Towards the Direct and Inverse Adaptive Mixed Finite Element Formulations for Nearly Incompressible Elasticity at Large Strains

Part of the Lecture Notes in Applied and Computational Mechanics book series (LNACM, volume 66)

Abstract

This contribution presents advanced numerical models for the solution of the direct and inverse problems of nearly incompressible hyperelastic processes at large strains. The discussed mixed finite element approach contributes to the numerical simulation of coupled multiphysics problems, including the calibration of appropriate material models (parameter identification). The presented constitutive approach is based on the multiplicative decomposition of the deformation gradient resulting in a two-field formulation with displacement components and hydrostatic pressure as primary variables. The ill-posed inverse problem of parameter identification analyzing inhomogeneous displacement fields is solved using deterministic trust-region optimization techniques.Within this context, a semi-analytical approach for sensitivity analysis represents an efficient and accurate method to determine the gradient of the objective function. Themixed boundary value problem is based on the spatial discretization of the weak formulations of the linear momentum balance and the incompressibility condition. Its linearization serves as basis for the solution of the direct problem, while the implicit differentiation of the weak formulations with respect to material parameters provides the necessary relations for the semi-analytical sensitivity analysis. Adaptive mesh refinement and mesh coarsening are realized controlled by a residual a posteriori error estimator. Efficiency and accuracy of the presented direct and inverse numerical techniques are demonstrated on a typical example.

