ReviewOptimum stacking sequence design of composite materials Part I: Constant stiffness design
Introduction
The advantage of composite materials is that they provide excellent mechanical properties. However, using this advantage requires the optimization of shape and size and the proper placement of fibers within the material, which gives a good opportunity to tailor the material properties; however, it increases the complexity of the design problem. This complexity exists, not only because of numerous design variables, but also because of having a multimodal and variable-dimensional optimization problem with unattainable or costly derivatives.
This paper classifies and compares optimization techniques used for optimal lay-up selection of laminated composite materials. The aim of the review is to provide a reference for the selection of the technique that is most appropriate to solve a given problem. More details about the algorithms are found in Haftka and Gurdal [1], Gurdal et al. [2], and other references provided in the following sections.
In the literature, different classifications for optimization of composite laminate have been suggested. For example, Fang and Springer [3] identified four categories for the optimization methods, namely: (1) analytical procedures, (2) enumeration methods, (3) heuristic schemes, and (4) non-linear programming. Abrate [4], on the other hand, decided to classify optimization problems with respect to their objective functions that can be either one or a combination of in-plane properties, flexural rigidity, buckling load, natural frequency, and thermal effects. Venkataraman and Haftka [5] suggested another classification for composite panels, which categorized the design methods into two groups: (1) single laminate design, and (2) stiffened plate design.
We follow the classification provided by Setoodeh et al. [6], in which there are two scenarios for the design of a composite structure:
- (I)
Constant stiffness, in which the composite part is considered as a single element with the same stacking sequence all over the domain. The design goal is to find an optimal stacking sequence that is uniform for the entire structure.
- (II)
Variable stiffness, in which the structure consists of multiple elements, each of them with a different stacking sequence. Here, material distribution and fiber orientation might change over the structural domain.
For the first scenario, which is the focus of this paper, we examine the following optimization methods:
- 1.
Gradient-based. These methods utilize gradient information of the objective(s) and the constraint(s) to find the direction and size of the step towards the optimum solution.
- 2.
Direct Search and Heuristic. In contrast to the previous group, these methods do not need any gradient information; rather they require only the values of the objective function. Heuristic and Enumeration methods are also included here.
- 3.
Specialized techniques. These methods are developed for solving a lay-up design problem, in which some properties of composite laminates are used to simplify the problem; thus, they cannot be applied to a general optimization problem.
- 4.
Hybrid methods. In this class fall methods that combine two or more optimization techniques to benefit from the strength of all constituent techniques.
In this paper, we review optimization techniques used for constant stiffness design of composite laminates. The scenario of variable stiffness design will be the focus of a future paper, which will also analyze other issues, such as multi-objective optimization, discretization techniques, design for manufacturing, sensitivity analysis, and design for uncertainty. The following sections describe the main optimization methods for constant stiffness design with respect to the four classes given above. This design scenario is simpler than that for the variable stiffness design, since generally there are fewer design variables involved.
Section snippets
Gradient-based methods
These algorithms are based on the gradient of the objective and the constraints, whose functions, when their mathematical closed-form expression is not available, can be approximated, although it may be computationally expensive. The solutions obtained with gradient-based methods are only local optima, but the advantage of these methods is the faster convergence rate as opposed to that achieved by direct and heuristic methods.
Direct search methods
While the analytical methods are known for a fast convergence rate, direct search methods have the advantage of requiring no gradient information of the objective functions and the constraints. This feature is a significant advantage because in composite laminate design derivative calculations or their approximations are often costly or impossible to obtain. Direct search methods systematically approach the optimum solution only by using function values from the preceding steps. As a result,
Specialized algorithms
In this category, fall methods developed explicitly for optimizing laminated composite materials. These strategies exploit a number of properties of the composite laminates to simplify the optimization process. Often developed for a particular application, these methods generally simplify the problem by restricting the design space in terms of allowable lay-up, loading condition, and/or the objective function. Since these methods are tailored to a specific design problem, they lose robustness
Hybrid methods
A hybrid method combines two or more optimization methods to benefit from the advantages of all of them in order to obtain a better convergence rate, to achieve a global optimum, to have a better accuracy, to make the optimization method more robust, or for other reasons.
In order to handle both continuous and discrete design variables in a general multimodal non-linear problem, Seeley and Chattopadhyay et al. [136] developed a combined simulated annealing and sequential programming optimization
Concluding remarks
The most popular methods for optimizing the stacking sequence of laminated composites with uniform stiffness have been reviewed in this paper. The optimization techniques are classified into four categories: gradient-based methods, direct search and heuristic methods, specialized techniques, and hybrid methods. A brief list of all these methods and their properties is shown in Table 1. The last column in this table shows the relative convergence rate of each method compared to others, when such
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