Space-time spectral element method for topology optimization of transient heat conduction

TL;DR

The paper addresses topology optimization for transient heat conduction described by $u_t=(κ(ρ)u_x)_x+f$ with objective $J=\int_0^T\int_a^b u^2 dx dt$ and a material-volume constraint. It proposes a space-time spectral element SBP–SAT discretization that enforces initial/boundary/interface conditions with SAT penalties, yielding a provably energy-stable monolithic scheme across heterogeneous subdomains. A discrete dual-consistent adjoint framework is developed to compute design sensitivities accurately, ensuring superconvergent functional estimates under standard smoothness. Numerical experiments show improved time-to-solution and accuracy-per-cost relative to low-order all-at-once solvers, highlighting the potential of the approach for large-scale time-dependent thermal topology optimization, with future work on scalable preconditioning and parallel-in-time strategies.

Abstract

We develop a space-time spectral element method for topology optimization of transient heat conduction. The forward problem is discretized with summation-by-parts (SBP) operators, and interface/boundary and initial/terminal conditions are imposed weakly via simultaneous approximation terms (SAT), 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. Dual consistency ensures that the discrete adjoint consistently approximates the continuous dual problem and, under standard smoothness assumptions, yields superconvergent functional estimates. We validate the resulting optimal designs 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.

Figures (9)

• Figure 1: Illustration of the partitioning of the space-time domain, $\Omega$, into non-overlapping elements, $\Omega_k$, with interfaces, $\Gamma_k$.
• Figure 1: Relative error of the MMAHeat optimal design (using MMSHeat as the reference). Left: spatial refinement by increasing $N_x$ with $N_t=30$; right: temporal refinement by increasing $N_t$ with $N_x=40$.
• Figure 2: Illustration of three subdomains discretized using LGL collocation nodes and coupled using boundary, initial, and interface SATs.
• Figure 2: Relative error of the optimal objective value (using MMSHeat as the reference). Left: versus spatial resolution $N_x$ per element with $N_t=30$; right: versus temporal resolution $N_t$ with $N_x=40$.
• Figure 3: Convergence of the space-time spectral element SBP-SAT scheme for a simple test example of the heat equation.

Theorems & Definitions (5)

• Theorem 1
• Proof 1
• Lemma 2
• Proof 2
• Proof 3: Proof of Theorem \ref{['thm:stab']}