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
ii
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

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: 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
Datemonthyear
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
ii
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
iii
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
iv
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.
v
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 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 toán tối ưu hóa dựa trên dân số khác, một trong
những nhược điểm của DE và Jaya là thời gian tính toán tối ưu chậm hơn nhiều so
với các phương pháp tối ưu hóa dựa trên độ dốc (gradient-based algorithms). Điều
này là do DE và Jaya mất rất nhiều thời gian để 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
vi
các bài toán 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 môi trường vận hành 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 độ tin cậy
(Reliability-Based Design Optimization) có thể được coi là một chiến lược toàn 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 toán Differentail Evolution cải tiến để hình thành thuật toán mới
gọi là thuật toán ABDE (Artificial Neural Network-Based Differential Evolution).
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 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 toán Jaya được gọi là thuật toán iJaya. Thuật toán Jaya cải tiến (iJaya) sau đó
được áp dụng để giải quyết bài toán 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.
vii
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
viii
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
ix
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
x
NOMENCLATURE
Latin Symbols
b The width of the composite beam
Cij Matrix of stiffness
m mb b sD ,D ,D ,D Material matrices of composite plate
,b sst stD D 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
xi
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
xii
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
xiii
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
xiv
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 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
xv
Figure 5. 3. A 72-bars space truss structure ............................................................. 74
Figure 5. 4. Structure of 120-bars arch space truss .................................................. 76
Figure 5. 5. Model of a reinforced composite plate ................................................. 77
Figure 5. 6. Models of square and rectangular reinforced composite plates ........... 79
Figure 5. 7. Model of reinforced composite plate for optimization ......................... 81
Figure 5. 8. Convergence curves of DE, IDE, Jaya and iJaya for the beam with P-P
condition .................................................................................................................... 89
Figure 5. 9. Convergence curves of DE, IDE, Jaya and iJaya for the beam with P-P
condition. ................................................................................................................... 91
Figure 5. 10. Comparison of different design approaches with different boundary
conditions. ................................................................................................................. 92
Figure 5. 11. Comparison of RBDO optimal results with different levels of reliability
................................................................................................................................... 97
CHAPTER 1
LITERATURE REVIEW
1.1 An overview on research direction of the thesis
Almost all design problems in engineering can be considered as optimization
problems and thus require optimization techniques to solve. However, as most real-
world problems are highly non-linear, traditional optimization methods usually do
not work well. The current trend is to use evolutionary algorithms and meta-heuristic
optimization methods to tackle such nonlinear optimization problems. Meta-heuristic
algorithms have gained huge popularity in recent years. These meta-heuristic
algorithms include genetic algorithms, particle swarm optimization, bat algorithm,
cuckoo search, differential evolution, firefly algorithm, harmony search, flower
pollination algorithm, ant colony optimization, bee algorithms, Jaya algorithm and
many others. The popularity of meta-heuristic algorithms can be attributed to their
good characteristics because these algorithms are simple, flexible, efficient, adaptable
and yet easy to implement. Such advantages make them versatile to deal with a wide
range of optimization problems, especially the structural optimization problems [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
2
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 Evo...lution. In this thesis, Differential
Evolution and Jaya algorithm are developed and applied to solve optimization
problem of two types of composite structure model. One model is Timoshenko
composite beam and another is reinforced composite plate. Theory related to these
two composite structures are presented in the following sections of this chapter.
2.2 Analysis of Timoshenko composite beam
Composite laminated Timoshenko beams can be treated as continuous models and
discrete models. The discrete models are easier to be implemented but difficult to
obtain the exact solution. It can only derive the approximate solution. In addition, the
discrete models such as finite element approaches are not so effective as the analytical
approaches of continuous models. Therefore, Liu [53] proposed an approach that
treated composite laminated Timoshenko beam as continuous model to achieve the
exact solution. The process to build up the analytical solution for the composite
laminated beam is simply presented as in the following section. For more details of
the method, readers are encouraged to refer to Liu’s work.
