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

Finally, the so-called calledSLMD-iJaya algorithm which is the combination of the improved Jaya algorithmand the Global Single-Loop Deterministic Methods SLDM has been proposed as anew t

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

Ho Chi Minh City, 01/2021

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 ofEXAMINATION COMMITTEE FOR PROTECTION OF DOCTORAL THESISHCM CITY UNIVERSITY OF TECHNOLOGY AND EDUCATION

Date……month……year……

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 beenpublished by any works

Ho Chi Minh City, January 2021

Lam Phat Thuan

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 toCivil Engineering 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

Almost all design problems in engineering can be considered as optimizationproblems and thus require optimization techniques to solve During the past fewdecades, many optimization techniques have been proposed and applied to solve awide range of various optimization problems Among them, meta-heuristicalgorithms have gained huge popularity in recent years in solving designoptimization problems of many types of structure with different materials Thesemeta-heuristic algorithms include genetic algorithms (GA), particle swarmoptimization (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 andmany others Among the methods mentioned above, the Differential Evolution isone of the most widely used methods Since it was first introduced in 1997 by Stornand Price [1], many studies have been carried out to improve and apply DE insolving structural optimization problems The DE has demonstrated excellentlyperformance in solving many different engineering problems Besides theDifferential Evolution algorithm, the Jaya algorithm recently proposed by Rao [2] in

2016 is also an effective and efficient methods that has been widely applied to solvemany optimization problems and showed its good performance It gains dominateresults when being tested with benchmark test functions in comparison with othermeta-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 thedesign optimization effectively Moreover, one of the most important issues inengineering design is that the optimal designs are often effected by uncertaintieswhich can be occurred 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

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 kỹ thuật tối ưu hóa đã được đề xuất và áp dụng để giải quyết một loạt các vấn

đề khác nhau Trong số đó, các thuật toán meta-heuristic đã trở nên phổ biến trongnhững năm gần đây trong việc giải quyết các vấn đề tối ưu hóa thiết kế của nhiềuloại cấu trúc với các vật liệu khác nhau Các thuật toán meta-heuristic này bao gồmGenetic Algorithms, Particle Swarm Optimization, Bat Algorithm, Cuckoo Search,Differential Evolutioin, Firefly Algorithm, Harmony Search, Flower PollinationAlgorithm, Ant Colony Optimization, Bee Algorithms, Jaya Algorithm và nhiềuthuật toán khác Trong số các phương pháp được đề cập ở trên, DifferentialEvolution là một trong những phương pháp được sử dụng rộng rãi nhất Kể từ khiđược Storn và Price [1] giới thiệu lần đầu tiên, nhiều nghiên cứu đã được thực hiện

để cải thiện và áp dụng DE trong việc giải quyết các vấn đề tối ưu hóa cấu trúc DE

đã chứng minh hiệu suất tuyệt vời trong việc giải quyết nhiều vấn đề kỹ thuật khácnhau Bên cạnh thuật toán Differential Evolution, thuật toán Jaya được Rao [2] đềxuất gần đây cũng là một phương pháp hiệu quả và đã được áp dụng rộng rãi để giảiquyết nhiều vấn đề tối ưu hóa và cho thấy hiệu suất tốt Nó đạt được kết quả vượttrội khi được thử nghiệm với các hàm test benchmark so với các phương pháp dựatrên dân số khác Tuy nhiên, giống như nhiều thuật toán tối ưu hóa dựa trên dân sốkhác, một trong những nhược điểm của DE và Jaya là thời gian tính toán tối ưuchậm hơn nhiều so với các phương pháp tối ưu hóa dựa trên độ dốc (gradient-basedalgorithms) Điều này là do DE và Jaya mất rất nhiều thời gian để đánh giá hàmmục tiêu của các cá thể trong bộ dân số Để khắc phục nhược điểm này, các mạng

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 phươngpháp tiếp 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

