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MODEL ORDER REDUCTION FOR PLANE ELASTICITY USING

EQUIVALENT MATERIAL DISTRIBUTION

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Michael K. Penner

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MODEL ORDER REDUCTION FOR PLANE ELASTICITY USING EQUIVALENT
MATERIAL DISTRIBUTION

BY

Michael K. Penner

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Mechanical Engineering

2002

ABSTRACT

MODEL ORDER REDUCTION FOR PLANE ELASTICIT Y USING EQUIVALENT
MATERIAL DISTRIBUTION

By

Michael K. Penner

A model order reduction technique is presented. This technique uses a multi-resolution
analysis on a non-homogenous material distribution with fine scale features to construct a
wavelet based reduced stiffness matrix. This reduced stiffness matrix is much smaller in
size than fine scale stiffness matrix. A topology optimization technique is implemented
to find a coarse, non-homogenous material distribution that has equivalent features to the
reduced stiffness matrix. Results are presented for three different types of problems
exhibiting different scales. The results show that fine scale materials can be represented
by a coarse scale material distribution while keeping the elastic characteristics of the two

systems approximately equal.

To my family and fiancée

iii

ACKNOWLEDGEMENT

I would like to thank my advisor Prof. Alejandro Diaz for his guidance and patience,
along with my committee, Professors Andre Benard and Farhang Pourboghrat. I would
also like to thank Sudarsanam Chellappa for his support and direction in solving the

problem.

iv

TABLE OF CONTENTS

LIST OF FIGURES vi
Chapter 1

INTRODUCTION ........................................................................... 1

Problem Statement ............................................................................ 1
Chapter 2

THE REDUCTION PROBLEM ........................................................... 7

Past Techniques ............................................................................... 7

Wavelet Stiffness Matrix ................................................................... 12

Wavelet Stiffness Matrix ................................................................... 12
Chapter 3

THE EQUIVALENT MATERIAL PROBLEM .......................................... 16

Equivalent Material ........................................................................... 16

Genetic Algorithm (GA) ..................................................................... 19
Chapter 4

EXAMPLES .................................................................................. 26

Scale 1 Solutions ............................................................................. 31

Scale 2 Solutions ............................................................................. 40

Scale 3 Solutions ............................................................................. 49
Chapter 5

CONCLUSIONS ............................................................................. 57
Reference ....................................................................................... 59

LIST OF FIGURES

Figure 1.1 Unit square with non-homogenous material .................................... 2
Figure 1.2 Flow Chart of Numerical Process ................................................ 5
Figure 2.1 Example of wavelet decomposition .............................................. 14

Figure 3.1 Distribution of p(x) computed using the 1St technique to create initial 22
population.

Figure 3.2 Distribution of p(x) computed using the 2"d technique to create initial 23
population

Figure 3.3 Distribution of p(x) computed using the 3rd technique to create initial 24
population

Figure 4.1 1A of material design ............................................................... 27
Figure 4.2 Symmetric material design ........................................................ 27
Figure 4.3 Geometry layouts for fine scale materials -Scale 1 ............................ 28
Figure 4.4 Geometry layouts for fine scale materials —Scale 2 ............................ 29
Figure 4.5 Geometry layouts for fine scale materials -Scale 3 ............................ 30
Figure 4.6 Fine scale material distribution ................................................... 32
Figure 4.7 Coarse scale material solution .................................................... 32
Figure 4.8 9th Mode shape from the target stiffness matrix ................................ 34
Figure 4.9 l3th Mode shape for the coarse material distribution .......................... 34
Figure 4.10 10th Mode shape from the target stiffness matrix ............................. 35
Figure 4.11 14"1 Mode shape from the target stiffness matrix ............................. 35
Figure 4.12 Fine scale material distribution .................................................. 36
Figure 4.13 Coarse scale material distribution ............................................... 36

vi

Figure 4.14 5th Mode shape from the target stiffness matrix .............................. 38

Figure 4.15 9th Mode shape for the coarse material distribution .......................... 38
Figure 4.16 6h Mode shape from the target stiffness matrix .............................. 39
Figure 4.17 10th Mode shape for the coarse material distribution ........................ 39
Figure 4.18 Fine scale material distribution .................................................. 40
Figure 4.19 Coarse scale material solution ................................................... 40
Figure 4.20 3rd Mode shape from the target stiffness matrix ............................... 42
Figure 4.21 4th Mode shape for the coarse material distribution .......................... 42
Figure 4.22 10h Mode shape from the target stiffness matrix ............................. 43
Figure 4.23 15th Mode shape for the coarse material distribution ........................ 43
Figure 4.24 12th Mode shape from the target stiffness matrix ............................. 44
Figure 4.25 21St Mode shape for the coarse material distribution ......................... 44
Figure 4.26 Fine scale material distribution .................................................. 46
Figure 4.27 Coarse scale material distribution solution .................................... 46
Figure 4.28 3rd Mode shape from the target stiffness matrix .............................. 47
Figure 4.29 4th Mode shape for the coarse material distribution .......................... 47
Figure 4.30 5lh Mode shape from the target stiffness matrix .............................. 48
Figure 4.31 5th Mode shape for the coarse material distribution .......................... 48
Figure 4.32 Fine scale material distribution ......................................... I ......... 49
Figure 4.33 Coarse scale material distribution solution .................................... 49
Figure 4.34 4‘h Mode shape from the target stiffness matrix .............................. 50
Figure 4.35 3rd Mode shape for the coarse material distribution .......................... 50
Figure 4.36 7th Mode shape from the target stiffness matrix .............................. 51

vii

Figure 4.37 7th Mode shape for the coarse material distribution .......................... 51

Figure 4.38 9th Mode shape from the target stiffness matrix .............................. 52
Figure 4.39 9‘h Mode shape for the coarse material distribution .......................... 52
Figure 4.40 Fine scale material distribution .................................................. 53
Figure 4.41 Coarse scale material solution ................................................... 53
Figure 4.42 3rd Mode shape from the target stiffness matrix .............................. 54
Figure 4.43 4‘h Mode shape for the coarse material distribution .......................... 54
Figure 4.44 5th Mode shape from the target stiffness matrix .............................. 55
Figure 4.45 7th Mode shape for the coarse material distribution .......................... 55

viii

Chapter 1

INTRODUCTION

With the need to increase the performance of a product, decrease design time and cost,
the use of computer aided engineering techniques such as finite element methods are
becoming ever more important. While this technique can be very effective on basic
geometries and materials, its application is spreading to include more complex designs
and materials including composites and foam like materials. This is causing the finite
element model to become very complex and also increasing the computation time to
solve these problems. This can have a negative effect on the design process, leaving some
ideas unexplored due to the high computation time required. Also with the use of
optimization programs that require the finite element computation to be done once per
iteration, the time it takes to solve the problem gets multiplied. The goal of this thesis is
to help develop model order reduction techniques that will decrease the complexity and
size of the model and decrease the computation time while still keeping a high degree of

accuracy of the solution.