2.2.1. Exact analytical displacement and stress
19
dx
Z1
ZN
ZN
Z2
Zl
Zk+1
b
h/2
h/2
Z
X
Y
(N)
(N-1)
(k)
(l)
(2)
(1)
Figure 2. 6. Composite laminated beam model
Consider a segment of composite laminated beam with N layers and the fiber
orientations of layers are of ( 1,..., ) i i N . The positions of layers are denoted by
( 1,..., )iz i 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 + dQ
M + dM
Nx +dNx
dx
q(x)
Nx
Q
M
Figure 2. 7. Free-body diagram
The displacement fields of the composite laminated beam calculated analytically
based on the first-order shear deformation theory (also called Timoshenko beam
theory) are:
20
3 20
1 4 5
1
( ) 4
6 2
o
q
u x B x C x C x C
(2.1)
4 3 20 0
1 2 6 7
1 1
( )
24 6 2 2
o
q q
w x A x AC x C AC x C x C
(2.2)
3 20
1 2 3
1
( ) 4
6 2
q
x A x C x C x C
(2.3)
where ( 1,...,7)iC i are indefinite integration constants determined by using the
boundary conditions of the composite laminated beams as shown in the following
section.
11 11
2 2
11 11 11 11 11 11 55
1
, ,
( ) ( )
A B
A B C
b B A D b B A D bKA
(2.4)
where
11 11 11 55, , ,A B D A are respectively extensional stiffness, bending-extensional
coupling stiffness, bending stiffness and extensional stiffness of the composite
laminate. K is the shear correction factor with the value of 5/6.
X
Y
Z=3
1
2
O
Figure 2. 8. The material and laminate coordinate system
The stress fields of the composite laminated beam include the plane stress
components and the shear stress components. According to the coordinate system
between the materials (123) and the beam/laminate (xyz) as depicted in Figure 2. 8,
in which the fiber orientation coincides with the 1-axis, the plane stress components
are expressed as follows
21
1
( )( )
2 1
12
,
x
kk
y k k
xy
T Q z z z
(2.5)
where the strain components 0, 0y xy , and
2 20 0
1 4 1 2
3 2
x
q q
B x C x C zA x C x C
(2.6)
( )kT is the coordinate transformation matrix and
( )k
Q is the matrix of material stiffness
coefficients
2 ( ) 2 ( ) ( ) ( )
( ) 2 ( ) 2 ( ) ( ) ( )
( ) ( ) ( ) ( ) 2 ( ) 2 ( )
cos sin 2sin cos
sin cos 2sin cos
sin cos sin cos cos sin
k k k k
k k k k k
k k k k k k
T
(2.7)
( ) ( ) ( )
11 12 61
( ) ( ) ( ) ( )
21 22 26
( ) ( ) ( )
16 26 66
k k k
k k k k
k k k
Q Q Q
Q Q Q Q
Q Q Q
(2.8)
The shear stress components in the material coordinate systems are
( )23 ( )
1
13
,
k yzk
ss k k
xz
T Q z z z
(2.9)
where the shear strain components 0 yz and
3 2 3 20 0
1 2 3 1 0 2 6
1 1 1
( )
6 2 6 2
xz
q
A x C x C x C A x AC x Cq AC x C
(2.10)
The coordinate transformation matrix ( )k
sT and the matrix of stiffness coefficients
( )k
sQ can be described as
( ) ( )
( )
( ) ( )
sin cos
cos sin
k k
k
s k k
T
(2.11)
22
( ) ( )
( ) 44 45
( ) ( )
45 55
k k
k
s k k
Q Q
Q
Q Q
(2.12)
In the above equations,
( )k
ijQ is the stiffness coefficients of the kth lamina in the
laminate coordinate system and are described clearly in [53].
2.2.2. Boundary-condition types
The indefinite integration constants in the above equations can be determined by
using different boundary conditions. In this thesis, four types of boundary conditions
are considered including pinned-pinned, fixed-fixed, fixed-free and fixed-pined.
