Abstract
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.
Highlights
Simulating the propagation of nonlinear acoustic waves is important for many branches of acoustics, including diagnostic ultrasound imaging, therapeutic ultrasound, underwater acoustics, and the study of sonic booms
For many problems that involve moderate or strong nonlinearity, full-wave modelling in three dimensions (3D) remains computationally challenging due to the very large grid sizes neededPortions of this work were presented in “Full-wave nonlinear ultrasound simulation in an axisymmetric coordinate system using the discrete sine and cosine transforms,” IEEE International Ultrasonics Symposium, Prague, Czech Republic, 21–25 July 2013
Collocation spectral method to calculate spatial gradients and a k-space corrected finite difference scheme to integrate in time, and can be considered as an extension to nonstandard finite difference methods
Summary
Simulating the propagation of nonlinear acoustic waves is important for many branches of acoustics, including diagnostic ultrasound imaging, therapeutic ultrasound, underwater acoustics, and the study of sonic booms. When the source geometry and the propagation medium are axisymmetric, the problem is reduced to two dimensions and the computational load is significantly reduced This geometry is of practical interest, for example, in the study of nonlinear resonantors and modelling the output of therapeutic ultrasound transducers.. The model is based on the k-space pseudospectral time domain (PSTD) method as previously used to solve the nonlinear wave equation in a Cartesian coordinate system.. The model is based on the k-space pseudospectral time domain (PSTD) method as previously used to solve the nonlinear wave equation in a Cartesian coordinate system.5,18 This formulation has advantages over FDTD and FE methods due to the reduced number of grid points needed per wavelength to reach convergence..
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