………………………………………………………….. 6
2 Literature Review
2.1 Truss Topology Optimization………………………………………………………………… 7
2.2 Topology Optimization Continuum Structures……………………………………………… 8
2.2.1 Microstructure Approaches for Conitnuum Topology Optimization……………………..…. 8
2.3 Continuum Topology Approach for Frequency Optimization……………………………… 10
3 Topology Optimization for Structural Stiffness
3.1 The Uniform Stress as an Criterion for Stiff Structure……………………………………... 11
3.2 Optimization of Stiff Structure Based on Minimizing the Mean Compliance……………... 19
4. Topology Optimization for Structural Frequency Using Classical Method
4.1 Problem Formulation……………………………………………………………………..… 24
4.2 Sensitivity Calculation of Frequency……………………………………………………..... 25

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4.3 Localized Eigen Mode……………………………………………………………………..... 27
4.4 Toology optimization for Maximizing Structural Frequency……………………………..… 29
4.5 Topology Optimization for Minimizing Structural Frequency……………………………... 33
5 Topology Optimization for Structural Frequency Using Modified Genetic
Algorithm
5.1 Introduction to Genetic Algorithm……………………………………………………...…… 38
5.2 Modified Genetic Algorithm to Frequency Optimization Problem………………………… 39
5.2.1 Initialization and Representation…………………………………………………….....….. 40
5.2.2 Fitness Evaluation………………………………………………………………………..…. 41
5.2.3 Selection……………………………………………………………………………..……… 41
5.2.4 Apply GA Operators……………………………………………………………………...…. 41
5.2.5 Connectivity Analysis……………………………………………………............................. 44
5.2.6 Elitism……………………………………………………………………………………..… 45
5.2.7 Termination………………………………………………………………………………... 45
5.3 Case Studies……………………………………………………………………………….... 45
6 Intutive Designs of the Low Frequency Structures……………………….. 50
7 Conclusions and Future Work
7.1 Summary of the Thesis……………………………………………………………………. 57
7.1.1 Optimization of Stiff Structure…………………………………………………………….. 57
7.1.2 Optimization of Structural Frequency……………………………………………………... 57
7.2 Contribution of the Thesis…………………………………………………………………. 58
7.3 Scope of Future Work…………………………………………………………………….. 58

vii
References 59
List of Figures
Figure No. Title Page No.
1.1 Sizing optimization…………………………………………………………………………... 3
1.2 Shape optimization……………………………………………………………………........... 4
1.3 Topology optimization………………………………………………………………………. 5
3.1 Flow chart for topology optimization algorithm based on uniform stress criterion………… 12
3.2 (a) Initial truss layout with boundary condition (b) Variation of load point displacment with
iteration……………………………………………………………………………………… 13
3.3 Topology configurations of truss Structure with iteration considering uniform stress as
optimality criterion……………………………………………………………………........... 14
3.4 The cantilever beam (a) initial design domain (b) schematic diagram of FE discretization… 15
3.5 Material distribution for volume fraction of 20% in given domain of the cantilever beam with
iteration…………………………………………………………………………………….... 16
iteration…………………………………………………………………………………….... 16
iteration……………………………………………………………………………………… 17
3.8 Variation of load point displacment for volume fraction of 20% in given domain of the
cantilever beam with iteration………………………………………………………………. 17
3.9 Variation of load point displacement for volume fraction of 50% in given domain of the
3.10 Variation of load point displacement for volume fraction of 80% in given domain of the
3.11 Optimal topologies for stiffness maximization of cantilever beam for volume fraction of 10%,
20%, 50%, 80%, and 90%........................................................................................................ 19
3.12 Optimal topological configuration for maximum stiffness for 50% and 80% volume fraction of
design domain……………………………………………………………………………….. 21

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3.13 Evolution history of objective function (mean compliance) for 50% volume fraction of design
domain………………………………………………………………………………………. 22
3.14 Evolution history of objective function (mean compliance) for 80% volume fraction of design
domain………………………………………………………………………………………. 22
4.1 First Eigen mode with mode localization…………………………………………………… 27
4.2 First Eigen mode shape after elimination of mode localization…………………………….. 28
4.3 (a-c) Admissible design domains of beam like 2-D structures (d-f) Objective function variation
with iteration for three different sets of boundary conditions (a) cantilever beam (b) fixed-fixed
beam (c) simply supported beam, respectively……………………………………………... 30
4.4 (a-c) First Eigen mode shape corresponding to optimal topology (d-f) Optimal topologies for
three different sets of boundary conditions defined in Figure 4.3(a-c)…………………..… 31
4.5 (a-c) Iteration histories of the first three natural frequencies associated with boundary
conditions (a) cantilever beam (b) fixed-fixed beam (c) simply supported
beam………………………………………………………………………………………... 32
4.6 (a-c) Admissible design domains of beam like 2-D structures (d-f) Objective function variation
with iteration for three different sets of boundary conditions (a) cantilever beam (b) fixed-fixed
beam (c) simply supported beam, respectively…………………………………………….. 34
4.7 (a-c) First Eigen mode shape corresponding to optimal topology (d-f) Optimal topologies for
three different sets of boundary conditions respectively defined in Figure 4.6(a-c)……… 35
4.8 (a-c) Iteration histories of the first three natural frequencies associated with boundary
conditions (a) cantilever beam (b) fixed-fixed beam (c) simply supported
beam……………………………………………………………………………………..…. 36
5.1 A flow chart to solve frequency optimization problem using modified GA……………….. 39
5.2 A material distribution in the design domain using binary string representation…………… 40
5.3 Topologies before and after application of crossover operator……………………………… 43
5.4 Topology before and after application of mutation operator……………………………….. 43
5.5 Topology (a) before connectivity analysis (b) after applying floating point analysis and (c) after
applying point connectivity analysis………………………………………………………... 44

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5.6 (a-c) Admissible design domain of beam like 2D structures with three sets of boundary
condition (i) cantilever beam (ii) fixed-fixed beam (iii) simply supported beam, (d-f) Initial
topology with randomly distributed material in given design domain corresponding to (a-
c)……………………………………………………………………………………………… 47
5.7 (a-c) Topology configuration at 25th
iteration , (d-f) Optimal topology configuration
corresponding to minimum first natural frequency of structure in figure 5.6 (a-c) for 50% of
volume fraction……………………………………………………………………………. . 48
6.1 (a) Structure with cantilever beam boundary condition (b) First mode shape corresponding to
natural frequency 20.354 Hz ………………………………………………………………. 51
6.2 (a) Flexible structure with two beam in parallel (b) First mode shape corresponding to natural
frequency 3.364 Hz…………………………………………………………………………. 52
6.3 (a) Folded beam with three beams in series (b) First mode shape corresponding to natural
frequency 0.416 Hz……………………………………………………………………….… 53
6.4 (a) Folded beam with five beams in series (b) First mode shape corresponding to natural
frequency 0.333 Hz………………………………………………………………………… 54
6.5 (a) Structure with three folded beams in parallel (b) First mode shape corresponding to natural
frequency 1.234 Hz………………………………………………………………………… 55

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1. Introduction
We are addressing two problems called stiff structure optimization and structural frequency
optimization. Though these two problems are different still they share common solution
procedure called topology optimization. We now discuss the background and motivation to the
both problems.
1.1 Background and Motivation of Stiff Structure optimization
Buildings have become taller and bridges longer in recent years. In the vertical direction against
gravity loads, we would usually wish to minimize deflection of the structure, i.e. we would want
to have adequate stiffness.
In machinery the components and support structure are subjected to loading. There are the
limitations on displacement of a component specified in design domain demand the structure
with sufficient stiffness. Considering the buildings, bridges, machine components and supports
where there is a limitation on deflection, to satisfy this objective the structure should have
maximum stiffness.
In this work we implemented the topology optimization approach to minimize the deflection or
maximize the stiffness of structure subjected to given loading and boundary condition.

