Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
권호 표지
DOI QR코드

DOI QR Code

REMARKS ON FINITE ELEMENT METHODS FOR CORNER SINGULARITIES USING SIF

  • Kim, Seokchan (Department of Applied Mathematics, Changwon National University) ;
  • Kong, Soo Ryun (Department of Applied Mathematics, Changwon National University)
  • Received : 2016.08.03
  • Accepted : 2016.09.05
  • Published : 2016.09.25
Download PDF
⟨ Previous Next ⟩

Abstract

In [15] they introduced a new finite element method for accurate numerical solutions of Poisson equations with corner singularities, which is useful for the problem with known stress intensity factor. They consider the Poisson equations with homogeneous Dirichlet boundary condition, compute the finite element solution using standard FEM and use the extraction formula to compute the stress intensity factor, then they pose a PDE with a regular solution by imposing the nonhomogeneous boundary condition using the computed stress intensity factor, which converges with optimal speed. From the solution we could get accurate solution just by adding the singular part. This approach works for the case when we have the accurate stress intensity factor. In this paper we consider Poisson equations with mixed boundary conditions and show the method depends the accrucy of the stress intensity factor by considering two algorithms.

Keywords

References

  1. I. Babuska, R.B. Kellogg, and J. Pitkaranta, Direct and inverse error estimates for finite elements with mesh refinements, Numer. Math., 33 (1979), 447-471. https://doi.org/10.1007/BF01399326
  2. H. Blum and M. Dobrowolski, On finite element methods for elliptic equations on domains with corners, Computing, 28 (1982), 53-63. https://doi.org/10.1007/BF02237995
  3. M. Bourlard, M. Dauge, M.-S. Lubuma, and S. Nicaise, Coefficients of the singularities for elliptic boundary value problems on domains with conical points III. Finite element methods on polygonal domains, SIAM Numer. Anal., 29 (1992), 136-155. https://doi.org/10.1137/0729009
  4. S. C. Brenner, Multigrid methods for the computation of singular solutions and stress intensity factor I: Corner singularities, Math. Comp., 68 (226), (1999), 559-583. https://doi.org/10.1090/S0025-5718-99-01017-0
  5. S. C. Brenner and L.-Y. Sung, Multigrid methods for the computation of singular solutions and stress intensity factors III: Interface singularities, Comput. Methods Appl. Mech. Engrg. 192(2003), 4687-4702. https://doi.org/10.1016/S0045-7825(03)00455-9
  6. Z. Chen and S. Dai, On the efficiency of adaptive finite element methods for elliptic problems with discontinuous coefficients, SIAM J. Sci. Comput., 24:(2002), 443-462. https://doi.org/10.1137/S1064827501383713
  7. Z. Cai and S.C. Kim, A finite element method using singular functions for the poisson equation: Corner singularities, SIAM J. Numer. Anal., 39:(2001), 286-299. https://doi.org/10.1137/S0036142999355945
  8. Z. Cai , S.C. Kim, S.D. Kim, S. Kong, A finite element method using singular functions for Poisson equations: Mixed boundary conditions, Comput. Methods Appl. Mech. Engrg. 195 (2006) 26352648 https://doi.org/10.1016/j.cma.2005.06.004
  9. M. Djaoua, Equations Integrales pour un Probleme Singulier dans le Plan, These de Troisieme Cycle, Universite Pierre et Marie Curie, Paris, 1977.
  10. M. Dobrowolski, Numerical Approximation of Elliptic Interface and Corner Problems, Habilitationsschrift, Bonn, 1981.
  11. G. J. Fix, S. Gulati, and G. I. Wakoff, On the use of singular functions with finite elements approximations, J. Comput. Phy., 13 (1973), 209-228. https://doi.org/10.1016/0021-9991(73)90023-5
  12. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer-Verlag, Berlin, 1983.
  13. P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, MA, 1985.
  14. F. Hecht, New development in FreeFem++, J. Numer. Math. 20 (2012), no. 3-4, 251265.
  15. S. Kim and H.C. Lee, A finite element method for computing accurate solutions for Poisson equations with corner singularities using the stress intensity factor, Computers and Mathematics with Applications, 71(2016) 2330-2337. https://doi.org/10.1016/j.camwa.2015.12.023
  16. V. Mazya and B. Plamenevskii, On the coefficients in the asymptotics of solutions of elliptic boundary-value problems near conical points, Soviet Math. Dokl., 15 (1974), 1570-1575.
  17. V. G. Mazya and B. A. Plamenevskii, Coefficients in the asymptotics of the solutions of an elliptic boundary value problem in a cone, J. Soviet Math., 9 (1978), 750-764. https://doi.org/10.1007/BF01085326
  18. P. Morin, R.H. Nochetto, and K.G. Siebert Data oscillation and convergence of adaptive FEM, SIAM J. Numer. Anal., 38 (2000), pp. 466.488.
  19. A. Schatz and L. Wahlbin, Maximum norm estimates in the finite element method on plane polygonal domains, Part 1, Math. Comp., 32 (141), (1978) 73-109. https://doi.org/10.1090/S0025-5718-1978-0502065-1
  20. A. Schatz and L. Wahlbin, Maximum norm estimates in the finite element method on plane polygonal domains, Part 2 (refinements), Math. Comp., 33 (146), (1979) 465-492. https://doi.org/10.1090/S0025-5718-1979-0502067-6
  21. Ch. Schwab, p- and hp-Finite Element Methods, Oxford University Press, Oxford, 1998.
  22. B. A. Szabo and I. Babuska, Finite Element Analysis, John Wiley & Sons, New York, 1991.
  23. A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Springer-Verlag, Berlin, Germany, 1994.
  24. Z. Yosibash, Singularities in elliptic boundary value problems and elasticity and their connection with failure initiation, Interdisciplinary Applied Mathematics, 37. Springer, New York, 2012.