Generalizations of the Finite Element Method

Marc Schweitzer

Open Mathematics, Tome 10 (2012), p. 3-24 / Harvested from The Polish Digital Mathematics Library

Résumé

This paper is concerned with the generalization of the finite element method via the use of non-polynomial enrichment functions. Several methods employ this general approach, e.g. the extended finite element method and the generalized finite element method. We review these approaches and interpret them in the more general framework of the partition of unity method. Here we focus on fundamental construction principles, approximation properties and stability of the respective numerical method. To this end, we consider meshbased and meshfree generalizations of the finite element method and the use of smooth, discontinuous, singular and numerical enrichment functions.

Détail

Publié le : 2012-01-01

Zbl 1245.65159

EUDML-ID : urn:eudml:doc:269166

Bibliographie

[1] Aragón A.M., Duarte C.A., Geubelle P.H., Generalized finite element enrichment functions for discontinuous gradient fields, Internat. J. Numer. Methods Engrg., 2010, 82(2), 242–268 | Zbl 1188.74051

[2] Babuška I., Banerjee U., Stable generalized finite element method, Comput. Methods Appl. Mech. Engrg. (in press), DOI: 10.1016/j.cma.2011.09.012 | Zbl 1239.74093

[3] Babuška I., Banerjee U., Osborn J.E., Meshless and generalized finite element methods: a survey of some major results, In: Meshfree Methods for Partial Differential Equations, Bonn, 2001, Lect. Notes Comput. Sci. Eng., 26, Springer, Berlin, 2003, 1–20 http://dx.doi.org/10.1007/978-3-642-56103-0_1

[4] 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., 1994, 31(4), 945–981 http://dx.doi.org/10.1137/0731051 | Zbl 0807.65114

[5] Babuška I., Lipton R., Optimal local approximation spaces for generalized finite element methods with application to multiscale problems, Multiscale Model. Simul., 2011, 9(1), 373–406 http://dx.doi.org/10.1137/100791051 | Zbl 1229.65195

[6] Babuška I., Melenk J.M., The partition of unity method, Internat. J. Numer. Methods Engrg., 1997, 40(4), 727–758 http://dx.doi.org/10.1002/(SICI)1097-0207(19970228)40:4<727::AID-NME86>3.0.CO;2-N | Zbl 0949.65117

[7] Béchet E., Minnebo H., Moës N., Burgardt B., Improved implementation and robustness study of the X-FEM for stress analysis around cracks, Internat. J. Numer. Methods Engrg., 2005, 64(8), 1033–1056 http://dx.doi.org/10.1002/nme.1386 | Zbl 1122.74499

[8] Belytschko T., Black T., Elastic crack growth in finite elements with minimal remeshing, Internat. J. Numer. Methods Engrg., 1999, 45(5), 601–620 http://dx.doi.org/10.1002/(SICI)1097-0207(19990620)45:5<601::AID-NME598>3.0.CO;2-S

[9] Belytschko T., Gracie R., Ventura G., A review of extended/generalized finite element methods for material modelling, Modelling and Simulation in Materials Science and Engineering, 2009, 17(4), #043001

[10] Belytschko T., Krongauz Y., Organ D., Fleming M., Krysl P., Meshless methods: an overview and recent developments, Comput. Methods Appl. Mech. Engrg., 1996, 139(1–4), 3–47 http://dx.doi.org/10.1016/S0045-7825(96)01078-X | Zbl 0891.73075

[11] Belytschko T., Lu Y.Y., Gu L., Crack Propagation by Element-free Galerkin methods, Engrg. Fracture Mech., 1995, 51(2), 295–315 http://dx.doi.org/10.1016/0013-7944(94)00153-9

[12] Belytschko T., Moës N., Usui S., Parimi C., Arbitrary discontinuities in finite elements, Internat. J. Numer. Methods Engrg., 2001, 50(4), 993–1013 http://dx.doi.org/10.1002/1097-0207(20010210)50:4<993::AID-NME164>3.0.CO;2-M | Zbl 0981.74062

