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An overview on research direction of the thesis
Design challenges in engineering are primarily optimization problems that necessitate effective optimization techniques Due to the highly non-linear nature of real-world issues, traditional optimization methods often fall short Consequently, there is a growing trend towards using evolutionary algorithms and meta-heuristic optimization methods to address these complex nonlinear problems Meta-heuristic algorithms, such as genetic algorithms, particle swarm optimization, and the firefly algorithm, have surged in popularity due to their simplicity, flexibility, efficiency, and ease of implementation, making them suitable for a diverse array of optimization challenges, particularly in structural optimization This field has garnered significant interest from researchers worldwide, leading to the development and application of numerous optimization techniques over the past decades, which can be broadly categorized into gradient-based and population-based approaches.
Gradient-based optimization methods, such as Sequential Quadratic Programming (SQP), Steepest Descent, Conjugate Gradient, and Newton's Method, are known for their rapid convergence to optimal solutions However, these methods can easily become trapped in local extrema and require gradient information to guide the search algorithm Additionally, their applicability is restricted to continuous design variables, which can hinder overall algorithm productivity The choice of initial solution or design parameters also plays a crucial role in the effectiveness of these approaches.
Population-based techniques, also known as meta-heuristic algorithms, significantly enhance the effectiveness of gradient-based algorithms in achieving global or local solutions Prominent examples include Genetic Algorithms (GA), Differential Evolution (DE), Particle Swarm Optimization (PSO), Cuckoo Search (CS), and Firefly Algorithm (FA) These methods are widely utilized in structural optimization due to their flexibility and efficiency in managing both continuous and discontinuous design variables Notably, solutions derived from these population-based algorithms often yield global optima, minimizing the influence of the initial design on the final outcome Among these techniques, Differential Evolution stands out as one of the most popular and effective methods since its introduction by Storn and Price.
Numerous studies have focused on enhancing Differential Evolution (DE) for structural optimization problems, showcasing its exceptional performance across various engineering challenges For instance, Wang et al utilized DE for optimal truss design involving both continuous and discrete variables, while Wu and Tseng implemented a multi-population DE with a self-adaptive strategy to address truss structure challenges Le-Anh et al improved DE for static and frequency optimization of laminated composite plates, and Ho-Huu et al introduced a novel DE version for optimizing truss shapes and sizes with discrete variables Additionally, the recently proposed Jaya algorithm by Rao has emerged as an efficient alternative for optimization tasks, outperforming several population-based methods in benchmark tests Despite its effectiveness, the original Jaya algorithm's performance has limitations, leading to various enhancements aimed at boosting its efficacy This thesis presents a refined version of the Jaya algorithm, designed to enhance population selection for improved convergence speed while maintaining accuracy and balancing exploration and exploitation.
While Differential Evolution (DE) and Jaya optimization methods offer benefits, they are slower than gradient-based techniques due to the extensive time required for fitness evaluations in population-based optimizations In structural optimization, this is particularly evident as the objective and constraint functions are often calculated using finite element analysis To enhance efficiency, integrating Artificial Neural Networks (ANN) with the DE algorithm presents a solution, as ANNs can quickly approximate outputs from input data after training This integration significantly accelerates the computation of objective and constraint function values in DE, thereby improving overall efficiency The effectiveness of ANNs, which trace back to foundational concepts introduced by Warren McCulloch and Walter Pitts in 1943, has been validated through various studies across diverse fields such as system identification, pattern recognition, data mining, and email filtering.
The development of optimal algorithms that integrate Artificial Neural Networks (ANN) with Differential Evolution (DE) is crucial for enhancing the design of reinforced composite plates, which are increasingly utilized in various fields such as construction, marine, and aviation due to their superior bending strength and lightweight properties These composite materials, particularly in the form of beams and plates, offer significant economic advantages by improving bending stiffness while minimizing material usage As the demand for cost-effective and efficient structural designs rises, optimizing these complex composite structures becomes essential However, the intricate behavior of composite materials, influenced by numerous geometric and material parameters, complicates the optimization process, making traditional gradient-based algorithms less effective Therefore, population-based methodologies emerge as a more suitable solution for addressing the optimization challenges associated with composite material structures.
In engineering design, uncertainties from sources like manufacturing processes, material properties, and operating environments significantly impact optimal designs, potentially leading to structural performance issues and increased risks To address these uncertainties, two primary approaches are employed: reliability-based design and robust design While robust design aims to minimize variance in outcomes despite changes in design variables and parameters, reliability-based design optimization focuses on enhancing the dependability of designs under uncertain conditions.
