Finite cell method: hh- and p-extension for embedded domain problems in solid mechanics

A simple yet effective modification to the standard finite element method is presented in this paper. The basic idea is an extension of a partial differential equation beyond the physical domain of computation up to the boundaries of an embedding domain, which can easier be meshed. If this extension is smooth, the extended solution can be well approximated by high order polynomials. This way, the finite element mesh can be replaced by structured or unstructured cells embedding the domain where classical h- or p-Ansatz functions are defined. An adequate scheme for numerical integration has to be used to differentiate between inside and outside the physical domain, very similar to strategies used in the level set method. In contrast to earlier works, e.g., the extended or the generalized finite element method, no special interpolation function is introduced for enrichment purposes. Nevertheless, when using p-extension, the method shows exponential rate of convergence for smooth problems and good accuracy even in the presence of singularities. The formulation in this paper is applied to linear elasticity problems and examined for 2D cases, although the concepts are generally valid.