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

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 of
EXAMINATION COMMITTEE FOR PROTECTION OF DOCTORAL THESIS
HCM 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

Lam Phat Thuan


ACKNOWLEDGEMENTS
This dissertation has been carried out in the Faculty of Civil Engineering, HCM City
University of Technology and Education, Viet Nam. The process of conducting this
thesis brings excitement but has quite a few challenges and difficulties. And I can say
without hesitation that it has been finished thanks to the encouragement, support and
help of my professors and colleagues.
First of all, I would like to express my deepest gratitude to Assoc. Prof. Dr. Nguyen
Hoai Son and Assoc. Prof. Le Anh Thang, especially Assoc. Prof. Dr. Nguyen Hoai
Son from GACES Group, Ho Chi Minh City University of Technology and
Education, Vietnam for having accepted me as their PhD student and for the
enthusiastic guidance and mobilization during my research.
Secondly, I would like also to acknowledge Msc. Ho Huu Vinh for his
troubleshooting and the cooperation in my study. Furthermore, I am grateful to Civil
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 Civil
Engineering, Ho Chi Minh City University of Technology and Education, for their

professional advice and friendly support.
Finally, this dissertation is dedicated to my parents who have always given me
valuable encouragement and assistance.
Lam Phat Thuan

ii


ABSTRACT
Almost all design problems in engineering can be considered as optimization
problems and thus require optimization techniques to solve. During the past few
decades, many optimization techniques have been proposed and applied to solve a
wide range of various optimization problems. Among them, meta-heuristic
algorithms have gained huge popularity in recent years in solving design optimization
problems of many types of structure with different materials. These meta-heuristic
algorithms include genetic algorithms (GA), particle swarm optimization (PSO), bat
algorithm (BA), cuckoo search (CS), differential evolution (DE), firefly algorithm
(DA), harmony search (HS), flower pollination algorithm (FPA), ant colony
optimization (ACO), bee algorithms (BA), Jaya algorithm and many others. Among
the methods mentioned above, the Differential Evolution is one of the most widely
used methods. Since it was first introduced in 1997 by Storn and Price [1], many
studies have been carried out to improve and apply DE in solving structural
optimization problems. The DE has demonstrated excellently performance in solving
many different engineering problems. Besides the Differential Evolution algorithm,
the Jaya algorithm recently proposed by Rao [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 gains dominate results when being tested with
benchmark test functions in comparison with other meta-heuristic methods. However,
like many other population-based optimization algorithms, one of the disadvantages
of DE and Jaya is that the computational time obtaining optimal solutions is much

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

iii


manufacturing processes, material properties and operating environments. These
uncertainties may cause structures to improper performance as in the original design,
and hence may result in risks to structures [3]. Therefore, reliability-based design
optimization (RBDO) can be considered as an important and comprehensive strategy
for finding an optimal design.
In this dissertation, an improved version of Differential Evolution has been first time
utilized to solve for optimal fiber angle and thickness of the reinforced composite.
Secondly, the Artificial Neural Network is integrated to the optimization process of
the improved Differential Evolution algorithm to form a new algorithm call ABDE
(ANN-based Differential Evolution) algorithm. This new algorithm is then applied to
solve optimization problems of the reinforced composite plate structures. Thirdly, an
elitist selection technique is utilized to modify the selection step of the original Jaya
algorithm to improve the convergence of the algorithm and formed a new version of
the original Jaya called iJaya algorithm. The improved Jaya algorithm is then applied
to solve for optimization problem of the Timoshenko composite beam and obtained
very good results. Finally, the so-called called (SLMD-iJaya) algorithm which is the
combination of the improved Jaya algorithm and the Global Single-Loop
Deterministic Methods (SLDM) has been proposed as a new tool set for solving the
Reliability-Based Design Optimization problems. This new method is applied to look

for optimal design of Timoshenko composite beam structures with certain level of
reliability.

