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Analysis and applications of a generalized finite element method with global-local enrichment functions. (English) Zbl 1169.74597
Summary: This paper presents a procedure to build enrichment functions for partition of unity methods like the generalized finite element method and the \(hp\) cloud method. The procedure combines classical global-local finite element method concepts with the partition of unity approach. It involves the solution of local boundary value problems using boundary conditions from a global problem defined on a coarse discretization. The local solutions are in turn used to enrich the global space using the partition of unity framework. The computations at local problems can be parallelized without difficulty allowing the solution of large problems very efficiently.
The effectiveness of the approach in terms of convergence rates and computational cost is investigated in this paper. We also analyze the effect of inexact boundary conditions applied to local problems and the size of the local domains on the accuracy of the enriched global solution.
Key aspects of the computational implementation, in particular, the numerical integration of generalized FEM approximations built with global-local enrichment functions, are presented.
The method is applied to fracture mechanics problems with multiple cracks in the domain. Our numerical experiments show that even on a serial computer the method is very effective and allows the solution of complex problems. Our analysis also demonstrates that the accuracy of a global problem defined on a coarse mesh can be controlled using a fixed number of global degrees of freedom and the proposed global-local enrichment functions.

MSC:
74S05 Finite element methods applied to problems in solid mechanics
74R10 Brittle fracture
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[1] Arnold, D.N.; Mukherjee, A.; Pouly, L., Locally adapted tetrahedral meshes using bisection, SIAM J. sci. comput., 22, 2, 431-448, (2000) · Zbl 0973.65116
[2] Babuška, I., The finite element method with Lagrange multipliers, Numer. math., 20, 179-192, (1973) · Zbl 0258.65108
[3] Babuška, I.; Andersson, B., The splitting method as a tool for multiple damage analysis, SIAM J. sci. comput., 26, 1114-1145, (2005) · Zbl 1149.65321
[4] Babuška, I.; Banerjee, U.; Osborn, J.E., Survey of meshless and generalized finite element methods: a unified approach, Acta numer., 12, May, 1-125, (2003) · Zbl 1048.65105
[5] Babuška, I.; Caloz, G.; Osborn, J.E., Special finite element methods for a class of second order elliptic problems with rough coefficients, SIAM J. numer. anal., 31, 4, 745-981, (1994) · Zbl 0807.65114
[6] I. Babuška, J.M. Melenk, The partition of unity finite element method, Technical Report BN-1185, Inst. for Phys. Sci. Tech., University of Maryland, June 1995.
[7] Babuška, I.; Melenk, J.M., The partition of unity finite element method, Int. J. numer. methods engrg., 40, 727-758, (1997) · Zbl 0949.65117
[8] Bansch, E., Local mesh refinement in 2 and 3 dimensions, Impact comput. sci. engrg., 3, 181-191, (1991) · Zbl 0744.65074
[9] R. Becker, P. Hansbo, A finite element method for domain decomposition with non-matching grids, Technical Report RR-3613, INRIA, 1999. · Zbl 1047.65099
[10] Belytschko, T.; Black, T., Elastic crack growth in finite elements with minimal remeshing, Int. J. numer. methods engrg., 45, 601-620, (1999) · Zbl 0943.74061
[11] Bernadi, C.; Maday, Y.; Patera, A., A new non-conforming approach to domain decomposition: the mortar element method, (), 13-51 · Zbl 0797.65094
[12] Carey, G.F.; Oden, J.T., Texas finite element series volume II - A second course, (1983), Prentice-Hall New Jersey
[13] Carmo, E.G.D.; Duarte, A.V.C., A discontinuous finite element-base domain decomposition method, Comput. methods appl. mech. engrg., 190, 825-843, (2000) · Zbl 0971.74071
[14] Diamantoudis, A.Th.; Labeas, G.N., Stress intensity factors of semi-elliptical surface cracks in pressure vessels by global – local finite element methodology, Engrg. fract. mech., 72, 1299-1312, (2005)
[15] Dolbow, J.; Moes, N.; Belytschko, T., Discontinuous enrichment in finite elements with a partition of unity method, Finite elem. anal. des., 36, 235-260, (2000) · Zbl 0981.74057
[16] C.A. Duarte, The hp Cloud Method. PhD dissertation, The University of Texas at Austin, December 1996. Austin, TX, USA.
