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DISCRETE CROSS-SECTION LAYOUT OPTIMIZATION OF
UNIFORM CELLULAR PANELS FOR BlOFlBER-POLYMER
COMPOSITES

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Aradhana Sharma

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DISCRETE CROSS-SECTION LAYOUT OPTIMIZATION OF UNIFORM
CELLULAR PANELS FOR BIOFIBER-POLYMER COMPOSITES

By

Aradhana Sharma

A THESIS
Submitted to
Michigan State University
In partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Civil and Environmental Engineering

2005

ABSTRACT
DISCRETE CROSS-SECTION LAYOUT OPTIMIZATION OF UNIFORM
CELLULAR PANELS FOR BIOFIBER-POLYMER COMPOSITES
By
Aradhana Sharma

Increased emphasis on sustainability and eco-efficiency of material resources for all
applications, including structural applications, has motivated exploration of natural fiber
composites, or biocomposites. Yet, due to their lower stiffness and strength properties,
biocomposites are unable to compete with existing construction materials for load-
bearing applications. However, as seen in nature, engineered or optimized designs can
lead to improved performance through efficient material and structural configurations.
This thesis presents two discrete optimization approaches to obtain optimum material
distribution in the cross-sectional layouts of continuous biocomposite cellular panels. The
first approach uses a gradient-based sizing optimization method and the second one uses
a genetic algorithm to select coupons with finite size features from a pre-defined library.
Unlike conventional topology optimization the presented methods enabled optimum
hybrid designs. The designs were evaluated through laboratory scale component testing.
The optimization results led to layouts that are easier to manufacture than those that can
be obtained by traditional topology optimization methods. As expected, the manufactured
optimum designs demonstrated improved performance. The material layout optimization
methods incorporating finite-size features presented in this study are thus thought to be
viable for optimizing material distribution in cross-sections of biocomposite cellular

panels and have led to designs feasible for manufacturing.

Copyright by
ARADHAN A SHARMA
2005

ACKNOWLEDGEMENTS

I would like to thank my advisor Dr. Rigoberto Burguefio for his excellent guidance,
patience, time and continuous encouragement throughout this work. It has been a great
learning experience to work with him. I would not have been to complete this work
successfully without his vision and support.

I would also like to thank my master’s committee members, Dr. Ronald Harichandran
and Dr. Ghassan Abu Lebdeh for their time and valuable guidance.

I also wish to thank Dr. Amar Mohanty and Dr. Majusri Misra for their support for
manufacturing biocomposite cellular beams.

I wish to thank Siavosh Ravanbaksh for his help in conducting my experiments at the
Civil Infrastructure Laboratory.

I am grateful to my fi‘iends and colleagues Christina Isaac, Mahmoodul Haq and Golrokh

Nossoni for their help and constant support throughout this work.

iv

TABLE OF CONTENTS

LIST OF TABLES ........................................................................................................ vii
LIST OF FIGURES ..................................................................................................... viii
LIST OF SYMBOLS ................................................................................................... xiii
1. INTRODUCTION ....................................................................................................... 1
1.1 General ....................................................................................................................... 1
1.2 Overview ................................................................................................................... 1
1.3 Biomimetics ............................................................................................................... 6
1.4 Objective and scope ................................................................................................... 8
2. STRUCTURAL OPTIMIZATION BACKGROUND .............................................. 10
2.1 General ..................................................................................................................... 10
2.2 Structural optimization ............................................................................................ 10
2.3 Optimization Design Process ................................................................................... 13
2.4 Design variables, Objective functions and Constraints ........................................... 14
2.4.1 Design variables ............................................................................................... 14
2.4.2 Objective function ............................................................................................ 15
2.4.3 Constraints ........................................................................................................ 16
2.4.4 Mathematical statement of optimization problem ............................................ 16
2.4.5 Multiobjective optimization ............................................................................. 22
2.4.6 Genetic algorithms ............................................................................................ 27
3. DISCRETE LAYOUT OPTIMIZATION OF CELLULAR STRUCTURES ........... 36
3.2 Sizing optimization of cellular cross-sections using gradient search ...................... 36
3.2.1 Design variables ............................................................................................... 37
3.2.2 Objective function ............................................................................................ 41
3.2.3 Constraints ........................................................................................................ 42
3.2.4 Implementation of sizing optimization problem in MATLAB ........................ 43
3.2.5 Scalar sizing optimization for maximization of stiffness ................................. 44
3.2.6 Scalar sizing optimization for maximization of specific stiffness .................... 47
3.2.7 Multiobjective sizing optimization using weighting coefficients method ........ 52
3.2.8 Limitations of sizing optimization using gradient based methods ................... 57
3.3 Layout optimization of beam/panel cross-sections using genetic algorithms ......... 58
3.3.1 Design variables ............................................................................................... 59
3.3.2 Objective function ............................................................................................ 62
3.3.3 Implementation of layout optimization in GALOPPS ..................................... 65

3.3.4 Multiobjective optimization of sectional stiffness for a square beam .............. 66
3.3.5 Multiobjective optimization of specific sectional stiffness for a square beam.69

3.3.6 Limitations of layout optimization using genetic algorithms ........................... 73
3.4 A coalescence approach for improved optimized layout designs ............................ 73
3.4.1 Steps for coalescence approach ........................................................................ 75
3.4.2 Two-dimensional coalescence for maximum effective stiffness ................. 76
3.5 Integrated coalescence approach for Baldwinian inheritance ................................. 78
3.6 Results for multiobjective optimization using Baldwinian approach ...................... 79
4. EXPERIMENTAL EVALUATION .......................................................................... 84
4.1 General ..................................................................................................................... 84
4.2 Structural layouts ..................................................................................................... 84
4.3 Materials and constituents ....................................................................................... 85
4.4 Manufacturing ......................................................................................................... 87
4.4.1 Automated manufacturing using VARTM ....................................................... 87
4.4.2 Material placement ........................................................................................... 88
4.4.3 VARTM: Setup and resin infusion process ...................................................... 90
4.5 Testing of beams ...................................................................................................... 92
4.6 Observed Behavior and Test Results ....................................................................... 96
4.6.1 Four-point bending ........................................................................................... 96
4.3.2 Three-point bending ....................................................................................... 105
5. CONCLUSIONS ..................................................................................................... 109
REFERENCES ............................................................................................................ 112

vi

LIST OF TABLES

Table 3.1 Results for scalar maximization of stiffness for a three-layer plate ............. 45
Table 3.2 Results for scalar maximization of stiffness for a five-layer plate ............... 46
Table 3.3 Results for scalar maximization of stiffness for a seven-layer plate ............ 46

Table 3.4 Results for scalar maximization of effective stiffness for a three-layer plate

....................................................................................................................................... 48
Table 3.5 Results for scalar maximization of effective stiffness for a five-layer plate .....
....................................................................................................................................... 49
Table 3.6 Results for scalar maximization of effective stiffness for a seven-layer plate
....................................................................................................................................... 49
Table 3.7 Pareto optimum set multiobjective optimization for a three layer plate ....... 53
Table 3.8 Pareto optimum set multiobj ective optimization for a five layer plate ........ 54
Table 3.9 Pareto optimum set multiobjective optimization for a seven layer plate ..... 55
Table 3.10 Results for multiobjective optimization of sectional stiffness .................... 68

Table 3.12 Results for coalescence approach in two dimensions for maximization of
stiffness ......................................................................................................................... 77

Table 3.13 Results for multiobjective optimization of sectional stiffness using
Baldwinian Approach ................................................................................................... 80

Table 3.14 Results for multiobjective optimization of specific sectional stiffiiess using
Baldwinian Approach ................................................................................................... 82

Table 4.2 Comparison of experimental and theoretical flexural stiffness from four-
point bending test ........................................................................................................ 103

Table 4.3 Comparison of failure strain and load measured at extreme tension fiber in
negative moment section ............................................................................................ 104

Table 4.4 Experimental stiffness from three-point bending test and theoretical stiffness
for base and optimal designs ....................................................................................... 108

vii

LIST OF FIGURES

Figure 1.1 Cellular structure of a plant stem [18] ........................................................ 2
Figure 1.2 Typical topology optimization solution for a cantilever subjected to point
load ............................................................................................................................... 3
Figure 1.3 Natural fibers .............................................................................................. 4
Figure 1.4 Transverse and longitudinal cross-sections in a beam considered for
material layout optimization ........................................................................................ 5
Figure 1.5 Sandwich structures and cellular structures in nature ................................ 7
Figure 2.1 Types of structural optimization ............................................................... 12
Figure 2.2 Optimum design process (adapted from [1]) ............................................ 14
Figure 2.3 Conceptual diagram for iterative steps of an optimization method .......... 19
Figure 2.4 Mapping from parameter space into objective function space ................. 24
Figure 2.5 Graphical representation of the weighted sum method ............................ 25
Figure 2.6 Schematic of a string population in a genetic algorithm (adapted from
[10]) ........................................................................................................................... 29
Figure 2.7 Two examples of one-point crossover (adapted from [6]) ....................... 32
Figure 2.8 Examples of bit mutation (adapted from [6]) ........................................... 34
Figure 3.1 Continuous panel system subjected to distributed load in one-way
bending ....................................................................................................................... 37
Figure 3.2 Cross-sectional view of a hierarchical cellular biocomposite panel ........ 37
Figure 3.3 A characteristic repeating coupon ............................................................ 38

Figure 3.4 Interpretation of a three layer repeating column layout in a three layer

plate ............................................................................................................................ 39
Figure 3.5 Design variables for the three—layer panel ................................................ 40
Figure 3.6 Design variables for the five-layer panel ................................................. 40

viii

Figure 3.7 Design variables for the seven-layer panel ............................................... 41
Figure 3.8 Constrained optimization using MATLAB’s “finincon” function ........... 44

Figure 3.9 Layout obtained for maximization of stiffness for three, five and seven
layer plates for constraint sets I, III and V respectively ............................................ 46

Figure 3.10 Layout obtained for maximization of stiffness for three layer plate for
constraint set 11 ........................................................................................................... 47

Figure 3.11 Layout obtained for maximization of stiffness for five and seven layer
plates for constraint set IV and VI respectively ......................................................... 47

Figure 3.12 Layout obtained for maximization of effective stiffness for a three layer
plate using constraint set I ......................................................................................... 49

Figure 3.13 Layout obtained for maximization of effective stiffness for five and
seven layer plates using constraint set 111 and V respectively ................................... 50

Figure 3.14 Layout obtained for maximization of effective stiffness for a three layer
plate using constraint set II ........................................................................................ 50

Figure 3.15 Layout obtained for maximization of effective stiffness for a five layer
plate using constraint set IV ...................................................................................... 51

Figure 3.16 Layout obtained for maximization of effective stiffiiess for a seven layer
plate using constraint set VI ...................................................................................... 51

Figure 3.17 Pareto optimum set for multiobjective optimization of moment of Inertia
and volume for a three layer plate ............................................................................. 56

Figure 3.18 Pareto optimum set for multiobjective optimization of moment of Inertia
and volume for a five layer plate ............................................................................... 56

Figure 3.19 Pareto optimum set for multiobjective optimization of moment of Inertia
and volume for a seven layer plate ............................................................................ 57

Figure 3.20 Overview of layout optimization using library of coupons with finite—
size features ................................................................................................................ 58

Figure 3.21 Layout of a general coupon with finite size features used for

optimization ............................................................................................................... 60
Figure 3.22 One-dimensional array of structures containing coupon data ................ 60
Figure 3.23 Library of hybrid and non-hybrid coupons ............................................ 61

ix

Figure 3.24 Determination of optimum layout for a given generation in GALOPPS

 

 

................................................................................................................................... 63

Figure 3.25 Transformed section dimensions of the hybrid coupon ......................... 64

Figure 3.26 Two-dimensional array of structures containing coupon data ............... 64

Figure 3.27 Organization of GALOPPS .................................................................... 65

Figure 3.28 Layout for maximization of f = I x ....................................................... 68
. . . . W1 1 x + “’2 1y

Figure 3.29 Layout for maxrmlzatlon of f = 2 ................................. 69

Figure 3.30 Layout for maximization of f = I), ....................................................... 69
o o o o 1“

Flgure 3.31 Layout for maxrmrzatlon of f = 7' .................................................. 72
. . . . Ix + 1y

Figure 3.32 Layout for max1m1zatlon of f = 2V .......................................... 72
. . . . 1y

Figure 3.33 Layout for maxrmrzation of f = 7 .................................................. 72

Figure 3.34 Coalescence in one dimension ............................................................... 74

Figure 3.35 Coalescence in two dimensions .............................................................. 75

Figure 3.36 Overview of coalescence algorithm for one dimensional coalescence ..76
Figure 3.37 Coalescence scheme used in one and two dimensions coalescence ....... 77
Figure 3.38 Layout for two-dimensional coalescence for maximization of stiffness78
Figure 3.39 Integrated coalescence approach for Baldwinian inheritance ................ 79

Figure 3.40 Layout for maximization of f = I x using Darwinian and Baldwinian
approaches ................................................................................................................. 81

Figure 3.41 Layout for maximization of f = I y using Darwinian and Baldwinian

approaches ................................................................................................................. 8 l

X

. . . I . . .
Figure 3.42 Layout for maxrrmzatlon of f = (—5—) obtamed from (a) Darwmian and

(b) Baldwinian approaches respectively .................................................................... 82
Figure 3.43 Layout for maximization of f = (—11—?) obtained from (a) Darwinian and
(b) Baldwinian approaches respectively ........................................................ i ............ 8 3
Figure 4.1 Cross-sectional layouts of base and optimal design used in experimental
evaluation ................................................................................................................... 85
Figure 4.2 Jute mat for top and bottom face sheets ................................................... 86
Figure 4.3 Industrial hemp fibers ............................................................................... 86
Figure 4.4 Preparation of mold for material placement ............................................. 90
Figure 4.5 Sample with vacuum bag, resin reservoir and resin trap .......................... 91

Figure 4.6 Simulation of a continuous panel system under symmetric loading and
boundary conditions in a four-point bending test ...................................................... 93

Figure 4.7 Bending moment diagram to be achieved for continuous beam test units

................................................................................................................................... 94
Figure 4.8 Cellular beam four-point bending test ...................................................... 95
Figure 4.9 Overview of cellular beam four-point bending test .................................. 96

Figure 4.10 Combined flexural and shear failure observed in four-point bending test

................................................................................................................................... 97
Figure 4.11 Crushing of cantilever tip in four-point bending test ............................. 97
Figure 4.12 Measurement of initial stiffness in a non-linear response ...................... 98
Figure 4.13 F orce-displacement response at midspan in four-point bending .......... 101

Figure 4.14 Normalized force-displacement response at midspan in four-point
bending ..................................................................................................................... 101

Figure 4.15 Force-displacement response at cantilever tip in four-point bending .. 102

Figure 4.16 Normalized force-displacement response at cantilever tip in four-point
bending ..................................................................................................................... 102

xi

Figure 4.17 Load-strain response at extreme tension fiber on maximum negative
moment section ........................................................................................................ 103

Figure 4.18 Normalized load-strain response at extreme tension fiber on maximum

negative moment section ....... 104
Figure 4.19 Cellular beam three-point bending test ................................................ 105
Figure 4.20 Cellular beam three-point bending test ................................................ 106
Figure 4.21 Load-displacement response at midspan for three-point bending ........ 106
Figure 4.22 Normalized load-displacement response at midspan for three-point
bending ..................................................................................................................... 107
Figure 4.23 Flexural failure in three-point bending ................................................. 107

xii

(k)

aL

aP

f1

f2

f3

f4

f5

g(X)

n2

113

LIST OF SYMBOLS

Step size

Fraction of span length

Fraction of applied load

Diameter of hole in the coupon

Displacement under applied load in a beam

Search direction

Small change in current design

Objective function moment of inertia

Objective fimction moment of inertia per unit volume

Vector of objective functions f] and f2

Vector of objective functions moment of inertia about x axis and moment
of inertia about y axis

Vector of objective functions moment of inertia per unit volume about x-
axis and moment of inertia per unit volume about y-axis

Constraint function

Iteration number

Modular ratio of top hybrid layer in a coupon (with respect to base coupon
elastic modulus)

Modular ratio of bottom hybrid layer in a coupon (with respect to base
coupon elastic modulus)

Probability of cross—over

Density of cellular material

xiii

Pbase

I'i

poptimal =

Prel

ps

Yi

2i

(Pi

F(x)

F(x)

Relative density of base cellular design

Radius of circular holes in a coupon

Relative density of optimized cellular design

Density of cellular material divided by density of solid from which cells
are made of

Density of solid material from which cells are made of
Weighting coefficient

Design variable

Distance of the centroid of the hole from the base of the coupon
Distance of a layer of holes in a plate from a reference axis
Fitness value of ith design

Elastic modulus

Vector objective function

Scalar objective function

Parameter space

Initial stiffness of a cellular beam

Moment of inertia

Moment of inertia about x-axis

Moment of inertia about y-axis

Cellular beam span

Load applied on the cellular beam

Fraction of fitness value of a design with respect to total fitness of all

designs

xiv

Volume of cellular beam
Feasible region

Objective function space

XV

1. INTRODUCTION

1.1 General

Fiber Reinforced Polymer (FRP) composites have emerged as attractive candidate
materials for civil infrastructure applications because of their light weight, high strength,
non-corrosive and non-magnetic characteristics compared to conventional construction
materials such as concrete and steel [24]. In spite of these attractive features their use has
been limited due to various factors like higher initial material costs, their use in non-
efficient structural forms and their environmental impact [18]. This chapter presents an
overview of this thesis, specific objectives and is concluded with a brief outline of the

thesis.

