Joe Alexandersen

This study presents a gradient-based topology optimisation framework for heat sinks embedded with phase-change material (PCM) that targets the mitigation of temperature oscillations under cyclic thermal loads. The approach couples transient thermal diffusion modelling in FEniCS with automatic adjoint sensitivities and GCMMA, and uses a simple analytical homogenisation to parametrise a composite of PCM and conductive material. With latent-heat buffering using PCM, the optimised layouts reduce the temperature variance by 41% when the full time history is used and by 32% when only the quasi-steady-state cycle is used. To improve physical manufacturability, explicit penalisation yields near-discrete designs with only ∼10% performance loss, preserving most oscillation reduction benefits. The results demonstrate that adjoint-driven PCM topology optimisation can systematically suppress thermal oscillations.

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.

Turbulent flows have high requirements for very fine meshes near the boundary to ensure accuracy. In the context of topology optimization (TO), such fine meshes become unrealistic and common approaches are hampered by low accuracy and overestimation of boundary layer thickness. Wall-functions are a natural way to ease the computational requirements, but they are not naturally imposed in density-based TO due to the diffuse design parametrization. We propose an implicit wall-function formulation for the Reynolds-Averaged Navier-Stokes (RANS), standard k-epsilon model that extracts wall-normal information directly from the gradient of the design variable and enables a penalty-based formulation for imposing wall-functions to the RANS equations, without the need for body-fitted meshes. The method provides a reliable route to high Reynolds number turbulent topology optimization, delivering boundary layer accuracy comparable to explicit-wall body-fitted analyses, while retaining the flexibility of density-based TO. Furthermore, because wall effects are modeled using wall-functions, accurate solutions are obtained on substantially coarser meshes, leading to significant reductions in computational cost. The approach is validated on three canonical benchmarks over Reynolds numbers up to Re = 2e5: a pipe-bend; a U-bend; and a Tesla-valve. Across all cases, the proposed method accurately recovers near-wall velocity profiles, closely matching verification simulations on body-fitted meshes with explicit wall-functions. In contrast, a conventional turbulent TO formulation, without the proposed wall-function treatment, mispredicts boundary-layer development and yields sub-optimal results.

This paper presents a method of performing topology optimization 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 optimization 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 minimized. It achieved a peak speedup relative to a sequential reference method of [Formula: see text] 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 optimization. This revealed that the Parallel Local-in-Time method was unstable for the considered test problem, but it was able to execute optimization iterations faster than the one-shot Parareal method. It was determined that the dominant bottleneck in the one-shot Parareal method was the time spent on computing coarse propagators.