Development of meta heuristic of optimization methods for mechanics problems

The modified algorithms are then combine with ANN and/or SLDM to develop new tools for solving design optimization problems and the RBDO problems of composite structures, such as stiffen

MAJOR: ENGINEERING MECHANICS

Ph.D THESIS LAM PHAT THUAN

DEVELOPMENT OF META-HEURISTIC OPTIMIZATION

METHODS FOR MECHANICS PROBLEMS

S K A0 0 0 0 2 9

MINISTRY OF EDUCATION AND TRAINING

HO CHI MINH CITYUNIVERSITY OF TECHNOLOGY AND EDUCATION

MINISTRY OF EDUCATION AND TRAINING

HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY AND EDUCATION

PHD THESIS MAJOR: ENGINEERING MECHANICS

Ho Chi Minh City, 05/2021

LAM PHAT THUAN

DEVELOPMENT OF META-HEURISTIC OPTIMIZATION

METHODS FOR MECHANICS PROBLEMS

THE WORK IS COMPLETED AT

HCM CITY UNIVERSITY OF TECHNOLOGY AND EDUCATION

LAM PHAT THUAN

DEVELOPMENT OF META-HEURISTIC OPTIMIZATION

METHODS FOR MECHANICS PROBLEMS

MAJOR: ENGINEERING MECHANICS - 13252010105

Supervisor 1: Assoc Prof NGUYEN HOAI SON

Supervisor 2: Assoc Prof LE ANH THANG

PhD thesis is protected in front of EXAMINATION COMMITTEE FOR PROTECTION OF DOCTORAL THESIS HCM CITY UNIVERSITY OF TECHNOLOGY AND EDUCATION

Date……month……year……

Originality Statement

I, Lam Phat Thuan, hereby assure that this thesis is my own work

The data and results stated in this thesis are honest and have not been published

by any works

Ho Chi Minh City, May 2021

Lam Phat Thuan

Acknowledgements

This thesis has been carried out in the Faculty of Civil Engineering, HCM City University of Technology and Education, Viet Nam The process of conducting this thesis brings excitement but has quite a few challenges and difficulties And I can say without hesitation that it has been finished thanks to the encouragement, support and help of my professors and colleagues

First of all, I would like to express my deepest gratitude to Assoc Prof Dr Nguyen Hoai Son and Assoc Prof Le Anh Thang, especially Assoc Prof Dr Nguyen Hoai Son from GACES Group, Ho Chi Minh City University of Technology and Education, Vietnam for having accepted me as their PhD student and for the enthusiastic guidance and mobilization during my research

Secondly, I would like also to acknowledge Msc Ho Huu Vinh for his troubleshooting and the cooperation in my study Furthermore, I am grateful to Faculty of Civil Engineering for their great support to help me have good environment to do my research

Thirdly, I take this chance to thank all my nice colleagues at the Faculty of Civil Engineering, Ho Chi Minh City University of Technology and Education, for their professional advice and friendly support

Finally, this thesis is dedicated to my parents who have always given me valuable encouragement and assistance

Lam Phat Thuan

Abstract

Almost all design problems in engineering can be considered as optimization problems and thus require optimization techniques to solve During the past few decades, many optimization techniques have been proposed and applied to solve a wide range of various optimization problems Among them, meta-heuristic algorithms have gained huge popularity in recent years in solving design optimization problems of many types of structure with different materials These meta-heuristic algorithms include genetic algorithms (GA), particle swarm optimization (PSO), bat algorithm (BA), cuckoo search (CS), differential evolution (DE), firefly algorithm (DA), harmony search (HS), flower pollination algorithm (FPA), ant colony optimization (ACO), bee algorithms (BA), Jaya algorithm and many others Among the methods mentioned above, the Differential Evolution is one of the most widely used methods Since it was first introduced in 1997 by Storn and Price, many studies have been carried out to improve and apply DE in solving structural optimization problems The DE has demonstrated excellently performance in solving many different engineering problems Besides the Differential Evolution algorithm, the Jaya algorithm recently proposed by Rao in 2016 is also an effective and efficient methods that has been widely applied to solve many optimization problems and showed its good performance It gains dominate results when being tested with benchmark test functions in comparison with other meta-heuristic methods However, like many other population-based optimization algorithms, one of the disadvantages

of DE and Jaya is that the computational time obtaining optimal solutions is much slower than the gradient-based optimization methods This is because DE and Jaya takes a lot of time evaluating the fitness of individuals in the population To overcome this disadvantage, Artificial Neuron Networks (ANN) are studied to combine with the meta-heuristic algorithms, such as Differential Evolution, to form a new approach which has the ability to solve the design optimization effectively Moreover, one of the most important issues in engineering design is that the optimal designs are often

manufacturing processes, material properties and operating environments These uncertainties may cause structures to improper performance as in the original design, and hence may result in risks to structures Therefore, reliability-based design optimization (RBDO) can be considered as an important and comprehensive strategy for finding an optimal design

