Higher-order extended FEM for weak discontinuities – level set representation, quadrature and application to magneto-mechanical problems.

*(English)*Zbl 1352.74377Summary: In this article, we present the application of bilinear and biquadratic extended FEM (XFEM) formulations to model weak discontinuities in magnetic and coupled magneto-mechanical boundary value problems. For properly resolving the location of curved interfaces and the discontinuous physical behaviour, the major part of the contribution is devoted to review and develop methods for level set representation of curved interfaces and numerical integration of the weak form in higher-order XFEM formulations. In order to reduce the complexity of the representation of curved interfaces, an element local approach that allows for an automated computation of the level set values and also improves the compatibility between the level set representation and the integration subdomains is proposed. Integration rules for polygons and strain smoothing are applied in conjunction with biquadratic elements and compared with curved integration subdomains. Eventually, a coupled magneto-mechanical demonstration problem is modelled and solved by XFEM. For demonstration purposes, a magneto-mechanical coupling due to magnetic stresses is considered. Errors and convergence rates are analysed for the different level set representations and numerical integration procedures as well as their dependence on the ratio of material parameters at an interface.

##### MSC:

74S05 | Finite element methods applied to problems in solid mechanics |

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

78M10 | Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory |

##### Keywords:

XFEM; higher order; level sets; Lasserre’s integration; strain smoothing; electromagnetics; elasticity; smart materials##### Software:

XFEM
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\textit{M. Kästner} et al., Int. J. Numer. Methods Eng. 93, No. 13, 1403--1424 (2013; Zbl 1352.74377)

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