The generalized finite element method: an example of its implementation and illustration of its performance

Abstract

SUMMARYThe generalized ÿnite element method (GFEM) was introduced in Reference [1] as a combination of the standard FEM and the partition of unity method. The standard mapped polynomial ÿnite element spaces are augmented by adding special functions which re ect the known information about the boundary value problem and the input data (the geometry of the domain, the loads, and the boundary conditions). The special functions are multiplied with the partition of unity corresponding to the standard linear vertex shape functions and are pasted to the existing ÿnite element basis to construct a conforming approximation. The essential boundary conditions can be imposed exactly as in the standard FEM. Adaptive numerical quadrature is used to ensure that the errors in integration do not a ect the accuracy of the approximation. This paper gives an example of how the GFEM can be developed for the Laplacian in domains with multiple elliptical voids and illustrates implementation issues and the superior accuracy of the GFEM versus the standard FEM.

Derivative recovery and a posteriori error estimate for extended finite elements

“…Other partition of unity enriched finite element methods are the generalized finite element method (GFEM) [37,38] and Nitsche's method [22], although the latter's kinematics is identical to that of the XFEM [5]. The key idea is to locally add arbitrary special functions to the finite element basis.…”

Multiscale enrichment based on partition of unity

“…Among the noteworthy SGEMs are the s-version of the finite element method [19,20,21,22] with application to strong [23,24] and weak [25,26,27,28] discontinuities, various multigrid-like scale bridging methods [29,30,31,32], the Extended Finite Element Method (XFEM) [33,34,35] and the Generalized Finite Element Method (GFEM) [36,37] both based on the Partition of Unity (PU) framework [38,39] and the Discontinuous Galerkin (DG) [40,41] method. Multiscale methods based on the concurrent resolution of multiple scales are often called as embedded, concurrent, integrated or hand-shaking multiscale methods.…”

“…For this discontinuous enrichment functions, the problem of linear dependency does not arise, but the issue of ensuring the integration errors to be significantly smaller than the approximation errors requires special attention [35]. In GFEM, which uses special handbook function [36,37], the resulting space is not linearly independent and the computational overhead (needed for either reorthogonalization or for using an indefinite solver) might be quite significant. Moreover, integration of coupling terms involving special handbook functions is challenging at best, and therefore it is not surprising that the method has been implemented for two dimensional problems only.…”

A modified perturbed Lagrangian formulation for contact problems

“…Here we only describe the basic aspects of the immersed boundary method. A more detailed description can be found in [36,49], for example. In the immersed boundary method, sometimes referred to as the Cartesian grid method, the underlying mesh consists of regular hexahedrals and this will be used in this work.…”