1.2 Overview

With increasing emphasis on sustainability and eco-efficiency in the selection of
materials [24] for all applications, including civil infrastructure applications, there is
more demand to use bio-based products as alternatives to conventional as well as new
materials like F RP. Natural-fiber-reinforced polymer composites, or biocomposites, seem
to be a cost-effective and environmentally fiiendly option to low to medium grade
synthetic F RP composites [24]. In spite of this appeal, their use has been limited to non-
primary or non-load-bearing applications due to their lower strength and stiffness.
Biocomposite materials with specific properties equivalent to entry-level structural
materials are feasible [24] for example specific modulus (modulus per unit density) of a
hemp and unsaturated polyester resin (UPE) composite with 25% fiber volume fraction is

about 5.07x106 N-mm/ g while specific modulus of a glass/UPE composite with 29% fiber

volume fraction is about 5.74 x106 N-mm/g [28]. However, this performance level is still
unable to compete with existing construction materials.

The performance of a structural component, however, depends not only on its material
but also its sectional and structural properties. Nature is abundant with examples where
low individual properties of simple materials are enhanced manifold through hybrid and
hierarchical designs that place such materials in specific locations for highest structural
efficiency [8][9][26][29]. Among nature’s most common efficient structures are
hierarchical cellular sandwich structures [9][12][18][29]. These structures consist of a
complex arrangement of cells of varied sizes arranged across the section such that dense

regions are integrally connected to regions with lower density region or core (Figure 1.1).

 

Figure 1.1 Cellular structure of a plant stem [18]

Varied cell sizes in specific locations along the cross-section lead to hierarchy in their
arrangement. Smaller cell sizes appear towards regions of high stress in the cross-section,
implying the importance of more material in those regions, while larger cell sizes in the
cross-section indicate removal of material where it is not needed. Thus, the different cell
sizes are arranged in the order of their importance. The feasibility of using biocomposites
for load bearing components has been positively evaluated by prior work on laboratory

scale (13mm x 102mm x 305mm) cellular plates made from industrial hemp fibers and

unsaturated polyester resin (UPE) using circular cells of varying diameter. Analytical and
experimental results showed that the cellular bio—panels can be as efficient as commercial
fiberglass panels [3][5][28].

Nature’s complex level of material and structural design is an outcome of millions of
years of evolution [28][31]. With the aid of high speed computational tools for
exploration of complex shapes through mathematical optimization, engineers have been
able to successfully simulate nature’s ingenious evolutionary processes to a great extent.
Structural optimization methods, in particular topology optimization methods, can be
employed to remove or redistribute material in a rational iterative manner from within
given structural domains subject to load and boundary conditions [9]. The optimal forms
thus obtained represent a contour diagram of material density from which the optimal
material distribution must be discerned either automatically, using image processing
algorithms, or intuitively, using engineering judgment, or by a combination of both. A
typical topology optimization solution for a cantilever subjected to a tip load is shown in

Figure 1.2.

 

 

 

 

 

Figure 1.2 Typical topology optimization solution for a cantilever subjected to point

load
It can be seen that the topology optimization solution has material distribution with
varying material density indicated by different colored cells which brings hierarchy in the

3

material distribution. Thus, topology optimization can be used to obtain hierarchical
designs, as in nature, to improve the low strength and stiffness properties of biocomposite
components. However, in spite of its power and efficiency, typical results from topology
optimization feature complex geometries that can be very difficult to manufacture using
biocomposite materials as explained next.

Polymer biocomposites consist of natural fibers embedded in a natural or synthetic
matrix. Natural fibers are typically short and intertwined, or clumped, (Figure 1.3(a) and
(b)). Thus, manufacturing methods like hand-layup, compression molding or vacuum
assisted resin transfer method (VARTM) are commonly used to manufacture
biocomposites using natural fibers. The nature of biocomposite materials thus poses
restrictions to the geometries and topologies that are feasible to manufacture with them.
The results from conventional optimization methods such as topology optimization
methods are thus difficult to use to obtain hierarchical forms for biocomposite

components.

      

(a) Industrial hemp fibers
Figure 1.3 Natural fibers

Innovative topology optimization methods have been recently proposed that use

introduction of finite-size features/perforations, some of which may be comparable to the

size of the structure, by using multiresolution analysis for material distribution [7]. Such
approaches are robust and sophisticated yet they are not ideally suited for the problem at
hand. Finite-size topology optimization routines are based on strain-energy minimization
formulations. Thus, this applies to the longitudinal response of a loaded element, i.e.
region B in Figure 1.4. In addition, the classical formulation of topology optimization is
not directly suitable for multiobjective optimization problems or hybrid materials, which
require re-formulation of the algorithm. Thus, in order to obtain cross-sectional layout
made up of regions with stiffer material layers along the longitudinal direction (shown by
area “A” in Figure 1.4), subjected to uniform stress distribution, as in the objective of this

study a different optimization approach was needed.

 

 

Figure 1.4 Transverse and longitudinal cross-sections in a beam considered for
material layout optimization
Considering the manufacturing limitations to components made from biocomposites, the
level of complexity of present layout optimization methods using finite-size features, and
their inability to have hybrid material distribution, two approaches are presented in this
thesis aimed at optimizing the sectional properties of biocomposite panels through hybrid

and hierarchical cellular geometries. Both approaches rely on creating cellular designs by

introducing finite size perforations in a panel’s cross-section to optimize the distribution
of material. The first approach treats the problem as a sizing optimization problem and
achieves hierarchical cellular geometries along beam/panel cross-sections through varied
perforation sizes. This approach uses a gradient based method for optimization. The
second approach treats the problem as a stochastic search for material distribution, or
layout optimization problem, by using a pre-defined library of hybrid and non-hybrid
coupons with finite-size features. The optimum layout for the beam/panel cross-section is
then obtained by selecting coupons from this library using a directed random search with
genetic algorithms.

The presented approaches for optimal cellular cross-sectional designs are not limited to
biocomposite materials and can be used for other materials. Thus, the cellular designs
based on the proposed approaches aim to maximize material and structural performance
by improving sectional properties with designs that have finite size features in their cross-

sections and which are thus easier to manufacture.

1.3 Biomimetics

Nature is abundant with structures in which design and firnction have been optimized
over millions of years of continuous evolution. Engineers are faced with similar design
constraints and objectives as nature. Thus, engineers have always been inspired by
biological structures and have borrowed ideas from nature. This inspiration led to an
altogether dedicated discipline for the study of same called ‘Biomimetics.’ 1n words of
J.F.V. Vincent [29] biomimetics is “the abstraction of good design from Nature.”

Though most of the materials used in nature are not of high performance their success

lies in their strategic arrangement. The most efficient structural forms commonly seen in

nature are cellular sandwich structures [8][9], which consist of a dense skin integrally
connected to a low density cellular core, as shown in Figure 1.4(a) [12][18][29]. Cellular
materials are made of interconnected network of solid struts or plates, which can either
have a general form of two-dimensional honeycombs (Figure 1.4(b)) or three-
dimensional foams (Figure l.4(c)). By integrally connecting the low density core and
face sheets delamination between the core and face sheets, which can lead to premature

failure of sandwich structures, can be avoided.

 

(a) A section through a bird’s wing (Thompson 1961) (b) Wood 2—D cellular structure

    

"_g__&-—"“

(c) Cancellous bone 3-D structure (Gibson and Ashby 1988)

Figure 1.5 Sandwich structures and cellular structures in nature

Thus, through hybridization of materials having lower stiffness with ones having higher

stiffness and arranging material in hierarchical cellular structures, Nature tries to improve

the performance of its “structures.” There is also a multiplicity of structures in nature that
represent different levels of aggregation of the load bearing materials. These hierarchical
organizations are present in all biological composites [29][30]. Thus, hybridization and

hierarchical arrangement of simple materials can lead to efficient load-bearing structures

[4]-

1.4 Objective and scope

The objective of this research is to develop optimized designs having finite size features
for beam/panel cross-sections with the aim of attaining improved performance for
biocomposite beams/panels as load-bearing structural components. The optimization
approaches developed to achieve this objective make use of introducing finite size
features in beam/panel cross-sections and distributing material in hybrid and hierarchical
arrangements. The designs thus obtained are easier to manufacture using biocomposite
materials. The optimized designs were also evaluated experimentally by manufacturing
and testing laboratory-scale beams. The beam components were tested to evaluate their
flexural performance as continuous beams to be used in one-way panel-like systems for
load-bearing structural applications. The improvement gained from the strategic material
and structural layout of biocomposite beams/panels in the optimad designs was
determined by comparing the specific stiffness of an optimized design to one with a base
cellular architecture. The organization and contents of the thesis is as follows:

Chapter 2 presents a background on structural Optimization and mathematical background
of various optimization methods. It also provides details for the specific optimization

methods used in this study and their computational implementation.

Chapter 3 discusses the two optimization approaches used in this research work,
formulation of the optimization problems to achieve the specific research objectives, and
key results obtained from them. Limitations observed in the two optimization approaches
are also discussed.

Chapter 4 describes the manufacturing and testing of cellular designs obtained from the
optimization approaches developed in this study for evaluation of the manufacturing
feasibility and improvements from the optimization process.

Chapter 5 summarizes the key conclusions of this study and briefly discusses

recommendations for future work in the same direction.

2. STRUCTURAL OPTIMIZATION BACKGROUND

2.1 General

Engineering design is a complex and iterative process which often aims at finding “the
best possible” or “optimum” solution that firlfills all design criteria within a given set of
conditions. In general, ‘best’ implies cost-effective, efficient, reliable and durable
systems. This conventional process becomes more and more complex as the design
domain, number of variables involved in the design problem and number of constraints or
conditions to be satisfied by the design, increases. This calls for a more systematic and
efficient approach than the conventional iterative design process of trial and error.
Optimization is one such mathematical tool that aims to solve the problems involved in
the conventional design process. This chapter contains an introduction to structural
optimization and discusses the general mathematical background of the various
optimization methods used in this research, namely gradient based methods and genetic

algorithms.

2.2 Structural optimization

The structural design process consists of several stages. The first stage can be considered
as the conceptual stage where the structural system and form are chosen. The second step
is the preliminary design stage, where the shape of the structure and the defining
geometry are decided. The final stage is the detailed design stage. The conventional
design process goes through a number of iterations before a satisfactory detailed design is

obtained. Use of optimization techniques at different stages of the structural design can

10

increase the efficiency and speed of the overall process and tremendously improve the
results [13].

Structural optimization can be classified into several broad categories. First in its level of
complexity is sizing optimization. It involves, for example, finding the optimum cross-
sectional properties of a truss or a frame or thickness of a plate structure (Figure 2.1(a)).
In these problems the design domain is fixed and doesn’t change during the optimization
process. Sizing optimization can be considered as the implementation of optimization at
the detailed design stage.

A more complex type of optimization is called shape optimization, which aims at finding
the optimal boundaries of two and three dimensional structural components. In this
problem the domain is not fixed but the topology is. Coordinates of some key points in
the boundary of the structure are usually considered as design variables (Figure 2.1(b)).
For example, in the case of truss structures shape optimization considers two sets of
design variables. First are geometric design variables, which control the position of the
truss joints and second are the topological design variables which decide which nodes are
connected by elements or members. This can be viewed as implementation of
optimization techniques at the preliminary design stage.

Sizing and shape optimization may not lead to optimal results as they do not begin with
an optimal topology. Topology optimization tries to overcome this shortcoming of sizing
and shape optimization (Figure 2.1(c)). Typically, topology optimization of two and three
dimensional continua aims at determining material distribution features such as holes and
their location. Topology optimization essentially consists of the gradual removal of small

portions of low stressed material that are being used inefficiently [13].

ll

 

 

 

(a) Sizing optimization (b) Shape optimization (c) Topology optimization

Figure 2.1 Types of structural optimization

An ideal optimization approach is to simultaneously optimize the geometry (size and '
shape) and topology of a structure. This is sometimes called layout optimization and can
be considered as optimization at the conceptual stage [13]. Various mathematical
optimization techniques have been developed to solve the structural optimization
problems just described at different design stages.

Structural topology optimization can be seen as a procedure for optimizing the rational
arrangement of the available material in the design space and eliminating material that is
not efficient. Many methods have been proposed for topology optimization [13], among
them are: (i) the ground structure approach, (ii) the homogenization method, (iii) the
bubble method and (iv) the fully stressed design technique. All these methods have been
shown to be robust in performance and can yield solutions with fine material resolution
and varied material density. However, practical designs with fine material resolution in

complicated patterns are difficult to manufacture. An alternative approach has been

12

proposed to overcome this problem by introducing multiple finite-size heterogeneities in
the structure for obtaining optimum layouts [7]. This approach uses a wavelet-based
decomposition of material distribution and a multiresolution analysis. To represent the
finite-sized perforations including perforations with dimensions comparable to the size of
the structure a numerical homogenization method is used in this approach to characterize
the “effective behavior” of perforated coupons of finite size. The method is again very
robust and sophisticated and successfully solves typical structural topology optimization
problems.

The structural optimization methods mentioned in section 2.2 are based on different
techniques of mathematical optimization. The following section discusses in detail some

of the mathematical optimization methods and their use in structural optimization.

2.3 Optimization Design Process

Success of an optimum design problem depends greatly on its mathematical formulation.
Formulation of an optimum design problem involves translating a verbal description of
the problem into a well-defined mathematical statement. The formulation process begins
by identifying a set of variables to describe the system called design variables. Once
numerical values are assigned to the design variables a design for the system can be
obtained. All systems are designed to perform within a given set of constraints which
include limitations on resources, material limits, acceptable response of the system,
member sizes, etc. If a design satisfies all of its constraints, it is referred to as a feasible
design or system. A criterion is needed to judge whether or not a given design is better

than another. This criterion is called the objective function, or cost function, and is a

13

function of the design variables. A typical design process using optimization as a tool to

achieve the desired design goal is summarized in the flow chart shown in Figure 2.2 [1].

 

Identify:
( 1) Design variables
(2) Objective function

 

 

(3) Constraints that must be satisfied

Y

[ Identify data to describe the system I
V
[ Estimate initial design I

V

F [ Analyze the system 1

7

Check if constraints are satisfied I

j

Does the design satisfy convergence L.

criteria?

 

 

 

 

 

 

 

 

 

 

 

No

 

V

 

Change the design using an optimization
method

 

 

 

 

 

 

Figure 2.2 Optimum design process (adapted from [1])

2.4 Design variables, Objective functions and Constraints

2.4.1 Design variables

The notion of improving or optimizing a structure implicitly presupposes some
freedom to change the structure. The potential for change is typically expressed in
terms of ranges of permissible changes to a group of parameters. In optimization
terminology such parameters are usually called design variables. Design variables

may be continuous or discrete. Continuous design variables have a range of variation

14

and can take any value in that range. Discrete design variables can take only isolated
values, typically from a list within a given range of permissible values. Once
numerical values are assigned to the design variables, a design for the system can be

obtained.

