  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.
