Structural Engineering and Mechanics

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Stabilization of pressure solutions in four-node quadrilateral elements

  • Lee, Sang-Ho (Department of Civil Engineering, Yonsei University) ;
  • Kim, Sang-Hyo (Department of Civil Engineering, Yonsei University)
  • Published : 1998.09.25
⟨ Previous Next ⟩

Abstract

Mixed finite element formulations for incompressible materials show pressure oscillations or pressure modes in four-node quadrilateral elements. The criterion for the stability in the pressure solution is the so-called Babu$\check{s}$ka-Brezzi stability condition, and the four-node elements based on mixed variational principles do not appear to satisfy this condition. In this study, a pressure continuity residual based on the pressure discontinuity at element edges proposed by Hughes and Franca is used to study the stabilization of pressure solutions in bilinear displacement-constant pressure four-node quadrilateral elements. Also, a solid mechanics problem is presented by which the stability of mixed elements can be studied. It is shown that the pressure solutions, although stable, are shown to exhibit sensitivity to the stabilization parameters.

Keywords

References

  1. Babuska, I. (1971), "Error bounds for finite element method", Numer. Math., 16, 323-333.
  2. Belytschko, T. and Bachrach, W.E. (1986), "Efficient implementation of quadrilateral with high coarsemesh accuracy", Comput. Meths. Appl. Mech. Engrg., 54, 279-301. https://doi.org/10.1016/0045-7825(86)90107-6
  3. Belytschko, T. and Bindeman, L.P. (1991), "Assumed strain stabilization of the 4-node quadrilateral with 1-point quadrature for nonlinear problems", Comput. Meths. Appl. Mech. Engrg., 88, 311-340. https://doi.org/10.1016/0045-7825(91)90093-L
  4. Brezzi, F. (1974), "On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers", RAIRO Ser. Rouge Anal. Numer., R-2, 129-151.
  5. Doherty, W.P., Wilson, E.L. and Taylor, R.L. (1969), "Stress analysis of axisymmetric solids utilizing higher order quadrilateral finite elements", SESM Report No. 69-3, Department of Civil Engineering, University of California, Berkeley, CA.
  6. Hughes, T.J.R. (1977), "Equivalence of finite elements for nearly incompressible elasticity", J. Appl. Mech., 44, 181-183. https://doi.org/10.1115/1.3423994
  7. Hughes, T.J.R. (1987), The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Prentice-Hall, Englewood cliff, NJ.
  8. Hughes, T.J.R. and Franca, L.P. (1987), "A new finite element formulation for computational fluid dynamics: VII. The Stokes problem with various well-posed boundary conditions: Symmetric formulations that converge for all velocity/pressure spaces", Comput. Meths. Appl. Mech. Engrg., 65, 85-96. https://doi.org/10.1016/0045-7825(87)90184-8
  9. Lee, R.L., Gresho, P.M. and Sani, R.L. (1979), "Smoothing techniques for certain primitive variable solutions of the Navier-Stokes equations", Internat. J. Numer. Meths. Engrg., 14, 1785-1804. https://doi.org/10.1002/nme.1620141204
  10. MacNeal, R.H. (1993), Finite Elements: Their Design and Performance, Marcel Dekker Inc., New York.
  11. Malkus, D.S. and Hughes, T.J.R. (1978), "Mixed finite element methods-reduced and selective integration technique: a unification of concepts", Comput. Meths. Appl. Mech. Engrg., 15, 63-81. https://doi.org/10.1016/0045-7825(78)90005-1
  12. Oden, J.T. and Carey, G. (1983), Finite Elements: Mathematical Aspects, Prentice-Hall, Inc., Englewood Cliffs, New Jersey.
  13. Oden, J.T., Kikuchi, N. and Song, Y.J. (1982), "Penalty-finite element methods for the analysis of Stokesian Flows", Comput. Meths. Appl. Mech. Engrg., 31, 297-329. https://doi.org/10.1016/0045-7825(82)90010-X
  14. Pian, T.H.H. and Sumihara, K. (1984), "Rational approach for assumed stress finite elements", Internat. J. Numer. Meths. Engrg., 20, 1685-1695. https://doi.org/10.1002/nme.1620200911
  15. Pitkaranta, J. and Saarinen, T. (1985), "A multigrid version of a simple finite element method for the Stokes problem", Math. Comp., 45, 1-14. https://doi.org/10.1090/S0025-5718-1985-0790640-2
  16. Pitkaranta, J. and Stenberg, R. (1984), "Error bounds for the approximation of the Stokes problem using bilinear/constant elements on irregular quadrilateral meshes", Report-MAT-A222, Helsinki University of Technology, Institute of Mathematics, Finland.
  17. Sani, R.L., Gresho, P.M., Lee, R.L., Griffiths, D.F. and Engleman, M. (1981), "The cause and cure (?) of the spurious pressures generated by certain FEM solutions of the incompressible Navior-Stokes equations: Part I and Part II", Internat. J. Numer. Meths. in Fluids, 1, 17-43 and 171-204. https://doi.org/10.1002/fld.1650010104
  18. Silvester, D.J. and Kechkar, N. (1990), "Stabilised bilinear-constant velocity-pressure finite elements for the conjugate gradient solution of the Stokes problem", Comput. Meths. Appl. Mech. Engrg., 79, 71-86. https://doi.org/10.1016/0045-7825(90)90095-4
  19. Simo, J.C. and Rifai, M.S. (1990), "A class of mixed assumed strain methods and the method of incompressible modes", Internat. J. Numer. Meths. Engrg., 29, 1595-1638. https://doi.org/10.1002/nme.1620290802
  20. Stolarski, H. and Belytschko, T. (1983), "Shear and membrane locking in curved $C^{\circ}$ elements", Comput. Meths. Appl. Mech. Engrg., 41, 279-296. https://doi.org/10.1016/0045-7825(83)90010-5
  21. Taylor, R.L., Beresford, P.J. and Wilson, E.L. (1976), "A non-conforming element for stress analysis", Internat. J. Numer. Meths. Engrg., 10, 1211-1219. https://doi.org/10.1002/nme.1620100602
  22. Taylor, C. and Hood, P. (1973), "A numerical solution of the Navior-Stokes equations using FEM techniques", Computers and Fluids, 1, 73-100. https://doi.org/10.1016/0045-7930(73)90027-3
  23. Timoshenko, S.P. and Goodier, J.N. (1970), Theory of Elasticity 3rd ed., McGraw-Hill book company, 41-46.
  24. Xue, W.M., Karlovitz, L.A. and Atluri, S.N. (1985), "On the existence and stability conditions for mixed-hybrid finite element solutions based on Reissner? variational principle", Intl. J. Solids Structures, 21, 97-116. https://doi.org/10.1016/0020-7683(85)90107-6