Reliability-Based Design Optimization (RBDO) is a strategic approach that ensures design feasibility amidst variable changes, focusing on achieving optimal designs While RBDO offers greater reliability than static optimization, its practical application is hindered by high computational costs To address this issue, various effective reliability analysis techniques have been developed, including sensitivity-based approximations, most probable point (MPP) methods, Monte Carlo simulations, and response surface models, all aimed at integrating optimization and reliability assessment Research also explores efficient decoupling strategies, categorized into nested double-loop methods, decoupled methods, and single-loop methods Double-loop approaches provide high accuracy by evaluating reliability at each optimization iteration but incur significant computational costs In contrast, decoupled methods sequentially address optimization and reliability analysis, reducing costs but still involving interrelated loops Single-loop methods further simplify the process by transforming RBDO into an approximate deterministic optimization (ADO) problem, effectively lowering computational expenses.
The application of reliability-based design optimization (RBDO) methods to laminated composite beams is currently limited This thesis explores the Single-Loop Deterministic Methods (SLDM), recently introduced by Li et al., and aims to combine it with a meta-heuristic optimization algorithm to create a new toolset, SLDM-iJaya, for addressing RBDO challenges in composite structures.
This thesis explores modifications to enhance the original Differential Evolution and Jaya algorithms, aiming to improve their convergence rates The improved algorithms will be 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 reinforced composite plates and Timoshenko beams.
Motivation of the research
The motivation for this thesis arises from a comprehensive analysis of existing literature and an assessment of the application potential of composite material structures and intelligent optimization techniques, particularly focusing on 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 dissertation
This thesis aims to develop advanced meta-heuristic optimization methods integrated with Artificial Neural Networks (ANN) to create a novel algorithm for optimizing composite material structures By modifying the original Differential Evolution and Jaya algorithms, the research enhances convergence towards global optimal solutions, allowing for effective design of reinforced composite plate structures Additionally, the thesis introduces a new toolkit that merges the improved Jaya algorithm with the Single-Loop Deterministic Method to address Reliability-Based Design Optimization (RBDO) challenges, specifically targeting the optimization of composite beam structures.
Research scope of the dissertation
The thesis focuses on the following main issues:
- Optimize truss, beam and stiffened plate structures using steel and composite materials.
-Study and improve population-based optimization methods to increase accuracy and efficiency in solving optimization problems.
Leverage the power of Neural Networks to create approximate models from data sets, integrating them with optimal algorithms to enhance performance and effectively address a variety of complex problems.
-Combine optimal algorithms with groups of reliability assessment methods to solve RBDO problems.
The chosen optimization problems are straightforward, primarily aimed at assessing the effectiveness, accuracy, and reliability of the proposed optimization methods Future research will focus on applying these optimal methods to more complex problems.
Outline
The dissertation contains seven chapters and is structured as follows:
Chapter 1 provides a comprehensive overview of meta-heuristic algorithms, composite material structures, and the significant role of artificial neural networks in optimization processes It outlines the organization of the thesis and highlights its novelty and objectives, offering a quick insight into the study's focus.
Chapter 2 offers a comprehensive overview of composite materials, highlighting fundamental concepts and their real-life applications It introduces the Timoshenko composite beam theory and the reinforced composite plate, which are the primary structures analyzed in the optimization problems addressed in this thesis.
Chapter 3 focuses on meta-heuristic optimization techniques, specifically the Differential Evolution and Jaya algorithms, detailing modifications made to enhance their performance Additionally, this 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 the historical evolution and foundational concepts of Artificial Neural Networks (ANN) It introduces the Neural Network Structure utilized in this thesis for approximating data derived from Finite Element Analysis Additionally, this chapter discusses the training algorithm, with a focus on the Levenberg-Marquardt method and the phenomenon of overfitting.
Chapter 5 demonstrates the effectiveness and efficiency 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 reinforced composite plates The iDE is utilized to optimize the weight of truss structures and the fiber angle and thickness of composite plates, showcasing its strong performance Additionally, the chapter highlights the improved Jaya algorithm's capability in optimizing the design of Timoshenko composite beams, with results indicating superior accuracy compared to existing methods A novel approach, SLDM-iJaya, combines the improved Jaya algorithm with single-loop methods to tackle reliability-based design optimization (RBDO) problems, yielding effective solutions for both standard optimization and RBDO challenges The chapter also explores the integration of Artificial Neural Networks with meta-heuristic optimization techniques, resulting in the development of the ANN-Based Differential Evolution (ABDE) algorithm, which effectively addresses the optimal design of reinforced composite plates and paves the way for future applications.
Finally, Chapter 6 closes the concluding remarks and give out some recommendations for future work.
Concluding remarks
This chapter provides an overview of meta-heuristic optimization methods, artificial neural networks, and composite material structures in optimization It highlights the novel contributions of this dissertation and outlines its organization across eight chapters Subsequent chapters will delve into fundamental theories, explore modifications to enhance meta-heuristic algorithms, and present applications supported by numerical results.
