Sequential permutation table method for optimization of stacking sequence in composite laminates
Introduction
Composite materials provide unmatched potential and large freedom in design to reduce their weight and/or to improve their performance. Thickness, a stacking sequence and ply orientations are often set as design variables to tailor global levels of stiffness and strength of composite laminates [1], [2]. A discrete set of ply laminates makes it a non-convex, nonlinear and high-dimensional complex mixed-discrete optimization problem [3] with many local optima, thus bringing new challenges for the design and optimization process.
Though the stacking-sequence optimization is a mixed-discrete and high-dimensional optimization problem with discrete variables, modern mathematical optimization methods combined with specific techniques appears to be capable of providing efficient solution to this problem. Buckling is one of the most important constraints in composite structural optimization since it is one of the most common failure forms of composite structures [2], [4]. A variety of optimization methods were applied to this problem [5], [6]. The most popular method, namely a genetic algorithm (GA) [7], [8], [9], is used in stacking-sequence design with buckling, strength and continuity constraints. The problem, described in [7], of optimization of the buckling load factor by changing the ply orientations, attracts many researchers developing a variety of heuristic methods [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20]. In particular, a permutation GA with a repair strategy was developed [12] for continuity constraints. Furthermore, a bi-level optimization scheme based on the permutation GA is proposed for the composite wing box design [14], [15]. An integrated approach, combining a shape optimization process with the GA was proposed in [21] for shape optimization and stacking-sequence design of composite laminates. Another popular method is evolutionary algorithm (EA), used in multiobjective optimization of composite laminates [22] and tapered composite structures [23]. More recently, a permutation search (PS) is proposed in [24], where a buckling load factor was expressed as discrete forms of function of ply orientations to reduce computational cost.
On the other hand, optimization of multi-panel composite structures is more difficult due to blending of adjacent panels and nature of ply drops in design. The GA coupled with a response surface method was used in [18] to optimize both single-panel and two-patch design examples. In recent years, a stacking-sequence table (SST) scheme was specialized for EA-based blending optimization [23]. Recently, a global shared-layer blending (GSLB) [25] method was proposed based on a shared-layer blending (SLB) method [26] for blending design.
Although, various heuristic methods are applied to the stacking-sequence optimization problem, they are sensitive to initial values and predefined parameters. For specific optimization problems, the predefined parameters should be adjusted properly to find the optimal solutions. Unsuitable initial values or predefined parameters may result in a low convergence rate or even fail to find the optimum. Additionally, the design space increases exponentially with the increasing number of design variables, making heuristic methods computationally expensive for solving optimization problems of large-scale composite structures. In order to overcome these limitations as well as concerns of convergence difficulties and of problem size of heuristic methods, a specific method is motivated by previous study for stacking-sequence optimization.
It is suggested that a stacking sequence of the laminate can be designed based on the classical laminate theory employing a specific designed algorithm [24], [27], [28]. In [27], a two-level method was developed to determine the feasible region of the lamination parameters. Recently, a bi-level optimization scheme for finding an optimal stacking sequence of composite laminates subjected to mechanical, blending and manufacturing constraints was developed in [28]. In this paper, based on a theory of lay-up optimization [27], [28], a sequential permutation table (SPT) method was proposed for stacking-sequence optimization.
The remainder of this paper is arranged as follows: in Section 2, an optimization problem is formulated. Then, the SPT method is developed in Section 3. In Section 4, performance of the method is compared with several heuristic methods via a benchmark problem. Finally, some conclusions and suggestions are provided in Section 5.
Section snippets
Statement of optimization model
Buckling analysis of a symmetric and balanced composite plate simply supported on four edges (Fig. 1a), can be formulated using an analytical method [16]. The classical plate theory of composite laminate is briefly summarized in Appendix A. Considering an orthotropic plate, where D16 = D26 = 0, buckling of the plate into m and n half-waves (along the x and y directions) occurs when the load amplitude factor λb reaches the following value [16]:where Fx
Design criterion
It is observed that feasible regions of lamination parameters are convex and their values reach a vertex with one ply orientation at all stacking positions [27], [28]. Such rules can be applied to flexural stiffness thanks to their linear dependence on lamination parameters. Furthermore, the buckling load factor λcb is a linear function of flexural stiffness parameters and reaches an extreme value with one ply orientation at all stacking positions. Therefore, λcb can be designed to vary
Numerical experiments
A benchmark problem proposed by Le Riche and Haftka [7], which was optimized with various methods in [8], [9], [10], [11], [13], [16], [19], [20], is used here to validate the SPT method. The material properties of a graphite-epoxy plate are given [7] as: E1 = 127.59 GPa, E2 = 13.03 GPa, G12 = 6.41 GPa, t = 0.0127 cm, υ12 = 0.3. In this work, four additional load cases with the aspect ratio a/b = 0.125 and 64 layers are added (Table 2). Furthermore, to verify the efficiency of this method, the orientations are
Conclusions
In this work, a sequential permutation table (SPT) method is proposed to optimize the stacking sequence of the composite laminate with buckling, strength and continuity constraints. The ply orientations are set to be the same with the predefined values at all stacking positions. The corresponding sequences are saved in a table, sorted according to the buckling load and failure load factors respectively. Corresponding sequence-design and ply-orientation selection strategies are also developed to
Acknowledgement
This work is supported by the National Natural Science Foundation of China and China Scholarship Council.
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