A Regularized Discrete Laminate Parametrization Technique with Applications to Wing-Box Design Optimization

In this paper we present a novel ply parametrization technique for layered composites. This parametrization takes into account important manufacturing considerations, including the discrete selection of ply angles, the discrete nature of the layup thickness, and the continuity of ply angles across adjacent structural elements. The proposed technique can be used as either an alternative or as an enhancement to SIMP-type methods, such as discrete material optimization (DMO). We apply the proposed laminate parametrization technique to several structural optimization problems, including compliance minimization, failure-margin maximization, and buckling-load maximization, that have thousands of design variables and hundreds of constraints. These large design problems can only be solved, within a reasonable computational time, using gradient-based optimization methods with efficient sensitivity methods. Unlike mixed-integer approaches, our proposed laminate parametrization technique is well suited for gradient-based design optimization. As a final example, we apply the proposed parametrization method to the sizing and lamination sequence design of a composite wing-box. These results demonstrate that the proposed parametrization is effective for a wide range of structural optimization problems. The parametrization of composite structures for design optimization is a challenging problem. Realistic structural design optimization problems require numerous manufacturing constraints. Additionally, these design problems often involve a selection amongst a discrete set of options. For instance, the selection of ply angles from a set of available manufacturing options, or the selection of a laminated composite thickness that must be a integer multiple of the laminae thickness. These discrete problems are not, in their most natural form, amenable to gradient-based optimization. On the other hand, methods for nonlinear mixed-integer programming are almost inevitably computationally expensive, especially for large numbers of design variables. Here, we examine a laminate parametrization for structural design studies that takes into account the discrete nature of some of the design variables. In order to avoid solving a large, nonlinear, mixed-integer program, we use a relaxation approach where the original discrete problem is transformed into a continuous analogue of the original problem. We then obtain solutions to the modified problem using gradient-based optimization. To illustrate the effectiveness of the proposed laminate parametrization, especially for non-symmetric laminates, we examine a series of structural optimization problems of increasing difficulty. Many authors have developed different laminate parametrization techniques. These generally fall into two categories: direct parametrizations that provide an explicit description of the physical laminate, and indirect parametrizations in which intermediate variables are employed and the lamination sequence is only available in a post-processing calculation. There are difficulties with using either of these approaches. Direct techniques often introduce many local minima in the design space, while indirect methods make it difficult to impose manufacturing constraints on the physical construction of the laminate. Here we use a direct parametrization of the laminate and accept the possibility of local minima. The remainder of the paper is structured as follows. In Section II we review relevant literature on laminate parametrization techniques. In Section III we describe our proposed laminate parametrization technique and develop a method for predicting material failure within the context of the laminate parametrization. In Section IV we present an approach for discrete thickness selection. In Section V we present results from a series of structural optimization studies. Finally, in Section VI, we present an application of the proposed laminate parametrization technique to a wing-box sizing and layup-determination study.