Abstract
The discontinuous Galerkin (dG) methodology provides a hierarchy of time discretization schemes for evolutionary problems such as elastoplasticity with the Prandtl-Reuß flow rule. A dG time discretization has been proposed for a variational inequality in the context of rate-independent inelastic material behaviour in Alberty and Carstensen in (CMME 191:4949–4968, 2002) with the help of duality in convex analysis to justify certain jump terms. This paper establishes the first a priori error analysis for the dG(1) scheme with discontinuous piecewise linear polynomials in the temporal and lowest-order finite elements for the spatial discretization. Compared to a generalized mid-point rule, the dG(1) formulation distributes the action of the material law in the form of the variational inequality in time and so it introduces an error in the material law. This may result in a suboptimal convergence rate for the dG(1) scheme and this paper shows that the stress error in the \(L^\infty (L^2)\) norm is merely \(O(h+k^{3/2})\) based on a seemingly sharp error analysis. The numerical investigation for a benchmark problem with known analytic solution provides empirical evidence of a higher convergence rate of the dG(1) scheme compared to dG(0).
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References
Alberty, J.: Zeitdiskretisierungsverfahren für elastoplastische Probleme der Kontinuumsmechanik. Ph.D. thesis, University of Kiel, FRG, Tenea Verlag (2001)
Alberty, J., Carstensen, C.: Numerical analysis of time-depending primal elastoplastic with hardening. SIAM J. Numer. Anal. 37, 1271–1294 (2000)
Alberty, J., Carstensen, C.: Discontinuous Galerkin time discretization in elastoplasticity: motivation, numerical algorithms and applications. Comp. Methods Appl. Mech. Eng. 191, 4949–4968 (2002)
Alberty, J., Carstensen, C., Zarrabi, D.: Adaptive numerical analysis in primal elastoplasticity with hardening. Comput. Methods Appl. Mech. Eng. 171, 175–204 (1999)
Bartels, Sören, Mielke, Alexander, Roubíček, Tomáš: Quasi-static small-strain plasticity in the limit of vanishing hardening and its numerical approximation. SIAM J. Numer. Anal. 50, 951–976 (2012)
Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, 15, Texts in Applied Mathematics. Springer, New York (1994)
Carstensen, C.: Domain decomposition for a non-smooth convex minimisation problem and its application to plasticity. NLAA 4, 1–13 (1997)
Carstensen, C.: Numerical analysis of the primal problem of elastoplasticity with hardening. Numer. Math. 82, 577–597 (1999)
Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)
Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. North-Holland, Amsterdam (1976)
Evans, L.C.: Partial Differential Equations, 19, Graduate Studies in Mathematics. American Mathematical Society, Providence (1998)
Han, W., Reddy, B.D.: Plasticity, 2nd edn. IAM, Springer, New York (2013)
Johnson, C.: Existence theorems for plasticity problems. J. Math. pures et appl. 55, 431–444 (1976)
Johnson, C.: On plasticity with hardening. J. Math. Anal. Appl. 62, 325–336 (1978)
Rindler, Filip, Schwarzacher, Sebastian, Süli, Endre: Regularity and approximation of strong solutions to rate-independent systems. Math. Models Methods Appl. Sci. 27, 2511–2556 (2017)
Seregin, G.A.: On the regularity of the minimizers of some variational problems of plasticity theory. St. Petersb. Math. J. 4, 989–1020 (1993)
Simo, J.C., Hughes, T.J.R.: Computational Inelasticity. Springer, New York (1998)
Suquet, P.M.: Discontinuities and plasticity, Nonsmooth Mechanics and Applications, CISM Courses, 302, pp. 279–341. Springer, New York (1988)
Temam, R.: Mathematical Problems in Plasticity. Gauthier-Villars, Paris (1985)
Thomee, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer, Berlin (2006)
Zeidler E.: Nonlinear Functional Analysis and Its Applications iii and iv. Springer, New York (1985,1988)
Acknowledgements
The work has been written, while the authors enjoyed the hospitality of the Hausdorff Research Institute of Mathematics in Bonn, Germany, during the Hausdorff Trimester Program Multiscale Problems: Algorithms, Numerical Analysis and Computation. The research of the first author (CC) has been supported by the Deutsche Forschungsgemeinschaft in the Priority Program 1748 ‘Reliable simulation techniques in solid mechanics. Development of non-standard discretization methods, mechanical and mathematical analysis’ under the project ‘Foundation and application of generalized mixed FEM towards nonlinear problems in solid mechanics’ (CA 151/22-1). The corresponding author’s research was supported by National Natural Science Foundation of China (Nos. 11571226, 11671313). The authors thank Simone Hock for a collaboration at the early stage of this research and the anonymous referees for valuable remarks that improved the presentation.
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Dedicated to Alexander Mielke on the occasion of his 60th birthday.
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Carstensen, C., Liu, D. & Alberty, J. Convergence of \(\text{ dG(1) }\) in elastoplastic evolution. Numer. Math. 141, 715–742 (2019). https://doi.org/10.1007/s00211-018-0999-6
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DOI: https://doi.org/10.1007/s00211-018-0999-6