1) Pinned-pinned (PP)
The boundary conditions, (0) 0ou , (0) 0ow , (0) 0xN , (0) 0yM , ( ) 0ow L ,
( ) 0yM L , ( ) 0
2
z
L
Q are applied. The seven indefinite integration constants are
then determined as follows
11 11 0
1
11 11
(2 3 )
,
6( )
B B D A q L
C
D A B B 2
0,C
311 11
3 0
11 11
3
,
72( )
D A B B
C q L
D A B B
4 0,C 5 0,C
311 11
6 0 0
11 11
3 1
,
272( )
B B D A
C Aq L Cq L
D A B B 7
0,C
(2.13)
2) Fixed-fixed (FF)
The boundary conditions, (0) 0ou , (0) 0ow , (0) 0 , ( ) 0ou L , ( ) 0ow L ,
( ) 0yM L , ( ) 0
2
z
L
Q are used to determine seven indefinite integration constants
0
1 ,
2
q L
C
2
2 ,
12
o
q L
C
3 0,C
2
0
4 ,
12
q L
C
5 0,C 6 0
1
,
2
C Cq L
7 0,C
(2.14)
3) Fixed-pinned (FP)
23
The boundary conditions, (0) 0ou , (0) 0ow , (0) 0 , ( ) 0ow L , ( ) 0xN L ,
( ) 0yM L ,
5
( ) 0
8
z
L
Q are employed and the seven the indefinite integration
constants are then determined
2
0 0
1
5 3
,
8
Aq L Cq
C
AL
2
0 0
2
3
,
8
Aq L Cq
C
A
3 0,C
2
0 0
4
7 9
,
24
Aq L Cq
C
A
5 0,C 6 0
5
,
2
C Cq L
7 0,C
(2.15)
4) Cantiliver/Fixed-free (CL)
The boundary conditions, (0) 0ou , (0) 0ow , (0) 0 , ( ) 0xN L ,
2
0(0)
2
y
q L
M
, ( ) 0yM L , ( ) 0zQ L , are applied to determine the seven the indefinite integration
constants
11 11 0
1
11 11
(3 3 2 )
,
6 ( )
AD b B Bb q L
C
b B B AD
2
11 0
2
11 11
(3 )
,
6 ( )
B Bb q L
C
b B B AD
3 0,C
2
11 0
4
11 11
(3 )
,
6 ( )
AD b q L
C
b B B AD
5 0,C 6 0 ,C Cq L 7 0,C
(2.16)
In the above boundary conditions, , ,x z zN Q M are the normal force along the x axis,
shear force along the z axis and bending moment about the y axis on the cross section
of the composite laminated beam, respectively, and are computed as in [53].
2.3 Analysis of reinforced composite plate
Reinforced composite plate is formed by a composite plate combining with a
stiffening Timoshenko composite beam, as illustrated in Figure 2. 9. The beam is
considered as a stiffener and is set parallel with the axes in the surface of the plate.
The centroid of the beam has a distance e from the middle plane of the plate. The
plate-beam system is discretized by a set of node. The degree of freedom (DOF) of
each node of the plate is [ , , , , ] Tx yu v wd , in which , ,u v w are the displacements
24
at the middle of the plate and , x y are the rotations around the y-axis and x-axis.
Each node of the beam has the DOF of [ , , , , ] Tst r s z r su u ud , where , ,r s zu u u are
respectively centroid displacements of beam and , r s are the rotations of beam
around r-axis and s-axis.
Figure 2. 9. A composite plate reinforced by an r-direction beam
The displacement compatibility between plate & beam is ensured by:
( ) ( ) ; ( ) ; ( ) r r s zu u r z r v z r w u r (2.17)
The strain energy of composite plate is given by:
0 0 0 0
1
d
2
T m T mb T mb T b T s
P b b b b
A
U Aε D ε ε D κ κ D ε κ D κ γ D γ
(2.18)
where
0 , ,bε κ γ are respectively membrane, bending and shear strains of composite
plate and are expressed as follows
0 , , , , , , , , , ,[ , , ] ; [ , , ] ; [ , ] .
T T T
x y y x b x x y y x y y x x x y yu v u v w wε κ γ (2.19)
m mb b sD ,D ,D ,D are material matrices of plate.
The strain energy of composite stiffener is given by
12 ( ) ( ) d
b T b b s T s s
st st st st st st st
l
U xε D ε ε D ε (2.20)
where ,b sst stε ε are respectively bending, shear strain of beam and are expressed as
follows:
, 0 , , , ,[ , , ] ; [ ]
b T s T
st r r r r r r s r st z r ru z uε ε
(2.21)
,b sst stD D are material matrices of composite beam.
