A comparison of pressure calculation methods that subtract near field singularities from analytical expressions developed for triangular apertures

Abstract

Previous simulation studies have shown that the fast nearfield method (FNM) is faster and more accurate than the impulse response and rectangular radiator methods for nearfield pressure calculations. The FNM achieves this advantage by subtracting a singularity from a one-dimensional integral expression, which improves the numerical convergence relative to these other techniques. A related approach subtracts a singularity from the two-dimensional (2D) Rayleigh integral, thereby separating the Rayleigh integral into two parts. In this approach, one integral is evaluated analytically and the remaining terms are evaluated numerically. Numerical calculations are performed with a seven point Gauss quadrature rule. This approach is evaluated for a triangular source and compared to results obtained from FNM calculations. For larger triangular sources, the 2D Rayleigh integral is subdivided into smaller triangular patches, and then the results are superposed. Results show that for an equilateral triangle with sides equal to one wavelength, this approach achieves a 10% peak error without subdividing the source, and a 1% peak error is obtained when the source is subdivided into four smaller triangles. Computation times with this approach are 0.140s and 0.2190s for 10% and 1% peak errors, respectively, whereas the FNM results for the same peak errors are 0.031s and 0.047s for a reduction in the computation time by a factor of 4.52 and 4.66, respectively. This shows that FNM calculations converge more rapidly than the method that subtracts the singularity from the 2D Rayleigh integral. Future efforts will replace this 2D Rayleigh integral calculation with the FNM in large-scale simulations of ultrasound devices designed for thermal therapy applications