Generalizations of the Finite Element Method

Marc Schweitzer

Open Mathematics (2012)

  • Volume: 10, Issue: 1, page 3-24
  • ISSN: 2391-5455

Abstract

top
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.

How to cite

top

Marc Schweitzer. "Generalizations of the Finite Element Method." Open Mathematics 10.1 (2012): 3-24. <http://eudml.org/doc/269166>.

@article{MarcSchweitzer2012,
abstract = {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.},
author = {Marc Schweitzer},
journal = {Open Mathematics},
keywords = {Generalized Finite Element Method; Extended Finite Element Method; Partition of Unity Method; generalized finite element method; extended finite element method; partition of unity method; stability; meshfree generalizations},
language = {eng},
number = {1},
pages = {3-24},
title = {Generalizations of the Finite Element Method},
url = {http://eudml.org/doc/269166},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Marc Schweitzer
TI - Generalizations of the Finite Element Method
JO - Open Mathematics
PY - 2012
VL - 10
IS - 1
SP - 3
EP - 24
AB - 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.
LA - eng
KW - Generalized Finite Element Method; Extended Finite Element Method; Partition of Unity Method; generalized finite element method; extended finite element method; partition of unity method; stability; meshfree generalizations
UR - http://eudml.org/doc/269166
ER -

References

top
  1. [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 Zbl1188.74051
  2. [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 Zbl1239.74093
  3. [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. [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 Zbl0807.65114
  5. [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 Zbl1229.65195
  6. [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 Zbl0949.65117
  7. [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 Zbl1122.74499
  8. [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. [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. [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 Zbl0891.73075
  11. [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. [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 Zbl0981.74062
  13. [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 Zbl0282.65087
  14. [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 Zbl1194.74367
  15. [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 Zbl1043.65136
  16. [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 Zbl0890.73060
  17. [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 Zbl1054.74047
  18. [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. [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 Zbl0992.65122
  20. [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 Zbl0991.65117
  21. [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 Zbl1110.74391
  22. [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 Zbl1032.76591
  23. [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 Zbl1129.65092
  24. [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 Zbl0952.65091
  25. [25] DeVore R.A., Lorentz G.G., Constructive Approximation, Grundlehren Math. Wiss., 303, Springer, Berlin, 1993 Zbl0797.41016
  26. [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 Zbl1027.74065
  27. [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. [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 Zbl1047.74056
  29. [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. [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 Zbl0918.73328
  31. [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. [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 Zbl1194.74385
  33. [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 Zbl1078.74637
  34. [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 Zbl0273.35004
  35. [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 Zbl1202.74169
  36. [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 Zbl1194.74402
  37. [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 Zbl1065.65138
  38. [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 Zbl0974.65090
  39. [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 Zbl1011.65069
  40. [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 Zbl1027.65168
  41. [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 Zbl1033.65102
  42. [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. [43] Grisvard P., Elliptic Problems in Nonsmooth Domains, Monogr. Stud. Math., 24, Pitman, Boston, 1985 Zbl0695.35060
  44. [44] Grisvard P., Singularities in Boundary Value Problems, Rech. Math. Appl., 22, Springer, Berlin, 1992 Zbl0766.35001
  45. [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 Zbl1261.76015
  46. [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 Zbl1035.65125
  47. [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. [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. [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 Zbl1181.74136
  50. [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. [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. [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 Zbl1032.74673
  53. [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 Zbl06161008
  54. [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 Zbl1082.65122
  55. [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 Zbl0881.65099
  56. [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 Zbl1217.74128
  57. [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 Zbl0955.74066
  58. [58] Mohammadi S., Extended Finite Element Method, Blackwell, Oxford, 2008 http://dx.doi.org/10.1002/9780470697795 Zbl1132.74001
  59. [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 Zbl05662215
  60. [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. [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 Zbl0229.65079
  62. [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 Zbl0976.74071
  63. [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 Zbl1180.65152
  64. [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 Zbl1156.74383
  65. [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 Zbl1202.74184
  66. [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 Zbl1216.74030
  67. [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 Zbl1157.65492
  68. [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 Zbl1016.65099
  69. [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. [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. [71] Schweitzer M.A., Generalized Finite Element and Meshfree Methods, Habilitation thesis, Univeristät Bonn, 2008 
  72. [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 Zbl1189.65284
  73. [73] Schweitzer M.A., Robust multilevel partition of unity method for problems with jumping coefficients, Technical report, Universität Stuttgart, 2011 
  74. [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 Zbl1217.65210
  75. [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 Zbl1218.65132
  76. [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. [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. [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 Zbl1113.74076
  79. [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 Zbl0983.65127
  80. [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 Zbl1120.76044
  81. [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 Zbl0997.74069
  82. [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 Zbl1169.76397
  83. [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 Zbl1054.74059
  84. [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 Zbl1059.65106
  85. [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. [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 Zbl1156.81313
  87. [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 Zbl1063.74102
  88. [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 Zbl1195.74201
  89. [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 Zbl1078.74665
  90. [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. [91] Catalogue of Finite Element Books, http://www.solid.ikp.liu.se/fe/index.html 
  92. [92] National Agency for Finite Element Methods and Standards, http://www.nafems.org 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.