25
Using the superposition principle, total energy strain of reinforced composite plate
is obtained:
1
siN
P st
i
U U U
(2.22)
where
stN is the number of stiffeners.
For static analysis, the global equations for the reinforced composite plate
K F can found in [61] for detail.
26
CHAPTER 3
Reliability-based optimization Methods with iJaya and
improved Differential Evolution
3.1 Overview of Metaheuristic Optimization
In meta-heuristic algorithms, meta- means ‘beyond’ or ‘higher level’. They generally
perform better than simple heuristics. All meta-heuristic algorithms use some tradeoff
of local search and global exploration. The variety of solutions is often realized via
randomization. Despite the popularity of meta-heuristics, there is no agreed definition
of heuristics and meta-heuristics in the literature. Some researchers use ‘heuristics’
and ‘meta-heuristics’ interchangeably. However, the recent trend tends to name all
stochastic algorithms with randomization and global exploration as meta-heuristic.
Randomization provides a good way to move away from local search to the search
on the global scale. Therefore, almost all meta-heuristic algorithms are usually
suitable for nonlinear modeling and global optimization. Meta-heuristics can be an
efficient way to produce acceptable solutions by trial and error to a complex problem
in a reasonably practical time. The complexity of the problem of interest makes it
impossible to search every possible solution or combination, the aim is to find good
feasible solution in an acceptable timescale. There is no guarantee that the best
solutions can be found, and we even do not know whether an algorithm will work and
why if it does work [62], [63]. The idea is to have an efficient and practical algorithm
that will work most the time and is able to produce good quality solutions. Among
the found quality solutions, it can be expected that some of them are nearly optimal,
though there is no guarantee for such optimality.
The main components of any meta-heuristic algorithms are: intensification and
diversification, or exploitation and exploration [64]. Diversification means to
generate diverse solutions so as to explore the search space on the global scale, while
intensification means to focus on the search in a local region by exploiting the
27
information that a current good solution is found in this region. This is in combination
with the selection of the best solutions [65]. The selection of the best ensures that the
solutions will converge to the optimality. On the other hand, the diversification via
randomization avoids the solutions being trapped at local optima, while increases the
diversity of the solutions. The good combination of these two major components will
usually ensure that the global solution is achievable.
Meta-heuristic algorithms can be classified in many ways. One way is to classify
them as: population-based and trajectory-based [63]. For example, genetic algorithm
(GA) and genetic programming (GP) are population-based as they use a set of strings,
so is the particle swarm optimization (PSO) which uses multiple agents or particles
[66]. On the other hand, simulated annealing (SA) [67] uses a single solution which
moves through the design space or search space, while artificial neural networks use
a different approach.
Modeling and optimization may have different emphasis, but for solving the real
world problems, we often have to use both modeling and optimization because
modeling makes the objective functions are evaluated using the correct
mathematical/numerical model of the problem of interest, while optimization can
achieve the optimal settings of design parameters. For optimization, the essential part
is the optimization algorithms. For this reason, we will focus on the algorithms,
especially meta-heuristic algorithms
3.1.1 Meta-heuristic Algorithm in Modeling
Various methodologies can be employed for nonlinear system modeling. Each
method has its own advantages or drawbacks. The need to determine both the
structure and the parameters of the engineering systems makes their modeling a
difficult task. In general, models are classified into two main groups: (1)
phenomenological and (2) behavioral [68]. The first class is established by taking into
account the physical relationships governing a system. The structure of a
phenomenological model is chosen on the basis of a prior knowledge about the
system. To cope with the complexity of design of the phenomenological models,
28
behavioral models are usually used. The behavioral models capture the relationships
between the inputs and outputs on the basis of a measured set of data. Thus, there is
no need for a prior knowledge about the mechanisms that produced the experimental
data. Such models are beneficial because they can provide very good results with
minimal effort [68]–[73]. Statistical regression techniques are widely-used behavioral
modeling approaches.
