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This is to certify that the
dissertation entitled
SHAPE OPTIMIZATION OF PLATE STRUCTURES

USING HOMOGENIZATION WITH APPLICATIONS
IN MECHANICAL DESIGN

 

presented by

CIRO ANGEL SOTO

has been accepted towards fulﬁllment
of the requirements for

Ph. D . degree in MECHANICAL ENGINEERING

All“ A - u A:
Majorfe‘fessor
2/ 2 1 / 1 9 9 4

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SHAPE OPTIMIZATION OF PLATE STRUCTURES
USING HOMOGENIZATION WITH APPLICATIONS
IN MECHANICAL DESIGN
By

Ciro Angel Soto

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department Of Mechanical Engineering

1993

ABSTRACT

SHAPE OPTIMIZATION OF PLATE STRUCTURES
USING HOMOGENIZATION WI I H APPLICATIONS
IN MECHANICAL DESIGN

By

Ciro Angel Soto

This work is about the problem of computing the Stiffest topology and shape of Mindlin
plates subject to several independent loading conditions. A structural shape is identified
here with a distribution of ribs symmetrically located above and below a core of ﬁxed
geometry. Small scale close ribs are used to build microscopically ribbed plates, and to
expand the space Of designs in order to guarantee attainability of the optimum.
Homogenization techniques are used to compute effective stiffness of such plates and to
establish a relation between the density of ribs and the plate stiffness, a fundamental piece
of information for shape Optimization algorithms based on homogenization techniques.
Effective properties are derived for two micro structures. The ﬁrst micro structure is made
up by two orthogonal layers of ribs, and the second one is built from a ﬁnite number of
layers of ribs arranged in arbitrary directions. The effective properties of multi-layer
material plates are expresed using only four variables or ‘moments’. A new formulation Of
the layout Optimization problem, using these variables is presented. This formulation splits
the Optimization problem in two subproblems, one where the optimum distribution Of
material used to build the ribs is computed, and another subproblem where the optimum
moments are computed for prescribed amounts of material and strain ﬁelds. Numerical
experiments and example problems with applications from the automotive industry are

included.

To Nazaret, who sacriﬁced her time for me.

To Alberto Mata, who never forgot our friendship.

ACKNOWLEDGMENTS

Special thanks to my advisor Professor Alejandro Diaz for all his help, support and advice.
I learned many things on the art Of research from you.

Thanks to the members Of my committee, Professors Noboru Kikuchi, Alan Haddow,
Thomas Pence and Charles MacClure, for expending their time in correcting this
document.

Thanks to Professor Robert Lipton whose ideas are spread throughout this dissertation.

Thanks to Professor Martin Bendsoe for his suggestions during the investigation. Thanks
also to Professor Klaus Schittkowski for letting me use his nonlinear programming solver.

This work would have been very difficult to do without the computational programs
MoniqueTM from RTB Corporation, and HyperMeshTM from Altair Engineering. Thanks
for letting me use them.

I will never forget my cheese sandwich lunches with Matthew Brach and Taner Onsay
sharing thoughts about how to solve the world’s problems. Thanks for those moments.

Thanks to Faramarz Farahanchi and Muqqafat Soomar for their kind company. Also
thanks to Neil Whitbeck for sharing with us his chalkboard thoughts on philosophy.

Very special thanks to my friends Ricardo Palma and Ender Parra for their invaluable help.
This effort is also yours.

PREFACE

When I began my doctoral studies I thought that getting a doctoral degree was going to be
an enjoyable and pleasant experience. I thought that I would get the degree and would go
back to work to have an easier life. The end has come and from the evaluation of these
years I would say that they were hard years for me and my family. However, difficult
times also teach, and I learned a lot. I learned that there are much more important things
than to live to work. I learned that our life style must be changed to a simpler life where
people can talk and share pleasant moments instead of ﬁghting for a better position in their
jobs or for making more money. I learned that our desires to have more in order to satisfy
our appetites can be replaced by turning to ourselves and ﬁnding out what we want to be,
instead of accepting what society forces us to be. I learned that there are many people that
die every day for not having the piece Of bread that we throw in the garbage. I learned that
my scientiﬁc research is not helpful to any of those hungry people who want a hope in their
life; and I learned that when we scientists neglect the welfare of Others in our attempts to
satisfy our egos by reaching new stages Of complex thinking and inventing sophisticated
theories, we betray the human race. I am happy of having learned all these things. N ow I
want to transform my knowledge and use all my energy to help any human being that needs
me.

C.S.

TABLE OF CONTENTS

LIST OF TABLES ................................................................................ viii
LIST OF FIGURES .............................................................................. ix

CHAPTER 1 INTRODUCTION

1.1 Introduction ........................................................................ 1
1.2 Historical Review .................................................................. 3
1.3 Outline .............................................................................. 9

CHAPTER 2 HOMOGENIZED PROPERTIES OF RANK-2 MINDLIN

 

PLATES
2.1 Homogenization of the Mindlin Plate Equation ................................ 12
2.1.1 Background ............................................................ 12
2.1.2 Mindlin Plate Model 18
2.1.3 Derivation of Effective Properties ................................... 20
2.1.4 Results Using Layered Materials ................................... 22
2.2 Comparison With Other Models For Ribbed Plates ........................... 29
2.2.1 Comparison of Effective Properties in Bending ................... 30

2.2.2 Comparison Based on the Mean Compliance of a Simply
Supported Plate ........................................................ 38

CHAPTER 3 HOMOGENIZED PROPERTIES OF RANK—L MINDLIN
PLATES — MOMENT REPRESENTATION

 

 

3.1 Effective Properties for Rank—L Material Plates 44
3.1.1 Set up of the Mindlin Plate Equatinnq 44
3.1.2 Computation Of Homogenized Stiffness Properties ............... 46
3.2 Moment Representation of the Homogenized Properties ..................... 57
3.2.1 Moment Representation of Bending Stiffness ..................... 61
3.2.2 Moment Representation of Shear Stiffness ......................... 62

vi

3.2.3 Moment Representation of Membrane Stiffness ................... 62
3.2.4 Feasible Set in the Moment Space ................................... 63
3.2.5 Determination of the Micro Structure Geometry ................... 64

CHAPTER 4 FORMULATION AND IMPLEMENTATION OF THE LAYOUT

 

 

 

OPTIMIZATION PROBLEM
4.1 Hierarchical Moment Formulation of the Stiffness Maximization
Problem ............................................................................. 70
4.1.1 Optimization Problem St ‘ ‘ 70
4.1.2 The Local Problem .................................................... 72
4.1.3 Equilibrium Equations ................................................ 73
4.1.4 The Global Problem ...................... 74
4.2 Numerical Implementation of the Optimization Problem ..................... 75
4.2.1 Solution of the Equilibrium Equation ............................... 75
4.2.2 Solution Of Optimization Problems 76
4.2.3 Convergence Criteria .................................................. 78
4.2.4 Algorithm Implementation ........ 79

 

CHAPTER 5 EXAMPLES

 

5.1 Comparison with Previous Work 82

5.2 Comparison of Micro Structure Models ........................................ 87

5.2.1 Uniform Transverse Load ............................................. 87

5.2.2 Point Loads on the Diagonals ........................................ 91

5.3 Parametric Study ................................................................... 95

5.3.1 Inﬂuence of the Thickness Ratio r=h2/h1 ........................... 95

5.3.2 Inﬂuence of the Amount of Material Vmax ......................... 99
5.3.3 Comparison of Cross Sections ...................................... 102
5.4 Industrial Examples ............................................................... 105
5.4.1 A Car Hood ............................................................ 105
5.4.2 A Car Door ............................................................. 110
CONCLUDING REMARKS .................................................................... 113
APPENDIX .................................................................................. 1 17
LIST OF REFERENCES ........................................................................ 119

vii

LIST OF TABLES

 

Table Page
2.1 Summary of homogenized properties of one-family, ribbed plates ........... 35
3.1 Effective stiffness for rank—L material plates 58

viii

Figure

1.1
1.2
1.3
1.4
1.5
1.6
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
2.13
2.14
3.1
3.2
5.1
5.2.
5.3.

LIST OF FIGURES

 

 

 

 

 

Egg
Problem statement ..................................................................... 2
Michell truss (Michell, (1904)) ...................................................... 3
Plate with properties varying in one direction. .................................... 6
Layered material conﬁguration. ..................................................... 8
Square cell geometry used in (Bendsoe and Kikuchi, ( 1988)). ................. 9
A layered material plate with three layers in arbitrary orientation ................ 10
Two-scale concept of microscopically periodic materials. ....................... 13
Layered materials ...................................................................... 14
Large scale geometry of the plate 15
Small scale domain .................................................................... 15
Graphical description of the homogenization procedure 16
Assembly strategies for the weak plate 17
Basic cell in a plate made using a rank-1 material. ............................... 23
Basic cell of a plate made of rank—2 material ....................................... 27
Geometry of plate cross section. .................................................... 29
Laminated plate model ................................................................ 32
Effect Of rib width on material properties ......................................... 36
Effect of rib height on material properties. ....................................... 37
Effect of rib width on mean compliance 40
Effect of rib height on mean compliance .......................................... 42
Rank-L layer plate (1:3 in this picture) ............................................ 44
Basic cell for a rank-1 material plate. ............................................... 49
Gray scale to identify the cross section of the plate in the optimum shapes. . . 81
Typical histogram for optimality condition 82
Example 1: geometry and loads. ................................................. 84

Fi ure

5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
5.13

5.14
5.15
5.16
5.18
5.19
5.20

5.21
5.22
5.23
5.24
5.25
5.26
5.28
5.29
5.30
5.31
5.32
5.33
5.34
5.35
5.36
5.37

Example 1:
Example 1:
Example 1:
Example 1:
Example 1:
Example 2:
Example 2:
Example 2:
Example 2:
Example 2:

structure.

Example 2:
Example 2:
Example 2:
Example 2:
Example 2:
Example 2:

structure.

Example 2:
Example 2:
Example 3:
Example 3:
Example 3:
Example 3:
Example 3:
Example 3:
Example 3:
Example 3:
Example 3:
Example 3:
Example 3:
Example 4:
Example 4:
Example 4:

ﬂge
optimum shapes with rank-2 materials. ........................... 84
Cheng and Olhoff (1981) result for the pinned plate ............. 85
Cheng and Olhoff (1981) result for the clamped plate ........... 85
iteration histories. .................................................... 86
optimality condition ratio histograms. ............................. 86
geometry and loads. ................................................. 88
optimum shapes. ..................................................... 88
distribution of one-layer materials in the plate. ................... 89
distribution of two-layer materials in the plate .................... 89
distribution of three-layer materials using the rank-L micro
......................................................................... 90
iteration histories. .................................................... 90
Optimality condition ratio histograms. ............................. 91
geometry and loads. ................................................. 92
distribution of one—layer materials in the plate. ................... 93
distribution Of two-layer materials in the plate .................... 93
distribution of three-layer materials using the rank-L micro
......................................................................... 94
iteration histories. .................................................... 94
optimality condition ratio histograms. ............................. 95
geometry and loads. ................................................. 96
Optimum shapes. ..................................................... 97
iteration histories. .................................................... 97
Optimality condition ratio histograms. ............................. 98
optimum shapes. ..................................................... 100
iteration histories. .................................................... 100
optimality condition ratio histograms. ............................. 101
geometry and loads. ................................................. 102
optimum shapes. ..................................................... 103
iteration histories. .................................................... 104
Optimality condition ratio histograms. ............................. 104
geometry and loads. ................................................. 106
Optimum Shapes. ..................................................... 107
iteration histories. .................................................... 108

5.38
5.39
5.40
5.41
5.42
5.44
5.45

Example 4:
Example 4:
Example 4:
Example 4:
Example 5:
Example 5:
Example 5:

Page

interpretation of the Optimum shape of the car hood ............. 108
optimum shape Obtained using the rank-L micro structure ...... 109
iteration histories. .................................................... 109
optimality condition ratio histograms. ............................. 110
geometry and loads. ................................................. 1 1 1
iteration histories. .................................................... 112
optimality condition ratio histograms. ............................. 112

CHAPTER 1
INTRODUCTION

1.1 Introduction

A natural way to improve structural designs is to optimize the use of material for prescribed
load conditions SO that the ‘performance’ of the system is maximized while the amount of
material is controlled. In this case, ‘performance’ can be measured in several ways. When
applied loads are static, it is convenient to characterize a given design by its ﬂexibility or
compliance. This measure is convenient from the mathematical View point and appropriate
from the engineering perspective. Mathematically, shape Optimization problems formulated
in terms of the mean compliance are easier to solve. From the engineering viewpoint,
compliance is one of the most popular ways to distinguish between designs simply because
it is a measure of the relation between the excitation and response of the structural system,

i.e., forces and displacements.

This dissertation deals with structural Shape optimization under static loads. Speciﬁcally, it
is concerned with the computation of the optimum layout and shape of plate structures that
minimize the mean compliance of the structure. This problem can be stated in the following
way: given a plate Of uniform thickness occupying the domain .00 ><(—h1,h1 ) (Figure
1.1), and given the loading and displacement boundary conditions, ﬁnd the distribution
(layout) Of ribs (QR in Figure 1.1) that minimizes the mean compliance (external work) of
the Structure. An isoperimetric constraint that prescribes an upper limit on the amount of
material used to build [2R is consider in the formulation. This problem may be written in a

compact form as

minimize C(u), (1.1.1)
I pd!) S Vmax
90
State Equations

where p is the area density of ribs on [20, and Vmax is a prescribed area fraction Of the
plate domain {20 that will be covered by the ribbed domain QR, i.e., VmaFQR/QO, and u
is the state variable (i.e., displacements). The thickness of the ribbed domain is ﬁxed to a
prescribed value 2h2.

    

| Q l/ i Q l S V
Cross R 0 max
Section

Figure 1.1. Problem Statement.

It is important to distinguish this problem from the sizing and boundary variation problems.
In Sizing and boundary variation problems the layout (topology) Of the structure is known a
priori (e. g., the number of holes in a plate), and the sizes of the plate cross section and/or
the position of the boundary of the domain are considered as design variables. In the
problem stated in equation (1 .1.1) the topology (layout) of QR is unknown (e.g., the
number and shape of holes in the plate are unknown). Hence, the problem is more general

and its solution is more difﬁcult to ﬁnd.

Structural Sizing optimization has had limited success because it starts from a fixed
topology that may not be optimum for the problem. New approaches that overcome this
limitation consider the use of composite materials with micro voids to Obtain the optimum
shape. The introduction of these materials expands the space of possible designs allowing
alternatives other than a single isotropic material to be allocated at a given point in the
domain. The presence of microscopically composite materials call for homogenization
techniques to compute average material properties that will be used in the solution

algorithm. Using composite materials with micro voids the shape optimization problem can

be transformed into a problem of optimization of material distribution where the solid
material volume fraction is the design variable. A more detailed description of this method
will be given in the following chapters.

1.2 Historical Review

In this brief historical review two main ﬁelds of research will be reviewed: structural shape
optimization and homogenization of composite materials. Both ﬁelds will be discussed in
relation to the solution of the optimization problem stated in (1.1.1).

One of the ﬁrst problems in Optimal design was proposed by Lagrange in 1770 when the
optimal shape of a column subject to an axial load was Obtained. Later, in 1856, the ﬁrst
topology optimization problem was solved by Saint-Venant, who found the stiffest cross
section of a prismatic beam subject to torsion -- the annular cross section. Another well-
known layout optimization problem was solved by Michell in (1904), who found the
optimum shape of trusses for minimum weight working with a continuum medium (Figure
1.2).

continuum
medium

 
 
 
 

A
v.- ‘ Load

 
 

Figure 1.2. Michell truss (Michell, (1904)).

In the late 60’s, Prager and Taylor (1968) proposed an optimality criteria method to
compute Optimal shapes of structures. The method consisted Of two main steps. The ﬁrst
one was the derivation of necessary conditions for optimality using, for example, Khun-
Tucker conditions. These conditions led to a differential equation of the form:

Strain Energy Density

. . . =constant (1.1.2)
Unit Stiﬂness Denszty

4

In a second step, equation (1.1.2) was integrated to compute the Optimal distribution of
elastic stiffness on the structure. When equation (1.1.2) cannot be integrated, an iterative
algorithm is built to solve the problem where condition (1.1.2) is used at each iteration step
to verify the optimality of the proposed solution. This approach is the basis of the
algorithm that is used in the present research to ﬁnd the optimum layout and shape of
structures.

During the 70's, problems of Optimum design of trusses and grillages were studied
extensively (Pedersen, (1970); Prager and Rozvany, (1977)). In 1973 Zienkiewicz and
Campbell wrote what appears to be the first paper on boundary variations using ﬁnite
element methods. They used the nodal coordinates of the discretized boundary as design
variables. Many researchers made important contributions using boundary variation
techniques during the 70's, among them Tvergaard, (1975); Kristensen and Madsen,
(1976); Dems and Mroz, (1978); Rousselet and Haug, (1980). These techniques are now
popular in commercial software to solve Optimal design problems where the topology of the
structure is known. Also in the 70's the problem of Optimal thickness of plates to
maximize the plate stiffness for a given amount of material was treated by several
researchers (Armand, (1974); Olhoff, (1975); Lurie and Cherkaev, (1976)), who found
that Optimum shapes may have inﬁnitely many thin ribs. This was the beginning of a new
era in the solution of layout optimization problems. Thereafter, composite materials were

considered as a way to model and guarantee the existence of solutions.

In the early 80's researchers reached the conclusion that in some cases Optimum layout of
structures could only be attained using non homogeneous anisotropic materials (Olhoff, et
al, ( 1980); Cheng and Olhoff, (1981); Armand, (1982); Kohn and Strang, (1982);
Bends¢e, (1982a)). They stated that problems of Optimum layout of ribs on plates required
that composite materials be used to obtain the rapidly varying thickness solutions Observed
earlier. The physical reason for this is that at each point of the domain the use of an
anisotropic material allows one to arrange the material more effectivelly compared to a
simple isotropic material. In other words, the use of an anisotropic materials expands the
space of solutions. In particular, if we build the plate with a microscopically porous
material whose porosity is potentially different at each point in the plate (i.e., a non
homogeneous plate) we are allowing the possibility of creation of holes (using high
porosity) or solid regions (using low porosity) at any location of the plate. Therefore, the
topology of the plate is not prescribed but automatically generated as a result Of the

Optimization process.

 

The problem of computing effective properties, and their bounds, in composites has been
addressed since the time of Maxwell, who in 1873 was Studying the effective conductivity
in composite materials. A microscopic non homogeneous composite material is
characterized by a change of properties in two scales. These materials have properties that
depend on two coordinates in different scales, a large scale coordinate, labeled x here, and

, x , .
a small scale, labeled y = — , where 8 1s a very small parameter that characterizes the order
8

of magnitude of the composite mixture. For instance, the elastic tensor of a composite

material microscopically mixed is E( x, y) = E( x, i;- ). In this work we will use a subset of

non homogeneous composite materials called pseudo-periodic. The properties of these
materials are such that E( x, y) = E( x, y+ p), Where p is the period in the small scale.

After applying a homogenization technique the homogenized (also called average or
eﬂective) properties E ( x) are computed, and the dependency on y is eliminated. Notice

that these properties still depend on the large scale coordinate x.

In 1889 Voigt found an upper bound (Ev) for effective mechanical properties (E) in

composites, given by

EV(x)= I E(x,y)dY, forﬁxed x (1.1.3)
Y

_1_
IIYII
E(x, y) is the mechanical property at location x, and Y is the small scale domain where the

properties change as a function of the coordinate y. A lower bound was found in 1906 by
Reuss, given by

-l

ER(x)= , forﬁxed x. (1.1.4)

1 1
— —— dY
"an; ax,»

These bounds are known in the literature as “elemental bounds” and are such that any
physically meaningful effective properties must satisfy the inequality E R S E S EV. In

1963 Hashin and Shtrikman found Optimal lower (E11513) and upper (E55) bounds in

mechanical properties of composites. Optimal bounds are those bounds that cannot be
improved with the only information Of volume fractions and individual properties of the

components. Hashin-Shtrikman bounds satisfy ER S E11115 S E S Egg S EV. These

bounds were found using physical arguments and variational principles.