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1.2 Background and Motivation of Structural Frequency Optimization
The quest for everlasting energy sources of micro devices has been intensiﬁed in the recent years
due to the limitations on the applications and deployments of conventional electrochemical
power sources arising from their short lifespan.
A completely autonomous energy source is particularly advantageous in low power systems with
restricted accessibility, such as remote micro-sensors and wireless devices. The ultimate
renewable energy source for micro devices should be equipped with an energy-harvesting
mechanism capable of capturing ambient energy and converting it into useable energy. In
evaluating the available energy harvesting technologies, solar cells are the most mature.
However, their dependence on sunlight restricts the locations where solar cells can be effective.
In contrast, environmental vibration is a particularly attractive energy source because of its
abundance in nearly all environments, spanning a wide frequency range. In many cases there is
the need of continuous energy to power the wireless sensor in remote area where solar and
conventional energy source are not available.
The seismic noise of frequency 0.5 Hz is available everywhere. Therefore, this vibration energy
can be efficiently utilized by generating a structure of natural frequency near to 0.5 Hz. Thus, in
this work, we want to study the feasibility of developing such a low frequency structure using
topology optimization approach.
1.3 Introduction to Topology Optimization
Topology optimization is aimed to find the optimum distribution of specified volume fraction of
material in selected design domain. The optimum distribution is often measured in terms of the
overall stiffness of the structure such that the higher the stiffness the more optimal the
distribution of the allotted material in the domain. Topology optimization can be regarded as an
extension of methods for sizing optimization and shape optimization.

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Sizing Optimization:
Sizing optimization performs optimization by holding a design’s shape and topology constant
while modifying specific dimensions of the design. Hence, the design variables control particular
dimensions of the design and value of the design variables define the values of the dimensions.
Optimization therefore occurs through the determination of the design variable values which
correspond to component dimensions providing optimum structure behavior.
Example of sizing optimization includes the calculation of an optimum truss member cross-
sectional area or column diameter. Figure 1.1 shows an example of sizing optimization of a beam
cross-section. Prior to optimization, the engineer must define the component’s material
properties and boundary conditions. Then specify the structure’s shape and topology and indicate
which dimension shall be optimized. In this example, the engineer specified that the beam would
have an I-beam type of shape and that the web height, flange width and flange thickness should
be optimized. Using the objective function and constraints, sizing optimization then determines
the optimal dimensions values.
Figure 1.1: Sizing optimization
Shape Optimization:
Shape optimization, performs optimization by holding a design’s topology constant while
modifying the design’s shape. Hence, the design variables control the design’s shape and the

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values of the design variables define the particular shape of the design. Optimization therefore
occurs through the determination of the design variable values which correspond to the
component shape providing optimal structural behavior.
Example of shape optimization includes the determination of the optimum shape of a rod in
tension, the optimum node locations in a 10-bar truss. Figure 1.2 shows an example of shape
optimization of a beam cross-section. Prior to optimization, the engineer must first define the
component’s material properties and boundary conditions. The engineer must then specify the
structure’s topology and indicate which portions of the component’s shape shall be optimized. In
this example, the engineer specified that the beam have no interior holes and selected the entire
boundary for shape optimization. Using the objective function and constraints provided by the
engineer, shape optimization determines the optimum size and shape of cross-section.
Figure 1.2: Shape optimization
Topology Optimization:
Topology optimization performs optimization by modifying the topology of a design. Hence, the
design variables control the design’s topology, and the values of the design variables define the
particular topology of the design. Optimization therefore occurs through the determination of the
design variable values which correspond to the component topology providing optimal structure

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behavior. Note that size and shape optimizations typically occur as byproducts of the topology
optimization process.
In topology optimization, region in space which the structure may occupy i.e. design space is
discretized with finite element considered as building blocks of structure then allowing each
building block to either exist or vanish, a unique design is created. During topology optimization,
a design’s building blocks are controlled by design variables, where the value of each design
variable determines the existence and characteristics of its corresponding building block. For
example, in the topology optimization of a cantilevered plate, the plate is typically discretized
into small rectangular elements, where each element is controlled by a design variable has a
value of 0 then the corresponding element is assumed to be a hole. Likewise, when a design
variable is equal to 1 then its corresponding element contains fully-dense material. Lastly, design
variables with intermediate values corresponding to elements containing material of intermediate
density. So, to create a hole at a particular location in a design, the design variable corresponding
to the element at that location is simply set equal to zero. Similarly, holes are removed from a
design by assigning non-zero values to the design variables corresponding to the elements.
Figure 1.3: Topology optimization
Figure 1.3 shows an example of topology optimization of a beam cross-section. Prior to
optimization, the engineer must first define the material properties and boundary conditions. The
engineer must then specify the structure’s design domain, representing the region which the
structure may occupy. In this example, the engineer specified that the beam cross-section must

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be contained within a rectangular design domain. Using the objective function and constrains
provided by engineer, topology optimization determines the optimal size, shape and topology of
the beam cross-section.
1.4 Objective of the Thesis
In this thesis we addressed two structural optimization problems: (i) stiff structure optimization
(ii) frequency optimization.
Stiffness optimization:
The stiff structure optimization problem is very well address in the literature. However, we
implement the same problem based on uniform stress criterion and compare with the mean
compliance optimization. In order to address this problem we use solid isotropic microstructures
with penalization (SIMP). This SIMP method will be discussed in the detail in the following
chapter.
Frequency optimization:
There are two types of frequency optimization problem are addressed in the thesis (i) first natural
frequency maximization (ii) first natural frequency minimization
The first problem has application in structural design whereas the second problem has
application in energy harvesting as pointed out earlier. The maximizing frequency is very well
address in the literature along with numerical issues. So in this thesis we implemented this
problem. In this thesis, we present frequency minimization problem whereas in the literature it
has not been addressed. The topology optimization of frequency optimization approached using
both classical gradient method and genetic algorithm. Apart from topology optimization we also
design low frequency structures.