2.4.2 Objective function

There can be many possible or feasible designs for a system. As some of them might
be better than the others, a criterion is required to constitute a basis of selection of one
of the several alternatives. Such a criterion is called an objective function or a cost
function. The objective function is a scalar fiinction of the design variables and its
least (or greatest) value is sought in an optimization process.

In a general optimization formulation, rarely does a single objective with several hard
constraints adequately represent the problem being faced. More often there is a vector

of objectives F (x) = {F1 (x), F2 (x), ..... F m (x)} that must be satisfied in some way. The

relative importance of these objectives is not generally known until the system's best
capabilities are determined and trade-offs between the objectives are fully
understood. As the number of objectives increases, trade-offs are likely to become
complex and less easily quantified. There is no general and reliable method for
solving multiobjective problems. However, several treatments for such situations are
possible. For example, a composite cost function for the problem can be defined as a
weighted sum of all the cost functions. A second way is to select the most important

criterion as the objective function and treat the rest as constraints [1].

15

2.4.3 Constraints

A constraint is a restriction to be satisfied in order for the design to be feasible. It

may take the form of a limitation imposed directly on a variable or group of variables

(explicit constraint), or it may represent a limitation on quantities whose dependence

on the design variables cannot be stated directly (implicit constraint) [1].

Many constraint functions include only first-order terms of the design variables.

These are called linear constraints. Most general problems, though, have non-linear

constraint functions. Design problems may have equality as well as inequality

constraints. Constraint functions are generally designated as follows:

g ,-(x) = 0 i=1, 2...m, for m such constraints

gI-(x) s 0 i=1, 2. . ...n, for n such constraints.

2.4.4 Mathematical statement of optimization problem
In algebraic terms, an optimization problem can be stated as [l]:
Minimize F (x)

Subjected to g,-(x) = 0 i=1, 2...m, for m such constraints

g,(x) S 0 i=1, 2. . ...n, for n such constraints

and explicit bounds on design variables

xi] S x,- S xiu 1=l ton

(2.1.1)

(2.1.2)

(2.2.1)

(2.2.2)

(2.2.3)

where x,, and xiu are respectively the smallest and largest allowed values for the ith

design variable x,- .

l6

Optimization methods in general can be classified into two broad categories:
(a) Optimality criteria methods or indirect methods
(b) Search methods or direct methods.
Each of these methods can be employed to solve both unconstrained as well as

constrained optimization problems and are described further in the following section.

2.4.4.1 Optimality criteria methods or indirect methods

Optimality criteria are the conditions a function must satisfy at its minimum point.
Optimization techniques seeking solutions to optimality conditions are referred to as
indirect methods [1]. As the number of design variables and constraints becomes large,
functions for the design problem (cost and constraint) become non-linear or the cost
and/or constraint functions are implicit in the design variables. This is commonly seen in
practical design optimization problems, so use of optimality criteria or indirect methods
becomes difficult due to the large number of equations involved in the solution. This
situation requires systematic numerical approaches for the optimum design of
engineering systems. An alternative to indirect methods are direct methods, which are
described in the next section.

2.4.4.2 Direct methods for unconstrained optimization

Direct methods seek an optimum solution by starting with an estimate of the initial design
and improve it until optimality conditions are satisfied [1]. Many numerical methods
based on search or direct methods are available to address the problems posed by
optimality criteria methods, or indirect methods, for solving non-linear optimization

problems. The key advantage of these numerical methods is that they are easy to

17

implement computationally. All unconstrained numerical optimization methods are based
on the iterative formulae given in Equations 2.3.1 and 2.3.2, and schematically shown in

Figure 2.3.
Vector form x(k+l) = xlk) + Axlk) k = 0, 1, 2, (2.3.1)
Component form xi(k+1) = xi“) + Axi(k) k = 0,1,2, (2.3.2)
In these equations, k represents the iteration number, i denotes the design variable

number, xlk) is any starting design and Ax“) represents a small change in the current
design. The iterative scheme noted above is continued until optimality conditions are
satisfied or an acceptable design is obtained (Figure 2.3).

The problem of obtaining the change in design Axis usually decomposed into two parts:

the finding direction and the step size determination, expressed as
Ax“) =akdl") (2.3.3)

where d (k ) is a desirable search direction of movement in the design space and - a“) is a
positive scalar called the step size in that direction. For an optimization problem with
several variables, the finding direction problem is solved first. A step size is then
determined by searching for the function minimum along a given direction in the design

space.

18

 

 

‘
*7

Figure 2.3 Conceptual diagram for iterative steps of an optimization method

Various numerical methods based on above concept have been developed to solve
unconstrained optimization problems, e. g., the steepest descent method, Newton
methods, and Quasi-Newton methods [1]. All these methods determine the search
direction in different ways. For example, the steepest descent method uses a negative
gradient vector to represent the direction of steepest descent for the cost function. For a
function fix) of n variables x,, x;, ...... x" the vector of partial derivatives of f(x) with
respect to variables x1, x2, ...... x,, at a given point x. is termed as the gradient vector.
Thus, the steepest descent method begins with an initial estimate for the minimum design
and computes the direction of steepest descent at that point. If the direction is non-zero
after moving along it, as long as possible to reduce the cost function, the new design
point is thus obtained. Once at the new design point the direction of steepest descent is
computed again and the entire process is repeated. This method uses only first order
derivative information in the representation of the cost function at a point to determine

the direction of travel. If second-order derivate information is used a better rate of

19

convergence can be expected. Newton’s method uses a second order Taylor’s series
expansion of the fiinction about the current design point. From it, the Hessian matrix is
obtained by differentiating each component of the gradient vector with respect to each x,-.
The Hessian matrix thus defines a quadratic rate of convergence for Newton’s method.
However, this approach suffers from certain drawbacks. For most engineering problems
calculation of the Hessian matrix is expensive due to the large number of calculations
involved. Like the steepest descent method, Newton’s method does not possess a learning
process, that is, each iteration starts with a new set of design variables without using
information from previous iterations. Also, if the Hessian of the function becomes
singular at any point then the method runs into difficulties. Quasi-Newton methods try to
overcome these difficulties by using computation of not only the first derivatives but they
use information from previous iterations [1]. This leads to higher rates of convergence
towards the minimum. Several other methods are also available based on same iterative
philosophy.

2.4.4.3 Direct methods for constrained optimization

The unconstrained optimization methods discussed above can also be used to solve
constrained optimization problems by transforming the constrained problem to an
unconstrained one by means of a composite function. The composite function is defined
using the cost and constraint functions. It also contains certain parameters called penalty
parameters that penalize the composite function for the violation of constraints. The
larger the violation, the larger is the penalty. Once the composite function is defined for a
set of penalty parameters, it is optimized using any of the unconstrained optimization

techniques. The penalty parameters are then adjusted based on given conditions and the

20

composite firnction is redefined and optimized. The process is continued until there is no
significant improvement in the estimate of the optimum point.
Constrained optimization problems may also be solved directly by so called “primal
methods.” Conceptually, algorithms for constrained and unconstrained optimization
problems are based on the same iterative philosophy. However, for the former,
constraints must be considered while determining the search direction as well as step size
for the constrained problem.
All algorithms need a design estimate to initiate the iterative process. The starting design
can be feasible or infeasible. Most of the direct search algorithms comprise of following
basic steps [1]:

1. Linearize the cost and constraint functions about the current design estimate.

2. Define a search direction determination subproblem using the linearized cost and

constraint functions.
3. Solve the subproblem that gives a search direction in the design space.
4. Calculate the step size required to minimize an appropriate descent function in the
search direction.

Linearization of the cost and constraint functions means that at each iteration the design
change is computed by solving an approximate “subproblem” obtained by writing linear
Taylor’s series expansions of the cost and constraint functions.
Based on the above philosophy many numerical methods or algorithms have been
developed to solve constrained optimization problems. Some examples are the sequential
linear programming method (SLP), the sequential quadratic programming method (SQP),

the constrained steepest descent method (CSD), and constrained quasi-Newton methods

21

(CON), [1]. SLP uses linear programming methods to obtain the design change. SQP
iteratively solves a quadratic programming subproblem and the step size is determined by
minimizing a descent function along the search direction. The quadratic programming
(QP) subproblem has a quadratic cost function and linear constraints. The solution of the
OP is used to form the search direction. The constrained steepest descent methods modify
the search direction so that it satisfies all the linearized constraints. Several other
methods, like the method of feasible directions, the gradient projection method, and the
generalized reduced gradient method, are also used to solve constrained optimization

problems.

2.4.5 Multiobjective optimization
In any general optimization formulation, rarely does a single objective with several hard
constraints adequately represent the problem being faced. More often there is a vector of

objectives F (x) = {Fl (x), F2 (x), ..... F m (x)} that must be collectively satisfied in some way.

The relative importance of these objectives is not generally known until the system's best
capabilities are determined and trade-offs between the objectives are fully understood. As
the number of objectives increases, trade-offs are likely to become complex and less
easily quantified. There is much reliance on intuition of the designer and his or her ability
to express preferences throughout the optimization cycle. Thus, requirements for a
multiobjective design strategy are to enable a natural problem formulation to be
expressed, yet be able to solve the problem and enter preferences into a numerically
tractable and realistic design problem.

Multiobjective optimization is concerned with the minimization of a vector of objectives
F (x) that may be the subject of a number of constraints or bounds.

22

Minimize F(x): {F1 (x), F2 (x), ..... F,,,(x)} (2.3.4)

n

x e K
gI-(x) = 0 i=1,...,me (2,3,5)
gi(x) S O i = me + l,....,m (2.3.6)

x, s x s x, (2.3.7)

The key point to be noted here is that, because F (x) is a vector, if any of the components
of F (x) are competing there is no unique solution to this problem. Instead, the concept

of noninferiority (also called Pareto optimality) must be used to characterize the
objectives. A noninferior solution is one in which an improvement in one objective

requires a degradation of another. To define this concept more precisely, consider a

feasible region Q in the parameter space x e K” that satisfies all the constraints, i.e.

Q = ire 26’} (2.3.8)

Subject to
gi(x) = 0 i =1,...,me (2.3.9)
gi(x) S 0 i = me + l,....,m (2.3.10)
x1 Sxqu (2.3.11)

This allows defining a corresponding feasible region for the objective function space A.
A ={ye Kn} where y = F(x) subject to er
The performance vector, F (x) maps the parameter space into the objective fiinction

space as is represented for a two-dimensional case in Figure 2.4.

23

 

 

A A
r Y

 

x2 F2

Figure 2.4 Mapping fiom parameter space into objective firnction space

A noninferior solution point can now be defined as a point, for some neighborhood of x' ,

such that there does not exist a Ax such that (x' + Ax) e Q and

17,-(x* + Ax)S F, (x) i=1,...,m (2.3.8)
Fj (x1: + Ax)S Fj (x) for somej (2.3.9)

If A and B are two such points under consideration then they are noninferior solution

points because an improvement in one objective, Fl , requires a degradation in the other
objective, F2 , i.e.:

F13 < 171.4,er > FZA-
Since any point in that is not a noninferior point represents a point in which improvement

can be attained for all of the objectives, it is clear that such a point is of no value.

Multiobjective optimization is, therefore, concerned with the generation and selection of

24

noninferior solution points. The techniques for multiobjective optimization are wide and
varied [l][ 14]. An overview of one of them is given next.

2.4.5.1 Weighted Sum Strategy

The weighted sum strategy converts the multiobjective problem of minimizing the

objective function vector into a scalar problem by constructing a weighted sum of all the

objectives.
Minimize f(x) = 2:1 w,.F,. (x) (2.3.10)
x e Q
F]
A
WT F (x) =c
A

 

A
7

F2

Figure 2.5 Graphical representation of the weighted sum method

The problem can then be optimized using a standard unconstrained optimization
algorithm. The problem here is in defining weighting coefficients to each of the
objectives. The weighting coefficients do not necessarily correspond directly to the
relative importance of the objectives or allow trade-offs between the objectives to be
expressed. Further, the noninferior solution boundary may be nonconcurrent so that
certain solutions are not accessible. This can be illustrated graphically. Consider the two-

objective case in Figure 2.5. In the objective function space F ,F 2 a line L is drawn. The

25

minimization of Eq. 2.3.10 can be interpreted as finding the value of c for which L just
touches the boundary of A as it proceeds outwards from the origin. Selection of weights,
therefore, defines the slope of L, which in turn leads to the solution point where L touches
the boundary of A . It is important to note that a convexity problem arises when the lower
boundary of is nonconvex. In this case the set of noninferior solutions between A and B
is not available.

Both indirect and direct methods for scalar as well as multiobjective optimization suffer
from certain limitations. First, both these methods are local in scope, that is, the optima
they seek are best in a neighborhood of the current point. So, if the function is not smooth
or if it is highly non-linear, these methods might give only a local minimum and not a
global minimum. Second, calculus-based methods depend upon the existence of
derivatives. Even if numerical approximations of derivatives are obtained, they are not
suitable for discontinuous and multi-modal functions. Thus, these methods are
insufficiently robust in rugged domains. Exhaustive search or enumerative schemes have
also been considered for such problems, in which the search algorithm looks at objective
function values at every point in space, one at a time. But for very large spaces the
exhaustive search methods turn out to be inefficient and computationally expensive.
Random search algorithms address to some extent the shortcomings posed by exhaustive
search methods; however, undirected random search can become computationally
expensive and inefficient, especially if the design domain is extremely large. The answer
to most of the above mentioned problems probably lies in a directed random search like

that provided by genetic algorithms.

26

2.4.6 Genetic algorithms

A genetic algorithm is a directed random search that tries to find the global optimum in
complex, multidimensional and multi-modal search spaces. It emulates natural evolution
wherein the operators used are inspired by the evolution process. Genetic algorithms use
random choices as a tool to guide a highly exploitative search through coding of a
parameter space [6]. There are two basic mechanisms that link a genetic algorithm to the
problem that it is solving. The first is a way of encoding solutions to the problem on
chromosomes and the second one is an evaluation function that returns a measurement of
the merit of any chromosome in the context of the problem.
An overview of a conceptual genetic algorithm is given as follows:
1. Initialize a population of chromosomes.
2. Evaluate each chromosome in the population.
3. Create new chromosomes by mating current chromosomes; apply mutation
and recombination as parent chromosomes mate.
4. Delete the members of the population to make room for the new
chromosomes.
5. Evaluate the new chromosomes and insert them into the population.
6. If a termination condition is met, stop and return the best chromosome; if not
go to 3.
The individuals of the population, i.e. the parameters to be optimized, are represented in a
string called “chromosome,” as genetic operators are suitable for this type of
representation. The string can be represented in terms of binary numbers or real numbers

(integers); the former representation is more commonly used. These strings are equivalent

27

to chromosomes in natural evolution, thus named chromosomes from here on. Each
chromosome represents a possible solution, i.e., values from a variable set. To create new
individuals two parents are required, which are then acted upon by genetic operators
(namely crossover operators) to yield “off springs” or “better solutions.” The crossover
operator creates an information exchange between the parent chromosomes. The
“offsprings” obtained by crossover possess features from both parents but sometimes
crossover can also result in offsprings that are completely different from their parents.
Thus, it is largely responsible for increasing diversity in a given population. There exists
a fitness value for every individual, which indicates the merit of an individual in the
evolution process. Parents are randomly chosen fi'om the population (by probabilistic
selection methods or operators) and individuals with high fitness value have higher
probability of getting selected as parents than individuals with low fitness value. The
genetic operation of mutation is randomly applied to the individuals. This causes a
random modification of a local part of the chromosomes, which may cause the new
chromoSomes to be different from their parents. Since individuals may pass away, as
their lifetime depends upon their fitness value, the mutation operator mainly tries to
provide supplementary diversity in a population to avoid loss of diversity. This diversity
allows the exploration of larger regions of the search space.

If all goes well throughout the process of simulated evolution an initial population of
chromosomes will improve as parents are replaced by better and better children. The best
individual in the population can thus be a highly-evolved solution to the problem. Finally,
when the termination condition is met the best individual is returned as the optimum

solution. The termination criterion can be a specified number of generations or other

28

stopping conditions such as the number of successive generations for which no

modification occurs in the problem.

