A fast method for calculating the pressure produced by a trapezoidal source applied to simulations of a sector-vortex ultrasound phased array

 

Abstract

A fast method for calculating the near field pressure distribution generated by a trapezoidal piston is derived and compared with similar results obtained from the impulse response approach.  The fast method, which defines an analytical integral expression for each side of the trapezoid, eliminates the singularities produced by the impulse response and thus the computation time and the peak numerical error is reduced. The results are demonstrated in a pressure field generated by an isosceles trapezoidal source with an upper base of 2 wavelengths, a lower base of 4 wavelengths, and height of 2 wavelengths.  The results show that the rapid method is consistently faster than the impulse response when the integrals from each method are computed with the same number of Gauss abscissas. For specified maximum errors of 10%, 1% and 0.1%, the rapid method is 2.795, 2.660, and 1.688 times faster than the impulse response in pressure field calculations evaluated in the axial plane that passes through the center of the trapezoid in the direction perpendicular to the two parallel bases of the trapezoid.  For specified maximum errors of 10%, 1% and 0.1%, the rapid method is 3.397, 2.624 and 1.920 times faster than the impulse response in pressure field calculations evaluated in the axial plane that passes through the center of the trapezoid in the direction parallel to the two parallel bases of the trapezoid.  The fast method is then applied to simulations of a sector-vortex ultrasound phased array, and various focal patterns are quickly evaluated.

 

Introduction

Usually the point source superposition method computes the field generated by the sector-vortex array, but this method converges slowly while generating relatively large errors. The impulse response approach is faster than point source methods, but the impulse response encounters some numerical problems near the edge of the transducer. The fastest and most accurate approach is the fast nearfield method (FNM), which also eliminates the numerical problems encountered in the impulse response.

 

Methods

For any (acute, right, or obtuse) triangle    ABC       the pressure generated by a triangular transducer above the point  C  has a unique expression based on the fast nearfield rectangular method [1]:

 

 

 

 

Where                                   describes the line that contains AB.

 

Fig. 1: The 3D coordinate system used in the computation of triangle field pressure.

 

The pressure field generated by a trapezoidal source is computed by superposing contributions from 4 triangles, with each triangle corresponding to one side of the trapezoid. The formula is given below and the corresponding coordinate system is given in Fig. 2. and Fig. 3.

 

 

 

 

 

 

 

 

 


Where                                                                                                                                                 , are the lines that contain AB and CD.

 

 

Fig. 2: The 3D (XYZ) coordinate system that defines individual triangles.

Fig. 3: The 2D (XY) coordinate system  wherein triangles are defined for pressure field computations.

 

Results

A. Single trapezoidal source

Pressure field results are demonstrated for an isosceles trapezoidal source with an upper base of 2 wavelengths, a lower base of 4 wavelengths, and a height of 2 wavelengths. The reference beam is generated by the fast nearfield method computed with 200 Gauss abscissas. The shape of the beam is shown in Fig. 4 and Fig. 5.    Comparisons of the maximum peak error and computation time as a function of Gauss abscissas between the fast nearfield method and the impulse response are illustrated in Fig. 6 and Fig. 7. Table 1 contains the computation time, the number of abscissas used, and the time ratio between the fast nearfield method and the impulse response for specified maximum errors of 10%, 1% and 0.1%.

 

 

Fig. 4: Simulated time-harmonic pressure field for a trapezoidal  source with an upper base of 2 wavelengths, a lower base of 4  wavelengths, and a height of 2 wavelengths. The pressure is evaluated in the x=0 plane.

Fig. 5: Simulated time-harmonic pressure field for a trapezoidal  source with an upper base of 2 wavelengths, a lower base of 4  wavelengths, and a height of 2 wavelengths. The pressure distribution is demonstrated in the z=0 plane.

 

 

Fig. 6: Comparison of maximum normalized errors obtained from the fast nearfield method (red solid line) and the impulse response as a function of the number of Gauss abscissas. The figure shows that the maximum error computed with the fast nearfield method is consistently smaller than that obtained with the impulse response.  The pressure field is evaluated in the x=0 plane

Fig. 7: Comparison of computation times required for simulations using the fast nearfield method (red solid line) and the impulse response as a function of the number of Gauss abscissas. The figure indicates that the fast nearfield method is always faster than the impulse response. The pressure field is evaluated in the x=0 plane.

 

 

 

10% peak error

1% peak error

0.1% peak error

 

FNM

Impulse Response

FNM

Impulse Response

FNM

Impulse Response

Time

0.078

0.218

0.094

0.250

0.157

0.265

Abscissas

6

7

7

8

12

9

Ratio

2.795

2.660

1.688

 

Table 1: Comparisons of computation time, number of abscissas used, and time ratio between the fast nearfield method and the impulse response for a specified maximum error of 10%, 1% and 0.1%. The results are computed in the x=0 plane.

 

 

B. Simulations of a sector-vortex ultrasound phased array

 

The array is constructed on a spherical shell. The sector-vortex array has two tracks and each track is divided into 32 sectors as illustrated in Fig. 8. The radius of the sphere is 80mm, the aperture radius is 61.4mm, the central hole radius is 19.3mm, and the partition radius is 42.2mm.

The    th sector on the   th track is driven by [2], [3]:

 

 

 

For                        ,          and                                                      , M is the vortex mode number,     is a constant.

 

Fig. 8: The configuration of the sector-vortex phased array

 

 

The various focus patterns generated by the sector-vortex array are illustrated in Fig. 9.

 

(a)

(b)

(c)

(d)

 

Fig. 9: The various focus patterns generated by the simulated sector-vortex phased array for each specified mode number M for a given number of Gauss abscissas 14. The pressure is simulated in the focal plane and the grid size is 201*201. (a) M=0, the computation time is 39.203s. (b) M=4, the computation time is 39.937. (c) M=8, the computation time is 40.328. (d) M=16, the computation time is 40.000s.

 

 

 

Conclusions

For a single trapezoidal source, the fast nearfield method achieves more accurate results and also is much faster than the impulse response for specified maximum errors of 10%, 1% and 0.1%. Various focal patterns of the sector-vortex array can be quickly generated when the fast nearfield method is applied to simulations of ultrasound phased arrays.

 

Reference

[1] R. J. McGough. Rapid calculations of time-harmonic nearfield pressures produced by rectangular pistons. J. Acoust. Soc. Am., 115 (5), pp. 1934-1941, 2004.

[2] C. Cain and S. Umemura, Concentric ring and sector vortex phased array applicators for ultrasound hyperthermia, IEEE Trans. Microwave Theory Tech., vol. MTT-34, no.5, pp. 542-551, 1986.

[3] S. Umemura  and C. Cain, The sector-vortex phased array: acoustic field synthesis for hyperthermia, IEEE Transactions on ultrasonics, ferroelectrics, and frequency control, vol. 36, no. 2, pp. 249-257, 1989.