The reliability of computational models of physical processes has received much attention and involves issues such as the validity of the mathematical models being used, the error in any data that the models need, and the accuracy of the numerical schemes being used. These issues are considered in the context of elastic, viscoelastic and hyperelastic deformation, when finite element approximations are applied. Goal oriented techniques using specific quantities of interest (QoI) are described for estimating discretisation and modelling errors in the hyperelastic case. The computational modelling of the rapid large inflation of hyperelastic circular sheets modelled as axisymmetric membranes is then treated, with the aim of estimating engineering QoI and their errors. Fine (involving inertia terms) and coarse (quasistatic) models of the inflation are considered. The techniques are applied to thermoforming processes where sheets are inflated into moulds to form thin-walled structures.

The reliability of computational models of physical processes has received much attention in recent years and involves issues such as the validity of the mathematical models being used, the error in any data that the models need, and the accuracy of the numerical schemes being used. These issues are considered in the context of elastic and hyperelastic deformation, when finite element approximations are applied. Goal oriented techniques using specific quantities of interest (QoI) are used for estimating discretisation and modelling errors. The computational modelling of the rapid large inflation of hyperelastic circular sheets modelled as axisymmetric membranes is treated, with the aim of estimating engineering QoI and their errors. Fine (involving inertia terms) and coarse (quasistatic) models of the inflation are considered. The techniques are applied to thermoforming processes where sheets are inflated into moulds to form thin-walled structures. Current work extending the modelling to thermoplastic materials (ie biomaterials) is discussed, particularly from the aspect of reliability.

The computational modelling of the rapid large inflation of hyperelastic circular sheets modelled as axisymmetric membranes is treated, with the aim of estimating engineering quantities of interest and their errors. Fine (involving inertia terms) and coarse (quasi-static) models of the inflation are considered and, using goal-oriented techniques, both modelling and discretization errors are estimated. Numerical results involving only discretization errors for the quasi-static problem and both modelling and discretization errors for the dynamic problem are presented.

Plastic packaging waste currently forms a significant part of municipal solid waste and as such is causing increasing environmental concerns. Such packaging is largely non-biodegradable and is particularly difficult to recycle or to reuse due to its complex composition. Apart from limited recycling of some easily identifiable packaging wastes, such as bottles, most packaging waste ends up in landfill sites. In recent years, in an attempt to address this problem in the case of plastic packaging, the development of packaging materials from renewable plant resources has received increasing attention and a wide range of bioplastic materials based on starch are now available. Environmentally these bioplastic materials also reduce reliance on oil resources and have the advantage that they are biodegradable and can be composted upon disposal to reduce the environmental impact.Many food packaging containers are produced by thermoforming processes in which thin sheets are inflated under pressure into moulds to produce the required thin wall structures. Hitherto these thin sheets have almost exclusively been made of oil-based polymers and it is for these that computational models of thermoforming processes have been developed. Recently, in the context of bioplastics, commercial thermoplastic starch sheet materials have been developed. The behaviour of such materials is influenced both by temperature and, because of the inherent hydrophilic characteristics of the materials, by moisture content. Both of these aspects affect the behaviour of bioplastic sheets during the thermoforming process.This paper describes experimental work and work on the computational modelling of thermoforming processes for thermoplastic starch sheets in an attempt to address the combined effects of temperature and moisture content. After a discussion of the background of packaging and biomaterials, a mathematical model for the deformation of a membrane into a mould is presented, together with its finite element discretisation. This model depends on material parameters of the thermoplastic and details of tests undertaken to determine these and the results produced are given. Finally the computational model is applied for a thin sheet of commercially available thermoplastic starch material which is thermoformed into a specific mould. Numerical results of thickness and shape for this problem are given.

