^{1,a)}and Robert J. McGough

^{1,b)}

### Abstract

Analytical two-dimensional (2D) integral expressions are derived for fast calculations of time-harmonic and transient near-field pressures generated by apodized rectangular pistons. These 2D expressions represent an extension of the fast near-field method (FNM) for uniformly excited pistons. After subdividing the rectangular piston into smaller rectangles, the pressure produced by each of the smaller rectangles is calculated using the uniformly excited FNM expression for a rectangular piston, and the total pressure generated by an apodized rectangular piston is the superposition of the pressures produced by all of the subdivided rectangles. By exchanging summation variables and performing integration by parts, a 2D apodized FNM expression is obtained, and the resulting expression eliminates the numerical singularities that are otherwise present in numerical models of pressure fields generated by apodized rectangular pistons. A simplified time space decomposition method is also described, and this method further reduces the computation time for transient pressure fields. The results are compared with the Rayleigh–Sommerfeld integral and the FIELD II program for a rectangular source with each side equal to four wavelengths. For time-harmonic calculations with a 0.1 normalized root mean square error (NRMSE), the apodized FNM is 4.14 times faster than the Rayleigh–Sommerfeld integral and 59.43 times faster than the FIELD II program, and for a 0.01 NRMSE, the apodized FNM is 12.50 times faster than the Rayleigh–Sommerfeld integral and 155.06 times faster than the FIELD II program. For transient calculations with a 0.1 NRMSE, the apodized FNM is 2.31 times faster than the Rayleigh–Sommerfeld integral and 4.66 times faster than the FIELD II program, and for a 0.01 NRMSE, the apodized FNM is 11.90 times faster than the Rayleigh–Sommerfeld integral and 24.04 times faster than the FIELD II program. Thus, the 2D apodized FNM is ideal for fast pressure calculations and for accurate reference calculations in the near-field region.

The authors would like to thank James F. Kelly and Xiaozheng Zeng for thoughtful discussions. This work was supported in part by NIH R01 CA093669, NIH R21 CA121235, NSF 0634786, and NIH P01 CA042745.

I. INTRODUCTION

II. METHODS

A. Steady-state apodized FNM expression

B. Transient apodized FNM expression

C. The Rayleigh–Sommerfeld integral

D. The FIELD II program

E. Apodization function

F. Input transient pulse

G. Time space decomposition

H. Error calculations

I. Computational resources

III. RESULTS

A. Time-harmonic pressure calculations

1. Reference pressure field

2. Error distributions

3. Time versus error comparisons for a time-harmonic input

B. Transient field calculations

1. Reference pressure field

2. Time versus error comparisons for a Hanning-weighted input pulse

IV. DISCUSSION

A. Large-scale computation

B. Time and error comparisons

C. Apodization functions

D. Time space decomposition

V. CONCLUSION

### Key Topics

- Particle velocity
- 7.0
- Antenna arrays
- 4.0
- Computer simulation
- 4.0
- Fourier transforms
- 3.0
- Sound pressure
- 3.0

## Figures

(Color online) Orientation of the computational grid relative to the rectangular source. The rectangular source, which has width and height , lies in the plane. The dashed lines define the extent of the computational grid in the plane. The extent of the computational grid is by in the and directions, respectively.

(Color online) Orientation of the computational grid relative to the rectangular source. The rectangular source, which has width and height , lies in the plane. The dashed lines define the extent of the computational grid in the plane. The extent of the computational grid is by in the and directions, respectively.

The decomposition of an apodized rectangular source into smaller rectangles, where each small rectangle is wide and high. The apodization function is defined as constant over each small rectangle.

The decomposition of an apodized rectangular source into smaller rectangles, where each small rectangle is wide and high. The apodization function is defined as constant over each small rectangle.

The apodization function evaluated on the face of a square piston. The maximum value of the apodization function is achieved when and .

The apodization function evaluated on the face of a square piston. The maximum value of the apodization function is achieved when and .

Simulated reference pressure field generated by an apodized rectangular source with each side equal to four wavelengths. The results are evaluated in the plane for a time-harmonic excitation.

Simulated reference pressure field generated by an apodized rectangular source with each side equal to four wavelengths. The results are evaluated in the plane for a time-harmonic excitation.

