Stable Generalized Finite Element Method (SGFEM)

By Ivo Babuska, Uday Banerjee. 2011

354 citations recorded by the source corpus.

Abstract The Generalized Finite Element Method (GFEM) is a Partition of Unity Method (PUM), where the trial space of standard Finite Element Method (FEM) is augmented with non-polynomial shape functions with compact support. These shape functions, which are also known as the enrichments, mimic the local behavior of the unknown solution of the underlying variational problem. GFEM has been successfully used to solve a variety of problems with complicated features and microstructure. However, the stiffness matrix of GFEM is badly conditioned (much worse compared to the standard FEM) and there could be a severe loss of accuracy in the computed solution of the associated linear system. In this paper, we address this issue and propose a modification of the GFEM, referred to as the Stable GFEM (SGFEM). We show that SGFEM retains the excellent convergence properties of GFEM, does not require a ramp-function in the presence of blending elements, and the conditioning of the associated stiffness matrix is not worse than that of the standard FEM. Moreover, SGFEM is very robust with respect to the parameters of the enrichments. We show these features of SGFEM on several examples.

Subfields: Computational Mechanics, Electrical and Electronic Engineering, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/c45fb0013fcd7674ea5ce397516b6a02f81b722a

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In Computational Mechanics

In Electrical and Electronic Engineering

In Mechanics of Materials

Citations in the corpus

A new generalized finite element method for two‐scale simulations of propagating cohesive fractures in 3‐D (2015, background): “The stable GFEM [114, 115] can also be used to address the issue of loss of accuracy on blending elements.”
An improved stable XFEM (Is-XFEM) with a novel enrichment function for the computational modeling of cohesive cracks (2015, background): “As already proved in Babuška and Banerjee [2], the scaled conditioning number of the system matrix for the standard GFEM/XFEM is O.”
A gradient-based shape optimization scheme using an interface-enriched generalized FEM (2015, neutral): “This contrasts with conventional G/XFEM where the generalized dofs are associated with duplicated nodes of the non-conforming elements [39] [40].”
A stable and optimally convergent generalized FEM (SGFEM) for linear elastic fracture mechanics (2013, methodology): “In the 1-D problems considered in [2, 3], the modification of enrichments used in the standard GFEM was shown to be sufficient for the method to [ May 4, 2013]”
Improved conditioning and accuracy of GFEM/XFEM for three-dimensional fracture mechanics (2015, methodology): “The work presented in [25] for 2-D fracture mechanics shows that a straightforward extension of ideas presented for 1-D problems in [4] is not sufficient.”
XLME interpolants, a seamless bridge between XFEM and enriched meshless methods (2014, background): “Another promising development for improving the condition number of geometric enrichment has been developed in [34].”
Cohesive and non‐cohesive fracture by higher‐order enrichment of XFEM (2012, methodology): “A promising robust solution is the stable GFEM recently developed by Babuska and Banerjee [39], who have shown a simple modification to the local PU-based enrichment approximation which yields a stiffness matrix with a condition number that scales as that of the FEM.”
The Discontinuity‐Enriched Finite Element Method (2017, neutral): “Equation (5) yields w1 = −xΓ∕L = −w while Equation (6) yields w2 = 1 + w1 = 1 − w.”
A primal-dual active set method and predictor-corrector mesh adaptivity for computing fracture propagation using a phase-field approach (2015, methodology): “Instead of modeling the discontinuities explicitly (like in the extended finite element method [9] or generalized finite elements [10]), the lower-dimensional crack surface is approximated by a phase-field function.”
Extended finite element method in computational fracture mechanics: a retrospective examination (2015, neutral): “It is coined SGFEM (stabilized generalized ﬁnite element method) (Babuška and Banerjee 2012).”