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

TL;DR

This work presents Space-Time Multigrid (STMG) methods for speeding up density-based topology optimisation of transient heat conduction on uniform space-time meshes. It introduces an effective anisotropy parameter, λ_eff, to guide semi-coarsening decisions in heterogeneous, high-contrast media, and investigates multiple rediscretisation schemes, identifying the resistivity-averaging approach as advantageous for problems with small features. Through several 1D test problems and a topology-optimisation-like experiment, STMG demonstrates robustness for both primal and adjoint solves, with causal interpolation generally preferred for smooth problems and resistivity-based coarse reassembly recommended when small spatial features are present. The results offer practical guidance for applying STMG to time-dependent, high-contrast heat-conduction topology optimisation and point to future extensions to higher-dimensional problems and more complex physics.

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.

Figures (12)

• Figure 1: Illustrations of the three different types of coarsening which are defined in Section \ref{['sec: stmg components def']}.
• Figure 2: Imposed heat load, $q(x,t)$, used for the test problems considered in this paper, as defined in Equation (\ref{['eq: heat load def']}).
• Figure 3: Estimated convergence factors of the space-time multigrid method defined in Section \ref{['sec: stmg components def']} when using two grids and the material parameters are uniform over space. These are plotted for each coarsening type as a function of $\lambda$, which is defined in Equation (\ref{['eq: lambda def']}). Specifically, it is the value of $\lambda$ on the finest grid.
• Figure 4: Design field, $\chi$, defined for problem 3 in Section \ref{['sec: eff lambda']}.
• Figure 5: Estimated convergence factors for the two-grid space-time multigrid method applied to the six test problems considered in Section \ref{['sec: eff lambda']}. The red crossed curves show the convergence factors when using $x$-coarsening, and the blue circled curves show the convergence factors when using $t$-coarsening. These are plotted as functions of $\log_2 (\lambda_\mathrm{eff} )$ on the finest level, where $\lambda_\mathrm{eff}$ is the quantity defined in Equation (\ref{['eq: lambda eff def 1']}).