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Models, algorithms and error estimation for computational viscoelasticity. (English) Zbl 1067.74066
Summary: This article reviews numerical algorithms for problems in solid polymer viscoelasticity for both small and large deformations. For the linear (small strain) case we review both the quasistatic and the dynamic problem and give recent results on a posteriori error estimation. For the large strain case we focus on the formulation and computational modelling of constrained membrane inflation, the application of which is to the thermoforming process.

MSC:
74S05 Finite element methods applied to problems in solid mechanics
74D05 Linear constitutive equations for materials with memory
74D10 Nonlinear constitutive equations for materials with memory
74K15 Membranes
74-02 Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids
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[1] W. Bangerth, R. Rannacher, Finite element approximation of the acoustic wave equation: error control and mesh adaptation, gaia.iwr.uni-heidelberg.de/Paper/Preprintl999-15.pdf, 1999 · Zbl 0948.65098
[2] Shaw, S.; Whiteman, J.R., Numerical solution of linear quasistatic hereditary viscoelasticity problems, SIAM J. numer. anal., 38, 1, 80-97, (2000) · Zbl 0991.74076
[3] S. Shaw, J.R. Whiteman, A posteriori error estimates for space-time finite element approximation of quasistatic hereditary linear viscoelasticity problems, Comput. Methods Appl. Mech. Engrg. See also BICOM Technical Report 03/2 at www.brunel.ac.uk/ icsrbicm, in press · Zbl 1076.74056
[4] S. Shaw, J.R. Whiteman, A posteriori error estimates for space-time finite element approximation of quasistatic hereditary linear viscoelasticity problems, BICOM TR03/2, www.brunel.ac.uk/ icsrbicm, 2003 · Zbl 1076.74056
[5] S. Shaw, J.R. Whiteman, Lp(0,t) error control using the derivative of the residual for a finite element approximation of a second-kind Volterra equation, BICOM TR03/1, www.brunel.ac.uk/ icsrbicm, 2003
[6] Eriksson, K.; Estep, D.; Hansbo, P.; Johnson, C., Introduction to adaptive methods for differential equations, Acta numer., Cambridge univ. press, 0, 105-158, (1995) · Zbl 0829.65122
[7] Johnson, C.; Hansbo, P., Adaptive finite element methods in computational mechanics, Comput. methods appl. mech. engrg., 101, 143-181, (1992) · Zbl 0778.73071
[8] Lockett, F.J., Nonlinear viscoelastic solids, (1972), Academic Press London · Zbl 0333.73034
[9] Shaw, S.; Whiteman, J.R., Negative norm error control for second-kind convolution Volterra equations, Numer. math., 85, 329-341, (2000) · Zbl 0955.65094
[10] Shaw, S.; Whiteman, J.R., Optimal long-time Lp(0,T) data stability and semidiscrete error estimates for the Volterra formulation of the linear quasistatic viscoelasticity problem, Numer. math., 88, 743-770, (2001), (BICOM Tech. Rep. 98/7 see: www.brunel.ac.uk/ icsrbicm) · Zbl 1002.74020
[11] R. Rannacher, Adaptive Galerkin finite element methods for partial differential equations, gaia.iwr.uni-heidelberg.de/Paper/Preprint1999-20.pdf, 1999
[12] Ranriacher, R., The dual-weighted-residual method for error control and rnesh adaption in finite element methods, (), 97-116
[13] Johnson, A.R., Modeling viscoelastic materials using internal variables, The shock and vibration digest, 31, 91-100, (1999)
[14] Johnson, A.R.; Tessler, A., A viscoelastic high order beam finite element, (), 333-345 · Zbl 0893.73064
[15] Johnson, A.R., A viscoelastic hybrid shell finite element, (), 87-96 · Zbl 0983.74066
[16] Johnson, A.R..; Tessler, A.; Dambach, M., Dynamics of thick viscoelastic beams, J. engrg. mater. technol., 119, 273-278, (1997)
[17] Le Tallec, P.; Rahier, C., Numerical models of steady rolling for nonlinear viscoelastic structures in finite deformations, Int. J. numer. methods engrg., 38, 1159-1186, (1994) · Zbl 0804.73063
[18] Eriksson, K.; Estep, D.; Hansbo, P.; Johnson, C., Computational differential equations, (1996), Cambridge University Press
[19] Johnson, C., Discontinuous Galerkin finite element methods for second order hyperbolic problems, Comput. methods appl. mech. engrg., 107, 117-129, (1993) · Zbl 0787.65070
[20] Hughes, T.J.R.; Hulbert, G.M., Space-time finite element methods for elastodynamics: formulations and error estimates, Comput. methods appl. mech. engrg., 66, 339-363, (1988) · Zbl 0616.73063
[21] Hulbert, G.M.; Hughes, T.J.R., Space-time finite element methods for second-order hyperbolic equations, Comput. methods appl. mech. engrg., 84, 327-348, (1990) · Zbl 0754.73085
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