Phát triển các phương pháp tối ưu hóa thông minh cho các bài toán cơ học

Among them, meta-heuristic algorithms have gainedhuge popularity in recent years in solving design optimization problems of many types of structure with different materials.. The DE hasd

MINISTRY OF EDUCATION AND TRAINING

HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY AND EDUCATION

LAM PHAT THUAN

DEVELOPMENT OF META-HEURISTIC OPTIMIZATION

METHODS FOR MECHANICS PROBLEMS

PHD THESIS MAJOR: ENGINEERING MECHANICS

THE WORK IS COMPLETED AT

HCM CITY UNIVERSITY OF TECHNOLOGY AND EDUCATION

LAM PHAT THUAN

DEVELOPMENT OF META-HEURISTIC OPTIMIZATION METHODS FOR MECHANICS

PROBLEMSMAJOR: ENGINEERING MECHANICS - 13252010105

Supervisor 1: Assoc Prof NGUYEN HOAI SON

Supervisor 2: Assoc Prof LE ANH THANG

PhD thesis is protected in front ofEXAMINATION COMMITTEE FOR PROTECTION OF DOCTORAL THESISHCM CITY UNIVERSITY OF TECHNOLOGY AND EDUCATION

Date……month……year……

ii

ORIGINALITY STATEMENT

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

The data and results stated in this dissertation are honest and have not been published by any works

Ho Chi Minh City, January 2021

This dissertation has been carried out in the Faculty of Civil Engineering, HCMCity University of Technology and Education, Viet Nam The process of conductingthis thesis brings excitement but has quite a few challenges and difficulties And Ican 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 NguyenHoai Son and Assoc Prof Le Anh Thang, especially Assoc Prof Dr Nguyen HoaiSon from GACES Group, Ho Chi Minh City University of Technology andEducation, Vietnam for having accepted me as their PhD student and for theenthusiastic guidance and mobilization during my research

Secondly, I would like also to acknowledge Msc Ho Huu Vinh for histroubleshooting and the cooperation in my study Furthermore, I am grateful to CivilEngineering Faculty 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 CivilEngineering, Ho Chi Minh City University of Technology and Education, for theirprofessional advice and friendly support

Finally, this dissertation is dedicated to my parents who have always given mevaluable encouragement and assistance

Lam Phat Thuan

ii

Almost all design problems in engineering can be considered as optimization problemsand thus require optimization techniques to solve During the past few decades, manyoptimization techniques have been proposed and applied to solve a wide range ofvarious optimization problems Among them, meta-heuristic algorithms have gainedhuge popularity in recent years in solving design optimization problems of many types

of structure with different materials These meta-heuristic algorithms include geneticalgorithms (GA), particle swarm optimization (PSO), bat algorithm (BA), cuckoosearch (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, theDifferential Evolution is one of the most widely used methods Since it was firstintroduced in 1997 by Storn and Price [1], many studies have been carried out toimprove and apply DE in solving structural optimization problems The DE hasdemonstrated excellently performance in solving many different engineering problems.Besides the Differential Evolution algorithm, the Jaya algorithm recently proposed byRao [2] 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 gainsdominate results when being tested with benchmark test functions in comparison withother meta-heuristic methods However, like many other population-based optimizationalgorithms, one of the disadvantages of DE and Jaya is that the computational timeobtaining optimal solutions is much slower than the gradient-based optimizationmethods This is because DE and Jaya takes a lot of time evaluating the fitness ofindividuals in the population To overcome this disadvantage, Artificial NeuronNetworks (ANN) are studied to combine with the meta-heuristic algorithms, such asDifferential Evolution, to form a new approach which has the ability to solve the designoptimization effectively Moreover, one of the most important issues in engineeringdesign is that the optimal designs are often effected by uncertainties which can beoccurred from various sources, such as

manufacturing processes, material properties and operating environments Theseuncertainties may cause structures to improper performance as in the originaldesign, and hence may result in risks to structures [3] Therefore, reliability-baseddesign optimization (RBDO) can be considered as an important and comprehensivestrategy for finding an optimal design.

In this dissertation, an improved version of Differential Evolution has been firsttime utilized to solve for optimal fiber angle and thickness of the reinforcedcomposite Secondly, the Artificial Neural Network is integrated to the optimizationprocess of the improved Differential Evolution algorithm to form a new algorithmcall ABDE (ANN-based Differential Evolution) algorithm This new algorithm isthen applied to solve optimization problems of the reinforced composite platestructures Thirdly, an elitist selection technique is utilized to modify the selectionstep of the original Jaya algorithm to improve the convergence of the algorithm andformed a new version of the original Jaya called iJaya algorithm The improvedJaya algorithm is then applied to solve for optimization problem of the Timoshenkocomposite beam and obtained very good results Finally, the so-called called(SLMD-iJaya) algorithm which is the combination of the improved Jaya algorithmand the Global Single-Loop Deterministic Methods (SLDM) has been proposed as anew tool set for solving the Reliability-Based Design Optimization problems Thisnew method is applied to look for optimal design of Timoshenko composite beamstructures with certain level of reliability

