Development of a quadratic finite element formulation based on the XFEM and NURBS

Abstract

SUMMARYThe FE-simulation of inhomogeneous structures, such as composite materials, biological tissues or foams, requires the generation of respective finite element meshes. With increasing complexity of the inner architecture of such structures, this becomes a time-consuming and laborious task. Additionally, the risk of forming bad-shaped elements that may lead to ill-conditioned numerical problems grows significantly. A solution to this problem provides the extended finite element method (XFEM). Thereby, the interface between different materials is represented by a local enrichment of the displacement approximation. As a consequence of this, the element boundary need not be aligned to the interface.In order to improve the accuracy of the interface approximation, the development of a plane element based on the XFEM and quadratic shape functions will be presented. This element allows for the description of curved material interfaces. The computation of the element stiffness matrix requires a numerical integration process that accounts for discontinuous fields. Regarding a linear element formulation, this can be achieved by an adapted triangulation of the element domain. However, in the case of a curved interface this solution is not applicable. Hence, non-uniform rational B-Spline (NURBS) surfaces are used to evaluate the integrals numerically.Finally, the results of different examples will show the general properties such as the accuracy of the numerical integration procedure and the convergence behavior of this element formulation.

A NURBS-based generalized finite element scheme for 3D simulation of heterogeneous materials

“…By incorporating directly the NURBS representation into the finite element formulation of the non-conforming elements intersected by the material interfaces, the NIGFEM greatly simplifies the mesh generation process while providing the same accuracy and convergence properties as those of conventional FEM based on conforming meshes. Similar ideas were presented in the context of regular FEM [30] and G/XFEM [31,32].…”

Fatigue crack growth analysis of a homogeneous plate in the presence of multiple defects using extended isogeometric analysis

“…Benson et al [8] analyzed the fracture mechanics problems using XIGA. Haasemann et al [17] analyzed a bi-material body with curved interfaces by XIGA. De Luycker et al [25] showed that the XIGA gives greater accuracy and higher convergence rate in solving the problems of fracture mechanics.…”

Crack analysis in media with orthotropic Functionally Graded Materials using extended Isogeometric analysis

“…De Luycker et al [42] analyzed a fixed mode crack with different ways of enrichment and imposition of displacement boundary conditions to achieve an optimal rate of convergence. Haasemann et al [46] combined the quadratic NURBS functions and XFEM for the analysis of a bimaterial body with a curved interface. Ghorashi et al [43] extended the conventional IGA to an enriched IGA approach for mixed mode fracture analysis of fixed and propagating cracks.…”