In this thesis, an improved version of Differential Evolution has been first time utilized to solve for optimal fiber angle and thickness of the stiffened composite Secondly, the Artificial Neural Network is integrated to the optimization process of the improved Differential Evolution algorithm to form a new algorithm call ABDE (ANN-based Differential Evolution) algorithm This new algorithm is then applied to solve optimization problems of the stiffened composite plate structures Thirdly, an elitist selection technique is utilized to modify the selection step of the original Jaya algorithm to improve the convergence of the algorithm and formed a new version of the original Jaya called iJaya algorithm The improved Jaya algorithm is then applied

to solve for optimization problem of the Timoshenko composite beam and obtained very good results Finally, the so-called called (SLMD-iJaya) algorithm which is the combination of the improved Jaya algorithm and the Global Single-Loop Deterministic Methods (SLDM) has been proposed as a new tool set for solving the Reliability-Based Design Optimization problems This new method is applied to look for optimal design of Timoshenko composite beam structures with certain level of reliability

Contents

Originality Statement i

Acknowledgements i

Abstract ii

Contents iv

Nomenclature vii

List of Tables xii

List of Figures xiii

Chapter 1: LITERATURE REVIEW 1

1.1 An overview on research direction of the thesis 1

1.2 Motivation of the research 6

1.3 Goals of the thesis 6

1.4 Research scope of the thesis 7

1.5 Outline 7

1.6 Contributions of the thesis 9

Chapter 2: FUNDAMENTAL THEORY OF COMPOSITE STRUCTURE IN DESIGN AND OPTIMIZATION 10

2.1 Introduction to Composite Materials 10

2.1.1 Basic concepts and applications of Composite Materials 10

2.1.2 Overview of Composite Material in Design and Optimization 16

2.2 Analysis of Timoshenko composite beam 18

2.2.1 Exact analytical displacement and stress 18

2.2.2 Boundary-condition types 22

2.3 Analysis of stiffened composite plate 23

Chapter 3: RELIABILITY-BASED OPTIMIZATION METHODS WITH IJAYA AND IMPROVED DIFFERENTIAL EVOLUTION 29

3.1 Overview of Metaheuristic Optimization 29

3.1.1 Meta-heuristic Algorithm in Modeling 30

3.2 Solving Optimization problems using improved Differential Evolution

43

3.2.1 Brief on the Differential Evolution algorithm [12], [128] 43

3.2.2 The modified algorithm Roulette-Wheel-Elitist Differential Evolution 44

3.3 Solving Optimization problems using improved Jaya algorithm 45

3.3.1 Jaya Algorithm 45

3.2.2 Improvement version of Jaya algorithm 47

3.4 Reliability-based design optimization using a global single loop deterministic method 48

3.4.1 Reliability-based optimization problem formulation 50

3.4.2 A global single-loop deterministic approach 51

Chapter 4: FUNDAMENTAL THEORY OF NEURAL NETWORK 55

4.1 Fundamental theory of Neural Network 55

4.1.1 Basic concepts on Neural Networks [145] 57

4.1.2 Neural Network Structure 58

4.1.3 Neural Network Design Steps 62

4.1.4 Levenberg-Marquardt training algorithm 63

4.1.5 Over fitting, Over training 65

4.2 Artificial Neural Network based meta-heuristic optimization methods 67 Chapter 5: DEVELOPMENTS OF META-HEURISTIC OPTIMIZATION METHODS 70

5.1 Verification of iDE algorithm 70

5.1.1 A 10-bars planar truss structure: 70

5.1.2 A 200-bars truss structure Error! Bookmark not defined 5.1.3 A 72-bar space truss structure 74

5.1.4 A 120-bar space truss structure: 77

5.2 Static analysis of the stiffened composite plate 79

5.3 The effective of the improved Differential Evolution algorithm 81

5.4 Optimization of stiffened composite plate 82

5.4.1 Thickness optimization of stiffened Composite plate 82

5.4.2 Artificial neural network-based optimization of stiffened composite plate 84

5.5 Deterministic optimization of composite beam 87

5.5.1 Optimal design with variables: b and h 89

5.5.2 Optimal design with variables: b and t i 92

5.6 Reliability-based optimization design of Timoshenko composite beam 95 5.6.1 Verification of SLDM-iJaya 96

5.6.2 Reliability-based lightweight design 97

Chapter 6: CONCLUSIONS AND RECOMMENDATIONS 102

6.1 Conclusions and Remarks 102

6.2 Recommendations and future works 106

REFERENCES 107

List of Publications 122

Nomenclature

Latin Symbols

st

y

q(x) Transversal force on the composite beam

z

u(x), w(x) Displacement field of the composite beam

P

st

 Shear correction factor

b

κ Bending strains of composite plate

xx

yy

γ Shear strains of composite plate

ABDE Artificial neural network-Based Differential Evolution

Evolutionary Algorithm

CS-DSG3 Cell-Smoothed Discrete Shear Gap technique using

triangle finite element

NNs Neural Networks

List of Tables

Table 5 1 Parameters for 10 bars truss 71

Table 5 2 The comparison results keep the solution from the improved DE algorithm with other methods for the 10-bar flattening problem 72