iv


TĨM TẮT
Hầu như các bài tốn thiết kế trong kỹ thuật có thể được coi là những bài tốn tối ưu
và do đó địi 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 tốn meta-heuristic đã trở nên phổ biến trong
những năm gần đây trong việc giải quyết các vấn đề tối ưu hóa thiết kế của nhiều loại
cấu trúc với các 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 Colony Optimization, Bee Algorithms, Jaya Algorithm và nhiều thuật
toán khác. Trong số các phương pháp được đề cập ở trên, Differential Evolution là
một trong những phương pháp được sử dụng rộng rãi nhất. Kể từ khi được 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 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 đã được 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 với các phương pháp dựa trên dân số khác.
Tuy nhiên, giống như nhiều thuật tốn tối ưu hóa dựa trên dân số khác, một trong
những nhược điểm của DE và Jaya là thời gian tính tốn 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 đánh giá hàm mục 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 các thuật toán meta-heuristic, như
Differential Evolution, để tạo thành một phương pháp tiếp cận mới giúp giải quyết

v


các bài tốn tối ưu hóa thiết kế một cách hiệu quả. Bên cạnh đó, một trong những vấn
đề quan trọng nhất trong thiết kế kỹ thuật là các thiết kế tối ưu thường bị ảnh hưởng
bởi những yếu tố ngẫu nhiên. Những yếu tố này có thể xảy ra từ nhiều nguồn khác
nhau, chẳng hạn như quy trình sản xuất, tính chất vật liệu và mơi trường vận hành và
có thể khiến các cấu trúc hoạt động khơng 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 độ tin cậy
(Reliability-Based Design Optimization) có thể được coi là một chiến lược tồn 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 Differential
Evolution đã được sử dụng để tìm góc hướng sợi tối ưu và độ dày của tấm gia cường
vật liệu composite. Thứ hai, Mạng nơ ron nhân tạo (ANN) được tích hợp vào quy
trình tối ưu hóa thuật tốn Differentail Evolution cải tiến để hình thành thuật toán mới
gọi là thuật toán ABDE (Artificial Neural Network-Based Differential Evolution).
Thuật tốn mới này sau đó được áp dụng để giải quyết các bài tốn tối ưu hóa của các
cấu trúc tấm composite gia cường. Thứ ba, một kỹ thuật lựa chọn tinh hoa (Elitist
Selection Technique) được sử dụng để hiệu chỉnh bước lựa chọn của thuật toán Jaya
ban đầu để cải thiện sự hội tụ của thuật toán và hình thành một phiên bản mới của
thuật tốn Jaya được gọi là thuật toán iJaya. Thuật toán Jaya cải tiến (iJaya) sau đó
được áp dụng để giải quyết bài tốn tối ưu hóa dầm Timoshenko vật 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 pháp vòng lặp đơn xác định
(Single-Loop Deterministic Method) đã được đề xuất như một 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 trúc dầm composite Timoshenk và

cho kết quả vượt trội.

vi


CONTENTS
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

vii


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

viii


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 ti ............................................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

ix


NOMENCLATURE
Latin Symbols
b

The width of the composite beam

Cij

Matrix of stiffness

Dm ,Dmb ,Db ,Ds

Material matrices of composite plate

Dbst , Dsts

Material matrices of composite beam

E

Young modulus

F

Loading vector

G

Shear modulus

h,t

The thickness of the composite beam/plate

K

Stiffness matrix of the plate

L

Length of the composite beam

m

Number of constraint satisfactions

N

Number of layers of composite materials

NP

Size of population

CR

Crossover control parameter

p

Vector of random parameters

Q

Matrix of material stiffness coefficients

S

Matrix of compliance

T

Coordinate transformation matrix

u(x), w(x)

Displacement field of the composite beam

x

Vector of design variables

X

Population set

wji

Vector of weights

Greek Symbols



Poison’s ratio

x




Natural frequency



Mass density



Stress field

 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



Strain field

 xx

Normal strain in x 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

x

Mean vector of x

j

Distance between feasible and infeasible design region

Abbreviations
2D

Tow dimension

3D

Three dimension

ANN

Artificial Neural Network

MLP

Multi-Layer Perceptron

DE

Differential Evolution

iDE

improved Differential Evolution

ABDE

Artificial neural network-Based Differential Evolution

xi

PSO

Particle Swarm Optimization

GA

Genetic Algorithm

FA

Firefly Algorithms

HS

Harmony Search

SLDM

Single Loop Deterministic Method

RBDO

Reliability Based Design Optimization

DOF

Degree Of Freedom

ADO

Approximate Deterministic Optimization

MPP

Most Probable Point

CS-DSG3

Cell-Smoothed Discrete Shear Gap technique using
triangle finite element

xii


LIST OF TABLES
TABLES

PAGE

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

xiii


LIST OF FIGURES
FIGURES

PAGE

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 LevenbergMarquardt 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

