Space-Time Multigrid Methods Suitable for Topology Optimisation of Transient Heat Conduction

TL;DR

This paper introduces Space-Time Multigrid methods tailored for efficient topology optimization in transient heat conduction, demonstrating robustness and effectiveness in solving both primal and adjoint problems.

Contribution

The paper develops novel space-time multigrid algorithms with specific coarsening strategies and smoothing techniques for transient heat conduction topology optimization.

Findings

The proposed methods are robust and converge in all optimization cycles.

Coarsening based on geometric mean of diffusivity improves performance.

The methods also effectively solve adjoint problems despite time direction differences.

Abstract

This paper presents Space-Time MultiGrid (STMG) methods which are suitable for performing topology optimisation of transient heat conduction problems. The proposed methods use a pointwise smoother and uniform Cartesian space-time meshes. For problems with high contrast in the diffusivity, it was found that it is beneficial to define a coarsening strategy based on the geometric mean of the minimum and maximum diffusivity. However, other coarsening strategies may be better for other smoothers. Several methods of discretising the coarse levels were tested. Of these, it was best to use a method which averages the thermal resistivities on the finer levels. However, this was likely a consequence of the fact that only one spatial dimension was considered for the test problems. A second coarsening strategy was proposed which ensures spatial resolution on the coarse grids. Mixed results were found for this strategy. The proposed STMG methods were used as a solver for a one-dimensional topology optimisation problem. In this context, the adjoint problem was also solved using the STMG methods. The STMG methods were sufficiently robust for this application, since they converged during every optimisation cycle. It was found that the STMG methods also work for the adjoint problem when the prolongation operator only sends information forwards in time, even although the direction of time for the adjoint problem is backwards.