Table 5 3 Parameter for 200-bars truss structure 74

Table 5 4 Results of the comparison between the solution from the improved DE algorithm and other methods for the problem of optimizing the 200-bar scaffold problem 75

Table 5 5 Parameters for 72-bars space truss structure 76

Table 5 6 Comparison between the solution from iDE algorithm with other methods for the 72-bars space truss problem 77

Table 5 7 Parameters for 120-bars arch space truss structure 78

Table 5 8 Results of comparison of solutions from the improved DE algorithm with other methods for the optimization problem of space bar of 120 bars 79

Table 5 9 Comparison of central deflection (mm) of the simply-supported square stiffened composite plates 80

Table 5 10 The optimal results of two problems 82

Table 5 11 Optimal thickness results for stiffened composite plate problems 84

Table 5 12 Sampling and overfitting checking error 85

Table 5 13 Comparison of the accuracy and computational time between DE and ABDE 86

Table 5 14 Material properties of lamina 89

Table 5 15 Comparison of optimal design with continuous design variables 90

Table 5 16 Comparison of optimal design with discrete design variables 92

Table 5 17 Comparison of optimization results of the mathematical problem 96 Table 5 18 Optimal results of reliability based lightweight design with different

List of Figures

Figure 2 1 Types of fiber-stiffened composites 12

Figure 2 2 Boeing 787 - first commercial airliner with composite fuselage and wings (Courtesy of Boeing Company.) 13

Figure 2 3 Composite mixer drum on concrete transporter truck weighs 2000 lbs less than conventional steel mixer drum 14

Figure 2 4 Pultruded fiberglass composite structural elements (Courtesy of Strongwell Corporation.) 15

Figure 2 5 Composite wind turbine blades (Courtesy of GE Energy.) 15

Figure 2 6 Composite laminated beam model 19

Figure 2 7 Free-body diagram 19

Figure 2 8 The material and laminate coordinate system 20

Figure 2 9 A composite plate stiffened by an r-direction beam 24

Figure 3 1 Source of inspiration in meta-heuristic optimization algorithms 34

Figure 3 2 Illustration of the feasible design region 52

Figure 4 1 Biological neuron 55

Figure 4 2 Perceptron neuron of Pitts and McCulloch 56

Figure 4 3 Applying a model based on field data 57

Figure 4 4 The relationship between Machine Learning and the neural network 58

Figure 4 5 A Multi-layer perceptron network model 59

Figure 4 6 Single node in an MLP network 59

Figure 4 7 Tanh and Sigmoid function 60

Figure 4 8 A multi-layer perceptron with one hidden layer Both layers use the same activation function g 61

Figure 4 9 Diagram for the training process of a neural network with the Levenberg-Marquardt algorithm 65

Figure 4 10 Dividing the training data for the validation process 67

Figure 4 11 Optimization process using Artificial Neural Network (ANN) based

Differential Evolution (ABDE) optimization algorithm 69

Figure 5 1 A 10-bars truss structure 71

Figure 5 2 A 200 bars truss structure 73

Figure 5 3 A 72-bars space truss structure 76

Figure 5 4 Structure of 120-bars arch space truss 78

Figure 5 5 Model of a stiffened composite plate 79

Figure 5 6 Models of square and rectangular stiffened composite plates 81

Figure 5 7 Model of stiffened composite plate for optimization 83

Figure 5 8 Convergence curves of DE, IDE, Jaya and iJaya for the beam with P-P condition 91

Figure 5 9 Convergence curves of DE, IDE, Jaya and iJaya for the beam with P-P condition 94

Figure 5 10 Comparison of different design approaches with different boundary conditions 95

Figure 5 11 Comparison of RBDO optimal results with different levels of reliability 100

Chapter 1: LITERATURE REVIEW

1.1 An overview on research direction of the thesis

Almost all design problems in engineering can be considered as optimization problems and thus require optimization techniques to solve However, as most real-world problems are highly non-linear, traditional optimization methods usually do not work well The current trend is to use evolutionary algorithms and meta-heuristic optimization methods to tackle such nonlinear optimization problems Meta-heuristic algorithms have gained huge popularity in recent years These meta-heuristic algorithms include genetic algorithms, particle swarm optimization, bat algorithm, cuckoo search, differential evolution, firefly algorithm, harmony search, flower pollination algorithm, ant colony optimization, bee algorithms, Jaya algorithm and many others The popularity of meta-heuristic algorithms can be attributed to their good characteristics because these algorithms are simple, flexible, efficient, adaptable and yet easy to implement Such advantages make them versatile to deal with a wide range of optimization problems, especially the structural optimization problems [1] Structural optimization is a potential field and has attracted the attention of many researchers around the world During the past decades, many optimization techniques have been proposed and applied to solve a wide range of various problems The algorithms can be classified into two main groups: gradient-based and popular-based approach Some of the gradient-based optimization methods can be named here as sequential linear programming (SLP) [2], [3], sequential quadratic programming (SQP) [4], [5], Steepest Descent Method, Conjugate Gradient Method, Newton's Method [6] The gradient-based methods are very fast in reaching the optimal solution, but easy trapped in local extrema and requires the gradient information to construct the searching algorithm Besides, the gradient-based approaches are limited

to continuous design variables and that decreases the productivity of the algorithm

