Level-set topology optimisation with unfitted finite elements and automatic shape differentiation

Abstract

In this paper we develop automatic shape differentiation techniques for unfitted discretisations and link these to recent advances in shape calculus for unfitted methods. We extend existing analytic shape calculus results to the case where the domain boundary intersects with the boundary of the background domain. We further show that we can recover these analytic derivatives to machine precision regardless of the mesh size using the developed automatic shape differentiation techniques, drastically reducing the burden associated with the analytic derivation of these quantities. In addition, we show that we can also recover the symmetric shape Hessian. We implement these techniques for both serial and distributed computing frameworks in the Julia package GridapTopOpt and the wider Gridap ecosystem. As part of this implementation we propose a novel graph-based approach for isolated volume detection. We demonstrate the applicability of the unfitted automatic shape differentiation framework and our implementation by considering the three-dimensional minimum compliance topology optimisation of a linear elastic wheel and of a linear elastic structure in a fluid-structure interaction problem with Stokes flow. The implementation is general and allows GridapTopOpt to solve a wider range of problems on unstructured meshes without analytic calculation of shape derivatives and avoiding issues that arise when material properties are smoothed at the domain boundary. The software is open source and available at https://github.com/zjwegert/GridapTopOpt.jl.