1.1 Problem Statement
The reduction process discussed in this thesis begins with a plane elasticity problem on a

unit square created with non-homogenous material as shown in Figure 1.1. In order to

resolve the details of the material distribution over the domain a fine scale resolution is

needed.

 

Figure 1.1 Unit square with non—homogenous material

The boundary conditions are periodic. Upon discretization and using wavelet Galerkin

methods this problem can be expressed as

Kyu?’ = FW (1)
where

K }v is the fine scale wavelet stiffness matrix
uyis the fine scale wavelet coefficients

F?) is the fine scale force wavelet coefficients

Equation (1) represents the fine scale problem. Since this process is based on a wavelet
discretization, the degrees of freedom associated with the problem are wavelet

coefficients.

Next, a reduction scheme based on a multi-resolution analysis (MRA) is applied to the

fine scale problem. This procedure starts with the (very large) fine scale stiffness matrix

(K W ) and creates a (much smaller) coarse scale stiffness matrix (K W ). The boundary
c

f

conditions remain periodic. The reduced system is
may = Few (2)
where

K 2.” is the coarse scale wavelet stiffness matrix
up is the coarse scale wavelet coefficients

Few is the coarse scale force wavelet coefficients
A transformation is now applied to the reduced wavelet stiffness matrix (K (W ) to
transform the wavelet degree of freedoms into nodal degrees of freedom. This procedure

is just a coordinate transform. The result is a coarse scale stiffness matrix (K 3) where

the degree of freedoms are now nodal displacements, not wavelet coefficients. Periodic

boundary conditions still remain in force. This new reduced system is
n n n
K c “c = Fe (3)

where

K 2 is the coarse scale nodal stiffness matrix
u :is the coarse scale nodal degree of freedom vector

F6" is the coarse scale nodal degree of freedom vector

It should be noticed that matrix K g , while it relates nodal degrees of freedom to nodal

forces, it is not a finite element matrix.

The work in this thesis is to setup and solve an optimization problem to identify a coarse

scale material distribution, E(x), on the unit square domain, such that the difference

K.(E<x»—K:°"“’|| (4)

 

 

is minimized, where K C (E (x)) is a standard finite element stiffness matrix. This

problem is solved using a genetic algorithm, assuming that the material is isotropic and
piecewise constant, i.e., E (x) = p(x)l§0 where p(x) e [0,1j is a piecewise constant

function and E 0 is a fixed material tensor.

This process can be seen in the flow diagram below.

Linear Elasticity
Wavelet
Discretized

 
   
    

Stiffness Matrix

K y for fine

material

Fine Scale Material
Distribution 64x64

 

    

‘ E

 

Multi-Resolution Scheme

 

 
 
      
  

       
 

  

Stiffness Matrix

tiffness Matrix
S K 3““ for coarse

K 25V for coarse material

 

material

Optimization using
genetic algorithm

 
     
   

Km)

Coarse Material
p(x)

 

1500 = p(x)E°

Figure 1.2 Flow Chart of Numerical Process

This thesis is divided into five chapters. The next chapter discusses in detail previous
techniques of model order reduction and the technique used to create the reduced
stiffness matrix used here. Chapter three talks about the equivalent material problem.
Chapter four shows the numerical solutions and the methods to Obtain these answers.

The last chapter has the conclusions of this work.

Chapter 2
THE REDUCTION PROBLEM

2.1 Past Techniques
In this section common modeling order reduction techniques will be discussed, also the

multi—resolution scheme used in this thesis will be shown.

(1) Static reduction. This is one of the simplest reduction methods. This process will
neglect the inertia terms. This will only produce an exact result at a static condition or
zero frequency. This can be seen in the following example taken from [3]. Start with
the following linear system

{F}= lKllx} ‘2'”
where
K is the stiffness matrix
F is a force vector

x is the nodal displacement vector

F1 KA B xm A
= 2.2
{F2} [3T CH x3} ( )

Assuming the F2 to be zeros, solving for F] produces

If [K] is symmetric

F1 = (A - BC_lB')xm (2.3)

with the reduced stiffness matrix being

K, = A- 3043' (2.4)

This creates a coordinate transform, which is equal to

I
= 7
T [_ C-lB'] (-5)

which will yield the following reduced stiffness and mass matrices

K1=T.KT

(2.6)
M, = T'MT

If M is partitioned as

[M]= [; 2:] (2.7)

the reduced mass matrix is

M, = Z — EC"B'—(C"B')(§'—C—‘C"B') (2.3)

(2) Dynamic reduction. This is a modified version of the static reduction. This will
produce an exact solution or response at a certain frequency. Selecting the correct
frequency for the response is not apparent. The following process taken from Paz [5]

outlines the procedure. Start with the eigenvalue problem in separated form

lxssl-wzwssl lxspl-leMspl {x3} _ {o}
[K,,]-w2[M,,] [K,,J-..2[M,,] lrxPiHioil ‘2'”)

where K is the stiffness matrix, M is the mass matrix, {X s} is the displacement or
eigenvector corresponding to the 5 degree of freedoms to be reduced, {Xp} are the

. . 2. . .
corresponding p degree of freedoms to remain, and a) rs the approximate eigenvalue at

each step. The first step is to assign a reasonable value to (02 for the first eigenvalue.

Then eliminate the first “5” displacements followed by elementary operations, at this

point, equation (2.9) can be shown as

113% [iélli’iiilifiil

with [T] as the transformation matrix that is defined in the following equation

{X}=ll:;il=llfi}w

and [D] is the dynamic matrix that satisfies the equation

[1?]=[5]+w2[1Ti] (2.12)
[1?] is the reduced stiffness matrix and [A7] is the reduced mass matrix which is found by
the following equation

[117 l = [TI’ [M IT] (2.13)

with [T] represented as

[T] = [[511] (2.14)

The transformation [7"] can be solved for, to obtain

[T]=[‘(Kss 'ngss)_l(Ksm “wonsm)] 0-5)

These expressions will lead to the equation
[IE]-w2[AT]J{XP}={0} (2.16)
Solving this equation will lead to an almost exact value of the first natural frequency and

eigenvector. The second eigenvalue will be a close approximation. By inserting this

close approximation value into the first equation and repeating the steps above, the output

will be a near exact solution for the second eigenvalue. This process can be repeated to

produce near exact solutions for the eigenvalues and eigenvectors for the lowest p values

[5].