Several alternative meta-heuristic approaches have been developed for the behavioral
modeling. Developments in computer hardware during the last two decades have
made it much easier for these techniques to grow into more efficient frameworks. In
addition, various meta-heuristics may be used as efficient tools in problems where
conventional approaches fail or perform poorly. Two well-known classes of the meta-
heuristic algorithms used in nonlinear modeling are Artificial Neural Networks
(ANNs) [74] and Genetic Programming (GP) [75]. ANNs have been used for a wide
range of structural engineering problems (e.g. [69], [76]). In spite of the successful
performance of ANNs, they usually do not give a deep insight into the process which
they obtain a solution. GP, as an extension of genetic algorithms (GAs), possess
completely new characteristics. This technique is essentially a supervised machine
learning approach that searches a program space instead of a data space. GP
automatically generates computerized programs that are represented as tree structures
and expressed using a functional programming language ([69]–[71], [75]). The ability
of generating prediction models without assuming the form of the existing
relationships is surely a main advantage of GP over regression and ANN techniques.
GP and its variants are widely used for solving real world problems (e.g., [69], [70]).
There are some other meta-heuristic algorithms have been used in the literature for
modeling such as Fuzzy Logic (FL) and Support Vector Machine (SVM).
3.1.1.1 Artificial Neural Networks
ANNs emerged as a result of simulation of biological nervous system. The ANN
method was developed in the early 1940s by McCulloch and co-workers [77]. The
first studies were focused on building simple neural networks to model simple logic
29
functions. At present, ANNs have been applied to problems that do not have
algorithmic solutions or problems with complex solutions. In this study, the
approximation ability of two of the most well-known ANN architectures, MLP and
RBF, are investigated
3.1.1.2 Genetic programming
GP is a symbolic optimization technique that creates computer programs to solve a
problem using the principle of Darwinian natural selection [75]. Friedberg [78] left
the first footprints in the area of GP by using a learning algorithm to stepwise improve
a program. Much later, Cramer [79] applied genetic algorithms (GAs) and tree-like
structures to evolve programs. The breakthrough in GP then came in the late 1980s
with the experiments of Koza [75] on symbolic regression. GP was introduced by
Koza as an extension of GA. The main difference between GP and GA is the
representation of the solution. The GP solutions are computer programs that are
represented as tree structures and expressed in a functional programming language
(like LISP) [75]. GA first creates a string of numbers that represent the solutions,
while in GP, the evolving programs (individuals) are parse trees than can vary in
length throughout the run rather than fixed-length binary strings. Essentially, this is
the beginning of computer programs that can program themselves [75]. Since GP
often evolves computer programs, the solutions can be executed without post-
processing, while coded binary strings typically evolved by GA require post-
processing. The optimization techniques, like GA, are generally used in parameter
optimization to evolve so as to find the best values for a given set of model
parameters. GP, on the other hand, provides the basic structure of the approximation
model, together with the values of its parameters [80]. GP optimizes a population of
computer programs according to a fitness landscape determined by a program ability
to perform a given computational task. The fitness of each program in the population
is evaluated using a predefined fitness function. Thus, the fitness function is the
objective function GP aims to optimize [81]. This classical GP approach is referred
to as tree-based GP. In addition to the traditional Tree-based GP, there are other types
30
of GP where programs are represented in different ways. These are linear and graph-
based GP[82], [83]. The emphasis of the present study is placed on the linear-based
GP techniques.
3.1.1.3 Fuzzy Logic
Fuzzy Logic (FL) is a process of mapping an input space onto an output space using
membership functions and linguistically specified rules [84]. The concept of ‘‘fuzzy
set’’ was preliminarily introduced by [85]. The fuzzy approach is more in-line with
human thought since it provides possible rules relating input variables to the output
variable. FL is well suited to implementing control rules that can only be expressed
verbally. It can also be used for the modeling of systems that cannot be modeled with
linear differential equations [86]. The essential idea in FL is the concept of partial
belongings of any object to different subsets of the universal set instead of full
belonging to a single set. Partial belonging to set can be described numerically by a
membership function [87]. A membership function is a curve, mapping an input
element to a value between 0 and 1, showing the degree to which it belongs to a fuzzy
set. Membership degree is the value of every element, varying between 0 and 1. A
membership function can have different shapes for different kinds of fuzzy sets, such
as bell, sigmoid, triangle, and trapezoid [84]. In FL, rules and membership sets are
used to make a decision. The idea of a fuzzy set is basic and simple: an object is
allowed to have a gradual membership of a set. It means the degree of truth of a
statement can range between 0 and 1, which is not limited to just two logic values
{true, false}.
