M. K. Warby

Finite element simulation of thermoforming processes for polymer sheets

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.

Numerical techniques for the treatment of quasistatic viscoelastic stress problems in linear isotropic solids

Abstract 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 predicated on exact analytical solutions for a simple model problem, and finally on design data for a real polymeric material.

Finite element model of viscoelastic membrane deformation

A numerical scheme is presented for modelling the axisymmetric inflation of a thin incompressible isotropic sheet up to a rigid obstacle under the action of a uniform pressure. A membrane model is assumed for the behaviour of the sheet. When the sheet makes contact with the obstacle it is further assumed that a condition of total sticking occurs. In the scheme the equilibrium and constitutive equations are kept separate and a modified H1-Galerkin technique using cubic Hermite approximations is employed. Numerical results are presented for the cases of both flat and elliptical obstacles and for both materials of the Mooney-Rivlin elastic type and of a viscoelastic generalization of the Mooney-Rivlin model.

Stability and convergence properties of Bergman kernel methods for numerical conformal mapping

SummaryIn this paper we study the stability and convergence properties of Bergman kernel methods, for the numerical conformal mapping of simply and doubly-connected domains. In particular, by using certain wellknown results of Carleman, we establish a characterization of the level of instability in the methods, in terms of the geometry of the domain under consideration. We also explain how certain known convergence results can provide some theoretical justification of the observed improvement in accuracy which is achieved by the methods, when the basis set used contains functions that reflect the main singular behaviour of the conformal map.

Numerical techniques for conformal mapping onto a rectangle

Abstract This paper is concerned with the problem of determining approximations to the function F which maps conformally a simply-connected domain Ω onto a rectangle R, so that four specified points on ∂Ω are mapped respectively onto the four vertices of R. In particular, we study the following two classes of methods for the mapping of domains of the form Ω≔ {z = x + iy:00 1 (x) 2 (x)} . (i) Methods which approximate F: Ω → R by F = S ∘ F , where F is an approximation to the conformal map of Ω onto the unit disc, and S is a simple Schwarz-Christoffel transformation. (ii) Methods based on approximating the conformal map of a certain symmetric doubly-connected domain onto a circular annulus.

The treatment of corner and pole-type singularities in numerical conformal mapping techniques

Abstract This paper is a report of recent developments concerning the nature and the treatment of singularities that affect certain numerical conformal mapping techniques. The paper also includes some new results on the nature of singularities that the mapping function may have in the complement of the closure of the domain under consideration.

Error Estimates with Sharp Constants for a Fading Memory Volterra Problem in Linear Solid Viscoelasticity

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.

Pole-type singularities and the numerical conformal mapping of doubly-connected domains

Abstract Let f be the function which maps conformally a given doubly-connected domain Ω onto a circular annulus, and let H(z)= f′(z) f(z) − 1 z . In this paper we consider the problem of determining the main singularities of the function H in compl( Ω∪∂Ω ). Our purpose is to provide information regarding the location and nature of such singularities, and to explain how this information can be used to improve the efficiency of certain expansion methods for numerical conformal mapping.

Numerical methods for treating problems of viscoelastic isotropic solid deformation

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 Poissons 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.

The determination of the poles of the mapping function and their use in numerical conformal mapping

Abstract Let ƒ be the function which maps conformally a simply-connected domain Ω onto the unit disc. This paper is concerned with the problem of determining the dominant poles of ƒ in compl(Ω ∪ ∂ Ω), and of using this information in order to obtain accurate numerical approximations to ƒ by means of the Bergman kernel method.