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Background

A decade after its introduction, the extended finite element method (XFEM) has now become an usefull numerical approach for analysis of crack propagation problems. In the XFEM, a standard displacement based finite element approximation is enriched by additional functions using the framework of partition of unity. Besides, the finite element mesh need not conform to the internal boundaries (cracks, material interfaces, voids, etc.), and hence a single mesh suffices for modeling as well as capturing the evolution of material interfaces and cracks. The striking advantages are that the finite element framework (sparsity and symmetry of the stiffness matrix) is retained.

Some papers on XFEM

[1] Belytschko T, Black T. Elastic crack growth in finite elements with minimal remeshing. International Journal for Numerical Methods in Engineering, 1999, 45(5):601-620.
[2] Moёs N, Dolbow J, Belytschko T. A finite element method for crack growth without remeshing. International Journal for Numerical Methods in Engineering, 1999, 46:131-150.
[3] Sukumar N, Moёs N, Moran B, et al. Extended finite element method for three-dimensional crack modelling. International Journal for Numerical Methods in Engineering, 2000, 48:1549-1570.
[4] Daux C, Moёs N, Dolbow J, et al. Arbitrary branched and intersecting cracks with the extended finite element method. International Journal for Numerical Methods in Engineering, 2000, 48:1741-1760.
[5] Belytschko T, Moёs N, Usui S, et al. Arbitrary discontinuities in finite elements. International Journal for Numerical Methods in Engineering, 2001, 50:993-1013.
[6] Moёs N, Gravouil A, Belytschko T. Non-planar 3D crack growth by the extended finite element and level sets part I: mechanical model. International Journal for Numerical Methods in Engineering, 2002, 53:2549-2568.
[7] Belytschko T, Hao C H, Xu J, et al. Dynamic crack propagation based on loss of hyperbolicity and a new discontinuous enrichment. International Journal for Numerical Methods in Engineering, 2003, 58:1873-1905.
[8] Réthoré J, Gravouil A, Combescure A. An energy-conserving scheme for dynamic crack growth using the extended finite element method. International Journal for Numerical Methods in Engineering, 2005, 63:631-659.
[9] Asadpoure A, Mohammadib S, Vafai A. Crack analysis in orthotropic media using the extended finite element method. Thin-Walled Structures, 2006, 44:1031-1038.
[10] Dumstorff P, Meschke G. Crack propagation criteria in the framework of X-FEM-based structural analyses. International Journal for Numerical and Analytical Methods in Geomechanics, 2007, 31:239-259.
[11] Abdelaziz Y, Hamouine A. A survey of the extended finite element. Computers and Structures, 2008, 86:1141-1151.
[12] Fries T P, Belytschko T. The extended/generalized finite element method: an overview of the method and its applications. International Journal for Numerical Methods in Engineering, 2010, 84: 253-304.

Click here to view a full list of papers.

Some books on FEM

[1] Logan, Daryl L. 2007. A first course in the finite element method (Nelson: Toronto, Ontario, Canada).
[2] Chandrupatla, T. R., A. D. Belegundu, T. Ramesh, and C. Ray. 2012. Introduction to Finite Elements in Engineering (Prentice Hall: New Jersey).
[3] Zienkiewicz, O. C., R. L. Taylor, and J.Z. Zhu. 2013. The finite element method: its basis and fundamentals, 7th Edition (Butterworth-Heinemann: Oxford, UK).
[4] Kim, Nam-Ho. 2015. Introduction to Nonlinear Finite Element Analysis (Springer: New York).

Some books on XFEM

[1] Mohammadi, Soheil. 2012. XFEM Fracture Analysis of Composites (Wiley: West Sussex, United Kingdom).
[2] Khoei, Amir R. 2015. Extended Finite Element Method: Theory and Applications (John Wiley & Sons: West Sussex, United Kingdom).

Some books on fracture mechanics

[1] Ingraffea, A. R., and P. A. Wawrzynek. 2003. Finite Element Methods for Linear Elastic Fracture Mechanics (Elsevier: UK).
[2] Anderson, T. L. 2005. Fracture Mechanics: Fundamentals and Applicaions (Taylor & Francis Group: Boca Raton, USA).

Some books on programming

[1] Press, William H., Saul A. Teukolsky, William T. Vetterling, and Brian P. Flannery. 1992. Numerical Recipes in Fortran: The Art of Scientic Computing (Cambridge University Press: New York).
[2] Chandra, Rohit, Ramesh Menon, Leo Dagum, David Kohr, Dror Maydan, and Jeff McDonald. 2000. Parallel Programming in OpenMP (Morgan Kaufmann Publishers: Burlington, Massachusetts).
[3] Attaway, Stormy. 2013. Matlab: A Practical Introduction to Programming and Problem Solving (Butterworth-Heinemann: Boulevard, Oxford).