Simon Shaw

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.

Discontinuous Galerkin finite element approximation of nonlinear non-Fickian diffusion in viscoelastic polymers

We consider discrete schemes for a nonlinear model of non-Fickian diffusion in viscoelastic polymers. The model is motivated by, but not the same as, that proposed by Cohen et al. in SIAM J. Appl. Math., 55, pp. 348–368, 1995. The spatial discretisation is effected with both the symmetric and non-symmetric interior penalty discontinuous Galerkin finite element method, and the time discretisation is of Crank-Nicolson type. We also discuss two means of handling the nonlinearity: either implicitly, which requires the solution of nonlinear equations at each time level, or through a linearisation based on extrapolating from previous time levels. The same optimal orders of convergence are proven in both cases and, to verify this, some numerical results are also given for the linearised scheme.

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.

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.

Applications and numerical analysis of partial differential Volterra equations: A brief survey

This article contains a concise survey of the numerical analysis of Volterra equations, and leads up to some recent results on a posteriori error estimation for finite element approximations.

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.

Spatial-Temporal Modelling and Analysis of Bacterial Colonies with Phase Variable Genes

This article defines a novel spatial-temporal modelling and analysis methodology applied to a systems biology case study, namely phase variation patterning in bacterial colony growth. We employ coloured stochastic Petri nets to construct the model and run stochastic simulations to record the development of the circular colonies over time and space. The simulation output is visualised in 2D, and sector-like patterns are automatically detected and analysed. Space is modelled using 2.5 dimensions considering both a rectangular and circular geometry, and the effects of imposing different geometries on space are measured. We close by outlining an interpretation of the Petri net model in terms of finite difference approximations of partial differential equations (PDEs). One result is the derivation of the “best” nine-point diffusion model. Our multidimensional modelling and analysis approach is a precursor to potential future work on more complex multiscale modelling.

Finite element approximation of Maxwell's equations with Debye memory.

Maxwells equations in a bounded Debye medium are formulated in terms of the standard partial differential equations of electromagnetism with a Volterra-type history dependence of the polarization on the electric field intensity. This leads to Maxwells equations with memory. We make a correspondence between this type of constitutive law and the hereditary integral constitutive laws from linear viscoelasticity, and we are then able to apply known results from viscoelasticity theory to this Maxwell system. In particular, we can show long-time stability by shunning Gronwalls lemma and estimating the history kernels more carefully by appeal to the underlying physical fading memory. We also give a fully discrete scheme for the electric field wave equation and derive stability bounds which are exactly analogous to those for the continuous problem, thus providing a foundation for long-time numerical integration. We finish by also providing error bounds for which the constant grows, at worst, linearly in time (excluding the time dependence in the norms of the exact solution). Although the first (mixed) finite element error analysis for the Debye problem was given by Li (2007), this seems to be the first time sharp constants have been given for this problem.