Geometric Integration Over Irregular Domains with topologic Guarantees

TL;DR

This paper introduces a new algorithm for accurately integrating over irregular, implicitly defined domains using a polyhedral reconstruction that preserves topology, enabling reliable finite element computations on level-set geometries.

Contribution

The paper presents a novel topologically guaranteed geometric integration algorithm based on Marching Cubes for domains described by level-set functions, with an open-source implementation.

Findings

Achieves second-order accuracy in numerical experiments.

Preserves topological properties of the implicit geometry.

Available as open-source software integrated into DUNE.

Abstract

Implicitly described domains are a well established tool in the simulation of time dependent problems, e.g. using level-set methods. In order to solve partial differential equations on such domains, a range of numerical methods was developed, e.g. the Immersed Boundary method, Unfitted Finite Element or Unfitted discontinuous Galerkin methods, eXtended or Generalised Finite Element methods, just to name a few. Many of these methods involve integration over cut-cells or their boundaries, as they are described by sub-domains of the original level-set mesh. We present a new algorithm to geometrically evaluate the integrals over domains described by a first-order, conforming level-set function. The integration is based on a polyhedral reconstruction of the implicit geometry, following the concepts of the Marching Cubes algorithm. The algorithm preserves various topological properties of the implicit geometry in its polyhedral reconstruction, making it suitable for Finite Element computations. Numerical experiments show second order accuracy of the integration. An implementation of the algorithm is available as free software, which allows for an easy incorporation into other projects. The software is in productive use within the DUNE framework.