Introduction to Composite Materials
2.1.1 Basic concepts and applications of Composite Materials
Structural materials are classified into four main categories: metals, polymers, ceramics, and composites Composites, which are formed by combining two or more distinct materials, often utilize combinations of the other three categories Initially, these materials were macroscopic, but advancements in technology have led to the development of nano-composites featuring nanometer-sized reinforcements like carbon nanoparticles, nano-fibers, and nanotubes, which exhibit remarkable properties The primary advantage of composites lies in their ability to provide desirable characteristics unattainable by individual constituent materials A common example is fibrous composites, where reinforcing fibers are embedded in a matrix material, though particle or flake reinforcements are less effective Natural examples of composites include wood, composed of fibrous cellulose in a lignin matrix, and mammalian bone, made of collagen fibrils within a protein-calcium phosphate matrix The effectiveness of fibrous reinforcement is attributed to the increased strength and stiffness of materials in fiber form compared to their bulk counterparts.
In 1920, Griffith discovered that the tensile strength of glass rods and fibers increased as their diameters decreased, due to a lower likelihood of failure-inducing surface cracks during production and handling Similar findings have been observed across various materials, highlighting the superior tensile strength and stiffness of fibers compared to bulk forms However, fibers have limitations; they cannot support longitudinal compressive loads, and their transverse properties often fall short of their longitudinal counterparts Therefore, fibers are typically ineffective as structural materials unless combined with a binder or matrix and reinforced transversely This necessity for strategic fiber orientation according to specific stress fields has led to the development of diverse composite materials.
1 In the continuous fiber composite laminate, individual continuous fiber/matrix laminae are oriented in the required directions and bonded together to form a laminate Although the continuous fiber laminate is used extensively, the potential for delamination, or separation of the laminae, is still a major problem because the interlaminar strength is matrix dominated Woven fiber composites do not have distinct laminae and are not susceptible to delamination, but strength and stiffness are sacrificed because the fibers are not as straight as in the continuous fiber laminate Chopped fiber composites may have short fibers randomly dispersed in the matrix Chopped fiber composites are used extensively in high-volume applications due to their low manufacturing cost, but their mechanical properties are considerably poorer than those of continuous fiber composites Finally, hybrid composites may consist of mixed chopped and continuous fibers, or mixed fiber types such as glass and carbon The design flexibility offered by these and other composite configurations is obviously quite attractive to designers, and the potential exists to design not only the structure but also the structural material itself.
Figure 2 1 Types of fiber-reinforced composites.
(a) Continuous fiber composite, (b) Woven composite, (c) Chopped fiber composite, (d) Hybrid composite.
Composite structural elements are increasingly utilized across various industries, including automotive, aerospace, marine, and architecture, as well as in consumer products like skis and golf clubs Military aircraft designers were pioneers in recognizing the advantages of composites due to their high specific strength and stiffness, which significantly enhance performance and maneuverability by reducing weight The use of composite materials also contributes to smoother surfaces that minimize drag Since the introduction of boron and graphite fibers in the 1960s, the application of advanced composites in military aircraft has rapidly expanded, with carbon fiber components like stabilizers and wing skins being common in fighter jets In commercial aviation, the adoption of composite materials has grown as costs decrease and technology advances, exemplified by the Boeing 787, which features a composite fuselage and wings, with up to 50% of its primary structure made from carbon fiber/epoxy composites The Airbus A350 XWB also showcases a similar focus on composite materials, highlighting the trend towards composites in modern airliner design.
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 use of composite components In cargo trucks, lighter composite materials allow for greater payload capacities, resulting in substantial economic benefits For instance, the composite concrete mixer drum depicted in Figure 2.3 is 2,000 lbs lighter than the traditional steel drum it has replaced.
Figure 2 3 Composite mixer drum on concrete transporter truck weighs 2000 lbs less than conventional steel mixer drum.
Composite materials, such as composite leaf springs, can achieve weight savings of over 70% compared to traditional steel components While experimental engine blocks made from graphite-reinforced thermoplastics have been developed, the ultimate aim is to create a ceramic composite engine that eliminates the need for water cooling Chopped glass fiber-reinforced plastics (FRPs) are widely used in body panels, focusing on stiffness and aesthetics Currently, composites are primarily utilized in secondary structures and appearance components in vehicles, with significant potential for primary structures yet to be realized The rise of electric vehicles necessitates lighter composite structures to offset the weight of heavy batteries, exemplified by the BMW Megacity electric vehicle, which features a carbon fiber composite passenger compartment integrated with an aluminum spaceframe Additionally, fiber-reinforced plastics are increasingly employed in civil infrastructure elements, while composite materials are favored for wind turbine blades due to their superior strength-to-weight and stiffness-to-weight ratios, along with excellent vibration damping and fatigue resistance This thesis focuses on the investigation and application of composite beam and reinforced composite plate structures in optimization design problems.
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 due to their numerous advantages over traditional materials like steel and aluminum A key benefit is their lower weight; for instance, high-strength Graphite/Epoxy and Glass/Epoxy have weight densities of 0.056 lb/in³ and 0.065 lb/in³, respectively, compared to aluminum's 0.10 lb/in³ Additionally, composites often exhibit superior stiffness and strength, allowing structural members to endure greater static loads and experience less deformation The stiffness of high-strength Graphite/Epoxy is approximately 22x10⁶ lb/in², significantly higher than aluminum's 10x10⁶ lb/in² These properties make composites an appealing choice for structural designers, who aim to optimize designs with minimal resource use, balancing weight, cost, strength, and stiffness Traditionally, engineers rely on experience to meet essential design requirements, but achieving an optimal balance between weight and stiffness can be challenging, as improvements in performance may compromise strength or stiffness standards.
