Higher order stable generalized finite element method (original) (raw)

References

  1. Abbas, S., Alizada, A., Fries, T.P.: The xfem for high-gradient solutions in convection-dominated problems. Int. J. Numer. Methods Eng. 82, 1044–1072 (2010)
    Article MATH MathSciNet Google Scholar
  2. Aragón, A.M., Duarte, C.A., Geubelle, P.H.: Generalized finite element enrichment functions for discontinuous gradient field. Int. J. Numer. Methods Eng. 10, 1–6 (2008)
    Google Scholar
  3. Babuška, I., Banerjee, U.: Stable generalized finite element method. Comput. Methods Appl. Mech. Eng. 201–204, 91–111 (2011)
    Google Scholar
  4. Babuška, I., Banerjee, U., Osborn, J.: Survey of meshless and generalized finite element methods. Acta Numer. 12, 1–125 (2003)
    Article MATH MathSciNet Google Scholar
  5. Babuška, I., Banerjee, U., Osborn, J.: Superconvergence in generalized finite element method. Numer. Math. 107, 353–395 (2007)
    Article MATH MathSciNet Google Scholar
  6. Babuška, I., Caloz, G., Osborn, J.: Special finite element methods for a class of second order elliptic problems with rough coefficients. SIAM J. Numer. Anal. 31, 945–981 (1994)
    Article MATH MathSciNet Google Scholar
  7. Babuška, I., Melenk, J.M.: The partition of unity finite element method. Int. J. Numer. Methods Eng. 40, 727–758 (1997)
    Article MATH Google Scholar
  8. Béchet, E., Minnebo, H., Moës, N., Burgardt, B.: Improved implementation and robustness study of the X-FEM method for stress analysis around cracks. Int. J. Numer. Methods Eng. 64, 1033–1056 (2005)
    Article MATH Google Scholar
  9. Belytschko, T., Black, T.: Elastic crack growth in finite elements with minimal remeshing. Int. J. Numer. Methods Eng. 45, 601–620 (1999)
    Article MATH MathSciNet Google Scholar
  10. Bochev, P., Lehoucq, R.: On the finite element solution of the pure Neumann problem. SIAM Rev. 47(1), 50–66 (2005)
    Article MATH MathSciNet Google Scholar
  11. Chahine, E., Laborde, P., Renard, Y.: A non-conformal extended finite element approach: integral matching XFEM. Appl. Numer. Math. 61, 322–343 (2011)
    Article MATH MathSciNet Google Scholar
  12. Chen, L., Rabczuk, T., Bordas, S.P.A., Liu, G.R., Zeng, K.Y., Kerfriden, P.: Extended finite element method with edge-based strain smoothing (Esm-XFEM) for linear elastic crack growth. Comput. Methods Appl. Mech. Eng. 209–212, 250–265 (2012)
    Article MathSciNet Google Scholar
  13. Chu, C.C., Graham, I.G., Hou, T.Y.: A new multiscale finite element method for high contrast elliptic interface problems. Math. Comput. 79(272), 1915–1955 (2010)
    Article MATH MathSciNet Google Scholar
  14. Dolbow, J., Harari, I.: An efficient finite element method for embedded interface problems. Int. J. Numer. Methods Eng. 78, 229–252 (2009)
    Article MATH MathSciNet Google Scholar
  15. Efendiev, Y., Hou, T.Y.: Multiscale Finite Element Method. Springer, Berlin (2009)
    Google Scholar
  16. Gerstenberger, A., Tuminaro, R.: An algebraic multigrid approach to solve XFEM based fracture problem. Int. J. Numer. Methods Eng. 00, 1–26 (2012). doi:10.1002/nme
    Google Scholar
  17. Gerstenberger, A., Wall, W.A.: An extended finite element method/lagrange multiplier based approach for fluid-structure interaction. Comput. Methods Appl. Mech. Eng. 197, 1699–1714 (2008)
    Article MATH MathSciNet Google Scholar
  18. Griebel, M., Schweitzer, M.A.: A particle-partition of unity method, Part II: efficient cover construction and reliable integration. SIAM J. Sci. Comput. 23(5), 1655–1682 (2002)
