The generalized finite element method

Strouboulis, T.; Copps, Kevin D.; Babuška, Ivo

doi:10.1016/s0045-7825(01)00188-8

Cited by 569 publications

(387 citation statements)

References 18 publications

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“…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.…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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Multiscale enrichment based on partition of unity

Fish¹,

Yuan²

2004

Numerical Meth Engineering

139

110

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A new Multiscale Enrichment method based on the Partition of Unity (MEPU) method is presented. It is a synthesis of mathematical homogenization theory and the Partition of Unity method. Its primary objective is to extend the range of applicability of mathematical homogenization theory to problems where scale separation may not be possible. MEPU is perfectly suited for enriching the coarse scale continuum descriptions (PDEs) with fine scale features and the quasi-continuum formulations with relevant atomistic data. Numerical results show that it provides a considerable improvement over classical mathematical homogenization theory and quasi-continuum formulations.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Multiscale enrichment based on partition of unity

Fish¹,

Yuan²

2004

Numerical Meth Engineering

139

110

View full text Add to dashboard Cite

show abstract

“…In the literature, numerical techniques such as the extended finite element method (X-FEM) [17,18], generalized finite element method [23], or the element partition method [24] are all particular instances of the partition of unity method. In the X-FEM, the emphasis has been on modeling discontinuities (such as cracks) with minimal enrichment.…”

Section: Extended Finite Element Methodsmentioning

confidence: 99%

Finite element-based model for crack propagation in polycrystalline materials

Sukumar

Srolovitz

2004

Mat. apl. comput.

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Abstract.In this paper, we use an extended form of the finite element method to study failure in polycrystalline microstructures. Quasi-static crack propagation is conducted using the extended finite element method (X-FEM) and microstructures are simulated using a kinetic Monte Carlo Potts algorithm. In the X-FEM, the framework of partition of unity is used to enrich the classical finite element approximation with a discontinuous function and the two-dimensional asymptotic crack-tip fields. This enables the domain to be modeled by finite elements without explicitly meshing the crack surfaces, and hence crack growth simulations can be carried out without the need for remeshing. First, the convergence of the method for crack problems is studied and its rate of convergence is established. Microstructural calculations are carried out on a regular lattice and a constrained Delaunay triangulation algorithm is used to mesh the microstructure. Fracture properties of the grain boundaries are assumed to be distinct from that of the grain interior, and the maximum energy release rate criterion is invoked to study the competition between intergranular and transgranular modes of crack growth.Mathematical subject classification: 74R10.

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“…The "Partition of Unity Method" (PUM) [32,3] uses a priori knowledge about the solution (discontinuities at interfaces) to obtain special PUM finite element spaces. In the "Generalized Finite Element Method" (GFEM) [48,46], the PUM and classical FEM basis functions are used together to improve the approximation. Starting from classical FEM and "enriching" the FE spaces by additional basis functions to incorporate discontinuities has been exploited with the "Extended Finite Element Methods" (XFEM) [4].…”

Section: Introductionmentioning

confidence: 99%

Composite finite elements for 3D image based computing

Liehr

Preußer

Rumpf

et al. 2008

Comput. Visual Sci.

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Abstract. We present an algorithmical concept for modeling and simulation with partial differential equations (PDEs) in image based computing where the computational geometry is defined through previously segmented image data. Such problems occur in applications from biology and medicine where the underlying image data has been acquired through e. g. computed tomography (CT), magnetic resonance imaging (MRI) or electron microscopy (EM). Based on a level-set description of the computational domain, our approach is capable of automatically providing suitable composite finite element functions that resolve the complicated shapes in the medical/biological data set. It is efficient in the sense that the traversal of the grid (and thus assembling matrices for finite element computations) inherits the efficiency of uniform grids away from complicated structures. The method's efficiency heavily depends on precomputed lookup tables in the vicinity of the domain boundary or interface. A suitable multigrid method is used for an efficient solution of the systems of equations resulting from the composite finite element discretization. The paper focuses on both algorithmical and implementational details. Scalar and vector valued model problems as well as real applications underline the usability of our approach.

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The generalized finite element method

Cited by 569 publications

References 18 publications

Multiscale enrichment based on partition of unity

Multiscale enrichment based on partition of unity

Finite element-based model for crack propagation in polycrystalline materials

Composite finite elements for 3D image based computing

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