In addition, the initial solution (or initial design parameters of the structure) also greatly affects the ability to achieve global or local solutions of gradient-based

algorithms The population-based techniques, also known as part of meta-heuristic algorithms, can be listed such as genetic algorithm (GA), differential evolution (DE), and particle swarm optimization (PSO), Cuckoo Search (CS), Firefly Algorithm (FA), etc [7] These methods are used extensively in structural problems because of their flexibility and efficiency in handling both continuous and discontinuous design variables In addition, the solutions obtained from population-based algorithms in most cases are global ones Therefore, the optimal result of the problem is not too much influenced by the initial solution (or initial design of the structure) Among the methods mentioned above, the Differential Evolution is one of the most widely used methods Since it was first introduced by Storn and Price [8], many studies have been carried out to improve and apply DE in solving structural optimization problems The

DE has demonstrated excellently performance in solving many different engineering problems Wang et al [9] applied the DE for designing optimal truss structures with continuous and discrete variables Wu and Tseng [10] applied a multi-population differential evolution with a penalty-based, self-adaptive strategy to solve the COP

of the truss structures Le-Anh et al [11] using an improved Differential Evolution algorithm and a smoothed triangular plate element for static and frequency optimization of folded laminated composite plates Ho-Huu et al [12] proposed a new version of the DE to optimize the shape and size of truss with discrete variables Besides the Differential Evolution algorithm, the Jaya algorithm recently proposed

by Rao [13] is also an effective and efficient methods that has been widely applied to solve many optimization problems and showed its good performance It gains dominate results when being tested with benchmark test functions in comparison with other population-based methods such as Homomorphous Mapping (HM), adaptive segregational constraint handling evolutionary algorithm (ASCHEA), simple multi-membered evolution strategy (SMES), genetic algorithm (GA), particle swarm optimization (PSO), differential evolution (DE), artificial bee colony (ABC), biogeography based optimization (BBO) Moreover, it has been also successfully

following literature [14]–[16] However, the performance of the original Jaya algorithm is not really high Therefore, there are many variations of the Jaya algorithm proposed to improve the original one In this thesis, a new improved version of the Jaya algorithm will be presented The new algorithm aims to improve the population selection technique for the next generation in order to improve the speed of convergence, while at the same time ensuring the accuracy and the balance between the exploration and exploitation of Jaya algorithm

Moreover, like many other population-based optimizations, one of the disadvantages

of DE and Jaya is that the optimal computational time is much slower than the gradient-based optimization methods This is because DE and Jaya takes a lot of time

in evaluating the fitness of individuals in the population Specifically, in the structural optimization problem, the calculation of the objective function or constraint function values is usually done by using the finite element to analyze the structural response

To overcome this disadvantage, artificial neuron networks (ANN) are proposed to combine with the DE algorithm Based on the idea of imitation of the brain structure, ANN is capable of approximating an output corresponding to a set of input data quickly after the network has been trained, also known as a learning process Thanks

to this remarkable advantage, the computation of objective function or constraint function values in the DE algorithm will be done quickly As a result, ANN will help significantly improve the efficiency of DE calculations The effectiveness and applicability of ANN since the early groundwork ideas put forward by Warren McCulloch and Walter Pitts [17] in 1943 have so far proved to be very convincing through numerous studies Application areas include system identification and control, pattern recognition, sequence recognition (gesture, speech, handwritten text recognition), data mining, visualization, machine translation, social networking filtering and email spam filtering, etc [18]–[23]

The next issue is the development of optimal algorithms integrated ANN with DE and applying the proposed algorithms to a practical structure to examine the effectiveness of the method At present, the structures made from composite material

are widely used in almost all fields such as construction, mechanical engineering, marine, aviation, etc In particular, beams and stiffened plates made of composite material are an outstanding form and are used increasingly by its superior advantages

By combining the advantages of composite materials and the stiffened beams structure, the stiffened composite plates have very high bending strength with very light weight Nowadays, stiffened composite plates have been widely used in many branches of structural engineering such as aircraft, ships, bridges, buildings, etc For its advantages in both bending stiffness and the amount of material in comparison with common bending plate structures, stiffened composite plate usually has higher economic efficiency in practical applications Owning to its high practical applicability, the need to optimize the design of the structure to save costs and increase the efficiency of use is also high However, because of the complexity of computing the behavior of this particular type of structure, finding a good algorithm for optimizing design parameters is essential to ensure computational efficiency Composite material structures have very complex behavioral equations, influenced

by many geometric and material parameters These characteristics of the composite mechanical system also lead to the complexity of the system of equations to describe the optimal problems, from the objective functions to the constrained equations So the use of gradient-based algorithms is not straightforward For such types of problems, population-based methodologies are a superior choice