When linguistic variables are used, these degrees may be managed by specific
functions. A fuzzy system consists of output and input variables. For each variable,
fuzzy sets that characterize those variables are formulated, and for each fuzzy set a
membership function can be defined. After that, the rules that relate the output and
input variables to their fuzzy sets are defined. Figure 1 depicts a typical fuzzy logic
system where a general fuzzy inference system has basically four components:
fuzzification, fuzzy rule base, fuzzy inference engine, and defuzzification [87]. In the
31
fuzzification stage, each piece of the input data is converted to degrees of membership
by a lookup in one or more several membership functions. The fuzzy rule base
contains rules including all possible fuzzy relations between the inputs and outputs.
These rules can be expressed as a collection of IF-THEN statements. All the rules
have antecedents and consequents. Fuzzy inference engine takes into account all the
fuzzy rules in the fuzzy rule base, and learns how to transform a set of inputs into
their corresponding outputs. The fuzzy inference process generates the resulting
fuzzy set, based on the input and the antecedents of the rules. Finally, the resulting
fuzzy outputs are converted from the fuzzy inference engine to a number through the
so-called defuzzification process [87].
3.1.2 Meta-heuristic Algorithm in Optimization
To find an optimal solution to an optimization problem is often a very challenging
task, depending on the choice and the correct use of the right algorithm. The choice
of an algorithm may depend on the type of problem, the available of algorithms,
computational resources, and time constraints. For large-scale, nonlinear, global
optimization problems, there is often no agreed guideline for algorithm choice, and
in many cases, there is no efficient algorithm. For hard optimization problems,
especially for nondeterministic polynomial-time hard, or NP-hard, optimization
problems, there is no efficient algorithm at all. In most applications, an optimization
problem can be commonly expressed in the following generic form [63], [88]:
where fi(x), hj (x) and gk(x) are functions of the design vector x = (x1, x2, ..., xn)T.
Here the components xi of x are called design or decision variables, and they can be
real continuous, discrete or the mixed of these two. The functions fi(x) where i = 1,
2,...M are called the objective functions, or simply cost functions, and in the case of
M = 1, there is only a single objective. The space spanned by the decision variables
minimize ( ), ( 1, 2,..., )
subject to
( ) 0, ( 1,2,..., ),
( ) 0, ( 1,2,..., )
n
i
j
k
x f x i M
h x j J
g x k K
32
is called the design space or search space. The equalities for hj and inequalities for gk
are called constraints. It is worth pointing out that we can also write the inequalities
in the other way ≥ 0, and we can also formulate the objectives as a maximization
problem. Various algorithms may be used for solving optimization problems. The
conventional or classic algorithms are mostly deterministic. As an instance, the
simplex method in linear programming is deterministic. Some other deterministic
optimization algorithms, such as Newton-Raphson algorithm, use the gradient
information and are called gradient-based algorithms. Non-gradient-based, or
gradient-free/derivative-free, algorithms only use the function values, not any
derivative [89]. Heuristic and Meta-heuristic are the main types of the stochastic
algorithms. The difference between Heuristic and meta-heuristic algorithms is
negligible. Heuristic means ‘to find’ or ‘to discover by trial and error’. Quality
solutions to a tough optimization problem can be found in a reasonable amount of
time, but there is no guarantee that optimal solutions are reached. It hopes that these
algorithms work most of the time, but not necessarily all the time. This is good when
good solutions which are easily reachable are need not necessarily the best solutions
[63], [90]. As discussed earlier in this chapter, meta-heuristic optimization algorithms
are often inspired from nature. According to the source of inspiration of the meta-
heuristic algorithms they can be classified into different categories as shown in
Figure 3. 1. The main category is the biology-inspired algorithms which generally
use biological evolution and/or collective behavior of animals. Science is another
source of inspiration for the meta-heuristics. These algorithms are usually inspired
physic and chemistry. Moreover, art-inspired algorithms have been successful for the
global optimization. They are generally inspired from artists’ behavior to create
artistic stuffs (such as musicians and architectures). Socially inspired algorithms can
be defined as another source of inspiration and the algorithm simulate the social
behavior to solve optimization.