In this paper we describe the computational simulation of the inflation phase of a thermoforming process under which a thin polymer sheet is deformed into a mould under the action of applied pressure. It is assumed that the sheet undergoes finite viscoelastic deformation which is treated using a hyperelastic model containing internal variables. The simplification is adopted that the sheet can be treated as a membrane and also that there is a total sticking contact condition when the sheet comes in contact with the mould. The computational model uses finite elements in space and incorporates mesh adaptivity based on a residual estimator in order to simulate the deformation accurately and efficiently. The internal variables satisfy an ordinary differential equation in time which is solved using a predictor–corrector scheme. The constitutive model is a generalisation of that of Le Tallec and Rahier [P. Le Tallec, C. Rahier, Numerical models of steady rolling for non-linear viscoelastic structures in finite deformations, Int. J. Numer. Methods Engrg. 37 (1994) 1159–1186]. In the simulation it is demonstrated how effectively the estimator works in controlling the meshes for some demanding mould shapes.

This article reviews numerical algorithms for problems in solid polymer viscoelasticity in both small and large deformation. 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.

The problem of modelling and the finite element simulation of thermoforming processes for polymeric sheets at various temperatures and for different loading regimes is addressed. In particular, the vacuum forming process for sheets at temperatures of approximately 200°C and the Niebling process for sheets at temperature of 100°C with high pressure loading are both described. Discussion is given to the assumptions made concerning the behaviour of the polymers and the physical happenings in the process in order that realistic models of the inflation part of each process may be produced. Stress–strain curves produced from experimental testing of BAYFOL® at various strain rates and temperatures are presented. A model for the elastic–plastic deformation of BAYFOL® is described and is used within the finite element framework to simulate the inflation part of the Niebling process. Numerical results for the deformation of sheets into a mould in the Niebling context are presented.

In this paper we describe the mathematical modelling and computational simulation of the high air pressure (HAP) thermoforming process which is used in the creation of thin walled polymeric structures. This involves, using data from material tests, an elastic-plastic constitutive equation valid for large deformations and a constrained deformation in which there is frictional contact between the polymeric sheet and a constraining surface (the mould surface). Despite a number of simplifying assumptions and some uncertainities in the mathematical model the finite element computations presented predict quite well the actual shape and thickness distribution which are found on sample products.

In the context of the equilibrium equations governing an Euler-Bernoulli beam and an assembly of such beams in a frame structure, this article considers the superconvergence of various parameters at various points of the finite element solutions and describes an a posteriori error estimator of the Bank Weiser type. The error estimator is shown to be consistent with the energy norm in all cases and, in the superconvergent cases that we consider, it is also shown to be asymptotically exact. As shown, asymptotic exactness can be obtained by merely using quadratics (instead of linears) for the compression and twisting terms and, as usual, cubics for the bending terms. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 169–197, 2001

In this paper we describe the pressure-driven inflation of an incompressible isotropic hyperelastic membrane into a rigid mould by a variational inequality and consider the existence of a solution in the case of various, suitably modified, strain energy functions of the Ogden form. The variational inequality description is applicable to the case of perfect sliding contact of the membrane with the mould and the modification to the strain energy function is according to tension field theory which rules out compressive stresses. The modified or relaxed strain energy functions obtained are shown, in our examples, to be polyconvex and in some cases convex. Using such properties, the main result of the paper is an existence theorem for a solution of the variational inequality. Copyright (C) 2000 John Wiley & Sons, Ltd.

discrete nite element approximations for linear parabolic integro-diierential equations with integrable kernels,Volterra projection onto nite element spaces and applications to integro-diierential and related equations, SIAM J. Nonsmooth data error estimates for approximation of an evolution equation with a positive-type memory term, Math. Error estimates with sharp constants for a fading memory Volterra problem in linear solid viscoelasticity, SIAM J. Error estimates for semi-discrete nite element methods for parabolic integro-diierential equations, Math. 21 where K(t; s) is singular kernel, for example 0 K(t; s) g(t; s)K (t ? s) with g(t; s) smooth and bounded and K 2 L 1 (0; 1). In this case K in (A2) should be replaced by K = max t>0 Z t 0 K(t; s)e (t?s) ds; max 0<< 1 in order to satisfy (A2). It is easy to see from a simple calculation that R(t) = O(e ?(?1)t) and Z t 0 R(t ? s)R(s)ds = O(te ?(?1)t); so that ^ R(t) = O(te ?(?1)t). This implies that (3.9) may not be the best convergence rate estimates. (R2) Since jju(t) ? u h (t)jj jju(t) ? u 1 (t)jj + jju 1 (t) ? u 1 h (t)jj + jju 1 h (t) ? u h (t)jj so that if the assumptions of Theorems 5.2-5.4, Theorem 3.2 and Theorem 4.4 are satissed we have jju(t) ? u h (t)jj = O(h r + te ?t + e ?t) for the semi-discrete approximation and jju(t n) ? u n h jj = O(h r + t + te ?t + e ?t) for the backward Euler scheme. Therefore if te ?t n + e ?t h r + t or t …

