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**Blending in the extended finite element method by discontinuous Galerkin and assumed strain methods.**
*(English)*
Zbl 1195.74175

Summary: In the extended finite element method (XFEM), errors are caused by parasitic terms in the approximation space of the blending elements at the edge of the enriched subdomain. A discontinuous Galerkin (DG) formulation is developed, which circumvents this source of error. A patch-based version of the DG formulation is developed, which decomposes the domain into enriched and unenriched subdomains. Continuity between patches is enforced with an internal penalty method. An element-based form is also developed, where each element is considered a patch. The patch-based DG is shown to have similar accuracy to the element-based DG for a given discretization but requires significantly fewer degrees of freedom. The method is applied to material interfaces, cracks and dislocation problems. For the dislocations, a contour integral form of the internal forces that only requires integration over the patch boundaries is developed. A previously developed assumed strain (AS) method is also developed further and compared with the DG method for weak discontinuities and linear elastic cracks. The DG method is shown to be significantly more accurate than the standard XFEM for a given element size and to converge optimally, even where the standard XFEM does not. The accuracy of the DG method is similar to that of the AS method but requires less application-specific coding.

### Keywords:

discontinuous Galerkin; DG; extended finite element method; XFEM; assumed strain; AS; partition of unity; PU; blending elements; dislocations; cracks; interfaces### Software:

XFEM
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\textit{R. Gracie} et al., Int. J. Numer. Methods Eng. 74, No. 11, 1645--1669 (2008; Zbl 1195.74175)

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