(3) Improved reduction system. This technique builds off the static condition by
including the inertia terms as psuedo static forces. This can be seen in the following
example taken from [6] and [7]. Start with the equations of motion assembled in the

following manner similar to the static condition

1““ “ii l‘m "mll‘ml l’ml
.. + = (2.17)
Msm M55 x: K3"! KS: x5 0

This expresses the mass [M] and stiffness [K] matrices ordered in terms of master and
slave degrees of freedom. The master (m) degrees of freedom represent the retained
values while the slave (3) degrees of freedoms represent the discarded values. Also
assume that there are no forces applied to the slave degrees of freedom.

Neglecting inertia terms in the second equation set

x’" I T (2 18)
= x = x .
X5 __ K511 KS," m S m

T: represents the static transformation from the full state vector to that of a master
coordinates. The reduced mass and stiffness matrices can be shown as

M R = TSTMTS

(2.19)
KR = TSTKTs

This is where the improved reduction system follows a different path than that of the

static reduction. This is done by including the inertia terms as pseudo-static forces in the

10

transformation from the static case. This will allow the reduced model to represent the

full model in low frequency responses. The improved reduction system starts with

Tm, = T, + SMT,M,‘,7‘K,, (2.20)
where
s — O O (2 21)
- 0 Kg} '

The reduced mass and stiffness matrices are

M IRS = TIRSMTIRS (222)
K IRS = Tarts KT IRS

(4) System equivalent reduction expansion process (SEREP). The SEREP process
utilizes eigenvectors to help deicide which nodes are kept. This method will produce
exact answers for lower natural frequencies [7]. The next few steps will outline the

procedure required to the reduction technique. First define the transformation noting that

the numbers of master degrees of freedom are more than that of the modes of interest.

<1) 1

T = [¢m][<r>;¢m]' ch; (2.23)
5

Chm and (I), are the modes of interest at the master and slave degrees of freedom.

Allowing the number of the numbers of master degrees of freedom to equal that of the
modes of interest the equation above can be simplified to result the following

transformation equation

(I) _
T = [ m ]¢ m1 (2.24)

11

Substituting equation (2.31) into IRS equation and applying to the transformation to <1)", ,

1 o 0 cm om
T<I>m = _, om + _, = (2.25)
‘ Kss Ksm Kss I (1’3 (1’3

All these methods have one goal: to be able reduce the finite element model and keep the

gives

results accurate. By reducing the model, the computation time is decreased allowing for
quicker results. Some of these techniques have been developed for specific set of
problems while others have been developed for use on a broad spectrum of problems.
The next section will discuss the techniques used in part by this thesis to create the

reduced stiffness matrix.
2.2 Wavelet Stiffness Matrix

Let Q be a domain occupied by a linearly elastic material, a square of size 2J x 2J with J>0
as a fixed integer that represents the level of discretization. Q is occupied by two

different materials. Let p(x) represent a piecewise constant function describing the

material distribution within S2. Let the material distribution within Q be of the form
E(x) = p(x) - E0 (2.26)

where

12

 

 

 

E E ‘n ‘
E1111 E1122 0 1111 l- V 2 H" l - V 2
E0 = E1122 E2222 0 E 2222 = 1 By 2 E1212 = ETC—"1%,;
0 0 51212 _ -
(2.27)

where E is the modulus of elasticity, v is the Poisson’s ratio and p6 [0,1] is piecewise

constant over the pixels [i,i+1]x[j,j+1].

Upon discretization and applying wavelet Galerkin techniques, the equilibrium equations

become

K F” (2.28)

w w _
fuf ‘
where

K ;’ is the (fine scale) wavelet stiffness matrix at level J

a?) are the (fine scale) vector of displacement wavelet coefficients

F; are the (fine scale) vector of force wavelet coefficients

Here we will describe a multi-resolution process that was developed by Diaz and
Chellappa [4]. Readers are encouraged to read this and other papers in the reference for
more details and insights into the multi-resolution process proposed here. Start with

equation (2.28):

KJuJ = F] (2.29)

13

where the new notation emphasizes the scale , i.e. operator K J is the stiffness matrix at

level J. u] is the displacement coefficients at level J and F J is the force coefficients at
level J. In two-dimensional elasticity 14’ is a vector of size (2*22").

Using a wavelet transformation W we decompose the displacement (signal) 14] into a

coarse component at scale J -l (u!-1 ) and a orthogonal complement of details at scale J -

w :uJ—-) uJ‘l ewJ‘l (2.30)

An example of this transformation can be seen in figure 2.1.

 

 

 

 

 

 

 

 

 

1 t = sin(2'pl‘x)‘sin(4'pl’x) 1 Coarse Components
0.5
0 .
0 5 L'
'10 0.5 1 -10 0.5 i
1 { Detail Components
0.5 l
0 NWWTLMMWW
-O.5 .
'10 0:5 1

Figure 2.1 Example of wavelet decomposition

Now equation (2.29) can be written as
KJ—l BT uJ—l _ fJ—l
J—l “- J—l (2'31)
B C W g

14

f J" and g1"1 are the coarse scale and detail components of the force. One should note

that equation (2.31) is similar to that of the static reduction (equation (2.2)) shown at the
beginning of the chapter. The only difference is that the master and slave degree of
freedoms shown in equation (2.31) are decided here by separating the detail and coarse

scales of the solution. Solving for the coarse scale problem will yield
1?] “lu’ ‘1 = F"1 (2.32)
where 1?] —l is

K!"1 = K‘]_1 — BTC‘IB (2.33)

This process can be repeated to give the operator I—{J-l at any reduced scale.

15

Chapter 3

THE EQUIVALENT MATERIAL PROBLEM

3.1 Equivalent Material

Let p(x) represent a piecewise constant function describing the material distribution
within the domain 52 where p(x) 6 [0,1]. The material tensor for each element in $2 is

defined as
_ 0
E(x) - .0005: (11)

where E 0 is a fixed tensor of elastic properties. The finite element stiffness matrix (K) is

created using the material distribution p(x) , i.e.

K(p<x))u = f (3.2)
where
K(p(x)) = the finite element stiffness matrix
u = the finite element nodal displacement vector

f = the finite element force vector

It should be noted that the material distribution within 82 will vary from element to
element. The stiffness matrix in equation (3.2) is called here the finite element stiffness

matrix.

Starting with equation (2.32) from the last chapter

Ku = F (3.3)

16

This (I? ) is a reduced matrix at scale J-l. if relates wavelet (coefficients) degrees of

freedom to wavelet (coefficient) forces. We now set out to find an equivalent finite

element matrix. To achieve this, first we must convert K to a matrix that relates nodal

displacements to nodal forces. Such matrix, K n is of the form

K" = B(r)T 31430) (3.4)

where

r = {r1, r2 ,. . .,rn }is a coarse scale material distribution

The stiffness matrix K n from equation (3.4) is called here the target stiffness matrix.

