Simultaneous optimization of shape and topology for solid structure (Proposal of a topology optimization method with variable design domain)

Abstract

Since the optimization results obtained by topology optimization depend on the size of the design domain, it is desirable to optimize the design domain as well during topology optimization. In this study, we propose a method to optimize the structural topology for solid structures by topology optimization based on the SIMP (Solid Isotropic Material with Penalization) method, while optimizing the design domain by shape optimization. In other words, we develop a simultaneous optimization method of shape and topology for structures. Under the constraints of volume and equilibrium equations, the compliance as the objective function is minimized. After formulating this design optimization problem, a sensitivity function is derived using the Lagrange multiplier method and adjoint method, which is applied to the H¹ gradient method to determine the optimal shape and topology. The introduction of the H¹ gradient method allows to optimize the shape and topology while simultaneously smoothing the outer shape and obtaining the topology without gray scale and checker board issues and also decreasing the objective function. The effectiveness of the method for topology optimization with the optimized design domain by shape optimization, or the simultaneous design optimization of solid structures is confirmed using numerical examples.