(luận án tiến sĩ) phát triển các phương pháp tối ưu hóa thông minh cho các 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 problemilizing the DLM-IDE and SLDM-IDE as in the work of
Ho-Huu et al. [3]. In particularly, the number of constraint function evaluations is
reduced nearly 16% for the case of using the DLM-iJaya to 20% for the case of using
the SLDM-iJaya. These reductions in term of computational costs are 52% and 36%,
respectively. These numbers strongly verified the efficiency and the effectiveness of
the proposed SLDM-iJaya method.
5.6.2 Reliability-based lightweight design
In this section, the proposed method is applied to solve the lightweight reliability-based
optimization design problem of the composite laminated beams with various types of
boundary conditions.
The optimal results are computed with three different required reliability indexes, r,i.
The results are listed in Table 5. 18. The optimal masses obtained, as shown in Table
5. 18, are compared between the DLM-iJaya and the SLDM-iJaya method. As we can
see easily, the optimal solutions of mass are agreed well, but the function count and the
CPU time by the SLDM-iJaya outperformed those of the DLM-iJaya in all cases. In
particularly, the CPU time consumed by the former is less than 5 times compared with
the latter for the case of the P-P condition. This number in term of function count is
even less than 10 times for the case of r,i = 4. The same performances occurred for the
rest boundary conditions of F-F, F-P and C-L. Moreover, the SLDM-iJaya method can
also achieve the same optimal masses obtained by the DLM-iJaya with smaller value
of reliability indexes, o,1’s. For example, in the case of F-P condition with the required
reliability index r,i = 2, the value of o,1 computed by the DLM-iJaya is 2.009 while
this number attained by the SLDM-iJaya is 1.997. These results prove that the SLDMiJaya method can acquire the optimal solutions with higher reliability and less effort.

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

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Figure 5. 11. Comparison of RBDO optimal results with different levels of
reliability
Figure 5. 11 depicted the comparison of optimal results of the RBDO problem with
different levels of reliability. The cyan line represents the result obtained from the
DO problem without the reliability analysis, and its optimal weight is undoubtedly
the smallest values. The top three line represents the optimal solutions achieved by
the RBDO problems with different values of reliability indexes of  =2,3 and 4,
respectively. As we can see, the larger the values of  the larger the values of masses,
which means if we want to improve the reliability of a structure, we need to reinforce
the structure with more materials and that make the weight of the structure increase.
This also implies that the optimization without reliability analysis can result in highrisk structure, although the cost is a lot saved. Therefore, in order to balance the costs
and safety, the RBDO method should be taken into account in the optimization of
structural design. And the SLMD-iJaya is one of the proven most effective method to
deal with such problems.

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CHAPTER 6

CONCLUSIONS AND RECOMMENDATIONS
6.1 Conclusions and Remarks
-

In this dissertation, the improved Differential Evolution is introduced and
apply to solve for optimal fiber angle and thickness of the reinforced
composite plate and the results showed its good effectiveness and accuracy.

-

The elitist selection technique is utilized to modify the selection step of the
original Jaya algorithm to improve the convergence of the algorithm. The
improved Jaya algorithm is then applied to solve for optimization problem of
the Timoshenko composite beam and obtained very good results.

-

In addition, 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 of the continuous
composite beam models. The results obtained is much better and more reliable
than those without the reliability factor.

-

The Artificial Neural Network is integrated to 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 two optimization problems of the reinforced composite
plate structures. The first one is optimizing the fiber angles of the reinforced
composite plate and the second one is optimizing the thickness of the
reinforced composite plate. The results obtained show a highly effective
performance of the proposed ABDE tool set. However, for problems with
available behavioral equations, creating data from FEM to train the model and
apply it to the optimization algorithm may take more time than solving directly
by the optimal algorithm.

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-

Besides, through several numerical results, some main conclusions can be
stated as follows:
 The optimal results of fiber orientations for both cases of square
and rectangular reinforced composite plate show that the solutions
by the iDE agree very well with those by the GA. The errors of
strain energy in both case are very low. The maximum one is just
about 0.2% for the case of rectangular plate. However, the
computational time of the iDE algorithm are much smaller. This
proved the accuracy and the effectiveness of the iDE method. The
results also show that the geometric parameters of the structures
also have influence to the optimal values of the problems.
Particularly, the optimal fiber orientations of the square and
rectangular plate are quite different under the same conditions
 The thickness optimization results show that, in the case of square
plate, the objective function reaches the lowest value with 4
reinforced beams (XX-YY) because the thickness of the plate
obtained is much smaller than the case of 2 reinforced beams. In
the case of rectangular plates, the best results are obtained in the
case of 2 reinforced beams arranged in Y-direction (Y-Y), with the
value of the plate thickness being the smallest. In other words, the
optimal option is usually achieved with the smallest plate thickness
combined with the more number of reinforced beams for the case
of uniform loads.
 The optimization design problems of Timoshenko composite beam

are solved with different types of boundary conditions (P-P, F-F,
F-P and C-L) using four different population-based algorithm
including DE, IDE, Jaya and iJaya and one gradient-based
algorithm from Liu’s work. The numerical results show that the
optimal mass obtained from iJaya are agreed well with other

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solutions. However, the iJaya algorithm consumed least time to
achieve the optimal solution in compared with other approaches.
Among the five methods, the SQP (implemented by fmincon promt
in MATLAB) algorithm used in Liu’s work reached the optimal
solution very fast but it could be stuck in the local optimum and
that affects the value the optimal solution. In comparison with
other global optimization method (DE, IDE, Jaya), the iJaya
method definitely outperforms them. The iJaya algorithm can be
considered as the most effective and the efficient algorithm
because it solved for the global solution at the highest speed of
convergence. In the case where the depth of the composite
laminated beam (h) are divided into thicknesses of the layers of the
beam to optimize. And once again, the iJaya method dominates the
other methods in both the number of function count and the CPU
time.
 The RBDO solutions for the benchmark problem using the SLDMiJaya has also presented and shown in Table 5. 17. The solutions
achieved by the SLDM-iJaya and DLM-iJaya are agreed well with
those obtained by the other methods. However, the computational
costs are significantly decreased when using the DLM-iJaya
method and the SLDM-iJaya method instead of utilizing the DLMIDE and SLDM-IDE. In particularly, the number of constraint
function evaluations is reduced nearly 16% for the case of using

the DLM-iJaya to 20% for the case of using the SLDM-iJaya.
These reductions in term of computational costs are 52% and 36%,
respectively. These numbers strongly verified the efficiency and
the effectiveness of the proposed SLDM-iJaya method.
 The proposed SLDM-iJaya method is applied to solve the
lightweight reliability-based optimization design problem of the

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