In the extended finite element method (X-FEM), a standard displacement based finite element approximation is enriched by additional (special) functions using the framework of partition of unity. It is a particular instance of the partition of unity finite element method (PUFEM) or the generalized finite element method (GFEM). In the X-FEM, 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 in two- and three-dimensions. The striking advantages are that the finite element framework (sparsity and symmetry of the stiffness matrix) is retained, and a single-field variational principle is used. The initial developments of the X-FEM took place at Northwestern University (NU); here's a touch of nostalgia on the NU-connection. If you're interested in learning about this method, an excellent source is the X-FEM page at Aachen.

Publications

• G. Hattori, R. Rojas-Díaz, A. Sáez, N. Sukumar and F. García-Sánchez (2012), "New Anisotropic Crack-Tip Enrichment Functions for the Extended Finite Element Method," Computational Mechanics, in press.
• S. E. Mousavi, J. E. Pask and N. Sukumar (2012), "Efficient Adaptive Integration of Functions with Sharp Gradients and Cusps in n-Dimensional Parallelepipeds," International Journal for Numerical Methods in Engineering, in press.
• R. Rojas-Díaz, N. Sukumar, A. Sáez and F. García-Sánchez (2011), "Fracture in Magnetoelectroelastic Materials using the Extended Finite Element Method," International Journal for Numerical Methods in Engineering, Vol. 88, Number 12, pp. 1238–1259.
• G. Hattori, R. Rojas-Díaz, A. Sáez, F. García-Sánchez and N. Sukumar (2011), "El Método de los Elementos Finitos Extendidos (X-FEM) para Medios Bidimensionales Fisurados Totalmente Anisótropos," Anales de Mecánica de la Fractura, Vol. 28, Number 2, pp. 451–455.
• S. E. Mousavi and N. Sukumar (2011), "Numerical Integration of Polynomials and Discontinuous Functions on Irregular Convex Polygons and Polyhedrons," Computational Mechanics, Vol. 47, Number 5, pp. 535–554.
• S. E. Mousavi, E. Grinspun and N. Sukumar (2011), "Higher-Order Extended Finite Elements with Harmonic Enrichment Functions for Complex Crack Problems," International Journal for Numerical Methods in Engineering, Vol. 86, Number 4–5, pp. 560–574.
• S. E. Mousavi, E. Grinspun and N. Sukumar (2011), "Harmonic Enrichment Functions: A Unified Treatment of Multiple, Intersecting and Branched Cracks in the Extended Finite Element Method," International Journal for Numerical Methods in Engineering, Vol. 85, Number 10, pp. 1306–1322.
• J. X. Shi, D. Chopp, J. Lua, N. Sukumar and T. Belytschko (2010), "Abaqus Implementation of Extended Finite Element using a Level Set Representation for Three-Dimensional Fatigue Crack Growth and Life Predictions," Engineering Fracture Mechanics, Vol. 77, Number 14, pp. 2840–2863. [Top 25 Hottest article (July 2010-present)]
• S. E. Mousavi and N. Sukumar (2010), "Generalized Gaussian Quadrature Rules for Discontinuities and Crack Singularities in the Extended Finite Element Method," Computer Methods in Applied Mechanics and Engineering, Vol. 199, Number 49–52, pp. 3237–3249.
• S. E. Mousavi and N. Sukumar (2010), "Generalized Duffy Transformation for Integrating Vertex Singularities," Computational Mechanics, Vol. 45, Number 2–3, pp. 127–140.
• R. Rojas-Díaz, N. Sukumar, A. Sáez and F. García-Sánchez (2009), "Crack Analysis in Magnetoelectroelastic Media using the Extended Finite Element Method," in Proceedings of the International Conference on Extended Finite Element Methods – Recent Developments and Applications, Eds. T.-P. Fries and A. Zilian, Aachen, Germany, pp. 181–186.
• W. Aquino, J. C. Brigham, C. J. Earls and N. Sukumar (2009), "Generalized Finite Element Method using Proper Orthogonal Decomposition," International Journal for Numerical Methods in Engineering, Vol. 79, Number 7, pp. 887–906.
• E. Giner, N. Sukumar, J. E. Tarancón and F. J. Fuenmayor (2009), "An Abaqus Implementation of the Extended Finite Element Method," Engineering Fracture Mechanics, Vol. 76, Number 3, pp. 347–368. Journal [Top 25 Hottest article (2009-present)]
• E. Giner, N. Sukumar, F. J. Fuenmayor and A. Vercher (2008), "Singularity Enrichment for Complete Sliding Contact using the Partition of Unity Finite Element Method," International Journal for Numerical Methods in Engineering, Vol. 76, Number 9, pp. 1402–1418.
• E. Giner, N. Sukumar, F. D. Denia and F. J. Fuenmayor (2008), "Extended Finite Element Method for Fretting Fatigue Crack Propagation," International Journal of Solids and Structures, Vol. 45, Number 22–23, pp. 5675–5687.