This matrix is the same dimension as the reduced wavelet matrix from equation (2.33).

The matrix (B) is such that if uwis a vector of wavelet coefficients, which results in a
strain field E(u"), B uwis a vector of nodal forces associated with this pre-strain, ie, B is

a (global) strain-displacement matrix.

The effective density for the fine scale and coarse scale problem can be defined as
1 ”°
efid =; 2216:.) (3.5)
1

where Nc is the number of elements

Here the effective density for the fine scale and coarse scale problems are called effdrm.B

and effdcoam respectively.

17

The objective of the equivalent material problem is to minimize the difference between
the topology finite element stiffness matrix (K(p)) and the target stiffness matrix. This

can be set up so that the energies of the two systems can be compared.

Find: pc={p., p2,... on}
Minimize: f(p)
SUbjCCI IO: xlowcr bound S p: S xuppcrbound

Cffdfinc z effdcoarse

The objective function is defined as

f(p) = Z (<1>,.TK"<1>, - <1>,TK (p)<1>, )2 + ”A — diag(A)||Wm
where

A =‘I’T(K" -K (10)}?

‘1’ = matrix of size (m, 2 * 22") with the iths eigenvectors
(D, = is the ith eigenvector of K "

m = dimension (1)
This problem is solved using a genetic algorithm. This method of solving was chosen
over other gradient-based methods, this was because the gradient-based methods were too
dependent on the staring guess. This means that the solutions from the gradient-based
methods were finding local minimums instead of global minimums. The genetic

algorithm was setup to maximize the fitness or the inverse of the objective function.

3.2 Genetic Algorithm (GA)

18

dependent on the staring guess. This means that the solutions from the gradient-based
methods were finding local minimums instead of global minimums. The genetic

algorithm was setup to maximize the fitness or the inverse of the objective function.

3.2 Genetic Algorithm (GA)
The genetic algorithms start with an initial population and employ the idea that only the
fittest members of the population will reproduce and make it to the next generation. The
evaluation or fitness of each member of the population is based on a function that is
created and is specific to the problem. The following summarized outline of a GA
program is shown below to illustrate the ideas mentioned here [13].
generate initial population, G(O)
evaluate G(0)
t =0
repeat
t=t + I
generate G( I) from G(t—I)
evaluate G( t)
until solution is found
This process is repeated until a solution that is fit enough is found.
There are six major components that make up a genetic algorithm [14]. They are
chromosome representation, selection function, genetic operators making up the
reproduction function, the creation of the initial population, termination criteria and
finally the evaluation function. They will be listed here and discussed in detail.
1. Chromosome Representation

This how each individual member of the population is represented and this will

determine how the GA is setup. This representation could be in many formats

19

including binary digits, floating-point numbers, integers, symbols, matrices, etc.
Much work has been done [Michalewicz 1994] comparing the performance
between different representations. In Michalewicz [Michalewicz 1994] it is
shown that floating-point numbers between the lower and upper bounds give

quicker and better results. This is the technique that was used in this research.

2. Selection Function

This will determine which and how many individuals contribute to the successive
generations. Based on performance a probabilistic selection is done with the
better-fit individuals having a better chance to get selected. The method used
here is a ranking selection function based on the normalized geometric
distribution. Ranking methods only need to map the solutions of a partially

ordered set. The normalized geometric ranking methods can be seen as

P[selecting the ith individual ]= q'(1 - q)'-1

with

q = the probability of selecting the best individual
r = the rank of the individual, where l is the best

P = the population size

1 q

q=l_(1_q)P

3. Genetic Operators

20

There are many crossover techniques used in GA’s these include simple
crossover, arithmetic crossover and heuristic crossover. The technique
implemented in this thesis was an arithmetic crossover and will be explained
below.

The arithmetic crossover produces two complimentary linear combinations of the

“parents”.

Mutation techniques can take on many forms. The technique used in this research

was non-uniform mutation.

4. Creation of the Initial Population

In most applications the initial population is created from a random set of values
with in the bounds of the problem. Once this is done the initial population gets
evaluated and the fitness of each member is then used to start the GA. Many
techniques can be implemented to create the initial population. This research

used a three-part technique to create the population.

The initial population was created using three different techniques. The first
technique creates the maximum number of black elements allowed by comparing
it to the effective density of the fine scale. An element is defined as one entry in
pc. A black element means that the entry will have a value of one. Once this is

done the black elements are then scaled down to achieve the actual effective

density.

21

 

Figure 3.1 Distribution of p(x) computed using the 1“ technique to create initial

population.

The second technique is a random distribution over all elements. If the effective
density of the random solution is below the prescribed effective density, the
elements with lower values are scaled up. If the density of the random solution is
above the effective density the elements with higher values are scaled down, this

uses an iteration technique to get the correct effective density.

22

 

Figure 3.2 Distribution of p(x) computed using the 2“d technique to create initial

population

The third technique (figure 3.3) involves using the images similar to that of the
reduced wavelet transform of the fine scale material distribution. These images
are created from taking the reduced wavelet transform of the fine scale material

distribution and adding noise as seen in the following figure.

23

 

Figure 3.3 Distribution of p(x) computed using the 3rd technique to create initial

population

This is done to partially seed the initial population. The density of this solution is
also equal to that of the effective density. Each of these three parts contributes
equally to the creation of the initial population. The initial population was created
using 85,000 members. This value was obtained from a trade off between size
and the time it took to solve the problem. Values calculated above this value

yielded little better results but increased the computation time.

24

5. Termination Function

The GA operates by evaluating all the members in a population, creating a new
population. This process is continued until some criteria are met. The usual
termination function (and the method used in this research) is the maximum of
generations allowed. The number used in this thesis was 65. Other techniques
include population convergence. This is when the sum of the deviations between
fitness values of members of the population becomes less then some specified
number, then the generations are terminated. The technique of terminating the
sequence after 65 generations was used instead of the population convergence

because of wanting to keep all the problems in the library consistent.

6. Evaluation Function

This is the function that evaluates the fitness of each member of the population.
This is done by using the objective function defined at the beginning of the
chapter. The fitness is defined as the inverse of the objective function. This done
because GAs aim to increase the fitness level, so by taking the fitness as being the

inverse of the objective function, it will be minimized.

