Biomedical Engineering Reference
In-Depth Information
equal to 1% throughout the entire grid defined for this pressure
calculation is N = 109. Although the aperture of the spherically
focused transducer simulated here is much larger than that simu-
lated for a circular piston in Figure 6.2, fewer abscissas are required
for the spherically focused transducer because the computational
grid is only adjacent to the spherically focused transducer at the
very edge of the grid where ( x,z ) = (± a ,− a ), whereas the pressure in
Figure 6.2 is computed across the entire face of the circular piston.
Using the same computer configuration described previously, the
computation time for the Rayleigh-Sommerfeld integral evaluated
on a 101-point by 201-point grid with N = 109 is 88.22 seconds (just
under 1.5 minutes).
6.2.3 Fast Near-Field Method
The fast near-field method (FNM), which evaluates one or more
one-dimensional integrals that subtract a singularity from the
integrand, addresses the main deficiencies of the Rayleigh-
Sommerfeld integral and the rectangular radiator approach. By
subtracting a singularity from the numerator of the integrand,
the otherwise slow convergence is eliminated near the piston
face and along the boundaries defined for impulse response
calculations. As a result, the errors in the near-field region are
much smaller with the fast near-field method than with these
other methods. Furthermore, the fast near-field method evalu-
ates one or more one-dimensional integrals (instead of a two-
dimensional integral), so the fast near-field method is O ( n ).
Thus, the time required for calculations with the fast near-
field method only increases linearly as the number of abscissas
increases. The rapid convergence achieved by subtracting the
singularity in one or more one-dimensional integrals enables
the fast near-field method to achieve smaller numerical errors in
less time compared to these other methods, especially for pres-
sure calculations in the near-field region.
In contrast to the Rayleigh-Sommerfeld integral, where the
integrand is consistently the same but the coordinates of the
source points, the expressions describing the incremental areas,
and the two-dimensional limits of integration are specific to
each different piston shape, the fast near-field method defines
unique one-dimensional integral expressions for each piston
shape. By optimizing the integral expressions for flat circular,
flat rectangular, and spherically focused transducers, near-
field calculations with the fast near-field method achieve much
smaller numerical errors in much less time while eliminating
the numerical problems inherent to the Rayleigh-Sommerfeld
integral near the piston face. The FNM expressions are evalu-
ated with Gauss quadrature, which converges much faster than
most other numerical integration schemes when applied to these
calculations.
6.2.2 rectangular radiator Method
The rectangular radiator method described in (Ocheltree and
Frizzell 1989) subdivides a rectangular transducer into small
subelements, calculates the pressures generated by the small
rectangular subelements, and superposes the results. The num-
ber of subdivisions is chosen such that the observation point is
in the far field of each subelement, and then the pressure from
each subelement is computed with the far-field approximation
in Equation 6.9. The main advantage of the rectangular radia-
tor method over the Rayleigh-Sommerfeld integral is the adap-
tive selection of the number of subelements, which reduces the
number of subelements far from the piston face and likewise
increases the number of subelements near the piston face as
needed. The number of subelements is determined by the sub-
division parameter F , which is defined such that the subelement
width Δ w (and height Δ h ) satisfies
4
λ
z
w
(6.15)
F
at each axial distance z from the piston face. In (Ocheltree and
Frizzell 1989), the choice of F = 10 is intended to guarantee that
the observation point is at least 10 far-field distances from the
subelement. However, this value of F is associated with discon-
tinuities in the computed pressure field that are clearly evident
when plotted on a linear scale. In an effort to avoid these numer-
ical artifacts, F = 20 was used in Moros et al. (1993).
The calculation times associated with this approach repre-
sent another limitation of the rectangular radiator method.
By subdividing the aperture in both directions, the rectan-
gular radiator method numerically evaluates a two-dimen-
sional integral. Thus, the Rayleigh-Sommerfeld integral and
the rectangular radiator method are inherently O ( n 2 ), where
n represents the number of subelements in each direction.
Consequently, the computation times associated with these
O ( n 2 ) methods increase quadratically as the number of subele-
ments defined for these calculations increases. When pressure
calculations are evaluated for several elements on a large grid
in the near-field region, this quadratic increase translates into
relatively long computation times.
6.2.3.1 Circular piston
The FNM expression for the time-harmonic pressure generated
by a circular piston (McGough et al. 2004) in cylindrical coor-
dinates is
a
π
ψ−
+− ψ
r
cos
a
jt
ω
przt
(, ,)
cU e
00
2
2
π
ra ar
2 os
0
(
)
22
2
jk
ra ar
+
2 os
ψ+
z
jkz
(6.16)
xe
e
d
ψ
,
where a is the radius of the circular piston, ( r,z ) is the coordinate
of the observation point, and ψ is the variable of integration.
When the pressure is evaluated on the z axis (i.e., when r = 0),
all ψ dependence is eliminated from the integrand, and the
analytical expression for the on-axis pressure in Equation 6.4 is
obtained. Numerical evaluations of Equation 6.16 discretize the
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