The intrinsic XFEM: a method for arbitrary discontinuities without additional unknowns

Abstract

SUMMARYA new method for treating arbitrary discontinuities in a finite element (FE) context is presented. Unlike the standard extended FE method (XFEM), no additional unknowns are introduced at the nodes whose supports are crossed by discontinuities. The method constructs an approximation space consisting of meshbased, enriched moving least-squares (MLS) functions near discontinuities and standard FE shape functions elsewhere. There is only one shape function per node, and these functions are able to represent known characteristics of the solution such as discontinuities, singularities, etc. The MLS method constructs shape functions based on an intrinsic basis by minimizing a weighted error functional. Thereby, weight functions are involved, and special mesh-based weight functions are proposed in this work. The enrichment is achieved through the intrinsic basis. The method is illustrated for linear elastic examples involving strong and weak discontinuities, and matches optimal rates of convergence even for crack-tip applications.

The extended/generalized finite element method: An overview of the method and its applications

“…This approach allows one to obtain a conforming approximation and to eliminate partially enriched elements, so that the partition of unity property is everywhere satisfied. The idea of a weighting the enrichment was also presented in [21] but with no reference to blending and also in [46] for governing the transition between finite elements and a moving least squares approximation. This approach has been extended in [108], where a more general ramp function was used and the influence of the weight function derivatives has been studied, as well as different enrichments.…”

“…Fries and Belytschko [46] proposed a discontinuity enrichment that does not require additional degrees of freedom at nodes whose elements are crossed by the discontinuity. An approximation space consisting of discontinuity enriched moving least squares [14] is constructed near discontinuities.…”

A locally extended finite element method for the simulation of multi-fluid flows using the Particle Level Set method

“…Beside local enrichment, global one, and in particular the XFEM have been also widely used to model multi-fluid flow [17][18][19][20]. Except for the intrinsic XFEM [21], all versions of the XFEM add enrichments to the global system and therefore the graph of the system needs to be updated as the interface moves.…”

A second order virtual node method for elliptic problems with interfaces and irregular domains in three dimensions

“…However, these can have adverse effects on conditioning and require the determination of the stabilization parameters. Instead of using Lagrange multipliers or stabilization, the methods of [66,43,58,80,71,44,74] alter the basis functions to either satisfy the constraints directly, or simplify the process of doing so. In this regard, such methods represent the finite element analogues of the IIM.…”