[13] Benzley S.E., Representation of singularities with isoparametric finite elements, Internat. J. Numer. Methods Engrg., 1974, 8(3), 537–545 http://dx.doi.org/10.1002/nme.1620080310 | Zbl 0282.65087

[14] Bordas S., Nguyen P.V., Dunant C., Guidom A., Nguyen-Dang H., An extended finite element library, Internat. J. Numer. Methods Engrg., 2006, 71(6), 703–732 http://dx.doi.org/10.1002/nme.1966 | Zbl 1194.74367

[15] Brenner S.C., Multigrid methods for the computation of singular solutions and stress intensity factors I: Corner singularities, Math. Comp., 1999, 68(266), 559–583 http://dx.doi.org/10.1090/S0025-5718-99-01017-0 | Zbl 1043.65136

[16] Brenner S.C., Sung L.-Y., Multigrid method for the computation of singular solutions and stress intensity factors II: Crack singularities, BIT, 1997, 37(3), 623–643 http://dx.doi.org/10.1007/BF02510243 | Zbl 0890.73060

[17] Brenner S.C., Sung L., Multigrid methods for the computation of singular solutions and stress intensity factors III: Interface singularities, Comput. Methods Appl. Mech. Engrg., 2003, 192(41–42), 4687–4702 http://dx.doi.org/10.1016/S0045-7825(03)00455-9 | Zbl 1054.74047

[18] Byskov E., The calculation of stress intensity factors using the finite element method with cracked elements, Internat. J. Fracture, 1970, 6(2), 159–167

[19] Cai Z., Kim S., A finite element method using singular functions for the poisson equation: corner singularities, SIAM J. Numer. Anal., 2001, 39(1), 286–299 http://dx.doi.org/10.1137/S0036142999355945 | Zbl 0992.65122

[20] Cai Z., Kim S., Shin B.-C., Solution methods for the Poisson equation with corner singularities: numerical results, SIAM J. Sci. Comput., 2001, 23(2), 672–682 http://dx.doi.org/10.1137/S1064827500372778 | Zbl 0991.65117

[21] Chessa J., Belytschko T., An extended finite element method for two-phase fluids, Trans. ASME J. Appl. Mech., 2003, 70(1), 10–17 http://dx.doi.org/10.1115/1.1526599 | Zbl 1110.74391

[22] Chessa J., Belytschko T., An enriched finite element method and level sets for axisymmetric two-phase flow with surface tension, Internat. J. Numer. Methods Engrg., 2003, 58(13), 2041–2064 http://dx.doi.org/10.1002/nme.946 | Zbl 1032.76591

[23] Dahmen W., Dekel S., Petrushev P., Multilevel preconditioning for partition of unity methods: some analytic concepts, Numer. Math., 2007, 107(3), 503–532 http://dx.doi.org/10.1007/s00211-007-0089-7 | Zbl 1129.65092

[24] De S., Bathe K.J., The method of finite spheres, Comput. Mech., 2000, 25(4), 329–345 http://dx.doi.org/10.1007/s004660050481 | Zbl 0952.65091

[25] DeVore R.A., Lorentz G.G., Constructive Approximation, Grundlehren Math. Wiss., 303, Springer, Berlin, 1993 | Zbl 0797.41016

[26] Duarte C.A., Babuška I., Mesh-independent p-orthotropic enrichment using the generalized finite element method, Internat. J. Numer. Methods Engrg., 2002, 55(12), 1477–1492 http://dx.doi.org/10.1002/nme.557 | Zbl 1027.74065

[27] Duarte C.A., Babuška I., Oden J.T., Generalized finite element methods for three-dimensional structural mechanics problems, Comput. & Structures, 2000, 77(2), 215–232 http://dx.doi.org/10.1016/S0045-7949(99)00211-4

[28] 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, Internat. J. Numer. Methods Engrg., 2001, 190(15–17), 2227–2262 | Zbl 1047.74056

[29] Duarte C.A., Kim D.-J., Analysis and applications of a generalized finite element method with global-local enrichment functions, Comput. Methods Appl. Mech. Engrg., 2007, 197(6–8), 487–504

