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

TL;DR

This paper introduces preconditioners for efficiently solving linear systems in 3D fluid topology optimization, enabling practical computation of multiple solutions using the deflated barrier method.

Contribution

It develops a nested block preconditioning approach with multigrid techniques tailored for the linear systems in 3D Stokes flow topology optimization.

Findings

Successfully computed multiple 3D solutions using the proposed iterative solver.

Reduced complex linear systems to simpler symmetric positive-definite matrices.

Demonstrated the effectiveness of the preconditioners in practical 3D examples.

Abstract

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, 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 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.