xiv


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

xv


CHAPTER 1

LITERATURE REVIEW
1.1 An overview on research direction of the thesis
Almost all design problems in engineering can be considered as optimization
problems and thus require optimization techniques to solve. However, as most realworld problems are highly non-linear, traditional optimization methods usually do
not work well. The current trend is to use evolutionary algorithms and meta-heuristic
optimization methods to tackle such nonlinear optimization problems. Meta-heuristic
algorithms have gained huge popularity in recent years. These meta-heuristic
algorithms include genetic algorithms, particle swarm optimization, bat algorithm,
cuckoo search, differential evolution, firefly algorithm, harmony search, flower
pollination algorithm, ant colony optimization, bee algorithms, Jaya algorithm and
many others. The popularity of meta-heuristic algorithms can be attributed to their
good characteristics because these algorithms are simple, flexible, efficient, adaptable

and yet easy to implement. Such advantages make them versatile to deal with a wide
range of optimization problems, especially the structural optimization problems [4].
Structural optimization is a potential field and has attracted the attention of many
researchers around the world. During the past decades, many optimization techniques
have been proposed and applied to solve a wide range of various problems. The
algorithms can be classified into two main groups: gradient-based and popular-based
approach. Some of the gradient-based optimization methods can be named here as
sequential linear programming (SLP) [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 easy trapped in local extrema and requires the gradient information to
construct the searching algorithm. Besides, the gradient-based approaches are limited
to continuous design variables and that decreases the productivity of the algorithm.
In addition, the initial solution (or initial design parameters of the structure) also


greatly affects the ability to achieve global or local solutions of gradient-based
algorithms. The population-based techniques, also known as part of meta-heuristic
algorithms, can be listed such as genetic algorithm (GA), differential evolution (DE),
and particle swarm optimization (PSO), Cuckoo Search (CS), Firefly Algorithm
(FA), etc [10]. These methods are used extensively in structural problems because of
their flexibility and efficiency in handling both continuous and discontinuous design
variables. In addition, the solutions obtained from population-based algorithms in
most cases are global ones. Therefore, the optimal result of the problem is not too
much influenced by the initial solution (or initial design of the structure). Among the
methods mentioned above, the Differential Evolution is one of the most widely used
methods. Since it was first introduced by Storn and Price [1], many studies have been
carried out to improve and apply DE in solving structural optimization problems. The
DE has demonstrated excellently performance in solving many different engineering
problems. Wang et al. [11] applied the DE for designing optimal truss structures with

continuous and discrete variables. 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. Le-Anh et al. [13] using an improved Differential Evolution
algorithm and a smoothed triangular plate element for static and frequency
optimization of folded laminated composite plates. Ho-Huu et al. [14] proposed a
new version of the DE to optimize the shape and size of truss with discrete variables.
Besides the Differential Evolution algorithm, the Jaya algorithm recently proposed
by Rao [2] is also an effective and efficient methods that has been widely applied to
solve many optimization problems and showed its good performance. It gains
dominate results when being tested with benchmark test functions in comparison with
other population-based methods such as homomorphous mapping (HM), adaptive
segregational constraint handling evolutionary algorithm (ASCHEA), simple multimembered evolution strategy (SMES), genetic algorithm (GA), particle swarm
optimization (PSO), differential evolution (DE), artificial bee colony (ABC),
biogeography based optimization (BBO). Moreover, it has been also successfully

2


applied in solving many optimal design problem in engineering as presented in
following literature [15]–[17]. However, the performance of the original Jaya
algorithm is not really high. Therefore, there are many variations of the Jaya
algorithm proposed to improve the original one. In this thesis, a new improved
version of the Jaya algorithm will be presented. The new algorithm aims to improve
the population selection technique for the next generation in order to improve the
speed of convergence, while at the same time ensuring the accuracy and the balance
between the exploration and exploitation of Jaya algorithm.
Moreover, like many other population-based optimizations, one of the disadvantages
of DE and Jaya is that the optimal computational time is much slower than the
gradient-based optimization methods. This is because DE and Jaya takes a lot of time
in evaluating the fitness of individuals in the population. Specifically, in the structural

