The extended/generalized finite element method: An overview of the method and its applications

Fries, Thomas‐Peter; Belytschko, Ted

doi:10.1002/nme.2914

Cited by 1,169 publications

(749 citation statements)

References 239 publications

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“…Please note that there is no partial enrichment, i.e. the complete domain is enriched so that no blending [3] is necessary. For the integrals which are evaluated at the boundary and include enrichment functions, we use a numerical high order Gauß integration scheme.…”

Section: Square-based Homogeneous Isotropic Pyramid With Crackmentioning

confidence: 99%

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Determination of singularity exponents in 3D elasticity problems using enriched base functions in the Scaled Boundary Finite Element Method

Hell

Becker

2016

Proc Appl Math and Mech

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The SBFEM is employed for the extraction of 3D stress singularity exponents and their associated deformation modes. It makes use of a separation of variables approach, which is substituted into the virtual work equation so that only parts of the boundary need to be discretized while the 3D boundary value problem is considered analytically in a scaling variable. The enrichment of the standard separation of variables representation for the displacements with a decomposition of the analytical crack displacement field yields a substantial improvement of the method's accuracy and convergence properties.

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Section: Square-based Homogeneous Isotropic Pyramid With Crackmentioning

confidence: 99%

“…As a remedy, we propose the use of enriched base functions on the discretized boundary. This enrichment approach, also known from the popular eXtended Finite Element Method [3], is applied to the SBFEM displacement representation.…”

Section: Introductionmentioning

confidence: 99%

Determination of singularity exponents in 3D elasticity problems using enriched base functions in the Scaled Boundary Finite Element Method

Hell

Becker

2016

Proc Appl Math and Mech

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“…Examples of such alternative methods include the diffuse element or element-free Galerkin methods [9,10], least-squares FEM [11], generalized or meshless finite different methods [12,13,14,15,16], generalized or extended FEM [17,18], and partition-of-unity FEM [19]. To reduce the dependency on mesh quality, these methods avoid the use of the piecewise-polynomial Lagrange basis functions found in the classical FEM.…”

Section: Introductionmentioning

confidence: 99%

Overcoming element quality dependence of finite elements with adaptive extended stencil FEM (AES‐FEM)

Conley

Delaney

Jiao

2016

Numerical Meth Engineering

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The finite element methods (FEM) are important techniques in engineering for solving partial differential equations, but they depend heavily on element shape quality for stability and good performance. In this paper, we introduce the Adaptive Extended Stencil Finite Element Method (AES-FEM) as a means for overcoming this dependence on element shape quality. Our method replaces the traditional basis functions with a set of generalized Lagrange polynomial (GLP) basis functions, which we construct using local weighted least-squares approximations. The method preserves the theoretical framework of FEM, and allows imposing essential boundary conditions and integrating the stiffness matrix in the same way as the classical FEM. In addition, AES-FEM can use higher-degree polynomial basis functions than the classical FEM, while virtually preserving the sparsity pattern of the stiffness matrix. We describe the formulation and implementation of AES-FEM, and analyze its consistency and stability. We present numerical experiments in both 2D and 3D for the Poisson equation and a time-independent convection-diffusion equation. The numerical results demonstrate that AES-FEM is more accurate than linear FEM, is also more efficient than linear FEM in terms of error versus runtime, and enables much better stability and faster convergence of iterative solvers than linear FEM over poor-quality meshes.

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“…The extended finite element method (XFEM) has developed to be a well-accepted and promising tool for the simulation of cracks [1,2]. The method features the consideration of inner-element discontinuities and singularities by adding customtailored enrichments to the classical finite element approximation space.…”

Section: Introductionmentioning

confidence: 99%

“…Recent developments in the field of the XFEM are discussed in the special issue in [3]. A recent overview on the XFEM is given in [2].…”

Section: Introductionmentioning

confidence: 99%

A new description of cracks in the XFEM

Fries

2011

Proc Appl Math and Mech

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This work outlines an approach that combines the advantages of explicit and implicit descriptions of the crack geometry in the context of the extended finite element method (XFEM). The XFEM is typically combined with an implicit description of the crack which is realized by the level-set method. This facilitates the definition of the enrichments in the XFEM. However, the update of the crack geometry during the propagation of a crack is simpler by means of explicit crack descriptions where the crack surface is meshed.

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The extended/generalized finite element method: An overview of the method and its applications

Cited by 1,169 publications

References 239 publications

Determination of singularity exponents in 3D elasticity problems using enriched base functions in the Scaled Boundary Finite Element Method

Determination of singularity exponents in 3D elasticity problems using enriched base functions in the Scaled Boundary Finite Element Method

Overcoming element quality dependence of finite elements with adaptive extended stencil FEM (AES‐FEM)

A new description of cracks in the XFEM

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