The problem characterizing nonageing linear isothermal quasi-static isotropic compressible solid viscoelasticity in the time interval [0,T] is described. This is essentially a Volterra equation of the second kind arrived at by adding smooth fading memory to the elliptic linear elasticity equations. We analyze the errors resulting from replacing the relaxation functions with practical approximations, in a semidiscrete finite element approximation, and in a fully discrete scheme derived by replacing the hereditary integral with the trapezoidal rule for numerical integration. The error estimates are sharp in the sense that if certain bounds on the data are independent of T, then so also are the constants involved in them. This is a consequence of bypassing the usual Gronwall lemmas with arguments that are more sensitive to the fading memory of the physical problem.

Mathematical models for treating problems of linear viscoelasticity involving hereditary constitutive relations for compressible solids are presented, and their discretisation using finite element methods in space together with quadrature rules in time to treat the hereditary integrals is described. Theoretical error estimates in appropriate Sobolev space settings are given, both as they arise as a result of using a Gronwall inequality, and also from employing a more sensitive comparison theorem which (for the quasistatic problem, and under physically reasonable assumptions on the relaxation function) yields much sharper constants in the estimates.The range of applicability of the mathematical models, and hence the numerical schemes and error estimates are discussed in the context of various materials, primarily polymeric materials, and extensions of the techniques to the modelling of manufacturing processes such as thermoforming are presented.

Mathematical models for treating problems of linear viscoelasticity involving hereditary constitutive relations for compressible solids are discussed, and their discretization using finite element methods in space together with quadrature rules in time to treat the hereditary integrals is described. The range of applicability of this type of formulation is reviewed in the context of geometric and constitutive linearity/nonlinearity, and the limitations imposed by the availability of physical data are discussed.One of the above models is a Volterra integral equation of the second kind. In this, when the kernel is separable, an established technique due to Goursat (1933) can be exploited to reformulate the problem as a system of ordinary differential equations. This approach will be described. For the special case of a linear viscoelastic, isotropic, homogeneous, synchronous (constant Poisson's ratio) solid this method results in a complete decoupling of the space and time dependence. In this case the problem can be solved at each time level by solving first a problem of linear elasticity and then a system of ordinary differential equations for each point in the spatial mesh at which the viscoelastic displacements are required. The advantages, disadvantages and limitations offered by this, and various other schemes for solving problems of viscoelasticity as outlined below, are discussed.

For quasistatic stress problems two alternative constitutive relationships expressing the stress in a linear isotropic viscoelastic solid body as a linear functional of the strain are available. In conjunction with the equations of equilibrium, these form the mathematical models for the stress problems. These models are first discretized in the space domain using a finite element method and semi-discrete error estimates are presented corresponding to each constitutive relationship. Through the use respectively of quadrature rules and finite difference replacements each semi-discrete scheme is fully discretized into the time domain so that two practical algorithms suitable for the numerical stress analysis of linear viscoelastic solids are produced. The semi-discrete estimates are then also extended into the time domain to give spatially H-1 error estimates for each algorithm.The numerical schemes are predicted on exact analytical solutions for a simple model problem, and finally on design data for a real polymeric material.

A fully discrete scheme for approximating a second order hyperbolic Volterra integrodiierential equation of the second kind is proposed. The discretization is accomplished by applying the nite element method in the space variables, a nite diierence replacement for the time derivative and the trapezoidal rule for the history integral. Using the Ritz{Volterra projection the error associated with the scheme is analyzed and an optimal error estimate derived. This bound depends crucially on the way the initial conditions are represented in the scheme in a similar way as for analogous discretizations of the wave equation.