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

Abstract

Topology optimization problems generally support multiple local minima, and realworld applications are typically three-dimensional. In previous work [I. P. A. Papadopoulos, P. E. Farrell, and T. M. Surowiec, Computing multiple solutions of topology optimization problems, SIAM Journal on Scientific Computing, (2021)], 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 positivedefinite 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 kernelcapturing relaxation method for the augmented momentum block. We present multiple solutions in three-dimensional examples computed using the proposed iterative solver.