Béchet, E.; Minnebo, H.; Moës, N.; Burgardt, B. Improved implementation and robustness study of the X-FEM for stress analysis around cracks. (English) Zbl 1122.74499 Int. J. Numer. Methods Eng. 64, No. 8, 1033-1056 (2005). Summary: Numerical crack propagation schemes were augmented in an elegant manner by the X-FEM method. The use of special tip enrichment functions, as well as a discontinuous function along the sides of the crack allows one to do a complete crack analysis virtually without modifying the underlying mesh, which is of industrial interest, especially when a numerical model for crack propagation is desired. This paper improves the implementation of the X-FEM method for stress analysis around cracks in three ways. First, the enrichment strategy is revisited. The conventional approach uses a ‘topological’enrichment (only the elements touching the front are enriched). We suggest a ‘geomatrical’ enrichment in which a given domain size is enriched. The improvements obtained with this enrichment are discussed. Second, the conditioning of the X-FEM both for topological and geometrical enrichments is studied. A preconditioner is introduced so that ‘off the shelf’ iterative solver packages can be used and perform as well on X-FEM matrices as on standard FEM matrices. The preconditioner uses a local (nodal) Cholesky based decomposition. Third, the numerical integration scheme to build the X-FEM stiffness matrix is dramatically improved for tip enrichment functions by the use of an ad hoc integration scheme. A 2D benchmark problem is designed to show the improvements and the robustness. Cited in 1 ReviewCited in 157 Documents MSC: 74S05 Finite element methods applied to problems in solid mechanics 74R10 Brittle fracture Keywords:X-FEM; convergence rate; integration scheme; preconditioner \(J\)-integral; crack propagation PDFBibTeX XMLCite \textit{E. Béchet} et al., Int. J. Numer. Methods Eng. 64, No. 8, 1033--1056 (2005; Zbl 1122.74499) Full Text: DOI HAL References: [1] Barsoum, International Journal of Fracture 10 pp 603– (1974) · Zbl 0351.73084 [2] Babu??ka, International Journal for Numerical Methods in Engineering 40 pp 727– (1997) [3] Strouboulis, Computer Methods in Applied Mechanics and Engineering 181 pp 43– (2000) [4] Moës, International Journal for Numerical Methods in Engineering 46 pp 131– (1999) [5] Belytschko, International Journal for Numerical Methods in Engineering 44 pp 601– (1999) [6] Strouboulis, International Journal for Numerical Methods in Engineering 47 pp 1401– (2000) [7] Berntson, ACM Transactions on Mathematical Software 17 pp 437– (1991) [8] Destuynder, Journal de Mécanique Théorique et Appliquée 2 pp 113– (1983) [9] Shih, International Journal of Fracture 30 pp 79– (1986) [10] Rice, Journal of Applied Mechanics 35 pp 379– (1968) · doi:10.1115/1.3601206 [11] Moës, International Journal for Numerical Methods in Engineering 53 pp 2549– (2002) [12] Gosz, Engineering Fracture Mechanics 69 pp 299– (2002) [13] Mécanique de la rupture fragile (in French). Masson: Paris, 1978. [14] Chessa, International Journal for Numerical Methods in Engineering 57 pp 1015– (2003) [15] Contribution à l’étude des méthodes des équations intégrales et à leur mise en oeuvre numérique en élastoplasticité (in French). Thèse de doctorat, Université de Nantes, 1999. [16] Moës, Computer Methods in Applied Mechanics and Engineering 192 pp 3163– (2003) [17] Gravouil, International Journal for Numerical Methods in Engineering 53 pp 2569– (2002) [18] Laborde, International Journal for Numerical Methods in Engineering (2004) [19] , , . Convergence and conditioning issues with X-FEM in fracture mechanics. WCCM6, 5-10 September 2004, Beijing. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.