2.4.6.1 Computer implementation of a genetic algorithm

Genetic algorithms process populations of strings. So, the primary data structure for a
simple genetic algorithm is a string population. One of the ways to implement
populations is to construct it as an array of individuals where each individual contains the
phenotype (the decoded parameter or parameters), the genotype (the artificial
chromosome or bit string), and the fitness (objective function) value along with other

auxiliary information. This structure is shown in Figure 2.6[10].

 

 

 

 

 

 

Tm- ___
INDIVIDUAL INDIVIDUALS l
‘NUMBER - ,1 + .
. ] STRING l x F(x) ,orHER,
e—# i l ‘
1 01111 15 225i
2 01001 %9 81
I i l ; ,
l l i I ‘ l
l l l 1
l 1 1 l l
II 00111 1 7 49

 

Figure 2.6 Schematic of a string population in a genetic algorithm (adapted from
[10])

Genetic operators may be applied to an entire population at each generation and as a

result they act on two non-overlapping populations. The offsprings are created from

29

members of the old population (data structure) and are placed in the new population. The
old population is then reset or regarded as the new population. A single overlapping
population can also be maintained by carefully keeping record of who replaces whom in

successive populations [6][10][27].

2.4.6.2 Initial population

Most genetic algorithms work with a fixed-size population so that the size of population
of designs determines the size of population in all future generations as well. Choosing
the population size is often a matter of trial and error. In general, the optimal size of a
population increases with problem size. The initial population is typically generated at
random. Once a population is generated, a series of genetic operators are applied that
produce a new generation and replace the initial population with a new one. The first of

these operators is selection [6][10][27].

2.4.6.3 Selection and fitness

The selection process in genetic algorithms mimics biology in giving more fit designs a
higher chance to breed and pass on their genes to future generations. Genetic algorithms,
therefore, need to define fitness and use it in a procedure that selects pairs of parent
designs that will be used to create child designs for future generations. For unconstrained
problems, the fitness of a design can be the problem objective function (for maximization
problems) or a constant minus the objective (for minimization problems). For constrained
problems the fitness must also consider constraint violations or constraint margins. Once

the fitness of all the designs in a population are defined, a common procedure for parent-

30

selection is to simulate a biased roulette wheel, with each design being assigned to a

sector of the roulette wheel with an area proportional to its fitness. That is, with ns
designs having fitness values of ¢,~, i = 1, ..... ,ns , the ith design gets a fraction R, of the

wheel, where

 

R,- = (2.3.17)

Defining fitness on the basis of the numerical value of the objective function carries the
disadvantage that towards the end of the optimization, where the differences between
competing designs becomes small, the selection pressure in favor of better designs
becomes small, and thus progress slows down. To overcome this difficulty many other
selection procedures such as stochastic remainder sampling, tournament selection,

stochastic universal sampling and linear ranking have been proposed [6][ 1 0] [27].

2.4.6.4 Crossover

Once pairs of parents are selected, the mating of a pair also involves a random process
called crossover. In a genetic algorithm, crossover recombines the genetic material of two
parent chromosomes to make two children. The simplest crossover operator is the one-
point crossover. One point crossover occurs when parts of two parent chromosomes are
swapped at a randomly selected point to create two children. One of the important
features of one-point crossover is that it can produce children that are radically different
from their parents. Two examples of one-point crossover are shown in Figure 2.7.

Another important feature of one-point crossover is that it will not introduce differences

31

for a bit in a position where both parents have the same value, as shown in second

example in Figure 2.7.

 

Parent 1: l l 1 1| 1 1 Child 1: l 1 l l 0 O
:>

Parent2: 0 0 0 0| 0 0 Child 2: 0 0 0 0 l 1

Parent 1: l 0 l | l 0 1 Child 1: l 0 l l 0 0
1::>

Parent2: 0 0 1 ll 0 0 Chile: 0 0 l 1 0 1

Figure 2.7 Two examples of one-point crossover (adapted from [6])

Crossover is usually implemented with some probability pc. If crossover is not
implemented, then one of the parents is cloned into the next generation. Implementation
of crossover begins with the generation of a single random number r uniformly
distributed between 0 and 1. If the number is less than pc crossover will be performed by
generating another random number for the cutoff point. The decisions on which parent to .
clone or which child design to select are not important, as the parents were selected at

random. The processes of selection and crossover are repeated until there are n, child

designs. Multiple point crossovers in which information between the two parents is

 

swapped among the string segments are also possible, but, because of the string mixing,
the crossover becomes a more random process and might degrade the performance 0f the

. . . . . e
algorrthm. However, a two-pornt crossover rs an exceptron and choosrng the end of th

32

string randomly as the second crossover point improves the performance of the
algorithm. When integer or real variables are represented by binary strings, crossover
may generate child designs that do not bear any resemblance to the parent designs.
Although such possibilities reduce the effectiveness of the crossover operator for
combining existing traits from parent designs into child designs, it contributes to the
exploration of new design alternatives [6][10][27]. The process of exploration of
alternative designs is usually handled by the mutation operator and it is preferable to let
the crossover operator take care of recombination and let the mutation operator handle
the exploration of new traits as described in next section.

2.4.6.5 Mutation

Mutation performs the important task of preventing premature loss of important genetic
information by introducing an occasional random alteration to a string. Inferior designs
may have some good traits that can get lost in the gene pool when these designs are not
selected as parents. Additionally, mutation is needed when an integer or real coding is
used because in most cases there is a low probability that all the possible genes are
represented in the initial population. Mutation is implemented by randomly changing the
value of a digit in the string with small probability (Figure 2.8). When real design _
variables are coded as binary numbers, there are situations where small changes in the
design can not be achieved by mutations. This occasionally results in slower progress for
the algorithm. To counteract this problem another type of binary coding called gray
coding is sometimes used, which doesn’t have abrupt changes in digits caused by small
changes in the value of the number. Alternatively, another mutation operator called local

mutation can be applied to the original variables rather than the coded ones. This local

33

mutation operator is also useful when actual values rather than binary numbers are used

for the variables in the string [6][10][27].

 

 

 

} Old Chromosome Random Numbers IBIiiw New Chromosome 1
' 1 0 1 0 : 0.80 0.102 0.266 0.373 ' - l l 0 l 0

l 1 ~

f 1 1 0 0 .0-12 0.096 0.005 0.840 r 0 . 1 1 1 0

1 0 0 1 0 ; 0.760 0.473 0.894 0.001 l l l 0 l l

l

 

Figure 2.8 Examples of bit mutation (adapted from [6])

2.4.6.6 Knowledge based evolution

In the course of evolutionary optimization solutions are often generated with low
phenotypic (physical) fitness even though the corresponding genotype may be close to an
optimum. Without additional information about the local fitness landscape such genetic
near misses would be overlooked under strong selection. Such issues are usually
addressed by hybrid genetic algorithms, which combine a global search using genetic
algorithms and a local search using individual learning algorithms. Hybrid algorithms can
either exploit active learning, using Lamarckian inheritance [6][10], or passive learning
through the Baldwinian effect [6][10].

Under Lamarckian algorithms [6][10], performance gains from individual learning are
mapped back into the genotype used for the production of the next generation. This is
analogous to Lamarckian inheritance in evolutionary theory — whereby characteristics
acquired during the parent’s lifetime are passed on to their offspring. Although

Lamarckian inheritance is rejected as a biological mechanism under the modem

34

 

synthesis, the algorithms based on the approach are able to perform mapping from the
learned phenotype to the genotype [6]. It is sometimes referred to as an active mechanism
and it results in change in the genetic makeup or genotype of the individual in the course
of evolution.

The Baldwinian evolution model [6] suggests that individual learning can alter the course
of evolution. According to this theory, evolution pressure favors individuals having
learning capabilities. Individuals able to learn are less dependent on their genetically
encoded traits. The frequency of the genes responsible for learning thus increases in
subsequent generations. Individuals able to learn a certain adaptation (allowmg learning
individuals to survive preferentially) are more likely to produce offsprings having this
learning capability. This implicit learning model is referred to as Baldwinian ejfect.

The Baldwinian effect highlights an indirect or passive mechanism allowing individual
learning to influence the rate of evolutionary progress. By increasing survival chances
and genetic diversity, individual learning capabilities increase the probability that the
population evolves genetically encoded traits that better fit a challenging environment. It
is referred to as a passive effect as learning capabilities enhance the chances of survival

of an individual but do not change the genetic makeup of the individual.

35

3. DISCRETE LAYOUT OPTIMIZATION OF CELLULAR STRUCTURES

3.1 General

The manufacturing limitations of structural forms fiom biocomposite materials with
hybrid material layouts make the implementation of conventional topology optimization
solutions difficult. Thus, two approaches were developed for Optimizing the sectional
properties of biocomposite panels with hybrid material designs and hierarchical
geometries. Both approaches consider finite size perforations in the cross-section of
biocomposite panels to optimize the distribution of material. This chapter firstly presents
the formulation of these optimization problems by identifying design variables and the
objective functions. It is followed by the description and implementation of the respective
optimization algorithms. The chapter is concluded by a discussion of the limitations

observed in the two approaches.

3.2 Sizing optimization of cellular cross-sections using gradient search

The sizing optimization problem aimed at maximizing the cross-sectional stiffness and
specific cross-sectional stiffness of panels subjected to one-way bending in a continuous
panel system is shown in Figure 3.1. Considering a representative strip fiom the panel as
shown in Figure 3.1, the indicated cross-section was optimized for maximum stiffness
and specific stiffness (stiffness per unit volume of material). Solving an optimization
problem involves transcribing a verbal description of the problem into a mathematical

statement. Hence, sizing optimization was started by identifying design variables, a scalar

36

objective function, a vector of objective functions (called multiple objective functions)

and corresponding constraints, which are presented in the following sections.

 

 

 

 

Distributed load
9'9- "‘ ,,,I:_-V_IT;~.,‘_ ~—-—.~_—. - '“ [T V " 1 T c ”T T _‘
1.31:; 17:11“. if: i?“ 1:” ___ L vlr 1:! 4 Lin’z'tr
‘ . ‘3‘: —: ‘- —-r»:-":“—‘ mw- —_. “rt—Z I 1'] '

 

 

0' Cross-section to be optimized

Figure 3.1 Continuous panel system subjected to distributed load in one-way
bending

3.2.1 Design variables
In order to implement the sizing optimization problem a numerical problem was

considered wherein a plate with cross-sectional dimensions of 100 mm x 80 mm was

considered (Figure 3.2).

 

 

 

+— 100 mm —- —-i

Figure 3.2 Cross-sectional view of a hierarchical cellular biocomposite panel

37

The cellular plate was assumed to be discretized into distinct layers where each layer was
perforated with circular holes such that same size holes were considered to be present
across a given layer. A linear bending-induced strain distribution was assumed across the
cross section. Thus, for simplicity each layer in the plate was considered to be made up of
a given number of repeating units or coupons. The repeating unit consisted of a circular
hole in the center of a square coupon where the hole diameter was a function of the

coupon size as shown in Figure 3.3.

 

Figure 3.3 A characteristic repeating coupon

Circular holes were considered as a specific case but the approach is not limited to
circular perforations and can be extended to other geometric shapes. A single column of
these characteristic coupons was considered to be repeated throughout the width of the
cross-section since the component was assumed to be under one-way bending. The
coupon column can be repeated along the cross-section in two possible ways as shown in
Figure 3.4(b) and Figure 3.4(c). The interpretation shown in Figure 3.4(b) has been

followed throughout the sizing optimization results.

38

 

 

 

 

 

 

Figure 3.4 Interpretation of a three layer repeating column layout in a three layer
plate

Variation of layer thickness changes the hole diameters as shown in Figure 3.4 for a
three-layer plate. Therefore, distances of different layers from a reference axis were
chosen as design variables (see Figure 3.5, Figure 3.6 and Figure 3.7). Three different
panels, each consisting of three, five and seven layers were considered for the numerical
example. The approach is not limited to the chosen numbers of layers and can clearly be
extended to different numbers of layers. The number of layers chosen for each of the
three panels was odd since symmetry about the midplane was intuitively expected
following that the material was assumed to be linear elastic. Consideration of different
number of layers was to add more resolution in the sizing optimization problem, thus
allowing the presence of greater hierarchy of perforations in the panels. The design
variables for three, five and seven layer panels are shown in Figure 3.5, Figure 3.6 and

Figure 3.7, respectively.
39

 

fl

 

 

 

 

Figure 3.5 Design variables for the three-layer panel

 

 

 

 

 

if“ - \ I
lllllllllllllllllllll 5—"“”;T——:r‘———T

.3 zl

L llllllllllll ,ITTT_””*“”j:jV ----- f 22 I
Z3 l

l

1 l” “““““““““““““““ l l l

L ......... Z4 1

I _T— T"_TT TTTTTTTTTTTT f‘”—L is 1

Figure 3.6 Design variables for the five-layer panel

40

 

 

 

 

. . , . . 22 l

L I". L“ i; ,, -"iT l: "_~; L :.~_“:T_ L ; :L: __r 23

" ’ " ‘ “ " 24 l ,
M_L_____L __L-L __L-L, ,L__L 1 1 l l
l l l l

, VI, ~ 25 l

I T T: _ T T T T2,," T T: T T: _ _ : _ T T: T T.: TyT—L Z6
+2 L .L'LL _ ___ _ TELL“ _ _:_ _ ‘;—-;..3-'.:; i ___.._.__l 217 7'8

_LL______: ,,,,,,,,,,,,,,,,,,,, -_ i g
I

 

 

 

 

Figure 3.7 Design variables for the seven-layer panel

3.2.2 Objective function

The objective functions used for the scalar optimization problem were the section
moment of inertia (Equation 3.1) and the section moment of inertia per unit volume
(Equation 3.2). These objective functions were used to quantify the sectional stiffness and
sectional specific stiffness (stiffness per unit weight).

f1 =1 and (3.1)

1
V (3.2)

f 2 =
For the multiobjective optimization problem a linear combination of moment of inertia
and cross-sectional volume with weighting coefficients was used (Equation 3.3). The
multiobjective formulation was considered to understand the trade-off in maximizing

moment of inertia while simultaneously minimizing weight. The multiobjective function

is given by:

41

f3 =W11+% (3.3)

where w1,w2 =weighting coefficients and wl + wz =1.

3.2.3 Constraints

The limits to which the movement of layers was confined were selected as the problem
constraints. For the numerical problem discussed in Section 3.2.1, for the three layer plate
two sets of constraints were studied as given below:
(I) -4 cm <22<0 and 0<23<4 cm
(11) -3 cm <22<1 and 1<23<3 cm.
Similarly, for the five layer plate the constraint sets were:
(III) -4 cm < zz<0, 0< 23<4 cm, 0< z4<4 cm
and 0< z5<4 cm
(IV) -3 cm <22<1, l<Z3<3 cm, l<z4<3 cm
and 1< z5<3 cm.
And for the seven layer plate the constraint sets were:
(V) -4 cm < 22<0, 0< 23<4 cm, 0< z4<4 cm,
0< Z5<4 cm, 0< z6<4 cm and 0< 27<4 cm
(VI) —3 cm <zz<l, l< 23<3 cm, l< z4<3 cm,
l< Z5<3 cm, l< 26<3 cm and 1<Z7<3 cm.
Since the overall plate thickness was kept constant for all cases, for the 3 layer plate zl=

z4=4 cm, for the 5 layer plate 2,: 26 = 4 cm and for the 7 layer plate 21: 28:4 cm.

42

3.2.4 Implementation of sizing optimization problem in MATLAB

The sizing optimization problem was solved using the optimization toolbox in MATLAB
[16] employing both scalar and multiobjective approaches for constrained optimization.
The function “finincon” was used to solve both the scalar and multiobj ective optimization
problems. MATLAB’s “frnincon” function uses a sequential quadratic programming
algorithm [1] to carry out constrained optimization (see Section 2.3.3). This function
requires the constraints and the objective function to be passed as its arguments, which
were written separately in script files and passed to it as functions. Figure 3.8 shows
schematically the input arguments and output options, for the “finincon” function. The
output options can be decided beforehand and can include information like the optimum
value of the design variables, or their value at the optimum solution, value of the
objective function at the optimum solution, value of the gradient and Hessian [1] (see
Section 2.3.2) at the optimum solution and as well as at each iteration. If the objective
function is highly non-linear and/or analytical calculation of the gradient is not possible,
“frnincon” can be made to use numerical gradients (calculated by the finite difference

method) by turning the Hessian (see Section 2.3.2) option “off.”

