A review on optimization of composite structures Part I: Laminated composites

Abstract

Composite, sandwich and functionally graded materials are frequently utilized for different applications. Thus, numerous optimization studies have been conducted on structures made up of these materials to improve their mechanical or thermal behavior such as buckling resistance, stiffness and strength along with reducing weight, cost and stress under various types of loadings. This work which is the first part of two sequential review papers attempts to review most of the studies carried out from 2000 on optimizing composite structures by representing a classification based on the type of structures. Important parameters of these optimization approaches namely objective functions, design variables, constraints and the applied algorithms are highlighted. The influential factors including boundary conditions, orientation of curved fibers, piezo electric patches, Shape Memory Alloy (SMA) fibers and stiffeners as well as the effect of design variables on the determined objectives are also noticed. Moreover, the examined outperformance of developed algorithms in case of accuracy and pace over their common counterparts is mentioned. The second part will be allocated to the publications on optimization of sandwich and functionally graded structures.

Introduction

For many reasons such as competitive marketplace, engineers are not only interested in designing a constructive structure, but they also seek for the best design. The process of finding the best possible design is called optimization which is highly applicable in various fields such as bridge engineering, building engineering, pipeline engineering and special mechanical structures such as components of vehicles, ships or aircrafts.

In the modern era today, optimization is one of the most crucial issues associated with engineering designs. Weight reduction, utilizing less expensive materials, increasing the strength, etc. have always been desired in designing different mechanical structures. The importance of optimization is even more in some progressive fields of engineering like aeronautics and automotive industries. Besides composite materials are getting more and more utilized in various industries due to their high strength to weight ratio. As a result, there are many optimization studies conducted on composite structures with two main perspectives. The first one involves the procedure of performing optimization, including various types of optimization problems and algorithms. The quality and quantity of objective functions and design variables examine the type of optimization problems. Algorithms represent the process by which design variables are optimized in order to achieve the objective functions. The next viewpoint deals with the types of composite structures, objective functions, design variables and constraints. A great insight into optimization of composite structures can be achieved by probing these two areas individually and combining them comprehensively.

Regarding the attribute of objective functions, design variables and constraints, optimization problems can be classified considering different aspects, which are described as follows.

• Trial and error and function-based optimization problems: Trial and error is a process in which the effect of design parameters on the objectives is not diagnosed. On the other hand, in some cases, there is a vivid formulation determining the correlation between design variables and objective functions. Such problems are called function-based optimization problems.

• One-directional and multi-directional optimization problems: This category is associated with the number of design variables. In case of a single design variable, the problem is called one-dimensional optimization problem while multi-directional problems deal with more than one design variable.

• Discrete and continuous optimization problems: This classification is based on the quality of design variables. When the design variables are discrete, the optimization problem is also indicated as discrete. The number of layers is an example of a discrete design variable. In contrast, in continuous optimization problems, the design variables are continuous.

• Constrained and unconstrained optimization problems: In some optimization approaches, along with changing design variables, certain constraints must be satisfied. These are called constrained optimization problems. On the contrary, in the procedure of unconstrained optimization studies, the design variables can freely change without any constraints.

• Local search and random search optimization problems: In local search problems, the optimization process starts at a particular point which has been diagnosed mathematically to be capable of obtaining better and more accurate results. These methods have a high convergency rate. However, there is the probability of miscalculating around extremum points as opposed to random search problems where better results are obtained following possible patterns. Therefore, it is complicated to predict the performance of the algorithm. The convergency rate of these methods is lower than local search problems while the chance of obtaining the global optima is higher.

• Single objective and multi objective optimization problems: This distinctive categorization has to do with the quantity of objective functions. When there is only one criterion or objective function to assess the optimum results, the optimization problem is referred to as single objective. In case of multi objective optimization however, several objective functions are often in conflict with each other and there are more than one optimum results regarding the priority of objectives. This set of optimum results is called Pareto front.

• Static and dynamic optimization problems: If the procedure of optimization depends on time, and the optimality changes after a period of time, it is known as a dynamic optimization problem in contrast with static type where the optimal design does not rely on time.

All the above mentioned features of optimization problems emerged in various types of algorithms classified into three main categories as follows.

• Blind search algorithms: These algorithms which are only capable of differentiating the objectives and producing some alternative results, need a wide search space. This leads to the drawback of low convergency rate.

