An a posteriori error analysis for dynamic viscoelastic problems

Fernández, J. R.; Santamarina, D.

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 45 (2011) no. 5, pp. 925-945.

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.

[1] J. Ahn and D.E. Stewart, Dynamic frictionless contact in linear viscoelasticity. IMA J. Numer. Anal. 29 (2009) 43-71. | MR | Zbl

[2] M. Barboteu, J.R. Fernández and T.-V. Hoarau-Mantel, A class of evolutionary variational inequalities with applications in viscoelasticity. Math. Models Methods Appl. Sci. 15 (2005) 1595-1617. | MR | Zbl

[3] M. Barboteu, J.R. Fernández and R. Tarraf, Numerical analysis of a dynamic piezoelectric contact problem arising in viscoelasticity. Comput. Methods Appl. Mech. Eng. 197 (2008) 3724-3732. | MR | Zbl

[4] A. Bergam, C. Bernardi and Z. Mghazli, A posteriori analysis of the finite element discretization of some parabolic equations. Math. Comp. 74 (2005) 1117-1138. | MR | Zbl

[5] C. Bernardi and R. Verfürth, A posteriori error analysis of the fully discretized time-dependent Stokes equations. ESAIM: M2AN 38 (2004) 437-455. | Numdam | MR | Zbl

[6] D.A. Burkett and R.C. Maccamy, Differential approximation for viscoelasticity. J. Integral Equations Appl. 6 (1994) 165-190. | MR | Zbl

[7] M. Campo, J.R. Fernández, W. Han and M. Sofonea, A dynamic viscoelastic contact problem with normal compliance and damage. Finite Elem. Anal. Des. 42 (2005) 1-24. | MR

[8] M. Campo, J. R. Fernández, K.L. Kuttler, M. Shillor and J.M. Viaño, Numerical analysis and simulations of a dynamic frictionless contact problem with damage. Comput. Methods Appl. Mech. Eng. 196 (2006) 476-488. | MR | Zbl

[9] 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. | MR | Zbl

[10] P. Clément, Approximation by finite element functions using local regularization. RAIRO Anal. Numér. 9 (1975) 77-84. | Numdam | MR | Zbl

[11] M. Cocou, Existence of solutions of a dynamic Signorini's problem with nonlocal friction in viscoelasticity. Z. Angew. Math. Phys. 53 (2002) 1099-1109. | MR | Zbl

[12] G. Del Piero and L. Deseri, On the concepts of state and free energy in linear viscoelasticity. Arch. Rational Mech. Anal. 138 (1997) 1-35. | MR | Zbl

[13] G. Duvaut and J.L. Lions, Inequalities in mechanics and physics. Springer Verlag, Berlin (1976). | MR | Zbl

[14] 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). | MR | Zbl

[15] M. Fabrizio and S. Chirita, Some qualitative results on the dynamic viscoelasticity of the Reissner-Mindlin plate model. Quart. J. Mech. Appl. Math. 57 (2004) 59-78. | MR | Zbl

[16] 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). | MR | Zbl

[17] J.R. Fernández and P. Hild, A priori and a posteriori error analyses in the study of viscoelastic problems. J. Comput. Appl. Math. 225 (2009) 569-580. | MR | Zbl

[18] W. Han and M. Sofonea, Quasistatic contact problems in viscoelasticity and viscoplasticity. American Mathematical Society-International Press (2002). | MR | Zbl

[19] C. Johnson, Y.-Y. Nie and V. Thomée, 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. | MR | Zbl

[20] M. Karamanou, S. Shaw, M.K. Warby and J.R. Whiteman, Models, algorithms and error estimation for computational viscoelasticity. Comput. Methods Appl. Mech. Eng. 194 (2005) 245-265. | MR | Zbl

[21] K.L. Kuttler, M. Shillor and J.R. Fernández, Existence and regularity for dynamic viscoelastic adhesive contact with damage. Appl. Math. Optim. 53 (2006) 31-66. | MR | Zbl

[22] P. Le Tallec, Numerical analysis of viscoelastic problems, Research in Applied Mathematics. Springer-Verlag, Berlin (1990). | MR | Zbl

[23] S. Migórski and A. Ochal, A unified approach to dynamic contact problems in viscoelasticity, J. Elasticity 83 (2006) 247-275. | MR | Zbl

[24] J.E. Muñoz Rivera, Asymptotic behaviour in linear viscoelasticity. Quart. Appl. Math. 52 (1994) 628-648. | MR | Zbl

[25] M. Picasso, Adaptive finite elements for a linear parabolic problem. Comput. Methods Appl. Mech. Eng. 167 (1998) 223-237. | MR | Zbl

[26] B. Rivière, S. Shaw and J.R. Whiteman, Discontinuous Galerkin finite element methods for dynamic linear solid viscoelasticity problems. Numer. Methods Partial Differential Equations 23 (2007) 1149-1166. | MR | Zbl

[27] R. Verfürth, A review of a posteriori error estimation and adaptive mesh-refinement techniques. Wiley and Teubner (1996). | Zbl

[28] R. Verfürth, A posteriori error estimates for finite element discretizations of the heat equation. Calcolo 40 (2003) 195-212. | MR | Zbl

[29] M.A. Zocher, S.E. Groves and D.H. Allen, A three-dimensional finite element formulation for thermoviscoelastic orthotropic media. Int. J. Numer. Methods Eng. 40 (1997) 2267-2288. | MR | Zbl