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

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

h,t The thickness of the composite beam/plate

Q Matrix of material stiffness coefficients

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

Greek Symbols

 Natural frequency

 xx Normal stress in x direction

 yy Normal stress in y direction

 xy Shear stress in xy direction

 yz Shear stress in yz direction

 xz Shear stress in xz direction

 yy Normal strain in y direction

xy Shear strain in xy direction

 yz Shear strain in yz direction

xz Shear strain in xz direction

ABDE Artificial 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

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

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 optimizationproblems and thus require optimization techniques to solve However, as most real-world problems are highly non-linear, traditional optimization methods usually donot 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, batalgorithm, cuckoo search, differential evolution, firefly algorithm, harmony search,flower pollination algorithm, ant colony optimization, bee algorithms, Jayaalgorithm and many others The popularity of meta-heuristic algorithms can beattributed 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 structuraloptimization problems [4] Structural optimization is a potential field and hasattracted 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-basedoptimization methods can be named here as sequential 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 arevery fast in reaching the optimal solution, but easy trapped in local extrema andrequires the gradient information to construct the searching algorithm Besides, thegradient-based approaches are limited to continuous design variables and thatdecreases the productivity of the algorithm In addition, the initial solution (or

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), FireflyAlgorithm (FA), etc [10] These methods are used extensively in structuralproblems because of their flexibility and efficiency in handling both continuous anddiscontinuous design variables In addition, the solutions obtained from population-based algorithms in most cases are global ones Therefore, the optimal result of theproblem is not too much influenced by the initial solution (or initial design of thestructure) 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[1], many studies have been carried out to improve and apply DE in solvingstructural optimization problems The DE has demonstrated excellentlyperformance in solving many different engineering problems Wang et al [11]applied the DE for designing optimal truss structures with continuous and discretevariables Wu and Tseng [12] applied a multi-population differential evolution with

a penalty-based, self-adaptive strategy to solve the COP of the truss structures Anh et al [13] using an improved Differential Evolution algorithm and a smoothedtriangular plate element for static and frequency optimization of folded laminatedcomposite plates Ho-Huu et al [14] proposed a new version of the DE to optimizethe 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 otherpopulation-based methods such as homomorphous mapping (HM), adaptivesegregational constraint handling evolutionary algorithm (ASCHEA), simple multi-membered evolution strategy (SMES), genetic algorithm (GA), particle swarmoptimization (PSO), differential evolution (DE), artificial bee colony (ABC),biogeography based optimization (BBO) Moreover, it has been also successfully

Le-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 thedisadvantages of DE and Jaya is that the optimal computational time is much slowerthan the gradient-based optimization methods This is because DE and Jaya takes alot of time in evaluating the fitness of individuals in the population Specifically, inthe structural optimization problem, the calculation of the objective function orconstraint function values is usually done by using the finite element to analyze thestructural response To overcome this disadvantage, artificial neuron networks(ANN) are proposed to combine with the DE algorithm Based on the idea ofimitation of the brain structure, ANN is capable of approximating an outputcorresponding to a set of input data quickly after the network has been trained, alsoknown as a learning process Thanks to this remarkable advantage, the computation

of objective function or constraint function values in the DE algorithm will be donequickly As a result, ANN will help significantly improve the efficiency of DEcalculations The effectiveness and applicability of ANN since the earlygroundwork ideas put forward by Warren McCulloch and Walter Pitts [18] in 1943have so far proved to be very convincing through numerous studies Applicationareas include system identification and control, pattern recognition, sequencerecognition (gesture, speech, handwritten text recognition), data mining,visualization, machine translation, social networking filtering and email spamfiltering, etc [19]–[24]