iv

TÓM TẮT

Hầu như các bài toán thiết kế trong kỹ thuật có thể được coi là những bài toán tối ưu và

do đó đòi hỏi các kỹ thuật tối ưu hóa để giải quyết Trong những thập kỷ qua, nhiều kỹthuật tối ưu hóa đã được đề xuất và áp dụng để giải quyết một loạt các vấn đề khácnhau Trong số đó, các thuật toán meta-heuristic đã trở nên phổ biến trong những nămgần đây trong việc giải quyết các vấn đề tối ưu hóa thiết kế của nhiều loại cấu trúc vớicác vật liệu khác nhau Các thuật toán meta-heuristic này bao gồm Genetic Algorithms,Particle Swarm Optimization, Bat Algorithm, Cuckoo Search, Differential Evolutioin,Firefly Algorithm, Harmony Search, Flower Pollination Algorithm, Ant ColonyOptimization, Bee Algorithms, Jaya Algorithm và nhiều thuật toán khác Trong số cácphương pháp được đề cập ở trên, Differential Evolution là một trong những phươngpháp được sử dụng rộng rãi nhất Kể từ khi được Storn và Price [1] giới thiệu lần đầutiên, nhiều nghiên cứu đã được thực hiện để cải thiện và áp dụng DE trong việc giảiquyết các vấn đề tối ưu hóa cấu trúc DE đã chứng minh hiệu suất tuyệt vời trong việcgiải quyết nhiều vấn đề kỹ thuật khác nhau Bên cạnh thuật toán Differential Evolution,thuật toán Jaya được Rao [2] đề xuất gần đây cũng là một phương pháp hiệu quả và đãđược áp dụng rộng rãi để giải quyết nhiều vấn đề tối ưu hóa và cho thấy hiệu suất tốt

Nó đạt được kết quả vượt trội khi được thử nghiệm với các hàm test benchmark so vớicác phương pháp dựa trên dân số khác Tuy nhiên, giống như nhiều thuật toán tối ưuhóa dựa trên dân số khác, một trong những nhược điểm của DE và Jaya là thời giantính toán tối ưu chậm hơn nhiều so với các phương pháp tối ưu hóa dựa trên độ dốc(gradient-based algorithms) Điều này là do DE và Jaya mất rất nhiều thời gian để đánhgiá hàm mục tiêu của các cá thể trong bộ dân số Để khắc phục nhược điểm này, cácmạng nơ ron nhân tạo (Artificial Neural Networks) được nghiên cứu để kết hợp với cácthuật toán meta-heuristic, như Differential Evolution, để tạo thành một phương pháptiếp cận mới giúp giải quyết

các bài toán tối ưu hóa thiết kế một cách hiệu quả Bên cạnh đó, một trong nhữngvấn đề quan trọng nhất trong thiết kế kỹ thuật là các thiết kế tối ưu thường bị ảnhhưởng bởi những yếu tố ngẫu nhiên Những yếu tố này có thể xảy ra từ nhiều nguồnkhác nhau, chẳng hạn như quy trình sản xuất, tính chất vật liệu và môi trường vậnhành và có thể khiến các cấu trúc hoạt động không đúng như trong thiết kế ban đầu,

và có thể dẫn đến rủi ro cho các cấu trúc [3] Do đó, tối ưu hóa thiết kế dựa trên độtin cậy (Reliability-Based Design Optimization) có thể được coi là một chiến lượctoàn diện, cần thiết để tìm kiếm một thiết kế tối ưu

Trong luận án này, lần đầu tiên một phiên bản cải tiến của phương pháp DifferentialEvolution đã được sử dụng để tìm góc hướng sợi tối ưu và độ dày của tấm giacường vật liệu composite Thứ hai, Mạng nơ ron nhân tạo (ANN) được tích hợp vàoquy trình tối ưu hóa thuật toán Differentail Evolution cải tiến để hình thành thuậttoán mới gọi là thuật toán ABDE (Artificial Neural Network-Based DifferentialEvolution) Thuật toán mới này sau đó được áp dụng để giải quyết các bài toán tối

