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 . 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 .Formally, the XFEM approximat...
SUMMARYThis paper investigates two approaches for the handling of hanging nodes in the framework of extended finite element methods (XFEM). Allowing for hanging nodes, locally refined meshes may be easily generated to improve the resolution of general, i.e. model-independent, steep gradients in the problem under consideration. Hence, a combination of these meshes with XFEM facilitates an appropriate modeling of jumps and kinks within elements that interact with steep gradients. Examples for such an interaction are, e.g. found in stress fields near crack fronts or in boundary layers near internal interfaces between two fluids. The two approaches for XFEM based on locally refined meshes with hanging nodes basically differ in whether (enriched) degrees of freedom are associated with the hanging nodes. Both approaches are applied to problems in linear elasticity and incompressible flows.
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