Models, algorithms and error estimation for computational viscoelasticity Marianna Karamanou * , Simon Shaw, M.K. Warby, J.R. Whiteman BICOM and Department of Mathematical Sciences, Brunel University, Uxbridge UB8 3PH, England Accepted 11 May 2004 Abstract This article reviews numerical algorithms for problems in solid polymer viscoelasticity in both small and large defor- mation. 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 model- ling of constrained membrane inflation, the application of which is to the thermoforming process. Ó 2004 Published by Elsevier B.V. Keywords: Thermoforming; Membranes; Finite strain; Viscoelasticity; Adaptivity; a posteriori error estimates 1. Introduction In this paper we describe computational models of both small strain and finite strain viscoelastic defor- mations of solid polymers. In the small strain case, which gives a linear model, we consider both quasistatic and dynamic problems and, for the former, we outline recent results on a priori and a posteriori error-norm estimation. For the dynamic problem our methodology has so far been confined to goal-oriented a poste- riori error estimation, and we outline our results for this in the context of a viscoelastic Timoshenko beam. The error estimates given can be used to drive adaptive algorithms. In these models, the viscoelasticity is modelled using a hereditary (Volterra) integral for the quasistatic problem, and internal variables for the dynamic problem. 0045-7825/$ - see front matter Ó 2004 Published by Elsevier B.V. doi:10.1016/j.cma.2004.05.013 * Corresponding author. E-mail addresses: marianna.karamanou@brunel.ac.uk (M. Karamanou), simon.shaw@brunel.ac.uk (S. Shaw), mike.warby@ brunel.ac.uk (M.K. Warby), john.whiteman@brunel.ac.uk (J.R. Whiteman). URL: www.brunel.ac.uk/~icsrbicm Comput. Methods Appl. Mech. Engrg. 194 (2005) 245–265 www.elsevier.com/locate/cma