Shape Optimization Using the Cut Finite Element Method

TL;DR

This paper introduces a cut finite element method for shape optimization in linear elasticity that avoids re-meshing, supports higher order elements, and uses a volume-based shape derivative approach.

Contribution

The paper develops a novel cut finite element approach for shape optimization that incorporates stabilization and higher order elements without re-meshing.

Findings

Effective shape optimization demonstrated in 2D cases

Supports higher order finite element approximations

No re-meshing needed during domain updates

Abstract

We present a cut finite element method for shape optimization in the case of linear elasticity. The elastic domain is defined by a level-set function, and the evolution of the domain is obtained by moving the level-set along a velocity field using a transport equation. The velocity field is the largest decreasing direction of the shape derivative that satisfies a certain regularity requirement and the computation of the shape derivative is based on a volume formulation. Using the cut finite element method no re--meshing is required when updating the domain and we may also use higher order finite element approximations. To obtain a stable method, stabilization terms are added in the vicinity of the cut elements at the boundary, which provides control of the variation of the solution in the vicinity of the boundary. We implement and illustrate the performance of the method in the two--dimensional case, considering both triangular and quadrilateral meshes as well as finite element spaces of different order.