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The penalty method for the Navier-Stokes equations

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Summary

The penalty finite element method as it applies to the Stokes and Navier-Stokes flow equations is reviewed. The main developments are discussed and selected but still extensive list of references is provided.

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References

  1. Courant, R. (1943), “Variational methods for the solution of problems of equilibrium and vibration”Bull. Am. Math. Soc.,49, pp. 1–23.

    Article  MATH  MathSciNet  Google Scholar 

  2. Reddy, J. N. (1982), “On penalty function methods in the finite element analysis of flowproblems”,Int. J. Num. Meth. Fluids,2, pp. 151–171.

    Article  MATH  Google Scholar 

  3. Herrman, L. R. (1965), “Elasticity equations for incompressible or nearly incompressible materials by a variational theorem”AIAA J.,3, pp. 1896–1900.

    MathSciNet  Google Scholar 

  4. Zienkiewicz, O. C. (1974), “Constrained variational principles and penalty function methods in finite element analysis” in G. A. Watson (Editor),Lecture Notes in Mathematics,363, pp. 207–214, Springer Verlag, Berlin.

    Google Scholar 

  5. Zienkiewicz, O. C., and Godbole, P. N. (1975), “A Penalty Function Approach to Problems of plastic flowof metals with large surface deformations”J. Strain Analysis,10, pp. 180–183.

    Article  Google Scholar 

  6. Zienkiewicz, O. C. and Godbole, P. N. (1978) “Viscous incompressible flow with special reference to non-newtonian (plastic) fluids”, in R. H. Gallagher, J. T. Oden, C. Taylor and O. C Zienkiewicz (Editors),Finite Elements in Fluids 1. pp. 25–55. Wiley, New York.

    Google Scholar 

  7. Hughes, T. J. R., Taylor, R. L. and Levy, J. F. (1978) “High Reynolds number, steady, incompressible flows by a finite element method”, in R. H. Gallagher, O. C. Zienkiewicz, J. T. Oden and M. Morandi-Cechi (Editors),Finite Elements in Fluids,3, pp. 52–72, Wiley, New York.

    Google Scholar 

  8. Hughes, T. J. R., Lin, W. K. and Brooks A. (1979) “Finite element analysis of incompressible viscous flows by the penalty function formulation”J. Comp. Physics,30 pp. 1–60.

    Article  MATH  Google Scholar 

  9. Bercovier, M. and Engelman, M (1979) “A finite element method for the numerical solution of viscous in compressible flows”J. Comp. Physics,30, pp. 181–201.

    Article  MATH  MathSciNet  Google Scholar 

  10. Heinrich, J. C. and Marshall R. S. (1981) “Viscous incompressible flowby a penalty function finite element method”Comp. Fluids,9, pp. 73–83.

    Article  MATH  Google Scholar 

  11. Sani, R. L., Eaton, B. E., Gresho, P. M., Lee, R. L. and Chan, S. T. (1981), “On the solution of the time-dependent incompressible Navier-Stokesequations via a penalty Galerkin finite element method”, in C. Taylor and B. Schreffler, EditorsProceeding of the 2nd International Conference on Laminar and Turbulent Flow, pp. 27–39, Pineridge Press, Swansea, U.K.

    Google Scholar 

  12. Engerman, M. S., Sani, R. L., Gresho, P. M. and Bercovier, M. (1982) “Consistent, vs reduced integration penalty methods for incompressible media using several old and newelements”,Int. J. Num. Meth. Fluids,2, pp. 25–42.

    Article  Google Scholar 

  13. Marshall, R. S., Heinrich, J. C. and Zienkiewicz, O. C. (1978), “Natural convection in a square enclosure by a finite element, penalty function method using primitive fluid variables”Num. Heat Transfer,1, pp. 315–330.

    Article  Google Scholar 

  14. Reddy, J.N. and Satake, A. (1980), “A comparison of a penalty finite element method with the stream function-velocity model of natural convection in enclosures”,ASME J. Heat Transfer,102, pp. 659–666.

    Article  Google Scholar 

  15. Heinrich, J.C. and Strada, M. (1982), “Penalty finite element analysis of natural convection at high Ray leigh numbers”, in R.H. Gallagher, D. H. Norrie, J. T. Oden and O. C. Zienkiewicz (Editors),Finite Elements in Fluids, 4, pp. 285–303, Wiley, New York.

    Google Scholar 

  16. Reddy, J.N. (Editor), (1982) “Penalty finite element method in mechanics”, AMD, 51, ASME, New York.

    Google Scholar 

  17. Heinrich, J.C. (1984) “A finite element model for double diffusive convection”,Int. J. Num. Meth. Eng.,20, pp. 447–464.

