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

Abstract

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

Theoretical and Numerical Investigation of the Finite Cell Method

“…Instead of classical hierarchic shape functions [25] NURBS, which have become very popular thanks to the isogeometric analysis [10], can also be successfully used within the FCM, see [22,17]. Local refinement strategies have been also developed for the FCM and it turned out that the hp-d method [15,5] presents a general framework for local improvement of accuracy within the FCM, see [21,11].…”

A high‐order enrichment strategy for the finite cell method

“…The continuous form is then obtained by performing either a low-or high-order interpolation, as φ(x) = n j=1 M j (x)Φ j . We showed in [5] that it is of advantage to employ Lagrange shape functions based on the Chen-Babuška points for M j since they yield superior interpolation properties with less error as compared to Lagrange shape functions defined on equidistant points. In addition to describing the location of the material interface, the level set function can be utilized to set up the enrichment function as F (x) = |φ(x)| [11].…”

“…In the case of the material interface, the exact solution exhibits a kink that cannot be represented by standard shape functions of the FCM if the interface is located within the element [5]. In order to overcome such a problem, several remedies have been proposed [5,7,13]. In [5], we showed that using the partition of unity method [1,8] together with the high-order representation of the material interface yields very accurate results, both in the displacements and the stresses.…”

“…To this end, we employ the finite cell method (FCM) where the mesh can be described independent from the geometry [3,10]. In the case of the material interface, the exact solution exhibits a kink that cannot be represented by standard shape functions of the FCM if the interface is located within the element [5]. In order to overcome such a problem, several remedies have been proposed [5,7,13].…”

“…In order to overcome such a problem, several remedies have been proposed [5,7,13]. In [5], we showed that using the partition of unity method [1,8] together with the high-order representation of the material interface yields very accurate results, both in the displacements and the stresses. There, the main idea was to extend the Ansatz with additional high-order shape functions that mimic the local phenomenon of interest.…”