optimization problem, the calculation of the objective function or constraint function
values is usually done by using the finite element to analyze the structural response.
To overcome this disadvantage, artificial neuron networks (ANN) are proposed to
combine with the DE algorithm. Based on the idea of imitation of the brain structure,
ANN is capable of approximating an output corresponding to a set of input data
quickly after the network has been trained, also known as a learning process. Thanks
to this remarkable advantage, the computation of objective function or constraint
function values in the DE algorithm will be done quickly. As a result, ANN will help
significantly improve the efficiency of DE calculations. The effectiveness and
applicability of ANN since the early groundwork ideas put forward by Warren
McCulloch and Walter Pitts [18] in 1943 have so far proved to be very convincing
through numerous studies. Application areas include system identification and
control, pattern recognition, sequence recognition (gesture, speech, handwritten text
recognition), data mining, visualization, machine translation, social networking
filtering and email spam filtering, etc. [19]–[24].
The next issue is the development of optimal algorithms integrated ANN with DE
and applying the proposed algorithms to a practical structure to examine the

3


effectiveness of the method. At present, the structures made from composite material
are widely used in almost all fields such as construction, mechanical engineering,
marine, aviation, etc. In particular, beams and reinforced plates made of composite
material are an outstanding form and are used increasingly by its superior advantages.
By combining the advantages of composite materials and the reinforced beams
structure, the reinforced composite plates have very high bending strength with very
light weight. Nowadays, reinforced composite plates have been widely used in many
branches of structural engineering such as aircraft, ships, bridges, buildings, etc. For
its advantages in both bending stiffness and the amount of material in comparison

with common bending plate structures, reinforced composite plate usually has higher
economic efficiency in practical applications. Due to its high practical applicability,
the need to optimize the design of the structure to save costs and increase the
efficiency of use is also high. However, because of the complexity of computing the
behavior of this particular type of structure, finding a good algorithm for optimizing
design parameters is essential to ensure computational efficiency. Composite material
structures have very complex behavioral equations, influenced by many geometric
and material parameters. These characteristics of the composite mechanical system
also lead to the complexity of the system of equations to describe the optimal
problems, from the objective functions to the constrained equations. So the use of
gradient-based algorithms is not straightforward. For such types of problems,
population-based methodologies are a superior choice.
Moreover, one of the most important issues in engineering design is that the optimal
designs are often effected by uncertainties which can be occurred from various
sources, such as manufacturing processes, material properties and operating
environments. These uncertainties may cause structures to improper performance as
in the original design, and hence may result in risks to structures [3]. There are two
groups of methods for dealing with uncertainties: reliability-based design and robust
design. Robust design focuses on minimizing variance in design results under
variations of design variables and parameters. Reliability-based design optimization

4


(RBDO) ensures that the design is feasible regardless of changes in design variables
and parameters. RBDO can be considered as a comprehensive strategy for finding an
optimal design. RBDO is the focus of this thesis. Although RBDO is more reliable
than static optimization, the biggest drawback of RBDO in practical application is the
high computational cost. To solve this problem, a lot of research has been done to
find effective reliability analysis techniques, such as: sensitivity-based approximation

approaches [25], [26], most probable point (MPP)-based approaches, Monte Carlo
simulations [27]–[29] and response surface model-based approaches [30]. These
techniques focus on nesting the optimization and the reliability assessment in one
process. Another RBDO research focus on exploring the efficient decoupling
strategies. These strategies can be divided into three groups: nested double-loop
methods, decouple-methods, and single-loop methods. Among these three categories,
the double-loop approaches may be the most accurate as it assesses the reliability in
every iteration during the optimization process. However, its limitation is the huge
cost of computation [31]–[33]. The decoupled methods solve the RBDO problem in
a different way by separating the optimization and reliability analysis and solve them
sequentially. Hence, the computational cost can be reduced considerably [31], [33]–
[35]. However, this approach still includes two interrelated loops that result in costly
computation. To overcome this drawback, the single-loop methods have been
proposed. In this approach, the RBDO problem is solved in a single-loop procedure
without reliability analysis. The strategy is to convert an RBDO problem into an
approximate

deterministic

optimization

(ADO)

problem

by

transforming

probabilistic constraints into approximate deterministic constraints. In so doing, the

computational cost significantly decreased [32], [36], [37]. Therefore, these methods
would be applicable to real-world problems. However, studies that deal with the
reliability-based design optimization of laminated composite beams are quite limited.
In this thesis, the Single-Loop Deterministic Methods (SLDM), which has been
recently proposed by Li et al. [38], will be studied to integrate with a meta-heuristic