• J. Shi, J. Lua, H. Waisman, P. Liu, T. Belytschko, N. Sukumar and Y. Liang (2008), "X-FEM Toolkit for Automated Crack Onset and Growth Prediction," 49th AIAA Conference, Schaumburg, IL, April 2008, pp. 1–22.
• N. Sukumar, D. L. Chopp, E. Béchet and N. Moës (2008), "Three-Dimensional Non-Planar Crack Growth by a Coupled Extended Finite Element and Fast Marching Method," International Journal for Numerical Methods in Engineering, Vol. 76, Number 5, pp. 727–748.
• A. Tabarraei and N. Sukumar (2008), "Extended Finite Element Method on Polygonal and Quadtree Meshes," Computer Methods in Applied Mechanics and Engineering, Vol. 197, Number 5, pp. 425–438.
• N. Sukumar, J. Dolbow, A. Devan, J. Yvonnet, F. Chinesta, D. Ryckelynck, Ph. Lorong, I. Alfaro, M. A. Martínez, E. Cueto and M. Doblaré (2005), "Meshless Methods and Partition of Unity Finite Elements," International Journal of Forming Processes, Vol. 8, Number 4, pp. 409–427. (proof)
• Z. Huang, Z. Suo, G. Xu, J. He, J. H. Prévost and N. Sukumar (2005), "Initiation and Arrest of an Interfacial Crack in a Four-Point Bend Test," Engineering Fracture Mechanics, Vol. 72, Number 17, pp. 2584–2601.
• N. Sukumar, Z. Y. Huang, J.-H. Prévost and Z. Suo (2004), "Partition of Unity Enrichment for Bimaterial Interface Cracks," International Journal for Numerical Methods in Engineering, Vol. 59, Number 8, pp. 1075–1102. [Abstract]
• N. Sukumar and D. J. Srolovitz (2004), "Finite Element-Based Model for Crack Propagation in Polycrystalline Materials," Computational & Applied Mathematics , Vol. 23, Number 2–3, pp. 363–380
• N. Sukumar (April 2003), "Meshless Methods and Partition of Unity Finite Elements," in Proceedings of the Sixth International ESAFORM Conference on Material Forming, Ed. V. Brucato, pp. 603–606.
• T. Belytschko, C. Parimi, N. Moës, N. Sukumar and S. Usui (2003), "Structured Extended Finite Element Methods for Solids Defined by Implicit Surfaces," International Journal for Numerical Methods in Engineering, Vol. 56, Number 4, pp. 609–635.
• N. Sukumar and J.-H. Prévost (2003), "Modeling Quasi-Static Crack Growth with the Extended Finite Element Method. Part I: Computer Implementation," International Journal of Solids and Structures, Vol. 40, Number 26, pp. 7513–7537 [Abstract]
• R. Huang, N. Sukumar and J.-H. Prévost (2003), "Modeling Quasi-Static Crack Growth with the Extended Finite Element Method. Part II: Numerical Applications," International Journal of Solids and Structures, Vol. 40, Number 26, pp. 7539–7552 [Abstract]
• D. L. Chopp and N. Sukumar (2003), "Fatigue Crack Propagation of Multiple Coplanar Cracks with the Coupled Extended Finite Element/Fast Marching Method," International Journal of Engineering Science, Vol. 41, Number 8, pp. 845–869 [Abstract]
• N. Sukumar, D. L. Chopp and B. Moran (2003), "Extended Finite Element Method and Fast Marching Method for Three-Dimensional Fatigue Crack Propagation," Engineering Fracture Mechanics, Vol. 70, Number 1, pp. 29–48 [Abstract]
• N. Sukumar, D. J. Srolovitz, T. J. Baker and J.-H. Prévost (2003), "Brittle Fracture in Polycrystalline Microstructures with the Extended Finite Element Method," International Journal for Numerical Methods in Engineering, Vol. 56, Number 14, pp. 2015–2037
• N. Sukumar, D. L. Chopp, N. Moës and T. Belytschko (2001), "Modeling Holes and Inclusions by Level Sets in the Extended Finite–Element Method," Computer Methods in Applied Mechanics and Engineering, Vol. 190, Number 46–47, pp. 6183–6200 [Abstract]
• N. Moës, N. Sukumar, B. Moran and T. Belytschko (2000), "An Extended Finite Element Method (X-FEM) for Two- and Three-Dimensional Crack Modeling," in ECCOMAS 2000, Barcelona, Spain, September 11–14, 2000
• N. Sukumar, N. Moës, B. Moran and T. Belytschko (2000), "Extended Finite Element Method for Three-Dimensional Crack Modelling," International Journal for Numerical Methods in Engineering, Vol. 48, Number 11, pp. 1549–1570
• C. Daux, N. Moës, J. Dolbow, N. Sukumar and T. Belytschko (2000), "Arbitrary Branched and Intersecting Cracks with the Extended Finite Element Method," International Journal for Numerical Methods in Engineering, Vol. 48, Number 12, pp. 1741–1760

• "Efficient Numerical Integration Schemes for Singularities, Sharp gradients and Cusps: Applications in Fracture and Quantum Mechanics (with J. E. Pask and S. E. Mousavi)," Symposium on Meshfree and Extended/Generalized Finite Element Methods, PACAM XII, Port of Spain, Trinidad, January 2012.

