Abstract. There are a number of interesting applications where modeling elastic and/or viscoelastic materials is fundamental, including uses in civil engineering, the food industry, land mine detection and ultrasonic imaging. Here we provide an overview of the subject for both elastic and viscoelastic materials in order to understand the behavior of these materials. We begin with a brief introduction of some basic terminology and relationships in continuum mechanics, and a review of equations of motion in a continuum in both Lagrangian and Eulerian forms. To complete the set of equations, we then proceed to present and discuss a number of specific forms for the constitutive relationships between stress and strain proposed in the literature for both elastic and viscoelastic materials. In addition, we discuss some applications for these constitutive equations. Finally, we give a computational example describing the motion of soil experiencing dynamic loading by incorporating a specific form of constitutive equation into the equation of motion.

Techniques for the nonparametric estimation of probability distributions are reviewed. Methods are divided into two categories for estimation problems: those applied to situations in which individual data is available, and those where only aggregate data is available. For each technique, the general ideas, strengths, and weaknesses of the corresponding methodology are discussed. In addition, the generation of estimates of not only the parameter distributions but also any structural parameters that are constant across the population is outlined. When discussing various techniques, particular consideration is given to the theory behind finite dimensional approximations (since the space of measures on a viable parameter space ‚ is an infinite dimensional space) in the development of computational methodologies. Additional issues, such as consistency of the estimation procedure, are considered when appropriate. Various sample problems on which the methods can be applied are referenced throughout the review.

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 that incorporates Kelvin-Voigt damping and internal variables, and develop a proof-of-concept methodology using simulated data. We first develop an estimation procedure for the material parameters. We use this procedure to determine confidence intervals for the estimated parameters, which indicates the efficacy of finding parameter estimates in practice. Confidence intervals are computed using asymptotic error theory as well as bootstrapping. We then develop a model comparison test to be used in determining if a particular data set came from a low input amplitude or a high input amplitude; this we anticipate will aid in determining when stenosis is present. These two thrusts together will serve as the methodological basis for our continuing analysis using experimental data currently being collected.Mathematics Subject Classification: 62F12; 62F40; 65M32; 74D05.

We compare the performance of three methods for quantifying uncertainty in model parameters: asymptotic theory, bootstrapping, and Bayesian estimation. We study these methods on an existing model for one-dimensional wave propagation in a viscoelastic medium, as well as corresponding data from lab experiments using a homogeneous, tissue-mimicking gel phantom. In addition to parameter estimation, we use the results from the three algorithms to quantify complex correlations between our model parameters, which are best seen using the more computationally expensive bootstrapping or Bayesian methods. We also hold constant the parameter causing the most complex correlation, obtaining results from all three methods which are more consistent than those obtained when estimating all parameters. Concerns regarding computational time and algorithm complexity are incorporated into a discussion on differences between the frequentist and Bayesian perspectives.

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 functions. If the spatial finite element space is of dimension D and polynomials of degree r are used in time, the block system has dimension (r + 1)D and is usually regarded as being too large when r > 1. Werder et al. found that the space-time coupling matrices are diagonalizable over for r ⩽100, and this means that the time-coupled computations within a time step can actually be decoupled. By using either continuous Galerkin or spectral element methods in space, we apply this DG-in-time methodology, for the first time, to second-order wave equations including elastodynamics with and without Kelvin–Voigt and Maxwell–Zener viscoelasticity. An example set of numerical results is given to demonstrate the favourable effect on error and computational work of the moderately high-order (up to degree 7) temporal and spatio-temporal approximations, and we also touch on an application of this method to an ambitious problem related to the diagnosis of coronary artery disease. Copyright © 2014 The Authors. International Journal for Numerical Methods in Engineering published by John Wiley & Sons Ltd.