ưu hóa của các cấu trúc tấm composite gia cường Thứ ba, một kỹ thuật lựa chọntinh hoa (Elitist Selection Technique) được sử dụng để hiệu chỉnh bước lựa chọncủa thuật toán Jaya ban đầu để cải thiện sự hội tụ của thuật toán và hình thành mộtphiên bản mới của thuật toán Jaya được gọi là thuật toán iJaya Thuật toán Jaya cảitiến (iJaya) sau đó được áp dụng để giải quyết bài toán tối ưu hóa dầm Timoshenkovật liệu composite và thu được kết quả rất tốt Cuối cùng, thuật toán mới SLMD-iJaya được tạo thành từ sự kết hợp giữa thuật toán Jaya cải tiến và phương phápvòng lặp đơn xác định (Single-Loop Deterministic Method) đã được đề xuất nhưmột công cụ mới để giải quyết các vấn đề Tối ưu hóa thiết kế dựa trên độ tin cậy.Phương pháp mới này được áp dụng để tìm kiếm thiết kế tối ưu của các cấu trúcdầm composite Timoshenk và cho kết quả vượt trội

vi

ORIGINALITY STATEMENT i

ACKNOWLEDGEMENTS ii

ABSTRACT iii

CONTENTS vii

NOMENCLATURE x

LIST OF TABLES xiii

LIST OF FIGURES xiv

CHAPTER 1 1

1.1 An overview on research direction of the thesis 1

1.2 Motivation of the research 6

1.3 Goals of the dissertation 6

1.4 Research scope of the dissertation 7

1.5 Outline 7

1.6 Concluding remarks 9

CHAPTER 2 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 reinforced composite plate 23

CHAPTER 3 26

3.1 Overview of Metaheuristic Optimization 26

3.1.1 Meta-heuristic Algorithm in Modeling 27

3.1.2 Meta-heuristic Algorithm in Optimization 31

3.2 Solving Optimization problems using improved Differential Evolution 41 3.2.1 Brief on the Differential Evolution algorithm [14], [129] 42

3.2.2 The modified algorithm Roulette-Wheel-Elitist Differential Evolution 43

3.3 Solving Optimization problems using improved Jaya algorithm 44

3.3.1 Jaya Algorithm 44

3.2.2 Improvement version of Jaya algorithm 45

3.4 Reliability-based design optimization using a global single loop deterministic method 46

3.4.1 Reliability-based optimization problem formulation 48

3.4.2 A global single-loop deterministic approach 49

CHAPTER 4 53

4.1 Fundamental theory of Neural Network 53

4.1.1 Basic concepts on Neural Networks [146] 55

4.1.2 Neural Network Structure 56

4.1.3 Neural Network Design Steps 60

4.1.4 Levenberg-Marquardt training algorithm 61

4.1.5 Over fitting, Over training 63

4.2 Artificial Neural Network based meta-heuristic optimization methods 65 CHAPTER 5 68

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5.1 Verification of iDE algorithm 68

5.1.1 A 10-bars planar truss structure: 68

5.1.2 A 200-bars truss structure 70

5.1.3 A 72-bar space truss structure 72

5.1.4 A 120-bar space truss structure: 75

5.2 Static analysis of the reinforced composite plate 77

5.3 The effective of the improved Differential Evolution algorithm 79

5.4 Optimization of reinforced composite plate 80

5.4.1 Thickness optimization of stiffened Composite plate 80

5.4.2 Artificial neural network-based optimization of reinforced composite plate 82

5.5 Deterministic optimization of composite beam 85

5.5.1 Optimal design with variables: b and h 86

5.5.2 Optimal design with variables: b and t i 89

5.6 Reliability-based optimization design of Timoshenko composite beam 93 5.6.1 Verification of SLDM-iJaya 93

5.6.2 Reliability-based lightweight design 95

CHAPTER 6 98

6.1 Conclusions and Remarks 98

6.2 Recommendations and future works 101

REFERENCES 103

LIST OF PUBLICATIONS 118

Loading vectorShear modulusThe thickness of the composite beam/plateStiffness matrix of the plate

Length of the composite beamNumber of constraint satisfactionsNumber of layers of composite materialsSize of population

Crossover control parameterVector of random parametersMatrix of material stiffness coefficientsMatrix of compliance

Coordinate transformation matrixDisplacement field of the composite beamVector of design variables

Population setVector of weights

Poison’s ratio

x

Normal strain in x directionNormal strain in y directionShear strain in xy directionShear strain in yz directionShear strain in xz directionMean vector of x

Distance between feasible and infeasible design region

Tow dimensionThree dimensionArtificial Neural NetworkMulti-Layer PerceptronDifferential Evolutionimproved Differential EvolutionArtificial neural network-Based Differential Evolution

PSO Particle Swarm Optimization

RBDO Reliability Based Design Optimization

ADO Approximate Deterministic Optimization

CS-DSG3 Cell-Smoothed Discrete Shear Gap technique using

triangle finite element

xii

LIST OF TABLES

Table 5 1 Parameters for 10 bars truss 69

Table 5 2 The comparison results keep the solution from the improved DE algorithm with other methods for the 10-bar flattening problem 70

