Representing arbitrary acoustic source and sensor distributions in Fourier collocation methods

Abstract

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.

Discrete Approximations of Acoustic Source Distributions

“…All results here employ c = 1500 m•s −1 , R = 20 mm and φ 0 = π/6 [9]. A first choice is that of the domain D used to represent the source distribution over a grid.…”

“…Here p(x, t) is the pressure, as a function of x ∈ R 3 and t ∈ R; here we consider an unbounded domain, so boundary conditions are not taken into account. We make here the further simplification [9] that the source term is separable as S(x, t) = g(x)f (t), for a fixed distribution g(x) and driving function f (t). Distributions are assumed normalised, so that…”

“…For all such methods, a 3D Cartesian grid is employed, and a major problem is the representation of source distributions of finite spatial extent. Key issues are staircasing effects [7], [5], [8] as well as the computational cost associated with the number of grid points required to represent a distribution [9]. In some approaches, grid point distributions are determined directly through geometric considerations, with additional local scaling techniques, alongside other local constraints (such as simply-connectedness of distributions) [10], [7].…”

TUSX: an accessible toolbox for transcranial ultrasound simulation

“…Increasing the resolution used also has the benefit of allowing for more accurate representations of curved (or otherwise non-orthogonal) acoustic sources within the grid, which is also crucial for simulation accuracy (Wise et al, 2019).…”

Modeling continuous source distributions in wave-based virtual acoustics

“…An example is loudspeaker modeling, where pistons of finite size are the key element. Such distributed models are used extensively to model focusing effects in transducers in ultrasound [23][24][25][26][27] and aeroacoustics 28 . In such work, the distribution is represented directly in terms of simple sources defined over a grid; staircasing issues become a major concern, and one possible remedy is through the use of spectral collocation techniques 25,27 , and direct approximation to the distribution in the wavenumber domain.…”