Development of meta heuristic of optimization methods for mechanics problems Development of meta heuristic of optimization methods for mechanics problems Development of meta heuristic of optimization methods for mechanics problems Development of meta heuristic of optimization methods for mechanics problems
LITERATURE REVIEW
An overview on research direction of the thesis
Most engineering design challenges can be framed as optimization problems, necessitating the use of optimization techniques for resolution Traditional methods often struggle with the complexities of real-world, highly non-linear problems As a result, there has been a growing trend towards employing evolutionary algorithms and meta-heuristic optimization methods to effectively address these challenges These algorithms, including genetic algorithms, particle swarm optimization, and ant colony optimization, have surged in popularity due to their simplicity, flexibility, efficiency, and ease of implementation, making them suitable for a variety of optimization scenarios, particularly in structural optimization Structural optimization has garnered significant interest from researchers, leading to the development of numerous techniques over the years Optimization methods are broadly categorized into gradient-based and population-based approaches While gradient-based methods, such as sequential linear programming and Newton's Method, can rapidly converge to optimal solutions, they often fall prey to local extrema and are limited to continuous design variables, which can hinder their overall effectiveness.
The initial design parameters of a structure significantly influence the effectiveness of gradient-based algorithms in achieving global or local solutions In contrast, population-based techniques, part of meta-heuristic algorithms, such as Genetic Algorithm (GA), Differential Evolution (DE), Particle Swarm Optimization (PSO), Cuckoo Search (CS), and Firefly Algorithm (FA), are widely utilized in structural optimization due to their flexibility and efficiency in managing both continuous and discontinuous design variables These methods typically yield global solutions, making them less sensitive to the initial design Among these approaches, Differential Evolution has gained prominence since its introduction by Storn and Price, leading to numerous studies aimed at enhancing its application in structural optimization challenges.
Differential Evolution (DE) has proven highly effective in addressing various engineering challenges, as demonstrated by multiple studies Wang et al utilized DE for optimal truss structure design, while Wu and Tseng employed a multi-population DE with a penalty-based strategy for truss structure optimization Le-Anh et al enhanced DE for static and frequency optimization of laminated composite plates, and Ho-Huu et al introduced a new DE variant for optimizing truss shapes and sizes In addition to DE, Rao's Jaya algorithm has emerged as an efficient tool for optimization, showcasing superior results against benchmark functions compared to other methods like genetic algorithms and particle swarm optimization Despite its strengths, the original Jaya algorithm's performance can be improved, leading to various enhanced versions This thesis presents a new improved Jaya algorithm that focuses on refining population selection to accelerate convergence while maintaining accuracy and a balanced exploration-exploitation dynamic.
One significant drawback of Differential Evolution (DE) and Jaya algorithms, similar to other population-based optimization methods, is their slower computational time compared to gradient-based optimization techniques This is primarily due to the extensive time required to evaluate the fitness of individuals within the population In structural optimization problems, the computation of objective or constraint function values typically involves using finite element analysis to assess the structural response.
To address the limitations of traditional methods, artificial neural networks (ANN) are integrated with the differential evolution (DE) algorithm, leveraging the brain's structural imitation Once trained, ANN can rapidly approximate outputs for given input data, significantly accelerating the computation of objective and constraint function values in the DE algorithm This integration enhances the efficiency of DE calculations, showcasing the effectiveness of ANN since its foundational concepts introduced by Warren McCulloch and Walter Pitts in 1943 Numerous studies have validated ANN's applicability in diverse fields such as system identification, control, pattern recognition, data mining, and filtering in social networking and email spam.
The focus of this study is on developing optimal algorithms that integrate Artificial Neural Networks (ANN) with Differential Evolution (DE), which will be applied to practical structures to evaluate their effectiveness Currently, composite materials are extensively utilized across various industries, including construction, mechanical engineering, marine, and aviation Notably, composite beams and stiffened plates are gaining popularity due to their exceptional advantages.
Stiffened composite plates, which combine the benefits of composite materials and stiffened beam structures, offer exceptional bending strength while remaining lightweight These plates are increasingly utilized across various sectors of structural engineering, including aircraft, ships, bridges, and buildings, due to their superior bending stiffness and material efficiency compared to traditional bending plate structures This results in higher economic efficiency in practical applications Given their high applicability, optimizing their design to reduce costs and enhance efficiency is crucial However, the complexity of the equations governing the behavior of these structures, influenced by numerous geometric and material parameters, complicates the optimization process Therefore, employing population-based methodologies is often more effective than gradient-based algorithms for addressing these optimization challenges.
Engineering design faces significant challenges due to uncertainties arising from manufacturing processes, material properties, and operating environments, which can lead to suboptimal performance and structural risks To address these uncertainties, two primary methods are utilized: reliability-based design optimization (RBDO) and robust design While robust design aims to minimize variance in outcomes, RBDO ensures design feasibility despite variable changes, making it a comprehensive strategy for optimal design However, RBDO's practical application is hindered by high computational costs Research has focused on effective reliability analysis techniques, including sensitivity-based approximations, most probable point methods, Monte Carlo simulations, and response surface models, which integrate optimization with reliability assessment Additionally, decoupling strategies have emerged, categorized into nested double-loop methods, decoupled methods, and single-loop methods Among these, double-loop approaches offer the highest accuracy but at a significant computational expense, while decoupled methods reduce costs by addressing optimization and reliability analysis sequentially.
