Static Aeroelastic Optimization of Composite Wings with Variable Stiffness Laminates

The application of composite material in load carrying structural components of an aircraft is rapidly gaining momentum. While part of the reason for this can certainly be attributed to an increasing confidence of designers in the new material as a result of growing experience, two other crucial points can be made. One, the continuous enhancements in the area of automated production technologies, which are an absolute necessity for ensuring consistent quality in a series production. Two, the progress in the development of computational methods to analyze and optimize composite structures in order to fully exploit their possible advantages over homogeneous materials. Nevertheless, it is still by virtue of challenges in their production as well as computational complexity, that full-fledged variable stiffness designs have not yet found their way in industrial large scale applications. Considering the complex path from the stiffness of a single laminate to the aeroelastic performance of an entire aircraft wing, it becomes clear that variable stiffness optimization is a non-trivial, laborious task. Not only does it require a large amount of design variables in order to achieve an adequate resolution, in addition the diverseness of responses impedes the problem definition. The research presented in this thesis aims at an advancement of the computational treatment, i.e. the development of a variable stiffness composite optimization framework, allowing for the consideration of static aeroelastic responses in the structural design of aircraft wings. Considering the different ways of optimizing composite structures, the strategy pursued in this thesis relates to a separation of the problem in three consecutive parts, the advantage being that each step can be handled with the most suitable optimization tools. The first part comprises an optimization based on laminate stiffnesses and is the main subject of this dissertation. It will be discussed in more detail below. The second part involves a stacking sequence optimization on the basis of the optimal stiffnesses derived in the first part. Part three deals with the optimal conversion of stacking sequences to fiber paths suitable for the chosen production technology. Parts two and three do not depend significantly on the physics of the problem. However, since it closely relates to the continuous optimization in part one, the stacking sequence optimization will also shortly be addressed. Part three is not dealt with in this thesis. The composite optimization framework consists of a successive convex subproblem iteration procedure, in which a gradient based optimizer consecutively solves a local approximation problem. Each response to be considered in the optimization, either as objective or as constraint, is approximated as a linear and/or reciprocal function of the laminate membrane and bending stiffness matrices A and D. Together with the laminate thicknesses h, they constitute the design variables in the optimization process. The distribution of design fields - each of which comprises its own set of A, D, h variables - within a structural entity like a wing skin, determines the variable stiffness resolution. Inside the optimization algorithm, stiffness matrices are parametrized by means of lamination parameters, resulting in a reduction in the amount of design variables on the one hand, and the constitution of a continuous, well-posed optimization problem on the other. The response sensitivities with respect to the design variables form an essential input in the assembly of response approximations. In this research, the finite element (FE) software Nastran is applied in order to generate sensitivities. Three major reasons account for this choice: one, the ability of specifying various types of responses, two, the time efficient implementation, and three, its prevalence in the aircraft industry. A Nastran FE model, suitable for the derivation of the required responses, is generated in a parametric model generation process. Aside from the structural FE representation, the model comprises a doublet lattice description for the computation of aeroelastic loads, and a mass model to incorporate non-structural masses like leading and trailing edge or fuel. Structural responses considered in the stiffness optimization are strength, buckling and mass. For strength, a failure criterion in lamination parameter space is adopted. Buckling is covered by a simply supported flat plate buckling model. Aside from the regular structural responses, the aeroelastic responses aileron effectiveness, divergence, and twist are also directly considered in the optimization process. While response values and sensitivities are an immediate result of Nastran, their approximations with respect to the design variables originate from a sensitivity convexification process, ensuring the approximation to incorporate as much reciprocal share as possible. The stiffness optimization fully relying upon the applied aeroelastic loads, a correction strategy by means of a higher order computational fluid dynamics (CFD) method is developed to enhance the doublet lattice aeroelastic loads. Eventually, the functionality of the stiffness optimization framework is verified by three applications, comprising different levels of complexity. Mostly wing skin weight serves as objective to be minimized, but also the maximization of aileron effectiveness for a prescribed weight is demonstrated. The possible advantages of unbalanced over balanced laminates are studied, as well as the influence of different sets of aeroelastic constraints on the achievable minimum wing skin weight. Finally, the modifications implied by an aero load correction are analyzed.