33
Nature - Inspired
Bio - Inspired Science - Inspired Art - Inspired Social - Inspired
Figure 3. 1. Source of inspiration in meta-heuristic optimization algorithms
Although there are different sources of inspirations for the meta-heuristic
optimization algorithms, they have similarities in their structures. Therefore, they can
also be classified into two main categories as Evolutionary Algorithms and Swarm
Algorithms.
3.1.2.1 Evolutionary Algorithms
The evolutionary algorithms generally use an iterative procedure, based on a
biological evolution progress to solve optimization problems. Some of the
evolutionary algorithms are described below:
a) Genetic Algorithm
Genetic algorithm (GA) is a powerful optimization method based on the principles of
genetics and natural selection. Holland [91] was the first to use the crossover and
recombination, mutation, and selection in the study of adaptive and artificial systems.
These genetic operators form the essential part of GA for problem-solving. Up to
now, many variants of GA have been developed and applied to a wide range of
optimization problems [92], [93]. One of the main advantages is that GA is a gradient-
free method with flexibility of dealing various types of optimization whether the
objective function is stationary or non-stationary, linear or nonlinear, continuous or
discontinuous, or with random noise. In GA, a population can simultaneously find
the search space in many directions because multiple offsprings in the population act
like independent agents. This feature idealizes the parallelization of the algorithms
for implementation. Further, different parameters and groups of encoded strings can
be manipulated at the same time. Despite several advantages of GAs, they have some
34
disadvantages pertaining to the formulation of fitness function, the usage of
population size, the choice of the important parameters, and the selection criteria of
new population. The convergence of GA can be seriously dependant on the
appropriate choice of these parameters.
b) Differential Evolution
Differential evolution (DE) was developed by Storn and Price [1]. It is a vector-based
evolutionary algorithm, and can be considered as a further development to genetic
algorithms. It is a stochastic search algorithm with self-organizing tendency and does
not use the information of derivatives. DE carries out operations over each component
(or each dimension of the solution). Solutions are represented in terms of vectors, and
then mutation and crossover are carried out using these vectors [72]. For example, in
genetic algorithms, mutation is carried out at one site or multiple sites of a
chromosome, while in differential evolution, a difference vector of two randomly-
chosen vectors is used to perturb an existing vector. Such vectorized mutation can be
viewed as a self-organizing search, directed towards optimality [62], [63]. This kind
of perturbation is carried out over each population vector, and thus can be expected
to be more efficient. Similarly, crossover is also a vector-based component-wise
exchange of chromosomes or vector segments. Solutions of DE are represented in
terms of vectors, and then mutation and crossover are carried out using these vectors.
For example, in genetic algorithms, mutation is carried out at one site or multiple sites
of a chromosome, while in differential evolution, a difference vector of two
randomly-chosen vectors is used to perturb an existing vector. Such vectorized
mutation can be viewed as a self-organizing search, directed towards optimality. This
kind of perturbation is carried out over each population vector, and thus can be
expected to be more efficient. Similarly, crossover is also a vector-based component-
wise exchange of chromosomes or vector segments. In this thesis, an improved
version of DE algorithm will be applied to solve for a optimization problem of
reinforced composite plate and show its effectiveness and efficiency.
