Accelerating frequency-domain numerical methods for weakly nonlinear focused ultrasound using nested meshes

TL;DR

This paper introduces a nested mesh approach for simulating weakly nonlinear focused ultrasound, significantly reducing computational resources by tailoring mesh resolution to harmonic wavelengths, enabling faster and more efficient simulations.

Contribution

The paper presents a novel nested meshing strategy that assigns separate meshes to each harmonic, reducing computational load and memory use in ultrasound simulations.

Findings

Achieved at least tenfold reduction in memory and computation time.

Validated the approach with numerical experiments on homogeneous media.

Demonstrated potential for application in inhomogeneous domains.

Abstract

The numerical simulation of weakly nonlinear ultrasound is important in treatment planning for focused ultrasound (FUS) therapies. However, the large domain sizes and generation of higher harmonics at the focus make these problems extremely computationally demanding. Numerical methods typically employ a uniform mesh fine enough to resolve the highest harmonic present in the problem, leading to a very large number of degrees of freedom. This paper proposes a more efficient strategy in which each harmonic is approximated on a separate mesh, the size of which is proportional to the wavelength of the harmonic. The increase in resolution required to resolve a smaller wavelength is balanced by a reduction in the domain size. This nested meshing is feasible owing to the increasingly localised nature of higher harmonics near the focus. Numerical experiments are performed for FUS transducers in homogeneous media in order to determine the size of the meshes required to accurately represent the harmonics. In particular, a fast \emph{volume potential} approach is proposed and employed to perform convergence experiments as the computation domain size is modified. This approach allows each harmonic to be computed via the evaluation of an integral over the domain. Discretising this integral using the midpoint rule allows the computations to be performed rapidly with the FFT. It is shown that at least an order of magnitude reduction in memory consumption and computation time can be achieved with nested meshing. Finally, it is demonstrated how to generalise this approach to inhomogeneous propagation domains.