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2. Literature Review
This chapter is dedicated to the literature review that aims to give an overview on the related
work to this thesis.
Topology optimization is roughly divided into two types of problems, i.e., for discrete structures
(or grid-like structures) and continuum structures, respectively (Rozvany, 2001a). Topology
optimization for continuum structures on the other hand, considers the structures to have large
volume fraction (i.e. material volume/available volume) that means structural material occupies a
large portion of the available space (Rozvany, 2001a).
2.1 Truss Topology Optimization
In discrete optimization the grid elements can be either bars with hinges at nodes or the beam
with rigid hinges. Here we consider only bars and it is known as truss topology optimization.
This deals with the simultaneous selection of hinge locations and cross-sectional dimensions of
truss members. However in general truss topology optimization considers the cross sectional
areas are variables whereas the hinge locations are fixed. It can be observed that the grid
structure in discrete topology optimization (with either beams or trusses) occupies low volume
fraction in contrast to the continuum topology optimization. The topology optimization of
discrete structures has long history than that for continuum structures ever since the seminal
work by Maxwell (1890) and Michell (1904). The area of truss topology optimization has been
extensively explored for several decades and the optimal layout theory has been developed by
Prager, Rozvany and Wang (Rozvany et al. 1976).

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In this context we implemented an algorithm based on stress criterion. The uniform stress
criterion for stiff structure can find in work by Rozvany (1995). There is an alternative approach
for stiff structure based on the minimizing the displacement and this can be found in the book by
Bendsøe and Sigmund (2003).
2.2 Topology Optimization of Continuum Structures
The topology optimization selects the best configurations involving topologies and geometries
for the design of continuum structures. This allows for not only hole-generation in the interior of
the design domain but also geometry and size changing.
Two classes of approaches can be identified for continuum topology optimization (Olhoff, 2001),
namely the macrostructure approach and the microstructure approach based on the material
assumed in the finite elements. The essential difference between these two classes is that the
macrostructure approach uses solid isotropic materials while microstructure approach uses the
concept of porous medium.
The macro structural approach was proposed by Rossow and Taylor (1973) to get an optimal
topology. In this approach, the design domain is assumed to be planar plate and discretized with
finite element mesh where the thickness of each element is a design variable. By assigning the
minimum allowable thickness to be close to zero, the topology and shape change is able to be
realized. However, this method cannot be directly applied to 3-D structure and hence,
microstructural approach is adapted by other works in literature.
2.2.1 Microstructure Approaches for Continuum Topology Optimization
The microstructure approaches for continuum topology optimization are based on the ability of
modeling porous materials with microstructures. We note that the microstructure consist of
variables such as density and orientation of fiber or cell. Based on these variables the effective
material properties have been calculated by homogenization theory.

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The calculation of effective material properties in the homogenization theory is done by
replacing the microstructure such as square cell with square hole by isotropic material with
equivalent effective material properties. This theory motivated to do the topology optimization
considering the density and orientation as the design variables. The topology optimization
approach developed based on this theory called the homogenization method (Rozvany et al.,
1995). The first numerical implementation presented by Bendsøe and Kikuchi (1988) toward
topology optimization. Later on various topology optimization problems using the
homogenization method have been addressed, for example: dynamics problems such as
eigenvalue problems (Ma et al., 1995), design of compliant mechanism (Ananthasuresh et al.,
1994).
In composite materials there can exists porous material i.e. materials with voids. Whereas in
topology optimization our main target (of course in the framework of finite element method) is
to bring element either to be zero density or full material density. In other words we do not want
to leave any material with intermediate density in the final design. Therefore keeping the
orientation and other variables in topology optimization does not help much and in addition the
algorithm becomes tedious. Thus, considering only the density as variable in homogenization
technique helps the topology optimization.
This method of determining the effective material properties using only the density was proposed
by Bendsøe (1989) and later on this method coined as SIMP by Rozvany et al. (1992). The SIMP
method describes the relation between the material Young’s Modulus and the relative density
(continuously varying from 0 to 1) through a power law. Due to its simplicity and computational
efficiency, the SIMP method has been widely accepted by researchers in the area of structural
topology optimization and used for many problems such as vibrating continua (Pedersen, 2000;
Du and Olhoff, 2007). For more details on topology optimization of stiff structure and compliant
mechanism can refer the book by Bendsøe and Sigmund (2003).

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2.3 Continuum Topology Approach for Frequency Optimization
One of the first applications of the topology optimization methods other than stiff structure was
design of compliant mechanism (Ananthasuresh et al., 1994; Sigmund, 1998)). Later on the
potential of topology optimization approach (homogenization method) to solve problem in
dynamics other than compliance problem in structural optimization was first introduced by Diaz
and Kikuchi (1992).
Diaz and Kikuchi (1992) dealt with the reinforcement (adding material to existing structure) of
given 2-D structures to find the shape and topology of structure that maximize a natural
frequency by homogenization approach. The maximizing the first natural frequency has an
application in earthquake resistant design of buildings. We note that the reinforcement is solved
with limited amount of material than can be reinforcement to the existing structure. Later Ma and
kikuchi (1993) solved the frequency maximizing problem without adding the material to
structure like stiff structure problem. Problem can be solved not only maximizing the single
frequency but also for multiple frequencies by assigning weights (Kosaka and Swan, 1999).
Another important issue in the frequency maximization called localized mode in topology
optimization was discussed by Pedersen (2000). He also proposed some remedies to overcome
this problem by changing the penalization to the stiffness term. Recently, Xie (2010) solved the
problem using evolutionary optimization where the material density considered as was in SIMP
model.
In conclusion all works in literature talk about maximization of frequency which is essential for
seismic-resistance building design and automobile manufacturing. In contrast, the problem of
energy harvester requires frequency minimization as ambience noise possesses very low
frequency, i.e., 0.5 Hz. Thus, we address the frequency minimization problem as main
contribution in our thesis.

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3. Topology Optimization for Structural Stiffness
The stiff structure problem is implemented using both truss and continuum elements.
Furthermore, the stiff structure optimization has been implemented in two ways (i) based on
uniform stress criterion (ii) based on minimizing the mean compliance.
3.1 The Uniform Stress as an Optimality Criterion for Stiff Structure
Let us consider two cylindrical shafts with same amount of material and length, one shaft being
hallow and another shaft being solid. It can be observed from basic solid mechanics that the
hollow shaft behave stiffer than solid shaft for the same twisting moment. We note that the
stiffness for the hallow shafts is more due to uniform stress when it is compared with solid shaft.
Similar kind of observation can be made in bending of ‘I’ cross-sectional beam and rectangular
cross-sectional beam. Thus, we conclude that uniform stress is one of the criteria for stiff
structure.
Motivated by these examples, we implemented an algorithm based on uniform stress criterion for
general structure as shown in Figure 3.1. We now present a few examples of optimal topologies
of both truss and continuum structure to demonstrate the algorithm.
Example 1: Truss Structure
The initial layout of truss structure (ground structure) is shown in Figure 3.2 (a) for a given
loading and boundary conditions. Initial truss dimensions, i.e., the length and area are 1 m and 2
m2
, respectively. The Young’s modulus of truss material is considered 1Pa.

