The Extended Finite Element Method

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

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.

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Non-smooth Solutions Properties: Discontinuities and Singularities

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.

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The Level-set Method

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 $ \Omega$ is devided into two subdomains $ \Omega^{+}$ and $ \Omega^{-}$ 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 $ r$ around $ (0,0)$, see Fig. 3. Then, this discontinuity may be defined by the level-set function

$\displaystyle \phi\left(x,y\right)=\sqrt{x^{2}+y^{2}}-r$ (1)

which is zero on the circle.

Fig. 3

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

(2)

where the sign is different on the two sides of the discontinuity and $ \left\Vert \,\cdot\,\right\Vert $ 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)

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Basics of the XFEM

General Formulation

Consider an $ n$ -dimensional domain $ \Omega\in\mathbb{R}^{n}$ which is discretized by $ n^{\mathrm{el}}$ elements, numbered from $ 1$ to $ n^{\mathrm{el}}$. $ I$ is the set of all nodes in the domain, and $ I_{k}^{\mathrm{el}}$ are the nodes of element $ k\in\left\{ 1,\ldots,n^{\mathrm{el}}\right\} $. A standard extended finite element approximation of a function is of the form

(5)

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 $ a_{i}$ which are defined at nodes in $ I^{\star}\subset I$. The local enrichment functions have the form

(6)

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 $ I^{\star}$, 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 $ I^{\star}$, 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 $ I^{\star}$ 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 $ I^{\star}$, 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.

Choice of the Enriched Nodes

For weak and strong discontinuities, the nodal subset $ I^{\star}$ 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 $ I^{\textrm{el }}$ is the set of element nodes.

Fig. 5

Global Enrichment Functions

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,

(10)

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,

(11)

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

Partition of Unity Functions

At this point, it is sufficient to set the partition of unity function $ N_{i}^{\star}$ 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 $ N_{i}$ are also linear.

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Special Issues in the XFEM

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



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Thomas-Peter Fries (fries@tugraz.at)