# Fast Nearfield Method

Thermal therapy simulations require the fast and accurate evaluation of continuous wave pressure fields over large computational volumes. The standard approach is point source superposition [1], where each radiating aperture is approximated as an ensemble of point sources radiating in a homogeneous, linear medium. Although conceptually simple, point source superposition is both inaccurate, especially in the near field region, and computationally expensive. To mitigate these problems, the impulse response approach [2,3] was developed in the 1960's and 1970's for simple geometries, such as baffled circular and rectangular pistons. The impulse response approach reduces the double integration required by point source superposition to a single integration. However, the impulse response approach, like point source superposition, is still inaccurate in the near field due to the singular nature of the integrand.

The fast near field method [4,5] (FNM) solves these problems by removing the singularity from the impulse response expressions. For baffled pistons with simple geometries, FNM reduces a double integral to a rapidly converging single integral, which is then evaluated using Gauss quadrature. Since the FNM integrand is smooth, Gauss quadrature achieves exponential convergence within the nearfield region, thereby generating machine accuracy with a small number of quadrature points. For a circular piston [4], the FNM utilizes a spatially bandlimited single-integral expression for all field points, whereas for the rectangular piston [5], the field is expressed as a sum of bandlimited single integrals. Expressions have also been developed for triangular pistons [6], apodized circular pistons [7], apodized rectangular pistons [8], and spherical shells [9]. Research is also being conducted on developing expressions for curved rectangular strips and continuum modeling of apodized arrays. The FNM, which converges much more rapidly than the point source superposition method or the impulse response, produces accurate numerical results in a fraction of the time required by other approaches. In addition, the FNM has been adapted to transient problems encountered in imaging applications [10]. For time-domain problems, the FNM avoids the temporal aliasing problems associated with the impulse response approach while also providing a fast and efficient method for computing transient fields.

For these reasons, we have chosen FNM as the basis for the computational engine of FOCUS. Since custom analytical expressions are available for all canonical geometries, the user models complex array geometries using circles, rectangles, triangles, spherical shells, and/or apodized elements. Since the field generated by each element is exact in a linear, homogeneous medium, the user does not have to worry about controlling the error in large acoustic calculations. When combined with the Angular Spectrum Approach, the FNM provides a fast, efficient, and conceptually simple model for both thermal therapy and imaging problems.

## References

- J. Zemanek, "Beam behavior within the nearfield of a vibrating piston", J. Acoust. Soc. Am., 49(1):181-191, 1971.
- F. Oberhettinger, "On transient solutions of the baffled piston problem", Journal of Research of the National Bureau of Standards-B. Mathematics and Mathematical Physics 65B:1, 1961.
- J. C. Lockwood and J. G. Willette, "High-speed method for computing the exact solution for the pressure variations in the nearfield of a baffled piston", J. Acoust. Soc. Am. 53:735,741, 1973.
- R. J. McGough, T. V. Samulski, and J. F. Kelly, "An efficient grid sectoring method for calculations of the near-field pressure generated by a circular piston", J. Acoust. Soc. Am., 115(5):1942-1954,2004.
- R. J. McGough, "Rapid calculations of time-harmonic nearfield pressures produced by rectangular pistons", J. Acoust. Soc. Am., 115(5):1934-1941
- D. Chen, J. F. Kelly, and R. J. McGough, "A fast nearfield method for calculations of time-harmonic and transient pressures produces by triangular pistons", J. Acoust. Soc. Am., 120(5), 2450-2459, 2006.
- J. F. Kelly and R. J. McGough, "An annular superposition integral for axisymmetric radiators", J. Acoust. Soc. Am. 121(2):759-765, 2007.
- D. Chen and R. J. McGough, "A 2D fast nearfield method for calculating nearfield pressures generated by apodized rectangular pistons", J. Acoust. Soc. Am. (under review)
- J. F. Kelly and R. J. McGough, "A time-space decomposition method for calculating the nearfield pressure generated by a pulsed circular piston", IEEE Trans. Ultrason. Ferroelectr. Freq. Contr., 53(6) 1150-1159, 2006