Higher‐order XFEM for curved strong and weak discontinuities

Abstract

The extended finite element method (XFEM) enables the accurate approximation of solutions with jumps or kinks within elements. Optimal convergence rates have frequently been achieved for linear elements and piecewise planar interfaces. Higher-order convergence for arbitrary curved interfaces relies on two major issues: (i) an accurate quadrature of the Galerkin weak form for the cut elements and (ii) a careful formulation of the enrichment, which should preclude any problems in the blending elements. For (i), we employ a strategy of subdividing the elements into subcells with only one curved side. Reference elements that are higher-order on only one side are then used to map the integration points to the real element. For (ii), we find that enrichments for strong discontinuities are easily extended to higher-order accuracy. In contrast, problems in blending elements may hinder optimal convergence for weak discontinuities. Different formulations are investigated, including the corrected XFEM. Numerical results for several test cases involving strong or weak curved discontinuities are presented. Quadratic and cubic approximations are investigated. Optimal convergence rates are achieved using the standard XFEM for the case of a strong discontinuity. Close-to-optimal convergence rates for the case of a weak discontinuity are achieved using the corrected XFEM.HIGHER-ORDER XFEM FOR CURVED DISCONTINUITIES 565 characteristics, the construction of an appropriate mesh is crucial for the success of the finite element (FE) approximation (i.e. element edges have to align with a discontinuity and a mesh refinement is required where the solution is expected to possess singularities or large gradients). Furthermore, if the discontinuity evolves with time, the nodes and elements must be moved or remeshed continuously [1][2][3][4]. For multiple discontinuities and three-dimensional problems, this becomes rapidly intractable. Therefore, a method for modelling arbitrary discontinuities in FEs on a fixed mesh without remeshing is desirable.An extension of the standard FEM, the extended finite element method (XFEM) [5, 6] has been found to yield accurate results and yet does not require the mesh to conform to discontinuities in the approximating function or its derivatives. The XFEM also avoids remeshing for moving discontinuities. This is accomplished by extending the piecewise polynomial approximation space of the FEM to include discontinuous function spaces in local regions of the computational domain where the solution exhibits jumps or kinks. This local enrichment of the approximation space is realized by means of the partition-of-unity concept [7,8]. As a result, optimal convergence rates are achieved for linear elements and (piecewise) planar interfaces. It is worthwhile to mention the close similarity between the XFEM and other partition-of-unity-based methods such as the partition-of-unity method (PUM) [7,8], the generalized finite element method (GFEM) [9-13] and the hp-cloud method [14][15][16].Formally, the XFEM approximat...

A high-order immersed boundary discontinuous-Galerkin method for Poisson's equation with discontinuous coefficients and singular sources

“…A higherorder X-FEM technique (see e.g. [26]) may show some improvement related to these concerns, but we found the approach based on the DG FEM as presented by [1] more natural to extend for our specific demands.…”

Efficient and accurate numerical quadrature for immersed boundary methods

“…Investigations in the field of XFEM show that it is beneficial to subdivide elements that are cut by the interface into a set of curved geometric entities. In [22], the cut elements are approximated by a set of Lagrangian elements with curved sides. The problem has also been addressed in [24], where an algorithmic approach to decompose cut elements into a set of curved triangles was presented.…”

“…Another technique to resolve the discontinuous integrand is to decompose the cut cells into Lagrangian elements in such a way that the edges of the resulting subcells align with the interface [22]. The use of triangular NURBS-Enhanced Finite Elements [23] in different XFEM settings has also been investigated in [24], based on an approach that is extended in this contribution.…”

Convergence of domain integrals for stress intensity factor extraction in 2‐D curved cracks problems with the extended finite element method

“…This introduces further approximations, although the associated error is deemed small for sufficiently refined meshes, as the ones used in this analysis. It is possible to improve the crack description using higher order elements [23], but this issue has not been considered in this work. …”