Hostname: page-component-848d4c4894-m9kch Total loading time: 0 Render date: 2024-06-06T07:41:23.727Z Has data issue: false hasContentIssue false
  • English
  • Français

An a posteriori error analysis for dynamic viscoelastic problems

Published online by Cambridge University Press:  26 April 2011

J. R. Fernández  and
D. Santamarina
J. R. Fernández
Affiliation:
Departamento de Matemática Aplicada I, ETSE de Telecomunicación, Universidade de Vigo, Campus As Lagoas Marcosende s/n, 36310 Vigo, Spain. jose.fernandez@uvigo.es
D. Santamarina
Affiliation:
Departamento de Matemática Aplicada, Escola Politécnica Superior, Campus Univ. s/n, Universidade de Santiago de Compostela, 27002 Lugo, Spain. duarte.santamarina@usc.es
Get access

Abstract


In this paper, a dynamic viscoelastic problem is numerically studied. The variational problem is written in terms of the velocity field and it leads to a parabolic linear variational equation. A fully discrete scheme is introduced by using the finite element method to approximate the spatial variable and an Euler scheme to discretize time derivatives. An a priori error estimates result is recalled, from which the linear convergence is derived under suitable regularity conditions. Then, an a posteriori error analysis is provided, extending some preliminary results obtained in the study of the heat equation and quasistatic viscoelastic problems. Upper and lower error bounds are obtained. Finally, some two-dimensional numerical simulations are presented to show the behavior of the error estimators.

Keywords

Viscoelasticitydynamic problemsfully discrete approximationsa posteriori error estimatesfinite elementsnumerical simulations
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 45 , Issue 5 , September 2011 , pp. 925 - 945
Copyright
© EDP Sciences, SMAI, 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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. 3.0.CO;2-P>CrossRef