Over the past thirty years, mathematical optimization has become a vital tool in structural design, focusing on maximizing or minimizing objective functions under specific constraints Recent studies have extensively explored the optimization of laminated composite structures, with notable works aimed at maximizing the first natural frequency, enhancing buckling load factors, minimizing weight, and maximizing strain energy For instance, various references detail the optimal design of laminated composite plates and beams, highlighting their effectiveness in improving performance metrics such as free vibration frequency.
The optimization design of continuous composite models for thin-walled composite box-beam helicopter rotor blades has been explored using various non-gradient-based algorithms, including particle swarm optimization and genetic algorithms These methods aim to either minimize weight or maximize buckling load while simultaneously reducing weight.
Lentz and Armanios derived precise 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.
The optimization methods for composite structures can be categorized into gradient-based and non-gradient-based algorithms Non-gradient-based algorithms, also known as random search algorithms, do not require gradient information, making them easier to implement compared to gradient-based algorithms, which rely on gradient calculations While gradient-based methods are more efficient and capable of identifying local optima, they have significant drawbacks, including a strong dependence on the initial point and a tendency to become trapped in local optima when multiple extremes exist Consequently, researchers often favor non-gradient-based methods, particularly meta-heuristic optimization techniques such as Particle Swarm Optimization (PSO), Genetic Algorithm (GA), Differential Evolution (DE), and Jaya algorithm, to achieve global solutions for laminated composite structure optimization problems This thesis focuses on the application of Differential Evolution and Jaya algorithm to optimize two composite structure models: the Timoshenko composite beam and the reinforced composite plate, with theoretical foundations discussed in 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 approaches, are simpler to implement, they often yield only approximate solutions and are less effective than the analytical methods associated with continuous models To address this limitation, Liu proposed a continuous model approach for composite laminated Timoshenko beams that allows for the derivation of exact solutions The following section outlines the process for developing the analytical solution for these beams, and readers are encouraged to consult Liu's work for a more comprehensive understanding of the method.
2.2.1 Exact analytical displacement and stress
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 (i 1, ,
The beam, characterized by a rectangular cross-section with width \( b \) and height \( h \), consists of multiple layers identified by their positions \( z_i \) (where \( i = 1, \ldots, N \)) Each beam segment \( dx \) experiences a transverse force, as illustrated 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:
(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.
11, B 11 , D 11 , A 55 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.
Figure 2 8 The material and laminate coordinate system
The stress fields in a composite laminated beam encompass both plane stress and shear stress components As illustrated in Figure 2.8, the coordinate system aligns the materials (123) with the beam/laminate (xyz), where the fiber orientation corresponds to the 1-axis The expressions for the plane stress components are defined accordingly.
12 xy where the strain components y 0, xy 0 , and
T (k) is the coordinate transformation matrix and Q(k ) is the matrix of material stiffness coefficients
sin (k) cos (k) sin (k) cos (k) cos 2 (k) sin 2
The shear stress components in the material coordinate systems are
13 xz where the shear strain components yz
T (k) and the matrix of stiffness coefficients
Qs can be described as
) ij is the stiffness coefficients of the kth lamina in the laminate coordinate system and are described clearly in [53].
The integration constants 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.
0 2 are applied The seven indefinite integration constants are then determined as follows
) 0 are used to determine seven indefinite integration constants 2
0 8) are employed and the seven the indefinite integration constants are then determined
2 , M y (L) 0 , Q z (L) 0, are applied to determine the seven the indefinite integration constants
Under the specified boundary conditions, the normal force along the x-axis (N), shear force along the z-axis (Q), and the bending moment about the y-axis (M) on the cross-section of the composite laminated beam are calculated as detailed in reference [53].
Analysis of reinforced composite plate
A reinforced composite plate is created by integrating a composite plate with a stiffening Timoshenko composite beam, which acts as a stiffener aligned parallel to the plate's surface axes, as shown in Figure 2.9.
The centroid of the beam is located at a distance e from the mid-plane of the plate The plate-beam system is divided into discrete nodes, with each node exhibiting a degree of freedom (DOF) represented by the vector d = [u, v, w, βx, βy].
T , in which u, v, w are the displacements 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 d [u , u
] T , wher e u r ,u s ,u z are respectively centroid displacements of beam and around r-axis and s-axis.
r , s are the rotations of beam
Figure 2 9 A composite plate reinforced by an r-direction beam
The displacement compatibility between plate & beam is ensured by: u u r (r) z r (r) ; v z s (r) ; w u The strain energy of composite plate is given by:
P 2 A 0 0 0 b b 0 b b w h e r e ε 0 , κ b , γ are respectively membrane, bending and shear strains of composite plate and are expressed as follows ε [u , v , u v ] T ; κ [ , , ] T ; γ
D m ,D mb ,D b ,D s are material matrices of plate.