The normalized error distribution describes the difference between the reference pressure field and the computed pressure field for an apodized source. The error distribution is plotted for (a) the apodized FNM evaluated with 16-point Gauss quadrature in each direction, (b) the apodized Rayleigh integral evaluated with 16-point Gauss quadrature in each direction, and (c) the FIELD II program evaluated with and 30 subdivisions in each direction.

The normalized error distribution describes the difference between the reference pressure field and the computed pressure field for an apodized source. The error distribution is plotted for (a) the apodized FNM evaluated with 16-point Gauss quadrature in each direction, (b) the apodized Rayleigh integral evaluated with 16-point Gauss quadrature in each direction, and (c) the FIELD II program evaluated with and 30 subdivisions in each direction.

Normalized root mean square error (NRMSE) plotted as a function of the computation time for time-harmonic calculations with the apodized FNM, the apodized Rayleigh–Sommerfeld integral, and the FIELD II program. This figure demonstrates that the apodized FNM achieves the smallest errors for a given computation time and that the apodized FNM requires the smallest amount of time to achieve a given error value.

Normalized root mean square error (NRMSE) plotted as a function of the computation time for time-harmonic calculations with the apodized FNM, the apodized Rayleigh–Sommerfeld integral, and the FIELD II program. This figure demonstrates that the apodized FNM achieves the smallest errors for a given computation time and that the apodized FNM requires the smallest amount of time to achieve a given error value.

Simulated reference transient field for an apodized square source excited by the Hanning-weighted pulse in Eq. (18) with and . The sides of the square source are equal to , where the wavelength is defined with respect to the center frequency . The apodization function is given by Eq. (17). The transient reference pressure, evaluated in the plane, is computed with abscissas in each direction using the Rayleigh integral. The results are plotted at (a) and (b) .

Simulated reference transient field for an apodized square source excited by the Hanning-weighted pulse in Eq. (18) with and . The sides of the square source are equal to , where the wavelength is defined with respect to the center frequency . The apodization function is given by Eq. (17). The transient reference pressure, evaluated in the plane, is computed with abscissas in each direction using the Rayleigh integral. The results are plotted at (a) and (b) .

Normalized root mean square error (NRMSE) plotted as a function of the computation time for transient pressure calculations evaluated with the apodized FNM, the apodized Rayleigh–Sommerfeld integral, and the FIELD II program. For the same computation time, the apodized FNM achieves the smallest errors. For the same error, the apodized FNM requires the least amount of time.

Normalized root mean square error (NRMSE) plotted as a function of the computation time for transient pressure calculations evaluated with the apodized FNM, the apodized Rayleigh–Sommerfeld integral, and the FIELD II program. For the same computation time, the apodized FNM achieves the smallest errors. For the same error, the apodized FNM requires the least amount of time.

## Tables

Terms that define the time space decomposition of the Hanning-weighted pulse for transient apodized FNM calculations.

Terms that define the time space decomposition of the Hanning-weighted pulse for transient apodized FNM calculations.

Terms that define the time space decomposition of the derivative of a Hanning-weighted pulse for transient calculations with the apodized Rayleigh–Sommerfeld integral.

Terms that define the time space decomposition of the derivative of a Hanning-weighted pulse for transient calculations with the apodized Rayleigh–Sommerfeld integral.

Simulation parameters for time-harmonic calculations that achieve normalized root mean square error (NRMSE) values of 0.1 and 0.01. The parameters listed include the number of Gauss abscissas or the corresponding FIELD II parameters, the resulting computation time, and the ratio of the computation time relative to the apodized FNM for the Rayleigh integral and the FIELD II program.

Simulation parameters for time-harmonic calculations that achieve normalized root mean square error (NRMSE) values of 0.1 and 0.01. The parameters listed include the number of Gauss abscissas or the corresponding FIELD II parameters, the resulting computation time, and the ratio of the computation time relative to the apodized FNM for the Rayleigh integral and the FIELD II program.

Simulation parameters for transient calculations that achieve normalized root mean square error (NRMSE) values of 0.1 and 0.01. The parameters listed include the number of Gauss abscissas or the corresponding FIELD II parameters, the resulting computation time, and the ratio of the computation time relative to the apodized FNM for the Rayleigh integral and the FIELD II program.

Simulation parameters for transient calculations that achieve normalized root mean square error (NRMSE) values of 0.1 and 0.01. The parameters listed include the number of Gauss abscissas or the corresponding FIELD II parameters, the resulting computation time, and the ratio of the computation time relative to the apodized FNM for the Rayleigh integral and the FIELD II program.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content