Keeping the grid size under control: Nyquist sampling in the Iterative nonlinear contrast source method

Abstract

Nonlinear propagation of ultrasound is applied in medical imaging and therapy, as well as in nondestructive testing. Development of dedicated equipment and protocols, and interpretation of results, is facilitated by reliable simulation of nonlinear ultrasound fields. Typical applications require accurate representation of higher harmonics with amplitudes 120 dB below the fundamental, over spatiotemporal domains that span several hundreds of wavelengths/periods of the fundamental. Finite-element and finite-difference schemes usually require 10–20 points per wavelength/period of the highest harmonic, without a way to restrict the number of harmonics involved in the solution. Solving the nonlinear wave equation with these methods may therefore require a prohibitively large computational grid. Alternatively, the nonlinear wave problem may be cast into an integral equation, which can be solved iteratively. This is the basis of our ever-expanding Iterative Nonlinear Contrast Source method. Most importantly, this approach allows to explicitly limit the wavenumber/frequency range included in the solution, and hence enables sampling at the Nyquist rate set by the highest desired frequency. As a consequence, the computational grid can be relatively small without introducing aliasing. This presentation elucidates how this is achieved by an appropriate combination of numerical filtering and windowing operations during each iteration.