From the mathematical viewpoint, a boundary value problem where composite materials are
used may be described by a partial differential equation with coefﬁcients changing rapidly
in the domain. In 1973, De Giorgi and Spagnolo introduced the theory of G—convergence a
theory that provided the mathematical basis for the proofs of existence of average or
eﬂective Operators in boundary value problems governed by partial differential equations
with rapidly varying coefﬁcients. In 1976 Babuska introduced the term homogenization to
denote the process Of computation of effective properties. Duvaut (1976) computed the
effective bending stiffness of Kirchhoff plates with material properties changing in one
direction (Figure 1.3) based on G—convergence theory. Due to the Kirchhoff assumption
(i.e., no transverse shear deformation), this set of homogenized properties did not include
stiffness in the direction of the transverse shear. In 1978, Bensoussan, Lions and
Papanicolaou published the book "Asymptotic Analysis for Periodic Structures" where they
presented a systematic approach to compute effective coefﬁcients in BVP’s with rapidly
varying coefficients. This is the approach used in this research to treat the Mindlin plate

equations.

Material 2
Materiall . .

   

Figure 1.3. Plate with properties varying in one direction.

Since 1980, mathematicians have become more involved not only in the field of
homogenization and the computation of optimal bounds, but also in the layout optimization
problem. In 1980, Banichuk, and independently Cea (1980), discussed the layout
optimization problem for a wide variety of problems, including plates, from a theoretical
viewpoint. One Of the pioneers in layout optimization in plate structures using
homogenization techniques was Bendsoe (1982a, 1982b) who used the results of Duvaut
(1976) for Kirchhoff plates to compute homogenized bending stiffness for plates with ribs
in one direction. He also computed homogenized bending stiffness for plates with ribs in

two orthogonal directions using a method he called “smear out”.

7

Bourgeat and Tapiéro (1983) proved the existence of homogenized coefﬁcients for the
Mindlin equations, and Lewinski and T elega (1988) derived the homogenized bending and
transverse shear stiffness of Mindlin plates with properties varying in one direction. Kohn
and Vogelius (1984) introduced a more general homogenized Kirchhoff plate model where
the ratio between the variation of the thickness and the mean plate thickness was
considered as a variable. This work clariﬁed aspects of the behavior of rapidly varying
thickness plates and will be discussed later in Chapter 1. In 1986 Kohn and Strang
published a series of three papers (1986a,b,c) where they studied the relations between
optimal design and homogenization using variational calculus and convex analysis.

Francfort and Murat (1986) found optimal bounds for layered materials (Figure 1.4) using
G—convergence theory. Layered materials are built from two isotropic materials mixed at
microscopic scale. The way layering materials are constructed is explained in more detail in
the body Of this document. There were some important conclusions in the work of
Francfort and Murat: i) The strongest (and the weakest) material in two- and three-
dimensional elasticity can be built using the layering scheme, i.e., Hashin-Shtrikman
bounds are attained with layered materials. ii) Homogenized properties of composite
materials built with layers in more than one direction can be computed analytically. These
conclusions had important consequences on the development of layout optimization
problems using homogenization. Layered materials are also called rank-n layered
materials, where n indicates the number of directions of layering.

In 1988 Bends¢e and Kikuchi put together previous research and solved the layout
optimization problem (1 . 1.1) in plane elasticity. They used homogenization techniques and
an optimality criterion approach and solved the state equation using ﬁnite element methods.
The composite material that they used was a microscopically porous material modeled using
a square cell with a rectangular hole in it (Figure 1.5). Since porous rank-1 layered
materials have no stiffness for in-plane shear deformations, Bendsoe and Kikuchi might
have considered this as a potential problem in the use of layered materials and decided to
use the square cell instead. Recent investigations (Allaire and Kohn, (1993)) have shown
that non optimal micro structures, such as those modeled with the square cell, give
optimum designs that make less use of composite materials in comparison with designs
Obtained using Optimum micro structures, such as those modeled with layered materials.

The microscopic cell introduced by Bends¢e and Kikuchi (1988) was parameterized with
two variables a and b (Figure 1.5). The global orientation of the material axes at each point

"PM

(a) rank- 1. (b) rank—2
with orthogonal layers.
Figure 1.4. Layered material conﬁguration.

of the domain, 9 in Figure 1.5, is considered as a third variable in the problem. This
results in a total of three design variables at each location of the domain. In 1989 Pedersen
showed that the Optimal orientation of the material axes, O, in two-dimensional elasticity
problems for a single load case coincided with the axes of principal strains. This simpliﬁed
the algorithm to ﬁnd the optimum orientation of the material axes in such particular case,
and provided a clue for more complicated cases, e.g., multiple load cases and plate

problems.

In 1991 Diaz and Belding applied the Bendsoe-Kikuchi approach to compute optimum
layouts of trusses. Suzuki and Kikuchi (1991b) used the homogenized stiffness properties
computed in two-dimensional elasticity to build a plate model based on classical laminate
theory. By 1992 the approach of Bendspe and Kikuchi was well known and researchers
became actively involved in the extension of this method to other problems, such as
multiple load case problems (Diaz and Bendsoe, (1992)), and maximization of natural
frequencies (Diaz and Kikuchi, (1992)). An extensive review of the Bendsoe-Kikuchi
approach is available in (Bendsoe, et al. (1992)).

Recently, Lipton (1993a) proposed a new representation of effective properties of
Kirchhoff plates. He derived a four—variable representation for layering material plates
with an arbitrary but ﬁnite number of layers oriented in any direction (see Figure 1.6 for
a case Of three layers). He called this the “moment representation” since it was based
on moments Of a probability measure. This compact formulation allows one to work
with mixtures of a more general geometry, including multiple layers in arbitrary
orientations. This representation works very well with a new formulation of the
optimization problem introduced by Jog, Haber and Bendsoe (1992). Working in

       

isotropic
material

1

Figure 1.5. Square cell geometry used in (Bendsoe and Kikuchi, (1988)).

two-dimensional elasticity and a single load case, they carried out analytical Optimization
over the widths and orientation of the layers in rank-2 layered materials (Figure 1.4 (b)) for
a given amount of material and a given ﬁeld of strains. This allowed them to pose the
layout optimization problem using composite materials that align themselves to have the
optimum orientation and optimum relative layer widths for the given strain ﬁeld and amount
of material. They called these materials “self-Optimizing materials”. In this new
formulation the original optimization problem is split into two hierarchically related
problems: one local optimization problem where the stiffest arrangement Of ribs is found
for a given set of strains and a certain amount of material, and a global optimization

problem where they solve for the best distribution of material in the plate domain.

1.3 Outline

The work is organized as follows:

In Chapter 2 the derivation of the homogenized stiffness properties of Mindlin plates using
materials layered in two orthogonal directions (Figure 1.4 (b)) is presented. This had been
done in part by (Lewinski and Telega, (1988)) using layers in only one direction. Here we
present the application of asymptotic expansions to the Mindlin equations to plates with
layers in two orthogonal directions and study the derived formulas in the context of Shape
optimization. An extensive comparison of models of effective properties Of plates is also

included in this chapter.

10

In Chapter 3 a “moment representation” of the stiffness properties of Mindlin plates is
derived. Following the work done by Lipton (1993a) in Kirchhoff plates, effective
stiffness formulas for layered material Mindlin plates with L layers of ribs are derived for
bending and shear deformations. These formulas are expressed in terms of only four
variables. This is the ﬁrst time this representation is derived and implemented for Mindlin
plates.

In Chapter 4 we present a formulation and implementation of the mean compliance
minimization problem that makes use of the results of Chapters 2 and 3. We focus our
attention on a new formulation called here the hierarchical moment formulation. This new
formulation splits the optimization problem in two: one problem where the optimum
distribution of material is computed, and another where the Optimum micro structure is
computed for a given amount of material and strain ﬁelds.

In Chapter 5 example problems are solved using the different micro structure models
developed and discussed in previous chapters. The examples include a comparison with
previous results of Cheng and Olhoff (1981); a comparison between the rank-2 (Chapter 2)
and rank-L (Chapter 3) micro structures; a parametric study of variables involved in the
optimization problem; and two example problems that show the applicability of the
approach to mechanical design problems with applications in industry.

 

Figure 1.6. A layered material plate with three layers in arbitrary orientation.

CHAPTER 2
HOMOGENIZED PROPERTIES OF RANK-2 IVIINDLIN PLATES

2.1 Homogenization of the Mindlin Plate Equation

2.1.1 Background

The problem Of homogenization in composite materials deals with the computation of
homogenized elastic properties (also called effective or average properties) of a mixture of
two or more materials. When the mixture is periodic and occurs at a scale of order 8,
which is small compared to the scale of the body, the material is called “microscopically
periodic” or “periodic in a small scale”. Figure 2.1 Shows this two-scale concept in a
generic three-dimensional body. These materials have properties that depend on two
coordinates in different scales, a large scale coordinate, labeled x here, and a small scale,

x . . . . . .
labeled y = -. For Instance, the elastic tensor of a composrte material microscopically
8

mixed is Eijkl( x, y) = E,-J-k1(x,%). The periodicity of the material is present when the
properties satisfy Eijkl( x, y) =Eijkl( x,y+ p), where p is the period in the small scale.

When the periodicity is a function of x, the material is called pseudo-periodic. Any
function that measures mechanical behavior of a body made of such materials, e.g.,
deformation or stress, also depends on both coordinates x and y. After applying a
homogenization technique the homogenized properties Eijkl( x) are computed, and the
dependency on y is eliminated. Homogenization techniques for periodic materials have
been used very successfully to obtain effective properties Of a variety Of mixtures of
materials in different geometric arrangements (e.g., Duvaut (1976), Bourgat (1977),
Guedes and Kikuchi (1990)). These properties can be used to determine the large scale
behavior of a body made of such materials using standard procedures (e.g., finite element
methods) without a detailed knowledge of the small scale behavior. This feature makes
homogenization methods very attractive in shape optimization problems where the shape of
the body is represented as a material property (e. g., Bends¢e (1989)).

11

   

Large scale

x3

X1

Figure 2.1. Two-scale concept of microscopically periodic materials.

Microscopically Periodic Materials
Microscopically periodic materials are characterized by a basic cell that is repeated

periodically in the domain occupied by the material. It is advantageous to use the simplest
geometry capable of generating a reasonable complete set of stiffness properties for the
small scale mixture of the materials involved. Layered materials characterized by a rank-n
cell are particularly useful in shape optimization Since their homogenized properties can be

expressed in an explicit form.

Given two materials with properties E + and E ', respectively, a rank-1 material is built
stacking alternatively E+ and E' in a single direction, as illustrated by Figure 2.2 (a). A
rank-2 material is made by stacking one of the two basic materials, say E+, and the rank-1
material in a different direction (not necessarily orthogonal to the ﬁrst one) as shown in
Figure 2.2 (b). The process may be continued to build a rank-n material stacking E+ and

rank-(n-l) layers in different directions.

In layout and shape Optimization problems we need to work with materials with E'=0 in
order to produce holes in the domain. This case in not considered in the G—convergence
theory (Allaire and Kohn, (1993)). The reason is that positive definiteness of the stiffness
tensor is a basic assumption in the computation of effective properties. It is then necessary
to work with non degenerate materials to perform homogenization techniques. Therefore, a
“hole” is represented by a very weak material with E" at 0, one in which E +/E ' >>1.
This technique is referred to as “regularization” in the literature. Once the Optimum

topology is Obtained, a second step is required, i.e., the limit as E_ —9 0 needs to be

13

taken. In practice, the limit process is not performed and optimum topologies with Et/E'
>>1 are considered as ﬁnal results. In this context E + is called the strong material and E‘
the weak material.

y2+
E‘ E‘
y]

(a) rank-1. (b) rank-2 (orthogonal layers)
Figure 2.2. Layered materials.

 

Notation and Deﬁnitions

Large Scale
The notation used here was strongly inﬂuenced by the work Of Guedes (1990).

As shown in Figure 1.1, the domain occupied by the plate is denoted by .Q, a bounded
open set in 9i3. In this derivation we assume that .Q is the parallelepiped
.Q = (0,Lx)><(0,Ly)><(—h2,h2), as shown in Figure 2.3. It is made up by three plies
arranged symmetrically about the neutral plane [20 c 5K2. The three plies are labeled the
core ply QC” =90 x[—h1,h1], and the two external plies .er =00 x(ih2,ihl).
Each ply may be made of a microscopically periodic material, provided that the symmetry
of the plate about .00 is preserved.

The coordinate system for any point in Q is denoted by N = {x1,x2, x3 } = { x, y, z}. The
material properties of each ply do not change with the thickness direction z. It is
convenient to introduce a system of coordinates for points in [20 CERZ, denoted by
x = {x1,x2 } = { x, y I. The boundary Of .00, denoted by F, is divided into two parts, Fd
where displacements are prescribed, and I" where tractions are applied. They are such
that 1" d ﬂFt = Q. It is also assumed that F satisﬁes Lipschitz conditions (smoothness).

 

/ x2=y
X3=Z

 

    

x1=x

 

 

\ i

Figure 2.3. Large scale geometry of the plate

'90 CSR2

Small Scale

A basic cell is deﬁned as the Open set Y X (—hQ,h2), Y = (0,1)x 9?, as shown in Figure
2.4. A microscopically Y-periodic plate is built repeating periodically Y x (—h2,112) in
[20, which means that periodicity is not present in the thickness direction. The system of
coordinates for any point in the small domain is 8 = {y1, y2, y3 }= {x1/8, x2 / 8 , x3 }. The
system of coordinates for any point in Y is labeled here y = {y1, y2 } = I x1 / 8, x2 / 8 } .

y2=x2/8

    

Y3 =x3

 

Figure 2.4. Small scale domain.

Graphical Illustration of the Homogenization Procedure
We present here a graphical description of the construction of the basic cell used in

homogenization of a ribbed plate. To simplify the exposition we use a rank—l material to
build a plate with ribs along only one direction.

15

We start with two different isotropic materials, a strong one, Et, and a weak one, E‘,
which are used to build two plates made by three plies. We label strong plate the one
built using the strong material in the three plies, as shown in Figure 2.5 (a). Similarly,
we label weak plate, the plate built using weak material in the lower and upper plies and
strong material in the central ply (Figure 2.5 (b)). The stiffness properties of these two
plates are computed using classical lamination theory (Jones (1975)), which yields the
reduced stiffness matrix of each plate as a function of h1, h2, E+ and E'.

A small scale cell is built by cutting a very thin slice (of order 8) from the strong and weak
plates and placing the slices next to each other, as illustrated in Figure 2.5 (c). The result

is a basic cell whose thickness is large (2h2) in comparison with the other two

dimensions. The strong plate slice represents the ribs and the weak plate slice models

the gap between the ribs. This cell is repeated in the domain [20 C 912 to build a large

 

 

 

(a) strong plate \ \ \ / / l/ (b) weak plate

   

2112 y1=x1/8

(c) basic cell

x1

((1) Plate with
one family of ribs
(rank— 1)

 

0(8)

Figure 2.5. Graphical description of the homogenization procedure.

16

scale plate of dimensions Lx XLy X2h2, as shown in Figure 2.5 (d). The stiffness

properties of this plate change rapidly in a small scale.

In order to build a plate made of a rank-2 material one combines a thin slice of the strong
plate (Figure 2.5 (a)) with a thin slice of the plate made of rank-l material. This stacking
process is made in a second direction, e.g., x2. The resulting plate has two families of

ribs, one running in the x1 direction and the other in the x2 direction.

A similar procedure may be followed to build plates with different cross section.
Depending on the type of material used in each ply of the weak plate it is possible to build
plates with internal ribs (honeycomb) or perforated plates. Figure 2.6 shows three
possible arrangements for symmetric plates. Since the assembly of the weak plate is
performed before the homogenization procedure, the homogenized formulas developed
here also work for honeycomb and perforated plates. The honeycomb configuration is
particularly attractive not only because it is stiffer compared to a ribbed plate of the same
weight, but also because the ﬂat top and bottom surfaces of the honeycomb plate are
often more suitable for industrial uses than the corrugated, uneven surfaces of ribbed
plates.

 

rank-2
isotropic
rank-2

RIBBED

 

 

isotropic
rank-2
isotropic

HONEYCOMB

 

 

rank-2
rank—2
rank-2

PERFORATED

 

 

 

 

 

Figure 2.6. Assembly strategies for the weak plate.

17

2.1.2 Mindlin Plate Model

The derivation of effective properties for the ribbed plate is based on Mindlin’s plate
model. It begins here with the presentation of the equilibrium statement for such plate in a
form that is particularly suitable to carry out the homogenization process, described in the
next section.

The strain energy in a symmetric plate including in-plane, bending and shear deformations
is given by the expression

 

 

U =% j 8t(u)M8(u) + Kt(u)BK(u) + y‘(u)Sy(u) d!) (2.1.1)
~90
where
u = {th vy, v2, 6x, 6), }t is the displacement vector,
dvx av): 313x &vy t . . .
8 = , , + 18 the In- lane strain vector,
{ (9x 3y 3y 3x I p
80 ac 30 89
y x x y t °
— — , , lS the curvature vector,
1 ax 3y ax 3y I
dv 8v t . .
y = {6), + Bxl’ —6x + U—yl } IS the transverse shear deformation vector,
vx, vy, vz are the displacements of the neutral plane in
the x, y and z directions, respectively and
0x and By are the rotations about the x and y axes using

the right hand convention.

The matrices M, B and S are the stiffness matrices for membrane, bending and transverse
shear deformations , respectively. M and B can be defined as M E D0 and B a D2,
where D]. (j=0 or 2) are computed using

18

,10 _ hr .
Dj=DJ€p szdz+2[DJ-]ep Iz’dz (nosumoverj), j=00r2. (2.1.2a)
‘ho ho

while the transverse shear stiffness matrix is computed as
0 cp — ep
S=2h1[ 5 ] +2(h2—h1)[ _ ] (2.1.2b)

The superscripts 'cp' and 'ep' refer to core and external plies, respectively. Because of
their dependence on the ply thicknesses, entries in D0, D2, and S will vary rapidly in
regions of rapid thickness variations.

The matrices 0101’ and D1317 in (2.1.23) are of the form

a. 5,, If].
Dj= 522 £26 (2.126)
sym D66

Matrices DO, D2 and S, reﬂect the plane stress assumption (O'z = 0), i.e.,

13,7: DU- -013—3 D303] (i,j¢3, i,j=1,...,6) (2.1.2d)

The entries Dij in (2.1.2d) are the stiffness coefﬁcients of an orthotropic material.

It is convenient to organize material properties into a set of matrices Wk! of dimension

(5x3) such that the strain energy U in (2.1.1) may be expressed as the sum

'a-M
u U'

3
X jw’“ j dguka-iudﬂ (2.1.3a)
g:

l
2 100

ilj,

where

{8;} ={ .1} (2.1.319)

19

are operators in the macroscopic coordinates (x, y, z}. Explicit formulas for the matrices
Wk! appear in the Appendix. The notation convention in (2.1.3b) is unusual, particularly

the deﬁnition of 83, but it is very convenient for the derivation of the ﬁnite element model.

With this notation we may write equilibrium equations for the plate in the following form:

Equilibrium Equations with Rapidly Vming Coefﬁcients

Let u={vx,vy,vz,6x,9y}t and ri={5vx,6vy,5vz,66x,66y]’ . The equilibrium

solution a8 e V3 is such that

5 3 5
z j mf‘agufajaidrz = 21/ j ffriidQ + j tiﬁidrt 1, v a e V‘ (2.1.4a)
i,k=1j,€=1_QO i=1 90 Ft
where Vi is the space of kinematically admissible functions, e.g.,

r_ . 1 5.._ d
V _ uE(H (90)) .u—O on the boundary r (2.1.419)

In (2.1.4), f 8 = {fvx, fvy, fvz,0,0}‘ are the body forces (per unit area), and t is the

vector of traction forces on the boundary 1"t . The superscript 8 in f 8 and u’3 denotes the
dependency on the parameter 8. Assuming f"3 and t are sufﬁciently smooth, the
variational problem (2.1.4) has a unique solution as, Due to the variation of f 8 and Wk!

at the microscopic level, the solution depends on x and y, that is, u8 = u5(x, y).

2.1.3 Derivation of Effective Properties

We proceed now with the derivation of homogenized plate stiffness properties using the
asymptotic expansion technique. Details of this technique are fairly well known and will
not be repeated here. A step-by-step description is found in Bensoussan, et al. (1978)
or Guedes (1990). Following these standard procedures the generalized displacement
vector we is expressed as

u‘8 = u(0)(x,y)+8u(1)(x,y)+82u(2)(x,y)+--~ (2.1.5a)

where the functions “(1') (x, y) , j=0,1,...,n, are Y-periodic. Equation (2.1.5) is replaced in

 

20

(2.1.4) and terms multiplying the same power of 8 are grouped together. Since 8 is
arbitrary, each group can be set, individually, equal to zero. Then, limits are taken letting
8 —> 0, and using the property

If g: QOxY—ﬂi, then lim j g(x, y) do 10] j g(x y) de0 (2.1.519)
84000 "Y”!2

This process leads to two boundary value problems at different scales. The small scale

problem is
5 3 Y kt Y 5 3 kg y
2 2 Wmamxp (x.y)ajﬁ.-(y)dr=2 2 1% ajuxyidr, v iieV‘,
i,p=1j,m=lY i=1 j=1 Y

k=1,---,5 and €=1,---,3 (2.1.6a)

where the unknown function I,“ (x, y) e V3 is a Y-periodic “characteristic function”

deﬁned on Y for a ﬁxed x. V5 is the space of Y-periodic functions in the small scale

domain Y
5
2 {£202} 6 (H1(Y)) .' ﬁ(y) is Y—periodic}, (2.1.6b)
y (9 _d_
and the Operator diy is deﬁned as {d- I: (By, By’ ,.0} Notice that the Y-periodicity of
1 2

the functions in (2.1.6b) takes place of the usual boundary conditions and insures that the
problem is well posed.