[30] Duarte C.A., Oden J.T., An h-p adaptive method using clouds, Comput. Methods Appl. Mech. Engrg., 1996, 139(1–4), 237–262 http://dx.doi.org/10.1016/S0045-7825(96)01085-7 | Zbl 0918.73328

[31] Duarte C.A., Oden J.T., H-p clouds ¶ an h-p meshless method, Numer. Methods Partial Differential Equations, 1996, 12(6), 673–705 http://dx.doi.org/10.1002/(SICI)1098-2426(199611)12:6<673::AID-NUM3>3.0.CO;2-P

[32] Duarte C.A., Reno L.G., Simone A., A high-order generalized FEM for through-the-thickness branched cracks, Internat. J. Numer. Methods Engrg., 2007, 72(3), 325–351 http://dx.doi.org/10.1002/nme.2012 | Zbl 1194.74385

[33] Fish J., Yuan Z., Multiscale enrichment based on partition of unity, Internat. J. Numer. Methods Engrg., 2005, 62(10), 1341–1359 http://dx.doi.org/10.1002/nme.1230 | Zbl 1078.74637

[34] Fix G.J., Gulati S., Wakoff G.I., On the use of singular functions with finite element approximations, J. Comput. Phys., 1973, 13(2), 209–228 http://dx.doi.org/10.1016/0021-9991(73)90023-5 | Zbl 0273.35004

[35] Fries T.-P., Belytschko T., The extended/generalized finite element method: an overview of the method and its applications, Internat. J. Numer. Methods Engrg., 2010, 84(3), 253–304 | Zbl 1202.74169

[36] Gracie R., Ventura G., Belytschko T., A new fast finite element method for dislocations based on interior dicontinuities, Internat. J. Numer. Methods Engrg., 2007, 69(2), 423–441 http://dx.doi.org/10.1002/nme.1896 | Zbl 1194.74402

[37] Griebel M., Oswald P., Schweitzer M.A., A particle-partition of unity method. VI. A p-robust multilevel solver, In: Meshfree Methods for Partial Differential Equations II, Lect. Notes Comput. Sci. Eng., 43, Springer, Berlin, 2005, 71–92 http://dx.doi.org/10.1007/3-540-27099-X_5 | Zbl 1065.65138

[38] Griebel M., Schweitzer M.A., A particle-partition of unity method for the solution of elliptic, parabolic, and hyperbolic PDEs, SIAM J. Sci. Comput., 2000, 22(3), 853–890 http://dx.doi.org/10.1137/S1064827599355840 | Zbl 0974.65090

[39] Griebel M., Schweitzer M.A., A particle-partition of unity method. II. Efficient cover construction and reliable integration, SIAM J. Sci. Comput., 2002, 23(5), 1655–1682 http://dx.doi.org/10.1137/S1064827501391588 | Zbl 1011.65069

[40] Griebel M., Schweitzer M.A., A particle-partition of unity method. III. A multilevel solver, SIAM J. Sci. Comput., 2002, 24(2), 377–409 http://dx.doi.org/10.1137/S1064827501395252 | Zbl 1027.65168

[41] Griebel M., Schweitzer M.A., A particle-partition of unity method. V. Boundary conditions, In: Geometric Analysis and Nonlinear Partial Differential Equations, Springer, Berlin, 2003, 519–542 | Zbl 1033.65102

[42] Griebel M., Schweitzer M.A., A particle-partition of unity method. VII. Adaptivity, In: Meshfree Methods for Partial Differential Equations III, Lect. Notes Comput. Sci. Eng., 57, Springer, Berlin, 2007, 121–147 http://dx.doi.org/10.1007/978-3-540-46222-4_8

[43] Grisvard P., Elliptic Problems in Nonsmooth Domains, Monogr. Stud. Math., 24, Pitman, Boston, 1985 | Zbl 0695.35060

[44] Grisvard P., Singularities in Boundary Value Problems, Rech. Math. Appl., 22, Springer, Berlin, 1992 | Zbl 0766.35001

[45] Groß S., Reusken A., An extended pressure finite element space for two-phase incompressible flows with surface tension, J. Comput. Phys., 2007, 224(1), 40–58 http://dx.doi.org/10.1016/j.jcp.2006.12.021 | Zbl 1261.76015

