A shape optimization problem is defined as an optimization problem to boundary shape of domain in which boundary value problem of partial differential equation is defined. A design variable is given by a domain mapping. Cost functions are defined as functionals of the design variable and the solution to the boundary value problem. The present paper described that the Frechet derivatives of cost functions with respect to domain variation do not have the regularity required in order to define a next domain, and that a gradient method can be considered for regularizing the derivatives.
This paper derives topological derivative, which is a derivative with respect to creation of a new infinity small hole for steady elastrodynamic problem. Based on the topological derivative and a regularization method, level set-based topology optimization method is formulated. In addition, topology optimization algorithm is constructed using Finite Element Method. Finally, three-dimensional numerical examples are shown to confirm the validity and utility of the proposed topology optimization method.