Krylov Subspace Methods for Topology Optimization on Adaptive Meshes

Topology optimization is a powerful tool for global and multiscale design of structures, microstructures, and materials. The computational bottleneck of topology optimization is the solution of a large number of extremely ill-conditioned linear systems arising in the finite element analysis. Adaptive mesh refinement (AMR) is one efficient way to reduce the computational cost. We propose a new AMR scheme for topology optimization that results in more robust and efficient solutions. For large sparse symmetric linear systems arising in topology optimization, Krylov subspace methods are required. The convergence rate of a Krylov subspace method for a symmetric linear system depends on the spectrum of the system matrix. We address the ill-conditioning in the linear systems in three ways, namely rescaling, recycling, and preconditioning. First, we show that a proper rescaling of the linear systems reduces the huge condition numbers that typically occur in topology optimization to roughly those arising for a problem with homogeneous density. Second, the changes in the linear system from one optimization step to the next are relatively small. Therefore, recycling a subspace of the Krylov subspace and using it to solve the next system can improve the convergence rate significantly. We propose a minimum residual method with recycling (RMINRES) that preserves the short-term recurrence and reduces the cost of recycle space selection by exploiting the symmetry. Numerical results show that this method significantly reduces the total number of iterations over all linear systems and the overall computational cost (compared with the MINRES method which is optimal for a single symmetric system). We also investigate the recycling method for adaptive meshes. Third, we propose a multilevel sparse approximate inverse (MSPAI) preconditioner for adaptive mesh refinement. It significantly improves the conditioning of the linear systems by approximating the global modes with multilevel techniques, while remaining cheap to update and apply, especially when the mesh changes. For convection-diffusion problems, it achieves a level-independent convergence rate. We then make a few changes in the MSPAI preconditioner for topology optimization problems. With these extensions, the MSPAI preconditioner achieves a nearly level-independent convergence rate. Although for small to moderate size problems the incomplete Cholesky preconditioner is faster in time, the multilevel sparse approximate inverse preconditioner will be faster for (sufficiently) large problems. This is important as we are more interested in scalable methods