Table 5 3 Parameter for 200-bars truss structure 72

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 73

Table 5 5 Parameters for 72-bars space truss structure 74

Table 5 6 Comparison between the solution from iDE algorithm with other methods for the the 72-bars space truss problem 75

Table 5 7 Parameters for 120-bars arch space truss structure 76

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 77

Table 5 9 Comparison of central deflection (mm) of the simply-supported square reinforced composite plates 78

Table 5 10 The optimal results of two problems 80

Table 5 11 Optimal thickness results for reinforced composite plate problems 82

Table 5 12 Sampling and overfitting checking error 83

Table 5 13 Comparison of the accuracy and computational time between DE and ABDE 84

Table 5 14 Material properties of lamina 87

Table 5 15 Comparison of optimal design with continuous design variables 88

Table 5 16 Comparison of optimal design with discrete design variables 90

Table 5 17 Comparison of optimization results of the mathematical problem 94

Table 5 18 Optimal results of reliability based lightweight design with different level of reliability 96

LIST OF FIGURES

Figure 2 1 Types of fiber-reinforced 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 reinforced by an r-direction beam 24

Figure 3 1 Source of inspiration in meta-heuristic optimization algorithms 33

Figure 3 2 Illustration of the feasible design region 50

Figure 4 1 Biological neuron 53

Figure 4 2 Perceptron neuron of Pitts and McCulloch 54

Figure 4 3 Applying a model based on field data 55

Figure 4 4 The relationship between Machine Learning and the neural network 56

Figure 4 5 A Multi-layer perceptron network model 57

Figure 4 6 Single node in an MLP network 57

Figure 4 7 Tanh and Sigmoid function 58

Figure 4 8 A multi-layer perceptron with one hidden layer Both layers use the same activation function g 59

Figure 4 9 Diagram for the training process of a neural network with the Levenberg-Marquardt algorithm 63

Figure 4 10 Dividing the training data for the validation process 65

Figure 4 11 Optimization process using Artificial Neural Network (ANN) based Differential Evolution (ABDE) optimization algorithm 66

Figure 5 1 A 10-bars truss structure 69

Figure 5 2 A 200 bars truss structure 71

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Figure 5 3 A 72-bars space truss structure 74

Figure 5 4 Structure of 120-bars arch space truss 76

Figure 5 5 Model of a reinforced composite plate 77

Figure 5 6 Models of square and rectangular reinforced composite plates 79

Figure 5 7 Model of reinforced composite plate for optimization 81

Figure 5 8 Convergence curves of DE, IDE, Jaya and iJaya for the beam with P-P condition 89

Figure 5 9 Convergence curves of DE, IDE, Jaya and iJaya for the beam with P-P condition 91

Figure 5 10 Comparison of different design approaches with different boundary conditions 92

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

97

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 problemsand thus require optimization techniques to solve However, as most real-worldproblems are highly non-linear, traditional optimization methods usually do not workwell The current trend is to use evolutionary algorithms and meta-heuristicoptimization methods to tackle such nonlinear optimization problems Meta-heuristicalgorithms have gained huge popularity in recent years These meta-heuristicalgorithms include genetic algorithms, particle swarm optimization, bat algorithm,cuckoo search, differential evolution, firefly algorithm, harmony search, flowerpollination algorithm, ant colony optimization, bee algorithms, Jaya algorithm andmany others The popularity of meta-heuristic algorithms can be attributed to their goodcharacteristics because these algorithms are simple, flexible, efficient, adaptable andyet easy to implement Such advantages make them versatile to deal with a wide range

of optimization problems, especially the structural optimization problems [4].Structural optimization is a potential field and has attracted the attention of manyresearchers around the world During the past decades, many optimization techniqueshave been proposed and applied to solve a wide range of various problems Thealgorithms can be classified into two main groups: gradient-based and popular-basedapproach Some of the gradient-based optimization methods can be named here assequential linear programming (SLP) [5], [6], sequential quadratic programming (SQP)[7], [8], Steepest Descent Method, Conjugate Gradient Method, Newton's Method [9].The gradient-based methods are very fast in reaching the optimal solution, but easytrapped in local extrema and requires the gradient information to construct thesearching algorithm Besides, the gradient-based approaches are limited to continuousdesign variables and that decreases the productivity of the algorithm In addition, theinitial solution (or initial design parameters of the structure) also

greatly affects the ability to achieve global or local solutions of gradient-basedalgorithms The population-based techniques, also known as part of meta-heuristicalgorithms, can be listed such as genetic algorithm (GA), differential evolution (DE),and particle swarm optimization (PSO), Cuckoo Search (CS), Firefly Algorithm (FA),etc [10] These methods are used extensively in structural problems because of theirflexibility and efficiency in handling both continuous and discontinuous designvariables In addition, the solutions obtained from population-based algorithms in mostcases are global ones Therefore, the optimal result of the problem is not too muchinfluenced by the initial solution (or initial design of the structure) Among the methodsmentioned above, the Differential Evolution is one of the most widely used methods.Since it was first introduced by Storn and Price [1], many studies have been carried out

