One-shot Parareal Approach for Topology Optimisation of Transient Heat Flow

TL;DR

This work tackles topology optimisation of transient heat conduction problems using parallel-in-time computation. It introduces a one-shot Parareal approach where the time history is iteratively refined through cumulative objectives and sensitivities solved with Parareal, coupled with MMA for design updates. The method achieves a peak speedup of about $5\times$ on a 2D test problem with 16 threads and yields final designs with objectives within $\pm 2\%$ of a sequential reference, with qualitative similarity in appearance. A comparison with Parallel Local-in-Time (PLT) shows PLT can reach higher speeds ($\approx 11.8\times$) but suffers from instability at larger thread counts, while the dominant bottleneck in the one-shot Parareal method lies in the coarse propagator evaluation. The results identify coarse-propagator efficiency as a key focus for improving time-parallel topology optimisation.

Abstract

This paper presents a method of performing topology optimisation of transient heat conduction problems using the parallel-in-time method Parareal. To accommodate the adjoint analysis, the Parareal method was modified to store intermediate time steps. Preliminary tests revealed that Parareal requires many iterations to achieve accurate results and, thus, achieves no appreciable speedup. To mitigate this, a one-shot approach was used, where the time history is iteratively refined over the optimisation process. The method estimates objectives and sensitivities by introducing cumulative objectives and sensitivities and solving for these using a single iteration of Parareal, after which it updates the design using the Method of Moving Asymptotes. The resulting method was applied to a test problem where a power mean of the temperature was minimised. It achieved a peak speedup relative to a sequential reference method of $5\times$ using 16 threads. The resulting designs were similar to the one found by the reference method, both in terms of objective values and qualitative appearance. The one-shot Parareal method was compared to the Parallel Local-in-Time method of topology optimisation. This revealed that the Parallel Local-in-Time method was unstable for the considered test problem, but it achieved a peak speedup of $12\times$ using 32 threads. It was determined that the dominant bottleneck in the one-shot Parareal method was the time spent on computing coarse propagators.

Figures (14)

• Figure 1: Sketch of an example of how the time axis is coarsened when using the Parareal method for $N_t = 15$, $N_\tau = 3$, and $M = 5$.
• Figure 1: Geometry and boundary conditions of the domain of the test case defined in \ref{['sec: test case def']}.
• Figure 1: Measured speedup of the one-shot Parareal method of topology optimisation. The peak is at 16 threads with a speedup of $4.95 \times$.
• Figure 1: A selection of nine different topologies that were found using the sequential reference method and the PLT method with different numbers of threads. The PLT method is clearly unstable when using large numbers of threads.
• Figure 2: Diagram of the flow of information between different time points and Parareal iterations during execution of the Parareal algorithm, which is governed by \ref{['eq: parareal corr form orig']}. Arrows labelled as $\mathcal{+G}$ indicate the addition of terms of the form $\mathcal{G}(\tau, \mathbf{u})$ and arrows labelled as $\mathcal{+F-G}$ indicate the addition of terms of the form $\mathcal{F}(\tau, \mathbf{u}) - \mathcal{G}(\tau, \mathbf{u})$.