hp‐adaptive extended finite element method

Abstract

SUMMARYThis paper discusses higher-order extended finite element methods (XFEMs) obtained from the combination of the standard XFEM with higher-order FEMs. Here, the focus is on the embedding of the latter into the partition of unity method, which is the basis of the XFEM. A priori error estimates are discussed, and numerical verification is given for three benchmark problems. Moreover, methodological aspects, which are necessary for hp-adaptivity in XFEM and allow for exponential convergence rates, are summarized. In particular, the handling of hanging nodes via constrained approximation and an hp-adaptive strategy are presented.

Analysis of the diffuse-domain method for solving PDEs in complex geometries

“…However, most of these methods typically require tools not frequently available in standard finite element and finite difference software packages. Examples of such approaches include the extended and composite finite element methods (e.g., [31,12,23,13,32,55,7,4]), immersed interface methods (e.g., [40,43,60,44,65]), virtual node methods with embedded boundary conditions (e.g., [3,73,34]), matched interface and boundary methods (e.g., [71,68,69,67,72]), modified finite volume/embedded boundary/cut-cell methods/ghost-fluid methods (e.g., [27,36,19,25,26,35,47,70,48,37,46,64,49,9,10,52,53,33,63]). In another approach, known as the fictitious domain method (e.g., [28,29,56,45]), the original system is either augmented with equations for Lagrange multipliers to enforce the boundary conditions, or the penalty method is used to enforce the boundary condi-tions weakly.…”

A NURBS-based generalized finite element scheme for 3D simulation of heterogeneous materials

“…The adopted hierarchical mesh refinement scheme leads to irregular meshes with hanging nodes. Utilization of irregular meshes with hanging nodes is also reported in other G/XFEM studies [36][37][38] generally to resolve steep gradients and enhance the precision of FE approximation locally.…”

“…A more detailed discussion about different types of hanging nodes and their treatment in G/XFEM analyses can be found in [38]. In general, there are two approaches in the analysis of hanging nodes based on whether or not additional degrees of freedom are introduced to the problem.…”

“…To enforce these algebraic constraints to the FE system of equations, we adopt the connectivity matrix method. Following [38], for each regular node a, π a is a zero vector with 1 in the position a. For hanging nodes a ∈ I c , π a is defined as…”

3-D local mesh refinement XFEM with variable-node hexahedron elements for extraction of stress intensity factors of straight and curved planar cracks

“…There are several other multiscale models available in literature, e.g., see Refs. [14][15][16][17][18], and the underlying idea of those methods is to impose constraints at nodes on mismatching interfaces to connect different scale meshes. Those methods however often require some modifications on the system matrix whenever the constraints are imposed [19].…”