A phase field method based on multi-level correction for eigenvalue topology optimization

Abstract

In topology optimization, finite element methods are usually required to solve state equations repeatedly. Considerable computational costs are required for iterative solvers to nonlinear state constraints, such as eigenvalue problems, especially in 3D with large-scale degrees of freedom. Recently, multi-level correction methods are developed to improve efficiency in solving nonlinear partial differential equations including eigenvalue problems. These methods involve a multi-step correction in a sequence of finite element spaces and can keep accuracy of the eigenpair approximation. The overall computational cost is nearly equivalent to that of solving linear source problems. In this paper, we propose a new phase field method based on multi-level correction for eigenfrequency topology optimization. Both eigenfrequency optimization problems associated with a scalar type elliptic operator and a vectorial type elastic operator are considered. The Allen–Cahn phase field equation discretized by a semi-implicit scheme and finite-elements can have topological and shape changes in arbitrary shaped design regions. The proposed optimization algorithm based on a multi-level correction solver for eigenvalue problems improves efficiency of the traditional phase field method. A variety of numerical examples in 2D and 3D are presented to illustrate the effectiveness and efficiency of the algorithm.