Large-Scale Topology Optimisation of Time-dependent Thermal Conduction Using Space-Time Finite Elements and a Parallel Space-Time Multigrid Preconditioner

Abstract

This paper presents a novel space-time topology optimisation framework for time-dependent thermal conduction problems, aiming to significantly reduce the time-to-solution. By treating time as an additional spatial dimension, we discretise the governing equations using a stabilised continuous Galerkin space-time finite element method. The resulting large all-at-once system is solved using an iterative Krylov solver preconditioned with a parallel space-time multigrid method employing a semi-coarsening strategy. Implemented in a fully parallel computing framework, the method yields a parallel-in-time method that demonstrates excellent scalability on a distributed-memory supercomputer, solving problems up to 4.2 billion degrees of freedom. Comparative studies show up to 52x speed-up over traditional time-stepping approaches, with only moderate increases in total computational cost in terms of core-hours. The framework is validated on benchmark problems with both time-constant and time-varying designs, and its flexibility is demonstrated through variations in material properties. These results establish the proposed space-time method as a promising approach for large-scale time-dependent topology optimisation in thermal applications.