The next issue is the development of optimal algorithms integrated ANN with DE

effectiveness of the method At present, the structures made from compositematerial are widely used in almost all fields such as construction, mechanicalengineering, marine, aviation, etc In particular, beams and reinforced plates made

of composite material are an outstanding form and are used increasingly by itssuperior advantages By combining the advantages of composite materials and thereinforced beams structure, the reinforced composite plates have very high bendingstrength with very light weight Nowadays, reinforced composite plates have beenwidely 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, reinforcedcomposite plate usually has higher economic efficiency in practical applications.Due to its high practical applicability, the need to optimize the design of thestructure 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 ofstructure, finding a good algorithm for optimizing design parameters is essential toensure computational efficiency Composite material structures have very complexbehavioral equations, influenced by many geometric and material parameters Thesecharacteristics of the composite mechanical system also lead to the complexity ofthe system of equations to describe the optimal problems, from the objectivefunctions to the constrained equations So the use of gradient-based algorithms isnot straightforward For such types of problems, population-based methodologiesare a superior 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

an optimal design RBDO is the focus of this thesis Although RBDO is morereliable than static optimization, the biggest drawback of RBDO in practicalapplication is the high computational cost To solve this problem, a lot of researchhas been done to find effective reliability analysis techniques, such as: sensitivity-based approximation approaches [25], [26], most probable point (MPP)-basedapproaches, Monte Carlo simulations [27]–[29] and response surface model-basedapproaches [30] These techniques focus on nesting the optimization and thereliability assessment in one process Another RBDO research focus on exploringthe efficient decoupling strategies These strategies can be divided into three groups:nested double-loop methods, decouple-methods, and single-loop methods Amongthese three categories, the double-loop approaches may be the most accurate as itassesses the reliability in every iteration during the optimization process However,its limitation is the huge cost of computation [31]–[33] The decoupled methodssolve the RBDO problem in a different way by separating the optimization andreliability analysis and solve them sequentially Hence, the computational cost can

be reduced considerably [31], [33]– [35] However, this approach still includes twointerrelated loops that result in costly computation To overcome this drawback, thesingle-loop methods have been proposed In this approach, the RBDO problem issolved in a single-loop procedure without reliability analysis The strategy is toconvert an RBDO problem into an approximate deterministic optimization (ADO)problem by transforming probabilistic constraints into approximate deterministicconstraints 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 oflaminated 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 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 compositematerials

- Study and improve population-based optimization methods to increase accuracyand 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 tosolve many different types of problems

- Combine optimal algorithms with groups of reliability assessment methods tosolve 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 andapplication in optimization process This chapter also give out theorganization of the thesis via the outline section and the novelty and goal ofthe thesis for quick review of what is studied in this thesis

 Chapter 2 provides an overview of composite material with basic conceptsand applications in real life The chapter also introduce theory of

composite beam and reinforced composite plate which are the main structureunder 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 andimprove 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 solvingRBDO problem

 Chapter 4 offers the introduction and the historical development of ArtificialNeural Network (ANN) This chapter gives out some basic concepts related

to ANN and introduce the Neural Network Structure which is used in thisthesis to approximate date generated from the Finite Element Analysis.Moreover, the training algorithm, especially the Levenberg-Marquardt and theoverfitting phenomenon are also presented in this chapter

 Chapter 5 illustrate the effectiveness and efficiency of the improve DifferentialEvolution and the improve Jaya in solving optimization problems Thestructures investigated in this section includes planar truss structure, spacetruss structure, Timoshenko composite beam and reinforced composite plate

In particular, the improve Differential Evolution (iDE) is applied to solve foroptimal weight of planar truss structures and space truss structures, then it isused to optimize the fiber angle and the thickness of reinforced compositeplates and show its good effectiveness and performance The last part of thischapter devotes to illustration of the improve Jaya algorithm in looking foroptimal design of the Timoshenko composite beam and the results obtainedprove its highly effective performance and accuracy compared with those ofothers’ author Moreover, this chapter also presents a new approach calledSLDM-iJaya which is formed by the combination of the improve Jayaalgorithm 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 otherauthors and show the effectiveness and the accuracy of the proposed SLDM-iJaya algorithm Afterward, this chapter presents the application of ArtificialNeural Network when it is integrated to a meta-heuristic optimizationmethod, such as Differential Evolution algorithm, to solve the optimizationproblems The integration form a new tool set call ABDE (ANN-BasedDifferential Evolution) algorithm and applied to solve for optimal design ofthe reinforced composite plate The results not only prove the effectiveness

of the proposed method but also open new aspect of applications for futureworks

 Finally, Chapter 6 closes the concluding remarks and give out somerecommendations 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 the dissertationwith eight chapters In the next chapters, fundamental theories, some approaches formodification to improve the solution of some meta-heuristic algorithm andapplication with numerical results will be presented