The strain energy of composite stiffener is given by st r s z r
U 1 (ε st 2 l s where ε b ,ε s are respectively bending, shear strain of beam and are expressed as follows: ε b st [ b s st st are material matrices of composite beam. st
Using the superposition principle, total energy strain of reinforced composite plate is obtained:
U U P U st i 1 where N s t is the number of stiffeners.
For static analysis, the global equations for the reinforced composite plate
F can found in [61] for detail.
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 variety, although there is no universally accepted definition of heuristics and meta-heuristics, leading to some interchangeability in their usage The recent trend categorizes stochastic algorithms that utilize randomization and global exploration as meta-heuristics This randomization facilitates a shift from local to global search, making meta-heuristics particularly effective for nonlinear modeling and global optimization They offer a practical approach to finding acceptable solutions to complex problems within a reasonable timeframe, despite the inherent limitations of not guaranteeing the discovery of the best solutions Ultimately, the goal is to develop efficient algorithms that consistently produce high-quality solutions, with some being nearly optimal, even if optimality cannot be assured.
Meta-heuristic algorithms are primarily composed of two key components: intensification (exploitation) and diversification (exploration) Diversification involves generating a variety of solutions to thoroughly explore the global search space, while intensification focuses on refining solutions within a localized area by leveraging information from promising candidates The selection of the best solutions is crucial, as it drives convergence towards optimality Simultaneously, diversification through randomization helps prevent the algorithm from getting stuck in local optima, thereby enhancing solution diversity A well-balanced integration of these components is essential for achieving global solutions effectively.
Meta-heuristic algorithms can be categorized into two main types: population-based and trajectory-based Population-based algorithms, such as genetic algorithms (GA), genetic programming (GP), and particle swarm optimization (PSO), utilize a group of solutions or agents to explore the search space In contrast, trajectory-based algorithms, like simulated annealing (SA), operate with a single solution that navigates through the design space Additionally, artificial neural networks adopt a unique methodology distinct from these classifications.
Modeling and optimization are crucial for addressing real-world problems, as modeling ensures that objective functions are assessed using accurate mathematical or numerical representations, while optimization identifies the best design parameter settings A key component of optimization is the use of algorithms, particularly meta-heuristic algorithms, which play a significant role in achieving optimal solutions.
3.1.1 Meta-heuristic Algorithm in Modeling
Nonlinear system modeling can utilize various methodologies, each with distinct advantages and disadvantages The challenge lies in determining both the structure and parameters of engineering systems, making modeling complex Generally, models are 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 the relationships between inputs and outputs derived from measured data, eliminating the need for prior understanding of the underlying mechanisms Behavioral models are advantageous due to their ability to yield accurate results with minimal effort, with statistical regression techniques being a popular approach within this category.
Recent advancements in computer hardware have facilitated the development of several alternative meta-heuristic approaches for behavioral modeling, making them more efficient than traditional methods Notably, Artificial Neural Networks (ANNs) and Genetic Programming (GP) are prominent meta-heuristic algorithms utilized in nonlinear modeling While ANNs have been successfully applied to various structural engineering challenges, they often lack transparency in their solution processes In contrast, GP, an extension of genetic algorithms, operates as a supervised machine learning technique that explores program spaces and generates models represented as tree structures using functional programming languages This unique capability allows GP to create predictive models without predefined relationship assumptions, offering a significant advantage over regression and ANN methods Additionally, other meta-heuristic algorithms, such as Fuzzy Logic (FL) and Support Vector Machine (SVM), have also been employed in modeling applications.
Artificial Neural Networks (ANNs) originated from the simulation of the biological nervous system and were first developed in the early 1940s by McCulloch and his team Initially, research concentrated on creating basic neural networks to model simple logic functions Today, ANNs are utilized for complex problems that lack straightforward algorithmic solutions This study explores the approximation capabilities of two prominent ANN architectures: Multi-Layer Perceptron (MLP) and Radial Basis Function (RBF).
Genetic Programming (GP) is a symbolic optimization technique that utilizes Darwinian natural selection principles to develop computer programs aimed at problem-solving The groundwork for GP was laid by Friedberg, who introduced a learning algorithm for progressive program enhancement Later, Cramer advanced the field by integrating genetic algorithms and tree-like structures to evolve programs A significant advancement in GP occurred in the late 1980s when Koza conducted pioneering experiments on symbolic regression, positioning GP as an extension of genetic algorithms, with distinct differences in their approaches.
Genetic Algorithms (GA) represent solutions through numerical strings, while Genetic Programming (GP) utilizes tree structures to express computer programs in functional programming languages like LISP Unlike GA's fixed-length binary strings, GP evolves programs as variable-length parse trees throughout the process This marks the inception of self-programming computer systems.
