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# XFEM

The Extended Finite Element Method (XFEM) is a numerical technique which extends the classical FEM approach by extending the solution space to better approach the real solution to a differential equation.

## History

In 1999, the extended Finite Element Method (XFEM), was born, in the team of Ted Belytschko, to help alleviate the above shortcomings of the FEM and has been used with great results 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 FEM is limited in its application. In this method, a solid, which contains a large number (billions) of material points is split into a finite (much fewer than billions) number of elements (the triangles on the figure, left) covering the domain (known as the mesh). The unknowns are calculated at all the vertices (known as nodes) of the elements and must be continuous (no jump) across the boundaries of the elements (edges of the triangles).

If a discontinuity (jump) is to be modelled, nodes have to be placed along the line of discontinuity and the element edges must conform to the discontinuity. If the discontinuity is evolving, the nodes and elements must be updated (remeshing). For multiple discontinuities and three dimensional problems, this becomes rapidly intractable.

## Principle

Enriched finite element methods is extend, or enrich the approximation space so that it is able to naturally reproduce the challenging feature associated with the problem of interest: the discontinuity, singularity, boundary layer, etc. It was shown that such an embedding of the problem's feature into the approximation space can significantly improve convergence rates and accuracy. Moreover, treating problems with discontinuities with eXtended Finite Element Methods suppresses the need to mesh and remesh the discontinuity surfaces, thus alleviating the computational costs and projection errors associated with conventional finite element methods.

## 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 is being taken up by industry, with a few plugins and actual core implementations available (ANSYS, ABAQUS, ...)