[17] Duarte, C.A.; Babuška, I., Mesh-independent directional p-enrichment using the generalized finite element method, Int. J. numer. methods engrg., 55, 12, 1477-1492, (2002) · Zbl 1027.74065
[18] C.A. Duarte, I. Babuška, A global – local approach for the construction of enrichment functions for the generalized fem and its application to propagating three-dimensional cracks, in: V.M.A. Leitão, C.J.S. Alves, C.A. Duarte, (Eds.), ECCOMAS Thematic Conference on Meshless Methods, Lisbon, Portugal, 11-14 July 2005. p. 8.
[19] 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, vol. I, Proceedings of the International Conference on Computational Engineering Science, Atlanta, GA, October 5-9, 1998, Tech Science Press, 1998, pp. 53-58.
[20] Duarte, C.A.; Babuška, I.; Oden, J.T., Generalized finite element methods for three dimensional structural mechanics problems, Comput. struct., 77, 215-232, (2000)
[21] Duarte, C.A.; Hamzeh, O.N.; Liszka, T.J.; Tworzydlo, W.W., A generalized finite element method for the simulation of three-dimensional dynamic crack propagation, Comput. methods appl. mech. engrg., 190, 2227-2262, (2001) · Zbl 1047.74056
[22] C.A. Duarte, D.-J. Kim, I. Babuška, Chapter: A global – local approach for the construction of enrichment functions for the generalized fem and its application to three-dimensional cracks, in: V.M.A. Leitão, C.J.S. Alves, C.A. Duarte, (Eds.), Advances in Meshfree Techniques, vol. 5 Computational Methods in Applied Sciences, The Netherlands, 2007, Springer. ISBN 978-1-4020-6094-6.
[23] C.A. Duarte, L.G. Reno, A. Simone, A high-order generalized FEM for through-the-thickness branched cracks, Int. J. Numer. Methods Engrg., in Press <http://dx.doi.org/10.1002/nme.2012>. · Zbl 1194.74385
[24] C.A.M. Duarte, J.T. Oden, Hp clouds – a meshless method to solve boundary-value problems. Technical Report 95-05, TICAM, The University of Texas at Austin, May 1995.
[25] Duarte, C.A.M.; Oden, J.T., An hp adaptive method using clouds, Comput. methods appl. mech. engrg., 139, 237-262, (1996) · Zbl 0918.73328
[26] Duarte, C.A.M.; Oden, J.T., Hp clouds—an hp meshless method, Numer. methods partial differ. equat., 12, 673-705, (1996) · Zbl 0869.65069
[27] Fish, J., The s-version of the finite element method, Comput. struct., 43, 539-547, (1992) · Zbl 0775.73247
[28] Flemisch, B.; Puso, M.A.; Wohlmuth, B.I., A new dual mortar method for curved interfaces: 2d elasticity, Int. J. numer. methods engrg., 63, 813-832, (2005) · Zbl 1084.74050
[29] Grisvard, P., Singularities in boundary value problems, Research notes in applied mathematics, (1992), Springer-Verlag New York · Zbl 0766.35001
[30] Hou, T.Y.; Wu, X.-H., A multiscale finite element method for elliptic problems in composite materials and porous media, J. comput. phys., 134, 169-189, (1997) · Zbl 0880.73065
[31] Lee, S.-H.; Song, J.-H.; Yoon, Y.-C.; Zi, G.; Belytschko, T., Combined extended and superimposed finite element method for cracks, Int. J. numer. methods engrg., 59, 1119-1136, (2004) · Zbl 1041.74542
[32] 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
[33] Moes, N.; Cloirec, M.; Cartraud, P.; Remacle, J.-F., A computational approach to handle complex microstructure geometries, Comput. methods appl. mech. engrg., 192, 3163-3177, (2003) · Zbl 1054.74056
[34] Moes, N.; Dolbow, J.; Belytschko, T., A finite element method for crack growth without remeshing, Int. J. numer. methods engrg., 46, 131-150, (1999) · Zbl 0955.74066
[35] 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 engrg., 53, 2549-2568, (2002) · Zbl 1169.74621
[36] Nazarov, S.A.; Plamenevsky, B.A., Elliptic problems in domains with piecewise smooth boundaries, De gruyter expositions in mathematics, vol. 13, (1994), Walter de Gruyter Berlin · Zbl 0806.35001
[37] Noor, A.K., Global – local methodologies and their applications to nonlinear analysis, Finite elem. anal. des., 2, 333-346, (1986)