To address the high computational costs associated with traditional reliability-based design optimization (RBDO) methods, single-loop approaches have been introduced, eliminating the need for reliability analysis This method transforms RBDO problems into approximate deterministic optimization (ADO) problems by converting probabilistic constraints into approximate deterministic ones, leading to significant reductions in computational expenses Consequently, these single-loop methods are more suitable for real-world applications However, research focusing on the reliability-based design optimization of laminated composite beams remains scarce.
This thesis explores the integration of Single-Loop Deterministic Methods (SLDM), recently introduced by Li et al., with a meta-heuristic optimization algorithm to develop a novel toolset, SLDM-iJaya This new approach aims to effectively address Reliability-Based Design Optimization (RBDO) challenges in composite structures.
This thesis explores enhancements to the Differential Evolution and Jaya algorithms to boost their convergence rates The improved algorithms are integrated with Artificial Neural Networks (ANN) and/or Stochastic Linear Decision Making (SLDM) to create innovative solutions for design optimization and Reliability-Based Design Optimization (RBDO) challenges in composite structures, including stiffened composite plates and Timoshenko beams.
Motivation of the research
The motivation behind this thesis stems from a comprehensive analysis of existing literature and an assessment of the application potential of composite material structures, along with intelligent optimization techniques, particularly reliability-based optimization methods.
- The development / improvement of existing algorithms to improve the efficiency of solving structural optimization problems with high accuracy and reliability
- Studying the advantages of Artificial Neural Network (ANN) to combine with optimal algorithms to improve the speed and the performance of solving structural optimization problems.
Goals of the thesis
This thesis explores the development of meta-heuristic optimization methods integrated with Artificial Neural Networks (ANN) to create a novel algorithm for addressing composite material structural optimization challenges It specifically aims to enhance the original Differential Evolution and Jaya algorithms to improve convergence towards global optimal solutions The improved meta-heuristic algorithms are then combined with ANN to effectively search for the optimal design of stiffened composite plate structures.
The thesis introduces an innovative toolset that integrates a meta-heuristic optimization algorithm with the Single-Loop Deterministic Method to effectively address Reliability-Based Design Optimization (RBDO) challenges Specifically, the original Jaya algorithm will be enhanced to boost convergence in finding optimal solutions for these optimization issues This refined Jaya algorithm will then be paired with the Single-Loop Deterministic Method to tackle the RBDO of composite beam structures.
Research scope of the thesis
The thesis focuses on the following main issues:
- To optimize truss, beam and stiffened plate structures using steel and composite materials
- To study and improve population-based optimization methods to increase accuracy and efficiency in solving optimization problems
Leveraging the power of Neural Networks to develop approximate models from data sets can significantly enhance performance when combined with optimal algorithms, enabling the effective resolution of various complex problems.
- To combine optimal algorithms with groups of reliability assessment methods to solve RBDO problems
The chosen optimization problems are straightforward, aimed at assessing the effectiveness, accuracy, and reliability of the proposed methods Future research will focus on applying these optimal techniques to more complex challenges.
Outline
The thesis contains six chapters and is structured as follows:
Chapter 1 provides a comprehensive overview of meta-heuristic algorithms and composite material structures, with a particular focus on the role of artificial neural networks in the optimization process This chapter outlines the organization of the thesis and highlights its novelty and objectives, offering a quick review of the key topics explored throughout the study.
Chapter 2 offers a comprehensive overview of composite materials, highlighting fundamental concepts and their real-world applications It also introduces the Timoshenko composite beam and stiffened composite plate theories, which are the primary structures examined in the optimization problems addressed in this thesis.
Chapter 3 focuses on meta-heuristic optimization techniques, specifically the Differential Evolution and Jaya algorithms, and explores modifications to enhance these algorithms Additionally, the chapter provides an overview of Reliability-Based Design Optimization (RBDO) and presents proposed methods for effectively addressing RBDO challenges.
Chapter 4 provides an overview of Artificial Neural Networks (ANN), detailing their historical development and fundamental concepts It introduces the specific Neural Network Structure utilized in this thesis for approximating data derived from Finite Element Analysis Additionally, this chapter discusses the training algorithm, with a particular focus on the Levenberg-Marquardt method, as well as the phenomenon of overfitting.
Chapter 5 demonstrates the effectiveness of the improved Differential Evolution (iDE) and the improved Jaya algorithm in addressing various optimization problems, including planar and space truss structures, Timoshenko composite beams, and stiffened composite plates The iDE is particularly effective in optimizing the weight of truss structures and the design parameters of composite plates, showcasing its strong performance Additionally, the improved Jaya algorithm excels in optimizing the Timoshenko composite beam design, yielding results that highlight its accuracy compared to existing methods This chapter introduces a novel approach, SLDM-iJaya, which combines the improved Jaya algorithm with single-loop methods to tackle the Reliability-Based Design Optimization (RBDO) problem for Timoshenko composite beams The chapter analyzes results from two optimization scenarios—one without a reliability index and the other focusing on RBDO—demonstrating the SLDM-iJaya algorithm's effectiveness and precision Furthermore, the integration of Artificial Neural Networks with meta-heuristic optimization methods, such as the Differential Evolution algorithm, leads to the development of the ABDE (ANN-Based Differential Evolution) algorithm, which successfully optimizes the design of stiffened composite plates and paves the way for future applications.
Finally, Chapter 6 closes the concluding remarks and give out some recommendations for future work.
Contributions of the thesis
This thesis presents an enhanced version of the Jaya algorithm, known as the improved Jaya algorithm (iJaya), which incorporates an elitist selection technique to refine the original selection step The iJaya algorithm has been effectively applied to design optimization problems in composite structures, yielding impressive results.