5


optimization algorithm to form a new tool set SLDM-iJaya for solving a RBDO
problems of composite structures.
In summary, in this thesis, some modifications will be investigated and propose to
improve the original algorithm of Differential Evolution and Jaya algorithm to
increase the convergence of DE and Jaya algorithm. The modified algorithms are then
combine with ANN and/or SLDM to develop new tools for solving design
optimization problems and the RBDO problems of composite structures, such as
reinforced composite plate, Timoshenko beams, etc.
1.2 Motivation of the research
The motivation to study the topics presented in the thesis comes from the analysis of
published literatures, and from the evaluation of the application potential of
composite material structures and intelligent optimization methods, especially the
reliability-based optimization methods. Therefore, the thesis is motivated by:
- The development / improvement of existing algorithms to improve the efficiency of
solving structural optimization problems with high accuracy and reliability.
- Studying the advantages of Artificial Neural Network (ANN) to combine with
optimal algorithms to improve the speed and the performance of solving structural
optimization problems.
1.3 Goals of the dissertation
Firstly, this thesis focuses on studying and developing meta-heuristic optimization
methods and combines them with the Artificial Neural Network, which has

advantages in approximating data, to build up a new algorithm for solving composite
material structural optimization problems. Particularly, the original Differential
Evolution or Jaya algorithm will be modified to improve the convergence in solving
for global optimal solution and then, the ANN will be integrated to the improved
meta-heuristic algorithms to form a new algorithm, which is used to look for optimal
design of reinforced composite plate structures.
Secondly, the thesis also proposes a new tool set, which is the combination of metaheuristic optimization algorithm and the Single-Loop Deterministic Method to deal

6


with Reliability-Based Design Optimization (RBDO) problems. In particular, the
original Jaya algorithm will be modified to improve the convergence in searching
optimal solutions of the optimization problems. Then, this improved version of Jaya
algorithm will be combined with Single-Loop Deterministic Method to solve the
Reliability-Based Design Optimization of composite beam structures.
1.4 Research scope of the 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
solve many 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 main purpose
of evaluating the effectiveness, accuracy and reliability of the proposed optimization
methods. The application of optimal methods proposed in the thesis for more complex

problems will be further studied in the future.
1.5 Outline
The dissertation contains seven chapters and is structured as follows:
 Chapter 1 presents an overview on meta-heuristic algorithms, composite
material structure and especially artificial neural networks and its role and
application in optimization process. This chapter also give out the organization
of the thesis via the outline section and the novelty and goal of the thesis for
quick review of what is studied in this thesis.
 Chapter 2 provides an overview of composite material with basic concepts and
applications in real life. The chapter also introduce theory of Timoshenko

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 to
Differential Evolution and Jaya algorithm and the approach to modify and
improve the original of the algorithm to obtain an improved version of its. This
chapter also gives out an overview and formulation for Reliability-Based
Design Optimization (RBDO) and the proposed methods for solving RBDO
problem.
 Chapter 4 offers the introduction and the historical development of Artificial
Neural Network (ANN). This chapter gives out some basic concepts related to
ANN and introduce the Neural Network Structure which is used in this thesis
to approximate date generated from the Finite Element Analysis. Moreover,
the training algorithm, especially the Levenberg-Marquardt and the overfitting
phenomenon are also presented in this chapter.
 Chapter 5 illustrate the effectiveness and efficiency of the improve Differential
Evolution and the improve Jaya in solving optimization problems. The

structures investigated in this section includes planar truss structure, space
truss structure, Timoshenko composite beam and reinforced composite plate.
In particular, the improve Differential Evolution (iDE) is applied to solve for
optimal weight of planar truss structures and space truss structures, then it is
used to optimize the fiber angle and the thickness of reinforced composite
plates and show its good effectiveness and performance. The last part of this
chapter devotes to illustration of the improve Jaya algorithm in looking for
optimal design of the Timoshenko composite beam and the results obtained
prove its highly effective performance and accuracy compared with those of
others’ author. Moreover, this chapter also presents a new approach called
SLDM-iJaya which is formed by the combination of the improve Jaya
algorithm and the single-loop methods for solving the RBDO problem of the
Timoshenko composite beam. This chapter illustrate the solutions for two

8