Local enrichment of the finite cell method for problems with material interfaces

This paper proposes an efficient, hierarchical high-order enrichment approach for the finite cell method applied to problems of solid mechanics involving discontinuities and singularities. In contrast to the standard extended finite element method, where new degrees of freedom are introduced for all finite elements located in the enrichment zone, we define the enrichment on a so-called overlay mesh which is superimposed over the base mesh. The approximation on the base mesh is obtained by means of the finite cell method where the hp-d method is employed to introduce the hierarchical extension on the overlay mesh. We present two different strategies for defining the enrichment on the superimposed overlay mesh. In the first approach, the enrichment is based on a local h-, p- or hp-refinement utilizing the finite element method on the overlay mesh. Alternatively, the enrichment is constructed by means of the partition of unity method introducing carefully selected enrichment functions suitable for the problem at hand. Our results reveal that the proposed method improves the accuracy of the finite cell method significantly with only a minimum number of additional degrees of freedom. In this paper we will focus on examples with material interfaces although the method can also be applied to problems involving strong discontinuities and singularities. Accurate stress distribution and an exponential rate of convergence are the two striking characteristics of the proposed method. Due to the hierarchical approach it paves the way to using different approaches for the approximation on the base and the overlay mesh and accordingly allows multiscale problems to be addressed as well.