[46] Hansbo A., Hansbo P., An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems, Comput. Methods Appl. Mech. Engrg., 2002, 191(47–48), 5537–5552 http://dx.doi.org/10.1016/S0045-7825(02)00524-8 | Zbl 1035.65125

[47] Huerta A., Belytschko T., Fernández-Méndez S., Rabczuk T., Meshfree methods, In: Encyclopedia of Computational Mechanics, 1, John Wiley & Sons, Chichester, 2004, chapter 10, 279–309

[48] Kim D.-J., Duarte C.A., Proença S.P., Generalized Finite Element Method with global-local enrichments for nonlinear fracture analysis, In: Mechanics of Solids in Brazil 2009, Rio de Janeiro, April 28–30, 2009, ABCM Symposium Series in Solid Mechanics, 2, Brazilian Society of Mechanical Sciences and Engineering, 2009, 317–330

[49] Laborde P., Pommier J., Renard Y., Salaün M., Higher order extended finite element method for cracked domains, Internat. J. Numer. Methods Engrg., 2005, 64(3), 354–381 http://dx.doi.org/10.1002/nme.1370 | Zbl 1181.74136

[50] Li H., A note on the conditioning of a class of generalized finite element methods, Appl. Numer. Math. (in press), DOI: 10.1016/j.apnum.2011.05.004

[51] Macri M., De S., Shepard M.S., Hierarchical tree-based discretization for the method of finite spheres, Comput. & Structures, 2003, 81(8–11), 789–803 http://dx.doi.org/10.1016/S0045-7949(02)00475-3

[52] Mariani S., Perego U., Extended finite element method for quasi-brittle fracture, Internat. J. Numer. Methods Engrg., 2003, 58(1), 103–126 http://dx.doi.org/10.1002/nme.761 | Zbl 1032.74673

[53] Mazzucato A.L., Nistor V., Qu Q., A non-conforming generalized finite element method for transmission problems, preprint available at http://www.math.psu.edu/mazzucat/preprint/GFEM.pdf | Zbl 06161008

[54] Melenk J.M., On approximation in meshless methods, In: Frontiers of Numerical Analysis, Durham, July 4–9, 2004, Universitext, Springer, Berlin, 2005, 65–141 | Zbl 1082.65122

[55] Melenk J.M., Babuška I., The partition of unity finite element method: basic theory and applications, Comput. Methods Appl. Mech. Engrg., 1996, 139(1–4), 289–314 http://dx.doi.org/10.1016/S0045-7825(96)01087-0 | Zbl 0881.65099

[56] Menk A., Bordas S.P.A., A robust preconditioning technique for the extended finite element method, Internat. J. Numer. Methods Engrg., 2011, 85(13), 1609–1632 http://dx.doi.org/10.1002/nme.3032 | Zbl 1217.74128

[57] Moës N., Dolbow J., Belytschko T., A finite element method for crack growth without remeshing, Internat. J. Numer. Methods Engrg., 1999, 46(1), 131–150 http://dx.doi.org/10.1002/(SICI)1097-0207(19990910)46:1<131::AID-NME726>3.0.CO;2-J | Zbl 0955.74066

[58] Mohammadi S., Extended Finite Element Method, Blackwell, Oxford, 2008 http://dx.doi.org/10.1002/9780470697795 | Zbl 1132.74001

[59] Mousavi S.E., Sukumar N., Generalized Duffy transformation for integrating vertex singularities, Comput. Mech., 2010, 45(2–3), 127–140 http://dx.doi.org/10.1007/s00466-009-0424-1 | Zbl 05662215

[60] Mousavi S.E., Sukumar N., Generalized Gaussian quadrature rules for discontinuities and crack singularities in the extended finite element method, Comput. Methods Appl. Mech. Engrg., 2010, 199(49–52), 3237–3249 http://dx.doi.org/10.1016/j.cma.2010.06.031

[61] Nitsche J., Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind, Abh. Math. Sem. Univ. Hamburg, 1971, 36, 9–15 http://dx.doi.org/10.1007/BF02995904 | Zbl 0229.65079