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Figure 3.1: Flow chart for topology optimization algorithm based on uniform stress criterion
Let be the stiffness matrix of given structure in finite element discretization (see Figure
3.2(a)). Let be a vector of cross-sectional areas of all elements. If the material properties
assumed to be fixed then the stiffness matrix is function of vector . Therefore, the vector
containing areas are design variables in the process of optimization.
Start
Define initial domain, Material
properties, loads, boundary
conditions and volume
Generate FE mesh
Initialize design variable in each
element (density)
Construct constitutive matrix for
elements
Carry out analysis and evaluate
objective function
Update design variable
(density)
Is there any change
in objective
function?
Stop
SIMP
Optimality
criterion
YES
NO
Plot density counter

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As mentioned in the previous flow chart (see Figure 3.1), the uniform stress is the optimality
criteria. In order to achieve the uniform stress, we adjust the areas (i.e., the vector A). We note
that the conservation of material has been taken care throughout the updating process. In other
words, the material is removed from the elements (i.e., reducing the cross-section of elements)
that are experiencing less stress and added to the elements (i.e., increasing the cross-section of
elements) that are experiencing high stress. This process is continued till attaining the
convergence. We note that the area of elements are not allowed to go zero in order to avoid
singularity in stiffness matrix rather the elements are allowed to very low value so that it doesn't
affect the overall performance.
The convergence criterion is the change in deflection of point where the load is being applied.
The convergence trend can be observed in Figure 3.2 (b). The topologies are presented with
progress (iterations) of optimization in Figure 3.3. The thickness of truss element represents its
cross section area.
Figure 3.2: (a) Initial truss layout with boundary condition (b) Variation of load point
displacment with iteration
.

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Iteration - 5 Iteration - 15
Iteration - 30 Iteration – 45
Figure 3.3: Topology configurations of truss Structure with iteration considering uniform stress
as optimality criterion.
Example 2: Cantilever Beam Problem
In this example also we consider the cantilever beam but in continuum framework, i.e., the
discretization is done with continuum elements. The maximization of stiffness is achieved
through the uniform stress criteria subjected to material resource constraint. The boundary
conditions and loading of beam are same throughout the optimization process. The cantilever
beam domain of size is 100 units into 60 units. We consider out-of-plane thickness is one unit. It
is well known in the topology optimization that the value of Young's modulus do not affect the
optimal topology. Therefore, we consider unit Young's modulus in all the following problems.
We assume 0.3 as Poisson’s ration in all the plane problems. The domain is discretized into 30 X
50 elements (vertical and horizontal directions, respectively) to obtain the optimal design. Each
element consists of density similar to the area in previous problem. Therefore, in this problem

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the densities are design variables. We note the densities are varied between zero and one using
SIMP approach. In this problem also the uniform stress criteria is used for getting the optimal
design. Of course, the von Mises stress is taken as measure of stress in the elements.
(a) (b)
Figure 3.4: The cantilever beam (a) initial design domain (b) schematic diagram of FE
discretization
A point load is applied vertically downwards on the cantilever beam as shown in Figure 3.4 (a).
Once the design variable for particular element reaches highest value ‘1’ the material addition is
not done in subsequent iteration. Similarly if the element reaches the lowest value ‘ ’ the
material removal is not done. Hence the material removal and addition is only done to the
elements those stay strictly between ‘1’ and ‘ ’. This process is continued till convergence is
obtained.
The problem is solved for 20%, 50% and 80% volume fraction. The evolutions of topologies
with iterations are shown in Figures 3.5, 3.6 and 3.7 for volume fractions 20%, 50% and 80%,
respectively. The convergence of displacement at the application of load is shown in Figures 3.8,
3.9 and 3.10 for volume fractions 20%, 50% and 80%, respectively.

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Iteration 10 Iteration 75
Figure 3.5: Material distribution for volume fraction of 20% in given domain of the cantilever
beam with iteration
beam with iteration

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beam with iteration
Figure 3.8: Variation of load point displacment for volume fraction of 20% in given domain of
the cantilever beam with iteration

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Figure 3.9: Variation of load point displacement for volume fraction of 50% in given domain of
the cantilever beam with iteration
Figure 3.10: Variation of load point displacement for volume fraction of 80% in given domain
of the cantilever beam with iteration

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Summary of Optimal Topologies for Different Volume Fractions
In summary of previous discussion, we presented the optimal topological configurations of given
design domain for the volume fractions of 10%, 20%, 50%, 80% and 90% are shown in Figure
3.11. Although 10% and 90% volume fractions were not discussed in previous section, we
presented here to compare with optimal topologies of 20%, 50% and 80% volume fractions.
Volume fraction 10% Volume fraction 20%
Volume fraction 50% Volume fraction 80% Volume fraction 90%
Figure 3.11: Optimal topologies for stiffness maximization of cantilever beam for volume
fraction of 10%, 20%, 50%, 80%, and 90%
3.2 Optimization of Stiff Structure Based on Minimizing the Mean
Compliance
Here we consider the problem of minimizing the mean compliance. The mean compliance is the
objective function which is defined by the sum of displacements with forces as weights. We also

20
imposed volume constraint while solving the problem. The formulation is presented below and
fmincon function of MATLAB (http://www.mathworks.in/) is used to minimize the objective
function.
∑ ,
,
Although the equilibrium equation is an equality constraint, we satisfy it implicitly by calculating
displacements and sensitivities in the problem formulation. We know from FE formulation that
the equilibrium equation.
,
Substituting value of ‘ ’ formulation can be written as follows:
∑ ,
∑ ,
,
Where ‘ ’ and ‘ ’ are the displacement and load vectors, respectively.
Clearly, the stiffness matrix depends on the vector ‘ ’ of the element-wise constant material
densities in the elements, numbered as . Using SIMP model, we can write
∑
Where, is the element stiﬀness matrix for element ‘ ’.

21
Here, the previous example 2, i.e., cantilever beam problem is solved for maximize the stiffness
by minimizing the mean compliance. The problem is solved for 50% and 80% volume fraction.
The optimal topological configuration of design domain for volume fraction of 50% and 80% are
shown in Figure 3.12. The convergence of objective function (mean compliance) is shown in
Figures 3.13 and 3.14 for volume fractions of 50% and 80%, respectively. Resultant topologies
are similar to stiff structure optimization solved by uniform stress criterion algorithm.
Volume fraction Optimal Topology
50%
80%
Figure 3.12: Optimal topological configuration for maximum stiffness for 50% and 80% volume
fraction of design domain

22
Figure 3.13: Evolution history of objective function (mean compliance) for 50% volume fraction
of design domain
Figure 3.14: Evolution history of objective function (mean compliance) for 80% volume fraction
of design domain

23
In conclusion in this chapter we presented the optimal topology of cantilever beam like 2-D
structure both with truss and continuum elements. Further, Optimal topologies are obtained for
maximum stiffness based on two criterions (i) uniformity of stress over domain (ii) minimizing
the mean compliance of structure. Both criterions are giving the same topology.

