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Well conditioned extended finite elements and vector level sets for three-dimensional crack propagation. (English) Zbl 1390.74173
Bordas, Stéphane P. A. (ed.) et al., Geometrically unfitted finite element methods and applications. Proceedings of the UCL workshop, London, UK, January, 6–8, 2016. Cham: Springer (ISBN 978-3-319-71430-1/hbk; 978-3-319-71431-8/ebook). Lecture Notes in Computational Science and Engineering 121, 307-329 (2017).
Summary: A stable extended finite element method (XFEM) is combined to a three dimensional version of the vector level set method G. Ventura et al. [Int. J. Numer. Methods Eng. 58, No. 10, 1571–1592 (2003; Zbl 1032.74687)] to solve non-planar three-dimensional (3D) crack propagation problems.
The proposed XFEM variant is based on an extension of the degree of freedom gathering technique P. Laborde et al. [Int. J. Numer. Methods Eng. 64, No. 3, 354–381 (2005; Zbl 1181.74136)] and K. Agathos et al. [Int. J. Numer. Methods Eng. 105, No. 9, 643–677 (2016)] which allows the use of geometrical enrichment in 3D without conditioning problems. The method is also combined to weight function blending and enrichment function shifting T.-P. Fries [Int. J. Numer. Methods Eng. 75, No. 5, 503–532 (2008; Zbl 1195.74173)] and G. Ventura et al. [Int. J. Numer. Methods Eng. 77, No. 1, 1–29 (2009; Zbl 1195.74201)] in order to remove blending errors and further improve conditioning. The improved conditioning results in a decrease in the number of iterations required to solve the resulting linear systems which for the cases studied ranges from 50% up to several orders of magnitude.
The propagating crack is represented using a 3D version of the level set method G. Ventura et al. [Int. J. Numer. Methods Eng. 58, No. 10, 1571–1592 (2003; Zbl 1032.74687)]. In this method at any propagation step, the crack front is represented as an ordered series of line segments and the crack surface as a sequence of four sided bilinear surfaces. Level set functions are obtained by projecting points on those surfaces and line segments thus employing only geometrical operations and avoiding the solution of differential evolution equations.
The combination of the aforementioned methods is able to handle crack propagation problems providing improved accuracy, reduced computational cost and simplified implementation.
For the entire collection see [Zbl 1392.65006].

74S05 Finite element methods applied to problems in solid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
74R10 Brittle fracture
74A45 Theories of fracture and damage
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