Keywords

Constitutive Model Deformation Gradient Posteriori Error Estimator Rectangular Element Corner Node 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Ainsworth, M., Oden, J.T.: A posteriori error estimators for the Stokes and Oseen equations. SIAM J. Numer. Anal. 34, 228–245 (1997)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Armero, F.: On the locking and stability of finite elements in finite deformation plane strain problems. Comput. Struct. 75, 261–290 (2000)CrossRefGoogle Scholar
  3. 3.
    Ball, J.M.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 3, 337–407 (1977)Google Scholar
  4. 4.
    Bank, R.E., Welfert, B.D.: A posteriori error estimates for the Stokes equations: a comparison. Comput. Meth. Appl. Mech. Engrg. 82, 323–340 (1990)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Bathe, K.-J.: Finite-Elemente-Methoden. Springer, Berlin (2002)CrossRefGoogle Scholar
  6. 6.
    Benedix, U.: Parameterschätzung für elastisch-plastische Deformationsgesetze bei Berücksichtigung lokaler und globaler Vergleichsgrößen (Parameter estimation for elasto-plastic material models considering local and global comparative quantities; in German). Dissertation, Report 4/2000, Institut für Mechanik der TU Chemnitz (2000)Google Scholar
  7. 7.
    Bramble, J.H., Pasciak, J.E.: A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems. Math. Comp. 50(181), 1–17 (1988)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Brezzi, F., Fortin, M.: Mixed and hybrid Finite Element Methods. Springer, New York (1991)MATHCrossRefGoogle Scholar
  9. 9.
    Brink, U., Stein, E.: On some mixed finite element methods for incompressible and nearly incompressible finite elasticity. Comp. Mech. 19, 105–119 (1996)MATHCrossRefGoogle Scholar
  10. 10.
    Bucher, A., Görke, U.-J., Kreißig, R.: A material model for finite elasto-plastic deformations considering a substructure. Int. J. Plast. 20, 619–642 (2004)MATHCrossRefGoogle Scholar
  11. 11.
    Bucher, A., Görke, U.-J., Steinhorst, P., Kreißig, R., Meyer, A.: Ein Beitrag zur adaptiven gemischten Finite-Elemente-Formulierung der nahezu inkompressiblen Elastizität bei großen Verzerrungen (A contribution to the adaptive mixed finite element formulations for nearly incompressible elasticity at large strains; in German). Preprint CSC/07-06, TU Chemnitz (2007)Google Scholar
  12. 12.
    Bucher, A., Meyer, A., Görke, U.-J., Kreißig, R.: A contribution to error estimation and mapping algorithms for a hierarchical adaptive FE-strategy in finite elastoplasticity. Comp. Mech. 36(3), 182–195 (2005)MATHCrossRefGoogle Scholar
  13. 13.
    Bucher, A., Meyer, A., Görke, U.-J., Kreißig, R.: A comparison of mapping algorithms for hierarchical adaptive FEM in finite elasto-plasticity. Comp. Mech. 39(4), 521–536 (2007)MATHCrossRefGoogle Scholar
  14. 14.
    Chen, J.S., Han, W., Wu, C.T., Duan, W.: On the perturbed Lagrangian formulation for nearly incompressible and incompressible hyperelasticity. Comput. Meth. Appl. Mech. Engrg. 142, 335–351 (1997)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Cook, R.D.: Improved two-dimensional finite element. J. Struct. Div. ASCE 100, 1851–1863 (1974)Google Scholar
  16. 16.
    Dennis, J.E., Schnabel, R.B.: Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice-Hall Inc., Englewood Cliffs (1983)MATHGoogle Scholar
  17. 17.
    Flory, P.J.: Thermodynamic relations for high elastic materials. Trans. Faraday Soc. 57, 829–838 (1961)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Gelin, J.-C., Ghouati, O.: An inverse method for material parameter estimation in the inelastic range. Comp. Mech. 16, 143–150 (1995)MATHCrossRefGoogle Scholar
  19. 19.
    Görke, U.-J., Bucher, A., Kreißig, R.: Ein Beitrag zur Materialparameteridentifikation bei finiten elastisch-plastischen Verzerrungen durch Analyse inhomogener Verschiebungsfelder mit Hilfe der FEM (A contribution to the identification of material parameters at large elasto-plastic strains analyzing inhomogeneous displacement fields using the finite element method; in German). Preprint SFB393/01-03, TU Chemnitz (2001)Google Scholar
  20. 20.
    Görke, U.-J., Bucher, A., Kreißig, R.: Zur Numerik der inversen Aufgabe für gemischte (u/p) Formulierungen am Beispiel der nahezu inkompressiblen Elastizität bei großen Verzerrungen (Towards the numerics of the inverse problem for mixed (u/p) formulations on the example of nearly incompressible elasticity at large strains; in German). Preprint CSC/07-07, TU Chemnitz (2007)Google Scholar
  21. 21.
    Herrmann, L.R.: Elasticity equations for nearly incompressible materials by a variational theorem. AIAA J. 3, 1896–1900 (1965)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Hood, P., Taylor, C.: Navier-Stokes equations using mixed interpolation. In: Oden, J.T., Gallagher, R.H., Zienkiewicz, O.C., Taylor, C. (eds.) Finite Element Methods in Flow Problems, pp. 121–132. University of Alabama in Huntsville Press (1974)Google Scholar