The SLMD-iJaya algorithm, a fusion of the enhanced Jaya algorithm and Global Single-Loop Deterministic Methods (SLDM), has been introduced to effectively address Reliability-Based Design Optimization challenges in continuous composite beam models.
The integration of Artificial Neural Networks into the optimization processes of the iDE and iJaya algorithms has led to the development of two innovative algorithms, ABDE and AB-iJaya These new algorithms have been successfully applied to optimize stiffened composite plate structures, demonstrating significantly improved performance in solving complex optimization problems.
FUNDAMENTAL THEORY OF COMPOSITE STRUCTURE IN
Introduction to Composite Materials
2.1.1 Basic concepts and applications of Composite Materials
Structural materials are categorized into four main types: metals, ceramics, polymers, and composites, with composites often comprising various combinations of these materials Over recent decades, advancements in composite technology have led to the development of "nano-composites" featuring nanometer-sized reinforcements, such as carbon nanoparticles and nanotubes, which exhibit extraordinary properties Composites are favored for their unique characteristics that surpass those of their individual components, exemplified by fibrous composites where reinforcing fibers are embedded in a matrix Natural composites, like wood and bone, illustrate the effectiveness of fibrous reinforcement, which enhances strength and stiffness Historical studies, such as those by Griffith in 1920, demonstrated that thinner fibers possess greater tensile strength due to reduced likelihood of surface defects However, fibers alone cannot withstand longitudinal compressive loads and require a binder for structural integrity Transverse reinforcement is achieved by orienting fibers at various angles based on the stress field, leading to diverse composite types, including continuous fiber laminates, woven fiber composites, chopped fiber composites, and hybrid composites Each type offers distinct advantages and design flexibility, enabling the creation of optimized structural materials tailored to specific applications.
Figure 2 1 Types of fiber-stiffened composites
(a) Continuous fiber composite, (b) Woven composite, (c) Chopped fiber composite,
Composite structural elements are increasingly utilized across various industries, including automotive, aerospace, marine, and architecture, as well as in consumer products like skis and tennis rackets Military aircraft designers were pioneers in recognizing the advantages of composites, which offer high specific strength and stiffness, crucial for enhancing performance and maneuverability by minimizing weight Additionally, composite construction contributes to smoother surfaces that reduce drag Since the introduction of boron and graphite fibers in the early 1960s, the use of advanced composites in military aircraft has rapidly expanded, with carbon fiber components such as stabilizers, flaps, wing skins, and control surfaces being integral to fighter aircraft design.
The use of composite materials in aircraft is on the rise due to decreasing material costs, advancements in design and manufacturing technology, and growing experience with these materials Notably, the Boeing 787 is the first commercial airliner to feature a composite fuselage and wings, with up to 50% of its primary structure, including the fuselage and wings, made from carbon fiber/epoxy composites or carbon fiber-stiffened plastics Similarly, the Airbus A350 XWB is another commercial airliner that heavily utilizes composite materials.
Figure 2 2 Boeing 787 - first commercial airliner with composite fuselage and wings (Courtesy of Boeing Company.)
The significance of structural weight in automotive vehicles is increasingly recognized, leading to a rise in the adoption of composite components In cargo trucks, lighter composite materials enhance payload capacity, resulting in substantial economic benefits For instance, a composite concrete mixer drum can weigh 2,000 lbs less than its traditional steel counterpart, exemplifying the advantages of weight reduction in the industry.
Figure 2 3 Composite mixer drum on concrete transporter truck weighs 2000 lbs less than conventional steel mixer drum (Courtesy of Oshkosh Truck Corporation)
Weight savings from composite materials, such as leaf springs, can exceed 70% compared to traditional steel Innovations in composite engine blocks, particularly those made from graphite-stiffened thermoplastics, aim to develop a ceramic composite engine that eliminates the need for water cooling Chopped glass fiber-reinforced plastics (FRPs) are widely utilized in automotive body panels, focusing on stiffness and aesthetics While composites have primarily been used in secondary structures and cosmetic parts, their potential for primary structures in vehicles remains largely untapped The rise of electric vehicles necessitates lighter composite structures to offset the weight of heavy batteries; for instance, the BMW Megacity electric vehicle features a carbon fiber composite passenger compartment integrated with an aluminum spaceframe Additionally, structural elements like I-beams and channel sections in civil infrastructure may also benefit from fiber-stiffened plastics Composites are increasingly favored in wind turbine blades due to their superior strength-to-weight and stiffness-to-weight ratios, as well as excellent vibration damping and fatigue resistance This thesis focuses on the investigation and optimization of composite beam and stiffened composite plate structures for various applications.
Figure 2 4 Pultruded fiberglass composite structural elements (Courtesy of
Figure 2 5 Composite wind turbine blades (Courtesy of GE Energy.)
2.1.2 Overview of Composite Material in Design and Optimization
The use of composite materials in structural design has surged in popularity over recent decades due to their numerous advantages over traditional materials like steel and aluminum A key factor driving this trend is their significant weight advantage; composites such as Graphite/Epoxy and Glass/Epoxy exhibit lower weight densities of 1550.07 kg/m³ and 1799.19 kg/m³, respectively, compared to aluminum's weight density of 2769.99 kg/m³ This reduction in weight enhances structural efficiency and performance, making composite materials an increasingly preferred choice in modern engineering applications.