Moreover, one of the most important issues in engineering design is that the optimal designs are often effected by uncertainties which can be occurred from various sources, such as manufacturing processes, material properties and operating environments These uncertainties may cause structures to improper performance as

in the original design, and hence may result in risks to structures [24] There are two groups of methods for dealing with uncertainties: reliability-based design and robust design Robust design focuses on minimizing variance in design results under variations of design variables and parameters Reliability-based design optimization

and parameters RBDO can be considered as a comprehensive strategy for finding an optimal design RBDO is the focus of this thesis Although RBDO is more reliable than static optimization, the biggest drawback of RBDO in practical application is the high computational cost To solve this problem, a lot of research has been done to find effective reliability analysis techniques, such as: sensitivity-based approximation approaches [25], [26], most probable point (MPP)-based approaches, Monte Carlo simulations [27]–[29] and response surface model-based approaches [30] These techniques focus on nesting the optimization and the reliability assessment in one process Another RBDO research focus on exploring the efficient decoupling strategies These strategies can be divided into three groups: nested double-loop methods, decouple-methods, and single-loop methods Among these three categories, the double-loop approaches may be the most accurate as it assesses the reliability in every iteration during the optimization process However, its limitation is the huge cost of computation [31]–[33] The decoupled methods solve the RBDO problem in

a different way by separating the optimization and reliability analysis and solve them sequentially Hence, the computational cost can be reduced considerably [31], [33]–[35] However, this approach still includes two interrelated loops that result in costly computation To overcome this drawback, the single-loop methods have been proposed In this approach, the RBDO problem is solved in a single-loop procedure without reliability analysis The strategy is to convert an RBDO problem into an approximate deterministic optimization (ADO) problem by transforming probabilistic constraints into approximate deterministic constraints In so doing, the computational cost significantly decreased [32], [36], [37] Therefore, these methods would be applicable to real-world problems However, studies that deal with the reliability-based design optimization of laminated composite beams are quite limited

In this thesis, the Single-Loop Deterministic Methods (SLDM), which has been

recently proposed by Li et al [38], will be studied to integrate with a meta-heuristic

optimization algorithm to form a new tool set SLDM-iJaya for solving a RBDO problems of composite structures

In summary, in this thesis, some modifications will be investigated and propose to improve the original algorithm of Differential Evolution and Jaya algorithm to increase the convergence of DE and Jaya algorithm The modified algorithms are then combine with ANN and/or SLDM to develop new tools for solving design optimization problems and the RBDO problems of composite structures, such as stiffened composite plate, Timoshenko beams, etc

1.2 Motivation of the research

The motivation to study the topics presented in the thesis comes from the analysis of published literatures, and from the evaluation of the application potential of composite material structures and intelligent optimization methods, especially the reliability-based optimization methods Therefore, the thesis is motivated by:

- The development / improvement of existing algorithms to improve the efficiency of solving structural optimization problems with high accuracy and reliability

- Studying the advantages of Artificial Neural Network (ANN) to combine with optimal algorithms to improve the speed and the performance of solving structural

optimization problems

1.3 Goals of the thesis

Firstly, this thesis focuses on studying and developing meta-heuristic optimization methods and combines them with the Artificial Neural Network, which has advantages in approximating data, to build up a new algorithm for solving composite material structural optimization problems Particularly, the original Differential Evolution or Jaya algorithm will be modified to improve the convergence in solving for global optimal solution and then, the ANN will be integrated to the improved meta-heuristic algorithms to form a new algorithm, which is used to look for optimal design of stiffened composite plate structures

Secondly, the thesis also proposes a new tool set, which is the combination of heuristic optimization algorithm and the Single-Loop Deterministic Method to deal with Reliability-Based Design Optimization (RBDO) problems In particular, the

meta-optimal solutions of the optimization problems Then, this improved version of Jaya algorithm will be combined with Single-Loop Deterministic Method to solve the

Reliability-Based Design Optimization of composite beam structures

1.4 Research scope of the thesis

The thesis focuses on the following main issues:

- To optimize truss, beam and stiffened plate structures using steel and composite materials

- To study and improve population-based optimization methods to increase accuracy and efficiency in solving optimization problems

- To exploit the ability to create approximate models from data sets of Neural Network to combine with optimal algorithms to improve the performance and the ability to solve many different types of problems

- To combine optimal algorithms with groups of reliability assessment methods to solve RBDO problems

- The problems selected for optimization are relatively simple with the main purpose

of evaluating the effectiveness, accuracy and reliability of the proposed optimization methods The application of optimal methods proposed in the thesis for more complex

problems will be further studied in the future

1.5 Outline

The thesis contains six chapters and is structured as follows:

 Chapter 1 presents an overview on meta-heuristic algorithms, composite material structure and especially artificial neural networks and its role and application in optimization process This chapter also give out the organization

of the thesis via the outline section and the novelty and goal of the thesis for quick review of what is studied in this thesis

 Chapter 2 provides an overview of composite material with basic concepts and applications in real life The chapter also introduce theory of Timoshenko composite beam and stiffened composite plate which are the main structure under investigated and studied in optimization problems of this thesis

 Chapter 3 devotes the presentation of meta-heuristic optimization related to Differential Evolution and Jaya algorithm and the approach to modify and improve the original of the algorithm to obtain an improved version of its This chapter also gives out an overview and formulation for Reliability-Based Design Optimization (RBDO) and the proposed methods for solving RBDO problem

