Multi-material topology optimization for lightweight design

Abstract

Multi-material topology optimization remains an active research topic. This work enriches the key techniques of multi-material topology optimization, such as the multi-material property determination, topology optimization problem statements, stiffness constraints, sensitivity analysis, and optimization algorithms. Firstly, the multi-material updating (also known as material interpolation) scheme is established, which is the fundamental basis of multi-material topology optimization. The interpolated properties contain Young’s modulus, constitutive matrix, weight, and cost. To avoid the limitations of handling the various materials with the conventional material updating scheme, we enhance the element interpolation to update the material properties. The element interpolation method transforms a multi-material topology optimization problem into a single-material problem, which has no limitation on handling the various materials and is easy to solve according to the current research literature. Secondly, the multi-material topology optimization problem statements for practical lightweight design are defined. Compliance minimization is the typical problem statement in academic research. However, in practical applications, compliance minimization is improper because compliance lacks clear physical meaning. Therefore, weight/cost minimization with stiffness constraints, which has clear physical meaning for product design, is proposed as the primary problem statement. Thirdly, sensitivity analysis based on the typical responses is provided. The typical responses in this thesis include compliance, weight, volume (or volume fraction), displacement, stress, and cost. The responses are generalized into one universal expression, and then their sensitivities are derived. The equations of the responses and sensitivities are summarized. Fourthly, the multi-material weight minimization considering stress constraints is discussed. The multi-material weight minimization considering stress constraints is addressed, with the transformation from the multi-material problem to the single-material problem using element interpolation. Finally, the Method of Moving Asymptotes is selected as the optimizer because of its stability and efficiency. Nevertheless, the traditional Method of Moving Asymptotes may fail when handling a high-resolution finite element model (i.e., the mesh size is small). The failure is caused by the null search direction due to the small sensitivities. Thus, I introduce the primal-dual interior-point method with a scaling mechanism into the Method of Moving Asymptotes to eliminate the small-sensitivity issue.