Keywords: extended finite element method, object oriented design.

This paper deals with the extended finite element method and its object-oriented implementation. The extended finite element method (XFEM), which belongs to the family of methods based on the partition of unity, is a numerical technique that extends the classical finite element method (FEM) approach by enriching the solution space with tailored functions.

The original idea of the partition-of-unity method [1] is that the approximation space spanned by a standard basis (e.g., by the standard finite element shape functions) that can be easily enriched by products of the standard basis functions with special functions. These functions can be selected by the user and constructed, e.g., from the analytical solution of the problem under some simplifying assumptions. This idea was first adapted for linear elastic fracture mechanics, with the enrichment constructed using the near-tip asymptotic solutions and simple Heaviside functions [2]. The method was later called the XFEM. It can efficiently handle three-dimensional cracks. Subsequent research has illustrated the more general use of the method for problems involving singularities, material interfaces, fluid-structure interactions, regular meshing of micro-structural features such as voids, resolution of boundary layers, and other problems where analytical solution is available.

The implementation of XFEM, particularly its seamless integration into an existing code, is a complex and challenging task. In this paper, a design of an object-oriented XFEM toolkit is presented and its integration into an existing object-oriented finite element environment [3,4] is discussed. The overall structure of the proposed XFEM toolkit consists of several top level classes, that represent enrichment, enrichment geometry, integration rule and the so called XFEM manager. The primary role of these core classes is to specify a general interface that defines the abstract services that are provided by the corresponding derived classes. This concept allows to a developer to implement high level functionality using general interfaces, without regarding details of a particular derived class. On the other hand, it allows a particular class to be implemented without deep knowledge of the whole code structure; it is only necessary to implement the required services, that constitute the general interface. The application of the developed object-oriented tool will be demonstrated on an example of the XFEM representation of the micro-structural features such as voids on arbitrary grids.

1

J.M. Melenk, I. Babuška, "The Partition of Unity Finite Element Method: Basic Theory and Applications", Comp. Meth. Appl. Mech. Engrg., 39, 289-314, 1996. doi:10.1016/S0045-7825(96)01087-0

2

N. Moës, J. Dolbow, T. Belytschko, "A finite element method for crack growth without remeshing", International Journal for Numerical Methods in Engineering, 46, 131-150, 1999. doi:10.1002/(SICI)1097-0207(19990910)46:1<131::AID-NME726>3.0.CO;2-J

3

B. Patzák, Z. Bittnar, "Design of object oriented finite element code", Advances in Engineering Software, 32 (10-11), 759-767, 2001. doi:10.1016/S0965-9978(01)00027-8

4

B. Patzák, "OOFEM home page", http://www.oofem.org, 2008.

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