Higher-order accurate integration of implicit geometries

A unified strategy for the higher-order accurate integration of implicitly defined geometries is proposed. The geometry is represented by a higher-order level-set function. The task is to integrate either on the zero-level set or in the sub-domains defined by the sign of the level-set function. In three dimensions, this is either an integration on a surface or inside a volume. A starting point is the identification and meshing of the zero-level set by means of higher-order interface elements. For the volume integration, special sub-elements are proposed where the element faces coincide with the identified interface elements on the zero-level set. Standard Gauss points are mapped onto the interface elements or into the volumetric sub-elements. The resulting integration points may, for example, be used in fictitious domain methods and extended finite element methods. For the case of hexahedral meshes, parts of the approach may also be seen as a higher-order marching cubes algorithm.