43

 

INPUT
°Objective function
-Constraint function (equality and inequality constraints)

°A starting point

 

 

-Objective function gradient

1

‘fmincon’

l

OUTPUT

 

 

 

 

 

 

-Optimum solution

°Objective function value at optimum solution
-Value of gradient at solution

°Value of Hessian at solution

Summary of all iterations

 

 

 

Figure 3.8 Constrained optimization using MATLAB’s “fmincon” function

3.2.5 Scalar sizing optimization for maximization of stiffness

The results obtained for the scalar sizing optimization for maximization of sectional
stiffness of three, five and seven layer plates subjected to constraint sets described in
Section 3.2.3 are presented in Tables 3.1 through Table 3.3. In case of three, five and
seven layer panels subjected to constraint sets I, III and V respectively, all the layers
merged together. Thus, only a single layer of perforations was seen in the middle of the
panel, leading to removal of maximum material from near the neutral axis and placement
of maximum material towards top and bottom of the panels. Results for constraint sets I,

III and V for the three, five and seven layer plates, respectively, show that in absence of

44

any restriction on movement of different layers the optimization process tries to remove
most material from near the neutral axis and places it more towards the top and bottom,
as seen in Figure 3.9. The layout for three layer panel subjected to constraint set 11
showed three distinct layers due to the lower bound on the design variables (Figure 3.10).
The layouts obtained for five and seven layer plates with constraint sets IV and VI
respectively were identical (Figure 3.11) and again showed material removal fi'om the
neutral axis and material placement towards top and bottom of the panel. For the seven
layer panel two layers were merged and the layout had only five layers due to the
imposed constraints and led to material placement towards the top and bottom of the
panel. Thus, it was clearly seen that whenever some restriction was imposed on the
movement of layers as seen in case of constraint sets 11, IV and V1 for three, five and
seven layer panels, respectively, smaller sized perforations were placed on the top and
bottom layers thus leading to removal of more material fiom the neutral axis. In the
absence of such restrictions layers with smaller perforations at the top and bottom sides
of the panel were avoided and the solutions tended towards a single large void layer at the
section neutral axis.

Table 3.1 Results for scalar maximization of stiffness for a three-layer plate

 

 

 

Constraint set 22 Z3 Moment of Inertia (f1)
mm mm mm4
1 -40 40 5.3 x 104
II 30 30 1.93 x 10“

 

 

 

 

 

 

45

 

Table 3.2 Results for scalar maximization of stiffness for a five-layer plate

 

 

 

Constraint set Z2 Z3 24 25 Moment of Inertia (f1)
mm mm mm mm I'l‘lIl’l4
III 0 -40 40 0 5.3 x 10‘
IV -30 -10 10 30 1.93x104

 

 

 

 

 

 

 

 

Table 3.3 Results for scalar maximization of stiffiress for a seven-layer plate

 

 

 

 

 

 

 

 

 

 

 

 

 

Constraint set 22 23 24 25 26 Z7 Moment of Inertia (7",)
mm mm mm mm mm mm mm4
v -40 0 -40 40 0 40 5.3 x 104
VI -10 -30 -30 30 30 10 1.93 x 10::
1 if
, - - L. 1
1 / ‘ -40 mm

+: i . _ x ,- ,

 

Figure 3.9 Layout obtained for maximization of stiffness for three, five and seven
layer plates for constraint sets I, III and V respectively

46

+2

 

Figure 3.10 Layout obtained for maximization of stiffness for three layer plate for
constraint set II

 

Figure 3.11 Layout obtained for maximization of stiffness for five and seven layer
plates for constraint set IV and VI respectively

3.2.6 Scalar sizing optimization for maximization of specific stiffness

The results obtained for the scalar sizing optimization for maximization of specific
sectional stiffness of three, five and seven layer plates subjected to constraint sets

described in Section 3.1.3 are presented in Tables 3.4 through Table 3.6. The results for

47

constraint sets I, III and V for three, five and seven layer plates respectively show that, in
the absence of any restriction on the movement of different layers, the optimization
process tries to remove most material from near the neutral axis and places it more
towards the top and bottom as seen in Figure 3.12 and Figure 3.13. In these figures, the
presence of top and bottom layer of perforations is due to the fact that optimization
algorithm tries to achieve a compromise solution between maximum moment of inertia
(or stiffness) and minimum volume. However, when some restriction is imposed on the
movement of layers, as seen in case of constraint sets 11, IV and V1 for the three, five and
seven layer plates, respectively (Figure 3.14, Figure 3.15 and Figure 3.16), smaller sized
perforation are seen near neutral axis as those layers were not allowed to coalesce
together or move further away due to the lower bound on their movement, leading to
more material removal from the neutral axis as was seen in case of constraint sets I, III

and V mentioned above.

Table 3.4 Results for scalar maximization of effective stiffness for a three-layer plate

 

 

 

Constraint set 22 z 3 Moment of Inertia per
mm mm unit volume (f2)
mm
1 -40 40 90
II -10 10 70

 

 

 

 

 

 

48

Table 3.5 Results for scalar maximization of effective stiffness for a five-layer plate

 

 

 

Constraint set Z2 Z3 Z4 25 Moment of Inertia
mm mm mm mm per unit volume
(f2) mm
III -25 -40 4O 25 90
IV -14 -10 10 14 60

 

 

 

 

 

 

 

 

Table 3.6 Results for scalar maximization of effective stiffness for a seven-layer plate

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Constraint Z2 Z3 Z4 25 26 27 Moment of Inertia
set mm mm mm mm mm mm per unit volume
()9) m

V -40 -25 -40 40 25 40 90

VI ~15 -13 -10 10 13 15 60
, 1 ,

I T I I -40 mm
- 4 ~44 4 ~ l 1
, i

Figure 3.12 Layout obtained for maximization of effective stiffiress for a three layer
plate using constraint set I

49

 

 

if. I ’ l" Ir l
T\ \V x .y v 5’ T i
..................................... ““_I
-L _ --L ~40 mm
‘ ‘ l l
1 1
l ' ~25 Tmm !
i i 1
MIL- ________ L ,__ _L-LL_.-T__T_._+
l T 1
T
; " ' . .. ‘ 1
+2 :3 ‘ 7 . T 25 mm 1
T “T T ’ " "T '''' 40 mm
L. __ L L , L L - L A _ L -- L __L
if] i
:1 ‘ 1 l
3 i >~ — , 4 - - ~~~ T

 

Figure 3.13 Layout obtained for maximization of effective stiffness for five and
seven layer plates using constraint set 111 and V respectively

 

 

 

l
~40 m
T “L.” “i if: T _ 7 ,L " __T i
’ ~10mrn
___|L* .: ____ fl 1.._+ _ _~ _ 7- _.J __. _ _ __:...v_—H_
i T \ ~ g , _‘_ a ’1' 10mm l
I - _ _ ._ L _ _ _ _ L_L T
+3 4&4. gym 5
T T 40 mm
‘ z 1 1
1 1 T
- l
.\ 12 .
‘ -,-L

 

 

Figure 3.14 Layout obtained for maximization of effective stiffness for a three layer
plate using constraint set 11

50

 

/,r-‘ —« ’r' x‘ _,r""" "~
[A] l
[l l 1‘
l 1
' l
l \ .
1 l r
l . r
i .
z ‘x. \‘
l ‘ A“, \ ,..— ~~--_.——/
I
l
— — r—T‘M r—A~~. ,r’ \\
x
/
. r
.1 1 1 '
_"________.l#__ k_'- ___L-_________;

 

Figure 3.15 Layout obtained for maximization of effective stiffness for a five layer

plate using constraint set IV

 

 

—. — -———.l

 

 

 

1 T' “W" T”
TT“TTTTTTTTTTTTTTTTTTTTT—T
r TTTT:***r-r*~-i*+ —r 1 3
. ~ +++++ -. ~— 1 1
' L - a A- l 1 cm1.3 cm 1
1 l :'

 

 

 

Figure 3.16 Layout obtained for maximization of effective stiffness for a seven layer

plate using constraint set VI

51

3.2.7 Multiobjective sizing optimization using weighting coefficients method

The Pareto optimum sets for multiobjective optimization for maximization of stiffness
and minimization of volume using the weighting coefficient method are presented in
Tables 3.7 through 3.9 and described graphically in Figures 3.17 through 3.19. Only
constraint sets I, II and V were considered for three, five and seven layer plates
respectively. For the three layer plate not much significant trade-offs could be observed
between the objective fiinctions probably due to lower resolution, or too few layers. This
implies that improvement in one function didn’t lead to significant degradation of the
other due to low resolution. However, with increase in the number of layers, as seen in
case of five and seven layer panels, improvement in one function value led to significant
degradation in the value of other and vice-versa. Thus, the increased resolution with
different possible perforation sizes led to improvements and degradations in the two

objective firnction values with different weighting coefficients.

52

Table 3.7 Pareto optimum set multiobjective Optimization for a three layer plate

 

 

 

 

 

 

 

 

 

 

 

Weighting coefficient Maximization of moment of inertia Minimization of volume
w Obj ective function Objective function
mm4 mm3
0.9 -l.14x106 1.7x 104
0.8 -1.l4x106 1.7x104
0.7 -l.l3x106 1.6x104
0.6 -1.05 x10‘5 1.4x 10‘r
0.5 -1.05x106 l.4x104
0.4 -1.05 x106 1.4x 104
0.3 -l.04x106 l.4x104
0.2 -5.6 x 105 1.2 x 10‘
0.1 -51 x 105 1.2x104

 

 

 

53

 

Maximization of Moment of Inertia

 

O I 1T T j

 

{g 5 10 15 20
a) -20 i
E 9
g] 1
.3 -40 T
t:
“5’ 1
0 ~60 ~
:2
“T 1
0 ~80
r:
.2 .
$3 -100 4
.§ ‘
a -1on
53 1

-140 ~

Minimization Of Volume

Figure 3.17 Pareto optimum set for multiobjective optimization of moment of Inertia
and volume for a three layer plate

 

0 8b

-100 ~
-200 -
-300-«
-400 ~

-500 3

 

 

 

-600
Minimization of Volume

Figure 3.18 Pareto optimum set for multiobjective optimization of moment of Inertia
and volume for a five layer plate

56

 

O

A
V

20 30 40 50 60 70 80

ron Of Moment of Inertia
1'» ['6 J—
O O O
O O O
L l

. -400 -

~500 .-

 

Maxrmrzat

 

 

-600 14444 --—— ~—--
Minimization of Volume

Figure 3.19 Pareto optimum set for multiobjective Optimization of moment of Inertia
and volume for a seven layer plate

3.2.8 Limitations of sizing optimization using gradient based methods

When using gradient-based Optimization techniques convergence depends on a good
initial guess or starting point for the problem. Thus, finding a global optimum largely
depend upon the initial guess or the optimization routine might just return a: local
Optimum value. Even if a good starting point has been provided, gradient optimization
methods can’t guarantee global Optimum as the search Often gets “stuck” in a local
optima. These problems were Observed in the gradient-based optimization approach used
for the sizing Optimization problem. This led to reformulation of the Optimization
problem as a material layout Optimization problem using genetic algorithms as described

in the next section.

57

3.3 Layout optimization of beam/panel cross-sections using genetic algorithms

The approach presented in this section is aimed at optimizing cross-sectional properties
of beam/panel cross-sections by using genetic algorithms. In this approach the entire
domain was discretized in a mesh, or grid, and then pre-defined material designs, or
coupons, were selected from a library. The coupons in this library had finite size features
and pre-defined geometric and material properties. A linear bending-induced strain
distribution was assumed across the beam/panel cross-section. Thus, for evaluation of the
Objective function the problem was simplified to a repeating column unit made up of
coupons. The entire approach is conceptually summarized in Figure 3.20. The
formulation of the layout Optimization problem using genetic algorithms is outlined in the

following section.

 

 

 

 

   
   

T" The l
f rectangular
T <— cross-section

 

A proposed library of coupons

Figure 3.20 Overview of layout optimization using library of coupons with finite-
size features

58

3.3.1 Design variables

Coupons in a pre-defined library were stored as an array of structures where each
structure contained all the geometric and material properties of a coupon. A characteristic
coupon with its finite-size features and geometric parameters is shown in Figure 3.21

where r,- denote the radii of different perforations or circular holes and y,- denote the

distances of centroids Of the perforations, or finite size features, from the base of the
coupon. The library was defined based on the above nomenclature and included both
non-hybrid and hybrid coupons (Figure 3.23). Various possible layouts, based on the
typical coupon, were incorporated in the coupon library ranging from coupons with only
a central hole to coupons with all possible holes present. For every non-symmetric
coupon a corresponding mirror image was also included. The hybrid coupons were
assumed to have a thin layer of comparatively stiffer material (i.e. having a modulus of
elasticity five times that of base coupon material) at the top, the bottom or both top and
bottom of the coupon. The moment of inertia of each coupon about its own centroidal
axis was calculated assuming it to be a composite section with internal circular voids, and
this value was then stored together with the other geometric and material parameters into

the coupon structure array (Figure 3.22).

59

 

 

 

 

 

Figure 3.21 Layout of a general coupon with finite size features used for
optimization

—Ffi___l__'l“ 1
0 l | 1 i 1 30 31 1 lDarrayofcouponIDs
!_L_LL

coupon[i].ri 1 coupon[i].Ai

 

._T.___.~‘_

ri 1 ‘ 1 Ai T [i Data structure for ith coupon
LL _ I_ ylm _ 1

1
_l_ __

coupon[i].yi coupon[i].li
Figure 3.22 One-dimensional array of structures containing coupon data

60

___—_a

COUPONO

..vm
. _._- A..L._H

COUPONS

 

r - .. -.-T -_-. _.___‘
W - ‘1
.

(«A 1

 

______4

COUPON16

.._..__;:;....1

COUPON24

 

COUPON28

 

.4

COUPON]

 

. Mg-“

r' "'12::

COUPON9

 

 

 

>——-. .—.—.—v—_ -——.———-j

COUPON25

,v—-—-—v~—-—1-- v-v———-1

COUPON29

 

;_4._ __..__.___
—\ ‘T
N/ .—

t-_.—_—.—»H———1

COUPON10

 

 

 

 

—- v
r 1
1 .—
1
!
' .
/ r

J I '1

‘4‘
{LT—“L .-._:—_', :ZZ'J

. ,1
a. 1* I
1' “’1

COUPONIS

 

F._._. — - — .'~_.~_.T

 

1
1
1
1
1

 

COUPON22

1
‘T' r—\

 

1
1

1 .
1 1
1 .

‘1 1

1

L '—-——-—_—-—— -—v~-v~<

COUPON30

 

L, ____
/“~.

1

|

1 A. x»

. ‘4’

l

 

 

 

L._._

eaaiaaii

 

 

 

 

His-IT

 

...A . _._...4 - -4.-. ‘.

COUPONIS

1
'— __ _._, ,_.___v___,__4
- _- ‘_.___._.+.__~

COUPON27

 

b- ——~—~—.——.—.~~—- .— a

COUPON31

Figure 3.23 Library of hybrid and non-hybrid coupons

61

The identification numbers for the coupons (0 through 31 as shown in Figure 3.23) were
used as the design variables. The variables were expressed as a “string of integers” or a
“chromosome of integers.” For evaluating the fitness of a particular layout of coupons
the chromosome was first unpacked fiom its binary representation into integers and those
integers were then used as the coupon identification numbers referencing the respective
geometric and material properties. Thus, coupons IDs acted as the design variables for

evaluation of the objective function.

