J. R. Whiteman

Graded mesh refinement and error estimates for finite element solutions of elliptic boundary value problems in non-smooth domains

This paper is concerned with the effective numerical treatment of elliptic boundary value problems when the solutions contain singularities. The paper deals first with the theory of problems of this type in the context of weighted Sobolev spaces and covers problems in domains with conical vertices and non-intersecting edges, as well as polyhedral domains with Lipschitz boundaries. Finite element schemes on graded meshes for second-order problems in polygonal/polyhedral domains are then proposed for problems with the above singularities. These schemes exhibit optimal convergence rates with decreasing mesh size. Finally, we describe numerical experiments which demonstrate the efficiency of our technique in terms of ‘actual’ errors for specific (finite) mesh sizes in addition to the asymptotic rates of convergence.

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 modelling of viscoelastic liquids using a finite-volume method

Abstract A new staggered-grid, finite-volume method for the numerical simulation of isothermal viscoelastic liquids is presented. The main features of this method are the use of a primitive variable formulation, the location of the velocities, pressures and stresses on different staggered grids and the use of a third-order difference scheme for the discretization of the constitutive equations. The method is applied to a sudden-expansion, viscoelastic-flow problem for a range of Weissenberg numbers. For one case the mesh has been refined to demonstrate that the method is robust and viscoelastic effects are not filtered out as the mesh size decreases. All the computations have been performed on a PC equipped with a floating-point coprocessor. Although the method has been defined for, and applied to, two-dimensional problems, the technique is readily extendable to three dimensions without excessive cost.

Discontinuous Galerkin finite element methods for linear elasticity and quasistatic linear viscoelasticity

Summary.We consider a finite-element-in-space, and quadrature-in-time-discretization of a compressible linear quasistatic viscoelasticity problem. The spatial discretization uses a discontinous Galerkin finite element method based on polynomials of degree r—termed DG(r)—and the time discretization uses a trapezoidal-rectangle rule approximation to the Volterra (history) integral. Both semi- and fully-discrete a priori error estimates are derived without recourse to Gronwalls inequality, and therefore the error bounds do not show exponential growth in time. Moreover, the convergence rates are optimal in both h and r providing that the finite element space contains a globally continuous interpolant to the exact solution (e.g. when using the standard ℙk polynomial basis on simplicies, or tensor product polynomials, ℚk, on quadrilaterals). When this is not the case (e.g. using ℙk on quadri-laterals) the convergence rate is suboptimal in r but remains optimal in h. We also consider a reduction of the problem to standard linear elasticity where similarly optimal a priori error estimates are derived for the DG(r) approximation.

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.

Superconvergence results on mildly structured triangulations

Many results on superconvergence for recovered gradients of piecewise linear Galerkin approximations on triangular mesh partitions to the weak solutions of elliptic boundary value problems in two dimensions have been proved in recent years. These were obtained first for ∞-regular (fully-structured) partitions in which the mid-points of the diagonals of all quadrilaterals formed by pairs of adjacent elements are coincident. This condition was then relaxed to allow for strongly-regular meshes, in which the distance between the above mid-points is O(h2), h being the mesh size parameter. In this paper these conditions are weakend still further to the case of globally mildly structured meshes, where the mid-point distance is O(h1+α), 0<α<1, and to meshes of this type where locally α=0. After a review of recovery and gradient superconvergence, a unified approach is presented in terms of a generic gradient recovery operator which possesses specific properties on rectangular domains. Then the well-known classic theorem of Oganesyan and Rukhovets is extended to the case of mildly structured triangulations of polygonal approximations of C3(d) domains. A class of gradient recovery operators is described on these mildly structured meshes and, using the extended Oganesyan–Rukhovets theorem, superconvergence is proved. We also obtain global superconvergence results for the recovered gradients over plane polygonal domains patchwise partitioned by fully-structured meshes. A feature of our results is that they allow local refinements of such meshes without loss of superconvergence. For the sake of completeness we have referenced the works of others in order to demonstrate the place of our work in the field.

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.

Error bounds for finite element solutions of mildly nonlinear elliptic boundary value problems

SummaryThe equivalence in a Hilbert space of variational and weak formulations of linear elliptic boundary value problems is well known. This same equivalence is proved here for mildly nonlinear problems where the right hand side of the differential equation involves the solution function. A finite element approximation to the solution of the weak problem ina finite dimensional subspace of the original Hilbert space is defined. An inequality bounding the error in this approximation over all functions of the space is derived, and in particular this holds for an interpolant to the weak solution. Thus this inequality, together with previously known, interpolation error bounds, produces a bound on the finite element solution to this nonlinear problem. An example of a mildly nonlinear Poisson problem is given.

Optimal long-time Lp(0, T) stability and semidiscrete error estimates for the Volterra formulation of the linear quasistatic viscoelasticity problem

Summary. The purpose of this article is to show how the solution of the linear quasistatic (compressible) viscoelasticity problem, written in Volterra form with fading memory, may be sharply bounded in terms of the data if certain physically reasonable assumptions are satisfied. The bounds are derived by making precise assumptions on the memory term which then make it possible to avoid the Gronwall inequality, and use instead a comparison theorem which is more sensitive to the physics of the problem. Once the data-stability estimates are established we apply the technique also to deriving a priori error bounds for semidiscrete finite element approximations. Our bounds are derived for viscoelastic solids and fluids under the small strain assumption in terms of the eigenvalues of a certain matrix derived from the stress relaxation tensor. For isotropic materials we can be explicit about the form of these bounds, while for the general case we give a formula for their computation.

Adaptive space-time finite element solution for Volterra equations arising in viscoelasticity problems

Abstract We give a short overview of our recent efforts towards constructing adaptive space–time finite element solvers for some partial differential Volterra equations arising in viscoelasticity theory.