• Heuristic search algorithms: The primary strategy which is exploited in these algorithms is to choose the node which is more probable to reach optimum results. This feature makes these methods more practical than blind search algorithms. Gradient Based Method (GBM) is the most popular heuristic type.

• Meta-heuristic algorithms: In most cases, a real optimization problem is involved with numerous effective parameters so that considering all of them leads to the exact optimal result in an unreasonably long period of time. Thus, in order to make a compromise between the accuracy of the results and computational costs, meta-heuristic algorithms have been proposed which lead to the vicinity of optimal point with substantially lower computational costs. The most common and applicable meta-heuristic algorithm is Genetic Algorithm (GA) which was firstly introduced by John Holland [1] in 1975. Ant Colony Optimization (ACO), Particle Swarm Optimization (PSO), Simulated Annealing (SA), Imperialistic Competitive Algorithm (ICA) and Firefly Algorithm (FA) are other most frequently used algorithms in optimization problems.

Advanced technology brought a significant improvement in mechanical structures by representing composite materials. Being lightweight as well as strong, these materials are perfectly suited for various industries. Although there have been some deficiencies in composites like sudden change in material properties which leads to various damage modes such as delamination and debonding, optimizing these structures is a rational idea to take advantage of them in different applications. These materials are constructed of two or more components providing enhanced properties compared to each component individually. Fiber-reinforced composites as an instance, are made up of fibers with high strength and stiffness embedded in the matrix which holds fibers and prevents fibers exposure from destructive environmental conditions such as humidity and moisture. Numerous developing applications of composite materials have been found in recent years as a result of utilizing boron, aramid and carbon fibers along with the ceramic and metal matrixes. In comparison with isotropic materials, composites have a wider range of parameters influencing the structural behavior. Ply orientations, fiber volume fraction, number of layers, stacking sequence, the material of fibers and matrix and the thickness of layers are examples of these effective parameters which can act as design variables in optimization problems. Most studies have considered fiber orientation angles and the thickness of layers as design variables. Moreover, objective functions vary tremendously among different optimization studies. The most frequent objective functions include buckling load, fundamental frequency, weight, load carrying capacity, deflection and stresses. In terms of mechanical perspective, composite structures are globally classified into three types of beams, plates and shells.

Since composite materials are getting more and more widespread among different industries, a bulk of review studies have been performed with the aim of classifying researches on these materials into particular categories. In an attempt to compile a collection of researches on vibration of composite beams, Hajianmaleki and Qatu [2] published a review paper to investigate various effective parameters such as shear deformation, piezoelectric, SMA fibers, damages, etc. on the vibrational behavior of composite beams. In a similar more comprehensive study, the works associated with buckling, bending and free vibration of composite beams were gathered into a review study carried out by Sayyad and Ghugal [3]. Another attempt was made by Sayyad and Ghugal [4] to review the articles on vibration characteristics of composite and sandwich plates. They also provided a number of numerical results obtaining fundamental frequency of square composite plates. Senthil et al. [5] reviewed most of the studies on the deficiencies such as debonding and delamination observed in structures constructed of composite materials. Damage prediction, growth initiation detection as well as corresponding numerical, analytical and experimental methods formed their review study. Ghiasi et al. [6], [7] reviewed over 200 papers on optimizing stacking sequence of constant-stiffness and variable-stiffness composite structures. Having the main focus on the process of optimizing and algorithms, they also discussed the level of efficiency and features of various methodologies. Some other review papers on mechanical behavior of composite structures, methods of analyzing these materials and advanced composite materials have been conducted in the literature [8], [9], [10], [11], [12], [13].

Although numerous review papers associated with composite structures have been published, there is a lack of a review study on optimizing constructions made up of these materials concentrating on the mechanical characteristics and results. The main purpose of this review paper is to highlight numerous features of works focused on optimization of composite structures. Type of structure, objective functions, design variables, constraints and algorithms constitute such features. The primary category is related to the type of structure as follows.

• Composite beams section which is divided into three parts namely conventional laminated beams, composite box-beams and composite bumper beams

• Composite plates which are sub-categorized into seven different classifications namely conventional laminated plates, smart composite plates with piezoelectric patches or SMA wires, stiffened composite plates, composite skew plates, variable-stiffness composite plates, perforated composite plates and patch repair composite plates

• Composite shells which are classified into five parts of composite cylindrical shell, composite conical shells, composite spherical shells, stiffened composite shells and general shaped composite shells.

• Other composite structures such as aero plane wings, turbine blades, crash structures, etc. made up of composite materials