to improve and apply DE in solving structural optimization problems The DE hasdemonstrated excellently performance in solving many different engineering problems.Wang et al [11] applied the DE for designing optimal truss structures with continuousand discrete variables Wu and Tseng [12] applied a multi-population differentialevolution with a penalty-based, self-adaptive strategy to solve the COP of the trussstructures Le-Anh et al [13] using an improved Differential Evolution algorithm and asmoothed triangular plate element for static and frequency optimization of foldedlaminated composite plates Ho-Huu et al [14] proposed a new version of the DE tooptimize the shape and size of truss with discrete variables Besides the DifferentialEvolution algorithm, the Jaya algorithm recently proposed by Rao [2] is also aneffective and efficient methods that has been widely applied to solve manyoptimization problems and showed its good performance It gains dominate resultswhen being tested with benchmark test functions in comparison with other population-based methods such as homomorphous mapping (HM), adaptive segregationalconstraint handling evolutionary algorithm (ASCHEA), simple multi-memberedevolution strategy (SMES), genetic algorithm (GA), particle swarm optimization(PSO), differential evolution (DE), artificial bee colony (ABC), biogeography basedoptimization (BBO) Moreover, it has been also successfully

applied in solving many optimal design problem in engineering as presented infollowing literature [15]–[17] However, the performance of the original Jayaalgorithm is not really high Therefore, there are many variations of the Jayaalgorithm proposed to improve the original one In this thesis, a new improvedversion of the Jaya algorithm will be presented The new algorithm aims to improvethe population selection technique for the next generation in order to improve thespeed of convergence, while at the same time ensuring the accuracy and the balancebetween 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 based optimization methods This is because DE and Jaya takes a lot of time inevaluating the fitness of individuals in the population Specifically, in the structuraloptimization problem, the calculation of the objective function or constraint functionvalues is usually done by using the finite element to analyze the structural response Toovercome this disadvantage, artificial neuron networks (ANN) are proposed to combinewith the DE algorithm Based on the idea of imitation of the brain structure, ANN iscapable of approximating an output corresponding to a set of input data quickly afterthe network has been trained, also known as a learning process Thanks to thisremarkable advantage, the computation of objective function or constraint functionvalues in the DE algorithm will be done quickly As a result, ANN will helpsignificantly improve the efficiency of DE calculations The effectiveness andapplicability of ANN since the early groundwork ideas put forward by WarrenMcCulloch and Walter Pitts [18] in 1943 have so far proved to be very convincingthrough numerous studies Application areas include system identification and control,pattern recognition, sequence recognition (gesture, speech, handwritten textrecognition), data mining, visualization, machine translation, social networkingfiltering and email spam filtering, etc [19]–[24]

gradient-The next issue is the development of optimal algorithms integrated ANN with DEand applying the proposed algorithms to a practical structure to examine the

3

effectiveness of the method At present, the structures made from composite materialare widely used in almost all fields such as construction, mechanical engineering,marine, aviation, etc In particular, beams and reinforced plates made of compositematerial are an outstanding form and are used increasingly by its superior advantages.

By combining the advantages of composite materials and the reinforced beamsstructure, the reinforced composite plates have very high bending strength with verylight weight Nowadays, reinforced composite plates have been widely used in manybranches of structural engineering such as aircraft, ships, bridges, buildings, etc For itsadvantages in both bending stiffness and the amount of material in comparison withcommon bending plate structures, reinforced composite plate usually has highereconomic efficiency in practical applications Due to its high practical applicability, theneed to optimize the design of the structure to save costs and increase the efficiency ofuse is also high However, because of the complexity of computing the behavior of thisparticular type of structure, finding a good algorithm for optimizing design parameters

is essential to ensure computational efficiency Composite material structures have verycomplex 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 objectivefunctions to the constrained equations So the use of gradient-based algorithms is notstraightforward For such types of problems, population-based methodologies are asuperior choice

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

in the original design, and hence may result in risks to structures [3] There are twogroups of methods for dealing with uncertainties: reliability-based design and robustdesign Robust design focuses on minimizing variance in design results undervariations of design variables and parameters Reliability-based design optimization

