Zackary R. Kenz

High-order space-time finite element schemes for acoustic and viscodynamic wave equations with temporal decoupling

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

Comparison of Frequentist and Bayesian Confidence Analysis Methods on a Viscoelastic Stenosis Model

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.

A comparison of nonlinear filtering approaches in the context of an HIV model

In this paper three different filtering methods, the Extended Kalman Filter (EKF), the Gauss-Hermite Filter (GHF), and the Unscented Kalman Filter (UKF), are compared for state-only and coupled state and parameter estimation when used with log state variables of a model of the immunologic response to the human immunodeficiency virus (HIV) in individuals. The filters are implemented to estimate model states as well as model parameters from simulated noisy data, and are compared in terms of estimation accuracy and computational time. Numerical experiments reveal that the GHF is the most computationally expensive algorithm, while the EKF is the least expensive one. In addition, computational experiments suggest that there is little difference in the estimation accuracy between the UKF and GHF. When measurements are taken as frequently as every week to two weeks, the EKF is the superior filter. When measurements are further apart, the UKF is the best choice in the problem under investigation.

Decomposition of permittivity contributions from reflectance using mechanism models

In this paper, we investigate the properties of a complex nonmagnetic material through the reflectance, where the permittivity is described by a mechanism model in which an unknown probability measure is placed on the model parameters. Specifically, we consider whether or not this unknown probability measure can be determined from the reflectance or the derivatives of the reflectance, and we also investigate the effect of measurement noise on the estimation. The numerical results demonstrate that if only the reflectance can be observed, then the distribution form cannot be recovered even in the case where the measurement noise level is small. However, if both the reflectance and the derivative of the reflectance can be observed, then the estimated distribution is reasonably close to the true one even in the case where the measurement noise level is relatively high.

Characterisation of Elastic and Acoustic Properties of an Agar-Based Tissue Mimicking Material

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 wavexa0propagation velocity, attenuation and viscoelastic constants, together with one dimensional quasi-static elastic moduli and Poisson’s ratio. Phase velocity and attenuation coefficients, determined by generating and detecting shear waves piezo-electrically in the range 300xa0Hz–2xa0kHz, were 3.2–7.5xa0ms−1 and 320xa0dBm−1. Quasi-static Young’s modulus, shear modulus and Poisson’s ratio, obtained by compressive or shear loading of cylindrical specimens were 150–160xa0kPa; 54–56xa0kPa and 0.37–0.44. The dynamic Young’s and shear moduli, derived from fitting viscoelastic internal variables by an iterative statistical inverse solver to freely oscillating specimens were 230 and 33xa0kPa and the corresponding relaxation times, 0.046 and 0.036xa0s. The results were self-consistent, repeatable and provide baseline data required for the computational modelling of wave propagation in a phantom.