On the use of multigrid preconditioners for topology optimization

Abstract

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.

Appendix. Large-scale 3D cantilever beam

Here we describe the application of our proposed hybrid GMG-AMG preconditioner to the 3D cantilever problem at a larger scale (512 × 256 × 256 element mesh, approximately 100e6 dofs). This time we only perform 16 optimization iterations for each penalty increment to improve runtime, but the penalty is still increased in increments of 0.25 to a final value of 4. The filter radius is again set to 1.5 times the element dimension, meaning that the physical dimension of the filter radius is reduced by a factor of 2.67. The final result is shown in Fig. 15.

Fig. 15

Optimized structure for 3D cantilever beam problem at an increased resolution

In Fig. 16 we show the performance of using the hybrid preconditioner for this problem. We restrict the coarse grid size to be less than 1000 dofs, corresponding to 7 levels in the geometric hierarchy, and we apply weighted point-block Jacobi as the smoother. It is important to note that in this case, the number of iterations to convergence remains substantially lower than in the smaller problem, never reaching even 100 iterations in a single optimization step. As a result, the hybrid scheme never converts levels of the multigrid preconditioner from geometric to algebraic coarsening. This means that 7 geometric levels are used through the entire simulation, with one small algebraic level at the end for ease of implementation.

Fig. 16

Performance of the hybrid preconditioner on the large 3D cantilever problem. a Time to set up the preconditioner. b Time to solve the linear system for displacements. c Combined time to set up and solve. d Iterations of the linear solver until convergence

It could be argued that this is an indication that AMG is not needed in larger scale problems; however, an important observation arises upon closer inspection of the optimized structures. Figure 17 shows the structure that develops along one edge of the domain for both the smaller and larger 3D problem and Fig. 18 shows the difference in the structures along the rear support. Note that in the small case, many more small structural elements develop in close proximity to each other in both of these regions. It is exactly these types of elements that cause difficulty for the GMG preconditioner, but they do not develop in the larger problem. Instead, these numerous structural elements are replaced with a single smooth feature that can only be captured on the higher-fidelity design space. As it is impossible to predict a priori what type of structure will develop from a given loading condition and mesh resolution this motivates the use of our hybrid preconditioner, which only applies algebraic coarsening as necessary. In the smaller problem the algebraic coarsening improves performance when geometric coarsening struggles, but in the larger problem the cheaper geometric strategy is used. In either case there is no need for the user to specify a fixed number of algebraic or geometric levels in the multigrid hierarchy.

Fig. 17

Comparison of structure along lateral edge of domain at coarse (left) and fine (right) resolution

Fig. 18

Comparison of structure along rear edge of domain at coarse (left) and fine (right) resolution