• Invited Participant, Mini-Workshop: Mathematical Models, Analysis, and Numerical Methods for Dynamic Fracture, Mathematische Forschungsinstitut Oberwolfach, Germany, April 24-30, 2011.

• "Extended Finite Elements for Crack Modeling: Numerical Integration Issues and Harmonic Enrichment Functions," Invited Lecture, Seminar in Computational Mechanics Organized by Laboratori de Calcul Numeric, Universitat Politecnica De Catalunya, Barcelona, Spain, July 2010.

• "Extended Finite Elements using Harmonic Enrichment Functions for Crack Modeling," Invited Lecture, Universidad Politecnica de Valencia, Spain, July 2010.

• "Extended Finite Element Method for Crack Modeling: Formulation and Recent Developments," Invited Seminar, Dpto. Mecanica de Estructuras, Universidad de Granada, Spain, June 2010.

• "Extended Finite Element Method for Crack Modeling: Formulation and Recent Developments," Invited Seminar, Universidad de Sevilla, Spain, June 2010.

• "Efficient Numerical Integration for Discontinuities and Vertex Singularities in the Extended Finite Element Method," Invited Lecture, Institute of Continuum Mechanics, Leibniz University Hannover, Hannover, Germany, October 6, 2009.

• "Generalized Gaussian Quadrature Rules for Discontinuities and Crack Singularities in the Extended Finite Element Method (with S. E. Mousavi)," ECCOMAS Thematic Conference - XFEM 2009 The extended finite element method: Recent developments and applications, Aachen, Germany, September 28-30, 2009.

• "X-FEM and FMM: A Paradigm for Planar and Non-Planar Crack Growth Simulations," Invited Presentation, Global Engineering and Materials, Inc., Baltimore, MD, December 5, 2007.

• "Three-Dimensional Non-Planar Crack Growth by a Coupled Extended Finite Element and Fast Marching Method," Invited Lecture, Department of Civil and Environmental Engineering, Cornell University, Ithaca, NY, October 29, 2007.

• "Mesh-Independent Crack Modeling using the Extended Finite Element Method," Invited Presentation, Cornell Fracture Group, Cornell University, Ithaca, September 27, 2007.

• "Three-Dimensional Non-Planar Crack Growth by a Coupled Extended Finite Element and Fast Marching Method (with D. L. Chopp, N. Moës, E. Béchet)," 9th U.S. National Congress on Computational Mechanics, San Francisco, CA, July 25, 2007.

• "Meshfree Methods and Partition of Unity Finite Elements: Basic Principles and Applications," Invited Lecture, Technical University of Kaiserslautern, Germany, September 2005

• "Modeling Arbitrary Crack Discontinuities in Finite Elements," Invited Lecture, Rockwell Scientific Company, Thousand Oaks, CA, November 2004

• "Partition of Unity and Polygonal Finite Element Methods," Invited Lecture, Universidad Politecnica de Valencia, Spain, April 2004

• "Modeling Strong and Weak Discontinuities within Displacement Based Finite Elements," Invited Lecture, Department of Applied Mechanics, Indian Institute of Technology Madras, Chennai, India, March 2004

• "Modeling Arbitrary Discontinuities in Finite Elements," Invited Lecture, Department of Mechanical Engineering, Indian Institute of Technology Bombay, Mumbai, India, March 2004

• "Partition of Unity Enrichment for Bimaterial Interface Cracks," (with J. H. Prévost, Z. Suo, and Z. Y. Huang) Seventh U.S. National Congress on Computational Mechanics, Albuquerque, NM, July 2003 [PDF]

• "Finite Element-Based Model for Crack Propagation in Polycrystalline Materials" Invited Lecture, 2nd Symposium on Computational Modeling of Multi-Scale Phenomena, Petropolis, Brazil, August 2002 [PDF]

• "Modeling Discontinuities and their Evolution within Finite Elements: Application to Material Interfaces, 3-D Cracks, and Microstructures," Invited Lecture, Department of Mechanical and Aerospace Engineering, Rutgers University, Piscataway, NJ, September 2001 [PDF]

• "Microstructural Effects in Brittle Fracture with the Extended Finite Element Method," (with D. J. Srolovitz, J.-H. Prévost, and T. J. Baker) Sixth U.S. National Congress on Computational Mechanics, Dearborn, MI, August 2001 [PDF] [PS]

• "Extended Finite Element Method: Introduction and Applications," MAE Material Group Seminar, Princeton University, NJ, December 2000

• "Modeling Holes and Inclusions by Level Sets in the Extended Finite Element Method," Mechanics Club Seminar, Northwestern University (February 25, 2000) [PDF] [PS]

• Code to partition a 2D finite element that is cut by a crack

• Code to partition a 3D finite element that is cut by a crack

• Getfem++

• Basics: Configure, make, and make install

• Finite element mesh generator: gmsh

• Fatigue crack growth of a single elliptical crack (GIF) [EPS]

• Fatigue growth of multiple coplanar penny cracks

• Non-planar crack growth of an inclined penny crack in tension (small Δt)

• Non-planar crack growth of an inclined penny crack in tension (larger Δt)