Octree-based integration scheme with merged sub-cells for the finite cell method: Application to non-linear problems in 3D

Abstract

Fictitious domain methods, such as the Finite Cell Method (FCM), allow for an efficient and accurate simulation of complex geometries by utilizing higher-order shape functions and an unfitted discretization based on rectangular elements. Since the mesh does not conform to the geometry, cut elements arise that are intersected by domain boundaries. For optimal convergence rates and the efficiency of the simulation in general, special integration schemes have to be used in such elements. In this contribution, the often used, robust octree-decomposition-based integration scheme is enhanced by a novel approach reducing the computational effort when evaluating the discontinuous integrals. This is realized by introducing an additional step, in which the local integration mesh is simplified using data compression techniques leading to fewer integration domains/points. An important advantage of the proposed method is that it can be added in a modular fashion to already existing codes. While it inherits all desired properties of the octree-decomposition-based integration scheme, it significantly reduces the number of integration points and has hardly any negative effect on the simulation accuracy. In this paper, the proposed integration scheme is introduced in detail, and investigated by means of numerical examples in the context of 3D non-linear problems.