  23. 23.
    Hughes, T.J.R.: The finite element method. Dover Publications, New York (2000)MATHGoogle Scholar
  24. 24.
    Ibrahimbegovic, A., Taylor, R.L., Wilson, E.L.: A robust quadrilateral membrane finite element with drilling degrees of freedom. Int. J. Num. Meth. Engng. 30, 445–457 (1990)MATHCrossRefGoogle Scholar
  25. 25.
    Johansson, H., Runesson, K.: Parameter identification in constitutive models via optimization with a posteriori error control. Int. J. Numer. Meth. Engng. 62, 1315–1340 (2005)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Kay, D., Silvester, D.: A-posteriori error estimation for stabilized mixed approximations of the Stokes equations. SIAM J. Sci. Comp. 21, 1321–1336 (1999)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Kreißig, R., Benedix, U., Görke, U.-J.: Statistical aspects of the identification of material parameters for elasto-plastic models. Arch. Appl. Mech. 71, 123–134 (2001)MATHCrossRefGoogle Scholar
  28. 28.
    Kreißig, R., Benedix, U., Görke, U.-J., Lindner, M.: Identification and estimation of constitutive parameters for material laws in elastoplasticity. GAMM-Mitteilungen 30(2), 458–470 (2007)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Lecampion, B., Constantinescu, A.: Sensitivity analysis for parameter identification in quasi-static poroelasticity. Int. J. Num. Anal. Meth. Geomech. 29(2), 163–185 (2005)MATHCrossRefGoogle Scholar
  30. 30.
    Le Tallec, P.: Numerical methods for nonlinear three-dimensional elasticity. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis, vol. III, pp. 465–622. Elsevier, Amsterdam (1994)Google Scholar
  31. 31.
    Mahnken, R., Kuhl, E.: Parameter identification of gradient enhanced damage models with the finite element method. Eur. J. Mech. A/Solids 18, 819–835 (1999)MATHCrossRefGoogle Scholar
  32. 32.
    Mahnken, R., Stein, E.: The parameter-identification for visco-plastic models via Finite-Element-Methods and gradient methods. Modelling Simul. Mater. Sci. Eng. 2, 597–616 (1994)CrossRefGoogle Scholar
  33. 33.
    Mahnken, R., Stein, E.: Parameter identification for finite deformation elasto-plasticity in principal directions. Comput. Meth. Appl. Mech. Engrg. 147, 17–39 (1997)MATHCrossRefGoogle Scholar
  34. 34.
    Mahnken, R., Steinmann, P.: Finite element algorithm for parameter identification of material models for fluid saturated porous media. Int. J. Num. Anal. Meth. Geomech. 25(5), 415–434 (2001)MATHCrossRefGoogle Scholar
  35. 35.
    Masud, A., Xia, K.: A stabilized mixed finite element method for nearly incompressible elasticity. J. Appl. Mech. 72, 711–720 (2005)MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Meyer, A.: Grundgleichungen und adaptive Finite-Elemente-Simulation bei ”Großen Deformationen” (Basic equations and adaptive finite element simulation at large strains; in German). Preprint CSC/07-02, TU Chemnitz (2007)Google Scholar
  37. 37.
    Meyer, A., Steidten, T.: Improvements and experiments on the Bramble-Pasciak type CG for mixed problems in elasticity. Preprint SFB393/01-12, TU Chemnitz (2001)Google Scholar
  38. 38.
    Nocedal, J., Wright, S.J.: Numerical Optimization. Springer (1999)Google Scholar
  39. 39.
    Ogden, R.W., Saccomandi, G., Sgura, I.: Fitting hyperelastic models to experimental data. Comp. Mech. 34, 484–502 (2004)MATHCrossRefGoogle Scholar
  40. 40.
    Rüter, M., Stein, E.: Analysis, finite element computation and error estimation in transversely isotropic nearly incompressible finite elasticity. Comput. Meth. Appl. Mech. Engrg. 190, 519–541 (2000)MATHCrossRefGoogle Scholar
  41. 41.
    Simo, J.C., Taylor, R.L.: Quasi-incompressible finite elasticity in principal stretches, continuum basis and numerical algorithms. Comput. Meth. Appl. Mech. Engrg. 85, 273–310 (1991)MathSciNetMATHCrossRefGoogle Scholar
  42. 42.
    Taylor, C., Hood, P.: A numerical solution of the Navier Stokes equations using the finite element technique. Comput. Fluids 1, 73–100 (1973)MathSciNetMATHCrossRefGoogle Scholar
  43. 43.
    Taylor, R.L., Pister, K.S., Herrmann, L.R.: On a variational theorem for incompressible and nearly-incompressible orthotropic elasticity. Int. J. Sol. Struct. 4, 875–883 (1968)MATHCrossRefGoogle Scholar
  44. 44.
    Verfürth, R.: A review of a posteriori error estimation and adaptive mesh-refinement techniques. Wiley and Teubner, Chichester and Stuttgart (1996)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Anke Bucher
    • 1
  • Uwe–Jens Görke
    • 2
  • Reiner Kreißig
    • 3
  1. 1.Fakultät Maschinen– und EnergietechnikHTWK LeipzigMarkkleebergGermany
  2. 2.Helmholtz Zentrum für UmweltforschungLeipzigGermany
  3. 3.Fakultät für MaschinenbauTU ChemnitzChemnitzGermany

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