Composites offer significant advantages over metals, particularly in terms of weight, stiffness, and strength For instance, high-strength Graphite/Epoxy exhibits a stiffness of approximately 15.467x10^9 kg/m², which is notably higher than Aluminum's stiffness of 7.030x10^9 kg/m² These properties make composites increasingly appealing for structural applications, as they can endure greater static loads and experience less deformation Structural designers aim to optimize designs while minimizing resource usage, balancing factors such as weight, cost, strength, and stiffness However, achieving an ideal design that meets both weight and stiffness requirements can be challenging, as improvements in performance may compromise the necessary strength or stiffness.
In the last thirty years, mathematical optimization has become a vital tool in structural design, focusing on maximizing or minimizing objective functions within certain constraints Recent studies have specifically targeted the optimization of laminated composite structures, with notable research aimed at enhancing the first natural frequency and maximizing the buckling load factor.
Recent studies have focused on optimizing laminated composite beams for various performance metrics, including minimizing free vibration frequency, reducing weight, and maximizing strain energy Notable references include works that address minimizing weight (Refs [49], [50], [53], [54]) and maximizing buckling load while minimizing weight (Ref [55]) Additionally, the optimization of continuous composite models has been explored using non-gradient-based algorithms, such as particle swarm and genetic algorithms, particularly for thin-walled composite box-beam helicopter rotor blades (Refs [56], [57]).
In their research, Lentz and Armanios derived exact solutions and analyzed the sensitivity of the first four frequencies using a continuous composite model They also developed a gradient-based algorithm aimed at achieving a lightweight design for solid composite laminated beams.
[58] described a gradient-based optimization scheme for obtaining the maximum coupling in thin-walled composite beams subject to hygrothermal and frequency constraints
Optimization methods for composite structures can be categorized into gradient-based and non-gradient-based algorithms, with the latter often referred to as random search algorithms Non-gradient-based algorithms do not require gradient information, making them easier to implement compared to gradient-based methods, which rely on gradients for their search processes While gradient-based algorithms are generally more efficient and capable of finding local optima, they have significant drawbacks, including a heavy reliance on the initial point chosen by the user and a tendency to get trapped in local optima when multiple local extremes exist As a result, researchers often favor non-gradient-based methods, particularly meta-heuristic approaches like Particle Swarm Optimization (PSO), Genetic Algorithm (GA), Differential Evolution (DE), and Jaya algorithm, for solving optimization problems in laminated composite structures to achieve global solutions This thesis focuses on the application of Differential Evolution and Jaya algorithm to optimize two types of composite structure models: the Timoshenko composite beam and the stiffened composite plate, with relevant theoretical foundations discussed in the subsequent sections.
Analysis of Timoshenko composite beam
Composite laminated Timoshenko beams can be analyzed using both continuous and discrete models While discrete models, such as finite element methods, are easier to implement, they often yield only approximate solutions and are less effective than the analytical approaches of continuous models To address this limitation, Liu proposed a method that treats composite laminated Timoshenko beams as continuous models, enabling the derivation of exact solutions The subsequent section outlines the process for developing the analytical solution for these beams, and readers seeking further insights are encouraged to consult Liu's original work.
2.2.1 Exact analytical displacement and stress d x
Figure 2 6 Composite laminated beam model
Consider a segment of composite laminated beam with N layers and the fiber orientations of layers are of i (i1, ,N) The positions of layers are denoted by
The beam, characterized by a rectangular cross-section with a width of b and a height of h, is illustrated in Figure 2.6 A segment of the beam, denoted as dx, experiences a transverse force q(x), as depicted in Figure 2.7.
The displacement fields of the composite laminated beam calculated analytically based on the first-order shear deformation theory (also called Timoshenko beam theory) are: dx z y x
x A q x C x C x C (2.3) where C i i ( 1, , 7)are indefinite integration constants determined by using the boundary conditions of the composite laminated beams as shown in the following section
A B C b B A D b B A D b A (2.4) where A B D A 11 , 11 , 11 , 55 are respectively extensional stiffness, bending-extensional coupling stiffness, bending stiffness and extensional stiffness of the composite laminate; is the shear correction factor with the value of 5/6
Figure 2 8 The material and laminate coordinate system
The stress fields in a composite laminated beam encompass both plane stress components and shear stress components As illustrated in Figure 2.8, the relationship between the material coordinate system (123) and the beam/laminate coordinate system (xyz) is established, with the fiber orientation aligning along the 1-axis The plane stress components can be defined accordingly.
T Q (2.5) where the strain components y 0, xy 0, and
T is the coordinate transformation matrix and Q ( ) k is the matrix of material stiffness coefficients
( ) ( ) ( ) ( ) 2 ( ) 2 ( ) cos sin 2sin cos sin cos 2sin cos sin cos sin cos cos sin
The shear stress components in the material coordinate systems are
T Q (2.9) where the shear strain components yz 0 and
(2.10) The coordinate transformation matrix T s ( ) k and the matrix of stiffness coefficients
In the above equations, Q ( ) ij k is the stiffness coefficients of the kth lamina in the laminate coordinate system and are described clearly in [53]
The constants of indefinite integration in the equations can be established by applying various boundary conditions This thesis examines four specific types of boundary conditions: pinned-pinned, fixed-fixed, fixed-free, and fixed-pinned.
Q L are applied The seven indefinite integration constants are then determined as follows
Q L are used to determine seven indefinite integration constants
Q L are employed and the seven the indefinite integration constants are then determined
, M L y ( )0,Q L z ( )0, are applied to determine the seven indefinite integration constants
Under the specified boundary conditions, the normal force (N), shear force (Q), and bending moment (M) acting on the cross-section of the composite laminated beam are defined along the x-axis, z-axis, and y-axis, respectively, and are calculated as detailed in reference [53].
