Lay-Up Optimization of Composite Stiffened Panels Using Linear Approximations in Lamination Space

Abstract

L AY-UP optimization of composite structures is discrete. The direct application of integer programming (e.g., [1]) or genetic algorithms (GAs) (e.g., [2]) requires large computational resources for demanding structural constraints [e.g.,finite element (FE) based], due to the high number of function evaluations involved in the optimization. Alternatively, the use of lamination parameters [3] as design variables enables an efficient continuous optimization (e.g., [4]), although discrete constraints on the laminate stacking sequence cannot be imposed. To overcome this difficulty, a two-step optimizationwas initially proposed byYamazaki [5] tomaximize the buckling and frequency performance of a composite plate. A gradient-based optimization using lamination parameters as design variables was applied to get near the optimum discrete design. Then, a GA was used to obtain the laminate stacking sequence that most closely matched the lamination parameters from the first step, while satisfying the discrete constraints. Autio [6] followed a similar approach and introduced some lay-up design rules as penalties in the fitness function. An alternative two-step approachwas formulated by Todoroki and Haftka [7] to maximize the buckling load of a composite plate. First, continuous optimization of lamination parameters was performed to identify the neighborhood of the optimum discrete design. Next, a response surface approximation was created in that neighborhood and used with a GA to determine the laminate stacking sequence. It was shown that the two-step approach provided good designs with relatively low loss in performance when compared with the global optima. The authors’ previous work [8], based on a two-step approach, was applied successfully to optimize anisotropic composite panels with T-shaped stiffeners. At the first step, continuous optimization of lamination parameters was used to get near the optimum discrete design. The composite stiffened panel was modeled by a single representative skin-stiffener assembly or superstiffener. Figure 1 shows the superstiffener components, geometry, material axis, and positive sign convention for the applied loading. The design variables were the cross-sectional dimensions and lamination parameters of the superstiffener. The design constraints were strength, buckling, and practical design rules. Four stiffener types (a– d), as shown in Fig. 2, were considered in the optimization. At the second step, a GA was used to identify the lay-ups for the superstiffener’s laminates, which were the closest in the lamination parameter space to the continuous optima and satisfied the discrete design constraints. However, it was noted that sometimes the optimum discrete designs were not the closest in the lamination parameter space to the continuous optima. This note seeks to address the problem that the discrete optimum may not be the closest in the lamination parameter space to the continuous optimum. Assuming that the continuous optimum is reasonably close to the discrete optimum, a first-order Taylor series about the continuous optimum is suggested to be sufficient to approximate the design constraints. Thus, a new fitness function based on constraint satisfaction, using the linear approximation of the design constraints, is proposed for the GA to determine the design lay-ups.