25

 

Chapter 4

EXAMPLES

This chapter illustrates the use of the model order reduction technique developed in the
previous chapters. In these examples the fine scale material distribution of different
geometries is present in the figures along with the coarse scale material distribution
solution. In all the examples cited the fine scale material is resolved by a 64x64 pixel
distribution, with the coarse scale distribution representing 8x8 pixel grid. As stated
before this represents three levels of reduction. In all cases the black material represents

the following elastic material tensor

.91 .3 O
Eblack = .3 .91 0 (4.1)
0 0 .769
and white material tensor shown as
9le — 6 .3 0
EMU-u, = .3 9le — 6 O (4.2)
0 0 70e — 6

The gray material is a linear interpolation between these two bounds.
For all the problems shown here a symmetry constraint was introduced. This constraint
was introduced because the fine scale material distribution had symmetry about the x and

y axis. Only allowing M1 of the design domain space to be solved and then duplicating or

26

repeating that area to fill in the rest of the design space accomplished the symmetry

constraint. This can be seen in figures (4.1) and (4.2).

 

Figure 4.1 V4 of material design

 

Figure 4.2 Symmetric material design

As stated before these examples were obtained by implementing a Genetic Algorithm
[14] technique to solve the inverse homogenization problem for the reduced stiffness
matrix. The details of these calculations are shown in the previous chapters. The rest of
this chapter will be divided into three sections, each section devoted to each one of the
scales. The geometries are laid out as shown in figure 4.3 for the first scale, figure 4.4 for

the second and figure 4.5 for the third scale.

27

 

 

In!

 

l-B

 

 

 

 

 

 

Figure 4.3 Geometry layouts for fine scale materials —Scale 1

28

(1-A)/2

 

 

 

(l-B)/2——_‘>

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 4.4 Geometry layouts for fine scale materials —Scale 2

29

(l-A)/4

 

 

 

 

 

We: If If Ii
Ii If IE If]
If If Ii If
If If f if

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 4.5 Geometry layouts for fine scale materials —Scale 3

The values of A and B vary depending on the scale that they are on. This can be seen in

the table list below.

Starting

0

 

Table 4.1 Values of A and B for the three scales

30

4.1 Scale 1 Solutions

The fine scale picture and coarse scale picture are shown in figures 4.6 and 4.7. The
fitness of this particular solution is shown to be 1.171. This represents a solution is 1.171
times better than that of the wavelet transform of the fine scale picture. The eigenvectors
that contributed to this solution are the 13th through the 28‘". These were picked so that
they exhibit the fine scale features of figure 4.6 and the location of movement was not in
an node that was surrounded by weak material. The rigid body mode shapes were not
included in the calculations. This procedure was done by examining each mode shape of

the reduced finite element solution and the wavelet transform of the fine scale material.

For the examples shown here the fine scale material will be presented along with the
coarse scale material solution. Following these figures mode shapes will be shown
demonstrating the fact that if the two systems are equivalent, similar mode shapes should

appear at approXimately the same frequency.

31

9Hd=0.78125

(A)
O

b
O

50

 

Figure 4.6 Fine scale material distribution

 

1 2 3

4 5
litness=1.1171

Figure 4.7 Coarse scale material solution

32

The following figures will demonstrate the accuracy of the solution above. This is done
by showing that certain mode shapes of each solution and noting that the energy

(eigenvalue) associated with that deformation shape is about equal.

33

Eigenvalue = 0.0835029

 

 

 

 

0 ' V v v \ , ‘
D .
k \ \ \ “w '
1 g \ \ \. k , ..
N N 2’ VI
2 \ \\ \\ R l’
’ \\ / / I
“C ' \ V,/ ’l b”
\ ‘\ i . 5
3 » fl ’1“ / f: ”I '1
5 i 1l l t I ly/I V
(E: 4 I ‘ j A i 1' 1' I
on i, . ,' " "I
5 "x / I'f / \ v \I; \ 4
/ ’71 l 1
6 / // // j V \3‘,’
kt ' \ -4
fl e! \ \ \s
4 / \\\ ‘
7 / / // /3' \ r
N \ \ \
\ x“) \
8 r 1 1 \‘I‘ 2
0 2 4 8

Figure 4.8 9th Mode shape from the target stiffness matrix

 

 

 

 

Eigenvalue = 0.080813
0 . . - . -
l
I V ‘1’ fl ’4 4
f
1 —- \ . -——-> ’ ' .
\ \ f' A
\' \Vl ’ ,’ 4
2 ‘ \ 1 \ <—- “4‘ 'l I -1
\ \ \ l‘ 1 A".
“‘1' \i‘ , f 4
4 1
3 "’ \ \ ’ /‘§\ '1
\ \ i 3 IT\
8 i \‘l; l 1 A
a 4 > I i V . f1“ 4
8 . , \ t.
1 s
5 t‘ I f l I ‘ \\' \' \ 1
' I", Ii "\i N
V \l/ \ \ It,
6 ' /r . V1 — -> \ \ \‘ -1
g . ./ k“ r\
1;- , :’ " . l
' v \ \ "t
7 r / /. / +— ‘
V . .' /
V V
8 . 1
O 4 5 8

Figure 4.9 13th Mode shape for the coarse material distribution

34

As seen in figures (4.8) and (4.9) the mode shape deformations are qualitatively the same,
with the eigenvalues about equal. The following figures will demonstrate this point for
more mode shapes.

Eigenvalue = 0.134917

 

 

 

 

o 7 7m I r v r fif I r
1..» _——) .3 \ ‘3, ,2 ,2,» ———> «
2. . , \ / 2
i
3 (2‘ <——- / ~ “ K «-
A
8
Ewe—W » «$6— ,
5r<——-<—-—\ ‘/ <——-- <---—
q
6, r x \ I 1 I .—
7“ ~—> "’T ’/ "T’ \ \y. ‘9
8 A 1 4
0 2 4 6 8

Figure 4.10 10th Mode shape from the target stiffness matrix

Eigenvalue = 0.122476

0 x. f "\ Vfi‘ \ "\

1*“"<-\\< —. /’e""

 

é...—

 

 

 

 

 

5'—> —9’ “~91. / I“) “—9- l
\ #‘

6_ f l _ _- 1 .

7 "5 é/ g/é (—

8o 2 i 6 8

Figure 4.11 14th Mode shape from the target stiffness matrix

35

Another example of scale one solution.

efld=0.875

 

 

1 2 3 4 5 6 7 8
fitness=1.0214

Figure 4.13 Coarse scale material distribution

36

Readers should note that the fine and coarse scale material distributions are very similar.
This is because the values of A and B correspond exactly to the wavelet transform of the

reduced material. Again noting the similarity between the energies of the two systems at

deformed shapes.