Analysis of stiffened composite plate
A stiffened composite plate is created by integrating a composite plate with a Timoshenko composite beam, as shown in Figure 2.9 There are two primary methods for analyzing beam structure behavior: the Timoshenko model and the Euler–Bernoulli model The Euler–Bernoulli beam theory disregards shear deformations, maintaining that plane sections remain flat and perpendicular to the longitudinal axis In contrast, the Timoshenko beam theory also keeps plane sections flat, but they are not necessarily perpendicular to the longitudinal axis.
Timoshenko beam and the Bernoulli beam is that the former includes the effect of the shear stresses on the deformation
This thesis employs the Timoshenko beam model, which acts as a stiffener aligned parallel to the plate's axes The beam's centroid is positioned at a distance 'e' from the plate's mid-plane The plate-beam system is discretized into a series of nodes, where each node in the plate has a degree of freedom (DOF) represented by d = [u, v, w, β_x, β_y]ᵀ, indicating displacements and rotations around the axes Conversely, each beam node features a distinct DOF defined as d_st = [u_r, u_s, u_z, β_r, β_s]ᵀ, where u_r, u_s, and u_z denote the centroid displacements of the beam, and β_r and β_s represent the rotations around the r-axis and s-axis, respectively.
Figure 2 9 A composite plate stiffened by an r-direction beam
The centroid displacements of beam are expressed as :
To connect the degrees of freedom (DOFs) of plates and beams, the displacement compatibility conditions proposed by Peng et al are applied The relationship is defined as \( st = \delta d d \), where \( \delta = 5N_s \times 5N_p \) serves as the transformation matrix for the beam and plate nodes.
N N are respectively the total nodes of plate and beam
2.3.1 Strain Energy Equation of stiffened composite plate
The strain energy of composite plate is given by:
U ε D ε ε D κ κ D ε κ D κ γ D γ A (2.19) where ε κ γ 0 , b , are respectively membrane, bending and shear strains of composite plate and are expressed as follows
D ,D ,D ,D are material matrices of plate and expressed by:
(2.21) where , are the shear correction factors; Q ij P are the material coefficients of kth layer in the plate coordinate; and N is the number of layers
The strain energy of composite stiffener is given by
b T b b s T s s st L st st st st st st
Where L is the length of the stiffener; ε ε b st , s st are respectively bending, shear strain of beam and are expressed as follows:
D D are material matrices of composite beam
D D (2.24) where 1 , 2 are the shear correction factors, A 11,B 11etc are the coefficients of matrix ([ ] [ ][ ] [ ])a b c 1 b and [a], [b] and [c] are defined as follows
The stiffener coefficients in equations (2.24) and (2.25) can be expressed by
N h st ij ij ij ij k h
Using the superposition principle, total energy strain of stiffened composite plate is obtained:
U U U (2.27) where N st is the number of stiffeners
2.3.2 Finite element analysis for stiffened composite plate
The displacement of the stiffened composite plate is calculated using the finite element method, applied independently to each plate and beam element Initially, a three-node triangular element is employed to approximate the displacement of the composite plate.
N p i i N p i i i i d N d x I d (2.28) where d i [ ,u v w i i , i , xi , yi ] T is the displacement field of plate at ith node; N ( ) i x is linear shape function of triangular element
The displacement of the stiffener (composite beam) is approximated by two-node bar element
Where d sti [u u u ri , si , zi , ri , si ] T is the vector of displacement field of beam at ith node; i ( )r is linear shape function of bar element
The local and global coordinate system is related by a transformation matrix T cos( ) sin( ) 0 0 0 sin( ) cos( ) 0 0 0
st x y u v d w Td (2.30) where is the angle between x-axis and r-axis
Substituting Eqs (2.28), (2.29) into Eq (2.27), total strain energy of the stiffened composite plate is derived as
U d Kd (2.31) where K is the stiffness matrix of stiffened composite plate and is defined by
K K are stiffness matrices of plate and beam, respectively, computed as follows
2 T b T b b T t T s s st L st st st st st st dl
K T B D B T T B D B T (2.34) where B m ,B b ,B s are obtained by substituting Eq (2.28) into Eq (2.20) and B b st , s
B st are obtained by substituting Eq (2.29) into Eq (2.23)
For static analysis, the global equations for the stiffened composite plate can be described as follows
In the equation (2.35), K is the stiffness matrix, dis the vector of displacement and f is the load vector
The CS-DSG3 element, introduced by T Nguyen-Thoi et al., is utilized to analyze the behavior of composite plates This method involves subdividing each three-node triangular element into three smaller sub-triangular elements, with the centroid of the triangle serving as the connection point for these sub-elements.
Figure 2 10 Three sub-triangles are connected through central point of the triangle in CS-DSG3 method
The Discrete Shear Gap technique is utilized within each sub-triangular element to compute the bending, shearing, and membrane components Following this, the cell-based strain smooth technique is employed to refine the strain fields of the sub-triangles Subsequently, the stiffness matrices for bending, shearing, and membrane are calculated based on the smoothed strain components The final steps of the Finite Element Method (FEM) procedure are then conducted as per standard practices.
