Computational analysis demonstrates a stable phase-field algorithm for two-material topology optimization on curved surfaces, suggesting robust design pathways for complex geometries.
Topology optimization on surfaces governed by a surface diffusion equation is studied from both mathematical analysis and numerical viewpoints. We aim to investigate the optimal distribution of two materials on a surface to minimize certain objective. Within the framework of the surface phase-field model, we prove the existence of a minimizer for the PDE-constrained optimization problem. We show the differentiability of the solution operator w.r.t. the phase-field function and conduct sensitivity analysis. A stabilized semi-implicit scheme for the gradient flow of variational inequality type is proposed, and the corresponding energy stability is shown. Moreover, the proposed scheme is proved to converge to the first-order necessary optimality condition of the considered optimization problem. Finally, numerical examples on different types of surfaces effectively validate the proposed optimization algorithm.
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Li et al. (2026) studied this question.
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