[62] Oden J.T., Duarte C.A., Clouds, Cracks and FEM’s, In: Recent Developments in Computational and Applied Mechanics, International Center for Numerical Methods in Engineering, CIMNE, Barcelona, 1997, 302–321 | Zbl 0976.74071

[63] Oh H.-S., Jae J.W., Hong W.T., The generalized product partition of unity for the meshless methods, J. Comput. Phys., 2010, 229(5), 1600–1620 http://dx.doi.org/10.1016/j.jcp.2009.10.047 | Zbl 1180.65152

[64] Pereira J.P., Duarte C.A., Guoy D., Jiao X., hp-generalized FEM and crack surface representation for non-planar 3-D cracks, Internat. J. Numer. Methods Engrg., 2009, 77(5), 601–633 http://dx.doi.org/10.1002/nme.2419 | Zbl 1156.74383

[65] Radtke F.K.F., Simone A., Sluys L.J., A partition of unity finite element method for obtaining elastic properties of continua with embedded thin fibres, Internat. J. Numer. Methods Engrg., 2010, 84(6), 708–732 http://dx.doi.org/10.1002/nme.2916 | Zbl 1202.74184

[66] Radtke F.K.F., Simone A., Sluys L.J., A partition of unity finite element method for simulating non-linear debonding and matrix failure in thin fibre composites, Internat. J. Numer. Methods Engrg., 2011, 86(4–5), 453–476 http://dx.doi.org/10.1002/nme.3056 | Zbl 1216.74030

[67] Riker C., Holzer S.M., The mixed-cell-complex partition-of-unity method, Comput. Methods Appl. Mech. Engrg., 2009, 198(13–14), 1235–1248 http://dx.doi.org/10.1016/j.cma.2008.04.026 | Zbl 1157.65492

[68] Schweitzer M.A., A Parallel Multilevel Partition of Unity Method for Elliptic Partial Differential Equations, Lect. Notes Comput. Sci. Eng., 29, Springer, Berlin, 2003 http://dx.doi.org/10.1007/978-3-642-59325-3 | Zbl 1016.65099

[69] Schweitzer M.A., Efficient implementation and parallelization of meshfree and particle methods ¶ the parallel multilevel partition of unity method, In: Frontiers of Numerical Analysis, Durham, July 4–9, 2004, Universitext, Springer, Berlin, 2005, 195–262

[70] Schweitzer M.A., A particle-partition of unity method. VIII. Hierarchical enrichment, In: Meshfree Methods for Partial Differential Equations IV, Lect. Notes Comput. Sci. Eng., 65, Springer, Berlin, 2008, 279–302 http://dx.doi.org/10.1007/978-3-540-79994-8_16

[71] Schweitzer M.A., Generalized Finite Element and Meshfree Methods, Habilitation thesis, Univeristät Bonn, 2008

[72] Schweitzer M.A., An algebraic treatment of essential boundary conditions in the particle-partition of unity method, SIAM J. Sci. Comput., 2008/09, 31(2), 1581–1602 http://dx.doi.org/10.1137/080716499 | Zbl 1189.65284

[73] Schweitzer M.A., Robust multilevel partition of unity method for problems with jumping coefficients, Technical report, Universität Stuttgart, 2011

[74] Schweitzer M.A., Stable enrichment and local preconditioning in the particle-partition of unity method, Numer. Math., 2011, 118(1), 137–170 http://dx.doi.org/10.1007/s00211-010-0323-6 | Zbl 1217.65210

[75] Schweitzer M.A., Multilevel particle-partition of unity method, Numer. Math., 2011, 118(2), 307–328 http://dx.doi.org/10.1007/s00211-010-0346-z | Zbl 1218.65132

[76] Shabir Z., Van der Giessen E., Duarte C.A., Simone A., The role of cohesive properties on intergranular crack propagation in brittle polycrystals, Modelling and Simulation in Materials Science and Engineering, 2011, 19(3), #035006 http://dx.doi.org/10.1088/0965-0393/19/3/035006

