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.

My interests are in modeling discontinuous phenomena using partition of unity methods. Initial contributions were in two- and three-dimensional crack modeling (strong discontinuities) using the X-FEM, and subsequent focus was on the modeling of material interfaces (weak discontinuities) using level sets and planar crack growth simulations by combining X-FEM and FMM. Current interests and work spans a combination of model development and application in a few distinct areas: use of the FMM to model non-planar 3D crack growth; crack propagation in polycrystalline materials; partition of unity-based finite element programming; and bone fracture to name a few.

Publications

• 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, in press.
• 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), "Inititation 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

• "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

• "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, 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)