# High-order Extended Finite Element Method: Influence of the geometrical representation

Kristell Dréau, Nicolas Chevaugeon, Nicolas Moes

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Kristell Dréau, Nicolas Chevaugeon, Nicolas Moes. High-order Extended Finite Element Method: Influence of the geometrical representation. 8th World Congress on Computational Mechanics (WCCM8), 5th European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2008), Jun 2008, Venise, Italy. hal-01008254

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Distributed under a Creative Commons Attribution 4.0 International License
HIGH ORDER EXTENDED FINITE ELEMENT METHOD :
INFLUENCE OF THE GEOMETRICAL REPRESENTATION

* K. Dréau¹, N. Chevaugeon¹ and N. Moës¹

¹ Institut de Recherche en Génie Civil et Mécanique, UMR CNRS 6183, École Centrale de Nantes,
B.P. 92101, 44321 Nantes Cedex 3, France

{kristell.dreau, nicolas.chevaugeon, nicolas.moes}@ec-nantes.fr

**Key Words:** X-FEM, level-set method, optimal rate of convergence, voids, material interfaces, curved cracks.

ABSTRACT

The Extended Finite Element Method (X-FEM) is a good alternative to the classical finite element method for solving problems with complex geometries. Within the classical finite element method, the mesh is required to conform to physical surfaces. Discontinuities such as holes, cracks and material interfaces can't cross mesh elements. Moreover, local refinements close to discontinuities and mesh modification to track the geometrical and topological changes in crack propagation problems for example, can be difficult. Also, robust methods to transfer the solution to the new mesh are needed.

Within the X-FEM, surfaces that are not represented explicitly by mesh boundaries can be implicitly
represented by the iso-values of a level-set function. This is particularly useful for moving interfaces
problems such as crack surfaces in crack propagation analysis [1]. The finite element approximation is
enriched by additional functions through the notion of partition of unity, to represent, for example, dis-
continuities at interfaces [2] or asymptotic expansions terms near crack tip [1] to improve convergence
rate. The enrichment functions are usually defined with the help of the level-set functions to access the
distance to the interface at any given point. Although interfaces do not have to be meshed, the correct
integration of the element stiffness matrix, for elements enriched by a discontinuous function along an
interface, need to be done carefully. The elements are split at the integration level along the iso-zero
value, and regular integration is done separately on each side of the integration partition. Difficulties to
represent the surface in the classical finite element method are, in the context of X-FEM, partly shifted
on the integration procedure. Nevertheless, X-FEM gives good results with linear elements and linear
level-set in elements : optimal rate of convergence is achieved with curved geometries. With shape
functions of higher order, if the error is reduced, the optimal rate of convergence can't be achieved
without improving the geometrical representation. The problem is of course related to the well known
fact that, when increasing the order of the approximation field, the order of representation of the ge-
ometry must be increase accordingly to get optimal rate of convergence, in the classical finite element
method. Iso-parametric elements have an energy error norm on regular problems with curved boundary
that converges as O(h^p), where h is the element size and p the polynomial basis order. To get optimal
rate of convergence when curved interfaces are represented with level-set, the obvious solutions would
be to increase the order of the level-set representation near the zero iso-values. In this case, the inte-
gration problem can become quite difficult : within an element, the iso-zero surface can have complex
shape and topology.

The alternative approach that we propose consists in representing the level-set on a finer mesh (figures
1 and 2), while keeping the level-set representation linear by element (without adding any degree of
freedom). We start with the same mesh for the field representation and the level set. Each element of
the initial mesh for the level-set that is crossed by the iso-zero is split (conformity of the mesh is not
required for the level-set representation), and value of the level-set are computed on the new nodes.
The procedure is done recursively up to a user defined maximum depth. Each element of the analysis
is then linked to an octree-like partition where the level-set is represented and on which the previously
described integration procedure is applied for each sub-element. The approximation of the zero surface
is then improved at each level of refinement. The algorithm was implemented and tested on simple
problems with curved boundaries. Numerical experimentation were done up to order three polynomial
approximation and we show that optimal convergence rate can be achieved (figure 3), with an optimal
depth of recursion. Engineering rules to choose the optimal depth as a function of *p* are presented.
Results on curved crack problems in two and three dimensions will be presented and compared to the
results given in the literature [3], [4].

<figure><img src="image_2.png" /><figcaption>Figure 1: Analysis mesh</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Figure 2: Integration mesh</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Figure 3: Energy error convergence</figcaption></figure>

REFERENCES

[1] N. Sukumar, N. Moës, B. Moran and T. Belytschko. "Extended finite element method for three-dimensional crack modelling". *Int. J. Numer. Meth. Engng.*, Vol **48**, 1549–1570, 2000.

[2] N. Sukumar, D.L. Chopp, N. Moës and T. Belytschko. "Modeling holes and inclusions by level sets in the extended finite-element method". *Comput. Methods Appl. Mech. and Engrg*, Vol. **190**, 6183–6200, 2001.

[3] F. L. Stazi, E. Budyn, J. Chessa and T. Belytschko. "An extended finite element method with higher-order elements for curved cracks". *Comput. Mech.*, Vol **31**, 38–48, 2003.

[4] P. Laborde, J. Pommier, Y. Renard and M. Salaün. "High-order extended finite element method for cracked domains". *Int. J. Numer. Meth. Engng.*, Vol **64**, 354–381, 2005.