(RBDO) ensures that the design is feasible regardless of changes in design variablesand parameters RBDO can be considered as a comprehensive strategy for finding anoptimal design RBDO is the focus of this thesis Although RBDO is more reliable thanstatic optimization, the biggest drawback of RBDO in practical application is the highcomputational cost To solve this problem, a lot of research has been done to findeffective reliability analysis techniques, such as: sensitivity-based approximationapproaches [25], [26], most probable point (MPP)-based approaches, Monte Carlosimulations [27]–[29] and response surface model-based approaches [30] Thesetechniques focus on nesting the optimization and the reliability assessment in oneprocess 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 iterationduring 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 byseparating 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 costlycomputation To overcome this drawback, the single-loop methods have been proposed Inthis approach, the RBDO problem is solved in a single-loop procedure without reliabilityanalysis The strategy is to convert an RBDO problem into an approximate deterministicoptimization (ADO) problem by transforming probabilistic constraints into approximatedeterministic 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 beamsare 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

5

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

In summary, in this thesis, some modifications will be investigated and propose toimprove the original algorithm of Differential Evolution and Jaya algorithm toincrease the convergence of DE and Jaya algorithm The modified algorithms arethen combine with ANN and/or SLDM to develop new tools for solving designoptimization problems and the RBDO problems of composite structures, such asreinforced 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 ofcomposite material structures and intelligent optimization methods, especially thereliability-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 withoptimal algorithms to improve the speed and the performance of solving structuraloptimization problems

1.3 Goals of the dissertation

Firstly, this thesis focuses on studying and developing meta-heuristic optimizationmethods and combines them with the Artificial Neural Network, which hasadvantages in approximating data, to build up a new algorithm for solvingcomposite material structural optimization problems Particularly, the originalDifferential Evolution or Jaya algorithm will be modified to improve theconvergence in solving for global optimal solution and then, the ANN will beintegrated to the improved meta-heuristic algorithms to form a new algorithm,which is used to look for optimal design of reinforced composite plate structures.Secondly, the thesis also proposes a new tool set, which is the combination of meta-heuristic optimization algorithm and the Single-Loop Deterministic Method to deal

with Reliability-Based Design Optimization (RBDO) problems In particular, theoriginal Jaya algorithm will be modified to improve the convergence in searchingoptimal solutions of the optimization problems Then, this improved version of Jayaalgorithm will be combined with Single-Loop Deterministic Method to solve theReliability-Based Design Optimization of composite beam structures.

1.4 Research scope of the dissertation

The thesis focuses on the following main issues:

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

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

- 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 solvemany different types of problems

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

- The problems selected for optimization are relatively simple with the mainpurpose of evaluating the effectiveness, accuracy and reliability of the proposedoptimization methods The application of optimal methods proposed in the thesis formore complex problems will be further studied in the future

1.5 Outline

The dissertation contains seven chapters and is structured as follows:

 Chapter 1 presents an overview on meta-heuristic algorithms, compositematerial structure and especially artificial neural networks and its role and application inoptimization process This chapter also give out the organization of the thesis via theoutline 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 andapplications in real life The chapter also introduce theory of Timoshenko

7

composite beam and reinforced 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 toDifferential Evolution and Jaya algorithm and the approach to modify and improve theoriginal 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 theproposed methods for solving RBDO problem

 Chapter 4 offers the introduction and the historical development of ArtificialNeural Network (ANN) This chapter gives out some basic concepts related to ANN andintroduce the Neural Network Structure which is used in this thesis to approximate dategenerated from the Finite Element Analysis Moreover, the training algorithm, especiallythe Levenberg-Marquardt and the overfitting phenomenon are also presented in thischapter

 Chapter 5 illustrate the effectiveness and efficiency of the improve DifferentialEvolution and the improve Jaya in solving optimization problems The structuresinvestigated in this section includes planar truss structure, space truss structure, Timoshenkocomposite beam and reinforced composite plate In particular, the improve DifferentialEvolution (iDE) is applied to solve for optimal weight of planar truss structures and spacetruss structures, then it is used to optimize the fiber angle and the thickness of reinforcedcomposite plates and show its good effectiveness and performance The last part of thischapter devotes to illustration of the improve Jaya algorithm in looking for optimal design ofthe Timoshenko composite beam and the results obtained prove its highly effectiveperformance and accuracy compared with those of others’ author Moreover, this chapteralso presents a new approach called SLDM-iJaya which is formed by the combination of theimprove Jaya algorithm and the single-loop methods for solving the RBDO problem of theTimoshenko composite beam This chapter illustrate the solutions for two

problems, the first one solving a common optimization problem without thereliability index, and the second is the RBDO problem The results obtainedfrom these two problems are compared and analyzed with those of other authorsand show the effectiveness and the accuracy of the proposed SLDM-iJayaalgorithm Afterward, this chapter presents the application of Artificial NeuralNetwork when it is integrated to a meta-heuristic optimization method, such asDifferential Evolution algorithm, to solve the optimization problems Theintegration form a new tool set call ABDE (ANN-Based Differential Evolution)algorithm and applied to solve for optimal design of the reinforced compositeplate The results not only prove the effectiveness of the proposed method butalso 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 Concluding remarks