The problem in the large scale is expressed in terms of the effective properties and does not
involve the small-scale coordinate y. Its solution is the unknown displacement u( 0)( x), a
solution to the problem

i.

3 5

Z [W]. “(xpgu(°)(x)ajai(x)do=2{ jﬁ(x)a,(x)dr2+ jt,(x)a,-(x)dr},

1 1313:1520 i=1 90 pt

v 12er (2.1.7a)

where WEJM (x) are the homogenized plate stiffness properties deﬁned by

 

21

3
1’ k!
We ||Y|—|l(Wiid" EZWJ’mamxp W (21.71»
p=1m=1

and

l
i( )=— {961 2.1.7
fx ”Yll‘lf Y ( c)

is the average value of the body force at x.

In the large scale problem all details of the micro structure are implicit in the homogenized
properties VIZ-lid (x). The explicit form of these coefﬁcients becomes available once

speciﬁc details of the micro structure are prescribed. To compute the homogenized
properties using (2.1.7b) one needs first to solve a series of small scale problems
(2.1.6) in the small scale domain Y. Depending on the particular details of the micro
structure used, this could be a very difﬁcult procedure. Fortunately, the process is
greatly simpliﬁed if one uses a layered material for which closed form expressions for
Brig/pd (in equations (2.1.6) and (2.1.7b)) are available. This is discussed in the next

section. The construction of the homogenized Mindlin plate model with periodically
varying stiffness was presented for the first time by Bourgeat and Tapiero (1983).
However, they did not make any choice for the geometry of the small scale domain. This
choice is made here picking layered materials to build the Y-periodic Mindlin plate.

2.1.4 Results Using Layered Materials

For simplicity, here and in what follows the discussion will be limited to the computation
of bending and transverse shear components of the homogenized properties WU“ (x).

They correspond to entries in the B and S reduced stiffness matrices in the expression for
the strain energy in (2.1.1). The homogenized version of these matrices will be labeled
B and S , respectively. The membrane component can be computed following the same
procedure. It is not surprising that they coincide with the effective properties found in
plane stress elasticity (Bendsoe, (1989)). Only the final formulas for membrane stiffness
will be presented in here.

Effeptive Prop_erties Using a Rank-1 Material
For a plate made of a rank-l material characterized by the basic cell (0,1) x SR x (-h1, hl ) as

shown in Figure 2.7, the characteristic functions 1;] in (2.6) are independent of y2.

 

I -a f
Figure 2.7. Basic cell in a plate made using a rank-1 material.

I
I a |

Entries in matrices B and S in (2.1.1) are computed by integration over 2 of the material
properties in the basic cell , as shown in (2.1.2a) and (2.1.2b). The computation of the
homogenized properties follows the procedure in (Bendspe (1989)). To illustrate this
procedure only the derivation of B( 2, 2)( x) E 174411( x) is presented in detail. It is the
effective properties associated with of D2( 2, 2 )( x, y ) a Wf11(x, y).

From (2.1.7b), using 82}, 1,56 = 0 (1;! does not depend on yz) and 33” =0, one obtains

— 1 41 1 Y 41
W411 =—llYlli(W41 - MM 214 )dY. (2.1.8)
Y

In order to obtain an expression for WflldlY 95:1“ appropriate indices k and Z and virtual

displacement functions v in the small scale problem (2.1.6) are chosen. In this case, let
k=4, 1 = l and ii = {0,0, 0, 124(y1),0}t. Then problem (2.1.6) becomes

1W441131Yxi 1(x. y)81Y124(y1 )dY = jwfllaly a4( y1 )dY for a fixed x (21.9)
Y Y

After integration by parts, and since the boundary integrals vanish because of the

periodicity of x”, the following equation is obtained
33/ WW 114%, y) = 83/ W4411 (2.1.10)

Integrating with respect to y1 it holds that

23

-1
41 Y 41 41 . 1 1
“1413114 =c1+W41 With c1=—— —— dY (2.1.11)
unlit/thy.)

With this expression for Will 1811,12” (2.1.8) becomes

-1
—41 1 1
W41 =— CldY= — dY . (2112)
ml, [Mien j
which is the ﬁnal result.

The procedure outlined above can be repeated to obtain each entry of the homogenized
reduced stiffness matrix. The equations needed for all entries in problem (2.1.6) are

—1
W513), 42:8 +W42 with c =_ W__5_412()’1)d
51 1965 2 51 2 I-l—lYllj—Sg-ll—- 5116101) YI—Il’llng 511(10’1)

—1
51 Y 51 51 .
“613115 =C3+W51 With C3=
l—lYlliW55+ 5116101)

-1
31 Y 31 31 .
14/313113 =C4+U61 With C4=— (2.113)
IIYIIIW— 3116101)

Due to the simple geometry of the rank-l cell, integrals in (2.1.13) may be simplified
considerably. The results can be expressed in a compact and convenient way in terms of
the arithmetic and harmonic averages, deﬁned as follows:

Arithmetic average. For all §e[0,1], the arithmetic average A of two real numbers
m1 and m2 is deﬁned here as the function

E(mitmaéhémr +(1-éf)mz (2.1.14a)

24

Harmonic average. For all §e[0,1], the harmonic average A of two positive real
numbers m1 and m2 is deﬁned here as the function

 

—1
A(m1,m2,§)=[§(m1)_l + (1 — army—1] = 5,1123}? am] (2.11419)

For positive m1 and m2 , A 2 A and A = A if and only if m1 = mg.

The homogenized stiffness properties in bending (B R1) and shear (SR1) of a plate made of
rank-1 materials may be expressed in compact form using these averages, as follows:

§R1(1,1)= A(B+(1, 1), B_(1, 1),a)

 

+ .—
BR1(1,2)=A(B+(1,1),B'(1,1),a)A[B (1’2) 3 (1’2) a]

B+(1,1)’ B"(1,1)’
_ (B+(1, 2 ))2 (B_(l, 2 ))2

 

 

—R1 — + —
B (2,2)=A B (2,2),B (2,2),a —A , - ,a +
i ) B+(1,1) B (1,1)
+ _
+22 31121311210 A(B+(1,1),B‘(1,1),a)
B (1,1) B (1,1)
ERIC, 3) = A(B+(3, 3), B'(3, 3), a) (bending)
5R1(1.1)=A(S+(1,1).S"(1.1).a)
SR1(2, 2) = A(S+(2,2),S-(2,2),a) (shear) (2.1.15a)
where the superscript “-” and “+” refer to the weak and strong materials, respectively,

and “a” is the area fraction Of the strong plate (Figure 2.7), i.e., the fraction of the unit cell
occupied by the strong material. The superscript “R1” indicates that a rank-1 layered

material is assumed.

Equations (2.1.15a) hold for ribbed, honeycomb and perforated plates. In each case the
matrices B and S must be deﬁned appropriately. Using matrices B and S for ribbed
plates, (2.1.15) yields the following expressions for a ribbed plate with ribs running
along only one direction

25

— 2

BR‘(1.1)=34(Eﬁnh3. Eﬁilhi+Ennrh23—h13). a)

_ _ 2 _
BR1(I.2)=BR1(2. 1)='3'A(E1122h23, EﬁnhiwnnrhS—hf’t a)
_ 2 — _
BR1(2,2)=3((1—122)A($mh§, 51222h13+52222(h§*h13), “Pr

2 3 _
v 4(1’2222’72, Eizzzhi +5222”; 4113): 0))
_ 2 .
BR1(3,3)=§A(E1212h3, Ehnhiwrmrhi-h?) a) (bending)

SR1(1,1) = 24(151313’72, 51313111 + E1.313012 — hr)» 0)
5 R1 (2, 2) = 2Z(E;323h2, E§323h1 + 132-323(h2 — h1 ), a) (shear) (2.1.15b)

Using matrices B andS for honeycomb plates, (2.1.15a) yield the following
expressions

_ 2 _

BR1(1»1)=§A(Eﬁ11h3» Errrrhfi'Eflrrthi-hiha)

_ _. 2 _

BR‘U 2)=BR1(2. 1178511221223. E1122h13+E11221h3—h13). a)
_ 2 _ _
BRIQZ):3((1—02)A($222hie 32222hi+113222(hi—hih “H

2 _
1) 43212221123, E22221113 +3222“; 4113), 0))
_ 2 .
BR‘(3,3)=§A(Ermh%. Ennhi+Etznth%-hi), a) (bendmg)

§R1(111)= 2451313112, E1313111 + E1313012 - ’11)» 0)
S‘Rl(2, 2) = 23 ($323112, 122—323h1 + 8:32th - hl ), a) (shear) (2.1.16)

Formulas in (2.1.15a) are the same as reported by Lewinski and Telega (1988) and
Lewinski (1991).

Formulas for bending stiffness in (2.1.15a) have the same structure of those derived from

plane elasticity equations using a rank-1 material (Bendsne (1989)). These formulas also
correspond to the membrane stiffness of ribbed plates, MR1:

MR‘(1.1)=2A(Eﬁnhn.Etmhr +El—111(h2 -11)..)

26

117R1(1.2) = 24(51'122’12: E11122111 + 151-12202 — hr): 0)
17 R112. 2) = 2((1— v2)Z(E§222h2, 19232th + 52222072 — hr), a) +
”2 4(51222’72’ E1222111 + E2222012 — hl): a))
17R1(3, 3) = 2A(E1+212h2, E1+212h1+ Isl—mm2 — h1 ), a) (2.1.17)

Effective Properties Using a Rank-2 Material

Once the plate made with a rank-1 material is built, one can build a plate using a rank-2
material following the procedure Showed in Figure 2.5. The formulas are easily derived
using (2.1.15a) and a stacking direction offset of 90 degrees, as shown in Figure 2.8.
This micro structure is fully described by two layer widths, a and b, and by the global
orientation of the micro structure, 6. A model for rank—L (Le N+) material plate will be
given in next Chapter using a different approach.

 

 

Figure 2.8. Basic cell of a plate made of rank-2 material.

In this case and using the approach of the previous section, we Obtain the following results:
+ 2 —R1 2
— — —— -— 2 M 1, 2
MR2(1,1)= A(M+(1,1),MR1(1,1),b)—A M—+£1—)—,_R—1()—
M (2,2) M (2, 2)

Z2[M+(1,2) 17Rl(1,2)

+ “'Rl
M+(2.2)'17R1(2.2)’b)A(M (2,2),M (2, 2), h)

+ —R1
‘Rz =1I7R2 2,1 =A M 2,2 ,17’“ 2,2 ,b A wilﬂh
M (1,2) ( ) _( ( ) ( )) M+(2,2)MR1(2,2)

27

17mm, 2) = A(M+(2, 2),1_1R1(2, 2),b)
17 R2 (3, 3) = A(M+(3, 3 ), M‘ “(3, 3 ), b) (membrane)

 

+ 2 —R1 2
§R2(1,1)=Z(3+(1,1),§R1(1,1),b)—Z[B (1’2) B (1'2) b]

B+(2,2)’ §R1(2,2)’

 

Zz B+(1,2) §R1(1,2)
B+(2,2)’§R1(2,2)’

b]A(B+(2, 2 ), E “(2, 2 ), b)

 

+ "Rl
BR2(1,2)=BR2(2,1)=A(B+(2,2),BR1(2,2),b)Z[B(1’2) 3 (1’2) b]

B+(2,2)’ §R1(2,2)’
§R2(2,2) =A(B+(2,2),BR1(2,2),b)
1‘3 R2 (3, 3) = 4(13‘73, 3 ), 13 “(3, 3 ), b) (bending)

SR2(1,1) =Z(s+(1,1),SR1(1,1),b)
SR2(2, 2) = 4(s+(2, 2), sRl(2, 2), b) (shear) (2.1.18)

where “b” is the area fraction of the strong plate in the second direction (Figure 2.8). The
superscript “R2” indicates that a rank-2 layered material is assumed. Layered materials of
second rank were also used by Lurie and Cherkaev (1986) for Kirchhoff (thin) plates,
where homogenized transverse shear stiffness (SR2) cannot be computed. In the case of
rotationally symmetric bending of thin plates the rank-2 micro structure was also used by
Rozvany, et al. (1987).

The effective properties in (2.1.18) summarize the result of this section. They express
the average properties of a plate whose cross section varies in the small scale in terms of
mixture of strong and weak materials E + and E'. Formulas (2.1.18) can be applied to
ribbed, honeycomb, and perforated plates, simply by assembling the plies in the weak
plate in a different order, as outlined in Figure 2.6. The properties of a ribbed plate are
obtained setting the stiffness of the weak material to zero. The formulas are used in this
manner in optimization formulations for shape and layout of plate structures, as outlined
below.

28

2.2 Comparison With Other Models For Ribbed Plates

Several models for the calculation of homogenized properties of plates of rapidly varying
thickness have appeared in the recent literature. Some of these models are suitable for
optimization of the plate geometry and its shape. In this section we present results that
can be used to compare four such models. To facilitate the presentation the study is
limited to plates with only one family of ribs, with cross sections as shown in Figure
2.9 and built using a rank-1 material. It should be observed that even though shape
optimization problems call for the use of a rank-2 material, when the optimum solution
prescribes that ribs be used, it is typical that ribs are aligned along a single direction in
ribbed and honeycomb plates.

”In:

 

u l-u

Figure 2.9. Geometry of plate cross section.

The homogenized plate models compared are

Mindlin Plate Model (M)

This is the model discussed in the previous section and summarized in
(2.1.15b).

Laminated Plate Model (L).

In this model the plate is made assembling three plies. The external plies
have homogenized properties computed using the plane elasticity equations.
This idea was discussed by Caillerie (1984) and implemented by Suzuki and
Kikuchi (1991b).

Kirchhoff Plate Models (K and K')
This refers to two models derived by Kohn and Vogelius (1984).

29

The following properties of isotropic materials will be used in the homogenized stiffness

 

 

 

 

. E1111 E1122 0 E 1 v 0
D's” = E2222 0 = 2 v 1 O (in plane stress) (2.2.1a)
1—v l-v
sym E1212 0 O T
- E 0 l 0
ngso = 1313 = E (in transverse shear) ( 2.2.] b)
0 E2323 2(1 + 1)) 0 1

where E is the Young's modulus, and v the Poisson ratio. The ribs are aligned along
direction “2” of the local material axes.

To facilitate the comparison we introduce the following arithmetic and harmonic averages
of the stiffness of two plates of thickness 2h1 and 2h2, respectively:

UBijkl=gE§Jlk1K(h§, h13, u) (for bending) (2.2.2a)
UBijkl =2Egk1/T(h2, hl, u) (for transverse shear) (2.2.2b)
LB-- -ZE-‘f A(3 h3 ) b d' ) (222)

ykl—3 Ukl— 1‘12, 1, u (for en mg , , c
B k,=2E,jfk,A(h2, hl, u) (for shear) (2.2.2d)

The reader may notice that UB corresponds to the classical Voigt bound for elastic
properties of a mixture of two materials while LB corresponds to the Reuss classical
bound. These should not be confused with optimal bounds found in the literature, such as
for example those in Francfort and Murat (1986). Bounds U3 and LB are used here only
to illustrate the relative magnitudes of properties computed using different modeling
assumptions.

2.2.1 Comparispn of Effective Properties in Bending
Homogenized Mindlin Plate Model (M)

This is the model developed in Section 2.1 using a rank-2 layered material. The
homogenized properties are the same as BR] and SR1 in (2.1.15b)

“.m’ .A—hb—' ' I _

3O

§M(1,1)=%A(El+111h23, E1+111hl3+Ef111(h2°’-h13): u)
§M(1,2)=§M(2’1)=%A(Eﬁ22h23, Eﬁ22h13+Ef122(h23-h13), u)
_ 2 _ _
BM(2.2)=-3-((1-v2)A(Ez+222hr§, 52222h13+32222(’123-h13), u)+

2 3 —
v 4( 53222172: 523221113 +52222( ’12? 4113 ): W)
_ 2 _
BM (3, 3) = 34(Ef212h23, Elezhf’ + 151212013 _ 1213), u) (2.2.3)

The superscript “M” (instead of R1) is used here to indicate that this is the result
associated with the “Mind ' ” model.

All entries in 17M , except entry (2,2), are at the lower bound LB (recall that ribs are
aligned along direction “2”) . Entry (2,2) is a convex combination of the lower and upper
bounds. For typical values of Poisson’s ratio, the bending stiffness along the rib
direction is essentially equal to the upper bound.

Homogenized Laminated Model (L)

In this model the plate is made by three plies as shown in Figure 2.10. In contrast to the
Mindlin model the properties of the external plies are the homogenized material properties
obtained from plane elasticity equations. The central ply is made of an isotropic material
and the external plies of a layered rank-1 material. The homogenized properties are given
by (2.1.16) and the assembly process is described by the following equation:

_ ho hl .
B L = DCP J zzdz + 21$" 1 zzdz = an); — h? )12R1 + h13D‘30] (2.2.4)
" ho

The homogenized plate properties of the laminated plate are derived using (2.2.4).
_ 2 -
3171,” = 3((h23 ‘h13)A(EfL111: E1111, u)+h1351+111)
_ ._ 2 _
BL(1, 2) = 31721) = 3(013 — hi3)A(E1+122r E1122» “)+ h13131+122)
— 2
BL(2, 2) = 3M; — h13) x

((1 - DZ)Z(32222: 52222, u) +0241 52322» 132222, W) + h1313222]

31

_ 2 _
BL(3,3)=§((h§—h13)A(Ef212, E1212, u)+h13E1+212) (2.2.5)

In model L (laminated) the assembly of the plies (Figure 2.10) is carried out after the
homogenized properties of each ply are obtained . As a result, in model L the thicknesses
h1 and h2 do not appear inside the averages /_l and X . In contrast, in model M
(2.2.3) the assembly of the plies occurs before the homogenization step. Core and rib
thicknesses and material properties are homogenized together . This is a fundamental
difference between models M (Mindlin) and L (laminated). The particular case of a plate
with uniform thickness (i.e., when u is set to 1 in Figure 2.9) is correctly predicted by

equations (2.2.5).

Homogenized Kirchhoff Plate Models (K and K')

Kohn and Vogelius (1984) developed expressions to compute homogenized stiffness
values for a thin ribbed plate using three-dimensional elasticity equations and the following
asymptotic expansion

Homogenization
using 2D elasticity
E Y2
YI
rank—1 cells
E 3’2
Y1 ..

x2

   
  
  
      

Assembly using
eq. (1.1 .2a)

   

2h1

2h2

  

x1

Large scale plate

Figure 2.10. Laminated plate model.

32

k
u8 = 2 ant/”Rx, x/e", z/e) a>O
n=0

The parameter “a” determines the length scale of the thickness variation. They found three
types of homogenized properties, depending on the value of the parameter “a”. Here we
summarize the results for a<1 and a>1. In both cases it is assumed from the start that
the “weak” material E‘ is absent.

For a < 1 the length scale of the variation of the plate thickness is larger than the mean
thickness. The homogenized plate stiffness properties in this case are (Kohn and
Vogelius, (1984))

PK =
E .405 113 u) E A(h23 h3 u) 0
2 1111— r 1’ 2 L122; 3,1, 2 3 3
‘3‘ E2222((1—v )A(h2,h1.u)+v mambo) 0
Sym E12123(h23,h13:“)
(2.2.6a)

This result can also be derived from the homogenization of Kirchhoffs equations by
Duvaut (1976). Therefore, in plates with ribs along only one direction the case a<1
corresponds to the standard homogenization of a Kirchhoff plate.

Comparing (2.2.6a) with the result from model M (2.2.3) we may conclude that in
plates with ribs along only one direction and no weak material (Eli/d = 0) models K

(Kohn and Vogelius (1984) with a<1) and M (Mindlin) result in the same averaged
properties 520,1), 52(1,2), and 52(22). The models, however, predict different
values of the twisting stiffness §2(3,3). In the Mindlin model 526,3) is a lower
bound while in the Kirchhoff model it is an upper bound. This difference will play a
role even in thin plates.

