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2016 | OriginalPaper | Buchkapitel

Crack Lip Contact Modeling Based on Lagrangian Multipliers with X-FEM

verfasst von : Yuan Jin, Olivier Pierard, Eric Wyart, Eric Béchet

Erschienen in: Advances in Discretization Methods

Verlag: Springer International Publishing

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Abstract

The eXtended Finite Element Method (X-FEM), developed intensively in the past 15 years has become a competitive tool for the solution of problems with evolving discontinuities and singularities. In the present study, we focus on the application of X-FEM on frictionless contact problems in the context of fracture mechanics. A promising approach in the literature counting for this problem consists in applying Lagrangian multipliers. Meanwhile, as pointed out in Ji and Dolbow (Int J Numer Methods Eng 61:2508–2535, 2004), a naive choice for Lagrangian multiplier space leads to oscillatory multipliers on the contact surface. This oscillation results from a non-uniform but mesh-dependent inf-sup condition. In this work, we adapt the algorithm proposed in Béchet et al. (Int J Numer Methods Eng 78:931–954, 2009) on crack lip contact by discretizing the displacement field with both scalar and vector tip enrichment functions (Chevaugeon et al., Int J Multiscale Comput Eng 11:597–631, 2013). The influence of the tip enrichment functions on the stability of the formulation is addressed. We show evidences that the vector enrichment functions can improve the conditioning of the problem without jeopardizing the simulation accuracy in the presence of contact.