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 separatematerials combined in a structural unit, are typically made from various combinations

of the other three materials In the early days of modern man-made compositematerials, the constituents were typically macroscopic As composites technologyadvanced over the last few decades, the constituent materials, particularly thereinforcement materials, steadily decreased in size Most recently, there has beenconsiderable interest in “nano-composites” having nanometer-sized reinforcementssuch as carbon nanoparticles, nano-fibers, and nanotubes, because of theextraordinary properties of these materials Composites are generally used becausethey have desirable properties that cannot be achieved by any of the constituentmaterials acting alone The most common example is the fibrous compositeconsisting of reinforcing fibers embedded in a binder or matrix material Particle orflake 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, Woodconsists mainly of fibrous cellulose in a matrix of lignin, whereas most mammalianbone is made up of layered and oriented collagen fibrils in a protein–calciumphosphate matrix [39] Fibrous reinforcement is very effective because many materialsare much stronger and stiffer in fiber form than they are in bulk form It is believedthat 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

to those published by Griffith have been reported for a wide variety of othermaterials The reasons for the differences between fiber and bulk behavior Therecan be no doubt that fibers allow us to obtain the maximum tensile strength andstiffness of a material, but there are obvious disadvantages of using a material infiber form Fibers alone cannot support longitudinal compressive loads and theirtransverse mechanical properties are often not as good as the correspondinglongitudinal properties Thus, fibers are generally useless as structural materialsunless they are held together in a structural unit with a binder or matrix material andunless some transverse reinforcement is provided Transverse reinforcement isgenerally provided by orienting fibers at various angles according to the stress field

in the component of interest The need for fiber placement in different directionsaccording 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 alaminate Although the continuous fiber laminate is used extensively, the potentialfor delamination, or separation of the laminae, is still a major problem because theinterlaminar strength is matrix dominated Woven fiber composites do not havedistinct laminae and are not susceptible to delamination, but strength and stiffnessare sacrificed because the fibers are not as straight as in the continuous fiberlaminate Chopped fiber composites may have short fibers randomly dispersed inthe matrix Chopped fiber composites are used extensively in high-volumeapplications due to their low manufacturing cost, but their mechanical properties areconsiderably poorer than those of continuous fiber composites Finally, hybridcomposites 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 potential

(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 forautomotive, aerospace, marine, and architectural structures in addition to consumerproducts such as skis, golf clubs, and tennis rackets [41] Military aircraft designerswere among the first to realize the tremendous potential of composites with highspecific strength and high specific stiffness, since performance and maneuverability

of those vehicles depend so heavily on weight Composite construction also leads tosmooth surfaces, which reduce drag Since boron and graphite fibers were firstdeveloped in the early 1960s, applications of advanced composites in militaryaircraft have accelerated quickly Carbon fiber composite structural elements such

as horizontal and vertical stabilizers, flaps, wing skins, and various control surfaceshave

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 theprimary structure - including the fuselage and wings - on the 787 consists of carbonfiber/epoxy composite materials or carbon fiber-reinforced plastics The AirbusA350 XWB is another composites-intensive commercial airliner similar to theBoeing 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

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 exceed70% compared with steel springs Experimental composite engine blocks have beenfabricated from graphite-reinforced thermoplastics, but the ultimate goal is aceramic composite engine that would not require water cooling Chopped glassFRPs have been used extensively in body panels where stiffness and appearance arethe principal design criteria So far, the applications of composites in automotivevehicles have been mainly in secondary structural elements and appearance parts,and the full potential of composite primary structures remains to be explored Withthe increased interest in electric vehicles comes a need for composite structures toreduce vehicle structural weight to compensate for the heavy batteries that are required.For example, the proposed BMW Megacity electric vehicle would have a carbonfiber 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 reinforced plastic Wind turbines (Figure 2 5)

are getting increased attention as environmentally attractive, alternative energysources, 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.)

