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An a posteriori error analysis for dynamic viscoelastic problems
Published online by Cambridge University Press: 26 April 2011
Abstract
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In this paper, a dynamic viscoelastic problem is numerically studied. The variationalproblem is written in terms of the velocity field and it leads to a parabolic linearvariational equation. A fully discrete scheme is introduced by using thefinite element method to approximate the spatial variable andan Euler scheme to discretize time derivatives. An a priori error estimatesresult is recalled, from which the linear convergence is derived under suitableregularity conditions. Then, an a posteriorierror analysis is provided, extending some preliminary resultsobtained in the study of the heat equation and quasistatic viscoelastic problems.Upper and lower error bounds are obtained. Finally, some two-dimensionalnumerical simulations are presented to show the behavior of the error estimators.
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- ESAIM: Mathematical Modelling and Numerical Analysis , Volume 45 , Issue 5 , September 2011 , pp. 925 - 945
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- © EDP Sciences, SMAI, 2011
References
Ahn, J. and Stewart, D.E., Dynamic frictionless contact in linear viscoelasticity. IMA J. Numer. Anal. 29 (2009) 43–71.
CrossRef
Barboteu, M., Fernández, J.R. and Hoarau-Mantel, T.-V., A class of evolutionary variational inequalities with applications in viscoelasticity. Math. Models Methods Appl. Sci. 15 (2005) 1595–1617.
CrossRef
Barboteu, M., Fernández, J.R. and Tarraf, R., Numerical analysis of a dynamic piezoelectric contact problem arising in viscoelasticity. Comput. Methods Appl. Mech. Eng. 197 (2008) 3724–3732.
CrossRef
Bergam, A., Bernardi, C. and Mghazli, Z., A posteriori analysis of the finite element discretization of some parabolic equations. Math. Comp. 74 (2005) 1117–1138.
CrossRef
C. Bernardi and R. Verfürth, A posteriori error analysis of the fully discretized time-dependent Stokes equations. ESAIM: M2AN 38 (2004) 437–455.
CrossRef
Burkett, D.A. and MacCamy, R.C., Differential approximation for viscoelasticity. J. Integral Equations Appl. 6 (1994) 165–190.
CrossRef
Campo, M., Fernández, J.R., Han, W. and Sofonea, M., A dynamic viscoelastic contact problem with normal compliance and damage. Finite Elem. Anal. Des. 42 (2005) 1–24.
CrossRef
Campo, M., Fernández, J. R., Kuttler, K.L., Shillor, M. and Viaño, J.M., Numerical analysis and simulations of a dynamic frictionless contact problem with damage. Comput. Methods Appl. Mech. Eng. 196 (2006) 476–488.
CrossRef
P.G. Ciarlet, The finite element method for elliptic problems, in Handbook of Numerical Analysis, P.G. Ciarlet and J.L. Lions Eds., Vol. II, North Holland (1991) 17–352.
Clément, P., Approximation by finite element functions using local regularization. RAIRO Anal. Numér. 9 (1975) 77–84.
Cocou, M., Existence of solutions of a dynamic Signorini's problem with nonlocal friction in viscoelasticity. Z. Angew. Math. Phys. 53 (2002) 1099–1109.
CrossRef
Del Piero, G. and Deseri, L., On the concepts of state and free energy in linear viscoelasticity. Arch. Rational Mech. Anal. 138 (1997) 1–35.
CrossRef
G. Duvaut and J.L. Lions, Inequalities in mechanics and physics. Springer Verlag, Berlin (1976).
C. Eck, J. Jarusek and M. Krbec, Unilateral contact problems. Variational methods and existence theorems, Pure and Applied Mathematics 270. Chapman & Hall/CRC, Boca Raton (2005).
Fabrizio, M. and Chirita, S., Some qualitative results on the dynamic viscoelasticity of the Reissner-Mindlin plate model. Quart. J. Mech. Appl. Math. 57 (2004) 59–78.
CrossRef
M. Fabrizio and A. Morro, Mathematical problems in linear viscoelasticity, SIAM Studies in Applied Mathematics 12. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1992).
Fernández, J.R. and Hild, P., A priori and a posteriori error analyses in the study of viscoelastic problems. J. Comput. Appl. Math. 225 (2009) 569–580.
CrossRef
W. Han and M. Sofonea, Quasistatic contact problems in viscoelasticity and viscoplasticity. American Mathematical Society-International Press (2002).
Johnson, C., Nie, Y.-Y. and Thomée, V., An a posteriori error estimate and adaptive timestep control for a backward Euler discretization of a parabolic problem. SIAM J. Numer. Anal. 27 (1990) 277–291.
CrossRef
Karamanou, M., Shaw, S., Warby, M.K. and Whiteman, J.R., Models, algorithms and error estimation for computational viscoelasticity. Comput. Methods Appl. Mech. Eng. 194 (2005) 245–265.
CrossRef
Kuttler, K.L., Shillor, M. and Fernández, J.R., Existence and regularity for dynamic viscoelastic adhesive contact with damage. Appl. Math. Optim. 53 (2006) 31–66.
CrossRef
P. Le Tallec, Numerical analysis of viscoelastic problems, Research in Applied Mathematics. Springer-Verlag, Berlin (1990).
S. Migórski and A. Ochal, A unified approach to dynamic contact problems in viscoelasticity, J. Elasticity 83 (2006) 247–275.
CrossRef
Muñoz Rivera, J.E., Asymptotic behaviour in linear viscoelasticity. Quart. Appl. Math. 52 (1994) 628–648.
CrossRef
Picasso, M., Adaptive finite elements for a linear parabolic problem. Comput. Methods Appl. Mech. Eng. 167 (1998) 223–237.
CrossRef
Rivière, B., Shaw, S. and Whiteman, J.R., Discontinuous Galerkin finite element methods for dynamic linear solid viscoelasticity problems. Numer. Methods Partial Differential Equations 23 (2007) 1149–1166.
CrossRef
R. Verfürth, A review of a posteriori error estimation and adaptive mesh-refinement techniques. Wiley and Teubner (1996).
Verfürth, R., A posteriori error estimates for finite element discretizations of the heat equation. Calcolo 40 (2003) 195–212.
CrossRef
Zocher, M.A., Groves, S.E. and Allen, D.H., A three-dimensional finite element formulation for thermoviscoelastic orthotropic media. Int. J. Numer. Methods Eng. 40 (1997) 2267–2288.
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