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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 M. Joulaian (B) · A. Düster Numerical Structural Analysis with Application in Ship Technology, approximation on the base and the overlay mesh and accordingly allows multiscale problems to be addressed as well.Keywords FCM · local enrichment · PUM · multiscale method · hp-d method · high-order method · XFEM

SUMMARYAn accurate and efficient simulation of wave propagation phenomena plays an important role in different engineering disciplines. In structural health monitoring, for example, ultrasonic guided waves are used to detect and localize damage and to assess the structural integrity of the component part under consideration. Because of the complexity of real structures, the numerical simulation of structural health monitoring systems is a computationally demanding task. Therefore, to facilitate the analysis of wave propagation phenomena, the authors propose to combine the finite cell method with the spectral element method. The ensuing novel method is referred to as the spectral cell method. Because it does not rely on body‐fitted meshes, the resulting approach eliminates all discretization difficulties encountered in conventional finite element methods. Moreover, with the aid of mass lumping, it paves the way for the use of explicit time‐integration algorithms. In the first part of the paper, we show that using a lumped mass matrix instead of the consistent one has no detrimental effect on the accuracy of the spectral element method. We introduce the spectral cell method in the second part, showing that, when applied to wave propagation analysis, the spectral cell method yields results comparable with other standard higher order finite element approaches.Copyright © 2014 John Wiley & Sons, Ltd.

a b s t r a c tWe present a numerical homogenization approach for hybrid metal foams, i.e. foams that are electrocoated to improve their mechanical properties. Based on the finite cell method, a spatial discretization of a µCT-scan of the microstructure of the hybrid metal foam under investigation is derived and the window method is applied to compute effective material properties. We demonstrate that this method offers the possibility to efficiently compute and study the influence of the coating thickness of hybrid metal foams.