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Note: On this page, the XFEM is described on a basic level. No references
are given here, see publications instead.

An outline of this page is as follows:

- Introduction
- Non-smooth Solutions Properties: Discontinuities and Singularities
- The Level-set Method
- Basics of the XFEM
- Special Issues in the XFEM

The extended finite element method (XFEM) is a numerical method that enables a local enrichment of approximation spaces. The enrichment is realized through the partition of unity concept. The method is useful for the approximation of solutions with pronounced non-smooth characteristics in small parts of the computational domain, for example near discontinuities and singularities. In these cases, standard numerical methods such as the FEM or FVM often exhibit poor accuracy. The XFEM offers significant advantages by enabling optimal convergence rates for these applications.

A discontinuity may be defined as a rapid change of a field quantity over a length which is negligable compared to the dimensions of the observed domain. In the real world, discontinuities are frequently found, some examples are shown in Fig. 1. In solids, stresses and strains are discontinuous across material interfaces, Fig. 1(a), and displacements are discontinuous at cracks, Fig. 1(b). Tangential displacements are discontinuous across shear bands, Fig. 1(c). In fluids, velocity and pressure fields may involve discontinuities at the interface of two fluids, Fig. 1(d). Furthermore, shocks and boundary layers, Fig. 1(e) and (f), can be interpreted as discontinuities.

Fig. 1 |

In reality, one may argue whether the length scale where a field variable
changes discontinuously can be exactly zero. In models, however, this
idealization is often justified. Then, one may classify two different
types of discontinuities, see Fig. 2: weak discontinuities, where
the field quantity has a kink (the *gradient* has a jump), and
strong discontinuities, where the field quantity has a jump.

Fig. 2 |

Other non-smooth solution properties that are frequently found in the real world and in models are singularities such as they occur at crack tips. Furthermore, oscillations e.g. in wave propagation are mentioned here. When using standard numerical methods (e.g FEM or FVM), for the approximation of non-smooth solutions, special care is required for the mesh construction. For example, the element edges must align with a discontinuity and a mesh refinement is needed near singularities. In contrast, the XFEM is able to achieve optimal convergence rates on structured meshes where arbitrary discontinuities and singularitities are present in element interiors.

The decription of discontinuities in the context of the XFEM is often realized by the level-set method. A level-set function is a scalar function within the domain whose zero-level is interpreted as the discontinuity. As a consequence, the domain is devided into two subdomains and on either side of the discontinuity where the level-set function is positive or negative, respectively.

For example, consider a two-dimensional domain with a circular discontinuity of radius around , see Fig. 3. Then, this discontinuity may be defined by the level-set function

(1) |

which is zero on the circle.

Fig. 3 |

Often, the signed distance function is used as a particular level-set function

where the sign is different on the two sides of the discontinuity and denotes the Euclidean norm,

(3) |

It is noted, that level-set functions are typically defined by discrete values at the nodes in the domain. They are then interpolated in the element interiors by standard finite element shape functions,

(4) |

Consider an
-dimensional domain
which is discretized by
elements, numbered from
to
.
is the set of all nodes in the domain, and
are the nodes of element
.
A standard *extended* finite element approximation of a function
is of the form

where for simplicity only one enrichment term is considered. The approximation consists of a standard finite element (FE) part and the enrichment. The individual variables stand for

The enrichment is built by *local enrichment functions*
and unknowns
which are defined at nodes in
.
The local enrichment functions have the form

and we call
*partition of unity functions* and
*global enrichment function*. The functions
are standard FE shape functions which are not necessarily the same than those of
the standard part of the approximation (5). These
functions build a partition of unity,

(7) |

in elements whose nodes are all in the nodal subset
,
see Fig. 4. In these elements, the global enrichment function
can be reproduced exactly; we call these elements *reproducing
elements.* In elements with only some of their nodes in
,
does not build a partition of unity,
,
see Fig. 4. As a consequence, the global enrichment function
cannot be represented exactly in these elements. Elements with only
some of their nodes in
are called *blending elements*.
Several publications discuss problems arising from blending elements.

Fig. 4 |

Equation (5) generally defines the XFEM. For a particular realization of the XFEM, the choice of the nodal subset , global enrichment function , and the partition of unity functions has to be defined. Here, we restrict ourselves to the case where strong or weak discontinuities are present in the solution of some model equations. More complicated cases such as an additional presence of singularities at crack tips etc. are not considered.

For weak and strong discontinuities, the nodal subset is built from all nodes of elements that are cut by the discontinuity, see Fig. 5. Whether or not an element is cut by the discontinuity can conveniently be determined on element-level by help of the level-set function

(8) | |||

(9) |

where is the set of element nodes.

Fig. 5 |

For weak discontinuities, where a solution shows a kink, or in other words, a jump in the gradient, the global enrichment function is typically chosen as the abs-function of the level-set function,

Along strong discontinuities, a jump is present in the solution. A typical choice for the global enrichment function is the sign-function (or Heaviside-function) of the level-set function,

It is noted that the sign- and Heaviside function lead to identical results as they span the same approximation space.

At this point, it is sufficient to set the partition of unity function equal to linear standard finite element shape functions. It is noted, that for the abs-enrichment, this leads to problems in the blending elements as long as the shape functions of the standard FEM part of the approximation are also linear.

On this page, for reasons of brevity, we did not work out the following issues related to the XFEM:

- The approximation does, in general, not have the Kronecker-delta property.
*Shifting*of the enrichment functions is usually employed in order to recover this property. - The resulting shape functions of the enrichment envolve kinks or jumps in elements cut by a discontinuity. A partitioning of the cut elements into subelements for integration purposes is, in general, required.
- There are often more than only one enrichment term. Formally, these additional enrichment functions are added to the approximation, so the extension is straightforward.
- A locally enriched approximation is also realized in the
*intrinsic XFEM*, which is formally equal to a standard FEM approximation (no additional enrichment terms). The intrinsic XFEM follows a totally different approach in order to realize the enrichment.

Thomas-Peter Fries (fries@tugraz.at)