From the geometric point of view models M(Mindlin) and K (Kohn and Vogelius (1984)
with a<1) represent different ribbed plates. When the size of the cell 8 tends to zero in the
homogenization process, the periodicity increases and ribs become thinner compared to the
plate thickness. Therefore, model M (Mindlin) corresponds to plates with slender ribs.
Instead, model K (Kohn and Vogelius (1984) with a<1) corresponds to the case when the

33

period is much greater than the plate thickness (“stubby” ribs). This fundamental
difference is only manifested in the twisting stiffness E 2(3,3). The comparison of these
two models is still useful since they are the most used plate models in the engineering
practice.

The second model derived by Kohn and Vogelius corresponds to values of the parameter
a > 1. In this case, the thickness varies in a scale smaller than the mean plate thickness.
The homogenized plate stiffness properties are

3 3
' 2 E1111h1 51122’11 0
BK =3 E2222((1—v2)A(h23,h13,u)+v2h13) 0 (2.2.6b)
Sym E1212h13

Notice that these equations, derived by Kohn and Vogelius from three-dimensional
elasticity, are not valid for plates of uniform thickness 2h2 (i.e., when u is set to 1 in
Figure 2.9). Setting u=1 in (2.2.6b) all entries in the stiffness matrix depend on h1, even
though the plate is of uniform thickness 2h2. Also notice that another way to derive

equations (2.2.6b) is setting Erik! = O in equations (2.2.5). However, this is considered

an improper operation since the homogenization procedure carried out in two-dimensional
elasticity is only possible under the assumption of positive deﬁniteness of the elasticity
tensor at each point of the microscopic cell. Using Egk, = 0 in equations (2.2.5) violates

this assumption.

A summary of the results for each model is presented in Table 2.1, where properties of
the weak material E' have been set to zero throughout. In order to show the fundamental
difference between model M and model L we also include the model L for the particular

(but improper) case when Eﬁkl = 0. These results are presented in graphical form in

Figures 2.11, 2.12, 2.13, and 2.14 , displaying the effect of changes in the rib
thicknesses and height on individual entries of the plate bending stiffness matrices.
The plate cross section is as shown in Figure 2.9. An isotropic “strong” material with
(E, v) = (100, 0.3) is used in computations.

Effect of Rib Width This experiment measures changes in the stiffness of the plate as the
width of the rib (u) increases. The core thickness OM) and the total plate thickness (2h2)

34

Table 2.1 Summary of homogenized properties of one-family, ribbed plates.

 

Mindlin Model (M):

FM =
2 E11114(h23'h13’“) E11224(h3.h13,u) 0
3 E2222((1—vbﬁhihﬁu)+v24(h23,hf’.u)) o

sym 13121240723, 1113.“)

Kirchhoff model for a<1 (K):

5K:

3 3 3 3
2 E11114(h2’h1»u) 511224(h2,h1’“) 0
3 E2222((1—v2 )A(h23, 1213, u)+v24(h23,h13,u)) 0

Sym E1212Z( ’12)) h13 : u)

Kirchhoff model for a>1 (K'):

3 3
2 Ellllhl E1122h1 0
3" = 3. 52222 ((1 — v2)A(h§’, h13, u) + (91113) 0
3
Sym E1212h1

Laminated model with rank-1 material (L):

3 3
51111172 51122111 0
E2222(u(1— (9)1123 + (1 — “(1 — v2))h13) 0

3
Sym E1212h1

§L=3
3

 

 

 

35

are ﬁxed. This is the way these formulas are used in layout and shape optimization of
plates (Suzuki and Kikuchi (1991b)).

Figure 2.11 shows results for B 2(2,2), and 15—32(12). As expected stiffness in the rib
direction (measured by B 2(2,2), Figure 2.11 (a) ) increases with the addition of material
for all models. Differences between models are not very signiﬁcant. Figure 2.11 (b)
illustrates the interesting observation that in models K' (Kirchhoff with a>1) and L
(laminated) B 2(1,2) is below the lower bound LB and does not change with the rib
width u. From Table 2.1 we may conclude that the same may be said about other terms
involving stiffness in the direction perpendicular to the ribs, [—3 2(1,1) and B 2(3,3). This
can be explained by the fact that a>l in the Kohn—Vogelius model implies that ribs are
modeled as beam elements with no transverse bending stiffness.

All models predict a monotonic increase in bending stiffness along the rib direction as
rib width u increases. This is an important property used in optimization algorithms based
on optimality criteria methods.

Effect of Rib Height This experiment measures the change in stiffness when ribs are
added to a core plate of constant thickness 2h. The height of the ribs is increased but the
amount of material used to build the ribs is kept constant, i.e., taller ribs are thinner. A
low amount of material, equivalent to 20% of the core material, was available to build the

ribs.

 

 

     

 

 

 

 

 

 

 

M, K and LB
K .
0 : : : 0 : l L and K:
O 0.25 0.5 0.75 1 O 0.25 0.5 0.75 1
Rib width, u Rib width, u
(a) (b)

Figure 2.11. Effect of rib width on material properties

36

Figure 2.12 shows results for B 2(2,2) and B 2(l,1). As expected, all models predict
a stiffening of the plate in the rib direction (as measured by B 2(2,2), Figure 2.12 (a) )
as rib height increases. But increasing rib height may have the opposite effect on the
stiffness in the direction perpendicular to the ribs, direction 1 . Figure 2.12 (b) shows
that in models M (Mindlin) and K (Kirchhoff, a<1) as the rib height increases, the
stiffness B 2(1,1) decreases, approaching the value of B 2°0r6(1,1) for very tall and
thin ribs. This is not the case in models L (laminated) and K' (Kirchhoff for a>1), where
B 2(1,1) is equal to the core stiffness for any rib height, as shown in Table 2.1. In
these models the addition of ribs does not have any effect on the stiffness of the plate along
directions perpendicular to the rib. This qualitative difference will be of importance later,
when results are presented that display the effect of changing the rib width on a plate built
with only one family of ribs.

The addition of ribs to a core of uniform thickness always stiffens the plate. Increasing rib
height while keeping a constant volume of rib material has a stiffening effect on values in
B deﬁned as upper bounds ( typically, B 2(2,2) in all models ) and a weakening effect
on entries deﬁned as lower bounds (typically, B 2(1,1) in all models ).

§(2,2) _
160 B(l,1)
Volume = 0.8, 2h1 =1 160

 

 

Volume = 0.8, 2111 =1 ‘

    

    
 

120" 120"

80" 80"

M, K, L and K'

 
 

M, K and LB

   

 

 

 

 

 

 

 

 

 

40‘ 40.. L and K'
LB]
0 : 1: = : 0 : : : :
0.8 1.2 1.6 2 2.4 0.8 1.2 1.6 2 2.4
(122 -h1)/h1 ('12 411 W11

(21) (b)
Figure 2.12. Effect of rib height on material properties.

37

2.2.2 Comparison Based on the Mean Compliance of a Simply Supported Plate

Here we use a simple plate structure to highlight the differences between the models
discussed in the previous section. The global stiffness of the plate is measured through the
computation of the mean compliance deﬁned as

C: jf‘d d9 (2.2.7)
90

where f is the vector of applied loads and d the vector of generalized displacements. This
is a measure that is commonly used in layout and shape optimization problems. In what
follows C is normalized using the compliance of the plate without ribs analyzed using
Kirchhoffs model K (a<1).

Consider a square plate simply supported on all sides subjected to a uniformly distributed
transverse load. The analysis of this plate is performed using a ﬁnite element
discretization and 4-noded quadrilateral elements. A single material is used in all
elements but the orientation of ribs may vary from element to element. The orientation of
ribs within each element, 9, is set to match the angle that minimizes the mean
compliance of a transversely orthotropic plate subjected to a uniform transverse load
(Pedersen, (1989)). Notice, however, that all plate stiffness components play a role in
the computation of the mean compliance, even when the orientation of the ribs is chosen to
take full advantage of the stiffest material direction.

Homogenized properties derived from plane elasticity equations do not provide for a
mechanism to compute the transverse shear terms S . A typical approach followed in
practice is to view the transverse shear term (S in (2.1)) as a penalty contribution in the
strain energy that enforces Kirchhoff's assumptions (31:0), that is,

_ 11 0
S =-— 0 < e «1 (2.2.8)
8 0 1

This view is adopted here in the analysis of the plate using the L (laminated) model. For
comparison, results are also shown of the analysis associated with model M (Mindlin),
modiﬁed to enforce the same condition. These results are identiﬁed as model M' in
Figures 2.13 and 2.14. All computations of Kirchhoff plates were done using the ﬁnite
element model for the Mindlin equations along with condition (2.2.8). This procedure

38

produces good results for the simple case studied in here. Comparisons with analytical
solutions for uniform thickness Kirchhoff plates validate this procedure.

Effect of Rib Width
Here we study variations in the mean compliance of a ribbed plate when the width of
the ribs is varied without changing the core or rib thicknesses.

For thin plates, Figures 2.13 (a) and 2.13 (b) show that plates whose effective properties
are computed using model M are consistently stiffer than plates analyzed using the
laminated model L. If the total amount of available material is ﬁxed, model M may
predict that it is possible to achieve a prescribed stiffness using a plate with thinner ribs
(u near 0). However, to achieve a similar stiffness using model L it may be necessary
to rely on plates of nearly uniform but higher width (u near 1). This suggests that in
an optimization problem where compliance is the cost function model M is more likely to
produce results that call for the use of ribs. Figures 2.13 (a) and 2.13 (b) indicate that the
difference will be more pronounced in plates with shorter ribs ((hz-h1)/h1 = 0.25).
Examples in Chapter 4 will show this phenomenon.

Differences between thick and thin plates depend on the contribution of the transverse
shear stiffness S, present only in model M. As expected, Figures 2.13 (c) and 2.13
(d) show qualitative changes only in model M and then only when shorter ribs are used
(Figure 2.13 (c)). In that case the effect of the choice of model on the stiffest solution
is unclear. Then it is more likely that differences between optima associated with models
M and L arise because of other factors, such as the effect of the transverse shear
contribution to the objective function on the optimum orientation angle 9.

Effect of Rib Height

Here we study variations in mean compliance when a ﬁxed amount of material is used to
add ribs of varying heights (kg) to a plate of uniform thickness 2h1.

Figure 2.14 shows the results for a thin and a thick plates and different amounts of rib
material. It is observed that although the ribs increase the stiffness of a plate of uniform
thickness, taller ribs are not always associated with lower mean compliance. This
occurs here because in some models (e. g. M and K), increasing rib height may actually
decrease stiffness in the direction transverse to the ribs. The non monotonic behavior
of the compliance as a function of rib height is more pronounced in the Mindlin model

39

 

Land K'

 

04'“ M and M'
_ Thin Plate (L=100)
(h2 - h1)/h1 = 0.25

‘.O
N
1

 

 

 

0 0.2 0.4 0.6 0.8 1
Rib width, u

 

 

Thick Plate (L=5)
02" (h2 - h1)/h1 = 0.25

 

 

 

o 0.2 0.4 0.6 0.3 1
Rib width, u
(C)

 

 

 

 

 

 

C
1 Thin Plate (L=100)
0.8" (1‘2 ' h1)/h1=2
0.6 -- L and K'
0.4 --
0.2 --
0 l l l 5
o 0.2 0.4 0.6 0.8 1
Rib width, u
(b)
C
1.2
Thick Plate (L=5)
1 M (h2 - mm = 2

 

 

 

 

 

0.4 0.6 0.8 1
Rib width, u

((1)

Figure 2.13. Effect of rib width on mean compliance.

40

because all transverse stiffness terms in that model decrease with rib height. Model
K (Kirchhoff with a<1) presents a similar behavior, less pronounced because the
twisting stiffness B (3,3) in model K increases with rib height. In contrast, the
compliances of plates modeled using models K' (Kirchhoff with a>1) and L (laminated)
varies monotonically with rib height, since twisting stiffness terms in these models are
independent M.

The presence of “weak” (lower bound) entries in the twisting stiffness entries in B makes
it possible that increasing the height of short ribs within a constant amount of material
result in more compliant plates. However, there appears to exist a rib height beyond
which this behavior is reversed and the addition higher and thinner ribs does result in
stiffer plates.

Regarding the differences between Mindlin and Kirchhoff model K (a>1), it is
interesting to note that these differences are not limited to the effect of transverse shear
deformations. When transverse shear contributions are neglected (as in model M'),
differences in the mean compliance associated with these models can only be explained
by differences in the twisting stiffness term B (3,3), as it is easily appreciated
comparing BM and B K in Table 2.1. Figures 2.13 and 2.14 suggest that the effect
of this term, far from being negligible, may be comparable to the effect of differences in

the treatment of transverse shear deformations.

41

 

 

    

 

 

 

3 1.5'
C Thin Plate (L=100) C N
2-5 Volume=0.8, 2h1=l L and K \
2 .. 1 ‘
1.5"
M and M. K M and M1
1 0.5“
Thin Plate (IFIOO)
051- Volume=0.2, 2h1=1
0 : . . O : 1
0.8 1.3 1.8 2.3 0.2 0.6 1 1.4
(’12 -h1)/h1 (h2-h1)/h1
(a) (b)
C M
1.6 - ‘/

    

 

0.8

 

 

 

1.2-

0.4 ~-

 

 

 

 

 

 

 

- \
m
./ \/r\
.. J M,

Thick Plate (L=5)

 

 

 

 

0,5" Thick Plate (L=5)
Volume=0.8, 2h1=1 0 Volume=0.2, 2h1=1
O l 1 : : :
0.8 1.3 1.8 2.3 0.2 0.6 1 1.4

(112 - h1)/h1
(C)

(hz - h1)/h1
((1)

Figure 2.14. Effect of rib height on mean compliance.

CHAPTER 3
HOMOGENIZED PROPERTIES OF RANK-L MINDLIN PLATES -
MOMENT REPRESENTATION

In this chapter formulas for stiffness properties of Mindlin plates made of layering materials
of rank L (Le N“) are developed (see Figure 3.1). The approach is different from the one
used in Chapter 2, and it is justiﬁed by the following observations:

0 These new formulas describe plates with L different layers of ribs
oriented in any direction and having arbitrary widths. This is a more
general model than the one presented in Chapter 2.

0 This formulation uses only four variables to describe the full anisotropy
of the plates. Even though there are L layers -- each one with a different
width and orientation -- it is possible to condense all the information in just
four variables once the amount of material is prescribed.

0 It will be seen later in this chapter that the stiffest micro structure for plate
problems is made of at most three layers, which is a special case of the
rank-L material plates.

The derivation will be presented for the bending and transverse shear deformation only.
Membrane effective properties where derived by Avellaneda and Milton (1989) and will be
presented here in their ﬁnal form.

N t ti n Convention
The following notation convention is adopted in this chapter: indices repeated twice imply

summation; Latin indices take values 1 and 2, while Greek indices take values 1, 2 and 3.
When index notation is used the convention is as follows: let a and b be vectors, C and D
be second order tensors (matrices), and E and F be fourth order tensors. Then,

42

43

 

w) -I-I-

cross section
Figure 3.1. Rank-L layer plate (L=3 in this picture).

a - b = aibi (a scalar)
C:D =C-jD-j (a scalar)
E: F: E: jkllfl'jkl (a scalar)
Da = Dija j (a vector)
a®b=aib- (amatrix)
BD= Bi] lekl (a matrix)

a ® a (8) a ‘8 a- — ai ajakal (a fourth order tensor)

3.1 Effective Properties for Rank-L Material Plates

Two plates of thickness 2h1 and 2h2 (hz > h) (see Figure 3.1) are mixed in a small scale to
build a plate of rapidly varying thickness. There are ribs in L different directions and of
arbitrary widths arranged in L different scales in a layering arrangement as shown in
Figure 2.5. The goal is to obtain an expression for the effective properties of this plate.

3.1.1 Set up of the Mindlin Plate Equations

As before, the plate occupies a domain

44

12=(xes)t3.- (x1,x2)er20c3t2, 4123x3912) (3.1.1)
where [20 is identiﬁed with the mid plane about which the plate is symmetric.

The plate is loaded by a total transverse load per unit area f applied over [20 , a transverse
shear force q applied over the boundary 1" 4 c_: 8120, and moments M={M1,M2} applied
over the boundary 1" M g 8.00. The notation is simpliﬁed replacing rotations (0);, ﬂy) by

(6 2,-6 1), and the transversal displacement vz by w. As before, the transverse shear strain
vector, y={ y1,y2}, is defined as

aw .
1’1 = b;- 91-, 1:12 (3.1.2)

Equilibrium of the Mindlin plate is satisfied by a solution u = (0 1,6 2, W)EV€ such that

((13%: vé+5y.y)do= jwfdr2+ jwqdr- (der, (3.1.3)
r20 120 1“! FM

for arbitrary functions a = (01,92,122) in the space of kinematically admissible solutions,

V"=(u=(61,92,w)eH1(r20)3. (91,92)=0 on FM, w=0 on r4 ) (3.1.4)

The operator V7 is deﬁned as ViGj E -;-(Vi6j +Vj0i). The tensors B(x1, x2) and S(x1,

x2) are, respectively, the bending and shear rigidity tensors obtained after integration
through the thickness of the material properties at location (x1, x2) 6.00 . Let Bk and

5" be the rigidity tensors corresponding to the cross section k, k=1,2. These tensors
have the symmetries

Bkmnpq = kaq M and Bkmnpq = qupmn = Bkmnqp (3.1.5a)

and

Skmn = Sknm (3.1.5b)

45

It is assumed that properties B2, and S2 correspond to a cross section of thickness
2h2 built using an isotropic material, i.e., for a material with Young’s modulus E and
Poisson’s ration v , 32 and S2 are of the form

(l-V)

 

B12111 = B22222 = 30 B12122 = V30 B12212 = 2 Bo, (3.1-6a)

$121 = s§~2 = so 5122 = o, (3.1.6b)
where

Boa3 Eh; 05—1-3112— (3.1.7)

 

1+v'

B2 is the “strong” cross section. In layout optimization B2 corresponds to the section
of the plate reinforced with stiffeners. B1 is the “weak” or unreinforced cross section, and
it is assumed that B1 < B2. No special orthotropy on B] is assumed.

3.1.2 Computation of Homogenized Stiffness Properties

As in the previous chapter, stiffness properties B and S depend on x and y=x/8, and are
Y-periodic, i.e., B = Bg(x, y=x/e), S = S€(x, y=x/8) , B3(x,y) = B€(x, y+Y),
S€(x,y) = S€(x,y+Y).

Accordingly, the rotations 6,- and displacement w also depend on the small scale
parameter y, therefore, 6 = 68(x, y=x/e) and w =w€(x, y=x/e) and the weak
equilibrium equations (3.1.3) is

j ({Be(x,y)vo€(x,y):vé+

{20 Y

S€(x,y)78(x,y)-}7/d7d[2= Jﬁzfdﬂ+ jwqdr- jé-Mdr (3.1.8)
{20 1“! PM

In (3.1.8) it was assumed that the external loads are independent of y and have made use

of the simpliﬁed notation

46

ﬂ. (1.] dYE ( (.1317. (3.1.9)
Y Y

We seek average tensors B and S that represent the limiting behavior of tensors Be and
SE, respectively, as 8 —> 0. That is, let us be the solution of (3.1.8) with B8 and Se.
We seek effective properties B and S such that, if us —) 17 as 8 —) 0, then 17 is the
solution of (3.1.8) with (Be, SE) replaced by (B, S). The computation of B and S is
again carried out using the asymptotic expansion in (2.1.5a). Only the ﬁrst two terms of
the expansion will be considered; this yields, in terms of the displacement vector,

u51x. y) = (618,628, we} = (01%», 931x). w0(x)/+ 21010:. y). 951x. y), Ma )1) (3.1.10)
The equilibrium statement for the small scale domain is:

For a ﬁxed point x e {20 ,

j B(1?+K*): v7, édY=o, for all Y-periodic 19(y)eH1(1()2 (3.1.11a)
Y

and
(5(7+y*).vy WdY=0, for all Y-periodic w(y)eH1(Y), (3.1.11b)
Y

where the following notation was introduced,

I?( x) E {7x 60( x) the mean value of the bending strain;
7(x) E wa°(x) — 0°(x) the mean value of the shear strain;
K *( x, y) E V), (91( x, y) the ﬂuctuating part of the bending strain, and

7 *(x, y)EVyw1(x, y)—61(x, y) the ﬂuctuating part ofthe shear strain.
(3.1.12)

Subscripts x and y in (3.1.12) indicate that the partial derivatives are with respect to the x
and y coordinates, respectively.

