On the use of multigrid preconditioners for topology optimization

Topology optimization for large-scale problems continues to be a computational challenge. Several works exist in the literature to address this topic, and all make use of iterative solvers to handle the linear system arising from the finite element analysis (FEA). However, the preconditioners used in these works vary, and in many cases are notably suboptimal. A handful of works have already demonstrated the effectiveness of geometric multigrid (GMG) preconditioners in topology optimization. We provide a direct comparison of GMG preconditioners with algebraic multigrid (AMG) preconditioners. We demonstrate that AMG preconditioners offer improved robustness over GMG preconditioners as topologies evolve, albeit with a higher overhead cost. In 2D the gain from increased robustness more than offsets the overhead cost. However, in 3D the overhead becomes prohibitively large. We thus demonstrate the benefits of mixing geometric and algebraic methods to limit overhead cost while improving robustness, particularly in 3D.