    Article MATH MathSciNet Google Scholar
  19. Haasemann, G., Kastner, M., Pruger, S., Ulbricht, V.: Development of quadratic finite element formulations based on XFEM and NURBS. Int. J. Numer. Methods Eng. 86, 598–617 (2011)
    Article MATH MathSciNet Google Scholar
  20. 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)
    Article MATH MathSciNet Google Scholar
  21. Hou, T.Y., Wu, X.H., Cai, Z.: Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients. Math. Comput. 68, 913–943 (1999)
    Article MATH MathSciNet Google Scholar
  22. Kästner, M., Haasemann, G., Ulbricht, V.: Multiscale XFEM-modeling and simulation of the inelastic material behaviour of textile-reinforced polymers. Int. J. Numer. Methods Eng. 86, 477–498 (2011)
    Article MATH Google Scholar
  23. 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)
    Article MATH Google Scholar
  24. Mayer, U.M., Gerstenberger, A., Wall, W.A.: Interface handling for three-dimensional higher-order XFEM-computations in fluid-structure interaction. Int. J. Numer. Methods Eng. 79, 846–869 (2009)
    Article MATH Google Scholar
  25. Melenk, J.M., Babuška, I.: The partition of unity finite element method: theory and application. Comput. Methods Appl. Mech. Eng. 139, 289–314 (1996)
    Article MATH Google Scholar
  26. Melenk, J.M., Babuška, I.: Approximation with harmonic and generalized harmonic polynomials in the partition of unity method. Comput. Assist. Mech. Eng. Sci. 4, 607–632 (1997)
    MATH Google Scholar
  27. Menk, A., Bordas, S.P.A.: A robust preconditioning technique for the extended finite element method. Int. J. Numer. Methods Eng. 85(13), 1609–1632 (2011)
    Article MATH MathSciNet Google Scholar
  28. Moës, N., Dolbow, J., Belytschko, T.: A finite element method for crack without remeshing. Int. J. Numer. Methods Eng. 46, 131–150 (1999)
    Article MATH Google Scholar
  29. Rabczuk, T., Bordas, S., Zi, G.: On three-dimensional modelling of crack growth using partition of unity methods. Comput. Struct. 88, 1391–1411 (2010)
    Article Google Scholar
  30. Schweitzer, M.A.: Stable enrichment and local preconditioning in the particle-partition of unity method. Numer. Math. 118, 137–170 (2011)
    Article MATH MathSciNet Google Scholar
  31. Souza, F.V., Allen, D.H.: Modeling the transition of microcracks into macrocracks in heterogeneous viscoelastic media using two-way coupled multiscale model. Int. J. Solids Struct. 48, 3160–3175 (2011)
    Article Google Scholar
  32. Strouboulis, T., Babuška, I., Copps, K.: The design and analysis of the generalized finite element method. Comput. Methods Appl. Mech. Eng. 181, 43–69 (2001)
    Article Google Scholar
  33. Strouboulis, T., Copps, K., Babuška, I.: The generalized finite element method. Comput. Methods Appl. Mech. Eng. 190, 4081–4193 (2001)
    Article MATH Google Scholar
  34. Sukumar, N., Moes, N., Moran, B., Belytschko, T.: Extended finite element method for three dimensional crack modelling. Int. J. Numer. Methods Eng. 48(11), 1549–1570 (2000)
    Article MATH Google Scholar
  35. Yvonnet, J., Quang, H.L., He, Q.C.: A XFEM/level set approach to modeling surface/interface effects and to computing the size-dependent effective properties of nanocomposities. Comput. Mech. 42, 119–131 (2008)
    Article MATH MathSciNet Google Scholar
  36. Zilian, A., Legay, A.: The enriched space-time finite element method (EST) for simultaneous solution of fluid-structure interaction. Int. J. Numer. Methods Eng. 75, 305–334 (2008)
    Article MATH MathSciNet Google Scholar

Download references