47

The equilibrium equation for the large scale domain is

)0{|—L'YlljB(K+K*)dY}:13dQ+joi{ —|i[s(7+y*)dY} y d!)—

llYl

jwfd12+ jwqdr— (M o Mara vueV‘ (3.1.I3a)
90 [‘4 PM

where I? and )7 are deﬁned as follows,
£(x)=vxé, ?(x)=wa—6l. (3.1.13b)

Equation (3.1.13) is the equilibrium equation of a plate with effective stiffness properties
B(x) and S(x) deﬁned as

§r=ij B(E+K*)dY (3.1.14a)
IIYIIY
s7 =I—LIYIIJ 5(7+y *)dY. (3.1.14b)

The geometric configuration of the basic cell Y is now prescribed in order to solve
(3.1.11).

Rank-1 material plate
In this section homogenized stiffness properties of a plate with ribs orthogonal to a generic

unit vector n are derived. The plate is built by repetition of the basic cell shown in Figure
3.2 and deﬁned as

Y = Y] U Y2,
=(012223):(y11y2)€s[)0,1]><91,y3 61-h1,h1],p 6 [0,1]}
Y2 =l(y1,y2.y3): (y11y2)610,p]><9i, y3 (El-1121121.)? €10.11}. (3.1.15)

It is assumed that stiffness properties, bending strains and transverse shear deformations
are constant in the sub domains Y1 and Y2 , i.e.,

B = 1,131+ 1232

48
s = 1151+ 95252
1?+ K4 = x1K1+ 252112

7+1” = 1171+ 76272 (3.1.16)

where x,- are deﬁnedas xi(Yj)=5i-, i.e.,

1 on the set occupied by plate with thickness 2h,-
Xi = (3.1.17)
0 elsewhere
Under these assumptions and using (3.1.14) it is possible to write
§E=(l-p)BlKl+pBZK2 (3.1.1861)
§ " = (1— (2)8171 + 103272 (3.1.18b)
and
1?=(1—p)1c1+px'2 (3.1.19a)
)7 =(1- (3)1" + 1372 (31191;)

where the superscripts ‘1’ and ‘2’ refer to the plate of thickness 2h1 and Mg, respectively,
and p is the area fraction of the thicker plate. The goal here is to solve for B and S in
terms of the constituents BI, 82, S 1, $2 and the cell geometry.

3’2

    
    

ya
12111

VI

 
 

4—1—1—
p 1—p

Figure 3.2. Basic cell for a rank-1 material plate.

Additional equations are required to solve for B and S . They are derived from the
continuity conditions at the interface of the two sub-domains. From static equilibrium on
the interface F 1’2 between Y1 and Y2, continuity of stresses is required. This can be

written in term of bending moments and shear forces as follows

49

321:2 - lel n = 0
( ) on 1‘12 (3.1.20a)

(523,2 _S171).n =0

For convenience, condition (3.1.20a)1 is expressed in the normal and tangential directions
along the interface F 1'2, i.e.,

(31x1 — 321(2)): . n = o (3.1.20b)
(le'l - 1321(2)): - t = o (3.1.20c)

where t is a unit vector tangent to the boundary 14,2. Additionally, there are continuity
conditions on the displacements. The solution u of the equilibrium equations in the small
scale domain Y is required to be Y-periodic and
3

u = {01,192,112 )6 (H1(Y)) (3.1.21)

This continuity is achieved if we assume 6], 92 and w are continuous across the interface
1" 1’2 , i.e.,
3

u = (61,62, wje (CO(Y)) (3.1.22)

This means that the displacements in Y1 and Y2 have to be the same on the interface 1" 1’2.
A sufﬁcient condition for a function f : Y—> 9i to be continuous on 11.2 is that the
gradients in Y1 and Y2 have the same component along 1112, that is,

Vf|2-1=Vf|1ot. (3.1.23)

Which in terms of this problem is written as

Via} — V1612 = alninj + aznitj (3 I 24)

Wt} —W3i2=a3n a1,a2,a3 69‘

Using the deﬁnition of bending strains and transverse shear strains the following kinematic
admissibility conditions are obtained from (3.1.24):

50

x1 —x2 =ﬂln®n+B2(t®n+n®t) (3.1.25a)
1 2 _
r -r - 133" (3.1.2512)

for some BI, [32, [33 6 SK. Equations (3.1.18), (3.1.19), (3.1.20) and (3.1.25) form a set

of equations from which the effective properties B and S can be obtained. What follows
is the solution of these equations.

Effgtive Promm'es in Bending
Using (3.1.19a) in (3.1.18a),

(132 — E )1? = (1 - (1))(132 — Bl)x'1 = (1 — pmb, (3.1.26)

where 77b is an arbitrary symmetric 2nd tensor often called the polarization tensor in the
literature (Lipton, (1993a)). Solving for K2 in (3.1.25a),

x2 =x1—131n8n-132(t8n+n8t). (3.1.27)
Using this expression in (3.1.20b) leads to,

[31x1 — 132(x1— 13111 ®n - 132(18 n+n 8 1))]: n8n = 0 (3.1.28)
Since 132 is isotropic, 132 (t 8 n 8 n 8 n) = 132 (n 8 t 8 n 8 n) = 0. Therefore,

_(BZ—B1)1c1:n®n _nbn-n

 

 

 

ﬁr (3.1.29)
30 30
with BO=32.':(n®n®n®n)=BZ1111=3 E2h23.
3l—v
Using (3.1.20c) and (3.1.27), [32 is obtained as,
2 _ 1 1 , .
[32: (B B)K .n®t _ nbnt (3.1.30)

132 .:(t®n®n®t+n®t®n®t)— (l—v)Bo°

51

Substituting the expressions of ,3] and [32 back in (3.1.27) gives,

K2 =Kl—i[ n®n®n®n+
BO

 

(11v)(t®n®n®t+n®t®n®t)]nb. (3.1.31)

which we re-write as

 

K2=Kl——1-[n®n®n®n+ 1 (n®t®n®t+
Bo 2(1—v)
t®n®n®t+n®t®t®n+t®n®t®n) ]n,,. (3.1.32)
where the following identity was used
nb(n®t®n®t+t®n®n®t)=nb(n®t®t®n+t®n®t®n), (3.1.33)

With equation (3.1.32), and using (3.1.26) to express E and K1 in terms of 111,, the

following version of (3.1. 19a) can be written,

(1— p1032 - 171411,, =132 — 311—111 — 7% Ab 111, (3.1.34)

where

 

Ab=n®n®n®n+ (t®n®t®n+

2(1—v)
n®t®n®t+n®t®t®n+t®n®n®t). (3.1.35)

and A}, has all the major symmetries. Since 77), is arbitrary (3.1.34) leads to the final
result when it is dropped

-1
E = 32 — (1 — p)[(BZ - 13‘)‘1 —B£A,,] . (3.1.36a)
0

This is the ﬁnal expression for the effective properties in bending. The fourth order tensor

52

A), can be written in a convenient way in terms of the normal vector n, and the Kronecker
delta 5,-1- as

1
Abijkl = 2—(1:v—)(nink5ﬂ + ninISJ-k + njnkdﬂ + ”Injaik — 2(1+ v) ninjnknl) (3.1.36b)

Effective Promrties in Transverse Shear
The effective shear stiffness is computed following a similar procedure. Using (3.1.19b)

in (3.1.18b), gives

(32 — s)? = (1— MS2 - 5111‘ = (1 — pm. (3.1.37)
where 773 is an arbitrary polarization vector. Solving for )2 in (3.1.25b),

312 = y] — 83 n, (3.1.38)
which can be used in (3.1.20a)2 to get

[Slyl — 52(3)] — [33 n )] . n = 0. (3.1.39)

From this equation B3 is solved to obtain

=(Sz-S1h/1-n =18"

 

83 So So (3.1.40)
where
50:82n'n=S121=I12-E—. (3.1.41)
1+ v
Using this expression of B3 back in (3.1.38), gives
72 =71 --S-1-n®nns (3.1.42)

0

53

With (3.1.42), and using (3.1.37) to express 7 and y‘ in terms of 17,, (3.1.19b) can be

written as

n®n

(1— (1)152 —§)‘1n. = ($2 —Sl)"n. — p 50

 

715- (3.1.43)

Since the vector 11, is arbitrary, it is possible to obtain the effective transverse shear
stiffness from (3.1.43):

—1
8:52 —(1—p)[(s2 —s1)‘1 711113] , (3.1.4412)
0
where
A, = n 8 n. (3.1.44b)

Next, effective stiffness properties for plates with ribs in more than one direction are

derived.

Homogenized promrties of a rank-L material plate
Plates with more than one layer of ribs are constructed recursively, that is, a rank-(i+1)

ribbed plate is built by mixing a plate of rank-(i) and a layer of strong plate in an given
direction. This process start with a rank-1 plate and is performed L times in order to create
rank-L plates with L different scales of ribs as shown in Figure 3.1. The properties of a
rank-L laminate can be derived by the recursive use of the formulas for the rank-1 laminate
(3.1.36). They are

—1

L
B: 132— ——(1 p) (132131—1— Elm/1M) (3.1.4511)

B
30)

L
291 =1, 91 2 0 (3.1.45b)
=1

54

where p now represents the total area fraction of the plate with thickness 2h2, and pi is
called ‘relative widths’ of layer i. Actual widths of each layer (w,- in Figure 3.1) can be
recovered from p and pi’s.

Next, formula (3.1.45) is proved by induction. For L=1, formula (3.1.45) reduces to
(3.1.36). Now assume that formula (3.1.45) is valid for L=K-1, i.e.,

K-11K-
le— ‘1—(32 BK—1)-1_(BZ_ Bl)—1_P_21:O pK-lAb(n1) (3.1.46)
11

where ‘K-l’ is used as superscript and p§_1 E p, p1K_1 '=' l- p. Consider now the
following mixture of two plates: one of properties B2 and another of properties BK_1,

corresponding to the effective properties of a rank-(K-l) laminate (3.1.46). Since B K ’1

< BZ, and no other particular assumption on the orthotropy of B1 was made to derive
(3.1.36), the same procedure and results apply. Taking care to use a different lamination
direction, identiﬁed by the normal vector nK, it holds now

p K(B2— BK)'1= (B2— 3K1) 1 —1;2)A,,(K. (3.1.47)

Using (3.1.46) in (3.1.47),

p1K—1p1K(B2_ BK)—-l_ (32- Bl)—l_

—[pK lip-(11,11 )+p1K 1 pKAb(nK )1. (3.1.48)
BO 1=1

New area fractions for material 1 and 2 in the rank-K mixture are deﬁned as

915101“th and 92595‘1pl<"+pl"192" (3.1.49)
yielding
K-l
p(B2- BK )‘1 (B2- B1)‘1— gz[2pgAb(n€)+pKAb(nK) ], (3.1.5011)
0 1:1

55

 

where
105-1 =K_p “1K9
121 5 pl“1 , t = 1,---, K —1 and pK1—— 2. (3.1.50b)
P2 P2

It is easy to show that (3.1.45b) is satisﬁed. Using p 5 p2, p1 = l— p, and changing K
by L, formula (3.1.45) is obtained. This ﬁnishes the proof.

The result for the shear stiffness of a rank-L laminate can be obtained following similar

steps,

—1

§=s2—(1—p) (s2-s1)'1—2p)1A,(n) . (3.1.51)
50121

As expected, when L=2, n1={1,0}1 and n2={0,1}1, formulas (3.1.45) and (3.1.51)
gives the same effective properties derived in Chapter 2 setting E'=0.

Membrane Effective Stiffness

The same procedure outlined here can be followed to obtain the in-plane effective stiffness
of plates with L layers of ribs. This was done in (Avellaneda and Milton, (1989)) and the
result is shown here for completeness.

-—l

L
117 = M2 — (1 — p) (M2 — M1)'1-K§—Zp)Am(ng) (3.1.5211)
0 1:1

where

1

Amijkl = mlnink5jl + "inlajk + njnk5i1+ "Injaik — 2(1+ v) ninjnknl), (3.1.52b)

215112
M .=.———, 3.1.52
0 (1—v2) ( C)

56

and MK are fourth order tensors of stiffness properties of a plate of thickness 2hk (k=1,2),
and have the same structure of (3.1.6a) replacing B0 by Mo. Table 3.1 summaries the
formulas presented in this section.

Other Effective Stiffness
To ﬁnish this Section some comments about other effective properties similar to (3.1.36)

are presented. For Kirchhoff plates, the bending stiffness is given by the same
expression in (3. 1.36) except that the tensor Ab is deﬁned as (Lipton, (1993a)):

Ab = (n (8) n ® n ® 12) for Kirchhoﬁ’s plates. (3.1.53)

All results presented in this section can be also applied to plates with honeycomb cross
sections (see Figure 1.6). The only change is in the properties of the “weak plate”. For
example, the bending stiffness tensor for honeycomb plates would be

(l—V)
2

1 = 31511113 4:13)
BO _ 3—(1_v2) . (3.1.54b)

 

1 1 1 1 1 1 1
B1111 =131111 = 30 31122 = V30 31212 = B0 (31-54“)

A similar argument can be used for perforated plates (Figure 2.6). In this case a weak
material with 0 < E' << 1 is used. As commented before, it is not possible to use h1=0
since the positive deﬁniteness property of B is lost, the G-convergence theory would not
be applicable, and the homogenization process would fail. Therefore, the problem is
“regularized” using a weak material to simulate transverse perforations.

3.2 Moment Representation of the Homogenized Properties

The effective properties derived in the previous section depend on the relative width (pi)
and orientation (m) of each layer i (i.e., two variables per layer). In this section we
introduce a transformation that allows us to express these average properties as a function
of at most four independent variables for any number of layers. Specifically, new
expressions are derived for (3.1.53) and (3.1.54) in terms of 9, m1, m2 and m3, which
are, respectively, the global angle of orientation of the whole micro structure, and three
“moments”, describing the internal geometry of the basic cell, i.e., relative widths and

 

57

Table 3.1. Effective stiffness for rank-L material plates.

 

Mem e
L -l
117 = M2 —(1—p) (M2 -M1)‘1 —i2p14m(n1)
MO [:1

1

Amijkl = -2(—1_—V-)-(nink5ﬂ + ninISJ-k + "jnkail + "Injaik — 2(1 + v) ninjnkn, )

Bending

—1
L
F: 32 —(1—19)[(B2 —Bl)‘1 -£211141(n€) ]
301:1

1
Abéikl = mminkdﬂ + ninldjk + njnkdi, + njnj6ik — 2(1+ V) ninjnknl)

Wises—hen

-1
L
§=52 -(1--10)[(S2 -Sl)_l —£Zp141(n‘)]

SO[=1
AS =n®n
L
291%, p120

 

 

58

orientations of layers. This transformation has been used in two—dimensional elasticity by
Avellaneda and Milton (1989) whose ﬁnal results will be presented here. Only formulas
for bending and transverse shear deformation are derived here.

Change of Basis
Instead of working with fourth order tensors, it is more convenient to work with their

matrix representation in a suitable basis. The corresponding (3x3) matrix representation of
fourth order tensors will be denoted using capital letters, e.g., Bag (a,B=1,2,3) is the
matrix representation of Bijkl- Membrane and bending strain tensors are expressed in the

following convenient basis

441: :2). 41-1: s.) 4112.1)

For in-plane deformations, tensor 53 correspond to radial expansion, and tensors 51 and
52 to in-plane shear deformations in two orthogonal axes. For bending deformations,
tensor 63 is an axisymmetric bending strain (i.e., the one produced by a transverse
displacement w(x,y) =(x2+y2)/2), and tensors 51 and 52 are twisting bending strains in two
orthogonal directions (i.e., those in w(x,y) =(x2 - y2+2xy)/2). The use of the basis (3.2.1)
allows one to state the following properties that can be easily proved and will be used in
this section. Properties will be stated for bending deformations. The version for membrane
deformations can be easily derived.

1) A generic bending strain tensor Ki]- is expressed as:

K“ij = 00551? = Klmggiég

___ 511122.155. 1 $821,341 Ell—”£52153. (3.2.2)

8 12

 

2) The bending strain energy density is:

l 1
#1; = EBijleinkl = 53th K111153115? K11551515- (32.3)

From (3.2.3), we deﬁne the following (3x3) matrices,

59

Ba)? 5 1711151565, (3.2.4,
Kap a 11,411,,131411. (3.2.5,

3) A planar rotation (9 of the unit vectors n6 in equation (3.1.35) produces the following
change in Ab :

R1 1 —R R R R 1 R— COS(9) SW9) 326
Abijkl( ’1)— pi qj rk slAbpqrs(n )1 — —Sin(6) (308(0), (. . ,
and its matrix representation, using the same transformation in (3.2.4), is
_ t 1’ _ t 1 .
Mb =Ab(R 12 )—Q Ab(n )Q, (3.27,
where
Cos(29) Sin(29) 0
Q: —Sin(26) Cos(26) 0. (3.2.8)

0 0 l

4) If Bijkl is a fourth order tensor with all minor and major symmetries, it can be proved

that
( 5311-1116511 = 51"} (1313111)”1 51'3- (3.2.9,

5) Using the transformation (3.2.4), an isotropic fourth order tensor such as that it.
(3.1.6), is of the form

 

2 E l—v 0 0
B=—h3 2 0 l—v 0 . (3.2.10,
3 l—v
0 O 1+v

Entries (1,1) and (2,2) in (3.2.10) are associated with the twisting bending in twc
orthogonal directions, and entry (3,3) with the axisymmetric bending.

60

3.2.1 Moment Representation of Bending Stiffness

Using transformation (3.2.4) and property (3.2.9) it is possible to write formula (3.1.45)
in a (3x3) matrix form as

—1

L
Bsz—(l—p) (32—31)‘1-%Zp)Ab(n1) . (3.2.11)
[=1

Using unit vectors of the form n! = {Cos(qog), Sin((pg ))1 , the formula for 11,, (113) gives,

after trigonometric simpliﬁcations,

 

 

 

 

F3—v—(1+U)Cos(41P1) _ 1+1) Sin(4(p)) Cos(2(pg)

4(1—v) 4(1—v) . 2
Ab = 3—v+(1+v)Cos(4(p)) Sm(2(o)) (3.2.12)

4(1—v) 2
. 1
t _

- symme no 2 _

or, deﬁning

L L
m1(p.rp) = 2111081241). "12(17, 9) = 2121514241).

 

 

(=1 (=1
L L
m3( p, (p) = 2 p)Cos(4tp) ), m4( p, (0) = Z pgSin(4(p)), (3.2.13)
[:1 [:1
then,
-3—U—(1+U)m3 _(1+‘U)m4 ﬂ-
L 4(1— v) 4(1— 1)) 2
3..—v+(1+v)m3 1112
M = A = — , 3.2.14
1’ a” 1’ 4(1— 0) 2 1 1
_ . 1
t _
L symme no 2 _

 

 

and the final form of the bending properties in matrix form is

—1
8 = B2 — (1 — p)[(82 — B1)‘1 — f-Mb] (3.21511)
0

61

 

 

where
2 E l—v O O
Bk——h,3 2 0 l—v 0 , k=l,2
3 l—v
O 0 1+1)
_2 2 E
30 3h21_v2

3.2.2 Moment Representation of Shear Stiffness

(3.2.15b)

(3.2.15c)

A similar procedure is performed to express the shear stiffness in terms of the moments.
Using 123 = {Cos((pg),Sin((pg)}t in As(n[) (eqn. (3.1.51)), and using the deﬁnition of

moments in (3.2.13), one obtains

1+m1 ﬂ

__ 2 2
MS— 212— 1”"1 .

2 2

and the ﬁnal formula for the shear properties in matrix form is

—]
s = 52 — (1 — p)[(s2 —sl)“1 —£—Ms]
50

k E 1 0 E
S =——h k=1,2, S =—
1+v k[0 I} 0 h?

.2.3 m Re r nt ion of M mbr Stiffne

(3.2.16)

(3.2.17a)

(3.2.1717)

Here the results obtained by Avellaneda and Milton (1989) for plane stress elasticity are

slightly modiﬁed to take into account the presence of ribs. The result is the moment

representation of the effective stiffness for membrane deformations.

—1
A7 = M2 -(1—— )o)[(M2 -—M1)‘1 mil—Mm] ,

 

 

 

 

 

 

M0
-3—‘U—(1+‘U)m3 _(1+‘U)M4 ﬂ-
4(1-v) 4(1—v) 2
3—v+(1+v)m3 mg
Mm: —,
4(1—v) 2
. l
smmetrlc —
_ y 2-
E l—v O 0
Mk=2hk1 2 0 l—v 0 k=1,2
"” 0 0 1+1)
E
M0=2h2 2
1-V . (3.2.18)

Notice that Mm=Mb since Am=Ab.

In summary, a representation of the homogenized stiffness of Mindlin plates with ribs in L
different directions with only four independent variables, m 1, m2, m3, and m4, that
describe the local orthotropy, i.e., the relative rib widths, pi, and the local orientation of the
layers, (pi.