24
4. Topology Optimization for Structural Frequency Using
Classical Method
In this chapter the problem of frequency optimization is addressed using topology optimization.
We solve the problem of both maximization and minimization. As mentioned in Chapter 1, the
maximization of frequency is useful in structural design whereas the minimization is useful for
the energy harvesting. We now present the precise definition of frequency optimization problem.
4.1 Problem Formulation
Let be natural frequency of structure. Let be square of natural frequency. Then the
problem is defined for maximize the natural frequency by
Objective function * +
( ) ( )
∑ ( )
( )
For minimize the natural frequency, the problem is defined by
Objective function * +
( ) (4.4)

25
∑ (4.5)
( )
In above equations denotes the total number of elements in the FE discretization. The design
variable ’ ’, represents the densities of the elements. An equation 4.3 and 4.6
specifies lower and upper limits and 1 for . To avoid singularity of the stiffness and mass
matrix is taken to be a small value for example . We also note that
should be chosen such that the performance of optimal design should not have considerable
affect. In equations 4.2 and 4.5, the symbol is the given volume of the design domain and is
the elemental volume.
The numerical algorithms of optimization problem usually developed for minimization.
Therefore, any maximization problem is converted to minimization by taking objective function
with negative sign. Clearly, both minimization and maximization problems need similar type of
computation for the sensitivities. Thus, we now present calculation of sensitivities for the
frequency.
4.2 Sensitivity Calculation of Frequency
The natural frequency of the designated mode is related to the eigenvalue as shown in the
following equation:
√
( )
Taking the differentiation of frequency with respect to the design variable , we get
√
( )

26
Clearly, it is sufficient to find the differentiation of eigenvalue with respect to design
variable. We now use the state equation to obtain the sensitivities of eigenvalue. The state
equation for the natural frequencies of a linearly elastic structure is given by
, ( ) ( ) ( )- ( ) ( )
Where, is the global stiffness matrix, is the global mass matrix, is natural frequency
and is the corresponding eigenvector.
Taking the diﬀerentiation of equation 4.9 with respect to design variable and then pre-
multiplying by transpose of eigenvector , we obtain
[ ] , - ( )
Substitution of equation 4.9 in 4.10 yields
[ ] ( )
The sensitivities of eigenvalue follow from the rearrangement of above equation, i.e.
[ ] ( )
Where , we note that the non-trivial solution to equation 4.9 should be obtained in
order to get . Let mass normalization for eigenvector i.e. Then equation
4.12 written as
[ ] ( )

27
Equation 4.13 represents the sensitivity of eigenvalue with respect to design variable. The
substitution of equation 4.13 in equation 4.8 yields the following sensitivities of frequency with
respect to design variables:
√
[ ] ( )
The sensitivities calculation is adopted from Bendsøe and Sigmund (2003).
There is a numerical artifact observed in problem of maximization known as mode localization
(Neves, 1994; Pedersen, 2000). We now discuss this issue using a numerical example.
4.3 Localized Eigen Mode
The mode localization problem often occurs in topology optimization where Eigen
frequency/Eigen modes are the objective or constraints (Neves et al., 1995; Pedersen 2000).
Figure 4.1: First Eigen mode with mode localization
Mode localization means the elements that are having low density (Young’s modulus) participate
in mode whereas the elements that have high density do not. This is attributed to the elements

28
that possess low stiffness to mass ratio (Pedersen, 2000). Figure 4.1 shows the mode localization
where the dark color indicates full density whereas white indicate low density material. From the
figure it is clear that only the low density elements are participating in the mode shape.
This numerical artifact can be eliminated by modifying the interpolation function for mass in
SIMP model while not changing the interpolation for the stiffness (Pedersen, 2000). The
modified SIMP model for mass interpolation is given by
( ) { (4.15)
Figure 4.2: First Eigen mode shape after elimination of mode localization
In equation 4.15, the mass is set to very low via a high value of the penalization power in sub
regions with low material density. Thus is chosen to be about , i.e. much larger than the
penalization power for the stiffness, which is kept unchanged at a value about
Consequently, the low stiffness to mass ratio is eliminated. Figure 4.2 shows the mode shape
after applying the modified interpolation for mass density and indicate that all elements are
participating in the mode shape. Thus, the mode localization problem avoided. We give the same
treatment for later problems.

29
4.4 Topology Optimization for Maximizing Structural Frequency
As mentioned in chapter 1 that the structural design requires first natural frequency as high as
possible to avoid failure. Therefore, now the problem is to maximize the first natural frequency
of given structure. Consider a beam like 2-D structure with three sets of boundary condition: (i)
cantilever beam, i.e., clamped at one end (ii) fixed-fixed beam, i.e., clamped at both ends (iii)
simply supported beam, i.e., the mid points of left and right ends are fixed. The length and width
of domain are 8 m and 1 m, respectively. The three types of boundary conditions are shown in
Figure 4.3(a-c).
The domain is discretized into 128 X 16 four node quadrilateral elements along length and width,
respectively. Clearly, there are 2048 elements, each element has its particular density ‘ ’. As
mentioned, density of each element is a design variable to maximize the first natural frequency.
The SIMP approach is used to interpolate both the stiffness and also mass density. The material
is assumed to be isotropic with Young’s modulus 150*109
N/m2
, Poisson’s ratio 0.17 and mass
density 2330 kg/m3
.
The design domain is solved for 50% volume fraction. The initial guess for the optimization is
taken as uniformly distributed over the whole design domain. The fmincon function in
MATALB is used for the optimization.
Figure 4.3 (d-f) shows the evolution history of objective function with optimization
corresponding to the structure in Figure 4.3 (a-c). Once the change in objective function value is
less than the tolerance then the optimization process is terminated. The optimal topology is
showed in Figure 4.4 (d-f) corresponding to structure in Figure 4.3 (a-c). Figure 4.4 (a-c) shows
the first Eigen mode shape of optimal topology for all three sets of boundary conditions. We note
that different scale factors along vertical and horizontal direction for plotting the mode shape as
shown in Figure 4.4 (a-c). Although the different scale factor skews the image, we considered to
make clarity in the mode shape.

30
(a) (d)
(b)
(e)
(c) (f)
Figure 4.3: (a-c) Admissible design domains of beam like 2-D structures (d-f) Objective function
variation with iteration for three different sets of boundary conditions (a) cantilever beam (b)
fixed-fixed beam (c) simply supported beam, respectively

31
(a) (d)
(b) (e)
(c) (f)
Figure 4.4: (a-c) First Eigen mode shape corresponding to optimal topology (d-f) Optimal
topologies for three different sets of boundary conditions defined in Figure 4.3(a-c)

32
(a)
(b)
(c)
Figure 4.5: (a-c) Iteration histories of the first three natural frequencies associated with boundary
conditions (a) cantilever beam (b) fixed-fixed beam (c) simply supported beam

33
At last the Figure 4.5 (a-c) shows the evolution histories of the first three natural frequencies
associated with boundary conditions (i) cantilever beam (ii) fixed-fixed beam (iii) simply
supported beam, respectively.
It can be observed from these three problems that the second and third natural frequencies
decrease when we maximize the first natural frequency. Furthermore, the difference between
first and second frequencies becomes very small.
4.5 Topology Optimization for Minimizing Structural Frequency
Structures with minimum first natural frequency have application in energy harvesting at low
frequency vibration. Therefore, now the problem is to minimize the first natural frequency of
given structure. Consider a beam like 2-D structure with three sets of boundary condition: (i)
cantilever beam, (ii) fixed-fixed beam, (iii) simply supported beam.
The structure with three sets of boundary conditions is shown in Figure 4.6(a-c). The domain of
size 6m X 1m is discretized using four node rectangular element. We consider 8 elements in the
vertical direction and 48 elements in the horizontal direction. The material properties are taken
same as previous problem. The design domain is solved for 50% volume fraction. The initial
guess for the optimization is taken as uniformly distributed over the whole design domain. The
fmincon function in MATALB is used for the optimization.
Figure 4.6 (d-f) shows the evolution history of objective function with optimization
corresponding to the structure in Figure 4.6 (a-c). Once the change in objective function value is
less than the tolerance then the optimization process is terminated. The optimal topology is
showed in Figure 4.7 (d-f) corresponding to structure in Figure 4.6 (a-c). Figure 4.7 (a-c) shows
the first Eigen mode shape of optimal topology for all three sets of boundary conditions. We note
that different scale factors along vertical and horizontal direction for plotting the mode shape as
shown in Figure 4.7 (a-c). Although the different scale factor skews the image, we considered to
make clarity in the mode shape.

