Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
To avoid confusion, the reader is reminded that in this chapter, the notation \(\varvec{\tilde{P}}\) refers to an improved first Piola-Kirchhoff stress solution rather than to the isochoric part of \(\varvec{P}\) introduced in Sect. 2.3.3.
References
Bank, R.E.: Hierarchical bases and the finite element method. Acta Numer. 1–43 (1996)
Bank, R.E.: A simple analysis of some a posteriori error estimates. Appl. Numer. Math. 26, 153–164 (1998)
Bank, R.E., Smith, R.K.: A posteriori error estimates based on hierarchical bases. SIAM J. Numer. Anal. 30, 921–935 (1993)
Blacker, T., Belytschko, T.: Superconvergent patch recovery with equilibrium and conjoint interpolation enhancements. Int. J. Numer. Meth. Engng. 37, 517–536 (1994)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
Whiteley, J.P., Tavener, S.J.: Error estimation and adaptivity for incompressible hyperelasticity. Int. J. Numer. Meth. Engng. 99, 313–332 (2014)
Wiberg, N.E., Abdulwahab, F.: Patch recovery based on superconvergent derivatives and equilibrium. Int. J. Numer. Meth. Engng. 36, 2703–2724 (1993)
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)
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)
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)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Rüter, M.O. (2019). Goal-oriented A Posteriori Error Estimates in Finite Hyperelasticity. In: Error Estimates for Advanced Galerkin Methods. Lecture Notes in Applied and Computational Mechanics, vol 88. Springer, Cham. https://doi.org/10.1007/978-3-030-06173-9_8
Download citation
DOI: https://doi.org/10.1007/978-3-030-06173-9_8
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-06172-2
Online ISBN: 978-3-030-06173-9
eBook Packages: EngineeringEngineering (R0)