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 sectorvortex ultrasound phased array, and various focal
patterns are quickly evaluated.
Usually the
point source superposition method computes the field generated by the
sectorvortex 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.
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. 
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 timeharmonic 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 timeharmonic 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 sectorvortex ultrasound phased array
The array is
constructed on a spherical shell. The sectorvortex 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
sectorvortex phased array 
The various focus patterns generated by the sectorvortex array are illustrated in Fig. 9.


(a) 
(b) 


(c) 
(d) 
Fig.
9: The various focus patterns generated by the simulated sectorvortex 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. 
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 sectorvortex array can be
quickly generated when the fast nearfield method is applied to simulations of
ultrasound phased arrays.
[1] R. J. McGough. Rapid calculations of
timeharmonic nearfield pressures produced by rectangular pistons. J.
Acoust. Soc. Am., 115 (5), pp. 19341941, 2004.
[2] C. Cain
and S. Umemura, “Concentric –ring and sector vortex phased array applicators for
ultrasound hyperthermia,” IEEE
Trans. Microwave Theory Tech., vol. MTT34, no.5, pp. 542551, 1986.
[3] S.
Umemura and C. Cain, “The sectorvortex phased array: acoustic field
synthesis for hyperthermia,” IEEE
Transactions on ultrasonics, ferroelectrics, and
frequency control, vol. 36, no. 2, pp. 249257, 1989.