Modeling holes and inclusions by level sets in the extended finite element method. (English) Zbl 1029.74049

Summary: We propose a methodology to model arbitrary holes and material interfaces (inclusions) without meshing the internal boundaries. The numerical method couples the level set method to the extended finite element method (X-FEM). In the X-FEM, the finite element approximation is enriched by additional functions through the notion of partition of unity. The level set method is used for representing the location of holes and material interfaces, and in addition the level set function is used to develop the local enrichment for material interfaces. Numerical examples in two-dimensional linear elastostatics are presented to demonstrate the accuracy and potential of the new technique.


74S05 Finite element methods applied to problems in solid mechanics
74B05 Classical linear elasticity
74E05 Inhomogeneity in solid mechanics
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[1] Barth, T.J.; Sethian, J.A., Numerical schemes for the hamilton – jacobi and level set equations on triangulated domains, J. comput. phys., 145, 1, 1-40, (1998) · Zbl 0911.65091
[2] Daux, C.; Moës, N.; Dolbow, J.; Sukumar, N.; Belytschko, T., Arbitrary cracks and holes with the extended finite element method, Int. J. numer. methods engrg., 48, 12, 1741-1760, (2000) · Zbl 0989.74066
[3] Dolbow, J.; Moës, N.; Belytschko, T., Modeling fracture in mindlin – reissner plates with the extended finite element method, Int. J. solids struct., 37, 48-50, 7161-7183, (2000) · Zbl 0993.74061
[4] C.A. Duarte, I. Babuška, J.T. Oden, Generalized finite element methods for three dimensional structural mechanics problems, in: S.N. Atluri, P.E. O’Donoghue (Eds.), Modeling and Simulation Based Engineering: Proceedings of the International Conference on Computational Engineering Science, vol. I, Technical Science Press, Atlanta, GA, October 1998, pp. 53-58
[5] Duarte, C.A.; Babuška, I.; Oden, J.T., Generalized finite element methods for three dimensional structural mechanics problems, Comput. struct., 77, 215-232, (2000)
[6] Krongauz, Y.; Belytschko, T., EFG approximation with discontinuous derivatives, Int. J. numer. methods engrg., 41, 7, 1215-1233, (1998) · Zbl 0906.73063
[7] Melenk, J.M.; Babuška, I., The partition of unity finite element method: basic theory and applications, Comput. methods appl. mech. engrg., 139, 289-314, (1996) · Zbl 0881.65099
[8] Moës, N.; Dolbow, J.; Belytschko, T., A finite element method for crack growth without remeshing, Int. J. numer. methods engrg., 46, 1, 131-150, (1999) · Zbl 0955.74066
[9] J.T. Oden, C.A. Duarte, O.C. Zienkiewicz, A new cloud-based hp finite element method, Technical Report TICAM Report 96-55, The University of Texas at Austin, Austin, TX, December 1996 · Zbl 0956.74062
[10] Oden, J.T.; Duarte, C.A.; Zienkiewicz, O.C., A new cloud-based hp finite element method, Comput. methods appl. mech. engrg., 153, 1-2, 117-126, (1998) · Zbl 0956.74062
[11] Osher, S.; Sethian, J.A., Fronts propagating with curvature-dependent speed: algorithms based on hamilton – jacobi formulations, J. comput. phys., 79, 1, 12-49, (1988) · Zbl 0659.65132
[12] Sethian, J.A., A marching level set method for monotonically advancing fronts, Proc. nat. acad. sci., 93, 4, 1591-1595, (1996) · Zbl 0852.65055
[13] Sethian, J.A., Level set methods and fast marching methods: evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science, (1999), Cambridge University Press Cambridge, UK · Zbl 0973.76003
[14] Strouboulis, T.; Babuška, I.; Copps, K., The design and analysis of the generalized finite element method, Comput. methods appl. mech. engrg., 181, 1-3, 43-69, (2000) · Zbl 0983.65127
[15] Strouboulis, T.; Copps, K.; Babuška, I., The generalized finite element method: an example of its implementation and illustration of its performance, Int. J. numer. methods engrg., 47, 8, 1401-1417, (2000) · Zbl 0955.65080
[16] N. Sukumar, D.L. Chopp, B. Moran, Extended finite element method and fast marching method for three dimensional fatigue crack propagation, J. Comput. Phys. (submitted) · Zbl 1211.74199
[17] Sukumar, N.; Moës, N.; Moran, B.; Belytschko, T., Extended finite element method for three-dimensional crack modeling, Int. J. numer. methods engrg., 48, 11, 1549-1570, (2000) · Zbl 0963.74067
[18] Szabó, B.; Babuška, I., Finite element analysis, (1991), Wiley New York
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