3.2.4 Feasible Set in the Moment Space

In order to perform the optimization in the moment space it is necessary to identify the set
of feasible solutions. In this context ‘feasible’ means that a point in the set must represent
a realizable material with average stiffness within the known bounds. The identiﬁcation of
this set is done using the approach presented by Avellaneda and Milton, (1989), in plane

elasticity. This requires the solution of the trigonometric moment problem (see theorem 2.6
in the text of Krein and Nudel’man, (1977)). Using the identity between tensors Ab and

Am it is possible to obtain the constraints on the moments for plate problems directly from
the work of Avellaneda and Milton. In the four-dimensional moment space, the feasible

set is given by

63

2m2 2m% m2 4111111127124
l+m3 l—m3 1—m3 1—m3 (3.2.19)

 

 

H={(m1,m2,m3,m4):

mf+m§sr —1_<.m3SII

From the layout optimization point of view, it is important to emphasize the difference
between the moment formulation and the formulation presented in Chapter 2. A shape
optimization problem based on the formulas given in Chapter 2 requires three design
variables, namely, a, b (the width of the layers) and 6 (the orientation of the cell), to
characterize a pm material plate with orthogonal layers. In the formulation of this
chapter we have ﬁve design variables, namely, p (the amount of material assigned to a
location), and the four moments m1, m2, m3, m4 (the description of the layering), to

 

characterize a rank-L material plate with arbitrary layer orientation. It was proved in
elasticity problems that for multiple load cases (i.e., multiple energy cases) the stiffest
micro structure can always be attained using at most rank-3 materials (Lipton, (1994))
which is an special case of rank-L materials. As it will be seen in section 3.2.5, this is also
the case of Mindlin plates.

It is important to mention that in Mindlin plates, even under a single load case, there are
always at least two sources of strain energies, one coming from the bending deformations
and another one coming from the transverse shear deformations. These two energies
always come together because of the nature of the Mindlin plate equations. If additionally
there are membrane deformations, there will be a third source of strain energy.

3.2.5 Determination of the Micro Structure Geomeg

Once the optimum moments are determined from the solution of the optimization problem
(as will be explained in Chapter 4), it is possible to obtain the geometry of the micro
structure, namely, the number of layers, layer widths and directions. Potentially, there are
2*L unknowns to be determined in order to describe the micro structure of the material,
i.e., L layer widths and L directions. Recent work, however, has shown that optimum
(stiffest) micro structures for multiple load cases are made of at most three layers (Lipton,
(1994)), therefore, we can reduce the number of unknowns to six, namely, pi, (pi,

i=1,2,3, which can be solved following the procedure described below.

64

First, a 3-dimensional set [-1 is obtained from the 4-dimensional set H making m4=0.
This can be done performing a rigid-body rotation, 6), of all layers in the micro structure.
Then the micro structure geometry of the three layers can be computed using the convexity
of 11 and the solution of the micro structure geometry of rank-2 materials on the boundary
of 11 that appears in (Lipton, (1993c)). The detailed procedure follows.

Noting that the layer angles are measured from an arbitrary reference, one can select a local
frame that is rotated an angle 9 from the ﬁxed frame. As this is just a planar rotation of
the layer direction vectors, it is possible to write, from (3.2.7),

12,, = Q’(9) M, Q(@). (3.2.20)

and select 9 so that MM 1, 2) = O. This angle is

2 2
1) + +
9=%ArcTan( m3 m4 m3). (3.2.21)

m4
With this choice,

'3-v—(1+u)n'a3

4(1—v)

M _ 3—v+(1+v)m3
b 4(1—v)

0

 

 

(3.2.22)

NIHNII§1N|_I

symmetric

l _

 

 

where the new moments are

L

L
231112.95) = 21216032131). mam <15) = 212152212131),

”=1 “=1 (3.2.23)

L
"73( P, 43) = 2P1COS(4¢1)
=1

and the new angles are ([3) =(pg +9. The relation between the moments in the 4-

dirnensional set and those in the 3-dimensional set is given by

65

r711 = m1C0s(29) — mQSin(26)
m1 = m2C0s(29) + mlSin(29) (3.2.24)
7713 = m3Cos(49) — m4Sm(49)

The new set of feasible moments, 1-1, is easily obtained from (3.2.19) setting m4=0 (see

Figure 3.3), i.e.,

2n”)?

 

~2
+2ﬂsr raf+ra§sl,—ism3si) (3.2.25)

{( l 2 3) 1 + "'13 1 — m3
It can be easily checked that the four points satisfying r7112 + 171% =1, namely, (1,0,1),
(0,1,—1), (0,—1,—1) and (-1,0,1), correspond to rank-1 materials with layer direction equal
to 09, +450, -450 and 90°, respectively. Additionally, it is known that all other points on

 

Figure 3.3. Convex set [-1.

the boundary 0711 correspond to rank-2 materials (Lipton, (1993c)). Making use of these
characteristics, it is possible to express any interior point in 11 as a convex combination of
a rank-l material point [i --one of the 4 points mentioned above-- and another point on the

boundary, 5, that is,

66

m=ad+(1—a)5, 0Sa_<1 (3.2.26)

The point I; is a rank-2 material point on 811, to be determined intersecting the line that
connects ii and m with the boundary 811. This is accomplished solving the non-linear

 

 

 

system of equations
“2 ~2
12h" +12b2" =1
+ _
~ b3 ~b3 ~ (3.2.27)
171-51: 192-52 = ’93- "3
”71—51 "72-52 "73-53

with — l S 1523 S 1. Once the point 5 is obtained from (3.2.27) it is possible to compute its
layer directions ((p1(b ) , (pg) = (pfg) + 5) and relative widths (pfb), pgb) =1— pfb) ) solving

the nonlinear system

p1 “0032(1)?” ) + (1 — p1(b) )Cos(2((p{”) + 6))

51 (
52 = p{’5’Sin(2(p{5)) + (1 — p{5))s.-n(2(.p{5) + 6))
1;, = pr’CoswaEU + (1 - p{5))cos(4(¢{5) + 5))

0 = p{5’51n(4¢{5’) + (1 — 12(5))Sin(4((o{5’ + 6)) (3.2.28)

A solution of the nonlinear system in (3.2.28) to compute the geometry of 5 can be found
in (Lipton, (1993c)). The ﬁnal step is to identify the layer directions, (0W) , and relative

widths, pg'ﬁ), of the interior point m using (3.2.26). In order to do this, we need to

pick one of the four points satisfying 11112 + 111% =1. We choose [i=(1,0,1), then (3.2.26)
becomes

3 ~ .. .. - - -
2 pgmmosaagm) ) = a + (1 — a )1 p1( b)Cos(2(p{b) ) + pgb)cos(2((p{b) + 6 ) )1

[=1

3 ~ ~ - - - ..
ngm151n(2¢gm)) = 0 + (1 — a )( p1( b)51n(2(p{ 1’) ) + pgb)sz°n(2(cp(”) + 5)); (3.2.29)
[=1

3 ~ ~ ~ ~ ~ ~
ngm)Cos(4tpgm)) = a +(1 — a){p{b)C0s(4§0{b))+ pgb)C0s(4(gol(b) + 5))}
[=1

67

Using equation (3.2.29), the identiﬁcation of terms results in

pgﬁi)=a,. pgm)=(1_a)p{b); pgrﬁ)=(l-a)p£b)
, (3.2.30)

(pgm) = 0,. (para) = (p92); (pgm) = (pfb) + 5

and the ﬁnal orientation of layers is computed using (p) = (ﬁg — 6'). Notice that different
geometries are obtained depending on which point (2' is used. This means that we are able
to identify at least four different rank-3 layering geometries that have the same moments,
i.e., the same stiffness properties. For example, the following two micro structures have
the same stiffness properties:

(9. pf”, 199’”, .097”, ¢{’ﬁ).¢£"”» 411"” )
micro structure 1: (33.7,0.27,0.42,0.31,-33.7,22.2,84.3)
micro structure 2: (33.7,0.30,0.34,0.36,-78.7,-12.7,38.8)

and correspond to m=(0.1,0.1,0.1,0.1).

This non-uniqueness of the layering arrangement is expected since the micro structure
geometry is determined solving a non-linear system of ﬁve equations and six unknowns,
namely, equations in (3.2.13) and (3.1.45b), and unknowns pi, (pi, i=1,2,3. The idea of
using a convex combination in (3.2.25) to obtain the micro structure geometry is due to
(Lipton, (1993d)).

CHAPTER 4
FORMULATION AND IMPLEMENTATION OF THE LAYOUT
OPTIMIZATION PROBLEM FOR MININIIZATION OF COMPLIANCE

A simple way to measure the ﬂexibility of a structure is to compute its mean compliance
(external work) for the given loading conditions. Since compliance is an inverse measure
of the stiffness, it is meaninful to use the mean compliance as a cost function in order to
maximize the global stiffness of the structure. From the mathematical viewpoint, the
most important advantage of this choice is that in the formulation of the optimization
problem the state variable (the displacement ﬁeld) and the adjoint variable turn out to be
the same, which makes the problem much easier to solve.

As it was said in the introduction, the idea behind the homogenization approach to shape
optimization is to ﬁll the domain with a microscopically porous material and to pose the
problem as one of optimum distribution of material. Using the functional relation
between the stiffness of the material and its volume (a relation computed via the
homogenization procedure), it is possible to compute the sensitivity of the stiffness with
respect to changes in the amount of material, which is indispensable for any topology
optimization approach. This is the basic idea that will be explored throughout this
chapter.

The layout and shape optimization problem presented here is based on the moment
formulation of the stiffness properties shown in Chapter 3 which allows us to build a
two-scale hierarchical statement of the optimization problem as in (Jog, et al., 1992).

In the first part of this chapter the problem of minimization of the mean compliance

(maximization of global stiffness) is considered. The second part is devoted to the

implementation aspects of the layout optimization problem.

68

69

4.1 Hierarchical Moment Formulation of the Stiffness Maximization Problem

The optimization problem in (Bendsoe and Kikuchi, (1988)) was stated in terms of three
design variables , namely, the rib widths (a(x), b(x)), and the global rotation of the micro
structure, 9(x). The strategy used in that case was to implement an iterative procedure
based on an optimality criterion approach to solve for all three variables simultaneously.
Here we split the optimization problem in two, one at the small scale level where the
micro structure of the plate is computed, and another at the large scale where the
distribution of material is computed. This is possible using standard variational
principles of mechanics that allow one to express the equilibrium state of the structural
system from a minimization problem. This idea was ﬁrst applied in static problems, in
two-dimensional elasticity, for a single load case, and using the rank-2 materials, by Jog,
Haber and Bendsoe (1992). Mindlin plates under multiple independent load cases are
now considered.

As will be seen in the following presentation, the formulation of Jog, Haber and Bends¢e
is more convenient than the Bendsoe-Kikuchi’s formulation when the effective properties
of rank-L material plates are represented through the moment formulation described in
Chapter 3. For the moment, it is sufﬁcient to know that the design variables are now the
internal geometry of the micro structure, given by the moments m(x)=(m1(x), m2(x),
m3(x), m4(x)) and the distribution of the material resource throughout the plate, p(x).
The notation in this section follows very close the tensorial notation used in section 3.2.

4.1.1 Optimization Problem Statement

Suppose there are P independent load cases for which a single optimum structural shape
is desired. This problem can be stated using a weighted average of the compliances C(up)
of each load case p, i.e.,

Find p(x) and m(x)=(m1(x), m2(x), m3(x), m4(x)), x6120 that

P
'n C({u}*)= w C(uf) (4.1.6a)
(anlkX 112-=1 p

where p, and m=lm1, m2, m3, m4} are the design variables in the set

70

X = XG x H, (4.1.6b)

X6 = {p(x) .- x e (20, jpdg s Vmax, 0 < pm," s p s1}, (4.1.6c)
00

where H ,is the four-dimensional moment space deﬁned in (3.2.19) and uf ={v1, v2, V3,
91, Ozjf a {vx, vy, vz, —-9y, 9x If represents the plate displacement ﬁelds at equilibrium.
Vmax<lf20| and Pmin are given data. The subscript ‘*’ in the displacement vector

emphasizes that it is the equilibrium solution and hence minimizes the potential energy of
the structure.

To identify the local and global problems the following series of equalities are used

min wp C(up )= min —2 wp 17(uf) (4.1.7)
(MM 210* (annex p211”

where H is the potential energy. Equality (4.1.7) uses the Clapeyron’s work theorem
(Fung, ( 1965)). Applying the principle of minimum potential energy:

P P
min —22w 17(1),}: )= min {—2 max zwp min [U(uP )—c(uP )1) (4.1.8)
(p, m)eX p= _1 peXG mer_ _1 upeV

where U is the strain energy in the plate and V the space of kinematically admissible
displacement ﬁelds. Now an exchange of the two internal optimization problems in
(4.1.8) is performed. This is possible due to a saddle point condition in the space
[V]1D x H similar to that proved by Lipton (1993b) for the three-dimensional elasticity
case (see also Ekeland and Teman, (1976), page 173). The space [V]1D is the cartesian
product (direct sum) of space V, P times, explicitly, [V]1D E VxVx---XV. After

exchanging the two internal optimization problems, we obtain

P
min {—2 min PEWP max [U(up)—c(up)]} (4.1.9)
PEXG (“IGIVIP p=1 meH

Finally, since material properties are independent from point to point in the domain, it is
possible exchange the maximization over the moments and the strain energy integral to

71

get

P
min C({u}*)= min —2{ min [J]. max

(p,m)eX peXG (u)e[V] meH

wp Mu”) 4:2 — C((uw 1
(20 p=1

(4.1.10)
where u is the strain energy density.

The ﬁnal arrangement in (4.1.10) suggests an iterative process based on a local problem
corresponding to the (point-wise) maximization of the strain energy density for ﬁxed
strains, and a global problem in which the amount of material p(x) is updated to

minimize the mean compliance for a ﬁxed micro structure.

4.1.2 The Local Problem
For ﬁxed p(x) 6 X0, and kinematically admissible displacement fields up, p=1,...,P, ﬁnd

m ={ "11. m2. m3, "14} to

P P
. . 1 — — —
maximize 2w], p(uP) =— zwp (M51185812+B.jk1K§K/Z +Sij7ipyf) (4.1.11a)

meH p=l 2 p=1

with

2 2 2
2m 2m m 4mmm
1 + 2 + 4 — 122431,

1+m3 l—m3 1—m3? 1—m3 (4.1.1112)

 

 

H={(m1,m2,m3,m4):

m3+m§sr -1Sm3SI}

Using deﬁnitions from Chapter 3, the weighted sum of the strain energy densities can be

written as

P
w, #(u”)=%(17(m): e((u)) + E(m) .- b((un + E(m) .- saw)» (4.1.12)
p=1

where formula (3.2.4) was applied to A7 and E . The following definitions were also
used in (4.1.12)

72

P
e((u/) a prEanmp).

p=1
P
Wu» 2 prKanmP),
=1
pP
s({u}) a prykmpm (up). (4.1.13)
p=1

where Edn and K 577 are computed with (3.2.5). Notice that all the information about
the different load cases is concentrated in the strains formulas (4.1.13). The solution
strategy for this problem will be discussed in section 4.2.

4.1.3 Equilibrium Equations

After solving the local optimization problem, the optimum micro structure for given
strain ﬁeld and resource p at each location of the plate is identiﬁed by m*(x). Next, for
ﬁxed m* the following equilibrium equation is solved. The equilibrium displacements
{u}={u1,-~,uP)e [VIP satisfy the equation

I (M(m’“. p) .- 411111121) + E(m“. p) .- b({u}. (11m
90

§(m*, p): s(/u),(12}))drz = C({ﬁ)) Vlﬁl,---,ﬁP}e[V]P (4.1.14)

It is important to realize that the material properties obtained in the solution of the local
problem depend upon all displacement fields simultaneously, that is,
117*: A7 *(m *(lu))), 8*: E *(m *({u})) and 8*: S *(m *((u})). This dependency
produces two difﬁculties. First, the strain energy density is no longer a pure quadratic
form of the displacements, and hence the equilibrium equation is non-linear. Second, the
strain energy density couples all displacement fields, meaning that the solution of the
equilibrium equation should be computed considering all load cases simultaneously.
More speciﬁcally, the Hessian matrix H * of the strain energy density -- which in linear
problems represents the stiffness matrix of the equilibrium equation-- consists of sub-
matrices 115* (i,j=1,..., P) given by

73

 

2
115*: 33311111“ *(u1,- --,uP)}, (i,j=1,..., P). (4.1.15)

In general, these sub-matrices are non-zero causing the coupling between different load
cases. In section 4.2 there is a discussion about how to solve this complex problem.

4.1.4 The Global Problem

For ﬁxed m* and ﬁxed equilibrium displacement ﬁelds {a j * find p(x) that

min (—2 11((u...(p,m*)}))= min (—2 min ( j u*({u})d.Q—- C({u}))
peXG peXc (u/eIVI” 90

X0 ={p(x):xe.QO, (pdgsvmax, O<pminSpSl} (4.1.16)
00

Using Lagrange multiplier techniques, an optimality criterion can be derived to solve
(4.1.16). Introduce the Lagrangian function,

[4(1), $1 amin’ amax) =
I (M(m’“. p): e((u1*)+ E(m*. p) .- b((u1*)+

90

E(m*.p):s((u}*)dn — 20((u/*) +

an] de-Vmax)+ jam,n(pm,,,—p)+amax(p—1)drz (4.1.17)
~90 90

where (D, “min and am are non-negative Lagrange multipliers. The ﬁrst variation of
the potential energy H *( p ) with respect to the density p is the sensitivity of an optimum
solution (deﬁned by m*) as the ‘parameter’ p changes. In this particular case, since the
constraints of the local problem do not depend on the density p , this sensitivity is simply
the ﬁrst variation of 11*(p ), leading to,

8M ( m ) 8S(m ___).

*
8p “W /)+

eu({ }*)+—— 8mm )
8p

SW1}*)- ¢+amin— amax=0.

(4.1.18)

74

Complementary conditions can be also obtained from (4.1. 17), i.e.,

a>(jde—Vmax)=o, amin(pm,-n—p)=0, amax(p—l)=0 (4.1.19)
90

4.2 Numerical Implementation of the Optimization Problem

The computational implementation of the hierarchical formulation presented above can
be organized in two main steps: the solution of the equilibrium equation and the solution
of the optimization problem. Here there is a presentation of these two topics in more
detail.

4.2.1 Splptipn pf the Equilibrium Equation

It was discussed in section 4.1 that the equilibrium equations that appear in the
hierarchical moment formulation of the optimization problem are non-linear and coupled
due to the non quadratic form of the strain energy density with respect to the strains. This
results in a Hessian matrix of the form

"111.: I112: 111;"
H“; ”:1 5'52 ‘ (4.2.1)
_Hfb’i‘ Hi}-

 

 

where the 8x8 sub-matrices 115* (i,j=l,..., P) are given by

 

2
115* = 3113311112“ *(u1,---,up )}, (i,j=l,..., P). (4.2.2)

where up, p=1,..., P is the p’h displacement ﬁeld and 11* is the strain energy density
computed for the Optimum moments m*=(m1, m2, m3, m4 )*.

75

In order to decouple the problem and to simplify the analysis the Hessian matrix is
approximated setting zero all off-diagonal sub-matrices and using the same sub-matrix in
all diagonal entries, that is, setting

1‘ * ‘ _
He 1;... . 9 M* 0 0
H*“PP’0x-a 9 . j ° , 119*; o 3* 0 (4.2.3)
6 ° ' if“ 0 o 5*

 

 

where A7 *, E * and S * are the stiffness properties computed for the optimum solution of
the local problem. This approximation is simply a linearization of the original problem
where the dependence of the material properties on the displacement ﬁelds is neglected.
The solution of the approximated problem requires much less computational effort than
the solution of the original one. A linear ﬁnite element code can be used in order to
obtain the solution of the approximated problem. In this work the program MoniqueTM
from RTB Corp. was used to solve the approximated equilibrium problem.

4.2.2 Splptipn of Optimization Problems

Two types of optimization problems were presented in section 4.1: the global
optimization problem on the material resource p, and the local optimization problem on
the moments. The optimization problem over the material resource is N-dimensional (N
is the number of ﬁnite elements in the mesh) and is usually solved using an optimality
criterion approach, while the optimization problem over the moments is four-dimensional
and can be solved using a non-linear programming optimization algorithm.

timali ri riaA r h- 10 a1 r le
Optimality criteria approaches ﬁnd the distribution of p (x) such that the optimality
conditions given in (4.1.18) are satisﬁed for every location x in the domain. This is done
using an iterative scheme where at each iteration the resource p (x) is updated according
to how far it is from the satisfaction of (4.1.18). If at a given point the optimality
conditions are satisﬁed, the material resource will not be changed, but if the optimality
conditions are violated, the material resource will be modiﬁed proportionally to some

measure of the violation.

