Topology optimization of Stokes eigenvalues by a level set method

Abstract

We propose a level set method for a Stokes eigenvalue optimization problem. A relaxed approach is employed first to approximate the Stokes eigenvalue problem and transform the original shape optimization problem into a topology optimization model. Then the distributed shape gradient is used in numerical algorithms based on a level set method. Single-grid and efficient two-grid level set algorithms are developed for the relaxed optimization problem. A two-grid mixed finite element scheme that has reliable accuracy and asymptotically optimal convergence is shown to improve the efficiency of the Stokes eigenvalue solver. Thus, it can save computational efforts of the whole optimization algorithm. Two and three-dimensional numerical results are reported to show effectiveness and efficiency of the algorithms.