数理計画法を用いたレベルセット法に基づくトポロジー最適化

Abstract

Several level set-based topology optimization methods have been recently proposed in which the boundaries of the optimal configuration are implicitly represented using the level set function. These methods naturally overcome numerical instability problems such as grayscale areas that typically appear in conventional topology optimization results. However, since most of them update design variables by solving certain partial differential equations, they lack the flexibility to deal with various constraints. This paper proposes a level set-based topology optimization method using mathematical programming, which facilitates the handling of optimization problems that have several constraint functionals. The level set function is updated using the Method of Moving Asymptotes (MMA), based on the sensitivities of the objective and constraint functionals. First, topology optimization using level set boundary expressions based on the concept of the phase field method is explained, and a new optimization algorithm for updating the level set function using mathematical programming is then developed. To confirm the validity and utility of our method, we apply it to multi-constraint optimization problems such as a minimum mean compliance problem that includes a stress constraint, and the design of compliant mechanisms including a displacement constraint.