[77] Shepard D., A two-dimensional interpolation function for irregularly-spaced data, In: Proceedings of the 1968 23rd ACM National Conference, Association for Computing Machinery, New York, 1968, 517–524 http://dx.doi.org/10.1145/800186.810616

[78] Simone A., Duarte C.A., Van der Giessen E., A generalized finite element method for polycrystals with discontinuous grain boundaries, Internat. J. Numer. Methods Engrg., 2006, 67(8), 1122–1145 http://dx.doi.org/10.1002/nme.1658 | Zbl 1113.74076

[79] Strouboulis T., Babuška I., Copps K., The design and analysis of the generalized finite element method, Comput. Methods Appl. Mech. Engrg., 2000, 181(1–3), 43–69 http://dx.doi.org/10.1016/S0045-7825(99)00072-9 | Zbl 0983.65127

[80] Strouboulis T., Babuška I., Hidajat R., The generalized finite element method for Helmholtz equation: theory, computation, and open problems, Comput. Methods Appl. Mech. Engrg., 2006, 195(37–40), 4711–4731 http://dx.doi.org/10.1016/j.cma.2005.09.019 | Zbl 1120.76044

[81] Strouboulis T., Copps K., Babuška I., The generalized finite element method, Comput. Methods Appl. Mech. Engrg., 2001, 190(32–33), 4081–4193 http://dx.doi.org/10.1016/S0045-7825(01)00188-8 | Zbl 0997.74069

[82] Strouboulis T., Hidajat R., Babuška I., The generalized finite element method for Helmholtz equation. II. Effect of choice of handbook functions, error due to absorbing boundary conditions and its assessment, Comput. Meth. Appl. Mech. Engrg., 2008, 197(5), 364–380 http://dx.doi.org/10.1016/j.cma.2007.05.019 | Zbl 1169.76397

[83] 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., 2003, 192(28–30), 3109–3161 http://dx.doi.org/10.1016/S0045-7825(03)00347-5 | Zbl 1054.74059

[84] Strouboulis T., Zhang L., Babuška I., p-version of the generalized FEM using mesh-based handbooks with applications to multiscale problems, Internat. J. Numer. Methods Engrg., 2004, 60(10), 1639–1672 http://dx.doi.org/10.1002/nme.1017 | Zbl 1059.65106

[85] Stüben K., An introduction to algebraic multigrid, An appendix to: Trottenberg U., Oosterlee C.W., Schüller A., Multigrid, Academic Press, San Diego, 2001, 413–532

[86] Sukumar N., Pask J.E., Classical and enriched finite element formulations for Bloch-periodic boundary conditions, Internat. J. Numer. Methods Engrg., 2009, 77(8), 1121–1138 http://dx.doi.org/10.1002/nme.2457 | Zbl 1156.81313

[87] Sukumar N., Prévost J.-H., Modeling quasi-static crack growth with the extended finite element method. I. Computer implementation, Internat. J. Solids Structures, 2003, 40(26), 7513–7537 http://dx.doi.org/10.1016/j.ijsolstr.2003.08.002 | Zbl 1063.74102

[88] Ventura G., Gracie R., Belytschko T., Fast integration and weight function blending in the extended finite element method, Internat. J. Numer. Methods Engrg., 2009, 77(1), 1–29 http://dx.doi.org/10.1002/nme.2387 | Zbl 1195.74201

[89] Ventura G., Moran B., Belytschko T., Dislocations by partition of unity, Internat. J. Numer. Methods Engrg., 2005, 62(11), 1463–1487 http://dx.doi.org/10.1002/nme.1233 | Zbl 1078.74665

[90] Xu J., Zikatanov L.T., On multigrid methods for generalized finite element methods, In: Meshfree Methods for Partial Differential Equations, Bonn, 2001, Lect. Notes Comput. Sci. Eng., 26, Springer, Berlin, 2003, 401–418 http://dx.doi.org/10.1007/978-3-642-56103-0_28

[91] Catalogue of Finite Element Books, http://www.solid.ikp.liu.se/fe/index.html

[92] National Agency for Finite Element Methods and Standards, http://www.nafems.org