37

Eigenvalue = 0.0876951

 

 

 

 

0 T '1 l I T f A I
1‘ V V I III/5 .74
1 .4 / / /
\ ‘\ \- T // I
\1 N ‘77 72'
2 .—-—> \ \ S / 4' / ‘
\ \\\ /4 l/j /”
3 r" \ ’ / '1
\ \ \ \\ \ji ,1
3 \ \l 1’ 1! ,A
9 4 L 1 , \ ' ' “
S i 1, 1 i.\ 1\
\ \ ‘\
5 . / /, MK 1
y i / ll \ “\\ x.
V V, \ \\ \\\
6 . / / / é" \ ‘
. / is
/ ,,/ ‘
7 / \ \\\ \\ .1
/ / / r
V/ l/
8o 2 3 e a

Figure 4.14 5th Mode shape from the target stiffness matrix

Eigenvalue = 0.0911849

 

 

 

 

0 . r -
l
v ‘3, V f :1 /
1 L". \ \ —> / "I -4
\ i, i 7’ .7
- ,/ / /fl
2 H \\ \\ \\ —? /l I I
\ s1 , [:4 4
.' I/ If
3 H \\ \ \\ .- 121 4
o \ \ ' A
.1, \11 1
s 4* 1 6 ’ \ ,
° ' l A i:
5 » / / V, .. \1 \\ \\ -
l- / /
if y ’1'. fi \
6 » i” <«-—— \ ~
”I” - / ,//
i/ i" is i\ \
\ .
7 h / // p/ *— d
y,
8 J 1 1
O 2 4 6 8

Figure 4.15 9th Mode shape for the coarse material distribution

38

Eigenvalue = 0.157605

 

 

 

 

o 1' ‘ I 7 *7 fi fi' V 4' f r
21' \ l I "
\ \
Q“ __._ \ <~ (~—

3r < 11/ / i
5 .
946—6——<——— \ «~— e— <~~ J
.‘9

6* r 3 j \ "

7.___> 9 ———" fl ’9 N » _._..’ .4

8 1 r

O 2 4 6 8

Figure 4.16 6th Mode shape from the target stiffness matrix

Eigenvalue = 0.151615

 

 

o .
(D \
1‘64—‘9 ‘7‘ > > ——-—-—>—-—,\ ——-—/ —~,
0
O

5~-—> -—.> "—9 .\ . / 1——--> ‘9 4

«\x
is
6- . , / (___ \ \ , .

 

 

 

Figure 4.17 10th Mode shape for the coarse material distribution

39

5.2 Scale 2 Solutions

20

(a)
O

elfd=0.6875

 

 

1 2 3

4 5
fitness=1.2516

Figure 4.19 Coarse scale material solution

40

The second scale results also yield interesting solutions; this can be seen in the figures
above. As seen in the solutions the use of gray material becomes more apparent and
necessary. Also in general, lower mode shapes are taken into account for the higher

scale materials. Again the mode shapes are inspected showing good results.

41

Eigenvalue = 0.108431

0 < v\ \ V \ Y\

 

1»\'\e‘é/<——\<~——</ .

1 1

2, l \ 1 . 1 \ ..
3E———> /"; ———>/7-'—‘?

X ’ \‘l

target
45

 

 

 

6 \ 1 ‘1
‘1 i
8 1 L 1
O 2 4 6 8

Figure 4.20 3rd Mode shape from the target stiffness matrix

 

Eigenvalue = 0.0746933
11 \ N a \ m- w
1.“7\<— / e—" \ \6— ‘£/ 1
A 1
21 1 l , 4
1 1

 

 

 

 

0
g 41-——>~-—-—>~——> .\ >—\ > > :4
? 7/
5 ———> \ ———>/' ——> \g ——>/' 1
i 1
6> l 1 I "
v i
L ,/ \ / \ ‘
7 2.2/*— (T_ é/"T—
8 M r 1
0 2 4 6 8

Figure 4.21 4th Mode shape for the coarse material distribution

42

Eigenvalue = 0.305737

 

1 I I , .
I I 1‘] ,1 I I i; '1'“
: , 1, 1 . ., v
'1‘ ‘11” V , \t 1'
1 I I 1 I \I I ‘1
I \Iv; (‘5 V \‘1':' \‘q‘,
2 r \ \ I \ \.
1 i 1 .1 ,1. ,
1' ,5 i 1‘
3 - .1. I ‘ ’ .1 i i
. 1 i 1': 1'1 -‘ 1 ‘~ ."‘1
‘ i I ’4‘ I J g A.
o I 1 I
a I 1 I 1 I I
£3 4F 2 A) I\ ‘i\
I i I I ‘1 A ,1 I '1 A
. . , . ..
5 __ ‘1 I I I i I
6 L ‘\ ‘\ I l \ \ I
7 ..
1 1 1 .1 1 1
i ' 1' 1
8 1 .L

 

.- .-

 

pL

 

Figure 4.22 10th Mode shape from the target stiffness matrix

Eigenvalue = 0.229096

 

 

 

 

 

. , f m‘r I T j
1 z 1 I v 1
‘v‘ 1 ' V v
‘\ l.’ 1 x
V “ "
‘ 7 . = 1 . 1 ' 1
11 I ‘i v I 1'
2 - ‘15, t’
.' j ‘1 .' l ,
l
1 I 1- , i. I 1
3 _ '. I I t '1 ‘ l
g . i 1 1 , i I 1
1 l ‘ 1 1‘
8 4 ’- IA! ."l* 1
o I I
i 1 I . I I A.
5 ~ * ~ ' . ‘ 1
6 I- ' - -1
7 1L ‘ a l ‘ 1 r ‘
V 1 l
v . \I it ; 1.
\‘1'. I u
8 4 A

Figure 4.23 15th Mode shape for the coarse material distribution

43

Eigenvalue = 0.408616

 

 

 

 

 

o a . 7\ [<— _. - -
I / - h\
‘ 1 ’ fl \ 1 / ‘7‘ J
1 \ ‘ I 1!, .‘1
2 1- I I \II I 0 I -1
.l, I »
$\ 'i I \IJ
3 1' \ k / \‘1’ \ fl 1! ‘
W ‘X
o— i
gum/(Xx \ 17 we“.
\\ I /’
5 " g \ A\\ / ‘9 \ J
l/ I A \11
y, K i x
1' I I
6 I 1 1 1
I k 1
I I; / [I ‘\ v
7 Iq’ ~> <“- , <
‘1‘ \ I
\\
8 1 1 1
0 2 4 6 8

Figure 4.24 12th Mode shape from the target stiffness matrix

 

 

 

 

Eigenvalue = 0.347141
0 \ \ Y \ V v 7 ,f ,7
A
h 7f
1 L é—’ \ I / ____).\ .
. / [1" ."'I\ A1 I -
.. ‘4 I t , 1:
2 L . 1 I .1
- 1 1
\ [if I R I I
3 t x ———>’ I <— } ~
\, /
, N 14
g 1'
g 41 -——> w>—~> <£—-<—<~- <— -
1.
0 1.
7‘ \
51 K -—-> \ 1 ,,..____< \1 «
\\‘\ I 1"] I‘d
\ l
V.
6 " ' I I 1
x I I 1 I
~\\ \I1' \1/ \ l //
7 _ \é—— . ‘v —-L—‘>" .
/ a \
fl" \
8 L J l
0 2 4 6 8

Figure 4.25 215‘ Mode shape for the coarse material distribution

The figures above showed one common trend: the target matrix was always stiffer than

the coarse scale material distribution. This problem can be solved by scaling up the

values in the coarse scale material.

