Level-set topology optimisation with unfitted finite elements and automatic shape differentiation
Journal:
arXiv
Published Date:
Apr 13, 2025
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. 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 wide range of problems 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.