c) Harmony Search
35
Harmony search (HS) algorithm is a music-inspired algorithm, based on the
improvisation process of a musician [94]. Previous reviews of the HS literature have
focused on applications in civil engineering such as engineering optimization [95],
design of structures [96], design of water distribution networks [97], design of steel
frames [98], [99], Groundwater Management Problems [100], Geotechnical
Engineering problems [101], etc. HS algorithm includes a number of optimization
operators, such as the harmony memory (HM), the harmony memory size (HMS), the
harmony memory considering rate (HMCR), and the pitch adjusting rate (PAR). In
the HS algorithm, the HM stores the feasible vectors, which are all in the feasible... case studies in structural design optimization using
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LIST OF PUBLICATIONS
Parts of this dissertation have been published in international journals, national
journals or presented in conferences. These papers are:
International Journal
1. T. Lam-Phat, V. Ho-Huu, S. Nguyen-Ngoc, S. Nguyen-Hoai, Trung Nguyen-Thoi.
Deterministic and reliability-based lightweight design of Timoshenko composite
beams. Engineering with Computers, 2020, https://doi.org/10.1007/s00366-020-
00946-8.
2. T. Lam-Phat, S. Nguyen-Hoai, V. Ho-Huu, Q. Nguyen, T. Nguyen-Thoi. An
Artificial Neural Network-Based Optimization of Reinforced Composite Plate Using
A New Adjusted Differential Evolution Algorithm. Proceedings of the International
Conference on Advances in Computational Mechanics 2017 pp 229-242 (Part of
the Lecture Notes in Mechanical Engineering book series (LNME))
Link: https://link.springer.com/chapter/10.1007/978-981-10-7149-2_16
3. Q. Nguyen, S. Nguyen-Hoai, T. Chuong-Thiet, T. Lam-Phat. Optimization of the
Longitudinal Cooling Fin by Levenberg–Marquardt Method. Proceedings of the
International Conference on Advances in Computational Mechanics 2017 pp 217-
227 (Part of the Lecture Notes in Mechanical Engineering book series (LNME))
Link: https://link.springer.com/chapter/10.1007/978-981-10-7149-2_15
4. T. Nguyen-Thoi, T. Rabczuk, T. Lam-Phat, V. Ho-Huu, P. Phung-Van (2014). Free
vibration analysis of cracked Mindlin plate using an extended cell-based smoothed
discrete shear gap method (XCS-DSG3). Theoretical and Applied Fracture
Mechanics. Vol.72, 150-163.
Link: https://www.sciencedirect.com/science/article/pii/S016784421400041X
National Journal
5. Lam Phat Thuan, Nguyen Nhat Phi Long, Nguyen Hoai Son, Ho Huu Vinh, Le Anh
Thang. Global Optimization of Laminzation Composite Beams Using An Improved
Differential Evolution Algorithm. Journal of Science and Technology in Civil
Engineering NUCE 2020. 14 (1): 54–64
6. Nguyen-Thoi, T., Ho-Huu, V., Dang-Trung, H., Bui-Xuan, T., Lam-Phat, T. (2013)
Optimization analysis of reinforced composite plate by sequential quadratic
programming. Journal of Science and Technology, Vol. 51(4B), p. 156-165.
7. Nguyen Thoi Trung, Bui Xuan Thang, Ho Huu Vinh, Lam Phat Thuan, Ngo Thanh
Phong. An Effective Algorithm For Reliability-Based Optimization Of Reinforced
119
Mindlin Plate. Vietnam Journal of Mechanics, VAST, Vol. 35, No. 4 (2013), pp.
335 – 346
International Conference
8. Thuan Lam-Phat, Son Nguyen-Hoai, Vinh Ho-Huu, Trung Nguyen-Thoi.
Optimization of reinforced composite plate using adjusted different evolution
algorithm. Proceeding of the international conference on computational methods
(Vol.3, 2016), Berkeley, CA, USA.
National Conference
9. Thuan Lam-Phat, Son Nguyen-Hoai, Vinh Ho-Huu, Trung Nguyen-Thoi.
Optimization analysis of reinforced composite plate by adjusted different evolution.
Hội nghị Khoa học – Công nghệ toàn quốc về cơ khí 2015
10. Lâm Phát Thuận, Nguyễn Hoài Sơn, Lê Anh Thắng, Hồ Hữu Vịnh. Tối ưu hóa góc
hướng sợi tấm Composite gia cường dùng thuật toán Differential Evolution kết hợp
mạng thần kinh nhân tạo. Hội nghị cơ học toan quốc lần thứ X, 8-9/12/2017)