 Chapter 4 offers the introduction and the historical development of Artificial Neural Network (ANN) This chapter gives out some basic concepts related to ANN and introduce the Neural Network Structure which is used in this thesis

to approximate date generated from the Finite Element Analysis Moreover, the training algorithm, especially the Levenberg-Marquardt and the overfitting phenomenon are also presented in this chapter

 Chapter 5 illustrate the effectiveness and efficiency of the improve Differential Evolution and the improve Jaya in solving optimization problems The structures investigated in this section includes planar truss structure, space truss structure, Timoshenko composite beam and stiffened composite plate In particular, the improve Differential Evolution (iDE) is applied to solve for optimal weight of planar truss structures and space truss structures, then it is used to optimize the fiber angle and the thickness of stiffened composite plates and show its good effectiveness and performance The last part of this chapter devotes to illustration of the improve Jaya algorithm in looking for optimal design of the Timoshenko composite beam and the results obtained prove its highly effective performance and accuracy compared with those of others’ author Moreover, this chapter also presents a new approach called SLDM-iJaya which is formed by the combination of the improve Jaya algorithm and the single-loop methods for solving the RBDO problem of the Timoshenko composite beam This chapter illustrate the solutions for two problems, the first one solving a common optimization problem without the reliability index,

problems are compared and analyzed with those of other authors and show the effectiveness and the accuracy of the proposed SLDM-iJaya algorithm Afterward, this chapter presents the application of Artificial Neural Network when it is integrated to a meta-heuristic optimization method, such as Differential Evolution algorithm, to solve the optimization problems The integration form a new tool set call ABDE (ANN-Based Differential Evolution) algorithm and applied to solve for optimal design of the stiffened composite plate The results not only prove the effectiveness of the proposed method but also open new aspect of applications for future works

 Finally, Chapter 6 closes the concluding remarks and give out some recommendations for future work

1.6 Contributions of the thesis

- In this thesis, an improved version of Jaya algorithm (iJaya) is introduced This algorithm is developed by modifying the selection step of the original Jaya algorithm using the elitist selection technique The iJaya algorithm is then applied to solve design optimization problems of composite structures and obtained very good results

- The so-called (SLMD-iJaya) algorithm which is the combination of the improved Jaya algorithm and the Global Single-Loop Deterministic Methods (SLDM) has been proposed to solve the Reliability-Based Design Optimization problems of the continuous composite beam models

- The Artificial Neural Network is integrated to optimization process of the iDE and iJaya algorithm to form a new algorithm called ABDE and AB-iJaya algorithm These new algorithm is then applied to solve optimization problems of the stiffened composite plate structures The results obtained show a highly effective performance

of the new algorithms

Chapter 2: FUNDAMENTAL THEORY OF COMPOSITE STRUCTURE IN

DESIGN AND OPTIMIZATION

2.1 Introduction to Composite Materials

2.1.1 Basic concepts and applications of Composite Materials

Structural materials can be classified into four categories: metals, ceramics, polymers, and composites Composites are typically made from various combinations of the other materials In the early days of modern man-made composite materials, the constituents were typically macroscopic As composites technology advanced over the last few decades, the constituent materials, particularly the reinforcement materials, steadily decreased in size Most recently, there has been considerable interest in “nano-composites” having nanometer-sized reinforcements such as carbon nanoparticles, nano-fibers, and nanotubes, because of the extraordinary properties of these materials Composites are generally used because they have desirable properties that cannot be achieved by any of the constituent materials acting alone The most common example is the fibrous composite consisting of reinforcing fibers embedded

in a binder or matrix material Particle or flake reinforcements are also used, but they are generally not as effective as fibers Some example of composite can easily find in the nature For example, Wood consists mainly of fibrous cellulose in a matrix of lignin, whereas most mammalian bone is made up of layered and oriented collagen fibrils in a protein–calcium phosphate matrix [39] Fibrous reinforcement is very effective because many materials are much stronger and stiffer in fiber form than they are in bulk form It is believed that this phenomenon was first demonstrated scientifically in 1920 by Griffith [40], who measured the tensile strengths of glass rods and glass fibers of different diameters Griffith found that as the rods and fibers got thinner, they got stronger, apparently because the smaller the diameter, the smaller the likelihood that failure-inducing surface cracks would be generated during fabrication and handling Results similar to those published by Griffith have been

fiber and bulk behavior There can be no doubt that fibers allow us to obtain the maximum tensile strength and stiffness of a material, but there are obvious disadvantages of using a material in fiber form Fibers alone cannot support longitudinal compressive loads and their transverse mechanical properties are often not as good as the corresponding longitudinal properties Thus, fibers are generally useless as structural materials unless they are held together in a structural unit with a binder or matrix material and unless some transverse reinforcement is provided Transverse reinforcement is generally provided by orienting fibers at various angles according to the stress field in the component of interest The need for fiber placement

in different directions according to the particular application has led to various types

