Simon Shaw

• PhD Numerical Analysis
• Brunel University London

Linear viscoelasticity can be characterized by a stress relaxation function. We consider a power‐law type stress relaxation to yield a fractional order viscoelasticity model. The governing equation is a Volterra integral problem of the second kind with a weakly singular kernel. We employ spatially discontinuous Galerkin methods, symmetric interior...

We consider linear scalar wave equations with a hereditary integral term of the kind used to model viscoelastic solids. The kernel in this Volterra integral is a sum of decaying exponentials (The so-called Maxwell, or Zener model) and this allows the introduction of one of two types of families of internal variables, each of which evolve according...

We consider a fractional order viscoelasticity problem modelled by a power-law type stress relaxation function. This viscoelastic problem is a Volterra integral equation of the second kind with a weakly singular kernel where the convolution integral corresponds to fractional order differentiation/integration. We use a spatial finite element method...

Deformations of viscoelastic materials such as soft tissues, metals at high temperature, and polymers can be described as Volterra integral equations of the second kind. We consider the viscoelasticity model problem involving with \textit{Dirichlet Prony} series kernel, which resulting constitutive relation with exponentially decaying faded memory....

We consider a fractional order viscoelasticity problem modelled by a power-law type stress relaxation function. This viscoelastic problem is a Volterra integral equation of the second kind with a weakly singular kernel where the convolution integral corresponds to fractional order differentiation/integration. We use a spatial finite element method...

We consider linear scalar wave equations with a hereditary integral term of the kind used to model viscoelastic solids. The kernel in this Volterra integral is a sum of decaying exponentials (The so-called Maxwell, or Zener model) and this allows the introduction of one of two types of families of internal variables, each of which evolve according...

We extend the formulation and a priori error analysis given by Claes Johnson (1993) from the acoustic wave equation to a Voigt and Maxwell–Zener viscodynamic system incorporating Rayleigh damping. The elastic term in the Rayleigh damping introduces a multiplicative T 1∕2 growth in the constant but otherwise the error bound is consistent with that o...

We formulate numerical schemes for viscoelastic wave propagation and present a stability analysis and an error analysis. • Stability bounds: Existence and uniqueness of discrete solutions. • Optimal broken energy norm error estimates. • Sub-optimal and optimal L 2 error estimates. • Numerical results.

We consider time domain formulations of Maxwell’s equations for the Lorentz model for metamaterials. The field equations are considered in two different forms which have either six or four unknown vector fields. In each case we use arguments tuned to the physical laws to derive data-stability estimates which do not require Gronwall’s inequality. Th...

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....

As a first step towards an acoustic localisation device for coronary stenosis to provide a non-invasive means of diagnosing arterial disease, measurements are reported for an agar-based tissue mimicking material (TMM) of the shear wave propagation velocity, attenuation and viscoelastic constants, together with one dimensional quasi-static elastic m...

A current thrust in medical research is the development of a non-invasive method for detection, localization, and characterization of an arterial stenosis (a blockage or partial blockage in an artery). A method has been proposed to detect shear waves in the chest cavity which have been generated by disturbances in the blood flow resulting from a st...

We revisit a method originally introduced by Werder et al. (in Comput. Methods Appl. Mech. Engrg., 190:6685–6708, 2001) for temporally discontinuous Galerkin FEMs applied to a parabolic partial differential equation. In that approach, block systems arise because of the coupling of the spatial systems through inner products of the temporal basis fun...

In [Comput. Methods Appl. Mech. Eng. 190, No. 49–50, 6685–6708 (2001; Zbl 0992.65103)] T. Werder et al. demonstrated that time discretizations of the heat equation by a temporally discontinuous Galerkin finite element method could be decoupled by diagonalizing the temporal Gram matrices. In this article we propose a companion approach for the heat...

In this paper, quasistatic models are developed for the slow flow of compressible fluids through porous solids, where the solid exhibits fading memory viscoelasticity. Problems of this type are important in practical geomechanics contexts, for example, in the context of fluid flow through unconsolidated reservoir sands and of wellbore deformation b...

Background: Turbulent flow downstream of atherosclerotic plaques produces low amplitude shear waves which travel through the chest and can be measured by skin sensors. This acoustic signature may provide a cheap non-invasive way to diagnose arterial disease. We report measurements of shearing oscillations and flow-induced turbulence in soft tissue-...