After the discretization of the plate domain into N finite elements, optimality conditions

represented by equation (4.1. 18) can be written for each element as

76

aMi’"(m)_ 61({u }*)+8-—B(m). b'iu({ }*)+8__S(m)

an (9p (9)0 Wu 1*): (DW'W‘ am"
I l l

i=1,...,N (4.2.4)

In elements where the material resource has not attained its lower or upper bound,
complementary conditions in (4.1.19) result in ami=amini=0. For these elements it is

possible to write (4.2.4) in the more convenient form

1 = (me),-

1. <15 :1, i=1,...,N (42.5)

where the sum of the three left-hand-side terms in (4.2.4) is usually called ‘mutual
energies’ and denoted here as (me),-. The ratio in (4.2.5) will be called here “optimality
condition ratio” which is equal to one (1) only for elements with Pmin < p < 1, and only
at a stationary point of the Lagrangian, which we assume corresponds to the optimum
solution of the shape optimization problem. Since the global stiffness increases
monotonically with the amount of material, the constraint on the total amount of material
available is always active and the Lagrange multiplier (D is greater than zero.

Using the Optimality condition ratio it is possible to construct an algorithm to update the
material resource at each element of the mesh. Notice that E,- is quadratic in the strains,
and therefore, high values of E,- (values greater than 1) are associated with high strains
that can be reduced adding more material. On the other hand, low values of E,- (values
less than 1) are associated with low strains that can be increased subtracting material.
This is the basis of the algorithm. Suppose that at certain iteration k of the optimization
algorithm there is an element with material resource pf and 51¢ 1. Then the proposed

value for the material resource in the next iteration will be

k g
k; - .
p," =(:{‘ ) pf: —[(——’:k)' ] pf, z=1,...,N (4.2.6)

where 0 < 6 < 1 is a “damping” factor that is used to improve the behavior of the
algorithm.

To apply (4.2.6), it is required that the Lagrange multiplier (PK be known. This

77

multiplier is computed such that the constraint on the total amount of material available is
active. This means that it is necessary to iterate in a inner loop using (4.2.6) until the
constraint in (4.1.16) is active. A bisection search method is suitable for these task.
Clearly, values of p computed using (4.2.6) must be maintained between the bounds of
the material resource, 10min and 1 (see (4.1.16)). Also, it is convenient to introduce a
maximum step size, 5, to obtain a stable convergence in the iteration process. A
practical implementation of (4.2.6) that takes into account these issues is

'max/(1—61pl‘,pm,-,,/ 11 51p? s max/(14111512....)

.. G . _
p1k+l= _5'1( ik) ptk 1f max/(1'— 5)pi 1pmin1<51k Pk1-<-Min{(1+5)Pi 11}
min{(l+5)p,k,1] if mm((1+5)Pi.1}55ikP1k

 

i=1,..., N (4.2.7)

This approach was used in the original paper of Bends¢e and Kikuchi (1988) and has
seen to be effective using values of Q’ between 0.7 and 0.85, and 6 between 0.45 and
0.55.

Nonlinear Programming Approach - Local Problem
In the hierarchical formulation of the optimization problem, the local problem over the

moments can be solved using conventional nonlinear programming techniques. In this
case we used the algorithm presented in (Schittkowski, (1985/6)) implemented in the
subroutine NLPQL and supplied by Professor Schittkowski.

4.2.3 Convergence Criteria

The convergence of the iterative algorithm is checked comparing two consecutive values
of the sequence {F }]6, where F is the objective function, and k is the iteration step.

Given a non-negative small tolerance t<<1, the used convergence criterion is:

|F(k)_F(k—1)|
If I FWD |<t, then CONVERGED else ITERATE (4.2.8)

 

Additional stopping criteria are used to prevent the program from running inﬁnite
iterations when the sequence does not satisfy inequality (4.2.8) in a reasonable number of

78

iterations. A limit in the number of iterations is the easiest way to stop the process in
these cases:

If k ZLimit of Iterations, then STOP,
else ONE MORE ITERATION. (4.2.9)

4.2.4 Algorithm Implementation

This is a sketch of the algorithm implemented to solve the layout and shape optimization
problem using the hierarchical moment formulation.

10 Read data: plate geometry, boundary conditions, vqax,
h1, h2, material properties.
20 Assign uniform distribution of pand moments.
30 Compute effective properties using homogenization.
4O Solve equilibrium equation using finite element
methods to get the strains.
50 Solve the local problem and obtain optimum values of
m1, m2, m3 and m4.
60 Compute optimality condition ratio for each element.
70 Update distribution of pin the global problem.
80 Check convergence:
If converged, PRINT RESULTS, else goto 30
9O Stopping criterion:
If k> Limit of Iteration, PRINT RESULTS and STOP,
else goto 30
100 Post—Processing: Plot optimum shape, and compute the

geometry of the micro structure.

After reading the data (10), a uniform distribution of the design variables is assigned to
the whole domain (20). Using explicit formulas developed in Chapter 3, effective
properties are computed for each ﬁnite element in the mesh (30). Then, the equilibrium
equations are solved calling the ﬁnite element program (40). Non-linear programming
techniques are used to solve the local problem in (50) to obtain m1* mg, m; and m: for

79

given density p and given strain fields. In (60) the sensitivity of the potential energy
with respect to the resource p is computed in order to obtain the optimality condition
ratio with (4.2.5). Step (70) corresponds to the updating of the distribution of p in the
global optimization problem using the optimality condition approach. Finally, the
convergence is checked (80) and the stopping criterion is in (90). Results are revised
off-line in a post-processing step (100).

CHAPTER 5
EXAMPLES

This chapter is devoted to the solution of some layout optimization example problems
using the homogenization approach and the stiffness properties for rank-2 and rank-L
derived in Chapter 2 and 3, respectively. Optimum shapes will be presented in a gray
scale (see Figure 5.1) where areas where the plate is of thickness 2h2 are denoted by
black, and the core plate of thickness 2h1 by white. Gray areas indicate the presence of
rapidly varying thickness between 2h1 and 2h2, i.e., a composite material plate. In the
iteration histories the mean compliance is normalized using the mean compliance of a
uniform thickness plate made with the same amount of material. The isotropic material
used in all examples has Young modulus E=2 1011 Pa and Poisson ratio v=0.3. The weak
material used to model the perforations and gaps between ribs has a Young modulus that
is 5% of the strong material.

2h2

 

no ribs full ribs

Figure 5.1. Gray scale to identify the cross section of the plate in the optimum shapes.

 

 

O timali ndi i n atisfa ti n
The simplest way to measure the satisfaction of the Optimality conditions is counting the

 

number of ribbed ﬁnite elements, i.e., elements with pm," < p < 1, for each value of the
optimality condition ratio (O.C.R.) 3,. With this information it is possible to construct a
histogram of the percentage of ribbed elements for each O.C.R. value as shown in Figure
5.2. Theoretically, when optimality conditions are satisﬁed, such histograms must show

80

81

a single peak of value 1 at 5',- =1. Figure 5.2 shows an optimality condition histogram
from a real problem where the micro structure was modeled using rank-2 materials with
rib widths a and b. Notice that peaks at 55:1 are less than 70% of the total ribbed
elements, but between 0.9 and 1.1 in the horizontal axis the percentage of ribbed elements
adds up to more than 90%. This can be considered as a good result.

 

i"
O

—o— fora
—O— fOI'b

.9
00
. 1

Percentage of Ribbed Elements
9 o
4> Ox

9
[Q
1

 

 

 

 

0.0
0.5 1.0 O.C.R 1.5

Figure 5.2. Typical histogram for optimality condition.

This chapter is divided in four sections. In section 5.1 the homogenized plate model
developed in Chapter 2 is compared with results from a previous work of Cheng and
Olhoff, (1981). In section 5.2 a parametric study is performed to show the inﬂuence of
some parameters involved in the optimization problem. In section 5.3 there is a
comparison of optimum shapes computed with the rank-2 and rank-L micro structures,
and ﬁnally, in section 5.4 some industrial examples are solved, showing the applicability
of the approach in mechanical design.

5.1 Comparison with Previous Work

In 1981 Cheng and Olhoff used the thickness of the plate as design variables to compute
the Optimum shape of square plates subject to a uniform transverse load. Their results are
compared here with the rank-2 plate model derived in Chapter 2. The rank-L plate model

82

developed in Chapter 3 gives very similar results to the rank-2 model, therefore, we
decided not to include the rank-L results in this example.

Figure 5.3 shows the geometry of the plate. The dimensions are Lx=Ly=500 mm, and
thicknesses, 2h2=10 mm, 2h1=6.67 mm. The area fraction prescribed to build the ribs is
Vmax=0.5, meaning that the ribbed area will cover 0.5*Lx*Ly=125,000 mmz. The
transverse uniform load is of unit magnitude. We study two different boundary
conditions, simply supported (pinned) and clamped on all sides. The optimum shapes
are: Figures 5.4(a) and (b) for the simply supported and clamped plate, respectively, using
rank—2 materials; Cheng and Olhoff’s results are presented in a three-dimensional view
in Figure 5 .5 and 5.6. Iteration histories for the rank-2 material plate are shown in Figure
5.7, and the histogram of the Optimality conditions is plotted in Figures 5.8(a) and (b) for
simply supported and clamped boundary conditions, respectively.

Optimum shapes computed with the rank-2 model are very similar to the results of Cheng
and Olhoff for both boundary conditions. The concentration of material in the corner of
simply supported plates is due to the stress concentration produced by the known “corner
effect” in plates. For the simply supported plate the normalized compliances computed
with both methods , i.e., Cheng-Olhoff and homogenization methods, are practically the
same. For the clamped boundary conditions, the rank-L plate model gave a more rigid
plate, C*=0.65, than the result reported by Cheng and Olhoff, C*=0.71., however, this
difference is only of 9%. Optimum shapes computed with thickness as design variable
(Cheng and Olhoff’s method) attain the same optimum shape obtained using composite
material plates because, for these cases, the optimum shape is formed by small areas of
composites, therefore, the advantages of using microscopically ribbed plates does not
appear. Histograms of the optimality conditions for the rank-2 material plate present
single high peaks, which indicates satisfaction of the optimality conditions at the
optimum shape computed.

 

 

2.1.6.671 #E—Jflo

Cross
Section

 

 

1 1 dimensions in mm

Figure 5.3. Example 1: geometry and loads.

   

(a) Pinned plate, C*=0.80. (b) Clamped plate, C*=O.65.

Figure 5.4. Example 1: optimum shapes with rank-2 materials.

84

 

 

 

Figure 5.5. Example 1: Cheng and Olhoff (1981) result for the pinned plate, C *=0.82.

 

Figure 5.6. Example 1: Cheng and Olhoff (1981) result for the clamped plate, C*=0.7l.

9
3°

.9
i?

O

Percentage of Ribbed Elements
. o
'1’

.C’

p—t

31‘

Normalized Compliance
.0 .O P
to O\ oo

.O
c

85

 

I"
O

.0
.1;
I

 

—°_ Simply Supported
- -—9— Clamped

 

 

O

10

 

I

20 Iteration 30

Figure 5.7. Example 1: iteration histories.

 

 

0.5

—O— fora
——O— forb

 

 

1.0

Percentage of Ribbed Elements
o o

 

O

O.C.R 1-5
(a) Simply supported plate.

p—r
O

.O

.9

 

 

 

—o— fora
8- + forb
~61
4-
.2-
.0-
0.5 1.0 O.C.R. 1.5

(b) Clamped plate.

Figure 5.8. Example 1: optimality condition ratio histograms.

86

5.2 Comparison of Micro Structure Models

In this section we solve two example problems and compare the optimum shapes given
by the homogenized plate models presented in chapters 2 and 3.

5.2.1 Uniform Transverse Load

A simply supported square plate subject to a uniform transverse load is studied (see
Figure 5.9). The thicknesses are 2h2=10 mm and 2h1=2 mm, and the prescribed area
fraction to build ribs is 50% of the plate area. Figures 5.10(a) and (b) show the optimum
shape for the rank-2 material plate and rank-L models, respectively.

The distribution of the number of layers that is obtained for each micro structure model is
shown in Figures 5.11-5.13. 'In these ﬁgures the black areas denote locations where the
material has the number of layers indicated, and white areas denote locations where the
material is isotropic of thickness 2h1 or 2h2, or where the material has a different number
of layers. Notice, for example, that the distribution of materials with only one layer of
ribs (see Figures 5.11(a) and (b)) is very similar for both micro structure models.
Materials with two layers (see Figures 5.12(a) and (b)) appear at the comers of the plate
for both models, but near the center of the plate the rank-2 model produces more
materials with two layers than the rank-L model. This difference is caused by the third
layer that can be used by the rank-L model (see Figure 5.13). If Figures 5.12(b) and 5.13
are combined, which correspond to the two-layer and three-layer materials obtained with
the rank-L model, respectively, we obtain a picture similar to that shown in Figure
5.12(a), which corresponds to the two-layer material obtained with the rank-2 model. In
other words, at locations where the rank-L model assigns three-layer materials, the rank-
2 model assigns the stiffest ribbed material that it can create, i.e., a two-layer material.

It can be seen that the two optimum shapes are very similar although the rank—L model
allows us to use stiffer micro structures with three layers. Also notice that there is little
use of three-layer materials in the rank-L solution. These two Observations indicate that
in this problem the difference in stiffness between the rank-2 and rank-3 materials may
be small, therefore, the rank-2 model is able to produce results as good as the rank-L
model.

87

The iteration histories for both models (see Figure 5.14) are very similar and the
optimum normalized compliance is C*=0.41. The optimality condition histograms (see
Figure 5.15) show that both models satisfy the optimality conditions quite well.

 

2h2=10

 

Cross
Sectron

 

 

, 1 drmensrons in mm

 

Figure 5.9. Example 2: geometry and loads.

 

(a) Using rank—2 micro structure. (b) Using rank-L micro structure.

Figure 5.10. Example 2: optimum shapes.

88

 

(a) Using the rank—2 model. (b) Using the rank-L model.

Figure 5.11. Example 2: distribution of one-layer materials in the plate.

   

(a) Using the rank-2 micro structure. (b) Using the rank-L micro structure.

Figure 5.12. Example 2: distribution of two-layer materials in the plate.

89

 

Figure 5.13. Example 2: distribution of three-layer materials using the rank-L micro
structure.

 

 

H
O

.9
oo

.3
ox

 

Normalized Compliance
o
21;

0.2 ‘ "—9— rank-2 micro structure
_°_ rank-L micro structure

0.0 ' I ' 1 '
0 10 20 Iteration 30

 

 

 

Figure 5.14. Example 2: iteration histories.

\O
O

 

 

.9
oo
1"
o

—°— fora

4

—°— forb

S3
O\
1

.9

OO

. 1

P

A

1
.0
A
1

.O
N
1

Percentage of Ribbed Elements
9 o
N Ch

Percentage of Ribbed Elements

 

 

 

 

 

 

 

 

0.0 ' 0.0 ‘—
0.5 1.0 O.C.R. 1.5 0.5 1.0 O.C.R. 1.5
(a) Using the rank-2 micro structure. (b) Using the rank-L micro structure.

Figure 5.15. Example 2: optimality condition ratio histograms.

5.2.2 Point Leeds on the Diagonals

In this example we study a simply supported square plate subject to four independent
load cases as shown in Figure 5.16. The thicknesses are 2h2=10 mm and 2h1=2 mm, and
the prescribed area fraction to build ribs is 50% of the plate area. Optimum shapes are
shown in Figures 5.17(a) and (b) for the rank-2 and rank-L models, respectively. Notice
that both optimum shapes are again very similar and the presence of three-layer materials
is not signiﬁcant (see Figure 5.20). Distributions of one-layer and two-layer materials are
shown in Figures 5.18 and 5.19. The same conclusions drawn from the previous example
can also be applied in this case, namely, the rank—2 micro structure is stiff enough to
attain an optimum (C*=0.32) very close to the optimum attained when a rank-L micro
structure is used (C*=0.31) (see Figure 5.21). Again, optimality condition ratio
histograms (see Figures 5.22(a) and (b)) show that at the Optimum solution both models
satisfy the optimality conditions very well.

From this example and the previous one may say that the availability of three-layer
materials in the rank-L micro structure did not make a substantial difference in the
Optimum solution obtained.

91

2h1=2 4; 2h2=10

Cross
Section

 

 

dimensions in mm

Figure 5.16. Example 2: geometry and loads.

 

(a) Using rank-2 micro structures. (b) Using rank-L micro structures.

Figure 5.17. Example 2: Optimum shapes.

92

 

(a) Using the rank-2 micro structure. (b) Using the rank—L micro structure.

Figure 5.18. Example 2: distribution of one-layer materials in the plate.

   

(a) Using the rank-2 micro structure.

Figure 5.19. Example 2: distribution of two-layer materials in the plate.

93

 

Figure 5 .20. Example 2: distribution of three-layer materials using the rank-L micro
structure.

 

 

 

 

 

 

8 0.6
c:
E
'6‘.
E
o
L)
E
.2 _°'— rank-2 model
E 0.4' C*=0.32
0
Z

l

""_ rank-L model
C*=0.3 1
02 . I . I w
0 10 20 30
Iteration

Figure 5.21. Example 2: iteration histories.

94

 

 

 

 

 

 

 

 

 

 

 

 

a 1.0 10
g . “‘0— for a *2
—o— 0
'5 0.8- forb E 0.8-
'1: LL] .
_g 'o
’13 06- .8 0.6“
«4.1 . E .
o a...
g 0.41 3 0.4-
H 01)
c: 5‘
§ _ a
a: 0.2 § 0.2‘
T “Ab—9* J k
0.0 v 0.0 . 1 -
0.5 1.0 O.C.R. 1.5 0.5 1.0 O.C.R. 1.5
(a) Using the rank-2 micro structure. (b) Using the rank-L micro structure.

Figure 5.22. Example 2: optimality condition ratio histograms.

5.3 Parametric Study

In these examples we perform a parametric study Of variables that appear in the
formulation of the layout optimization problem. The effect of the thickness ratio
r=h2/h1, the amount of material Vmax and the type of cross section (ribbed, honeycomb or

perforated) on the optimum shape is studied.

5.3.1 Inﬂu nce of the Thi kness Ratio r=h 1

In this example a simply supported rectangular plate (Figure 5.23) is considered to study
the effect of the thickness ratio on the optimum shape of the plate. The total thickness is
ﬁxed at 2h2=25 mm and the core plate thickness 2111 is computed such that three values
of the thickness ratio are considered, namely, r=h2/h1=1.5, 3 and 6. The prescribed
amount of material used to build ribs is 20%.

Optimum shapes using the rank-2 material plate are shown in Figure 5.24. Notice that
for high values of r=h2/h1, the optimum shape presents more areas with ribs (gray areas)

95

in comparison with the optimum shapes for low values of r. This happens because a
required level of stiffness can be obtained with less material when the ribs are higher. In
other words, when the ratio between the stiffness of the strong plate and the stiffness of
the weak plate is high more gray areas are expected in the optimum shape. Iteration
histories in Figure 5.25 show that tall ribs (r=6) can achieve a reduction of 83% of the
compliance of a uniform thickness plate, while short ribs (r=1.5) can only reduce 21%.

The three optimum shapes shown in Figures 5.24 produce important reductions in the
compliances and have good iteration histories (see Figure 5.25). When the histograms of
the optimality condition ratio are computed (see Figure 5 .26) one can see that for the rib
width “a” there are always several peaks far from the main peak. A closer look at the
results showed that the number of elements satisfying am," < a < 1 was less than 10% of
the total number of elements in the mesh, meaning that the poor satisfaction of the
Optimality conditions for “a” is not so important. Additionally, it was found that the
number of elements satisfying a > am,” was the same as before, that is, about 90% of the
elements had a: am)", meaning that the width “b” was giving the optimum shape and

“a” did not participate much in the optimum solution.

uniform load
7/

    

 

 

l.. , 800mm . _ .. ..l

2h 1=variable

”’1 i _U—U‘ji‘Z 2h2=25 mm

Cross
Section

Figure 5.23. Example 3: geometry and loads.

 

96

 

 

 

(a) For hz/h1=1.5

 

(b) For h2/h1=3.0

 

 

 

 

 

(c) For h2/h1=6.0

Figure 5.24. Example 3: Optimum shapes.