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Literatur
1.
Zurück zum Zitat Amdouni, S., Moakher, M., Renard, Y.: a stabilized lagrange multiplier method for the enriched finite-element approximation of contact problems of cracked elastic bodies. Comput. Methods Appl. Mech. Eng. 270, 178–200 (2014) Amdouni, S., Moakher, M., Renard, Y.: a stabilized lagrange multiplier method for the enriched finite-element approximation of contact problems of cracked elastic bodies. Comput. Methods Appl. Mech. Eng. 270, 178–200 (2014)
2.
Zurück zum Zitat Béchet, E., Minnebo, H., Moës, N., Burgardt, B.: Improved implementation and robustness study of the X-FEM for stress analysis around cracks. Int. J. Numer. Methods Eng. 8, 1033–1056 (2005)CrossRefMATH Béchet, E., Minnebo, H., Moës, N., Burgardt, B.: Improved implementation and robustness study of the X-FEM for stress analysis around cracks. Int. J. Numer. Methods Eng. 8, 1033–1056 (2005)CrossRefMATH
3.
Zurück zum Zitat Béchet, E., Moës, N., Wohlmuth, B.: A stable Lagrangian multiplier space for stiff interface conditions within the extended finite element method. Int. J. Numer. Methods Eng. 78, 931–954 (2009)CrossRefMATH Béchet, E., Moës, N., Wohlmuth, B.: A stable Lagrangian multiplier space for stiff interface conditions within the extended finite element method. Int. J. Numer. Methods Eng. 78, 931–954 (2009)CrossRefMATH
4.
Zurück zum Zitat Brezzi, F., Bathe, K.J.: A discourse on the stability conditions for mixed finite element formulations. Comput. Methods Appl. Mech. Eng. 53, 27–57 (1990)MathSciNetCrossRefMATH Brezzi, F., Bathe, K.J.: A discourse on the stability conditions for mixed finite element formulations. Comput. Methods Appl. Mech. Eng. 53, 27–57 (1990)MathSciNetCrossRefMATH
6.
Zurück zum Zitat Chevaugeon, N., Moës, N., Minnebo, H.: Improved crack tip enrichment functions and integration for crack modeling using the extended finite element method. Int. J. Multiscale Comput. Eng. 11, 597–631 (2013)CrossRef Chevaugeon, N., Moës, N., Minnebo, H.: Improved crack tip enrichment functions and integration for crack modeling using the extended finite element method. Int. J. Multiscale Comput. Eng. 11, 597–631 (2013)CrossRef
7.
8.
Zurück zum Zitat Ferté E., Massin, P., Moës, N.: Interface problems with quadratic X-FEM: design of a stable multiplier space and error analysis. Int. J. Numer. Methods Eng. 11, 834–870 (2014)MathSciNetCrossRef Ferté E., Massin, P., Moës, N.: Interface problems with quadratic X-FEM: design of a stable multiplier space and error analysis. Int. J. Numer. Methods Eng. 11, 834–870 (2014)MathSciNetCrossRef
9.
Zurück zum Zitat Fries, T.P.: A corrected XFEM approximation without problems in blending elements. Int. J. Numer. Methods Eng. 75, 503–532 (2008)MathSciNetCrossRefMATH Fries, T.P.: A corrected XFEM approximation without problems in blending elements. Int. J. Numer. Methods Eng. 75, 503–532 (2008)MathSciNetCrossRefMATH
10.
Zurück zum Zitat Geniaut, S., Massin, P., Moës, N.: A stable 3D contact formulation using X-FEM. Eur. J. Comput. Mech. 16, 259–275 (2007)MATH Geniaut, S., Massin, P., Moës, N.: A stable 3D contact formulation using X-FEM. Eur. J. Comput. Mech. 16, 259–275 (2007)MATH
11.
Zurück zum Zitat Giner, E., Tur, M., Tarancón, J.E., Fuenmayor, F.J.: Crack face contact in XFEM using a segment to segment approach. Int. J. Numer. Methods Eng. 82, 1424–1449 (2010)MATH Giner, E., Tur, M., Tarancón, J.E., Fuenmayor, F.J.: Crack face contact in XFEM using a segment to segment approach. Int. J. Numer. Methods Eng. 82, 1424–1449 (2010)MATH
12.
Zurück zum Zitat Gupta, V., Duarte, C.A., Babuška, I., Banerjee, U.: A stable and optimally convergent generalized FEM (SGFEM) for linear elastic fracture mechanics. Comput. Methods Appl. Mech. Eng. 266, 23–39 (2013)MathSciNetCrossRefMATH Gupta, V., Duarte, C.A., Babuška, I., Banerjee, U.: A stable and optimally convergent generalized FEM (SGFEM) for linear elastic fracture mechanics. Comput. Methods Appl. Mech. Eng. 266, 23–39 (2013)MathSciNetCrossRefMATH
13.
Zurück zum Zitat Gupta, V., Duarte, C.A., Babuška, I., Banerjee, U.: Stable GFEM (SGFEM): improved conditioning and accuracy of GFEM/XFEM for three-dimensional fracture mechanics. Comput. Methods Appl. Mech. Eng. 289, 355–386 (2015)MathSciNetCrossRef Gupta, V., Duarte, C.A., Babuška, I., Banerjee, U.: Stable GFEM (SGFEM): improved conditioning and accuracy of GFEM/XFEM for three-dimensional fracture mechanics. Comput. Methods Appl. Mech. Eng. 289, 355–386 (2015)MathSciNetCrossRef
14.
Zurück zum Zitat Ji, H., Dolbow, J.E.: On strategies for enforcing interfacial constraints and evaluating jump conditions with the extended finite element method. Int. J. Numer. Methods Eng. 61, 2508–2535 (2004)CrossRefMATH Ji, H., Dolbow, J.E.: On strategies for enforcing interfacial constraints and evaluating jump conditions with the extended finite element method. Int. J. Numer. Methods Eng. 61, 2508–2535 (2004)CrossRefMATH
15.
Zurück zum Zitat Laborde, P., Pommier, J., Renard, Y., Salaün, M.: High order extended finite element method for cracked domains. Int. J. Numer. Methods Eng. 64, 354–381 (2005)CrossRefMATH Laborde, P., Pommier, J., Renard, Y., Salaün, M.: High order extended finite element method for cracked domains. Int. J. Numer. Methods Eng. 64, 354–381 (2005)CrossRefMATH
