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On finite element formulations for nearly incompressible linear elasticity

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Abstract

In this paper we present a mixed stabilized finite element formulation that does not lock and also does not exhibit unphysical oscillations near the incompressible limit. The new mixed formulation is based on a multiscale variational principle and is presented in two different forms. In the first form the displacement field is decomposed into two scales, coarse-scale and fine-scale, and the fine-scale variables are eliminated at the element level by the static condensation technique. The second form is obtained by simplifying the first form, and eliminating the fine-scale variables analytically yet retaining their effect that results with additional (stabilization) terms. We also derive, in a consistent manner, an expression for the stabilization parameter. This derivation also proves the equivalence between the classical mixed formulation with bubbles and the Galerkin least-squares type formulations for the equations of linear elasticity. We also compare the performance of this new mixed stabilized formulation with other popular finite element formulations by performing numerical simulations on three well known test problems.

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References

  1. Arnold DN, Brezzi F and Fortin M (1984). A stable finite element for the Stokes equations. Calcolo 21: 337–344

    Article  MATH  MathSciNet  Google Scholar 

  2. Ayub M and Masud A (2003). A new stabilized formulation for convective-diffusive heat transfer. Numer Heat Transf Part B 44: 1–23

    Article  Google Scholar 

  3. Babuska I (1971). Error bounds for finite element methods. Numerische Mathematik 16: 322–333

    Article  MATH  MathSciNet  Google Scholar 

  4. Baiocchi C, Brezzi F and Franca L (1993). Virtual bubbles and Galerkin-least-squares type methods (Ga.L.S.). Comput Methods Appl Mech Eng 105: 125–141

    Article  MATH  MathSciNet  Google Scholar 

  5. Bischoff M, Ramm E and Braess D (1999). A class of equivalent enhanced assumed strain and hybrid stress finite elements. Comput Mech 22: 443–449

    Article  MATH  Google Scholar 

  6. Brezzi F and Fortin M (1991). Mixed and hybrid finite element methods, Springer series in computational mathematics, vol 15. Springer, New York

    Google Scholar 

  7. Brooks AN and Hughes TJR (1982). Streamline-upwind/Petrov-Galerkin methods for convection dominated flows with emphasis on the incompressible Navier–Stokes equations. Comput Methods Appl Mech Eng 32: 199–259

    Article  MATH  MathSciNet  Google Scholar 

  8. Felippa CA (2000). On the original publication of the general canonical functional of linear elasticity. J Appl Mech 67: 217–219

    Article  MATH  Google Scholar 

  9. Franca LP and Russo A (1997). Unlocking with residual-free bubbles. Comput Methods Appl Mech Eng 142: 361–364

    Article  MATH  MathSciNet  Google Scholar 

  10. Hughes TJR (1980). Generalization of selective integration procedures to anisotropic and nonlinear media. Int J Numer Methods Eng 15: 1413–1418

    Article  MATH  Google Scholar 

  11. Hughes TJR (1987). The finite element method: linear static and dynamic finite element analysis. Prentice-Hall, Englewood Cliffs

    MATH  Google Scholar 

  12. Hughes TJR (1995). Multiscale phenomena: Green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods. Comput Methods Appl Mech Eng 127: 387–401

    Article  MATH  Google Scholar 

  13. Hughes TJR, Franca L and Balestra M (1986). A new finite element formulation for computational fluid dynamics: V. Circumventing the Babuska–Brezzi condition: a stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations. Comput Methods Appl Mech Eng 59: 85–99

    Article  MATH  MathSciNet  Google Scholar 

  14. Hughes TJR, Franca L and Hulbert G (1989). A new finite element formulation for computational fluid dynamics: VIII. The Galerkin/least-squares method for advective-diffusive equations. Comput Methods Appl Mech Eng 73: 173–189

    Article  MATH  MathSciNet  Google Scholar 

  15. Kasper EP and Taylor RL (2000). A mixed-enhanced strain method Part I: Geometrically linear problems. Comput Struct 75: 237–250

    Article  Google Scholar 

  16. Keener JP (2000). Principles of applied mathematics. Perseus Books, Cambridge

    MATH  Google Scholar 

  17. Klaas O, Maniatty A and Shephard MS (1999). A stabilized mixed finite element method for finite elasticity. Formulation for linear displacement and pressure interpolation. Comput Methods Appl Mech Eng 180: 65–79