RELIABILITY-BASED OPTIMIZATION METHODS WITH
Overview of Metaheuristic Optimization
Meta-heuristic algorithms, which operate at a higher level than simple heuristics, typically outperform them by balancing local search with global exploration These algorithms often incorporate randomization to enhance solution diversity and facilitate movement away from local search toward a broader global search While there is no universally accepted definition of heuristics and meta-heuristics, the trend is to categorize stochastic algorithms that utilize randomization and global exploration as meta-heuristics They are particularly effective for nonlinear modeling and global optimization, providing practical solutions to complex problems within a reasonable timeframe Although there is no assurance of finding the best solutions, an efficient meta-heuristic algorithm can consistently yield high-quality solutions, with some being nearly optimal, despite the inherent uncertainty in their performance.
Meta-heuristic algorithms primarily consist of two key components: intensification and diversification Intensification focuses on exploiting promising regions of the search space, while diversification aims to explore a broader range of potential solutions globally By generating diverse solutions, diversification helps prevent the algorithm from getting stuck in local optima Additionally, selecting the best solutions promotes convergence towards optimality A balanced integration of intensification and diversification is crucial for achieving a global solution effectively.
Meta-heuristic algorithms can be categorized into two main types: population-based and trajectory-based methods Population-based algorithms, such as genetic algorithms (GA) and genetic programming (GP), utilize a group of solutions or strings Similarly, particle swarm optimization (PSO) operates with multiple agents or particles to explore the solution space effectively.
[65] On the other hand, simulated annealing (SA) [66] uses a single solution which moves through the design space or search space, while artificial neural networks use a different approach
Modeling and optimization are crucial for addressing real-world problems, as they complement each other; modeling ensures that objective functions are assessed with accurate mathematical or numerical representations, while optimization identifies the best settings for design parameters A key component of optimization is the use of algorithms, particularly meta-heuristic algorithms, which play a significant role in achieving effective solutions.
3.1.1 Meta-heuristic Algorithm in Modeling
Nonlinear system modeling can be approached through various methodologies, each with distinct advantages and disadvantages The complexity of determining both the structure and parameters of engineering systems makes this task challenging Models are generally categorized into two main types: phenomenological and behavioral Phenomenological models are based on physical relationships and require prior knowledge of the system, while behavioral models focus on capturing input-output relationships from measured data, eliminating the need for prior understanding of the underlying mechanisms Behavioral models, including widely-used statistical regression techniques, are advantageous as they can yield effective results with minimal effort.
In recent years, various alternative meta-heuristic approaches have emerged for behavioral modeling, significantly aided by advancements in computer hardware These techniques have proven to be more efficient, especially in scenarios where traditional methods struggle Two prominent meta-heuristic algorithms utilized in nonlinear modeling are Artificial Neural Networks (ANNs) and Genetic Programming (GP) While ANNs have been effectively applied to numerous structural engineering challenges, they often lack transparency in their solution processes In contrast, GP, an evolution of genetic algorithms, offers distinct advantages as a supervised machine learning method that explores program space rather than data space This allows GP to automatically create predictive models represented as tree structures, enabling it to generate solutions without preconceived notions about existing relationships, a key benefit over regression and ANN methods.
Genetic Programming (GP) and its variants are extensively applied to address real-world challenges Additionally, other meta-heuristic algorithms like Fuzzy Logic and Support Vector Machines have also been utilized in modeling within the literature.
Artificial Neural Networks (ANNs) originated from the simulation of the biological nervous system, with their development beginning in the early 1940s by McCulloch and colleagues Initially, research concentrated on creating simple neural networks to model basic logic functions Today, ANNs are utilized for complex problems that lack straightforward algorithmic solutions This study examines the approximation capabilities of two prominent ANN architectures: Multi-Layer Perceptron (MLP) and Radial Basis Function (RBF) networks.
Genetic Programming (GP) is an advanced optimization technique that utilizes Darwinian natural selection principles to generate computer programs aimed at solving specific problems Initially pioneered by Friedberg, who implemented a learning algorithm for program enhancement, GP gained significant traction in the late 1980s through Koza's groundbreaking work on symbolic regression Unlike Genetic Algorithms (GAs), which represent solutions as fixed-length binary strings, GP represents solutions as tree structures expressed in functional programming languages like LISP This allows GP to evolve programs that can execute without the need for post-processing, unlike GA-evolved binary strings While GAs are primarily used for parameter optimization, GP not only optimizes parameters but also defines the structure of the approximation model itself The fitness of each program in GP is assessed using a predefined fitness function, which serves as the objective for optimization Traditional GP is categorized as tree-based GP, but there are also linear and graph-based variations, with this study focusing on linear-based GP.
3.1.2 Meta-heuristic Algorithm in Optimization
Finding an optimal solution to an optimization problem can be quite challenging, largely influenced by the selection and proper application of the appropriate algorithm The choice of algorithm is contingent upon the problem type, available algorithms, computational resources, and time limitations For large-scale, nonlinear, global optimization problems, there is often a lack of consensus on which algorithm to use, and efficient algorithms are frequently unavailable In particular, NP-hard optimization problems do not have any efficient algorithms Typically, optimization problems can be expressed in a generic form.
In the context of optimization, the design vector \( x = (x_1, x_2, , x_n)^T \) comprises decision variables \( x_i \), which can be real continuous, discrete, or a mix of both The functions \( f_i(x) \) for \( i = 1, 2, , M \) are referred to as objective functions or cost functions, playing a crucial role in evaluating the performance of different design configurations.