    Article  MATH  MathSciNet  Google Scholar 

  18. Reddy, J.N. and Padhye, V.A. (1988), “A penalty finite element model for axisymmetric flows of non-newtonian fluids”,Num. Meth. Partial Diff. Eqns.,4, pp. 33–56.

    Article  MATH  MathSciNet  Google Scholar 

  19. Codina, R., Cervera, M. and Oñate, E. (1993), “A penalty finite element method for non-newtonian creeping flow”,Int. J. Num. Meth. Eng.,36, pp. 1395–1412.

    Article  MATH  Google Scholar 

  20. Spilker, R.L. and Maxian, T.A. (1990), “A mixed penalty finite element formulation of the linear biphasic theory for soft tissues”,Int. J Num. Meth. Eng.,30, pp. 1063–1082.

    Article  MATH  MathSciNet  Google Scholar 

  21. Simon, B.R. (1992), “Multiphase poroelastic finite element models for soft tissue structures”,Int. Appl. Mech. Rev.,45, pp. 191–218.

    Article  Google Scholar 

  22. Baker, A.J. (1985), “On a penalty finite element CFD algorithm for high speed flow”,Comp. Meth. Appl. Mech. Eng.,51, pp. 395–420.

    Article  MATH  Google Scholar 

  23. Salonen, E.M. (1976), “An iterative penalty function method in structural analysis”,Int. J. Num. Meth. Eng.,10, pp. 413–421.

    Article  MATH  MathSciNet  Google Scholar 

  24. Felippa, C.A. (1978), “Iterative procedure for improving penalty function solutions of algebraic systems”,Int. J. Num. Meth. Eng.,12, pp. 165–185.

    Article  Google Scholar 

  25. Zienkiewicz, O.C., Vilotte, J.P., Toyoshima, S. and Nakazawa, S. (1985), “Iterative method for constrained and mixed approximation: An inexpensive improvement for F.E.M. performance”,Comp. Meth. Appl. Mech. Eng.,51, pp. 3–29.

    Article  MATH  MathSciNet  Google Scholar 

  26. Codina, R. (1993), “An iterative penalty method for the finite element solution of the stationary Navier-Stokes equations”,Comp. Meth. Appl. Mech. Eng.,110, pp. 237–262.

    Article  MATH  MathSciNet  Google Scholar 

  27. Malkus, D.S. and Hughes, T.J.R. (1978), “Mixed finite element methods-reduced and selective integration techniques: A unification of concepts”,Comp. Meth. Appl. Mech. Eng.,15, pp. 63–81.

    Article  MATH  Google Scholar 

  28. Malkus, D.S. (1981), “Eigenproblems associated with the discrete LBB condition for incompressible finite elements”,It. J. Eng. Sci.,19, pp. 1299–1310.

    Article  MATH  MathSciNet  Google Scholar 

  29. Malkus, D.S. and Olsen, T. (1984), “Obtaining error estimates for optimally constrained incompressible finite elements”,Comp. Meth. Appl. Mech. Eng.,45, pp. 331–353.

    Article  MATH  MathSciNet  Google Scholar 

  30. Oden, J.T. (1982), “RIP methods for stokesian flows”, in R.H. Gallagher, D.H. Norrhe, J.T. Oden, and O.C. Zienkiewicz (Editors),Finit. Elements in Fluids, 4, pp. 305–318, Wiley, New York.

    Google Scholar 

  31. Oden, J.T., Kikuchi, N. and Song, Y. (1982), “Penalty-finite element methods for the analysis of stokesian flows”,Comp. Meth. Appl. Mech. Eng.,31, pp. 297–329.

    Article  MATH  MathSciNet  Google Scholar 

  32. Carey, G.F. and Krishnan, R. (1982), “Penalty approximation of Stokes flow”,Comp. Meth. Appl. Mech. Eng.,35, pp. 169–206.

    Article  MATH  MathSciNet  Google Scholar 

  33. Carey, G.F. and Krishnan, R. (1984), “Penalty finite element method for Navier Stokes equations”,Comp. Meth. Appl. Mech. Eng.,42, pp. 183–224.

    Article  MATH  MathSciNet  Google Scholar 

  34. Carey, G.F. and Krishnan, R. (1985), “Continuation techniques for a penalty approximation of the Navier—Stokes equations”,Comp. Meth. Appl. Mech. Eng.,48, pp. 265–282.

    Article  MATH  MathSciNet  Google Scholar 

  35. Carey, G.F. and Krishnan, R. (1987), “Convergence of iterative methods in penalty finite element approximations of the Navier-Stokes equations”,Comp. Meth. Appl. Mech. Eng.,60, pp. 1–29.

