Geometrical discretisations for unfitted finite elements on explicit boundary representations

TL;DR

This paper introduces an automatic framework for discretising PDEs on explicit boundary representations using unfitted finite elements, overcoming limitations of traditional implicit geometrical methods and enabling application to complex industrial geometries.

Contribution

It presents a novel computational framework that connects boundary meshes with functional representations, facilitating numerical integration on explicit geometries in unfitted finite element methods.

Findings

Successfully applied to nearly 5,000 boundary meshes from Thingi10K.

Enables straightforward numerical integration on explicit boundary representations.

Supports discretisation of PDEs on complex geometries without body-fitted meshes.

Abstract

Unfitted (also known as embedded or immersed) finite element approximations of partial differential equations are very attractive because they have much lower geometrical requirements than standard body-fitted formulations. These schemes do not require body-fitted unstructured mesh generation. In turn, the numerical integration becomes more involved, because one has to compute integrals on portions of cells (only the interior part). In practice, these methods are restricted to level-set (implicit) geometrical representations, which drastically limit their application. Complex geometries in industrial and scientific problems are usually determined by (explicit) boundary representations. In this work, we propose an automatic computational framework for the discretisation of partial differential equations on domains defined by oriented boundary meshes. The geometrical kernel that connects functional and geometry representations generates a two-level integration mesh and a refinement of the boundary mesh that enables the straightforward numerical integration of all the terms in unfitted finite elements. The proposed framework has been applied with success on all analysis-suitable oriented boundary meshes (almost 5,000) in the Thingi10K database and combined with an unfitted finite element formulation to discretise partial differential equations on the corresponding domains.