In this chapter, an overview of meta-heuristic optimization methods, artificial neuralnetwork, composite material structure in optimization is given out This chapter alsopresents the novelty points of this dissertation, and the organization of thedissertation with eight chapters In the next chapters, fundamental theories, someapproaches for modification to improve the solution of some meta-heuristicalgorithm and application with numerical results will be presented

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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 divided into four basic categories: metals, polymers,ceramics, and composites Composites, which consist of two or more separate materialscombined in a structural unit, are typically made from various combinations of theother three materials In the early days of modern man-made composite materials, theconstituents were typically macroscopic As composites technology advanced over thelast 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 beachieved 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 matrixmaterial Particle or flake reinforcements are also used, but they are generally not aseffective as fibers Some example of composite can easily find in the nature Forexample, Wood consists mainly of fibrous cellulose in a matrix of lignin, whereas mostmammalian bone is made up of layered and oriented collagen fibrils in a protein–calcium phosphate matrix [39] Fibrous reinforcement is very effective because manymaterials are much stronger and stiffer in fiber form than they are in bulk form It isbelieved that this phenomenon was first demonstrated scientifically in 1920 by Griffith[40], who measured the tensile strengths of glass rods and glass fibers of differentdiameters 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-inducingsurface cracks would be generated during fabrication and handling Results similar tothose published by Griffith have been reported for a wide variety of other materials.The reasons for the differences between fiber and bulk behavior There can be no doubtthat 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 alonecannot support longitudinal compressive loads and their transverse mechanicalproperties are often not as good as the corresponding longitudinal properties Thus,fibers are generally useless as structural materials unless they are held together in astructural unit with a binder or matrix material and unless some transversereinforcement is provided Transverse reinforcement is generally provided by orientingfibers at various angles according to the stress field in the component of interest Theneed 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 continuousfiber laminate is used extensively, the potential for delamination, or separation of thelaminae, is still a major problem because the interlaminar strength is matrix dominated.Woven fiber composites do not have distinct laminae and are not susceptible todelamination, but strength and stiffness are sacrificed because the fibers are not asstraight as in the continuous fiber laminate Chopped fiber composites may have shortfibers randomly dispersed in the matrix Chopped fiber composites are used extensively

in high-volume applications due to their low manufacturing cost, but their mechanicalproperties are considerably poorer than those of continuous fiber composites Finally,hybrid composites may consist of mixed chopped and continuous fibers, or mixed fibertypes such as glass and carbon The design flexibility offered by these and othercomposite configurations is obviously quite attractive to designers, and the potentialexists to design not only the structure but also the structural material itself

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(a) (b)

Figure 2 1 Types of fiber-reinforced 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 asskis, 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 highspecific stiffness, since performance and maneuverability of those vehicles depend soheavily on weight Composite construction also leads to smooth surfaces, which reducedrag 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

been used in fighter aircraft for many years Composites applications in commercialaircraft have been steadily increasing as material costs come down, as design andmanufacturing 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/epoxycomposite materials or carbon fiber-reinforced plastics The Airbus A350 XWB isanother 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 ofcomposite automotive components continues to grow In cargo trucks, the reducedweight 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

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Figure 2 3 Composite mixer drum on concrete transporter truck weighs 2000 lbs

less than conventional steel mixer drum

Weight savings on specific components such as composite leaf springs can exceed 70%compared with steel springs Experimental composite engine blocks have beenfabricated from graphite-reinforced thermoplastics, but the ultimate goal is a ceramiccomposite engine that would not require water cooling Chopped glass FRPs have beenused extensively in body panels where stiffness and appearance are the principal designcriteria So far, the applications of composites in automotive vehicles have been mainly

in secondary structural elements and appearance parts, and the full potential ofcomposite primary structures remains to be explored With the increased interest inelectric vehicles comes a need for composite structures to reduce vehicle structuralweight to compensate for the heavy batteries that are required For example, theproposed BMW Megacity electric vehicle would have a carbon fiber compositepassenger 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 reinforced plastic Wind turbines (Figure 2 5) are getting increased

attention as environmentally attractive, alternative energy sources, and their blades

are typically made from composites due to their high strength-to-weight ratio, highstiffness-to-weight ratio, excellent vibration damping, and fatigue resistance Otherapplications of structural composites are numerous In this thesis, composite beamand reinforced composite plate structure are chosen to investigate and apply incomputing 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.)

