Space-time spectral element summation-by-parts method for heterogeneous transient heat conduction arising in topology optimization

Abstract

We develop a stable space-time spectral element method for transient heat conduction in heterogeneous multi-material domains, motivated by topology optimization. The method treats space and time simultaneously, using summation-by-parts (SBP) operators in both directions and simultaneous approximation terms (SATs) to impose boundary, initial, terminal, and material-interface conditions weakly, yielding a stable monolithic space-time scheme on heterogeneous domains. Stability is proven under specific conditions on the SAT parameters, scaled with the spatial mesh resolution and material properties. We compute design sensitivities using a discrete space-time adjoint scheme that is dual-consistent with the primal SBP-SAT scheme. Numerical experiments demonstrate spectral convergence of the forward and adjoint solution errors and of the error in the functional output for smooth manufactured solutions in heterogeneous domains. Also under mesh refinement, the observed errors indicate higher-order convergence of the optimal objective value compared with the optimized design. We validate the resulting optimal design by comparison with an independently computed reference optimal design and report time-to-solution and cost-of-accuracy curves, comparing against low-order time-marching and all-at-once solvers for the forward and adjoint systems. The proposed scheme attains high accuracy with fewer space-time degrees of freedom and remains stable, reducing time-to-solution and memory compared with an alternative all-at-once solver. This makes it a future candidate for large-scale topology optimization of time-dependent thermal systems.