In optimization problems with a single objective (M = 1), the design space is defined by decision variables, while constraints are represented by equalities (h j) and inequalities (g k) These inequalities can also be expressed as ≥ 0, and objectives can be framed as maximization problems Various algorithms, including deterministic methods like the simplex method and gradient-based algorithms such as the Newton-Raphson algorithm, are employed to solve these problems In contrast, non-gradient-based algorithms rely solely on function values without using derivatives Stochastic algorithms, including heuristic and meta-heuristic methods, aim to find quality solutions through trial and error, often yielding satisfactory results in a reasonable timeframe, though they do not guarantee optimal solutions These approaches are beneficial for obtaining good solutions when the best solutions are not strictly required.
Meta-heuristic optimization algorithms are frequently inspired by natural phenomena These algorithms can be categorized into various types based on their sources of inspiration, as illustrated in the accompanying figure.
Biology-inspired algorithms leverage biological evolution and animal collective behavior, while science-based meta-heuristics draw inspiration from physics and chemistry Additionally, art-inspired algorithms excel in global optimization by mimicking the creative processes of artists, including musicians and architects Furthermore, socially inspired algorithms simulate social behaviors to effectively tackle optimization challenges.
Bio - Inspired Science - Inspired Art - Inspired Social - Inspired
Meta-heuristic optimization algorithms draw inspiration from various sources, yet they share structural similarities These algorithms can be broadly categorized into two primary types: Evolutionary Algorithms and Swarm 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
Solving Optimization problems using improved Jaya algorithm
The Jaya algorithm, developed by Ventaka Rao, is a straightforward population-based global optimization technique that requires no algorithm-specific parameters This easy-to-implement method focuses on avoiding poor solutions while striving for the best outcomes during the optimization process It utilizes common controlling parameters, such as population size and the number of generations, to efficiently achieve its optimization goals The process can be summarized in four simple steps, highlighting its user-friendly nature.
An initial population of NP individuals is randomly generated within the search space, where each candidate represents a vector of n design variables, denoted as x i (x 1 , x 2 , , x n ) These vectors are created while adhering to specified lower and upper bounds.
, l , [0,1] u , l , , (1, 2, , ), (1, 2, , ) j i j i j i j i p x x rand x x j n i N (3.6) where x u j and x l j are respectively the upper and lower bounds of the design variable x j ; rand[0,1] generates a random number within the interval [0,1]
In an optimization problem, the objective function is denoted as f(x), which is used to evaluate the fitness of individuals within a population The individuals with the highest and lowest fitness values are identified as the best candidate (x_best) and the worst candidate (x_worst), respectively During the k-th iteration, the value of the j-th variable for the i-th candidate, represented as x_j,i,k, is stochastically modified to produce a new vector.
, , j best k x are the values of j th variable corresponding to the worst ( worst x ) and the best (x best ) candidate in the whole population at k th iteration
In the context of optimization, the random numbers \( r_1 \) and \( r_2 \) are generated within the range of [0,1] The expression \( r_1 \times (x_{j,\text{best},k} - x_{j,i,k}) \) reflects the inclination of the design variable towards the optimal solution, while the term \( r_2 \times (x_{j,\text{worst},k} - x_{j,i,k}) \) indicates the strategy to steer clear of the least favorable solution.
Next, if the value of x ' j i k , , is out of the range between lower bound and upper bound, an operation is carried to reflect it back to the allowable region
In each iteration, a candidate solution \( j_{ik}^x \) is accepted if it improves the objective function value; if not, the existing value of \( j_{ik}^x \) is retained All candidates that are accepted by the end of the iteration are preserved and serve as inputs for the subsequent iteration.
3.2.2 Improvement version of Jaya algorithm (iJaya)
The original Jaya algorithm selects the next generation's population based on paired comparisons of fitness function values, which can lead to the elimination of potentially strong individuals when matched against stronger competitors This method may overlook individuals that, while not the best in their pair, could outperform winners in other pairs across the entire population Consequently, relying solely on the best individuals from combined parent and offspring populations can reduce the algorithm's exploration capabilities, increasing the risk of premature convergence.
To enhance the selection of candidates for the next generation while maintaining algorithm exploration, a new selection procedure is implemented This procedure divides the population into two groups: one-third of the total population (N/3) and two-thirds (2N/3) During the selection phase, one-third of the next population is chosen from the first group using the standard selection technique, while the remaining individuals are selected from the second group through an elitist approach Specifically, this involves combining two-thirds of the parent and offspring populations and selecting half of the top individuals for the next generation This method effectively balances exploitation and exploration, leading to improved solution quality and convergence rates for the algorithm.
The new selection procedure can be illustrated by the following figure
Figure 3 2 Selection procedure of the iJaya.
Reliability-based design optimization using a global single loop
Engineering design faces significant challenges due to uncertainties stemming from manufacturing processes, material properties, and operating environments, which can adversely affect optimal designs and lead to structural risks These uncertainties can be modeled using Gaussian distribution, making reliability-based design optimization (RBDO) a vital strategy for achieving optimal designs RBDO methods can be categorized into three groups: double-loop methods (DLM), decoupled methods, and single-loop methods While DLM integrates reliability analysis within the optimization loop, resulting in high computational costs, decoupled methods separate these tasks but still incur significant expenses due to interrelated loops To mitigate these costs, single-loop methods convert RBDO problems into approximate deterministic optimization (ADO) problems, drastically reducing computation time and enhancing applicability in real-world scenarios Additionally, selecting an effective optimization algorithm is crucial, with population-based methods, particularly the Jaya algorithm, proving to be more efficient than gradient-based approaches The Jaya algorithm has demonstrated superior performance against various benchmark functions and has been successfully applied to numerous engineering design problems, although its potential in RBDO applications remains largely untapped.