34
(a) (d)
(b) (e)
(c) (f)
Figure 4.6: (a-c) Admissible design domains of beam like 2-D structures (d-f) Objective function
variation with iteration for three different sets of boundary conditions (a) cantilever beam (b)
fixed-fixed beam (c) simply supported beam, respectively

35
(a) (d)
(b) (e)
(c) (f)
Figure 4.7: (a-c) First Eigen mode shape corresponding to optimal topology (d-f) Optimal
topologies for three different sets of boundary conditions, respectively defined in Figure 4.6(a-c).

36
(a)
(b)
(c)
Figure 4.8: (a-c) Iteration histories of the first three natural frequencies associated with boundary
conditions (a) cantilever beam (b) fixed-fixed beam (c) simply supported beam

37
At last Figure 4.8 (a-c) shows the evolution histories of the first three natural frequencies
associated with boundary conditions (i) cantilever beam (ii) fixed-fixed beam (iii) simply
supported beam, respectively.
It can be observed from three problem that all first three natural frequency decreases with
iteration. Furthermore, the difference between second and third frequencies becomes very small.
From Figure 4.7(d) (optimal topology of structure with cantilever beam boundary condition) it
can be seen that the material is distributed towards the free end. This indicates that the first
natural frequency is minimum when the fixed boundary is connected to the material with the
most flexible structure.
Similarly in Figure 4.7 (e) and (f) (optimal topology of structure with fixed-fixed beam and
simply supported beam, respectively) it can be seen that the material is distributed at the center
of design domain, i.e., away from the fixed boundary. It also keeps a little gap in between
distributed material at center in order to attain maximum flexibility. Of course, similar to
previous case, flexible structure is required to connect the lumped masses to the supports.
In conclusion the low frequency structure is related to the flexible structure with heavy mass far
from the support. In other words as the stiffness of the connecting structure between mass and
support approaches small value then the frequency also approaches small value.

38
5. Topology Optimization for Structural Frequency Using
Modified Genetic Algorithm
5.1 Introduction to Genetic Algorithm
Genetic algorithms (GAs) are search and optimization procedures that are motivated by the
principles of natural genetics and natural selection. Some fundamental ideas are borrowed
from the nature to construct a search algorithm that is robust and requires minimum problem
information.
The working principle of GA is different from the classical optimization techniques. GA has
three characteristic operators, namely selection, crossover and mutation. In each iteration, or
generation, these operators are applied on a population of possible solutions, or individuals in
order to improve their fitness. Initially, the population is created randomly, and the selection,
crossover and mutation continue until a stopping criterion is reached, e.g. the exceeding of a
certain number of generations, or the absence of further improvements among the individuals.
There are many advantages with the GA technique, primarily its simplicity and broad
applicability. It can easily be modified to work on a wide range of problems, as contrary to
traditional search methods that are specified on a certain type of problem. The technique is
relatively robust and it does not tend to get stuck in local optima. Furthermore, due to the use
of function evaluations rather than derivatives, it can handle non-derivative functions,
nonlinear functions and discrete variables and is able to work in highly complex search
spaces (Deb, 1997).
The major limitation of GA is large requirement of function evaluations as compared to the
classical optimization methods. Explicit diversity preserving mechanism is required to avoid
evolution of similar solutions in the GA population.

39
Figure 5.1: A flow chart to solve frequency optimization problem using modified GA
5.2 Modified Genetic Algorithm to Frequency Optimization Problem
GA is used for frequency optimization problem in which natural frequency of given structure
is minimized. Beam like 2-D structure or design domain with different boundary conditions
are solved and topologies are presented in this chapter. Flow chart of modified GA is shown
in Figure 5.1. In subsequent sections, GA and its modifications are described for frequency
optimization problem.
Start
Initial Population
Fitness
FE Analysis
Iteration?
Selection
Crossover
Mutation
Connectivity
Analysis
Fitness of Child
Population
FEM Analysis
Combine parent and
child population
Choose Best
(survivor)
No
Yes
Terminate

40
5.2.1 Initialization and Representation
In the traditional GA, the string consists of a fixed-length binary string. The length of binary
string depends on the number of variables used in the given optimization problem. In
traditional GA, the initial population is generated by assigning 0 or 1 randomly to each bit of
the binary string.
For frequency optimization problem, a 2-D design domain is discretized into small, square
elements, where each element represents either material or void. The state of the individual
elements defines the distribution of material or void within the domain and therefore establish
the topology.
Figure 5.2 shows distribution of material in 2-D design domain. It can be observed that a
binary string is converted into 2-D representation in which each element is either assigned
with 1 or 0 values. Therefore, the material is assigned to those elements in which 1 are
assigned as shown in Figure 5.2(b).
By following 2-D representation, an initial population of size 20 is generated randomly.
These individuals satisfy volume constraint wherein the volume of material distribution in the
design domain is less than or equal to 50% of total volume of the design domain.
11011001110110100111
(a)
1 1 0 1 1
0 0 1 1 1
0 1 1 0 1
0 0 1 1 1
(b)
Figure 5.2: A material distribution in the design domain using binary string representation.

41
5.2.2 Fitness Evaluation
To find the structural performance of topology, we consider minimization of first natural
frequency of given structure. The fitness value is evaluated for each individual (topology) is
defined as
( ) {
( )
( )
Where, ( ) is the objective function value for feasible solution calculated using finite
element methodology, is the objective function value of the worst feasible solution in
the population and ( )is the constraint violations for infeasible solutions.
5.2.3 Selection
The fitness is calculated for all individuals, good individuals are chosen via reproduction to
form a mating pool. There exist many different ways to choose individuals for the mating
pool, but the main idea is that the fitter individuals have larger probability to be chosen. The
mating pool has the same size as the population, but good individuals are more frequent due
to duplication.
The binary tournament selection operator is used for selection in this frequency optimization
problem. In the binary tournament selection operators, individuals from the populations are
chosen at randomly in a pair and fitter individual get selected in the mating pool. With a
population size of N, N tournaments are held to fill the mating pool. This way, no copy of the
worst individual is selected.
5.2.4 Apply GA Operators
In this step, objective is to generate a second generation population of solutions from those
selected in selection operator through a combination of genetic operators, that is, crossover
and mutation.
For each new solution to be produced, a pair of "parent" solutions is selected for breeding
from the mating pool. A new solution is created which typically shares many of the