Genetic Programming (GP) develops computer programs that can be executed directly, unlike Genetic Algorithms (GA), which often require post-processing for their evolved binary strings While GA is primarily used for parameter optimization to identify optimal values within a model, GP not only determines the structure of the approximation model but also optimizes its parameters GP evaluates a population of programs based on a fitness landscape, which is defined by each program's ability to perform specific computational tasks The fitness of these programs is assessed through a predefined fitness function, which serves as the objective for optimization in GP This traditional method of GP is commonly known as tree-based GP.
In the realm of genetic programming (GP), various forms exist, including linear and graph-based representations This study primarily focuses on linear-based GP methodologies.
Fuzzy Logic (FL) maps input spaces to output spaces through membership functions and linguistically defined rules, aligning closely with human reasoning Introduced by Zadeh, the fuzzy set concept allows for partial membership of objects in various subsets, represented numerically by a membership function that assigns values between 0 and 1, indicating the degree of belonging to a fuzzy set Membership functions can take various shapes, including bell, sigmoid, triangle, and trapezoid, facilitating the modeling of complex systems that cannot be expressed with linear equations In FL, decision-making relies on rules and membership sets, emphasizing that the truth of a statement can vary continuously rather than being confined to binary logic values.
A fuzzy system utilizes linguistic variables to manage degrees through specific functions, comprising input and output variables Each variable is represented by fuzzy sets, which are characterized by defined membership functions The system operates based on a fuzzy rule base that establishes relationships between these input and output variables through IF-THEN rules The fuzzy logic system includes four key components: fuzzification, fuzzy rule base, fuzzy inference engine, and defuzzification In the fuzzification stage, input data is transformed into degrees of membership using membership functions The fuzzy rule base contains all possible fuzzy relations, while the fuzzy inference engine processes these rules to convert inputs into corresponding outputs The inference process results in a fuzzy set, which is then translated into a numerical value during the defuzzification stage.
3.1.2 Meta-heuristic Algorithm in Optimization
Finding an optimal solution to an optimization problem can be quite challenging and largely depends on selecting the appropriate algorithm based on the problem type, available algorithms, computational resources, and time constraints For large-scale, nonlinear, global optimization problems, there is often a lack of clear guidelines for algorithm selection, and efficient algorithms are frequently unavailable This challenge is particularly pronounced in hard optimization problems, such as NP-hard problems, where no efficient algorithms exist Generally, optimization problems can be expressed in a common generic form: minimize x.
subject to f i (x), (i 1, 2, , M ) h j (x) 0,( j 1, 2, , J ), g k (x) 0,(k 1, 2, , K ) where f i (x), h j (x) and g k (x) are functions of the design vector x = (x1, x2, , xn) n
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 f i (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 is called the design space or search space The equalities for h j and inequalities for g k 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.
Figure 3 1 Source of inspiration in meta-heuristic optimization algorithms
Solving Optimization problems using improved Differential Evolution 41
An optimization problem can be expressed as follows:
hi (x) 0 i 1, ,l min f (x) s.t x g j (x) 0 j 1, , m where x is the vector of design variables; h i (x) 0 and gj (x)
0 are equality and inequality constraints; l, m are the number of inequality and equality constraints, respectively; cost, etc. f
The objective of design optimization is to minimize the objective function, which may depend on weight, by identifying the optimal values of design variables within the design space Various optimization methods, including gradient-based and population-based approaches, are employed to tackle these challenges This paper specifically utilizes Differential Evolution to determine the optimal fiber orientations and thicknesses of reinforced composite plates.
3.2.1 Brief on the Differential Evolution algorithm [14], [129]
The Differential Evolution algorithm, initially introduced by Storn and Price, has gained popularity for addressing various optimization challenges This algorithm operates through a structured approach that includes four key phases.
Create an initial population by randomly sampling from the search space
Generate a new mutant vector v i from each current individual x i based on mutation operations.
Create a trial vector u i by replacing some elements of the mutant vector v i via crossover operation.
Compare the trial vector u i with the target vector x i One with lower objective function value will survive in the next generation
To improve the effectiveness of the algorithm, the Mutation phase and the Selection phase are modified to increase the convergence rate as follow:
In the mutation phase of Differential Evolution (DE), randomly selecting parent vectors from the current population can hinder the exploitation of optimal solutions To enhance convergence speed, individuals for mutation should be prioritized based on their fitness levels This approach preserves valuable information from high-fitness parents in the offspring, improving overall solution quality Instead of random selection, the Roulette wheel selection method, as proposed by Lipowski and Lipowska, is employed to choose individuals, ensuring that beneficial traits are effectively passed on to subsequent generations.
In the selection phase of the Differential Evolution (DE) algorithm, the Elitist operator introduced by Padhye et al is utilized instead of the traditional selection method This elitist approach involves merging the trial vectors from the children population (C) with the target vectors from the parent population (P) to form a combined population (Q) From this combined population, the top-performing individuals are selected to ensure that the best solutions are preserved for the subsequent generation This strategy guarantees that the most optimal individuals are retained, enhancing the overall effectiveness of the evolutionary process.
