Abstract
The paper considers an hp-finite element discretization for a model problem of elastoplasticity with linear kinematic hardening. The use of biorthogonal basis functions for the discretization of a mixed formulation enables the decoupling of the constraints resulting from the involved plasticity functional. This yields a nonlinear equation which allows the application of various solution schemes, e.g. a semi-smooth Newton solver. Numerical experiments demonstrate the robustness of a semi-smooth Newton solver based on the proposed decoupling with respect to mesh size, polynomial degree and projection parameter.
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
References
Alberty, J., Carstensen, C.: Numerical analysis of time-depending primal elastoplasticity with hardening. SIAM J. Numer. Anal. 37, 1271–1294 (2000)
Bammer, P., Banz, L., Schröder, A.: hp-FEM for a mixed variational approach in elastoplasticity (in preperation)
Bammer, P., Banz, L., Schröder, A.: A posteriori error estimates and a semi-smooth Newton solver for hp-FEM in elastoplasticity (in preperation)
Banz, L., Schröder, A.: Biorthogonal basis functions in hp-adaptive FEM for elliptic obstacle problems. Comput. Math. Appl. 70, 1721–1742 (2015)
Carstensen, C.: Numerical analysis of the primal problem of elastoplasticity with hardening. Numer. Math. 82, 577–597 (1999)
Carstensen, C., Schröder, A., Wiedemann, S.: An optimal adaptive finite element method for elastoplasticity. Numer. Math. 132, 131–154 (2016)
Christensen, P.W.: A nonsmooth Newton method for elastoplastic problems. Comput. Methods Appl. Mech. Eng. 191, 1189–1219 (2002)
Gruber, P., Kienesberger, J., Langer, U., Schöberl, J., Valdman, J.: Fast solvers and a posteriori error estimates in elastoplasticity. In: Numerical and Symbolic Scientific Computing, pp. 45–63. Springer, NewYork (2012)
Gwinner, J.: hp-FEM convergence for unilateral contact problems with Tresca friction in plane linear elastostatics. J. Comput. Appl. Math. 254, 175–184 (2013)
W. Han.: Finite element analysis of a holonomic elastic-plastic problem. Numer. Math. 60, 493–508 (1991)
Han, W., Reddy, B.D.: On the finite element method for mixed variational inequalities arising in elastoplasticity. SIAM J. Numer. Anal. 32, 1778–1807 (1995)
Han, W., Reddy, B.D.: Plasticity: Mathematical Theory and Numerical Analysis, 2nd edn. Springer, New York (2013)
Haslinger, J., Panagiotopoulos, P.D.: Finite element method for hemivariational inequalities. In: Theory, Methods and Applications. Springer, Berlin (1999)
Hintermüller, M., Rösel, S.: A duality-based path-following semismooth Newton method for elasto-plastic contact problems. J. Comput. Appl. Math. 292, 150–173 (2016)
Kienesberger, J., Valdman, J.: An efficient solution algorithm for elastoplasticity and its first implementation towards uniform h- and p- mesh refinements. In: Numerical Mathematics and Advanced Applications, pp. 1117–1125. Springer, Berlin (2006)
A. Schröder, S. Wiedemann.: Error estimates in elastoplasticity using a mixed method. Appl. Numer. Math. 61, 1031–1045 (2011)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Bammer, P., Banz, L., Schröder, A. (2023). hp-Finite Elements with Decoupled Constraints for Elastoplasticity. In: Melenk, J.M., Perugia, I., Schöberl, J., Schwab, C. (eds) Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2020+1. Lecture Notes in Computational Science and Engineering, vol 137. Springer, Cham. https://doi.org/10.1007/978-3-031-20432-6_7
Download citation
DOI: https://doi.org/10.1007/978-3-031-20432-6_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-20431-9
Online ISBN: 978-3-031-20432-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)