42
characteristics of its "parents". This process continues until a new population of solutions of
appropriate size is generated.
These processes ultimately result in the next generation population of individuals that is
different from the initial generation. Generally the average fitness of the population increase
by this procedure since only the better solutions from the last generation are selected for
breeding.
There exist different crossover operators like single point crossover, two point crossover,
operator but the main idea is that two random individuals from the mating pool are chosen as
parents, and some portion of their strings are switched to create two children.
A 2-D crossover is adopted for frequency optimization problem which is discussed in the
following section. Two individuals (topologies) are selected randomly and the crossover site
is selected as follows.
The rectangular box is selected randomly with size limit (m-1) in horizontal direction and (n-
1) in vertical direction. Where m and n are the no of elements in horizontal and vertical
direction’s respectively of discretized design domain. Figure 5.3 shows the topology before
and after crossover operator for a random size of rectangular box.
After performing crossover operator, now mutation operator is applied on the population. In
mutation, there is always a small probability for each bit in the string to change from 0 to 1 or
vice versa. If so, the child is mutated:
0000000000000 001000000000
Mutation
The purpose of this feature is to maintain the diversity amongst the individuals and to prevent
the algorithm from getting stuck in a local minimum (Man, 1996). The mutation probability
is kept low.
For frequency optimization problem, mutation operator is applied with probability of 0.01.
For all elements, the random number is generated between 0 and 1. If the random number is

43
less than or equal to 0.01, then the bit of the element is mutated. This process repeated for all
the elements of the given structure. As shown in Figure 5.4, color element is mutated.
1 1 0 0 1 1
0 1 1 0 1 0
0 0 1 1 1 0
1 1 1 1 0 0
1 0 1 1 1 1
1 0 0 1 0 1
1 0 1 1 1 1
1 0 1 1 0 0
1 1 1 1 1 1
0 0 1 0 1 0
1 1 1 1 1 1
1 0 1 0 1 0
1 1 0 0 1 1
0 1 1 1 1 0
0 0 1 1 1 0
1 1 1 0 0 0
1 0 1 1 1 1
1 0 0 1 0 1
1 0 1 1 1 1
1 0 1 0 0 0
1 1 1 1 1 1
0 0 1 1 1 0
1 1 1 1 1 1
1 0 1 0 1 0
Fig 5.3: Topologies before and after application of crossover operator
1 0 0 0
1 1 1 1
0 1 1 1
0 1 0 1
1 0 0 0
1 1 0 1
0 1 1 1
0 1 0 1
Figure 5.4: Topology before and after application of mutation operator

44
5.2.5 Connectivity Analysis
Once the child population is produced from parent population by applying selection,
crossover and mutation operators the connectivity analysis is applied to make our structure
feasible and practical. Two types of connectivity analysis are performed during optimization.
1. Floating point analysis
In this analysis the material element is considered as floating material element if that material
element does not share any edge and corner with any other material element in design
domain.
The floating point analysis sets to void all material elements in the topology which are
floating material elements. Figures 5.5 (a) and (b) show the topology before and after
implementation of floating point analysis.
(a) (b) (c)
Figure 5.5: Topology (a) before connectivity analysis (b) after applying floating point
analysis and (c) after applying point connectivity analysis
2. Point connectivity analysis
In this analysis material element is considered as point connected if that material element
shares only edge in the whole design domain with other material elements.

45
The point connectivity analysis sets to void point connected material element by transferring
this material to neighbor void element that share an edge with point connected material
element in topology. There are many edge connected void elements to point connected
material element. So that location at which this point connected elements material is
transferred is based on objective function value. Figure 5.5 (b) and (c) shows the topology
before and after implementation of point connectivity analysis.
5.2.6 Elitism
Elitism means that the good individuals are copied into the new generation population. In
frequency optimization problem, after performing connecting analysis the child solutions are
obtained. The fitness is evaluated for child population using FE analysis. Both parents and
child individuals are combined and out of total 40 individuals the best 20 individuals are
selected based on fitness value for next generation. Up to elitism the one generation of GA is
over.
5.2.7 Termination
The modified GA gets terminated when the termination condition is met. The termination
condition is set to 300 generations.
5.3 Case Studies
In this section beam like 2-D structure with different boundary conditions is optimized using
modified GA. The objective function is to minimize the first natural frequency of the
structure.
To calculate the structural performance of each topology (individuals) the design domain is
discretized with 16 x 8 no of four node quadratic elements in horizontal and vertical direction
respectively. Element’s corresponding to solid is assigned a high value of Young’s modulus
of order 150*109
N/m2
and mass density 2330 Kg/m3
. The void elements are assigned to a
very low Young’s Modulus of order 10-2
and mass density of order 10-1
.

46
The admissible design domain of beam like 2-D structures with three sets of boundary
condition: (i) cantilever beam, i.e., clamped at one end (ii) fixed-fixed beam, i.e., clamped at
both ends (iii) simply supported beam, i.e., the mid points of left and right ends are fixed is
shown in Figure 5.6 (a-c).
The given admissible design domain or structure is solved for volume fraction of equal to or
less than 50% of design domain volume. Figure 5.6 (d-f) shows the random distribution of
volume equal to or less than 50% of design domain. The white section represents that there is
void and black region signifies material. The modified GA is applied to obtain optimal
topology to minimize the first natural frequency of these structures.
The change in topological configuration with iteration can be seen in Figures 5.7 (a), (b) and
(c) corresponding to initial design domain in Figures 5.6 (a), (b) and (c), the changes indicate
that along with iteration the material gets distributed away from the support. It is observed
that the topology does not change after 300 iterations, the algorithm is terminated and the
optimal topology configuration is obtained as shown in Figures 5.7 (d), (e) and (f)
corresponding to initial design domain in Figures 5.6 (a), (b) and (c).
It can be seen in Figure 5.7(d) (optimal topology of 2-D beam like structure with cantilever
beam boundary condition) that the material is distributed on the other side of the fixed
boundary. This indicates that the first natural frequency is minimum when the fixed boundary
is connected to the material with the most flexible link.
Similarly in Figure 5.7 (e) and (f) (optimal topology of 2-D like beam structure with fixed-
fixed ends simply supported ends respectively) it can be seen that the material is distributed
at the center of admissible design domain which is away from the fixed boundary.

47
(a) (d)
(b) (e)
(c) (f)
Figure 5.6: (a-c) Admissible design domain of beam like 2-D structures with three sets of
boundary condition (i) cantilever beam (ii) fixed-fixed beam (iii) simply supported beam, (d-
f) Initial topology with randomly distributed material in given design domain corresponding
to (a-c)

48
(a) (d)
(b) (e)
(c) (f)
Figure 5.7: (a-c) Topology configuration at 25th
iteration , (d-f) Optimal topology configuration
corresponding to minimum first natural frequency of structure in figure 5.6 (a-c) for 50% of
volume fraction

49
In this chapter, the modified GA is applied to solve frequency optimization problem and case
study i.e. a beam like 2-D structure with three sets of boundary conditions (i) cantilever beam
(ii) fixed-fixed beam (iii) simply supported beam is solved for minimizing the first natural
frequency. The optimal topology configuration for first set of boundary condition indicates
that the material is distributed on the other side of the fixed boundary. This concludes that to
minimize the first natural frequency we need to provide as much as flexible connection
between boundary and material in given admissible design domain. For second and third sets
of boundary condition, the optimal topology configuration indicates that the material
distributed at center of admissible design domain should be connected to clamped boundary
with the most flexible link to get minimum first natural frequency.

