Goal-oriented A Posteriori Error Estimates in Finite Hyperelasticity

  • Marcus Olavi RüterEmail author
Part of the Lecture Notes in Applied and Computational Mechanics book series (LNACM, volume 88)


Coming full circle in this chapter, expansions of the goal-oriented error estimation procedures presented in the preceding chapter to the finite hyperelasticity problem within both Newtonian and Eshelbian mechanics are derived for compressible and (nearly) incompressible materials. These error estimation procedures represent the most challenging ones presented in this monograph from both theoretical and numerical points of view. As a consequence, attention is focused on the derivation of error approximations rather than upper- or lower-bound error estimates. In the nonlinear case, a natural norm, such as the energy norm does not exist. The estimation of the general error measures introduced in the preceding chapter, on the other hand, does not necessarily rely on norm-based error estimators and thus allows for the derivation of a more versatile approach in a posteriori error estimation that can be employed in this chapter. Throughout this chapter, we confine ourselves to Galerkin mesh-based methods although similar error estimation procedures can also be developed for Galerkin meshfree methods.


  1. Bank, R.E.: Hierarchical bases and the finite element method. Acta Numer. 1–43 (1996)MathSciNetCrossRefGoogle Scholar
  2. Bank, R.E.: A simple analysis of some a posteriori error estimates. Appl. Numer. Math. 26, 153–164 (1998)MathSciNetCrossRefGoogle Scholar
  3. Bank, R.E., Smith, R.K.: A posteriori error estimates based on hierarchical bases. SIAM J. Numer. Anal. 30, 921–935 (1993)MathSciNetCrossRefGoogle Scholar
  4. Blacker, T., Belytschko, T.: Superconvergent patch recovery with equilibrium and conjoint interpolation enhancements. Int. J. Numer. Meth. Engng. 37, 517–536 (1994)CrossRefGoogle Scholar
  5. Bonet, J., Huerta, A., Peraire, J.: The efficient computation of bounds for functionals of finite element solutions in large strain elasticity. Comput. Methods Appl. Mech. Engrg. 191, 4807–4826 (2002)MathSciNetCrossRefGoogle Scholar
  6. Brink, U., Stein, E.: A posteriori error estimation in large-strain elasticity using equilibrated local Neumann problems. Comput. Methods Appl. Mech. Engrg. 161, 77–101 (1998)MathSciNetCrossRefGoogle Scholar
  7. Carstensen, C., Funken, S.A.: Averaging technique for FE - a posteriori error control in elasticity. Part I: Conforming FEM. Comput. Methods Appl. Mech. Engrg. 190, 2483–2498 (2001)MathSciNetCrossRefGoogle Scholar
  8. Kvamsdal, T., Okstad, K.M.: Error estimation based on superconvergent patch recovery using statically admissible stress fields. Int. J. Numer. Meth. Engng. 42, 443–472 (1998)MathSciNetCrossRefGoogle Scholar
  9. Larsson, F., Hansbo, P., Runesson, K.: Strategies for computing goal-oriented a posteriori error measures in non-linear elasticity. Int. J. Numer. Meth. Engng. 55, 879–894 (2002)MathSciNetCrossRefGoogle Scholar
  10. Ródenas, J.J., Tur, M., Fuenmayor, F.J., Vercher, A.: Improvement of the superconvergent patch recovery technique by the use of constraint equations: The SPR-C technique. Int. J. Numer. Meth. Engng. 70, 705–727 (2007)CrossRefGoogle Scholar
  11. Rüter, M., Heintz, P., Larsson, F., Hansbo, P., Runesson, K., Stein, E.: Strategies for goal-oriented a posteriori error estimation in elastic fracture mechanics. In: Lund, E., Olhoff, N., Stegmann, J. (eds.) Proceedings of the 15th Nordic Seminar on Computational Mechanics, NSCM 15, Aalborg, Denmark, pp. 43–46 (2002)Google Scholar
  12. Rüter, M., Stein, E.: Analysis, finite element computation and error estimation in transversely isotropic nearly incompressible finite elasticity. Comput. Methods Appl. Mech. Engrg. 190, 519–541 (2000)MathSciNetCrossRefGoogle Scholar
  13. Rüter, M., Stein, E.: On the duality of global finite element discretization error-control in small strain Newtonian and Eshelbian mechanics. Technische Mechanik 23, 265–282 (2003)Google Scholar
  14. Rüter, M., Stein, E.: Adaptive finite element analysis of crack propagation in elastic fracture mechanics based on averaging techniques. Comp. Mat. Sci. 31, 247–257 (2004)CrossRefGoogle Scholar
  15. Rüter, M., Stein, E.: On the duality of finite element discretization error control in computational Newtonian and Eshelbian mechanics. Comput. Mech. 39, 609–630 (2007)MathSciNetCrossRefGoogle Scholar
  16. Stein, E., Rüter, M.: Finite element methods for elasticity with error-controlled discretization and model adaptivity. In: Stein, E., de Borst, R., Hughes, T.J.R. (eds.) Encyclopedia of Computational Mechanics, vol. 3, 2nd edn., pp. 5–100. John Wiley & Sons, Chichester (2017)Google Scholar
  17. Whiteley, J.P., Tavener, S.J.: Error estimation and adaptivity for incompressible hyperelasticity. Int. J. Numer. Meth. Engng. 99, 313–332 (2014)MathSciNetCrossRefGoogle Scholar
  18. Wiberg, N.E., Abdulwahab, F.: Patch recovery based on superconvergent derivatives and equilibrium. Int. J. Numer. Meth. Engng. 36, 2703–2724 (1993)MathSciNetCrossRefGoogle Scholar
  19. Zienkiewicz, O.C., Zhu, J.Z.: A simple error estimator and adaptive procedure for practical engineering analysis. Int. J. Numer. Meth. Engng. 24, 337–357 (1987)MathSciNetCrossRefGoogle Scholar
  20. Zienkiewicz, O.C., Zhu, J.Z.: The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique. Int. J. Numer. Meth. Engng. 33, 1331–1364 (1992a)MathSciNetCrossRefGoogle Scholar
  21. Zienkiewicz, O.C., Zhu, J.Z.: The superconvergent patch recovery and a posteriori error estimates. Part 2: Error estimates and adaptivity. Int. J. Numer. Meth. Engng. 33, 1365–1382 (1992b)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Civil and Environmental EngineeringUniversity of CaliforniaLos AngelesUSA

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