[38] Oden, J.T.; Carey, G.F., Texas finite element series volume IV - mathematical aspects, (1983), Prentice-Hall New Jersey
[39] J.T. Oden, C.A. Duarte, Chapter: Clouds, Cracks and FEM’s, in: B.D. Reddy, (Ed.), Recent Developments in Computational and Applied Mechanics, Barcelona, Spain, 1997. International Center for Numerical Methods in Engineering, CIMNE, pp. 302-321. · Zbl 0976.74071
[40] Oden, J.T.; Duarte, C.A.; Zienkiewicz, O.C., A new cloud-based hp finite element method, Comput. methods appl. mech. engrg., 153, 117-126, (1998) · Zbl 0956.74062
[41] Oden, J.T.; Duarte, C.A.M., Chapter: solution of singular problems using hp clouds, (), 35-54 · Zbl 0891.73067
[42] Park, C.; Felippa, C.A.; Rebel, G., A simple algorithm for localized construction of non-matching structural interfaces, Int. J. numer. methods engrg., 53, 2117-2142, (2002) · Zbl 1169.74653
[43] Park, J.W.; Hwang, J.W.; Kim, Y.H., Efficient finite element analysis using mesh superposition technique, Finite elem. anal. des., 39, 619-638, (2003)
[44] M.L. Parks, E. de Sturler, G. Mackey, D.D. Johnson, S. Maiti, Recycling Krylov subspaces for sequences of linear systems, Technical Report UIUCDCS-R-2004-2421, UILU-ENG-2004-1722, University of Illinois at Urbana-Champaign, Urbana, IL, March 2004. · Zbl 1123.65022
[45] J.P. Pereira, X. Jiao, C.A. Duarte, A robust geometry engine for modeling 3D crack problems with the generalized finite element method, in: Seventh World Congress on Computational Mechanics, Los Angeles, CA, USA, 16-22 July 2006. Invited abstract.
[46] Quarteroni, A.; Valli, A., Domain decomposition methods for partial differential equations, ISBN: 0198501781, (1999), Oxford University Press · Zbl 0931.65118
[47] Simone, A.; Duarte, C.A.; van der Giessen, E., A generalized finite element method for polycrystals with discontinuous grain boundaries, Int. J. numer. methods engrg., 67, 8, 1122-1145, (2006) · Zbl 1113.74076
[48] Smith, B.; Bjorstad, P.; Gropp, W., Domain decomposition: parallel multilevel methods for elliptic partial differential equations, ISBN: 0521602866, (2004), Cambridge University Press
[49] Strouboulis, T.; Babuška, I.; Copps, K., The design and analysis of the generalized finite element mehtod, Comput. methods appl. mech. engrg., 81, 1-3, 43-69, (2000) · Zbl 0983.65127
[50] 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
[51] Strouboulis, T.; Copps, K.; Babuška, I., The generalized finite element method, Comput. methods appl. mech. engrg., 190, 4081-4193, (2001) · Zbl 0997.74069
[52] Strouboulis, T.; Zhang, L.; Babuška, I., Generalized finite element method using mesh-based handbooks: application to problems in domains with many voids, Comput. methods appl. mech. engrg., 192, 3109-3161, (2003) · Zbl 1054.74059
[53] Strouboulis, T.; Zhang, L.; Babuška, I., P-version of the generalized FEM using mesh-based handbooks with applications to multiscale problems, Int. J. numer. methods engrg., 60, 1639-1672, (2004) · Zbl 1059.65106
[54] Sukumar, N.; Chopp, D.; Moes, N.; Belytschko, T., Modeling holes and inclusions by level sets in the extended finite element method, Comput. methods appl. mech. engrg., 190, 6183-6200, (2001) · Zbl 1029.74049
[55] Sukumar, N.; Moes, N.; Moran, B.; Belytschko, T., Extended finite element method for three-dimensional crack modelling, Int. J. numer. methods engrg., 48, 11, 1549-1570, (2000) · Zbl 0963.74067
[56] Szabo, B.; Babuška, I., Finite element analysis, (1991), John Wiley and Sons New York
[57] Wohlmuth, B.I., A comparison of dual Lagrange multiplier spaces for mortar finite element discretizations, Math. model. numer. anal., 36, 6, 995-1012, (2002) · Zbl 1024.65111
[58] Zienkiewicz, O.C.; Taylor, R.L., The finite element method, vol. I, (1981), McGraw-Hill · Zbl 0484.73056
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