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.Composite materials such as Graphite/Epoxy and Glass/Epoxy have smaller weightdensity compared to metallic materials For example, the weight densities of high-strength Graphite/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 Inaddition to their weight advantage per unit volume, some composites provide betterstiffness and strength properties compared to metals That is, structural membersmade out of composite materials may undergo smaller deformations, and carrylarger static loads than their metallic counterparts Stiffness of high strengthGraphite/Epoxy is around 22x106 lb/in2 compared to Aluminum’s stiffness of10x106 lb/in2 These advantages on weight and stiffness and strength propertiesmake composites more attractive than alloys [42] Structural designers always seekthe best possible design while using the least amount of resources The measure ofgoodness of a design depends on the application, typically related to strength orstiffness, while resources are measure in terms of weight or cost Therefore, the bestdesign often means either the lowest weight (or cost) with limitations on thestiffness (or strength) properties Traditionally, engineers have based onexperience to achieve such design For a given application, first a set of essentialrequirements are obtained Next, structural modifications that are likely to improvethe performance or reduce the weight or the cost are implemented However, thisapproach is often difficult to satisfy both requirement of weight and stiffness at thesame time because implementation that improves the performance may yielddesigns that violate the strength or stiffness requirements

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,many works have been published for optimization of laminated compositestructures For example, the optimum design of laminated composite For example,the optimum design of laminated composite plates for maximizing the first naturalfrequency can be found in [43]–[45], or those for maximizing the buckling loadfactor in Refs [46]– [48], or those for minimizing the weight in Refs [49], [50], and

or those for maximizing strain energy in Ref [13] The optimal design of laminatedcomposite 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 thebuckling load and minimize the weight at the same time in Ref [55] Theoptimization design of the continuous composite models using the different nongradient-based algorithms (particle swarm algorithm and genetic algorithm) for thethin-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 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 thesearching algorithm Therefore, the non-gradient-based algorithms are easier to becarried out than the gradient-based algorithms Compared with the random search

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 onthe 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, thesolution obtained by these methods is easily trapped in local optimal solutions if theproblem has more than one local extreme [13] Therefore, researchers prefer to usethe non-gradient based methods, especially meta-heuristic optimization methodssuch as Particle Swarm Optimization (PSO), Genetic Algorithm (GA), DifferentialEvolution (DE), Jaya algorithm, for solving the optimization problems of laminatedcomposite structures to obtain the global solution In this thesis, DifferentialEvolution and Jaya algorithm are developed and applied to solve optimizationproblem of two types of composite structure model One model is Timoshenkocomposite beam and another is reinforced composite plate Theory related to thesetwo 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 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

Z 1

(1) (2)

(l) (k) (N)

dx

dx

Z

Figure 2 6 Composite laminated beam model

Consider a segment of composite laminated beam with N layers and the fiberorientations of layers are of i (i  1, ,

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

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

 cos2  (k) sin2  (k) 2sin (k) cos (k) 

T (k)   sin2  (k) cos2  (k) 2sin (k) cos (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 byusing different boundary conditions In this thesis, four types of boundaryconditions are considered including pinned-pinned, fixed-fixed, fixed-free andfixed-pined

1) Pinned-pinned (PP)

The boundary conditions, u o (0)  0 , w o (0)  0 , N x (0)  0, My (0)  0 , w o (L) 

0 ,

M y (L)  0 ,

Q z ( )  0 2 are applied The seven indefinite integration constants are

then determined as follows