ACM Transactions on Mathematical Software (TOMS), 10 October 2017, Vol.44(2), pp.1-20
Implicitly described domains are a well-established tool in the simulation of time-dependent problems, for example, using level-set methods. To solve partial differential equations on such domains, a range of numerical methods was developed, for example, the Immersed Boundary method, the Unfitted Finite Element or Unfitted Discontinuous Galerkin methods, and the 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 (Bastian et al. 2008a).
Level-Sets ; Cut-Cell Methods ; Geometry Reconstruction ; Implicit Domains ; Marching Cubes ; Numerical Quadrature ; Surface Reconstruction ; Sciences (General) ; Computer Science