# Extended finite element method

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Extended finite element method

The extended finite element method (XFEM), also known as generalized finite element method (GFEM) or partition of unity method (PUM) is a numerical technique that extends the classical finite element method (FEM) approach by extending the solution space for solutions to differential equations with discontinuous functions.

History

The extended finite element method (XFEM) was developed in in 1999 by Ted Belytschko and collaborators, to help alleviate the above shortcomings of the finite element method and has been used to model the propagation of various discontinuities: strong (cracks) and weak (material interfaces). The idea behind XFEM is to retain most advantages of meshfree methods while alleviating their negative sides.

Rationale

The extended finite element method was developed to ease difficulties in solving problems with localized features that are not efficiently resolved by mesh refinement. One of the initial applications was the modelling of fractures in a material. In this original implementation, discontinuous basis functions are added to standard polynomial basis functions for nodes that belonged to elements that where intersected by a crack to provide a basis that included crack opening displacements. A key advantage of XFEM is that in such problems the finite element mesh does not need to be updated to track the crack path; a downside is that cracks can only follow mesh edges. Subsequent research has illustrated the more general use of the method for problems involving singularities, material interfaces, regular meshing of microstructural features such as voids, and other problems where a localized feature can be described by an appropriate set of basis functions.

Principle

Enriched finite element methods extend, or enrich, theapproximation space so that it is able to naturally reproduce thechallenging feature associated with the problem of interest: thediscontinuity, singularity, boundary layer, etc. It was shown thatfor some problems, such an embedding of the problem's feature into the approximationspace can significantly improve convergence rates and accuracy.Moreover, treating problems with discontinuities with eXtendedFinite Element Methods suppresses the need to mesh and remesh thediscontinuity surfaces, thus alleviating the computational costs and projection errorsassociated with conventional finite element methods, at the cost of restricting the discontinuities to mesh edges.

Existing XFEM codes

There exists several research codes implementing this technique to various degrees.

* getfem++
* xfem++
* openxfem++

XFEM was also implemented in code ASTER and in Morfeo and is being taken up by industry, with a few plugins and actual core implementations available (ANSYS, ABAQUS, SAMCEF, OOFELIE, etc.).

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