3.2.2 The modified algorithm Roulette-Wheel-Elitist Differential Evolution
Generate the initial population Evaluate the fitness for each individual in the population while do
Calculate the selection probability for each individual for i =1 to NP do {NP: Size of population}
In the mutation phase, a Roulette wheel selection is employed where a random index \( j_{rand} \) is generated from the design variable count \( D \) For each design variable \( j \) from 1 to \( D \), if a random number between 0 and 1 is less than the crossover control parameter \( CR \) or if \( j \) equals \( j_{rand} \), a new trial vector component \( u_{i,j} \) is created using a combination of existing vectors \( x_{r1,j} \), \( x_{r2,j} \), and \( x_{r3,j} \), scaled by a randomly chosen factor \( F \) within the interval [0,1] If the condition is not met, \( u_{i,j} \) retains the value from the original vector \( x_{i,j} \) After evaluating the trial vector \( u_i \), the selection phase is conducted using an elitist selection operator The process continues until completion, ultimately yielding the results.
Solving Optimization problems using improved Jaya algorithm
Jaya algorithm is a population-based global optimization technique, developed by
Ventaka Rao has introduced a straightforward algorithm that is easy to implement and does not rely on specific parameters The core principle of this algorithm is to consistently steer clear of failure by distancing itself from the worst solutions while moving towards the optimal solution during the search process Its simplicity lies in the fact that it only requires common controlling parameters for effective execution.
(population size and number of generations) to accomplish to optimization task The optimization process using this algorithm can be summarized in four following simple steps as in [128].
An initial population of NP individuals is randomly generated within the search space, where each candidate represents a vector of n design variables \( x_i \) These vectors are created while adhering to specified lower and upper bounds.
(x 1 , x 2 , , x n ) x x l rand[0,1] x u x l , j (1, 2, ,n),i (1, 2, , N ) j ,i j ,i j ,i j ,i p (3.5) where x u and l 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), and the fitness of each individual in the population is assessed using the values of f(x_i) 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, if x_{j,i,k} represents the j-th variable for the i-th candidate, it undergoes stochastic modification to produce a new vector x'_{j,i,k} This new value is calculated using the formula x'_{j,i,k} = x_{j,i,k} + r_{1,j,k} × (x_{j,best,k} - x_{j,i,k}) - r_{2,j,k} × (x_{j,worst,k} - x_{j,i,k}).
In the kth iteration, the values of the jth variable are represented by the worst candidate (x) and the best candidate (x) within the entire population The random numbers, denoted as r_wors and t_b_e_1,j,k, are generated within the range of [0,1] This randomness plays a crucial role in the optimization process.
indicates the tendency of the design variable moving toward the best solution and the te rm r
x shows the tendency of the solution avoiding the worst o n e
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
,k i s a c c e p t e d i f i t gives better objectiv e function value, otherwis e, the val ue of x is chosen
All accepted candidat es at the end of iteration are maintain ed j ,i ,k and beco me the input to the next iterati on.
Impr ovem ent versio n of
In the original Jaya algorithm, the next generation of the population is determined through paired comparisons of fitness values This method can lead to the unintended elimination of strong individuals if they are compared against even stronger counterparts.
Despite an individual's performance being inferior to its competitor within a pair, it may still outperform winners from other pairs in the broader population To identify the best individuals for the next generation, we implement the elitist selection technique proposed by Padhye et al or an alternative selection step in the Jaya algorithm The selection process begins at the kth iteration, where a set of 2N individuals, referred to as set A, is generated This set A is created by merging all candidates from set X with the modified candidates from set X’.
NP best individuals out of 2NP candidates from set A are selected to construct the population for the (k+1)th generation The elitist selection technique is depicted as follows
1 Input: initial population set X and the modified set X’
3 Select Npbest out of 2NP individuals of A and assign to X new
This method ensures that the top individuals from the population are preserved for future generations, allowing the algorithm to achieve optimal solutions more efficiently and with improved convergence rates.
Reliability-based design optimization using a global single loop
Engineering design often faces 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 issues, reliability-based design optimization (RBDO) has emerged as a strategic approach for achieving optimal designs RBDO methods can be categorized into three groups: double-loop methods (DLM), decoupled methods, and single-loop methods Double-loop methods integrate reliability analysis within the optimization process, resulting in high computational costs In contrast, decoupled methods separate the two tasks, reducing costs but still involving complex computations Single-loop methods simplify the process by converting RBDO problems into approximate deterministic optimization (ADO) problems, significantly lowering computational demands and making them suitable for real-world applications Additionally, selecting the right optimization algorithm is crucial for effective RBDO solutions, with population-based methods being favored over gradient-based approaches The Jaya algorithm, introduced by Rao, has shown promising results in optimization tasks, outperforming various population-based methods and demonstrating its effectiveness in engineering design problems, although its application to RBDO remains underexplored.