Looking at another example from scale 2, again noticing the energy similarity in the

mode shapes.

45

1 0
20
50 - -

Figure 4.26 Fine scale material distribution

efid=0.6875
8

8

 

 

1 2 3 4 5 6 7 8
fitness=1 .2488

Figure 4.27 Coarse scale material distribution solution

46

Eigenvalue = 0.0945083

 

 

 

 

0. . ' c /. 7r \ ' .
1 \.~' "I; l "\ \:.,
'1 V f g ‘\ v
1 f \ a ,‘ l ‘— 'I ‘1
\\ 'tfi A A f/
k; W i .I 1 $71
2 1 ‘ I ”k | x ’ ..
a, t 9
y ‘9' \ i V
3 ‘ i *’ ‘ ,‘ ’ '" a i
/ A ' : ’ 1‘ 1.
*6 g V l l i
9 ; 9,! ‘l ' l. .
\ ." l‘ i \ \‘I
V ‘ l ‘ '
5 t1 \ 3 / ’ \ *— /I ‘
\x A A A /
f \t " Q l V
6 1 ‘ A; ‘ / ‘
‘ \ ‘ 9,1 ,
7 I, \ g 1/ V!
l. I *- l ,0 ..
I
[I \l
V \
8 L 4 g
0 2 4 6 8

Figure 4.28 3rd Mode shape from the target stiffness matrix

 

 

 

 

 

Eigenvalue = 0.0742874
0 i. - . -
If. '9 1 l l. 1".
.‘ W 3‘/ \vl‘ /
1 K \ $— \V ‘ ———-y / .1
. ’l|\ / l \ I?
2' l ‘ i4 v ‘ "' r
R 7’. i l l .N T
I I’ \l.‘ \1/ '. \
3 k 5' ——> ‘l' I r <— \‘ J
\ I? \\ i /’ A
o \i .9
g 4 r E i \Y/ "
8 \ 1“ l i. ! 4"
\ ‘f V /
. _ " __ / .
5 “x R < / \‘ll ‘\ > A.
: : .\9 -
l . 9' . !
6 R. . ! f l \ .4
7; l l f l\
\3. w ~:g/ \
7 . I a.) < T y (a \ ..
\‘ xi; Ii
8 L x 1
0 2 4 6 8

Figure 4.29 4th Mode shape for the coarse material distribution

47

Eigenvalue = 0.133876

 

 

 

 

0 V f /v /'I\ v O; if I‘.
' . ; v
/ / 0,: \ ..
.9 .
2 —--'>———~> ‘ é-I—‘é—d é..— . —-——> ..
99.
\ l ,7!
3 \ ‘ \ X. A /’ u
l' 1‘
\i V / ll“ : .tx
a. 9 .
c» 4 r . - , / I l l .1
5 .9" \ {it
5 r V \Jl" fl ’ ' \ .4
/ ' \
. 2 A
6 ' é-r a a "" "’9’ ' e ‘
9 l /
\ l
7 r \ \ I {49/ ..
\ V’,
8 r r
O 2 4 6 8

Figure 4.30 5‘h Mode shape from the target stiffness matrix

 

 

 

 

 

 

Eigenvalue = 0.105763
0 v f
i 5 fi‘ .
v l 9' f i ‘F I
1 ‘93:“ 9/ l \\
* x :9 \ "9 ’ ‘
/ .
19 g \z
, \f/ -
2 -~’—\——- ‘9 ——> > >< .
9,9
I 4
\\ .
3 ’ \\ L . / —-‘> \ <
1‘ \ ’ //
Ill . T \ _ i/
l . v-
8 4 ‘ 9 l l l ‘
° 7f l ' 9s i’ ‘ ‘i'
5 ——=—» ’l \ <—- x .
I / ! \\
i5 i 1“;
- / V9 \
6 > ;\< \ <--— f“ —7
l
R l /fl4
7 ~35 \ l (m‘ \ 3 / "
\— ' /
xx . .9
\l/
8 A: 1 r
0 2 4 6 8

Figure 4.31 5‘h Mode shape for the coarse material distribution

48

5.3 Scale 3 Solutions

efld=0.90625
8

A
O

50

 

 

1 2 3 4 5 6 7 8
fitness=1.016

Figure 4.33 Coarse scale material distribution solution

49

Eigenvalue = 0.202476
° . \ f ' \ ‘ '

1 —\ \ .l

. é...<——
2/4/ 6”

 

Sleé/Ke/RRR

target
35
\
\
|\

 

 

 

Figure 4.34 4.1. Mode shape from the target stiffness matrix

Eigenvalue = 0.174474

 

O 1 ! 5 r a f -
v V
‘C R
, (A '5‘ \ ‘ ‘\
f,
‘1‘—

» __ \\~\\\

2 g///{/’< J

1 ‘ l l .
l

’7
'7 .
5-r—> ——>/ / //74

31////

p

coarse
p

 

 

 

Figure 4.35 3rd Mode shape for the coarse material distribution

50

Eigenvalue = 0.291125

 

 

 

 

0 7 ' V ' Y 45’ '\ ‘“§
9.
\ \ / 1
x. r 9'9 ,4
‘94 9' I //
2 I \ I I K ' "
l . .
Q | ‘1' hr ‘5
3 . ,V \ / l \ 4
/ f \
l " ‘1
76 "l’
9 4 <—-—— ./ 4 —-> —--» r<£- .
Q
.- b
\ ,9
5 r \ “ '4’ 'l
A 9 \ I. 9/
:5 V
6 ,_ 11 \ 1 I ' I q
,9, l l J
7 N / l \ I \V \ '1
x
. l .1
8 L \y
0 2 4 6 8

Figure 4.36 7.1. Mode shape from the target stiffness matrix

 

 

 

 

Eigenvalue = 0.31518
0 \ ' 9 (’fi ,/ ’v \
A
1 > \\ , I], —-9 ‘ / "
A A 1 \~ 3 9/
1 l - N 7%
i l l ‘1"
2 > ‘ 9 l l .l
[in l
,77 I is: l , 1'
3 / l \ , ‘ f
—>», ‘9‘— , / i \\ 1
x! ,' as
a 1’
g 4 k-->>——},)~ <—-<—- «<—— - ————>-l
o f 1‘
\\ 1 ,7.’
5‘ \\ 1 9/ ‘ 91“. i,
a g 99’ .9 ‘ 9
6 1’ l . l
r l i o -'
t . 1 11.
.9; i \*.‘ I I k
7 r /, V \ / l \\ 4
9 l ‘9
V
8 1 r
0 2 4 6 8