16.
Zurück zum Zitat Li, F.Z., Shih, C.F., Needleman, A.: A comparison of methods for calculating energy release rates. Eng. Fract. Mech. 21, 405–421 (1985)CrossRef Li, F.Z., Shih, C.F., Needleman, A.: A comparison of methods for calculating energy release rates. Eng. Fract. Mech. 21, 405–421 (1985)CrossRef
17.
Zurück zum Zitat Melenk, J.M., Babuška, I.: The partition of unity finite element method: basic theory and applications. Comput. Methods Appl. Mech. Eng. 1–4, 289–314 (1996)MathSciNetCrossRefMATH Melenk, J.M., Babuška, I.: The partition of unity finite element method: basic theory and applications. Comput. Methods Appl. Mech. Eng. 1–4, 289–314 (1996)MathSciNetCrossRefMATH
18.
Zurück zum Zitat Moës, N., Dolbow, J., Belytschko, T.: A finite element method for crack growth without remeshing. Int. J. Numer. Methods Eng. 1, 131–150 (1999)CrossRefMATH Moës, N., Dolbow, J., Belytschko, T.: A finite element method for crack growth without remeshing. Int. J. Numer. Methods Eng. 1, 131–150 (1999)CrossRefMATH
19.
Zurück zum Zitat Moës, N., Gravouil, A., Belytschko, T.: Non-planar 3D crack growth by the extended finite element and level sets - Part I: mechanical model. Int. J. Numer. Methods Eng. 53, 2549–2568 (2002)CrossRefMATH Moës, N., Gravouil, A., Belytschko, T.: Non-planar 3D crack growth by the extended finite element and level sets - Part I: mechanical model. Int. J. Numer. Methods Eng. 53, 2549–2568 (2002)CrossRefMATH
20.
Zurück zum Zitat Moës, N., Béchet, E., Tourbier, M.: Imposing Dirichlet boundary conditions in the extended finite element method. Int. J. Numer. Methods Eng. 67, 1641–1669 (2005)MathSciNetCrossRefMATH Moës, N., Béchet, E., Tourbier, M.: Imposing Dirichlet boundary conditions in the extended finite element method. Int. J. Numer. Methods Eng. 67, 1641–1669 (2005)MathSciNetCrossRefMATH
21.
Zurück zum Zitat Moran, B., Shih, C.F.: Crack tip and associated domain integrals from momentum and energy balance. Eng. Fract. Mech. 27, 615–641 (1987)CrossRef Moran, B., Shih, C.F.: Crack tip and associated domain integrals from momentum and energy balance. Eng. Fract. Mech. 27, 615–641 (1987)CrossRef
22.
Zurück zum Zitat Moran, B., Shih, C.F.: A general treatment of crack tip contour integrals. Int. J. Fract. 35, 295–310 (1987)CrossRef Moran, B., Shih, C.F.: A general treatment of crack tip contour integrals. Int. J. Fract. 35, 295–310 (1987)CrossRef
23.
Zurück zum Zitat Pereira, J.P., Duarte, C.A., Jiao, X.: Three-dimensional crack growth with hp-generalized finite element and face offsetting methods. Comput. Mech. 46, 431–453 (2010)MathSciNetCrossRefMATH Pereira, J.P., Duarte, C.A., Jiao, X.: Three-dimensional crack growth with hp-generalized finite element and face offsetting methods. Comput. Mech. 46, 431–453 (2010)MathSciNetCrossRefMATH
24.
Zurück zum Zitat Shih, C.F., Moran, B., Nakamura, T.: Energy release rate along a three-dimensional crack front in a thermally stressed body. Int. J. Fract. 30, 79–102 (1986) Shih, C.F., Moran, B., Nakamura, T.: Energy release rate along a three-dimensional crack front in a thermally stressed body. Int. J. Fract. 30, 79–102 (1986)
25.
Zurück zum Zitat Siavelis, M., Guiton, M.L.E., Massin, P., Moës, N.: Large sliding contact along branched discontinuities with X-FEM. Comput. Mech. 1, 201–219 (2012)MathSciNetMATH Siavelis, M., Guiton, M.L.E., Massin, P., Moës, N.: Large sliding contact along branched discontinuities with X-FEM. Comput. Mech. 1, 201–219 (2012)MathSciNetMATH
26.
Zurück zum Zitat Stolarska, M., Chopp, D.L., Moës, N., Belyschko T.: Modelling crack growth by level sets in the extended finite element method. Int. J. Numer. Methods Eng. 51, 943–960 (2001)CrossRefMATH Stolarska, M., Chopp, D.L., Moës, N., Belyschko T.: Modelling crack growth by level sets in the extended finite element method. Int. J. Numer. Methods Eng. 51, 943–960 (2001)CrossRefMATH
27.
Zurück zum Zitat Wyart, E.: Three-dimensional crack analysis in aeronautical structures using the substructured finite element/extended finite element method. Ph.D Thesis, Universié catholique de Louvain Faculté des Sciences Appliquées (2007) Wyart, E.: Three-dimensional crack analysis in aeronautical structures using the substructured finite element/extended finite element method. Ph.D Thesis, Universié catholique de Louvain Faculté des Sciences Appliquées (2007)
28.
Zurück zum Zitat Yau, J.F., Wang, S.S., Corten, H.T.: A mixed-mode crack analysis of isotropic solids using conservation laws of elasticity. J. Appl. Mech. 47, 335–341 (1980)CrossRefMATH Yau, J.F., Wang, S.S., Corten, H.T.: A mixed-mode crack analysis of isotropic solids using conservation laws of elasticity. J. Appl. Mech. 47, 335–341 (1980)CrossRefMATH
Metadaten
Titel
Crack Lip Contact Modeling Based on Lagrangian Multipliers with X-FEM
verfasst von
Yuan Jin
Olivier Pierard
Eric Wyart
Eric Béchet
Copyright-Jahr
2016
DOI
https://doi.org/10.1007/978-3-319-41246-7_6

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