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The intrinsic XFEM: a method for arbitrary discontinuities without additional unknowns. (English) Zbl 1129.74045
Summary: We present a new method for treating arbitrary discontinuities in a finite element (FE) context. 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 mesh-based, 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.

MSC:
74S05 Finite element methods applied to problems in solid mechanics
74R10 Brittle fracture
74B05 Classical linear elasticity
Software:
XFEM
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