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Efficient modeling of internal cracks for Laplace problem by XFEM using Joukowski mapping

: Nakasumi, Shogo; Schweitzer, Marc Alexander


International journal for numerical methods in engineering 119 (2019), No.1, pp.1-17
ISSN: 0029-5981
Journal Article
Fraunhofer SCAI ()
Efficient modeling; Laplace problem; XFEM; elliptic; extended finite element method; finite element methods; inverse problem; partial differential equations; partition-of-unity; potential flow

In this study, the extended finite element method (XFEM) is applied to the two-dimensional Laplace equation with an internal discontinuity. The real part of a complex velocity potential from potential flow theory is used to repre- sent the enrichment function in this technique. The Joukowski mapping, which maps a circle to a line, is mainly used to obtain a solution around an airfoil in two-dimensional potential flow; here, we extend that solution to model magnetic flux around an internal crack. The effectiveness of the proposed method is veri- fied using numerical examples of single and multiple cracks. The L 2 error norm is used to evaluate the accuracy of the proposed method in comparison with XFEM using previously proposed enrichment functions (Heaviside and analyt- ical f orms for a single crack tip). The proposed method gives better results than those of the existing XFEM in the case of a coarse mesh.