Preconditioners for computing multiple solutions in three-dimensional fluid topology optimization

Topology optimization problems generally support multiple local minima, and real-world applications are typically three-dimensional. In previous work [I. P. A. Papadopoulos, P. E. Farrell, and T. M. Surowiec, SIAM J. Sci. Comput., 43 (2021), pp. A1555-A1582], the authors developed the deflated barrier method, an algorithm that can systematically compute multiple solutions of topology optimization problems. In this work, we develop preconditioners for the linear systems arising in the application of this method to Stokes flow, making it practical for use in three dimensions. In particular, we develop a nested block preconditioning approach which reduces the linear systems to solving two symmetric positive-definite matrices and an augmented momentum block. An augmented Lagrangian term is used to control the innermost Schur complement, and we apply a geometric multigrid method with a kernel-capturing relaxation method for the augmented momentum block. We present multiple solutions in three-dimensional examples computed using the proposed iterative solver.