3.3.2 Objective function

A multiobjective optimization was implemented to achieve the desired optimum layout of
the beam/panel cross-section. Two sets of objective functions were considered for
calculating the fitness of a given layout. The first one was a linear combination of
moments of inertia along the x and y axes (quantifying sectional stiffness) and the second
one was a linear combination of moments of inertia per unit volume along the x and y

axes (quantifying specific sectional stiffness) as expressed in equations (3.4) and (3.5),

 

respectively:
f1: wllx +w21y (3.4)
W11 +W21
fz=£ xV y] (3.5)

where for each case wl , wz =weighting coefficients and wl + wz = 1.

Encoding and decoding of the design variables and objective function is described

schematically in Figure 3.24.

62

Parent 1 Parent 2

   

. [2/2/0/3/1/0/2/2] Parent 1 [010/010/000 /01 l/ 001/000/010/010] Parent 1
1 _.
b — 7 ’1 [O/O/O/O/O/O/O/O] Parent 2 [000/ 000/000 l000/ 000/000/ 000/ 000] Parent 2

lTwo point Cross over

 

 

1 _. _A 4 [010/010/000/000/001/000/010/010] Child 1

[000/000/000/011/000/000/000/OOO] Child 2

.5; l Mutation
[010/010/000/000/00 0/000/010/010] Child I

. [OOO/OOO/OOO/Ol l/OO l/OOO/OOO/OOO] Child 2

14 1

 

 

Optimum layout

 

 

1 Unpacking the chromosome Compute objective function ,compare

[2/2/0/0/0/0/2/2] *— [Ol0/0l0/000/000/000/000/010/010] ‘- and select design with larger value of
objective function.

 

 

 

Figure 3.24 Determination of optimum layout for a given generation in GALOPPS

The moment of inertia of hybrid coupons was calculated by considering the entire coupon
to be a composite section. The moment of inertia and volume calculations were thus
based on a transformed section to the base coupon elastic modulus. Thus, the width of the
top and bottom stiffer layers was increased by a factor of their modular ratio 112 and n3,
respectively, as shown in Figure 3.25. Consequently, in the optimization of specific
stiffness the use of hybrid coupons with stiffer material not only resulted in a higher
stiffness value but it also led to higher coupon volume, thus penalizing the use of a stiffer

(and most likely more expensive) material in hybrid coupons.

63

     
 

Hybrid
material
layer

Figure 3.25 Transformed section dimensions of the hybrid coupon

In order to check the robustness of the proposed approach and to increase the size of
design domain a square cross-section was also considered. The objective functions used
were same as those outlined in Section 3.2.2 (Equations 3.4 and 3.5). However, in this
case the design space was represented as a two-dimensional array instead of a one-

dimensional array as shown in Figure 3.26.

 

A[il[j]
Unpacked ' V F v R 1‘ 7 +—fi—1 _
chromosome1 0 1 (3 _ _1_ k 1 k 1 26 26 0 1 0 2 D array of coupon IDs
A[i][i] 1
coupon[k].m 1 coupon[k].Ak
1’ 7 __ T' T _ '7
rm 1 1 1 1 Ak 1 1k 1 Data structure for coupon k
yk .
1 L .‘ i _ _,

coupon[k].yk coupon[k].Ik

Figure 3.26 Two-dimensional array of structures containing coupon data

64

3.3.3 Implementation of layout optimization in GALOPPS

The genetic algorithm package GALOPPS (“Genetic Algorithm Optimized for Portability
and Parallelism” System) [11] was used for solving the cross-sectional material layout
optimization problem. GALOPPS is written in the C programming language and was
compiled and linked using visualstudionet [20]. The objective function to be optimized
in GALOPPS is written in a special file called the application file. A standard
application file template is available in this package and the user can define his/her own
objective function within it. Input parameters can be provided through the command line
or through an input file. Standard input file templates are also available with the package.
Similarly, output can be obtained in a file and the desired output details can be specified.

The basic organization of GALOPPS is outlined in Figure 3.27.

 

1 GALOPPS

1 , l

 

 

 

 

 

 

 

 

 

HEADER FILES SOURCE FILES WORK FILES

DIRECTORY DIRECTORY DIRECTORY

oContains functions to be 0Contains functions °Contains application

used during optimization. defining genetic routine for the specific
operators optimization problem.
0Contains other files °Contains input file for the
required compiling specific problem and stores
and linking. output file if generated in

the optimization process.

 

 

 

 

 

 

Figure 3.27 Organization of GALOPPS

As mentioned earlier, visualstudionet was used to compile and link the above mentioned
routines from GALOPPS. Project properties contain the information about input and

output files and their location, e. g. their residing directory. Besides the application routine

65

a number of other source files are required to be added in the source files directory of the

project. The source files required for any visualstudio project for GALOPPS are [8]:

checkhdr.c checkrd.c checkwt.c ffscanf.c
filestat.c generate.c memory.c random.c
report.c statistic utility.c user_in.c

Finally, an application file containing the objective function and appropriate source files
to specify the selection method, the crossover operator, the mutation operator, the
inversion operator and the files named mainone.c and startup.c for a single population
operator, need to be added to the source file directory of a project. After all these required
files are added to the project, the application file is compiled and run to generate the

desired output containing the solution.

3.3.3.1 Parameters used in GALOPPS for the genetic algorithm

A single population size of 200 was considered for all cases of the layout optimization
problem. The number of maximum generations in a single run was selected as the
termination condition. All of the optimization cases were solved as unconstrained
optimization problems. A two-point cross-over was used as the recombination operator
and bit-mutation as the type of mutation operator. For parent selection a stochastic

universal sampling method was used [11] [10].

3.3.4 Multiobjective optimization of sectional stiffness for a square beam
Multiobjective optimization of f = wllx + wz I), was carried out using the library of

coupons described in Figure 3.23. Ix is the moment of inertia of the cross-section about

66

the x-axis and I), is the moment of inertia of the cross-section about y-axis. Three sets of

values were used for the weighting coefficients W1 and w; as shown in Table 3.10. The
results are also summarized in Table 3.10. As expected, the results show that for
maximization of sectional stiffness about the x-axis the coupons with hybrid material
layer on the top and bottom were selected and placed such that the stiffer material was
placed away from the section neutral axis and thus lead to a higher stiffness value
(Figures 3.28 through 3.30). As there were no coupons in the library having stiffer layers
on the top and bottom layers about y-axis the layout obtained for all three cases was same

as shown in Table 3.10.

67

Table 3.10 Results for multiobj ective optimization of sectional stiffness

 

W1

CHROMOSOME

W2

FITNESS VALUE (mm‘)

 

[8 8 8 8
24242424
16161616

8888]

8.5x 101°

 

0.5

[8 8 8 8
24242424
0.5
16161616

8888]

9.2x 10lo

 

 

 

[8 8 8 8
24242424
16161616

8888]

 

9.8x 10lo

 

 

 

4.1..
1

~— ..:<.::‘ «
(«m 11

l
“‘81

 

;~.,<-

'2
.‘- . .\
' "K I“
, , ., _, 1 '4
‘ . g \, l . ‘: '2“ ‘
, .4 ~ ‘ Iv . I.- ._ 1
* xv l ,» r.
_ / .
.. .
x

1
it

1
1

.1

____, v >
1

1

.‘sl

 

 

\
K.
r a .
.1. .
J

.' '11 ,- ‘q‘_
11!
Ni

3.1

1'1”
\

1
.111 ..

11.
1‘1

. (>11; ',

11 l

. ,.

1‘ ‘1" I
.' . .55 ‘- ‘.~ 11 ”’37"

A, 1 ',.‘ ‘ 1 - '."' .1', '
l: /’ ‘ 11'; "‘ l
x“\ 3 1 \‘ '
1‘ \. ,1 1». 4

, . . -.;‘ 1 I

‘x \ -,'-‘_41\.,. 1‘ ‘1 1
\’ f t_'," {/1

I ' '

,t

.1 I‘
\
’ r
{I

’(\
K
1 l
I
C
._‘ x.\
)1.
. ‘.‘ ’
5234/ I
1 ;\
"t

 

\
\x
I a .
a . A i a.
’4'?“ l , "g‘ -. 1‘

.. ._ .. .._'__._. ”hi—,1."

. ‘:‘.-:’, i
J ‘4
I

,;‘.

. _‘ " ',_, A, "'Q
r '/-\' ~ "- ,l\ .. \

 

1
1
1
~ 1
\ 1 r. .1
.1‘1.“

Figure 3.28 Layout for maximization of f = I x

68

 

 

\
/
K
1
n 't’ ‘
.>-<.~
\1/

_ '1,“ (U-.- -
1
)

WlIx +W21y
2

 

Figure 3.29 Layout for maximization of f =

I A j .
1 ' .3 r
t _ t l . ' _ ‘ v . .
1 " \ . \ , " ‘ A
.. ' \ .: ,‘, V“ ~ \
Y"“" 4‘1‘ "7 —:." 7k '1 LJ \.‘ H “ A7
‘ ..‘\_ L~ >H; x 1 ~ 5
1' .'
\ I
3.! :1, ‘ f/ 5‘ I \

Figure 3.30 Layout for maximization of f = 1},

3.3.5 Multiobjective optimization of specific sectional stiffness for a square beam

Wllx +W21y

V was carried out using the library of

 

Multiobjective optimization of f =

coupons described in Figure 3.23 and the results are summarized in Table 3.11. Again Ix

69

is the moment of inertia of the cross-section about x-axis, [y is the moment of inertia of

the cross-section about y-axis and V is the volume of the component. The results for
maximization of specific sectional stiffness demonstrated that coupons with hybrid
material layer on top and bottom were selected to place more material away from the
neutral axis. Conversely, coupons with more perforations, or less material, were placed
closer to the neutral axis of the section to reduce the overall weight of the section, thus
achieving a compromise solution for maximum stiffness per unit weight. For
maximization of specific sectional stiffness about the x-axis, coupons with stiffer top-
only and bottom-only layers were placed on the top and bottom of the section
respectively. On the other hand, “plain” coupons were placed near the neutral axis
leading to a maximum stiffness per unit volume (Figure 3.31). With weighting coefficient
values of 0.5, coupons with more material were placed along the periphery of the cross-
section while coupons with less material were placed close to the neutral axis to get
maximum effective stiffness (Figure 3.32). In the case of maximization of specific
sectional stiffness about the y-axis all of the selected coupons had top and bottom layers
of stiff material as there were no coupons in the library with both top and bottom stiff
layers about y-axis (Figure 3.33). In this case also coupons with maximum perforations
were placed about y-axis to create a region with less material. Thus, the presence of all
hybrid coupons in the maximization of effective stiffness about y-axis led to higher
stiffness value, while the presence of coupons with maximum perforations towards the
neutral axis led to minimization of volume. This arrangement thus led to a compromise

solution for maximum specific stiffiiess about the y-axis.

70

Table 3.11 Results for multiobjective optimization of specific sectional stiffness

 

wl wz CHROMOSOME FITNESS VALUE (mm)

 

[24242424
2 2 2 2
1 0 3.6x103
2 2 2 2

16161616]

 

[8 8 8 8
8 26 26 8
0.5 0.5 3.711103
8 18 18 8

8888]

 

[8 1010 8
8 10 10 8
o 1 3.911103
8 10 1o 8

81010 8]

 

 

 

 

 

 

71

 

Ix+Iy

Figure 3.32 Layout for maximization of f = 2V

 

Figure 3.33 Layout for maximization of f =

72

3.3.6 Limitations of layout optimization using genetic algorithms

There were several limitations in approach just presented using a standard genetic
algorithm. The approach works only for one constant scale of coupons and it is difficult
to incorporate multiple scales of coupons in the problem. In other words, according to the
present approach beam/panel cross-section can be discretized only into coupons of the
same size and there is no option for different sized coupons. Also, as the domain of the
problem increases sometimes there are chances of loss of diversity of solutions in the
optimization process and good chromosomes might be lost leading to a solution that is

not the global optimum.

3.4 A coalescence approach for improved optimized layout designs

In general, it was observed that optimization of specific stiffness resulted in the selection
of coupons having more perforations. However, if more perforations could be associated
with greater difficulty in manufacturing then a small improvement in the coupon stiffness
might not be advantageous from a manufacturing point of view. Coalescence is an idea to
obtain an improvement of optimum designs obtained by merging neighboring coupons
that qualify to do so. It is an idea for “post-processing” of the optimum layout obtained.
Coupons qualify for coalescence depending upon a pre-defined heuristics of merging.
This idea can be implemented in two ways:
1. One-dimensional coalescence: Since initially a column of coupons is considered
to be repeating itself throughout the panel cross-section, as only one-way bending

is considered, two adjacent coupons within the repeating column units could be

73

compared for coalescence and the scheme can be then extended to the remaining

cross-section (Figure 3.34).

 

Before coalescence After coalescence

Figure 3.34 Coalescence in one dimension

2. Two-dimensional coalescence: instead of extending the idea of a repeating
column unit throughout the cross-section, coupons from two adjacent columns
can be compared together such that if four adjacent coupons qualify for merging

then they can be coalesced together (Figure 3.35).

74

 

Before coalescence After coalescence

Figure 3.35 Coalescence in two dimensions

3.4.1 Steps for coalescence approach

For the coalescence approach new coupons were added to the library of existing coupons.
The new coupons consisted of a quarter of a circular hole as shown in Figure 3.36.
Heuristics of coalescence was decided by considering coalescence of four neighboring
coupons with ID=0 as shown in Figure 3.37. If four neighboring coupons had an ID equal
to 0 they were coalesced and new layout was represented by replacing all four coupons
by new coupons (coupons 32 to 35) as shown in Figure 3.37. If all four neighboring
coupons did not have ID equal to 0 they did not qualify for coalescence and the next
layout was checked. An overview of the two-dimensional coalescence algorithm is given

in the flow chart shown in Figure 3.36.

75

 

Create a new library of coupons containing all existing coupons and
additional coupons to be used for representing coalesced coupons.

1

 

 

 

 

Establish heuristics determining which coupons can be merged together
and replaced by which coupons from this new library representing the
coalescence.

 

L
F

 

 

 

Store the best fit chromosome obtained from optimization in a one dimensional
array.

 

 

V

 

 

 

Compare the adjacent elements of the array to find candidate coupons for
coalescence based on heuristics by using sets of ‘if—else’ condition statements.

 

 

1

 

 

Evaluate the objective function by using the ‘new’ best fit chromosome
obtained from coalescence.

 

 

 

 
      
 

 

Create new coalescence
heuristics for better new
chromosome.

 
 

Is new objective
function greater
than old one?

 

 

 

 

Report the ‘new’ chromosome as new
solution.

 

 

 

 

l
C Stop D

Figure 3.36 Overview of coalescence algorithm for one dimensional coalescence

76

11“,. ,«i'

'{V "t; 1
I. 3., ‘1. 1
1 ; ' ~. . 1
1 .,.-"’"‘>;:- \1
COUPON 0

K—..—.—“—.~.__w_. .
‘ , .
r "I“: -

I

_ f, .

1 1’ . l 1
\ l

1 ;1 1‘ 1

14 3
COUPON 0

 

i1" ‘1. 1 ' ' 1
1 14 1 RI -. ; .
1 " 1 5’ 1
A ..-Q.‘ ~§1 12:11 . 1
COUPON 0 COUPON 32

——.—

r“ 771 *‘TT
1 3,3471 , ‘ 1 "1' ‘1
1 7-" \L. -. n7);
If," ‘ . l:“)-t‘::: x": 1 Rl _ f . : . _ / isz'tzd
COUPON 0 COUPON 34

— RI
. -’- 1
;__L-.i:.1
COUPON 33

 

1 . ,’1

1
\
‘

1

1

1 x

1 ,‘\ I", . I

1 x". "“— R1
.4 . I

-.g
«N ......

‘1
‘. -';- “t

 

COUPON 35

Figure 3.37 Coalescence scheme used in one and two dimensions coalescence

3.4.2 Two-dimensional coalescence for maximum effective stiffness

The coalescence approach when applied in one dimension to obtain optimum layouts for
maximization of stiffness and effective stiffness didn’t result in any improvement in the
fitness value. However, better fitness values did result when applied in two dimensions

(Figure 3.3 8) for maximization of stiffness as shown in Table 3.12.