    Article  MATH  MathSciNet  Google Scholar 

  36. Ladyszhenskaya, O.A. (1969), “The mathematical theory of viscous incompressible flow”, Gordon and Breach, New York.

    Google Scholar 

  37. Babuška, I. (1978), “The finite element method with penalty”,Num. Meth.,27, pp. 221–228.

    Google Scholar 

  38. Brezi, F. (1974), “On the existence, uniqueness and approximation of saddle point problems arising from Lagrange multipliers”,RAIRO,8, pp. 129–151.

    Google Scholar 

  39. Teman, R. (1977), “Navier-Stokes Equations”, North-Holland, Amsterdam.

    Google Scholar 

  40. Brooks, A.N. and Hughes, T.J.R. (1982), “Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations”,Comp. Meth. Appl. Mech. Eng.,32, pp. 199–259.

    Article  MATH  MathSciNet  Google Scholar 

  41. Yu, C.C. and Heinrich, J.C. (1986), “Petrov-Galerkin methods for the time-dependent convective transport equation”,Int. J. Num. Meth. Eng.,23, pp. 883–901.

    Article  MATH  MathSciNet  Google Scholar 

  42. Carey, G.F. and Oden, J.T. (1986), “Finite elements: Fluid mechanics”, Vol. VI, Prentice-Hall, Englewood Cliffs, N.J.

    MATH  Google Scholar 

  43. Lee, R.L., Gresho, P.M. and Sani, R.L. (1979), “Smoothing techniques for certain primitive variable solutions of the Navier-Stokes equations”,Int. J. Num. Meth. Eng.,14, pp. 1785–1804.

    Article  MATH  MathSciNet  Google Scholar 

  44. Sohn, J.L. and Heinrich, J.C. (1990), “A Poisson equation formulation for pressure calculations in penalty finite element models for viscous incompressible flows”,Int. J. Num. Meth. Eng.,30, pp. 349–361.

    Article  MATH  Google Scholar 

  45. Dyne, B.R. and Heinrich, J.C. (1993), “Physically correct penalty-like formulations for accurate pressure calculation in finite element algorithms of the Navier-Stokes equations”,Int. J. Num. Meth. Eng.,36, pp. 3883–3902.

    Article  MATH  Google Scholar 

  46. Vionnet, C.A. and Heinrich, J.C., “Physical role of a static pressure field in the computation of viscous incompressible flows”,J. Comp. Phys., (submitted).

  47. Vionnet, C.A. and Heinrich, J.C. (1993), “An application of small gap equations in sealing devices”, inFifth Annual Thermal Fluids Workshop, pp. 499–512, NASA Lewis Research Center, Conference Publ. 10122.

  48. Vionnet, C.A. and Heinrich, J.C. (1993), “Penalty modeling of flows influenced by Body Forces”, in K. Morgan, E. Oñate, J. Periaux, J. Peraire and O.C. Zienkiewicz (Editors),Finite Elements in Fluids, pp. 174–183, CIMNE, Barcelona, Spain.

    Google Scholar 

  49. De Bremaecker, J.C. (1986), “Penalty solution of the Navier-Stokes equations”,Computer Fluids,15, pp. 275–280.

    Article  Google Scholar 

  50. Ramshaw, J.D. and Mesina, G.L. (1991), “A hybrid penalty-pseudocompressibility method for transient incompressible fluid flow”,Computers Fluids,20, pp. 165–175.

    Article  MATH  Google Scholar 

  51. Heinrich, J.C. and Yu, C.C. (1988), “Finite elements simulation of buoyancy-driven flows with emphasis on natural convection in a horizontal circular cylinder”,Comp. Meth. Appl. Meth. Eng.,69, pp. 1–27.

    Article  MATH  Google Scholar 

  52. Bhat, M.S., Poirier, D.R. and Heinrich, J.C., “Permeability of cross flowthrough columtiar dendritic alloys”,Metall. Trans., (accepted).

  53. Nagelhout, D.F., Heinrich, J.C., and Poirier, D.R., “Mesh generation flowcalculation in highly contorted geometries”,Comp. Meth. Appl. Mech. Eng., (submitted).

  54. Rouse, H. (1936), “Discharge characteristics of the free overall”,Civil Eng.,6, pp. 257–263.

    Google Scholar 

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  1. Department of Aerospace and Mechanical Engineering, The University of Arizona, 35721, Tucson, AZ, USA

    J. C. Heinrich & C. A. Vionnet

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Heinrich, J.C., Vionnet, C.A. The penalty method for the Navier-Stokes equations. ARCO 2, 51–65 (1995). https://doi.org/10.1007/BF02904995

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