Geodesic
Utilization of closest point projection surface representation in extended finite element method
Matematičeskoe modelirovanie, Tome 31 (2019) no. 6, pp. 18-42.

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Currently X-FEM approach is widely used computational technique for solution of solid mechanics problems given a number of large scale fractures. Its main advantage is a possibility to use computational meshes which do not fit fractures midsurface geometry and possibility for an exact accounting for singular asymptotic of the solution in the neighborhood of the fractures fronts. One of the key components of the method is representation of the fractures midsurface in the algorithm. Usually implicit representation is used based on the level set method. Such approach is efficient and robust and allows to perform simulation in the presence with evolutionary fractures. In this paper we present a variant of the X-FEM approach which utilizes fracture midsurface representation based on closest point projection method and provides, up to authors’ opinion, certain benefits comparably to the classical X-FEM. We provide a short overview of the X-FEM technique. The proposed algorithm is described in details and its advantages are formulated. We consider questions of numerical integration of function defined at closest point projection surfaces, local recovery of the level set function and computation of local basises in the neighborhood of the points at the surface and its boundary. Algorithmic details of the proposed method are described. Finally we present some numerical results which demonstrate application of the proposed algorithm and its capabilities.
Keywords: hydraulic fracture problem, poroelastic medium, incomplete coupling principle. 
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     author = {E. B. Savenkov and V. E. Borisov and B. V. Kritsky},
     title = {Utilization of closest point projection surface representation in extended finite element method},
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E. B. Savenkov; V. E. Borisov; B. V. Kritsky. Utilization of closest point projection surface representation in extended finite element method. Matematičeskoe modelirovanie, Tome 31 (2019) no. 6, pp. 18-42. http://geodesic.mathdoc.fr/item/MM_2019_31_6_a1/

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