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. 2008 Sep;124(3):1526-37.
doi: 10.1121/1.2950081.

A 2D fast near-field method for calculating near-field pressures generated by apodized rectangular pistons

Duo Chen  1 , Robert J McGough
Affiliations

A 2D fast near-field method for calculating near-field pressures generated by apodized rectangular pistons

Duo Chen et al. J Acoust Soc Am. 2008 Sep.

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.

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Figures

Figure 1
Figure 1
(Color online) Orientation of the computational grid relative to the rectangular source. The rectangular source, which has width a and height b, lies in the z=0 plane. The dashed lines define the extent of the computational grid in the x=a∕2 plane. The extent of the computational grid is 2b by 0.99a2∕4λ in the y and z directions, respectively.
Figure 2
Figure 2
The decomposition of an apodized rectangular source into smaller rectangles, where each small rectangle is Δμ wide and Δν high. The apodization function f(μ,ν) is defined as constant over each small rectangle.
Figure 3
Figure 3
The apodization function f(μ,ν)=sin(μπ∕a)sin(νπ∕b) evaluated on the face of a 4λ×4λ square piston. The maximum value of the apodization function is achieved when μ=2λ and ν=2λ.
Figure 4
Figure 4
Simulated reference pressure field generated by an apodized rectangular source with each side equal to four wavelengths. The results are evaluated in the x=2.0λ plane for a time-harmonic excitation.
Figure 5
Figure 5
The normalized error distribution η(x,y,z;k) describes the difference between the reference pressure field and the computed pressure field for an apodized 4λ×4λ source. The error distribution η 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 fs=48 MHz and 30 subdivisions in each direction.
Figure 6
Figure 6
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.
Figure 7
Figure 7
Simulated reference transient field for an apodized square source excited by the Hanning-weighted pulse in Eq. 18 with f0=1.5 MHz and W=2.0λ. The sides of the square source are equal to 4λ, where the wavelength is defined with respect to the center frequency f0. The apodization function is given by Eq. 17. The transient reference pressure, evaluated in the x=2.0λ plane, is computed with 100 000 Gauss abscissas in each direction using the Rayleigh integral. The results are plotted at (a) t=1.5625 μs and (b) t=3.0625 μs.
Figure 8
Figure 8
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.
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