Table 3.12 Results for coalescence approach in two dimensions for maximization of

 

 

stiffness
Initial chromosome Initial fitness Chromosome Fitness value after
value (mm4) after Coalescence
coalescence (m4)
[8, 0, O, 8 [8, 40, 41, 8
4.20 x 10'0 4.22 x 1010
8, O, O, 8] 8, 42, 43, 8]

 

 

 

 

 

 

77

 

ffifwri;L-'A~+- -- i:7 ‘1':* ,.'A A,# ::_—.‘._7
1 r J- 1 11,-. 7 - ,
1 . - ' 7‘ ’, : , 1 ,_‘_.e(‘ .. i ‘1’).‘1‘, ’A '
1 5“] 1+. g: I 1 I ‘k‘. I ‘3 1
r . 1‘ 1 .s‘ 1 4"", . “.I I l 1 ‘\‘
: .4 , Irv—~15. -. : .~ a t. i .- .. ' 1 T/x-x ' - 1
r ‘. _‘ j / . \1 -‘ ‘- 1
1r -4 - 'v‘ --_ gA 11", ‘- v .-.--’_.‘\1
1“- "*"‘ ‘M "N 7 K i’ .- . .. _
'--.. 1 .- .41 : "1 ' ”fit" I 1' 1
1";;..1'» 1 gig." . ’ |
~ ' , x 1 - . 1
:1: f ‘ j . 1 1 a/ \~.\_ 1
, ‘w " . W’ ; I “'1‘“ I
. - . / . '. “<7 _
.- ' “ -5 - I __.J- '7'—'_“\1
1 2“ 1 '4‘.’ - ' A, - '
1 4‘ 1"“ ,"1- :2, 1 F37“: - .. 1' 1
1 -‘ , -. \ , ' - __ m‘ 1 . - 3
" ‘ . ‘ ' 1,7"; \ ' 1 ‘~ . ~. 1
l/ 1 1 1 k 1 :
_ "* m’vf 1:31: 5‘ 1' -—1 r~~4 ”———_ A:
- «I.» , . .1 x . _, , A ~¢
1 -_ E , ' xx.» '1 . i V I 1 - 1
l -‘ I ’ ‘ 1 ~ -_ ‘ r} ’r -.'.-l > ’4 ‘
I x ' 1 x e '
:11“. ‘- 1 1 I (t 1 1 g
" "I" 1 . 1 1 v. . ‘1 I "y'\1,.\~lvu
/ 1 \ 1 1 \ / - \
I" —— _- V —~ ~44— 4‘ f u, ' \

 

1
1

Figure 3.38 Layout for two-dimensional coalescence for maximization of stiffiress

3.5 Integrated coalescence approach for Baldwinian inheritance

To solve the problem of loss of diversity among solutions with increase in the complexity
of the optimization problem, and hence size of solution space, the idea of using
knowledge based evolution was introduced. This consisted of utilizing knowledge gained
from the coalescence of optimum layouts to influence the evolution process to search for
layouts competing with improved fitness values obtained from coalesced designs. The
idea was to explore the existence of designs with higher fitness values and passively
influence the genetic algorithm to look for better designs that matched the new fitness
value obtained through coalescence [6][14]. Figure 3.39 shows an integrated approach
incorporating Baldwinian inheritance (a passive mechanism enhancing genetic diversity
and survival chances of individuals through their learning capabilities) within the existing

genetic algorithm module. The results obtained for layout optimization using the

78

integrated coalescence approach for stiffness and stiffness per unit volume are

summarized in Section 3.6 next.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

GA OR EVOLUTION MODULE COALESCENCE MODULE
Chromosome : : .
encoding ——‘—‘—-' Does chromosome qualify No
i i -+ m:
i for coalescence
l Fitness evaluation] 1 Yes
i t 4 { Coalesce coupons and evaluate new fitness 1
Best fit chromosome at the end of l

 

 

 

: maximum generations : :
5 5 5 Is new fitness>Fitness from No Keep

 

 

 

 

 

GA module H fitness
from
1 Yes GA

 

 

 

 

module

 

[ F itness=new fitness 1

 

 

Figure 3.39 Integrated coalescence approach for Baldwinian inheritance

3.6 Multiobjective optimization using Baldwinian approach

The results for optimization of sectional stiffness using Baldwinian inheritance showed
no change from the prior optimization results (Darwinian approach) without Baldwinian
inheritance (Table 3.13). It was due to the fact that the problem was not very complex
and thus the solution space was small. Consequently, a global optimum could be
achieved even without Baldwinian inheritance. In the optimization of specific sectional
stiffness, however, due to increase in the complexity of the optimization problem in the
absence of Baldwinian inheritance the original optimization process could not converge
to the global optimum. When external knowledge was used to force the genetic algorithm

to look for better designs it resulted in the exploration of new designs (Table 3.14) with

79

higher fitness than before. The results for maximization of sectional stiffness and specific

sectional stiffness are presented in Figure 3.40 through Figure 3.43.

Table 3.13 Results for multiobjective optimization of sectional stiffness using

 

 

 

 

 

 

 

 

 

Baldwinian Approach
W1 wz Chromosome Fitness Chromosome from Fitness from
from Darwinian from Baldwinian Baldwinian
approach Darwinian approach approach (mm4)
approach (m4)
[8 8 [8 8
24 24 24 24
1 0 4.3x10'0 4.32:1010
16 16 l6 l6
8 8] 8 8]
[8 8 [8 8
8 8 8 8
0 1 1.7x10ll 1.7x10“
8 8 8 8
8 8] 8 8]

 

80

 

 

Figure 3.40 Layout for maximization of f = I x using Darwinian and Baldwinian
approaches

 

 

 

Figure 3.41 Layout for maximization of f = I), using Darwinian and Baldwinian

approaches

81

Table 3.14 Results for multiobjective optimization of specific sectional stiffness using

 

 

 

 

 

 

 

 

 

 

 

Baldwinian Approach
w] wz Chromosome Fitness Chromosome from Fitness from
from Darwinian from Baldwinian integrated
approach GA (mm) approach coalescence
(Inn!)
[2424 [2424
2 2 8 8
1 0 3.6x 103 3.7x 103
2 2 8 8
16 16] 16 16]
[8 8 [10 10
8 8 10 10
0 1 1.3x 103 1.4x103
8 8 10 10
8 8 ] 10 10 ]
I—ivx: , - z Wham
- V i . x
..... v --x Lime.”

(a) (b)

I . .
Figure 3.42 Layout for maximization of f = [3%] obtained from (a) Darwmian and
(b) Baldwinian approaches respectively

82

f‘\
1
l 1/
t
l

f‘\
l

. 1 ~/‘
>.(“\q
l/

’\
l
l
/
2‘ \
I/_..__._.__.‘:SJ/ _-, S1 r. -

(a) (b)
1y

Figure 3.43 Layout for maximization of f = 7 obtained from (a) Darwinian and

(b) Baldwinian approaches respectively

83

 

4. EXPERIMENTAL EVALUATION

4.1 General

The results obtained by using finite size features for optimization of beam/panel cross-
sections enabled designs with optimum material distribution that could potentially be
manufactured at laboratory-scale using biocomposite materials. To evaluate the results of
the computational simulation an experimental study was conducted that was aimed at
comparing the flexural performance of hybrid and hierarchical designs with respect to a
base cellular design. The experimental program comprised of manufacturing and testing
beams with cross-sectional layouts based on optimization studies. The structural layouts
investigated for evaluation of the optimization approach are described in the following
sections along with the manufacturing and testing details. The chapter concludes with

presentation of the results and inferences of the experimental evaluation program.

4.2 Structural layouts

Two types of structural layouts were investigated in the experimental evaluation program
namely a base cellular design and an optimized hierarchical cellular design. The cellular
base design consisted of equal-sized cells repeated at equal distance, thus making the
cells periodic. The optimized design consisted of two different cell sizes distributed in the
cross-section in a manner similar to the analytical solutions obtained with the discrete

layout optimization approach. The two designs are schematically shown in Figure 4.1(a)

and (b).

84

 

 

 

 

I __ :1
:- l ‘ 11. “F
Z l i
!
, " a l
1 51 mm ‘ 51 mm
1 ., a E,1. ‘
___~-» ...,. fl ! v" w I Y
1 l
*1 25mm K” ——-—7 25mm 5'“—
(a) Base Design (b) Optimized Design

Figure 4.1 Cross-sectional layouts of base and optimal design used in experimental
evaluation

Mechanical properties of cellular structures are greatly determined by their relative
density pm], i.e., the density of the cellular material divided by the density of the solid
from which the cells are made of (pm; =p'/p, ). The relative density of the base design

was 0.53 while relative density of the optimum design was 0.7.

4.3 Materials and constituents

The biocomposite system used for manufacturing the cellular and hybrid, hierarchical
beams consisted of industrial hemp fibers (Flaxcrafi, Inc., Cresskill, NJ) (Figure 4.3) and
unsaturated polyester resin (UPE). The catalyst used was methyl ethyl ketone peroxide
(MEKP). It was used in a proportion of 1% by weight of resin and cobalt napthalate (CN)
was used as promoter in a proportion of 0.03% by weight of resin. The base design had a

fiber weight fraction (ratio of fiber weight to total weight) of 23% while the optimal

85

 

design had a fiber weight fraction of 22%. For manufacturing purposes the beams were
built with integral top and bottom mats of Hessian jute (UIRA, Calcutta, India) as shown
in Figure 4.2. Average properties of some hybrid material systems from prior
experimental studies by our research group (Quagliata 2003) with biocomposite fibers are

provided in Table 4.1.

 

Figure 4.2 Jute mat for top and bottom face sheets

 

Figure 4.3 Industrial hemp fibers

86

Table 4.1 Average properties of some material systems with natural fibers and UPE

 

 

 

 

 

 

 

 

 

 

 

 

 

resin [28]
MOE Tensile Tensile Std.
Std. Dev. Std. Dev.
SAMPLE Str. Elong. Dev.
(N/mmz) (N/mmz)
(N/mmz) (N/mmz) (%) (%>
Green
Hemp/UPE- 5230 820 16.79 4.19 0.33 0.07
25% wt.
Woven
4030 640 22 1.3 0.68 0.07
jute/UPE-
Green
Hemp/UPE top
10,280 6120 28.02 4.5 0.39 0.24
& bottom jute-
27% wt.
4.4 Manufacturing

4.4.1 Automated manufacturing using VARTM

Manufacturing automation of the cellular biocomposite beams was evaluated by
implementing a vacuum assisted resin transfer molding (VARTM) manufacturing setup.
The method was selected as it has been found to be an effective process for
manufacturing cellular structures using natural fibers [25]. VARTM is essentially a

closed-mold process that can be used to manufacture fiber reinforced polymer composites

87

(FRP). The process consists of the placement of dry fibers inside a bagging film where
resin is drawn into the sample by using a vacuum pump. The VARTM process can be
effectively used for manufacturing of large FRP composite parts with high mechanical

properties and complex shapes. Thus, it is ideal for making large structural components.

4.4.2 Material placement

A mold with removable face plates was used for manufacturing all of the beams. The
dimensions of the mold were 813 mm x 54 mm x 38 mm. The same mold was used for
making both beams by using removable face plates for different cross-sectional cellular
designs. The cellular sandwich base design was manufactured by using equal-size
circular periodic holes in the longitudinal direction of the cross-section of the beam. The
beam was provided with top and bottom face sheets of jute. The optimal design also
consisted of circular cells in the longitudinal direction of the beam, arranged in varying
sizes through the depth of beam. The dimensions of the test units for the above mentioned
layouts and cell-sizes were decided considering the available circular metal rods which
were available in standard sizes: in 3 m (1/8”) intervals. The effective cell sizes were
then defined by the thickness of rubber tubes, also with diameters available in 3 m
(1/8”) intervals, which were used to avoid bonding between the polymer and the metal
rods. To manufacture the three designs at first the bottom of the mold was lined with a
non-porous Teflon ply (Figure 4.4(a)) in order to release the beam from the mold after
curing. A breather ply was placed on top of the Teflon ply to absorb excess resin from the
sample (Figure 4.4(b)). A resin transfer media (high density polyethylene, green house
shade cloth) was placed on top of breather ply to obtain a uniform flow of the resin

through the sample (Figure 4.4(c)). A porous bleeder ply (polyester cloth) was used on

88

top of the resin transfer media to achieve a smooth surface after curing and absorb excess
resin from the sample (Figure 4.4(d)). Finally a layer of porous peel ply was used to on
top of the bleeder ply (Figure 4.4(e)) to allow excess resin to be squeezed out of the
sample. After setting all of the above mentioned layers, a single layer of jute mat was
placed in the mold such that it would wrap around the entire beam (Figure 4.4(0). The
bottom layer of short industrial hemp fibers was then placed on top of the jute mat and
then the face plates were attached to the mold. After placing the face plates the circular
rods (covered with rubber tubing) were inserted along the longitudinal beam cross-
section. The fibers were then tightly placed in between the rod spacing. After placing the
top-most layer of hemp fibers the top of the sample was wrapped completely with the jute
mat and all the plies used in the bottom of the mold were mirrored on the top as well. The
entire set up was then placed inside a vacuum bag and sealed with sealant tape around the
mold. The ends of the rods and other sharp corners of the mold were covered with a layer
of resin transfer media and breather cloth to prevent them from puncturing the vacuum
bag when under pressure. Resin was infused using a vacuum pump through VARTM

method as described in Section 4.4.3.

89

 

(a) Non-porous Teflon ply (b) Breather ply

 

(c) Resin transfer media (d) Porous bleeder ply

      

(e) Porous peel ply (f) Jute mat layer

Figure 4.4 Preparation of mold for material placement

4.4.3 VARTM: Setup and resin infusion process

The VARTM setup was prepared on a movable cart with a single vacuum pump on the
bottom of the cart and a mold containing the sample on the top of the cart. Before the
sample was sealed completely inside the vacuum bag using sealant tape, a vacuum port
was placed on top of the sample and secured to the bag using additional sealant tape
around the port. The vacuum port was covered with a small piece of resin transfer media

and breather cloth to prevent the vacuum bag from blocking the port. The resin port was

90

connected to the other end of the sample, also from the top, and sealed again using tape.
The cover of the sealant tape was then removed and the entire setup was then sealed. The
unsaturated polyester resin was then added to the resin reservoir (Figure 4.5(a)) and a
resin trap was connected to a pressure gauge (Figure 4.5(b)). A two-way inlet valve was
used to control the flow of resin into the sample. The sample was compressed under
vacuum (635 mm of mercury) before the resin was injected into the sample. Once the
sample was compressed under pressure the inlet valve was opened and resin was drawn
into the sample by the vacuum pressure. Once the sample was completely impregnated
with the resin the valve was closed to let the sample absorb the already present resin and

allow excess resin to be removed.

 

(a) Flow of resin from reservoir (b) Excess resin collected in the resin trap

Figure 4.5 Sample with vacuum bag, resin reservoir and resin trap

The system was kept under vacuum pressure for approximately an hour to let the resin

gel. Afier this, the vacuum pressure was removed and the ports were detached to prepare

91

the sample for elevated temperature curing. The sample was oven cured inside the mold
and with all bagging materials for 6 hours at100°C . The curing temperature was kept at

100°C to avoid melting the resin transfer media, which is made of high density
polyethylene. Steel plates were placed on top of the sample during curing to keep the

sample compressed and thus improve the density and quality of the sample.

4.5 Testing of beams

The biocomposite beams were optimized for a continuous panel type application, thus,
their performance in flexure under both positive and negative moment demands was
investigated. The panel system was assumed to be modeled as a continuous beam
supported at equal intervals and loaded under concentrated loads at mid-spans of each
panel length (see Figure 4.6(a)). To simplify the test set-up only a portion of the
continuous beam was considered, after reducing the system due to symmetry conditions
as shown in Figure 4.6(b) and Figure 4.6(c). The moment diagrams for the simplified
systems are shown in Figure 4.7. In order to obtain the desired design moments as shown
in Figure 4.7(b), the beams were tested as a simply supported beam with a cantilevered
overhang as shown in Figure 4.7(c). The load levels at the mid-span and the cantilever tip
were selected to match the desired design moments. The cantilever length aL (Figure
4.6(c)) was calculated as a fraction of the simply supported length L and a was equal to
0.19. The beam was loaded at the cantilever tip and at the middle of the simply supported

span. The load at the cantilever tip ,BP (Figure 4.6(c)), was calculated as a fraction of the

load at the middle of the simply supported span, P , where ,6 was equal to 0.837.