Despite numerous studies on the optimal design of composite laminated beams, there remains a scarcity of research incorporating reliability analysis, which often leads to limitations For instance, Yangjun et al utilized a double-loop method to tackle the reliability-based design optimization of a single-span adhesive bonded steel-concrete composite beam under various loading scenarios Similarly, Fabrizio et al employed probabilistic analysis for the design optimization of an aircraft's composite floor beam, aiming for a design that is resilient to system variations and minimizes failure risk While the reliability-based design optimization (RBDO) methods in these studies enhance result reliability, they frequently converge on local optima, and the beam models are derived from numerical methods, resulting in persistent drawbacks and inaccuracies in achieving optimal solutions.
This thesis introduces an innovative approach to Reliability-Based Design Optimization (RBDO) by integrating a Single-Loop Deterministic Method (SLDM) with an Improved Jaya Algorithm (iJaya) The proposed algorithm is utilized to address RBDO challenges in composite laminated beams, specifically employing a continuous Timoshenko beam model This combination effectively reduces computational costs, achieves global optimal solutions, and efficiently manages both discrete and continuous design variables in RBDO problems.
3.4.1 Reliability-based optimization problem formulation
The mathematical model of a typical RBDO problem can be described as follows [130]:
The objective function is defined as \( f(d, \mu_x, p) \), where \( d \) represents the vector of deterministic design variables constrained by lower and upper bounds \( d_{low} \) and \( d_{up} \) The vector \( x \) consists of random design variables, with \( \mu_x \) denoting their mean vector and \( \mu_{x_{low}} \) and \( \mu_{x_{up}} \) as the bounds for \( \mu_x \) Additionally, \( p \) is the vector of random parameters, while \( g_i(d, x, p) \) represents the constraint functions, with \( R_i \) indicating the desired reliabilities for constraint satisfaction The number of constraints is denoted by \( m \) The probability operator, \( \text{Prob}[.] \), ensures that the likelihood of satisfying the constraints \( g_i(d, x, p) \leq 0 \) meets or exceeds the specified reliability \( R_i \) For simplicity, all random design variables and parameters are assumed to be statistically independent, following a normal distribution, represented by \( \theta = \{x_1, x_2, \ldots, x_n, p_1, p_2, \ldots, p_n\} \), with their mean and standard deviation vectors indicated as \( \mu = \{\mu_x, \mu_p\} \) and \( \sigma = \{\sigma_x, \sigma_p\} \), respectively.
3.4.2 A global single-loop deterministic approach
The single-loop deterministic method (SLDM) has been recently introduced for reliability-based design optimization (RBDO), transforming probabilistic constraints into approximate deterministic ones, which significantly reduces computational costs However, SLDM is restricted to continuous design variables and often results in local extrema To address these limitations, Vinh et al developed a global single-loop deterministic approach applicable to RBDO problems involving both continuous and discrete design variables, utilizing the improved differential evolution algorithm known as IDE In this thesis, the Jaya algorithm, along with its enhanced version iJaya, is employed to seek global optimal solutions for the design optimization of composite beam structures.
* Formulation of approximate deterministic constraints
The initial phase of Stochastic Linear Decision Making (SLDM) involves establishing an approximate deterministic feasible region by adjusting the boundary of the probabilistic constraint by a distance of β from its original position, as illustrated in Figure 3.3.
Figure 3 3 Illustration of the feasible design region
Figure 3 illustrates the feasible design region, where the red curve represents the limit-state function, while the green curves indicate the boundary of the transformed deterministic constraint function The dotted area highlights the deterministic feasible design region This transformation guarantees that the minimum distance from any point on the red curve to the green curve is denoted as βj, and it has been demonstrated that the resulting solutions meet the probabilistic constraint requirements.
After, the feasible design region is formed Suppose that μ θ is a point lying on
( ) g i d,μ θ , then the Most Probable Point (MPP), θ MPP corresponding to μ θ can be determined by moving μ θ backward to g i ( d,θ )a distance
As shown in [129], θ MPP on the failure in a standard normal space can be defined by
MPP u i j u j i u i j j g j n n n q g ( 3.11 ) where the subscript u denotes the standard normal distribution space and the derivatives ( g i / u j ) * are evaluated at u j , MPP
The relationship of random parameters in the original design space and the normal standard space is depicted as follows
By combining equation (3.11),(3.12),(3.13), the relationship between μ θ and θ MPP in the original design space is denoted by
MPP j i j j j i j u j i j j g g ( 3.14 ) where the derivatives (g i / j ) * are evaluated at j , MPP
According to Li et Al [38], the derivatives (g i / j ) * can be approximately assessed at j , and then equation (3.14) can be rewritten as
MPP j i j j j i j u j i j j g g ( 3.15 ) where the derivatives (g i / j ) # are evaluated at j
Once θ MPP in the original design space has been determined, the feasible domain of the RBDO problem can be denoted by approximate deterministic constraints as
( ) i ( i )0 g i d,μ θ g d,μ θ σ n θ (3.16) where n( θ θ g i (μ θ ) / θ θ g i (μ θ ) ) is the approximately normalized gradient vector evaluated at μ θ on g i (d,θ)
Then, the RBDO problem in Equation (3.10) can be reformulated by an ADO problem as follows:
The derivatives, denoted as ∇θ gi(μθ), can be obtained directly from an explicit limit-state function However, in practical applications, structural behaviors are typically analyzed using numerical methods, leading to implicit limit-state functions To compute the derivatives ∇θ gi(μθ) in this study, the finite difference method, a numerical derivative technique, is employed.