Despite numerous studies on the design optimization of composite laminated beams, there remains a scarcity of research incorporating reliability analysis, often leading to significant limitations Yangjun et al employed a double-loop method to tackle the reliability-based design optimization of a single-span adhesive bonded steel–concrete composite beam across various loading scenarios Similarly, Fabrizio et al utilized probabilistic analysis for the design optimization of a composite floor beam in aircraft, aiming for a design resilient to system variations and reduced failure risk While the RBDO methods used in these studies enhance result reliability, they frequently yield optimal solutions confined to local regions, with beam models derived from numerical methods Consequently, these approaches still exhibit notable drawbacks and inaccuracies in achieving optimal solutions.
This thesis introduces a novel 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 using a continuous Timoshenko beam model Key enhancements to the original Jaya algorithm include the implementation of a special rounding technique for design variables, which improves its handling of discrete variables Additionally, during the population selection for the next generation, the current population set is combined with a modified set to form a compound set, from which the top Np individuals with the best fitness values are retained This modification ensures the retention of valuable information for subsequent generations, thereby enhancing the algorithm's effectiveness The combination of SLDM and iJaya is shown to significantly reduce computational costs, achieve global optimal solutions, and efficiently manage both discrete and continuous design variables in RBDO problems.
The mathematical model of a typical RBDO problem can be described as follows[135]: find d,μ x min f (d,μ x , μ p ) subject toProb.[g i (d, x, p) 0] R i , i 1, 2,3, , m d low d d up ,μ x low μ μ x x up
The objective function is represented by \( f(d, x, p) \), where \( d \) is the vector of deterministic design variables, and \( d_{\text{low}} \) and \( d_{\text{up}} \) denote the lower and upper bounds of \( d \), respectively Additionally, \( x \) represents the vector of random design variables, with \( \mu_x \) as its mean vector, and \( \mu_{\text{low}} \) and \( \mu_{\text{up}} \) as the bounds for \( \mu_x \) The random parameters are encapsulated in vector \( p \), while \( g_i(d, x, p) \) defines the constraint functions, with \( R_i \) indicating the desired reliabilities for these constraints The total number of constraint satisfactions is denoted by \( m \), and the probability operator is expressed as \( \text{Prob}[g_i(d, x, p) \leq 0] \geq R_i \), signifying that the probability of meeting the reliability requirement \( R_i \) for constraint satisfaction is maintained.
0 should be greater than or equal to the desired
3.4.2 A global single-loop deterministic approach
The single-loop deterministic method (SLDM) has emerged as an effective technique for addressing reliability-based design optimization (RBDO) challenges by transforming probabilistic constraints into approximate deterministic ones, significantly reducing computational costs However, SLDM is restricted to continuous design variables and often yields local extrema in optimal solutions To address these limitations, Vinh et al developed a global single-loop deterministic approach, utilizing the improved differential evolution algorithm (IDE) to tackle RBDO problems in truss structures with both continuous and discrete design variables This thesis introduces the Jaya algorithm, along with its enhanced version, iJaya, to pursue global optimal solutions for the design optimization of composite beam structures.
* Formulation of approximate deterministic constraints
The initial stage of SLDM involves creating 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 3.2.
Figure 3 2 Illustration of the feasible design region.
Figure 3.2 illustrates the feasible design region, where the red curve represents the limit-state function, and the green curves indicate the boundary of the transformed deterministic constraint function The dotted area signifies the deterministic feasible design region This transformation guarantees that the minimum distance from any point on the red curve to the green curve is βj, and it has been demonstrated that the resulting solutions comply with the probabilistic constraint [38].
After, the feasible design region is formed Suppose that μ θ is a point lying on gi (d,μθ ) , then the Most Probable Point
(MPP), θ MPP corresponding to μ θ can be determined by moving μθ backward to g i (d,θ) a distance
As shown in [134], θ MPP on the failure in a standard normal space can be defined by
(3.10) where the subscript u denotes the standard normal distribution space and the derivatives (g / u
The relationship of random parameters in the original design space and the normal standard space is depicted as follows
By combining equation ( 3.10),( 3.11),( 3.12), the relationship between in the original design space is denoted by μ θ and θ MPP where the derivatives (g i / j ) * are evaluated at θ j ,
According to Li et Al [38], the derivatives (g i / j ) * can be approximately assessed at j , and then equation ( 3.13) can be rewritten as θ j (g i / j ) # j ,MPP j i j j ( j ( g / i j u # ) ) 2
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 g i (d,μ θ ) g i (d,μ θ i σ θ n) 0 wher e n ( θ θ g i
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 ( 3.9 ) can be reformulated by an ADO problem as follows: find d,μ x min f (d,μ x ) subject to g i (d,μ x + i σ θ n) 0, i 1, 2,3, , m d low d d up ,μ x low μ μ x x up
The derivatives, denoted as θ g i (μ θ ), can be obtained directly from an explicit limit-state function However, in practical applications, structural behaviors are frequently analyzed using numerical methods, resulting in implicit limit-state functions To compute the derivatives θ g i (μ θ ), this study employs the finite difference method, a numerical approach for calculating derivatives.