Space-time unfitted finite elements on moving explicit geometry representations

TL;DR

This paper introduces a variational unfitted finite element method for PDEs on moving geometries, enabling large boundary displacements without remeshing, using a background mesh and geometric intersection algorithms.

Contribution

The novel scheme handles large boundary movements on explicit geometries without remeshing, utilizing a reference configuration and intersection algorithms for 3D geometries.

Findings

Method achieves optimal convergence rates.

Applicable to fluid problems around rotating geometries.

Efficient handling of large boundary displacements.

Abstract

This work proposes a novel variational approximation of partial differential equations on moving geometries determined by explicit boundary representations. The benefits of the proposed formulation are the ability to handle large displacements of explicitly represented domain boundaries without generating body-fitted meshes and remeshing techniques. For the space discretization, we use a background mesh and an unfitted method that relies on integration on cut cells only. We perform this intersection by using clipping algorithms. To deal with the mesh movement, we pullback the equations to a reference configuration (the spatial mesh at the initial time slab times the time interval) that is constant in time. This way, the geometrical intersection algorithm is only required in 3D, another key property of the proposed scheme. At the end of the time slab, we compute the deformed mesh, intersect the deformed boundary with the background mesh, and consider an exact transfer operator between meshes to compute jump terms in the time discontinuous Galerkin integration. The transfer is also computed using geometrical intersection algorithms. We demonstrate the applicability of the method to fluid problems around rotating (2D and 3D) geometries described by oriented boundary meshes. We also provide a set of numerical experiments that show the optimal convergence of the method.