Extended Finite Element Method
Introduction
The extended finite element method1 (XFEM) uses the partition of unity framework2 to model strong and weak discontinuities independent of the finite element mesh. This allows discontinuous functions to be implemented into a traditional finite element framework through the use of enrichment functions and additional degrees of freedom.
As the nodal displacement is now a function of both the traditional and enriched degrees of freedom it is common practice to shift the enrichment functions3 such that the enrichment functions vanish at the nodes. The Heaviside nodes are shown with circles and the crack tip nodes are shown with squares. Thus when the system of equations is solved the enriched degrees of freedom are only used to interpolate within a particular element.
Therefore the XFEM approximation with the shifted enrichment function takes the following form.
The extended finite element method has been implemented in various degrees of complexity in several open source codes such as getfem++, openxfem++, and MXFEM. Furthermore, several commercial implementations are available in packages such as ABAQUS and Morfeo. Please contact me with information of other implementations so that they can be added to the list.
Crack
Cracks are modeled using a combination of two enrichment functions. One for the complicated behavior at the crack tip and a Heaviside step function to represent the discontinuity across the body of the crack. The Heaviside function takes a value of 1 above the crack and -1 below the crack, thus putting a displacement discontinuity across the body of the crack in elements whose support is cut by the crack. For the crack tip the enrichment functions originally introduced by Fleming4 for use in the element free Galerkin method. They were later adobted by Belytschko5 for use in XFEM. These four functions span the crack tip displacement field. Also note that the first function is discontinuous across the crack within the element containg the crack tip. For information on modeling bimaterial or branching cracks please refer to the papers by Sukumar6 and Daux7.
Material Interface
Moes8 introduced a modified absolute value enrichment of the following form where zeta is the value of the level set function at node I. Note that this function vanishes at the nodes itself and therefore does not need to be shifted as is typical of other enrichment functions.
Void
The enrichment function for a void7,9 takes a different form from the traditional enrichment. Instead a step function is used such that V(x) takes a value of 1 in the domain and 0 inside the void. In practice, integration is not performed at gauss points within the domain where V(x) is zero.
References
1. Moes, N., Dolbow, J., Belytschko, T. (1999) "A finite element method for crack growth without remeshing," International Journal for Numerical Methods in Engineering, 46, 131-150.
2. Babuska, I., Melenk, J. (1997) "The partition of unity method," International Journal for Numerical Methods in Engineering, 40, 727-758.
3. Belytschko, T., Moes, N., Usui, S., Parimi, C. (2001) "Arbitrary discontinuities in finite elements," International Journal for Numerical Methods in Engineering, 50, 993-1013.
4. Fleming, M., Chu, A., Moran, B., Belytschko, T. (1997) "Enriched element-free Galerkin methods for crack tip fields," International Journal for Numerical Methods in Engineering, 40, 1483-1504.
5. Belytschko, T., Black, T. (1999) "Elastic crack growth in finite elements with minimal remeshing," International Journal for Numerical Methods in Engineering, 45, 601-620.
6. Sukumar, N., Huang, Z.Y., Prevost, J.H., Suo, Z. (2004) "Partition of unity enrichment for bimaterial interface crack," International Journal for Numerical Methods in Engineering, 59, 1075-1102.
7. Daux, C., Moes, N., Dolbow, J., Sukumar, N., Belytschko, T. (2000) "Arbitrary branched and intersecting cracks with the extended finite element method," International Journal for Numerical Methods in Engineering, 48, 1741-1760.
8. Moes, N., Cloirec, M., Cartraud, P., Remacle, J.F. (2003) "A computational approach to handle complex microstructure geometries,"Computer Methods in Applied Mechanics and Engineering, 192, 3163-3177.
9. Sukumar, N., Chopp, D.L., Moes, N., Belytschko, T. (2001) "Modeling holes and inclusions by level sets in the extended finite-element method," Computer Methods in Applied Mechanics and Engineering, 190, 6183-6200.