50
6. Intuitive Designs of the Low Frequency Structures
As pointed out in previous chapters, a flexible structure is requires to connect the heavy mass
and support. Of course, heavy mass is far from the support. Furthermore, all boundary condition
indicates the same result i.e., the connecting structure should be as flexible as possible.
Therefore, now our task is to come up with a design based on the observation.
We know that the cantilever beam is flexible among the cases that we dealt previously.
Therefore, we present few designs based on the observation on optimal topology of cantilever
beam. In all the designs we consider the solid mass occupies 50% of the domain towards the free
end and flexible structure occupies 50% of the domain towards the fixed end. We also present
the performance of cantilever beam in order to compare the result of our designs.
Example 1: Cantilever Beam
Here, we consider domain of size 200 mm and 20 mm along length and width respectively (see
Figure 6.1 (a)). The out of plane thickness is taken as one mm. The Young’s modulus, Poisson’s
ration and density of material are taken as 150 kPa, 0.17 and 2330 kg/mm3
. This problem has
been solved using ANSYS finite element analysis (FEA) package using 4 node quadrilateral
element assuming plane stress condition. We observed the first natural frequency as 20.354 Hz
and the mode shape is shown in Figure 6.1 (b). Later design will be compared with this example.

51
(a)
(b)
Figure 6.1(a) Structure with cantilever beam boundary condition (b) First mode shape
corresponding to natural frequency 20.354 Hz
In all the following examples the domain is considered as 200 mm and 20 mm along length and
width, similar to the cantilever beam. The only difference is the material distribution or design of
structure that connects mass and support. As mentioned, the 50% of domain occupies the solid
mass towards the free end.

52
Example 2:
Here, we consider thin beam like structures connected to solid material on the top and bottom as
shown in Figure 6.2(a). The each beam consists the width 2 mm. We found the frequency
corresponding to first natural mode is 3.364 Hz. The mode shape is shown in Figure 6.2(b).
(a)
(b)
Figure 6.2: (a) Flexible structure with two beam in parallel (b) First mode shape corresponding to
natural frequency 3.364 Hz
Example 3:
As one can observe two beams are parallel in previous example. We now improved the
flexibility of connecting structure by introducing folded beam like structure as shown in Figure
6.3(a). The folded beam can be thought of three beam are in series. Thus, improves the

53
flexibility. The first frequency of the structure is observed to be 0.416 Hz whereas the mode
shape is presented in Figure 6.3(b). In this example also beam thickness in folded beam is 2 mm.
We also provided 7 mm gap between beams and 10 mm gap between heavy mass and right
extreme of beam.
(a)
(b)
Figure 6.3: (a) Folded beam with three beams in series (b) First mode shape corresponding to
Example 4:
It clears that the no of folds in the folded beam increases then the frequency goes down. This fact
is verified in this example by taking five beams in series as shown in Figure 6.4 (a). Here also we

54
consider 2mm thick beam in supporting structure. As the domain size is constant we decrease the
gap between beams to 2.5 mm and other parameter remains same. The frequency is observed to
be 0.3333 Hz whereas mode shape is shown in Figure 6.4(b). Now the problem comes with the
manufacturing constraints and contact. The problem of contact is not taken care in linear
analysis. While increasing the beam in supporting structure one should take care of this
constraint.
(a)
(b)
Figure 6.4: (a) Folded beam with five beams in series (b) First mode shape corresponding to

55
Example 5:
Previous two examples were not symmetric structures. Here we provided a symmetric structure
that is flexible than the first example. Three folded beams structure is attached in parallel in
order to get symmetric and as well as flexible structure (see Figure 6.5(a)). Here, we decrease the
gap between beams to 1.6 mm and other parameter remains same. The frequency is observed to
be 1.234 Hz whereas mode shape is shown in Figure 6.4(b).
(a)
(b)
Figure 6.5: (a) Structure with three folded beams in parallel (b) First mode shape corresponding
to natural frequency 1.234 Hz

56
From above examples we can attain the stiffness as low as possible by decreasing the beam
thickness in supporting structure and increasing the number of beams in series. However
manufacturing constraints will not allow vary low thickness beam. Furthermore, contact also
becomes a problem while designing these flexible structures. Therefore, designer should take
care of these factors while designing the low frequency structures.

57
7. Conclusions and Scope of Future Work
In this chapter, we summarize the thesis and highlight the contribution of the thesis.
7.1 Summary of the Thesis
We addressed two problems in this thesis: (i) optimal design of stiff structure (ii)
optimization of structural frequency. The problems are solved in a common frame work of
topology optimization.
7.1.1 Optimization of Stiff Structure
We addressed the optimization of stiff structure in two ways (i) based on uniform stress
criterion (ii) based on minimization of mean compliance. This problem is very well addressed
in the literature and we have implemented in thesis. The results in both approaches shows that
they are equivalent i.e., topologies obtain in both approach are same. We also note that this
problem is known to be convex and hence final results are independent of initial guess.
Therefore, the final topologies whatever we obtained were the optimal topologies for those
problems.
7.1.2 Optimization of Structural Frequency
We addressed two structural frequency optimization problems: (i) maximization of structural
frequency (ii) minimization of structural frequency. The maximization problem is well
addressed in the literature and we also implemented and tested few results. In literature a
numerical artifact called mode localization and their remedies is presented in the context of
maximization of frequency. The remedies of this problem is implemented in our program and
presented few test cases. We note that the maximization problem is important in structural
design. We present the minimization problem that is useful in design of energy harvester.

58
This problem has not been addressed in the literature. We observed that the mode localization
do not affect the minimization problem unlike in the case of minimization problem. We
solved the minimization problem using both classical optimization and genetic algorithms.
The results are found to be similar. We came up with some intuitive design based on the
optimal topologies that were obtained from the optimization.
7.2 Contribution of the Thesis
 The stiff structure optimization has been implemented using both uniform stress
criterion and mean compliance.
 The results obtained from both methods are shown to be equal through numerical
examples.
 The maximization of frequency has been implemented by accounting the removal of
mode localization.
 An effective example was presented to show the effect of mode localization.
 Few examples were presented for maximization of frequency and they agree with
examples in literature.
 Minimization of the frequency problem is addressed in this thesis.
 For minimization of frequency three examples have been presented.
 Few intuitive designs of low frequency structures are presented these were designed
based on optimal topologies.
7.3 Scope of Future Work
We state future work in the minimization of the frequency as it is the main contribution. The
beam thickness in supporting structures (see previous chapter) were limited by manufacturing
processes. Therefore, it is important to account manufacturing constraints while designing
low frequency structures. One should also take care of contact as stiffness of supporting
structure is very less. These two issues could be addressed along with experiments in future
work.

59
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[12] Michell, A.G.M., “The limits of economy in frame structures”, Philosophical
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