Zammarchi, M.; Fantoni, F.; Salvadori, A.; Wawrzynek, P. High order boundary and finite elements for 3D fracture propagation in brittle materials. (English) Zbl 1439.74377 Comput. Methods Appl. Mech. Eng. 315, 550-583 (2017). Summary: The quasi-static propagation of fracture in brittle materials was studied in several recent publications. A variational formulation [the third author, J. Mech. Phys. Solids 56, No. 5, 2092–2116 (2008; Zbl 1162.74439); the third author and A. Carini, Int. J. Solids Struct. 48, No. 9, 1362–1369 (2011; Zbl 1236.74260); the third and second author, “Minimum theorems in 3D incremental linear elastic fracture mechanics”, Int. J. Fract. 184, No. 1, 57–74 (2013; doi:10.1007/s10704-013-9818-3)] led to three-dimensional crack tracking strategies [the third and second author, “Weight function theory and variational formulations for three-dimensional plane elastic cracks advancing”, Int. J. Solids Struct. 51, No. 5, 1030–1045 (2014; doi:10.1016/j.ijsolstr.2013.11.029); “On a 3D crack tracking algorithm and its variational nature”, J. Eur. Ceram. Soc. 34, No. 11, 2807–2821 (2014; doi:10.1016/j.jeurceramsoc.2013.12.010); “Fracture propagation in brittle materials as a standard dissipative process: general theorems and crack tracking algorithms”, J. Mech. Phys Solids 95, 681–696 (2016; doi:10.1016/j.jmps.2016.04.034)]. One of the complexities of this new class of algorithms is the evaluation of high-order terms of the expansion of the crack opening and sliding. In this paper, new types of finite and boundary elements are formulated that capture the near crack-front asymptotical displacement behavior up to the order \(3 / 2\). The use of these elements with the quasi-static crack propagation algorithms of the above references is demonstrated for a simple crack configuration. MSC: 74R10 Brittle fracture 74S15 Boundary element methods applied to problems in solid mechanics 74S05 Finite element methods applied to problems in solid mechanics 65M38 Boundary element methods for initial value and initial-boundary value problems involving PDEs 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs Keywords:fracture mechanics; high order elements; FEM; BEM Citations:Zbl 1162.74439; Zbl 1236.74260 Software:XFEM PDFBibTeX XMLCite \textit{M. Zammarchi} et al., Comput. Methods Appl. Mech. Eng. 315, 550--583 (2017; Zbl 1439.74377) Full Text: DOI References: [1] Salvadori, A.; Fantoni, F., Minimum theorems in 3D incremental linear elastic fracture mechanics, Int. J. Fract., 184, 1, 57-74 (2013) [2] Salvadori, A.; Fantoni, F., Weight function theory and variational formulations for three-dimensional plane elastic cracks advancing, Int. J. 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