92

;_._—.. ___.

..._L_;,._L+_L*+_L__g

(a) Continuous panel system with multiple supports
P P

,z” i.‘ '~\ I’M.

'\_ __

1'“— L l L —"|

(b) Reduced two-span panel arrangement

[31) P
l

l l

"— , w-

.r/
,4

____\

i

(c) Reduced two-span arrangement for test units

Figure 4.6 Simulation of a continuous panel system under symmetric loading and
boundary conditions in a four-point bending test

93

 

-0,107pL 70.161 PL

 

‘ 3+
0.116PL 0.17PL

(a) Bending moment diagram of symmetry-reduced continuous panel system

-0.l6l PL,

 

l +

l

,/"
0.17 PL

(b) Bending moment diagram for test unit

Figure 4.7 Bending moment diagram to be achieved for continuous beam test units

The continuous flexural test setup (see Figure 4.8 and Figure 4.9) was mounted on an
MTS loading fiame. Load was applied to the beams using the test frame loading ram and
a loading fixture that applied two point loads at 457 mm apart. The total span was 787
mm. The samples were supported using a fixture from a steel I-beam and adjustable roller
supports. All test units were loaded monotonically up to failure in displacement control at
a deformation rate of 0.01mm/sec for the base design and a rate of 0.025mrn/sec for the

optimal design.

94

 

0.837 P P

 

 

 

! TOP strain gage I Top strain gage 11
Linear transducer v Linear transducer 1
xi" E ~44 ‘“ Bottom strain gage I ‘76 97“" Bottom strain gage II T
13mm r6127mm—i—e—— 330mm f 330mm —-———*i 13mm

 

 

-— 787mm 4

Figure 4.8 Cellular beam four-point bending test

For the four-point bending test the deflections of the beams at the center span and at the

cantilever tip were measured using externally mounted linear transducers. Strains at the

top tensile and compression fibers were measured using strain gages at the sections with

maximum moments (mid-span under the load and at the intermediate support) as shown

in Figure 4.8 and Figure 4.9. The strain gage readings, external displacements, and

applied loads were simultaneously recorded with a digital data acquisition system. The

results obtained from the four-point bending test and comparison of the flexural

performance of the base and optimized designs are discussed in the next section.

95

 

Figure 4.9 Overview of cellular beam four-point bending test

4.6 Observed Behavior and Test Results

4.6.1 Four-point bending

The load-deflection response fi'om the four point bending test of both the base and
optimal beams is shown in Figure 4.13 and Figure 4.14 respectively. The beams failed
under combined flexural and shear demands near the intermediate support (near the
cantilever end). This failure was most probably initiated by tensile cracking at the top of
the beam as shown in Figure 4.10. Due to manufacturing limitations, namely poor resin
injection, the end of the beam did not have adequate strength. Thus, significant crushing

of the cantilever end was observed as seen in Figure 4.1 1.

96

 

failure observed in four-point bending test

ined flexural and shear

4.10 Comb

Figure

 

Figure 4.1 1 Crushing of cantilever tip in four-point bending test

97

Only certain void sizes could be attained for both base and optimal beams as the metal
rods used for the cells are only available in standard sizes. Therefore, it was not possible
to achieve the same relative density in both cellular base design and optimal design.
Hence, to compare the performance of both designs, their response was normalized with
respect to their relative densities. The actual and normalized load displacement responses
and load strain responses are shown in Figures 4.13 through 4.18. In order to determine
the initial stiffness from the load displacement response of the two beams the linear
response region was identified qualitatively as indicated in Figure 4.12. Two points were
then selected on the tangent drawn along the linear variation where one of the points was
the origin. Initial stiffiress was calculated as force per unit displacement using Equation
4.1 where P and 5 were the load and displacement at the middle of the simply supported

span as shown schematically in Figure 4.12

 

P
K = — 4.1
6 ( )
y
l
1
i
e? 1 . K
‘8 ‘ L
3 1 Non-linear response
, / <~~— Proportional limit
‘/ —————— x
Displacement (5)

Figure 4.12 Measurement of initial stiffness in a non-linear response

98

The measured beam stiffness values were related to the beam sectional stiffness by
assuming elastic material response and relating beam displacements to the flexural
stiffness (E1) through simple beam theory. Theoretical values for E1 were obtained by
calculating the moment of inertia for the geometric layout of the beams and taking a
value of E (5230 N/mmz) from previous work (see Table 4.1) [28]. The measured and
theoretical values are summarized in Table 4.2, where it can be seen that the optimal
design beam was about 1.4 times stiffer than the base design as calculated fi'om the
normalized load-displacement response (Figure 4.14). The normalized flexural stiffness
value for the base design was 21% higher than the theoretical value, while the normalized
flexural stiffness value for the optimal design was 25% higher than the theoretical value.
The difference in the observed and theoretical values can be attributed to various reasons
such as manufacturing errors, like non-circular cells and non-uniform surface leading to
non-uniform cross-section, or wrong value of modulus for the elasticity which was taken
from previous work. The observed load-strain responses at the maximum negative
moment section for the two beams are shown in Figure 4.17 and observed values are
summarized in Table 4.3. The normalized failure strain and load response is shown in
Figure 4.18. As mentioned earlier in this section, a combined flexure-shear failure was
observed near the intermediate support (near cantilever end) in both the beams. Failure
was characterized by the appearance of inclined cracks which started at the top of the
section and proceeded towards the bottom. A drop in the load in the load-displacement
response was also observed at this instance. Also no apparent signs of damage were
observed near the maximum positive moment section. Thus, it was concluded that both

the beams experienced a combined flexure-shear failure near the negative moment

99

 

section. The difference in the failure strain values in the negative moment section was
most likely due to malfunctioning of the strain gage in the base cellular beam which was
observed fi'om the load-strain response. Otherwise the failure strain in the base cellular

beam should have been similar to that of the optimal beam.

100

 

Displacement (in.)
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

 

 

 

 

 

 

 

1.75.:Al LLAJJAJLJALI l AlllllllllLAIo.“
3 + Base Design
1.50 i + Optimal Desrgn 0.35
I 0.30
1.25 2‘ phase =0.53
1
A 1 popfimal=o.7 0.25 A
g 1.00 : a.
'o 1
g .
.l 0.75:
i
0.50 a
1
. 0.837P P
0.25 l g
'1 .L—‘——_ - 9 - _.
—~‘T";I“" r r":
6

 

 

Figure 4.13 Force-displacement response at midspan in four-point bending

Displacement (in.)
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

 

 

 

 

 

 

 

Inn. AJLJAA ALLLALL4IIIAALLALAILLA LLLlAAAA'o.60
2-50 + Base Design 0.55
A 2.25 + Optimal Desrgn 0.50 A
'2’ E
2- 2.00 0.45 3.
9; phase =0.53 2:
1.75 _ 0.40
2 poptimal-o'7
x 1.50
'3 1.25
3
u 1.00
1:
o
g 0.75 a!»
E 0.837? p a
0.50 l
o 1; . , E
z 0.25 a _ . “2:. 2
5
0.00

Figure 4.14 Normalized force-displacement response at midspan in four-point
bending

101

Displacement (in.)

 

 

 

 

 

0.00 0.05 0.10 0.15 0.20 0.25
9L A A A l A A A A l A A A l A L A A l A A A A I 0.25
1 0 + Base Design
' + Optimal Design

pbase =0.53
i poptimal=0'7
5
1:
a
O
.1

0.837 P P

...a p...—

 

Figure 4.15 F orce-displacement response at cantilever tip in four-point bending

 

 

 

 

 

 

 

Displacement (in.)
0.00 0.05 0.10 0.15 0.20 0.25
l A A A A l A 1 A A 1 A A A 4 L A A A l A A L I 0.35
1 4 + Base Design
+ Optimal Design 030
’73.. . 9 - f 3
2-
& pm, =0.53 g
poptimal=0' 7
1: 13
.8 .8
E _
E 0.837 P P E
O l 1 O
2 L—_ _ . a z
5 {if _>‘ ~‘ ——*“—‘B *‘ "i—fi

Figure 4.16 Normalized force-displacement response at cantilever tip in four-point
bending

102

Table 4.2 Comparison of experimental and theoretical flexural stiffness from four-

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

point bending test
Theoretical Measured Measured
Cellular Error Error
E1 E1 Normalized EI
design (%) (%)
N-mm4 N-mm4 N-mm4
Base 826 850 24% 1000 21%
Optimal 1130 1510 33% 1420 25%
Strain (microstrains)
0 1000 2000 3000 4000 5000 0000
‘ A A A A l A A A A l A A A A l A A A l A g A_A L A A A A F 0.30
1'25 j -—A— Base Design :
+ Optimal Design -
: 0.25
1-00 ‘ Strain gage location: r
J Top left, above the support 30.20
.5. 0.751 phase =0'53 ' =3
U ‘ popu'malzo'7
G A
3 .
0.50 ~
, 0.837 P P
“5 1 _L _ 5.0- - -A s l _
0.00 é

Figure 4.17 Load-strain response at extreme tension fiber on maximum negative

moment section

103

Strain (microstrains)

 

 

 

 

 

 

 

0 1000 2000 3000 4000 5000 6000
1.75 2‘ A A A A l A A A A l A A A A L+ A P; l A L 4 A l A A A A L 0.40
3 + Base Design
150 _ + Optimal Desrgn 0.35
e . m g . '=
S- 1 25 J Straln e loca on. 0.30 $3
9.: ' : Top left, above the support 9;,
E 1.00; phase ‘ '
g . poptimal=o'7
3 0.75:
g 050 j g
= ' T 0.837P p =
8 ‘ 1 1 g
I , so
3 0.25 - _ " L 3
z 7 « ~— ..................... = z

 

0.00 k
0

Figure 4.18 Normalized load-strain response at extreme tension fiber on maximum
negative moment section

Table 4.3 Comparison of failure strain and load measured at extreme tension fiber in
negative moment section

 

 

 

Measured Normalized
Failure strain
Cellular design load load
(microstrains)
(N) (N)
Base 4100 623 1201
Optimal 4950 1245 1690

 

 

 

 

 

 

104

 

4.3.2 Three-point bending

After the end crushing of the cantilever tip started during the four-point bending test
loading the beam all the way up to failure wouldn’t have yielded any useful information.
Thus, in order to obtain the response up to failure the simply supported span of both base
and optimal beams was tested in three-point bending after removing the cantilever part.
The beams had a span of 660 mm and the load was applied at the center of the span. The
test set-up for three-point bending is shown in Figure 4.19. Both the beams were loaded
monotonically up to failure in displacement control at a rate of 0.01 min/sec for base
design and at a rate of 0.025 mm/sec for optimized design. The deflection was measured

under the centrally applied load using an externally mounted extensometer as shown in

 

 

 

 

 

Figure 4.20.
P
Extensometer
, A, A, , , , i
. __ 7 A2 _L
FT U :
25 mm « é 610mm 4 :25 mm
- ~ ~ ~———-—-- ——-—----~~-~—— 660 mm —~--

 

Figure 4.19 Cellular beam three-point bending test

The results of the three-point bending test of cellular beams show that both the beams,
base and optimal, behaved non-linearly up to failure (Figure 4.21) afier a brief initial

linear response. The normalized load-displacement response (Figure 4.22) showed that

105

the optimal beam was about 1.12 times stiffer than the base design (Table 4.4). The beam

experienced flexural failure under the central load (Figure 4.23).

 

Figure 4.20 Cellular beam three-point bending test

Displacement (in.)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 11

 

+ Base Design
+ Optimal Design

1.8 /
pm, =0.53

popm=0.7

 

 

 

 

 

°°i~

0.4

Figure 4.21 Load-displacement response at midspan for three-point bending

106

Displacement (in.)
0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.7 0.8 0.9 1.0 1.1

IAAAAIAAAAI

 

 

 

 

 

 

 

 

 

 

. “ 0.7
3.0 1 . :
3 + Base Desugn .
2.7 -j + Optimal Design L 05
E 2.4-j E
5. 2...: p... =0-53 :°-5 5
A _ poptimal=0'7 : a
g 1-8 : :0.4 i
g 1.5-; : 'g
3 z -0.3 3
'8 1'23 I 'U
1 _ O
._u_ 0.9 1 - 0.2 E
a 1 i a
E 0.6; I" . E
O 2 [ 1.0.1 2
z 0.33 — _ , . 1’
0.0 ""l""l""l""l"-IIl'frrfirvrét'rr'vwrrr'v--o.°
3 0 9 12 15 10 21 24 27 30

Displacement (mm)

Figure 4.22 Normalized load-displacement response at midspan for three-point
bending

 

Figure 4.23 Flexural failure in three-point bending

107

Table 4.4 Experimental stiffness from three-point bending test and theoretical stiffness
for base and optimal designs

 

 

 

Measured Measured
Cellular Theoretical Error - , Error
EI Normalized E1
design EI (%) (%)
N-mm4 N-mm4
Base 826 353 57% 686 17%
Optimal 1 130 6250 45% 7690 32%

 

 

 

 

 

 

 

 

The sectional stiffness (E1) values obtained from the normalized load displacement
responses in four and three point bending for the base and optimal designs were not in
close agreement as expected. This could be attributed to various reasons such as
significant deformation in the cross-sections of the beams after the four-point bending
test (due to local deformations from the test set-up) and manufacturing errors (like non-
circular cells or non-uniform surface leading to non-uniform beam dimensions).

In spite of the discrepancies between the experimental and calculated values, the test
results of both four-point bending and three point bending indicated that the optimal

design had better flexural performance than the selected base design.

108

 

5. CONCLUSIONS

Results from the analytical studies on the sizing and discrete layout optimization
approaches showed that both optimization methods are viable for optimizing material
distribution in cross-sections of biocomposite panels. Both methods led to designs with
finite-sized features and thus the designs were easier to manufacture as compared to most
of the conventional topology optimization methods. These two methods were also able to
handle multiple objectives and constraints. The discrete layout optimization method was
also successful in using hybrid material in the cross-section, unlike conventional topology
optimization methods. The experimental evaluation program also showed how an
optimized cellular biocomposite beam had almost 1.4 times higher flexural stiffness than
a base design through a four-point continuous bending test, and about 1.12 times higher
flexural stiffness in a three-point bending test. The failure load of the optimized beam
was 1.9 times higher than the base periodic cellular design. In particular, the findings in
this study have led to the following conclusions regarding optimized hierarchical cellular
biocomposite beams:

o The sizing and discrete layout optimization approaches using finite-size
features led to designs that had higher stiffness and specific stiffness, were
feasible to manufacture with natural fibers and could successfully handle
multiple objectives, multiple constraints and hybrid material in the cross-
section.

0 The optimal hybrid and hierarchical cellular beam designs with finite-
sized features obtained from discrete layout optimization were shown to

have higher stiffness and thus demonstrated better flexural performance

109

 

than a periodic cellular beam when tested in continuous and simple
bending.
Based on the findings and limitations observed in the optimization approaches presented
in this thesis following recommendations are provided for future work:

0 Different geometric shapes can be used as perforations in the cellular
designs, besides circular perforations, using both sizing and discrete layout
optimization methods.

0 Other objective functions, such as strain energy should be explored using
both the sizing optimization and the discrete layout optimization
approaches. Non-uniform strain distribution in the cross-section should
also be incorporated in the problem formulation. A finite-element module
can be coupled with both optimization approaches for calculation of
objective fimctions such as strain energy.

0 An algorithm with multi-start capability needs to be used for the sizing
optimization approach so that more initial starting points can be explored
to enhance chances of reaching a global optimum.

o The layout optimization algorithm can be modified to incorporate different .
scales or coupon sizes in the original coupon library instead of relying on
the coalescence approach alone. The coalescence concept can still be used
to continuously merge repeating cells throughout the optimization process.

0 If different scales can be successfully incorporated in the optimization
process, as mentioned above, then further heuristics for coupon

subdivision or splitting can also be investigated in addition to coalescence.

110

 

More knowledge-based operators could be used in the layout optimization
approach to preserve the diversity of the designs, thus enhancing the

chances of finding a global optimum.

lll

 

10.

11.

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