Blood turbulence in the wake of plaque build-up in an atherosclerotic coronary artery induces wall shear stresses which travel through the thorax and produce an audible ‘bruit’ at the chest wall. Detection and identification of this (very weak) signal has the potential to provide a low-cost and non-invasive diagnostic technology. This presentation...

We propose two fully discrete mixed and Galerkin finite element approximations to a system of equations describing the slow flow of a slightly compressible single phase fluid in a viscoelastic porous medium. One of our schemes is the natural one for the backward Euler time discretization but, due to the viscoelasticity, seems to be stable only for...

Non-invasive detection, localization and characterization of an arterial stenosis (a blockage or partial blockage in the artery) continues to be an important problem in medicine. Partial blockage stenoses are known to generate disturbances in blood flow which generate shear waves in the chest cavity. We examine a one-dimensional viscoelastic model...

We re-visit a method originally introduced byWerder et al. (in Comput. Methods Appl. Mech. Engrg., 190:6685—6708, 2001) for the numerical solution of parabolic partial differential equations by space-time discontinuous Galerkin finite element methods. In that approach block systems arise due to the coupling of the spatial systems through inner prod...

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 hyperelast...

The problem of non-local nonlinear non-Fickian polymer diffusion as modelled by a diffusion equation with a nonlinearly coupled boundary value problem for a viscoelastic 'pseu-dostress' is considered (see, for example, DA Edwards in Z. angew. Math. Phys., 52, 2001, pp. 254—288). We present two numerical schemes using the implicit Euler method and a...

The problem of nonlinear non-Fickian polymer diffusion as modelled by a diffusion equation with an adjoined spatially local evolution equation for a viscoelastic stress is considered (see, for example, Cohen, White & Witelski, SIAM J. Appl. Math. 55, pp. 348–368, 1995). We present numerical schemes based, spatially, on the Galerkin finite element m...

We consider an initial-boundary value problem for the Korteweg-de Vries equation on the negative quarter-plane. The normalized Korteweg-de Vries equation considered is given by u τ +uu x +u xxx =0,x<0,τ>0, where x and τ represent dimensionless distance and time, respectively. In particular, we consider the case when the initial and boundary conditi...

Maxwell's 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 Maxwell's equations with memory. We make a correspondence between this type of constitutive law and the he...

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, b...

This project has been concerned with the reliable computation of engineering Quantities of Interest (QoI), and the application of these mathematical methods to problems of interest to the US Army (informed by our long-standing contact with Dr A R Johnson, ARL, VTD, LaRC). In the project we have applied state-of-the-art mathematical theory to a prac...

We consider the usual linear elastodynamics equations augmented with evolution equations for viscoelastic internal stresses. A fully discrete approximation is defined, based on a spatially symmetric or non-symmetric interior penalty discontinuous Galerkin finite element method, and a displacement-velocity centred difference time discretisation. An...

The problem of computing accurate Coriolis distortion modes in mass flow meters is discussed. This is illustrated by several numerical results, and it is tentatively suggested that the problem is due mainly to computer rounding error rather than any fundamental weakness in the finite element method or the eigensolvers. An empirically evaluated succ...

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, White, and Witelski in SIAM J. Appl. Math., 55 (1995), pp. 348-368. The spatial discretization is effected with both the symmetric and nonsymmetric interior penalty discontinuo...

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 model...

We give a space–time Galerkin finite element discretisation of the quasistatic compressible linear viscoelasticity problem as described by an elliptic partial differential equation with a fading memory Volterra integral. The numerical scheme consists of a continuous Galerkin approximation in space based on piecewise polynomials of degree p>0 (cG(p)...

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 Volt...

This paper is concerned with enhancing the a posteriori energy-error estimators developed in Part I in order to accomodate transition elements in the nite element mesh. The resulting estimators are then used in an adaptive nite element model employing transition elements and the subsequent results discussed and compared with those in Part I. A majo...

This paper considers a posteriori error estimation for nite element solution of Reissner-Mindlin type thick plates, modelled using rst-order shear deformation theory (hereinafter referred to FSDT). We outline the formulation given by Reddy in [10] below in Section II. and then introduce a standard weak formulation in Section III.. From this the nit...

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.

We consider a piecewise constant finite element approximation to the convolution Volterra equation problem of the second kind: find u such that u = f + phi * u in a time interval [0, T]. An a posteriori estimate of the error measured in the W-p(-1) (0,T) norm is developed and used to provide a time step selection criterion for an adaptive solution...