 

 

 

 

 

 

 

8 1.0
5 5
'7'; Ragn- r=1.5 C*=0.79
g 08‘ ___... ...
u .
-o
“g 0.61
E ' 1:3-0 -- c*=o.41
<23 04‘ """"""
02"R. 1=60 c*=o.17
0.0 ' I ' I '
0 10 20 Iteration 30

Figure 5.25. Example 3: Iteration histories.

Percentage of Ribbed Elements

\0
\)

 

 

_°— fora
—'_ forb

1.0
(a) For h2/h1=1.5.

 

O.C.R.

 

Percentage of Ribbed Elements
c
c»

 

 

_°_ fora

—’_ forb

 

(b) For h2/h1=3.0.

 

I"
O

.o
00
I

Q
A
1

Percentage of Ribbed Elements
9 o
N O\

 

.9
90
u:

—°— fora
—°— forb

 

1,0 O.C.R.

(c) For h2/h1=6.0.

 

1.5

Figure 5.26. Example 3: optimality condition ratio histograms.

 

98

5 .3.2 Inﬂuence of the Amount of Material Vmax

In this example we study the variation of the optimum shape with changes in the
prescribed area fraction of ribs, Vmax. A clamped rectangular plate (Figure 5.27) of
dimensions 800mm x 500mm subject to a uniform transverse load is studied for four
different amounts of material, Vmax=5%, 10%, 20% and 40% of the plate area. The
thicknesses are ﬁxed at 2h2=5mm and 2h1=lmm. Optimum shapes are given in Figure
5.28.

It is important to realize that each time more material is added, this material is not only
used to reinforce areas that already had ribs -- which would imply that the topology does
not change --, but is also used to reinforce new areas that did not have ribs previously.
This produces new topologies as the material is added to the plate. More black areas with
thickness 2112 appear as more material is available to allocate ribs. The concentration of
material on the sides of the plate is a characteristic of the clamped plates that also was
seen in the example of comparison with Cheng and Olhoff results above. The iteration
histories (see Figure 5.29) indicate that the more material is used, the less improvement
on the minimization of the compliance is achieved. This is consistent with the limit case
of Vmax=100% that corresponds to C*=1. Notice that a reduction of almost 90% in the
compliance is achieved when the amount of material is less or equal to 10%. Optimality
condition histograms showed in Figure 5.30 indicate a good satisfaction of the optimality
conditions. Notice that when Vmaxs 10% there are no plots for the optimality conditions
for a. This is because the optimum shape only uses one-layer materials of width b, and
a=0, therefore, there are no points to plot for the rib width a.

ﬂiform load

 

   

cross section

Figure 5.27. Example 3: geometry and loads.

 

 

99

 

       

 

 

 

(a) For Vmax-=5%. (b) For Vmax=10%.

 

 

(c) For Vmax=20%. (d) For Vmax=40%.

Figure 5.28. Example 3: optimum shapes.

 

 

 

 

 

8 0.6
_§ - Vmax=5%
E“ 0.5 '“ Vmax=10%
.D 0 4‘ —‘— Vmax=40%
E ‘
E 0.3 1
E .
0
Z 0 2 J
’ 1
0.1 ‘
0.0 ' I ' I '

 

 

0 10 20 Iteration 30

Figure 5.29. Example 3: iteration histories.

 

 

 

é

 

:9, 1.0 3
c: c.
o . '—'— for b o
g g
E 0'8 no plot E

f "c
E ' 22
“a ‘ “a
g 0.4 - goo.
§ §
an: 0.2' i an: 0.

00 “I u
0.5 1. OCR 1.5
(a) For Vmax=5%.
a 1.0 £1
E - _°_ for a a
5 0.8- —'_ for b £0
E ‘ 33
-° u :0
a 0.6 g0.
‘4—4 . q_‘
o o
o _ a)
330.4 330.
c: c:
8 8
£02“ $0
0.0“ O
0.5 1.0 O.C.R. 1.5
(c) For Vmax=2%.

 

 

 

 

I"
O

.9
00
n l

 

 

 

 

 

 

in

'o

60
n l

4:.

N

 

.9
u.

C
L 1 . 1 . l

—’— forb

no plot
for a

 

 

 

1.0 O.C.R.
(b) For Vmax=10%.

p—L
{It

 

A Q
l . l

N
l

 

0-

0.5

—°"" fora
—_°_ forb

 

 

1.0 O.C.R. 1.5
(d) For Vmax=40%.

Figure 5.30. Example 3: optimality condition ratio histograms.

101

5.3.3 Comparison of Cross Sections

In this example, the effect of the cross section on the optimum shape is studied. A
simply supported plate subject to ﬁve simultaneous point loads (see Figure 5.31) is
considered for the experiment. The area fraction of ribs is Vmax=50% and the
dimensions of the ribbed cross section are 2h2=15 mm and 2h1=5 mm. The
honeycomb cross section has the same external thickness 2h2 and the height of the
internal ribs is such that the amount of material is the same as in the ribbed plate. The
perforated cross section has a thickness of 2h2=20mm to preserve the amount of
material for the three cross sections studied.

When the ratios between the bending stiffness of the strong plate and the stiffness of the
weak plate are computed for each cross section, the results are, 1.4 for the honeycomb,
27.0 for the ribbed, and 0° for the perforated cross section . Therefore we can see this
experiment as if we were studying three different thickness ratios r=h2/h1 such that the
thicknesses produce the three stiffness ratios computed above. The optimum shapes in
Figure 5.32 are consistent with the results obtained when we studied the effect of the ratio
h2/h1, namely, large areas of rapidly varying thickness for the relatively stiffer cross
section (the perforated one) and small areas of rapidly varying thickness for the
relatively less stiff cross section (the honeycomb one). The ribbed cross section gives an
intermediate result between the perforated and the honeycomb cross sections. Iteration
histories and optimality condition histograms are shown in Figures 5.33 and 5.34,
showing a good convergence and satisfaction of the optimality conditions. For the
honeycomb cross section, Figure 5. 34(c), peaks far away from the main peak (for the
width “ a”) are not important since only 5% of the elements of the mesh satisfy am” < a

< 1 and only few of them do not satisfy optimality conditions.

2h1 i I l I l :E: Ribbed Cross

Section

 
 

 

0h
lt— 2Honeycomb
2Cross Section
h 11:—

 

Figure 5.31. Example 3: geometry and loads.

102

   

(a) For the ribbed cross section. (b) For the honeycomb cross section.

 

(c) For the perforated cross section.

Figure 5.32. Example 3: optimum shapes.

Percentage of Ribbed Elements

103

 

.9 t"
00 C

.9

Normalized Compliance
O\

.o
4:.

 

  

Honeycomb, C*=0.32

Perforated, C*=0.23

Ribbed, C*=O.51

 

 
 
   
 
 

DDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

  

 

T

10

I ' I

Iteration

Figure 5 .33. Example 3: iteration histories.

 

 

 

1.0 521.0
. "—0— for a g . —°_ for a
0.8“ ' forb £03— . _ forb
E .
0.6- 530.6-
. Q—d- .
O
0.4- £00.4-
3 l
8
0.2- 50.2-
0.0 > 0.0-
0.5 1.0 O.C.R. 1.5 0.5 1.0 O.C.R

(a) For the ribbed cross section.

 

 

 

(b) For the honeycomb cross section.

Figure 5.34. Example 3: optimality condition ratio histograms.

 

 

 

I"
O

.9
00

.O
A

.o
N

'o\

104

 

l

l

l

1

—°— fora
__‘— forb

Percentage of Ribbed Elements
o

L

 

 

 

.9
QC
u:

1.0 1.5
O.C.R.

(c) For the perforated cross section.

Figure 5 .34. (cont) Example 3: optimality condition ratio histograms.

5.4 Industrial Examples

5.4.1 A Car Hood

In this example we compute the optimum reinforcement of a car hood subject to three
independent load cases, a frontal load and two lateral loads as shown in Figure 5.35. The
two lateral loads are applied independently but always together such that the resulting
optimum shape is symmetric. These loads simulate the crash of the car from the front
and from the sides. The weighted compliance is computed using equal weights. In this
case we use a perforated cross section with total thickness 2h2=1 mm and the prescribed
amount of material covers 30% of the area of the hood shell.

When the frontal and lateral loads are studied separately we obtain the two optimum
shapes shown in Figures 5.36(a) and (b), respectively. Notice that more material is
allocated where the loads are applied.

When the three load cases are considered, the optimum shape obtained is that shown in
Figure 5.36(c). Since we are using a perforated cross section, large gray areas (rapidly

varying thickness plate areas) appear in the optimum shape.
the normalized compliance (Figure 5.37) show that the reduction in compliance for the
three load cases (the frontal and the two lateral loads) is almost 70% of the uniform
thickness plate compliance. If we were to build the hood, we could interpret the optimum
reinforcement in Figure 5.36(c) as is shown in Figure 5.38. The prescribed amount of

105

material is violated in Figure 5.38, but the essence of the optimum shape is preserved.

This problem was also studied using the rank-L micro structure model of Chapter 3. In
this case, we only studied the situation when the three loads where applied. The optimum
the iteration history in Figure 5.40. Both micro
structures, the rank-2 and the rank—L, give basically the same optimum shape and the
Again, the use of a stiffer micro structure did not
Optimality condition

shape is shown in Figure 5.39 and

same ﬁnal normalized compliance.

improve the optimum shape or the normalized compliance.

histograms are shown in Figure 5.41.

O
\‘
l
'0
O
‘3'
\ \
O‘
.9

     

  
  

’4 ...

\
l
O
'o
\‘ 0:01...
‘i 1‘30"
O N
00M
0'0 '1"
l

O

 

Q
0'0
.0
o

O
0
i
Q t
;s
0
.0
0

(If,
.7 lg"!
#191147};
"a ’I
A

Figure 5.35. Example 4: geometry and loads.

The iteration histories of

106

 

(a) Frontal load. (b) Two lateral loads.

 

(c) Frontal and lateral loads.

Figure 5 .36. Example 4: optimum shapes.

107

 

 

 

 

o 1.50
g l' _9— Frontal
'7‘; 1,25% —‘— Lateral
g - —'_ Frontal+Lateral
U _
,0 1.00
<1) .
.5
3 0.75'
E .
0
Z 0.50-

0.25"

0.00

O 10 20 Iteration 30

Figure 5.37. Example 4: iteration histories.

 

Figure 5.38. Example 4: Interpretation of the optimum shape of the car hood.

108

 

Figure 5.39. Example 4: optimum shape obtained using the rank—L micro structure.

 

 

 

 

 

g 1.4 1,
.9
2‘12 . '°—' Rank-2 micro structure
81.0- —'— Rank-L micro structure
E .
z 0.8“
a I
Em
z .
().4“l v. U V 3 v :AVAVAV .‘ ‘v
0.2"
i
0.0 ' l ' I u
0 10 20 Iteration 30

Figure 5.40. Example 4: iteration histories.

109

 

 

 

 

 

 

 

 

 

 

 

.9 1.0 a 1.0
g . g, .
£08— i—F f0” 208'
a ' . —*- forb ”é .
.3; E

22‘ 0.6“ a 0-6‘
95 . c8 .
o _ o
30-4 £90.41
§ § ‘
g 0.2- 5'? 0.2-

. ‘ . L

O.C.R. 1.5
(b) Using the rank-L micro structure.

0.5 1.0 O.C.R. 1.5 0.5 1.0
(a) Using the rank-2 micro structure.

 

Figure 5.41. Example 4: optimality condition ratio histograms.

5.4.2 A Car Door

Figure 5.42 shows the geometry of a car door that is subject to a point load at the center
of the main panel and is supported at four points. The thicknesses are 2h2=1 mm and
2h1=0.67 mm, and the prescribed amount of material is Vmax-=35%. The frame of the
window has ﬁxed thickness equal to 2h2 and is not considered in the optimization
problem. The problem was solved using the rank-2 and the rank-L micro structures. The
optimum shapes obtained are shown in Figures 5.43(a) and (b). One more time,
optimum°shapes computed with both micro structures are very similar. The optimum
normalized compliance is C*=0.64 for both shapes. Histograms in Figures 5.45
indicate that optimum shapes satisfy optimality conditions satisfactorily.

110

 

geometry and loads.

Figure 5.42. Example 5

 

ank-L micro structure.

ing r

(b) Us

tructure.

11an0 S

(a) Using rank-2

hapes.

optimum s

0

Figure 5.43. Example 5

111

 

 

. —°— Rank—2 micro structure
—'— Rank-L micro structure

Normalized Compliance
:3

 

 

 

0.0 . . . . .
O 10 20 Iteration 30

Figure 5.44. Example 5: iteration histories.

 

 

 

h—t
O
i"
O
A...”

—°— fora
_'—' forb

O
00
I
9
oo
1

L

Percentage of R1bbed Elements
. .0 .O . .
4? °.‘ . .
Percentage of R1bbed Elements
c
O\

O
[Q
l

.O
A
I

p
N
l

 

 

 

 

 

 

 

0.0 ' l 0.0 ' I
0.5 1.0 O-C-R. 1.5 0.5 ' 1.0 O.C.R. 1.5

(a) Using the rank-2 micro structure. (b) Using the rank—L micro structure.

Figure 5.45. Example 5: Optimality condition ratio histograms.

CONCLUDING REMARKS

General

As in most dissertations, a signiﬁcant portion of the work presented here is an extension of
previous research. In this case, the effort -- often independent -- of M. Bends¢e, A. Diaz,
N. Kikuchi, R. Lipton and K. Suzuki was the starting point of our research. A substantial
amount of time was spent collecting diverse knowledge on the subject and putting it
together to clarify many aspects that were not fully understood when this research began.
There were also other aspects in the subject of shape optimization of plates that had not
been addressed in the literature and were developed in our investigation. Conclusions
about speciﬁc topics studied in this research were given in Chapters 2, 3, 4 and 5. Here
we only present general conclusions emphasizing the new results introduced in this work.

In Chapter 2 we presented the mathematical derivation of the effective properties for
bending and transverse shear deformations of plates made of two orthogonal layers of ribs
(rank-2 material plates) using the method of asymptotic expansions on the Mindlin’s plate
equations. An extension of this work to plates with other cross sections (honeycomb and
transversely perforated) was also described.

The use of rank-2 material to solve shape optimization problems in plates is one of the main
contributions of Chapter 2. Previous work used other micro structures whose effective
properties were computed using approximate numerical procedures. The formulas
developed in Chapter 2 represent an important improvement in the computation of effective
properties of plates in topology optimization algorithms based on homogenization
techniques.

The difference between homogenized plate models based on two-dimensional elasticity and

plate equations was also addressed in this work. We concluded that effective plate
properties computed when the homogenization techniques are applied to the two-

112

 

113

dimensional elasticity equations are in general less stiff than those obtained when the
homogenization techniques are applied directly to plate equations. The implication of this
difference from the shape optimization viewpoint is that when we use homogenization
techniques in the plate equation, optimum topologies prescribe more composite areas.

A comparison of several micro structures discussed in the literature was presented to show
the inﬂuence of different geometric parameters on individual bending stiffness coefﬁcients.
The comparison helps understand relations among popular models used in the literature.
Such study was not available in the open literature.

In Chapter 3 we presented a mathematical derivation of the effective properties for bending
and transverse shear deformations of plates made of materials with a ﬁnite number of layers
of ribs oriented in L arbitrary directions. These properties were then presented in a
compact form called the “moment representation”, which uses only four variables
(moments) to describe the effective properties of rank-L material plates once an amount of
material is prescribed. The important characteristic of these plates is that they attain optimal
bounds for multiple energy conditions, i.e., under prescribed strain ﬁelds, they are the
stiffest plates. This allowed us to solve the layout and shape optimization problem using
the stiffest micro structure at each location of the plate.

The derivation of bending and transverse shear effective stiffness properties of rank-L
material plates is the main contribution of Chapter 3. The moment representation of these
effective properties is new for Mindlin plates. This representation allowed us to solve the
shape optimization problem using the stiffest micro structure with only five design
variables, namely, the material resource p, and four moments m1, m2, m3, m4.

In Chapter 4 we presented the formulation of the layout and shape optimization problem in
plate structures using homogenization techniques. The stiffest distribution of the
reinforcing material was obtained using the weighted sum of the compliances as the
objective function. The optimization problem was split into two problems taking advantage
of the moment representation of the effective properties developed in Chapter 3 and using
known variational principles. This formulation was called “hierarchical moment
formulation” and allowed us to solve the shape optimization problem using optimal micro
structures at every location in the plate. Implementation of the hierarchical moment
formulation for multiple load cases using the moment representation of the effective

properties is new for Mindlin plates.

114

Implementation issues of the optimization problem using the hierarchical moment
formulation were also discussed. From the results obtained in the problem examples we
conclude that the proposed strategy to solve the nonlinear equilibrium problem works very
well.

In Chapter 5 several example problems were solved in order to show the capabilities of the
homogenization approach to solve layout and shape optimization problems. Optimum
shapes obtained with the rank-L model showed that, for the problems studied, materials
with three layers of ribs are seldom prescribed. This indicates that even though rank-3
materials are, in general, stiffer than rank-2 materials, the difference between them in terms
of stiffness may be small. Because of this small difference, optimum shapes and
compliances computed with the rank-2 model are very similar to those computed with the
rank-L model.

Examples in the automobile industry showed that the optimization algorithm using
homogenization works well in practical engineering situations which makes very attractive

this approach.

Future Research

It is clear from the examples that the homogenization approach is a very powerful method
to solve the layout and shape optimization problem in plate structures. Some improvements
are still necessary, and they are cited here as future research directions.

From the examples solved with both the rank-2 and rank-L micro structures, we observed
that there is not much difference in the results obtained using these two models. More
research should be done to know the reasons of this observation and to understand the
behavior of rank-L materials.

One of the drawbacks of the homogenization method applied to shape optimization is the
lack of smoothness of the boundaries of the optimum shape due to the discretization of the
domain. This can be addressed, in part, using ﬁner meshes, but the cost of the solution of
the equilibrium equation increases. Better strategies to solve this problem should be
investigated.

 

115

For practical applications it is important to consider more realistic constraints in the
optimization problem. Local stresses and displacements, buckling and manufacturing
constraints should be incorporated in the formulation of the problem.

There is always the question about the quality of the optimum found using optimality
criterion methods. These methods use ﬁrst order conditions that do not guarantee that the
minimum is a global one. Some research is required in this subject.

This work intended to contribute to the enrichment of the scientiﬁc knowledge in the ﬁeld
of optimum structural design. We hope that this work will help future students and
researchers to ﬁnd new routes to expand the knowledge frontier that we reach in this
research.

APPENDIX

F— " ’ " ~ 7 , nu

 

 

 

APPENDIX

Matrices of the stiffness coefﬁcients in (2.1.3a)

"D0(l,1) D0(1,3) 0‘
000,3) D0(1,2) o

 

 

 

W”: 0 0 0 W12:
0 0 0
_ 0 0 0_
fD0(1,3) D0(3.3) 0'
000,3) 00(2,3) 0
W”: 0 0 0 W”:
0 0 0
_ 0 0 0_
r 0 0 0‘
0 o 0
W41: 0 0 0 W42—
Dz(3.3) 02(23) 0
_—D2(1,3) —D2(3,3) 0_
' 0 o 0‘
0 o 0
W51: 0 0 0 W52—
—Dz(1.3) —Dz(1,2) 0
_ 020,1) 020,3) 0_

 

 

116

 

 

 

 

"Do(l.3) D0(3,3) 0
006,3) 00(2,3) 0

0 0 0

O 0 0
_ 0 0 0_
"00(12) 00(2,3) 0‘
Dom) 000.2) 0

0 0 0

O 0 O
_ 0 0 0_
— 0 0

0 0

0 O
020.3) 020.2)
_-Dz(1.2) -Dz(2.3)
P O O

0 O

0 O
-D2(3.3) -Dz(2,3)
_ Dz(l.3) 020.3)

OOOOOOOOOO

 

 

117

F- —

 

0 0 0 r o 0
0 0 0 0 0
W43: —S(1,2) —S(2,2) 0 W53: S(1,1) 50,2)
0 0 S(2,2) 0 0
_ 0 o —S(1,2)_ _ 0 0
' 0 0 0 ‘ f 0 0
0 0 0 0 0
W31: 5(1,1) 5(1,2) 0 W32: S(1,2) 5(22)
0 0 —S(1,2) 0 0
_ o 0 5(1,1)_ _ 0 0

 

 

 

W13 = W23 = W33 = [01(5x3)

The major symmetry is present in these matrices, i.e.,

k€_ zj
Wij - a

but the minor one is not since

kt 2k kt
Wij 1* Wij ¢ Wji

This results from grouping together all terms from membrane, bending and transverse

shear deformations into a single array my.

-I

0
0
0
—S(1,2)
S(1, 1) .
o 1
o
0
—S(2, 2)
5(1, 2) _

 

 

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