Figure 4.37 7.1. Mode shape for the coarse material distribution

51

Eigenvalue = 0.364857

 

 

 

 

0 ' ' fl ' \, 's I
.9. 9 9
7' i; '
1 N / \ K / \' a
a J \
2 ’\> ——-9 I ‘2‘ 6“ <~ ’ \ 1
\ ,1
3 K) \ 1‘ / g \ '|\ -” d
éX w. 5" ’ 1‘ 'T‘
g. 4_ \ .9 , \ l . .
.99 \, .
l 1 \
5 9 9 V a, / 9' \ .
/ l \
V
6 p. k I \ \> .--> r' ‘\ ..
\
7. \ \ / ‘1, \ l / .
1‘
8 r r
0 2 4 6 8

Figure 4.38 9m Mode shape from the target stiffness matrix

 

 

 

 

Eigenvalue = 0.365677
0 - - a .
A T ‘
:1 . \ V \ / V
1 —-—> ’ \ <—- , " 9 ‘
/ \
14 xx
2 ———9———> <——é——— <——-—— . —~> J
A
V l /,’
3 ”—9 \\ / e- \ l‘ / "
\\ y, P ’1‘ F ’9‘
0 l
9 4 , ' I 1 i .
8 ' l i 1
0 l l I l'\
V \1.’ //’/’ l V\
5 r , ‘1 \ —-~> ~ ~
g/ l 4‘. I
V
6 r (‘— 9 -—a» ——>—— ‘7 <———- 9
\ Al '7
7 r / “—3’ 1
.9 /
\ w ’
8 4 1. Fr
0 2 4 6 8

Figure 4.39 9th Mode shape for the coarse material distribution

52

Showing another scale 3 problem.

8

efld=0.875

p
O

50

 

Figure 4.40 Fine scale material distribution

 

1 2 3

6 7 8

4 5
fitness=1 .0226

Figure 4.41 Coarse scale material solution

 

0.1 65655

Eigenvalue

 

 

 

54

 

 

 

 

Again looking at the eigenvectors of the two systems and comparing the eigenvalues.

.092

X
m
a
1 q ‘1 4 a J 8 m
J u .1 < 4
S
S
till 6
I I I .I’ .10: I'll K\ |1\|n E ‘\I\: 11' n 11.: “mm. It \I‘H’U‘
- ....... ll... \l 1 1.91-1
.1 .1
t
1 S
4... i - l i il -l- l- . l t I. . 1
i AM i 1m ii A111 r 6 e mm 1. .Mrill.lMK.....tl.u..Vl..liWVl . 1W... liVltllWVlill-\u
g 3 \
”a 8
\. t 8
A... - - i/ ll A l Ail. All. C m 111V {:9 91-“? --l-,w.v -..|.V 1-1V
=
. m e - .
. . . , . . 4 m 1U. . \ .\ 9 \ .
f a“
m. m
.nV 1.1in i x - r .\ . . ,. Amflllll
-. 1-1-1 .11 ls... l... a 9 F.» .-ilmlli ,
h 0'
S E
J-.. v. . I!!! n e I” Art.” ll
- - H.511-.- Veilillillilf melill/hilil-..Wilfiufl 2 w TI .1. rfuti 110.111 Arm III 11/ I! I 1’4 l
11' W 9-H lath“: -/ Aflj
1-1- v xiv . l- in ..... ll - I i
v x x i v // 1.11/ - \v \\ AI
IV \Jv 3 \i.
a
O
C
r
U
05
E

Figure 4.43 4m Mode shape for the coarse material distribution

Eigenvalue = 0.24342

 

 

 

 

0 ‘ r - *4 V j
l "l‘ l.
I VI 4 I K
1 l \ \ / l \ k / ..
r *4
’ /
2r \ —~> ”’3’ ‘\ K“ e/ 4
7!
3i- ./ ’9- \ .‘ // <.__. \ 4
\ |._ ,
I? W
‘5 l T
9 4 ’ I l l ,‘ ' ‘ 1
53 l v E ;'
”a v
5- ‘\ \ V f .
/ l' \ ‘5
l ‘3
xi;
6 I k. 6 6". I ‘9 -—> ”—9 ..
A
.\ 4
7 l' // (’0‘ \ / ”'4; \ -1
i
8 _1 1
0 2 4 6 8

Figure 4.44 5‘h Mode shape from the target stiffness matrix

 

 

 

 

 

 

Eigenvalue = 0.291453
0 v . I a -
\ l l I
\\\ v : . v [4
1 r- ‘ fi:—'— / \|/ \\\ .__7\. I/ q
1’4 xi, \
"I,
2 r 9% 9—— . :> >; Ix ‘
A
x: ; 7r
. _ ‘ ‘ /
3 / <— — \ fix ”‘3’ \ ‘
g T l A \
E 4 _ l .. .
23 i i
V /’31 1 V9 \i/
5% \ ~—-> / \ (=— / ~
\ /
\‘k X4
/
6 y ’3" > >3 O ‘-< < 4 q
\
fl \
7 '- / *9 \ l / é“ \\ 1
n. .z
VI
8 1 L l
O 2 4 6 8

Figure 4.45 7th Mode shape for the coarse material distribution

55

As shown in the figures for the third scale the use of gray material is increased from the
second scale. The eigenvectors taken into account start with the 3rd mode shape. As
stated before, this is the first non-rigid body mode shape.

This chapter presented fine scale material distributions along with the coarse scale
material solutions. These solutions had varying success, which can be seen by comparing
the energies associated with the modes shapes for the solutions. The last chapter is going

to discuss the conclusions of this research.

56

Chapter 5

CONCLUSIONS

A model order reduction technique was shown. This technique uses a multi-resolution
analysis on a non-homogenous material distribution with fine scale features to construct a
wavelet based reduced stiffness matrix. A Equivalent material problem was posed; find a
material distribution that represents a reduced wavelet stiffness matrix. This problem

was successfully solved using a genetic algorithm.

Results for three different scales are shown along with mode shapes of the reduced matrix
and the coarse scale material solution. These mode shapes were matched, and then the
energy (eigenvalues) compared. If the energy between the two systems were equal it
could be said that the fine scale material distribution could be represented by the coarse
scale solution for that frequency. This fact has been shown with good accuracy that this
procedure can be done not for just one frequency, but a range of frequencies, making this
correlation between the fine scale and the coarse scale correct for a wide range of loading

conditions.

57

Acknowledgement

This work was supported, in part, by the National Science Foundation through grant
DMIQ912520. This support is gratefully acknowledged

58

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[12]

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