Constitutive models for viscoelastic deformation are presented. Numerical schemes, based on these, are described for quasistatic problems, and a posteriori error estimates are presented. Numerical results on test problems illustrate spatial adaptivity.

. 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...

. We give a space-time Galerkin finite element discretization of the linear quasistatic compressible viscoelasticity problem as described by an elliptic partial differential equation with a Volterra (memory) term. The discretization consists of a continuous piecewise linear approximation in space with a discontinuous piecewise constant or linear ap...

. We give a space-time Galerkin finite element discretization of the linear quasistatic compressible viscoelasticity problem as described by an elliptic partial differential equation with a Volterra (memory) term. The discretization consists of a continuous piecewise linear approximation in space with a discontinuous piecewise constant or linear ap...

The major goal was to develop a framework for the adaptive finite element solution of quasistatic viscoelasticity problems in the context of the practical utility of the internal variable formulation, as used by Dr. A. R. Johnson of the Vehicle Technology Center, NASA, Langley, and the theoretical utility of the hereditary integral formulation, as...

This paper is concerned with the modelling of problems involving viscoelastic materials which, even in their simplest form, exhibit behaviour characteristic of both classical Hookean solids and Newtonian fluids. The resulting effects are important when the material is deforming under an applied load. This load could, for example, be due to external...

this article we describe the mathematical models obtained when the problem of linear quasistatic solid viscoelasticity is modelled by: (i) a hereditary integral with Prony series; and, (ii) an evolution equation for internal strain variables. We then briefly describe the form of the a posteriori error estimates recently obtained for the second of t...

This article should be viewed in the context of [7] where we gave a priori and a posteriori

During this second three month phase of the seed project we have consolidated the phase 1 work on the Volterra formulation of linear quasistatic viscoelasticity problems, and three papers have now been submitted to research journals. Another paper is expected to be submitted before the summer. The visit of Dr. A.R. Johnson during March 1-8 was high...

.36> s)" kl (u(s)) ds; where (D ijkl (t)) 3 i;j;k;l=1 is a tensor of stress relaxation functions. Use of this constitutive equation in the equilibrium equations leads to a deformation model of the type: find the displacement u such that Au(x; t) = f(x; t) + Z t 0 B(t Gamma s)u(x; s) ds; where x is a point in the body, f is a given load, A and B are...

. A fully discrete scheme for approximating a second order hyperbolic Volterra integrodifferential equation of the second kind is proposed. The discretization is accomplished by applying the finite element method in the space variables, a finite difference replacement for the time derivative and the trapezoidal rule for the history integral. Using...

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...

This article first compares the mathematical models obtained when a problem of linear quasistatic solid viscoelasticity is modelled by: (i) a hereditary integral with Prony series; and, (ii) an evolution equation for internal strain variables. We then confirm that while the models are the same, the numerical approximations arising from them will, i...

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. 1 Introduction Modern techniques for the numerical solution of problems involving partial differential equations are concerned not only with the discretizatio...

We give a brief indication of how elliptic, parabolic and hyperbolic partial differential equations with memory arise when modelling viscoelastic materials. We then point out the urgent need for adaptive solvers for these problems and, employing the methodology of Eriksson, Johnson et al. (e.g., SIAM J. Numer. Anal. 28 (1991)), we given ana posteri...

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 Sobole...

We consider a discontinuous Galerkin finite element method applied in time to a model Volterra equation of the second kind. A residual-based computable Galerkin-error estimate is derived for . This estimate does not explicitly contain the time step and therefore the time step control must be based on a heuristic criterion, the estimate can then be...

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 formul...

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 spa...

The paper is concerned with the numerical solution of problems of linear elasticity and viscoelasticity. First, in the context of finite element solution of problems of linear elasticity, we discuss various aspects of finite element gradient recovery, superconvergence, a posteriori error estimation and adaptivity. These form the basis for the spati...

We consider a piecewise constant finite element approximation to the convolution Volterra equation problem of the second kind: find ! # " % $ ' &) (such that 1 0 3 2 5 4 7 6 5 8 in a time interval 9 " % $ ' & A @ . An a posteriori B C D " ! $ ' & E (error estimate involving the derivative of the residual weighted with the time steps is developed, a...