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2.1.2 Overview of Composite Material in Design and Optimization

Use of composite materials in structural design has gained popularity over the pastfew decades because of several advantages that these materials offer in comparisonwith traditional structural materials, such as steel, aluminum, and various alloys.One of the primary reasons for their popularity is their weight advantage Compositematerials such as Graphite/Epoxy and Glass/Epoxy have smaller weight densitycompared to metallic materials For example, the weight densities of high-strengthGraphite/Epoxy and Glass/Epoxy are 0.056 lb/in3 and 0.065 lb/in3, respectively,compared to the weight density of Aluminum which is 0.10 lb/in3 In addition totheir weight advantage per unit volume, some composites provide better stiffnessand strength properties compared to metals That is, structural members made out ofcomposite materials may undergo smaller deformations, and carry larger static loadsthan their metallic counterparts Stiffness of high strength Graphite/Epoxy is around22x106 lb/in2 compared to Aluminum’s stiffness of 10x106 lb/in2 These advantages

on weight and stiffness and strength properties make composites more attractivethan alloys [42] Structural designers always seek the best possible design whileusing the least amount of resources The measure of goodness of a design depends

on the application, typically related to strength or stiffness, while resources aremeasure in terms of weight or cost Therefore, the best design often means either thelowest weight (or cost) with limitations on the stiffness (or strength) properties.Traditionally, engineers have based on experience to achieve such design For agiven application, first a set of essential requirements are obtained Next, structuralmodifications that are likely to improve the performance or reduce the weight or thecost are implemented However, this approach is often difficult to satisfy bothrequirement of weight and stiffness at the same time because implementation thatimproves the performance may yield designs that violate the strength or stiffnessrequirements

Over the past three decades, mathematical optimization, which deals with eithermaximization or minimization of an objective function subjected to constraintfunctions, has emerged as a powerful tool for structural design In recent years, manyworks have been published for optimization of laminated composite structures Forexample, 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], orthose for minimizing the weight in Refs [49], [50], and or those for maximizing strainenergy in Ref [13] The optimal design of laminated composite beams to minimize thefree 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 atthe same time in Ref [55] The optimization design of the continuous compositemodels using the different non gradient-based algorithms (particle swarm algorithmand genetic algorithm) for the thin-walled composite box-beam helicopter rotor bladeshave been investigated [56], [57] Liu

[54] derived the exact solutions and sensitivity of the first four frequencies using thecontinuous composite model and developed the gradient-based algorithm to achievethe lightweight design of the solid composite laminated beams Lentz and Armanios[58] described a gradient-based optimization scheme for obtaining the maximumcoupling in thin-walled composite beams subject to hygrothermal and frequencyconstraints

The optimization methods for the composite structures, as mentioned above, can beclassified into gradient-based and non-gradient-based algorithms The non-gradient-based algorithms are also called random search algorithms The random searchalgorithms can implement the optimization design without the gradient information.However, the gradient-based algorithms require the gradient to construct the searchingalgorithm Therefore, the non-gradient-based algorithms are easier to be carried outthan the gradient-based algorithms Compared with the random search algorithms, thegradient-based algorithms are more efficient and can find the optimum

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design (at least the local optimum) if the gradient can be computed efficiently andaccurately [53] However, the gradient-based optimization methods possess two maindrawbacks related to local optimization methods Firstly, they depend too much on theinitial 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 oreven impossible for local search methods to find the optimum solution Secondly, sincelocal search methods use gradient information for searching the solution, the solutionobtained by these methods is easily trapped in local optimal solutions if the problemhas more than one local extreme [13] Therefore, researchers prefer to use the non-gradient based methods, especially meta-heuristic optimization methods such asParticle Swarm Optimization (PSO), Genetic Algorithm (GA), Differential Evolution(DE), Jaya algorithm, for solving the optimization problems of laminated compositestructures to obtain the global solution In this thesis, Differential Evolution and Jayaalgorithm are developed and applied to solve optimization problem of two types ofcomposite structure model One model is Timoshenko composite beam and another isreinforced composite plate Theory related to these two composite structures arepresented in the following sections of this chapter.

2.2 Analysis of Timoshenko composite beam

Composite laminated Timoshenko beams can be treated as continuous models anddiscrete models The discrete models are easier to be implemented but difficult toobtain 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 theanalytical approaches of continuous models Therefore, Liu [53] proposed anapproach that treated composite laminated Timoshenko beam as continuous model

to achieve the exact solution The process to build up the analytical solution for thecomposite laminated beam is simply presented as in the following section For moredetails of the method, readers are encouraged to refer to Liu’s work

2.2.1 Exact analytical displacement and stress

2

(2) Z

1

(1)

d

x

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

z i (i 1, , N ) The beam has rectangular cross section with the width b and the

length h as depicted in Figure 2 6 The beam segment dx is subjected to the

transversal force as shown in Figure 2 7.

q(x)

Q + dQ Q

dx

Figure 2 7 Free-body diagram

The displacement fields of the composite laminated beam calculated analyticallybased on the first-order shear deformation theory (also called Timoshenko beamtheory) are:

Figure 2 8 The material and laminate coordinate system

The stress fields of the composite laminated beam include the plane stresscomponents 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 componentsare expressed as follows

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(2.6)