The finite cell method (FCM) is a combination of a fictitious domain approach with finite elements of high order. The physical domain is embedded into an easy to mesh bigger computational domain. Finite cell meshes are then generated ignoring boundaries of the physical domain and result in rectangular ‘cells’ as support for the higher order shape functions. The geometry of the problem is only considered during the integration of the cell matrices. To this end an indicator function is introduced being equal to 1 inside the physical domain. Outside it is set (in order to avoid conditioning problems) to a very small value. The contribution of the fictitious domain is thus penalized, shifting the effort of meshing towards the numerical integration of the cell matrices. Combining these ingredients, an exponential rate of convergence can be observed for smooth problems, when performing a p-extension. Since the quality and efficiency of the finite cell approximation strongly depends on the numerical integration scheme, this presentation will first discuss a new algorithmic subdivision approach, extending the conventional octree-based integration with the ability of resolving ‘kinks’ or corners of the physical domain. In combination with the blending function method, this algorithm yields a nearly exact decomposition of the cut cells. Our approach is able to resolve close-to-degenerate cases, but remains algorithmically simple at the same time. Several further recent developments of the FCM will be discussed. A new two- and three-dimensional hierarchical refinement strategy yields exponential rate of convergence in energy norm even for singular problems, and its simple algorithmic structure allows an easy extension to transient problems with local refinement and de-refinement. Finally, a re-interpretation of the fictitious domain in the sense of a ‘third material’ opens the way for a modified contact formulation without the necessity of contact search.
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