of composites, as shown in Figure 2 1 In the continuous fiber composite laminate, individual continuous fiber/matrix laminae are oriented in the required directions and bonded together to form a laminate Although the continuous fiber laminate is used extensively, the potential for delamination, or separation of the laminae, is still a major problem because the interlaminar strength is matrix dominated Woven fiber composites do not have distinct laminae and are not susceptible to delamination, but strength and stiffness are sacrificed because the fibers are not as straight as in the continuous fiber laminate Chopped fiber composites may have short fibers randomly dispersed in the matrix Chopped fiber composites are used extensively in high-volume applications owning to their low manufacturing cost, but their mechanical properties are considerably poorer than those of continuous fiber composites Finally, hybrid composites may consist of mixed chopped and continuous fibers, or mixed fiber types such as glass and carbon The design flexibility offered by these and other composite configurations is obviously quite attractive to designers, and the potential exists to design not only the structure but also the structural material itself

Figure 2 1 Types of fiber-stiffened composites

(a) Continuous fiber composite, (b) Woven composite, (c) Chopped fiber composite,

(d) Hybrid composite

Composite structural elements are now used in a variety of components for automotive, aerospace, marine, and architectural structures in addition to consumer products such as skis, golf clubs, and tennis rackets [41] Military aircraft designers were among the first to realize the tremendous potential of composites with high specific strength and high specific stiffness, since performance and maneuverability

of those vehicles depend so heavily on weight Composite construction also leads to smooth surfaces, which reduce drag Since boron and graphite fibers were first developed in the early 1960s, applications of advanced composites in military aircraft have accelerated quickly Carbon fiber composite structural elements such as horizontal and vertical stabilizers, flaps, wing skins, and various control surfaces have

(d) (c)

aircraft have been steadily increasing as material costs come down, as design and manufacturing technology evolves, and as the experience with composites in aircraft continues to build For example, the Boeing 787 Figure 2 2 is the first commercial airliner with a composite fuselage and wings As much as 50% of the primary structure - including the fuselage and wings - on the 787 consists of carbon fiber/epoxy composite materials or carbon fiber-stiffened plastics The Airbus A350 XWB is another composites-intensive commercial airliner similar to the Boeing 787

Figure 2 2 Boeing 787 - first commercial airliner with composite fuselage and

wings (Courtesy of Boeing Company.) Structural weight is also very important in automotive vehicles, and the use of composite automotive components continues to grow In cargo trucks, the reduced weight of composite components translates into increased payloads, which can have

a significant economic impact For example, the composite concrete mixer drum shown in Figure 2 3 weighs 2000 lbs less than the conventional steel mixer drum that

it replaced

Figure 2 3 Composite mixer drum on concrete transporter truck weighs 2000 lbs less than conventional steel mixer drum (Courtesy of Oshkosh Truck Corporation) Weight savings on specific components such as composite leaf springs can exceed 70% compared with steel springs Experimental composite engine blocks have been fabricated from graphite-stiffened thermoplastics, but the ultimate goal is a ceramic composite engine that would not require water cooling Chopped glass FRPs have been used extensively in body panels where stiffness and appearance are the principal design criteria So far, the applications of composites in automotive vehicles have been mainly in secondary structural elements and appearance parts, and the full potential of composite primary structures remains to be explored With the increased interest in electric vehicles comes a need for composite structures to reduce vehicle structural weight to compensate for the heavy batteries that are required For example, the proposed BMW Megacity electric vehicle would have a carbon fiber composite passenger compartment integrated with an aluminum spaceframe I-beams, channel sections, and other structural elements (Figure 2 4) used in civil infrastructure may

be made of fiber stiffened plastic Wind turbines (Figure 2 5) are getting increased attention as environmentally attractive, alternative energy sources, and their blades

high stiffness-to-weight ratio, excellent vibration damping, and fatigue resistance Other applications of structural composites are numerous In this thesis, composite beam and stiffened composite plate structure are chosen to investigate and apply in computing and solving optimization design problems

Figure 2 4 Pultruded fiberglass composite structural elements (Courtesy of

Strongwell Corporation.)

Figure 2 5 Composite wind turbine blades (Courtesy of GE Energy.)

2.1.2 Overview of Composite Material in Design and Optimization

Use of composite materials in structural design has gained popularity over the past few decades because of several advantages that these materials offer in comparison with traditional structural materials, such as steel, aluminum, and various alloys One

of the primary reasons for their popularity is their weight advantage Composite materials such as Graphite/Epoxy and Glass/Epoxy have smaller weight density compared to metallic materials For example, the weight densities of high-strength Graphite/Epoxy and Glass/Epoxy are 1550.07 kg/m3 and 1799.19 kg/m3, respectively, compared to the weight density of Aluminum which is 2769.99 kg/m3