    Article  MATH  Google Scholar 

  18. Malkus DS and Hughes TJR (1978). Mixed finite element methods—reduced and selective integration techniques: a unification of concepts. Comput Methods Appl Mech Eng 15: 63–81

    Article  Google Scholar 

  19. Masud A and Bergman LA (2005). Application of multiscale finite element methods to the solution of the Fokker-Planck equation. Comput Methods Appl Mech Eng 194: 1513–1526

    Article  MATH  MathSciNet  Google Scholar 

  20. Masud A and Hughes TJR (2002). A stabilized mixed finite element method for Darcy flow. Comput Methods Appl Mech Eng 191: 4341–4370

    Article  MATH  MathSciNet  Google Scholar 

  21. Masud A and Khurram RA (2004). A multiscale/stabilized finite element method for the advection-diffusion equation. Comput Methods Appl Mech Eng 193: 1997–2018

    Article  MATH  Google Scholar 

  22. Masud A and Khurram RA (2006). A multiscale finite element method for the incompressible Navier–Stokes equations. Comput Methods Appl Mech Eng 195: 1750–1777

    Article  MATH  MathSciNet  Google Scholar 

  23. Masud A and Xia K (2005). A stabilized mixed finite element method for nearly incompressible elasticity. J Appl Mech 72: 711–720

    Article  MATH  MathSciNet  Google Scholar 

  24. Masud A and Xia K (2006). A variational multiscale method for inelasticity: application to superelasticity in shape memory alloys. Comput Methods Appl Mech Eng 195: 4512–4531

    Article  MATH  MathSciNet  Google Scholar 

  25. Nakshatrala KB, Turner DZ, Hjelmstad KD and Masud A (2006). A stabilized mixed finite element formulation for Darcy flow based on a multiscale decomposition of the solution. Comput Methods Appl Mech Eng 195: 4036–4049

    Article  MATH  MathSciNet  Google Scholar 

  26. Pian THH and Sumihara K (1984). Rational approach for assumed stress elements. Int J Numer Methods Eng 20: 1685–1695

    Article  MATH  Google Scholar 

  27. Simo JC and Rifai MS (1990). A class of mixed assumed strain methods and the method of incompatible modes. Int J Numer Methods Eng 29: 1595–1638

    Article  MATH  MathSciNet  Google Scholar 

  28. Stakgold I (1998). Green’s functions and boundary value problems. Wiley, New York

    MATH  Google Scholar 

  29. Stolarski H and Belytschko T (1987). Limitation principles for mixed finite elements based on the Hu–Washizu variational formulation. Comput Methods Appl Mech Eng 60: 195–216

    Article  MATH  MathSciNet  Google Scholar 

  30. Taylor RL, Beresford PJ and Wilson EL (1976). A non-conforming element for stress analysis. Int J Numer Methods Eng 10: 1211–1219

    Article  MATH  Google Scholar 

  31. de Veubeke BF (1965) Displacement and equilibrium models in the finite element method. In: Zienkiewicz OC, Hollister GS (eds), Chap 9, pp 145–197. Wiley, New York. Int J Numer Methods Eng 52:287–342, 2001 (reprinted)

  32. Wolfram S (2001). Mathematica, Version 4.1. Wolfram Research, Inc., Champaign

    Google Scholar 

  33. Zienkiewicz OC and Taylor RL (1989). The finite element method, vol 1. McGraw-Hill, New York

    Google Scholar 

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Authors and Affiliations

  1. Department of Civil and Environmental Engineering, University of Illinois, Urbana, IL, 61801, USA

    K. B. Nakshatrala, A. Masud & K. D. Hjelmstad

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  1. K. B. Nakshatrala
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  2. A. Masud
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  3. K. D. Hjelmstad
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Correspondence to K. D. Hjelmstad.

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Nakshatrala, K.B., Masud, A. & Hjelmstad, K.D. On finite element formulations for nearly incompressible linear elasticity. Comput Mech 41, 547–561 (2008). https://doi.org/10.1007/s00466-007-0212-8

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  • DOI: https://doi.org/10.1007/s00466-007-0212-8

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