Accurately representing acoustic source distributions is an important part of ultrasound simulation. This is challenging for grid-based collocation methods when such distributions do not coincide with the grid points, for instance when the source is a curved, two-dimensional surface embedded in a three-dimensional domain. Typically, grid points close to the source surface are defined as source points, but this can result in “staircasing” and substantial errors in the resulting acoustic fields. This paper describes a technique for accurately representing arbitrary source distributions within Fourier collocation methods. The method works by applying a discrete, band-limiting convolution operator to the continuous source distribution, after which source grid weights can be generated. This allows arbitrarily shaped sources, for example, focused bowls and circular pistons, to be defined on the grid without staircasing errors. The technique is examined through simulations of a range of ultrasound sources, and comparisons with analytical solutions show excellent accuracy and convergence rates. Extensions of the technique are also discussed, including application to initial value problems, distributed sensors, and moving sources.

A Green's function solution is derived for calculating the acoustic field generated by phased array transducers of arbitrary shape when driven by a single frequency continuous wave excitation with spatially varying amplitude and phase. The solution is based on the Green's function for the homogeneous wave equation expressed in the spatial frequency domain or k-space. The temporal convolution integral is solved analytically, and the remaining integrals are expressed in the form of the spatial Fourier transform. This allows the acoustic pressure for all spatial positions to be calculated in a single step using two fast Fourier transforms. The model is demonstrated through several numerical examples, including single element rectangular and spherically focused bowl transducers, and multi-element linear and hemispherical arrays.

A full-wave model for nonlinear ultrasound propagation through a heterogeneous and absorbing medium in an axisymmetric coordinate system is developed. The model equations are solved using a nonstandard or k-space pseudospectral time domain method. Spatial gradients in the axial direction are calculated using the Fourier collocation spectral method, and spatial gradients in the radial direction are calculated using discrete trigonometric transforms. Time integration is performed using a k-space corrected finite difference scheme. This scheme is exact for plane waves propagating linearly in the axial direction in a homogeneous and lossless medium and significantly reduces numerical dispersion in the more general case. The implementation of the model is described, and performance benchmarks are given for a range of grid sizes. The model is validated by comparison with several analytical solutions. This includes one-dimensional absorption and nonlinearity, the pressure field generated by plane-piston and bowl transducers, and the scattering of a plane wave by a sphere. The general utility of the model is then demonstrated by simulating nonlinear transcranial ultrasound using a simplified head model.

A class of nonstandard pseudospectral time domain (PSTD) schemes for solving timedependent hyperbolic and parabolic partial differential equations (PDEs) is introduced. These schemes use the Fourier collocation spectral method to compute spatial gradients and a nonstandard finite difference scheme to integrate forwards in time. The modified denominator function that makes the finite difference time scheme exact is transformed into the spatial frequency domain or k-space using the dispersion relation for the governing PDE. This allows the correction factor to be applied in the spatial frequency domain as part of the spatial gradient calculation. The derived schemes can be formulated to be unconditionally stable, and apply to PDEs in any space dimension. Examples of the resulting nonstandard PSTD schemes for several PDEs are given, including the wave equation, diffusion equation, and convection-diffusion equation.

Over the last century, several different theoretical models have been proposed for the calculation of the transverse modulus of fibres or cylinders from compression experiments. Whilst they all give similar results, the differences are significant enough to cause errors in computer simulation predictions of composite properties, and hence the issue warrants further investigation. Two independent approaches were applied to clarify this. Firstly, using an experimental approach, compression tests have been carried out on model elastic cylinders of poly(methyl methacrylate) as well as cuboids machined from the cylinders. The transverse modulus of this hard elastic material was determined directly from compression experiments on the cuboids and by analysis using different models for the cylinder compression data. Since machining was shown to change the modulus by virtue of relieving stresses in the samples, comparison was made between cuboids and machined cylinders. The transverse modulus obtained by direct compression of the cuboids was statistically equivalent to that obtained from the cylinders using the Morris model and was within 8% of the value obtained using the model derived by Jawad and Ward, as well as the mathematically equivalent models derived by Phoenix and Skelton and Lundberg. Finally, the separate and independent approach of finite element numerical modelling was also utilised. The finite element approach gave results that lie between the Jawad and Ward and Morris models. The close agreement in the outcomes of the finite element modelling and the experimental approach leads to the conclusion that the most accurate of the different analytical models are the equations by Morris as well as those due to Jawad and Ward, Phoenix and Skelton and Lundberg.