In addition to their weight advantage per unit volume, some composites provide better stiffness and strength properties compared to metals That is, structural members made out of composite materials may undergo smaller deformations, and carry larger static loads than their metallic counterparts Stiffness of high strength Graphite/Epoxy

is around 15.467x109 kg/m2 compared to Aluminum’s stiffness of 7.030x109 kg/m2 These advantages on weight and stiffness and strength properties make composites more attractive than alloys [42] Structural designers always seek the best possible design while using the least amount of resources The measure of goodness of a design depends on the application, typically related to strength or stiffness, while resources are measure in terms of weight or cost Therefore, the best design often means either the lowest weight (or cost) with limitations on the stiffness (or strength) properties Traditionally, engineers have based on experience to achieve such design For a given application, first a set of essential requirements are obtained Next, structural modifications that are likely to improve the performance or reduce the weight or the cost are implemented However, this approach is often difficult to satisfy both requirement of weight and stiffness at the same time because implementation that improves the performance may yield designs that violate the strength or stiffness requirements

Over the past three decades, mathematical optimization, which deals with either maximization or minimization of an objective function subjected to constraint functions, has emerged as a powerful tool for structural design In recent years, many works have been published for optimization of laminated composite structures For example, the optimum design of laminated composite For example, the optimum design of laminated composite plates for maximizing the first natural frequency can

be found in [43]–[45], or those for maximizing the buckling load factor in Refs [46]–[48], or those for minimizing the weight in Refs [49], [50], and or those for maximizing strain energy in Ref [11] The optimal design of laminated composite beams to minimize the free vibration frequency was found in Refs [51], [52], or those to minimize the weight in Refs.[53], [54], or those to maximize the buckling load and minimize the weight at the same time in Ref [55] The optimization design

of the continuous composite models using the different non gradient-based algorithms (particle swarm algorithm and genetic algorithm) for the thin-walled composite box-beam helicopter rotor blades have been investigated [56], [57] Liu [54] derived the exact solutions and sensitivity of the first four frequencies using the continuous composite model and developed the gradient-based algorithm to achieve the lightweight design of the solid composite laminated beams Lentz and Armanios [58] described a gradient-based optimization scheme for obtaining the maximum coupling in thin-walled composite beams subject to hygrothermal and frequency constraints

The optimization methods for the composite structures, as mentioned above, can be classified into gradient-based and non-gradient-based algorithms The non-gradient-based algorithms are also called random search algorithms The random search algorithms can implement the optimization design without the gradient information However, the gradient-based algorithms require the gradient to construct the searching algorithm Therefore, the non-gradient-based algorithms are easier to be carried out than the gradient-based algorithms Compared with the random search algorithms, the gradient-based algorithms are more efficient and can find the optimum

design (at least the local optimum) if the gradient can be computed efficiently and accurately [53] However, the gradient-based optimization methods possess two main drawbacks related to local optimization methods Firstly, they depend too much on the initial point provided by users As a result, if the initial point is not chosen well, especially for the optimization problems with many design variables, it is very hard

or even impossible for local search methods to find the optimum solution Secondly, since local search methods use gradient information for searching the solution, the solution obtained by these methods is easily trapped in local optimal solutions if the problem has more than one local extreme [11] Therefore, researchers prefer to use the non-gradient based methods, especially meta-heuristic optimization methods such

as Particle Swarm Optimization (PSO), Genetic Algorithm (GA), Differential Evolution (DE), Jaya algorithm, for solving the optimization problems of laminated composite structures to obtain the global solution In this thesis, Differential Evolution and Jaya algorithm are developed and applied to solve optimization problem of two types of composite structure model One model is Timoshenko composite beam and another is stiffened composite plate Theory related to these two composite structures are presented in the following sections of this chapter

2.2 Analysis of Timoshenko composite beam

Composite laminated Timoshenko beams can be treated as continuous models and discrete models The discrete models are easier to be implemented but difficult to obtain the exact solution It can only derive the approximate solution In addition, the discrete models such as finite element approaches are not so effective as the analytical approaches of continuous models Therefore, Liu [53] proposed an approach that treated composite laminated Timoshenko beam as continuous model to achieve the exact solution The process to build up the analytical solution for the composite laminated beam is simply presented as in the following section For more details of

the method, readers are encouraged to refer to Liu’s work

2.2.1 Exact analytical displacement and stress

Figure 2 6 Composite laminated beam model Consider a segment of composite laminated beam with N layers and the fiber orientations of layers are of i(i1, ,N) The positions of layers are denoted by ( 1, , )

Figure 2 7 Free-body diagram The displacement fields of the composite laminated beam calculated analytically based on the first-order shear deformation theory (also called Timoshenko beam theory) are:

dx

z

y

x

3 2 0

X

Y Z=3

1 2

O

Figure 2 8 The material and laminate coordinate system The stress fields of the composite laminated beam include the plane stress components and the shear stress components According to the coordinate system

between the materials (123) and the beam/laminate (xyz) as depicted in Figure 2 8,

in which the fiber orientation coincides with the 1-axis, the plane stress components are expressed as follows:

( ) ( )

In the above equations, Q( )ij k is the stiffness coefficients of the kth lamina in the

laminate coordinate system and are described clearly in [53]

2.2.2 Boundary-condition types

The indefinite integration constants in the above equations can be determined by using different boundary conditions In this thesis, four types of boundary conditions are considered including pinned-pinned, fixed-fixed, fixed-free and fixed-pined