Biomedical Engineering Reference
In-Depth Information
Pressures computed with the FNM expression for a spheri-
cal shell with an aperture radius
a
= 20λ and a radius of cur-
vature
R
= 20 λ
are evaluated on the same 101- by 201-point
grid shown in Figure 6.6. The FNM expression in Equation 6.18
converges to an error less than or equal to 1% throughout the
grid with
N
= 30 Gauss abscissas. The result is calculated with
optimized C++ code called from MATLAB
®
on the same com-
puter in 0.3588 seconds. With additional abscissas, the peak
FNM errors are even smaller at the expense of a small increase
in computation time. Thus, relative to the Rayleigh-Sommerfeld
integral, the fast near-field method achieves equal or smaller
errors in less time for calculations of pressures generated by
spherically focused transducers. For this combination of piston
and grid parameters, where the peak errors obtained with both
methods are equal to 1%, the FNM is more than 200 times faster
than the Rayleigh-Sommerfeld integral.
6.2.4 angular Spectrum approach
The angular spectrum approach accelerates computations of dif-
fracting pressure fields evaluated in parallel planes using 2D fast
Fourier transforms (FFTs). The angular spectrum approach is
especially valuable for simulating large 3D pressure fields generated
by ultrasound phased arrays. When applied properly, the angular
spectrum approach achieves a substantial reduction in the compu-
tation time without significantly increasing the numerical error.
The angular spectrum approach is analytically equivalent to
the Rayleigh-Sommerfeld integral, which convolves the spatial
distribution of the normal particle velocity in the plane contain-
ing the transducer with the Green's function for a point source
surrounded by a rigid baffle. The Green's function for a point
source surrounded by a rigid baffle radiating into a semi-infinite
half-space is
e
−
jkR
6.2.3.4 Gauss Quadrature
FNM calculations are typically evaluated with Gauss quadrature
(Davis and Rabinowitz 1975), which converges rapidly for the
expressions in Equations 6.16, 6.17, and 6.18. The Gauss abscissas
g
i
and weights
w
i
are computed with the
N
point Gauss-Legendre
quadrature rule for the interval (−1,1), then the Gauss abscissas
are linearly mapped (Abramowitz and Stegun 1972) to the limits
of integration (
γ
min
,
γ
max
) according to
hxyz
(, ,)
=
,
(6.23)
u
2π
R
2 2 2
=++
is the distance from a point source
located at (0,0,0) to an observation point located at (
x ,y,z
), the
subscript
u
designates a normal particle velocity input, and
lower case
h
indicates that the Green's function is evaluated in
the spatial domain. Similarly, the normal particle velocity for an
ultrasound transducer or an array of transducers strictly located
in the
z
= 0 plane is represented by
u
(
x ,y
,0). The Green's function
h
u
(
x ,y,z
) convolved with the normal particle velocity
u
(
x ,y
,0) is
proportional to the pressure
where
Rxyz
γ=
γ−γ
+
γ+γ
max
in
max
in
g
(6.19)
i
i
2
2
and the Gauss weights
w
i
are scaled by (
γ
max
−
γ
min
)/2. When
applied to FNM calculations for pistons exhibiting cylindrical
symmetry (i.e., circular pistons and spherical shells),
γ
min
= 0 and
γ
max
= π, so the linearly mapped Gauss abscissas are
jt
ω
px yzt
(, ,,)
=ωρ
j
euxy
(
, 0)
**
hxyz
(, ,),
(6.24)
0
xy u
,
where
**
x
,
y
indicates 2D convolution with respect to the spatial
variables
x
and
y
. The Fourier transform of Equation 6.24, evalu-
ated with respect to
x
and
y
in a plane orthogonal to the
z
axis at
a distance
z
from the transducer face, is
ψ
ππ
i
=+
2
2
(6.20)
i
and the scaled Gauss weights are given by
d
ψ
i
= (π/2)
w
i
. FNM
calculations for rectangular pistons define limits of integration
that vary based on the location of the observation point, and
separate mappings are defined for the variables of integration
x
i
′
and
y
i
′
. Integration with respect to the
x
i
′
variable is performed
with abscissas defined by
jt
ω
Pk kz
(,
,)
=ρω
j
eUk kHkkz
(,
,)
0
(,
,).
(6.25)
xy
0
xy
ux y
In Equation 6.25, the spatial frequencies are represented by
k
x
and
k
y
, the 2D Fourier transform of the normal particle velocity
is
Uk k
F=
, the 2D Fourier transform of the
Green's function evaluated in a plane orthogonal to the
z
axis
is
Hkkz
(,
,0)
{ (, ,0)}
u xy
xy
xy
,
x gx
i
′=+
(6 . 21)
F=
, and convolution in the spatial
domain is replaced by multiplication in the spatial frequency
domain (i.e.,
k
-space). The analytical expression for
H
u
(
k
x
,
k
y
,
z
) is
(,
,)
{(,,)}
hxyz
i
ux y
x yu
,
′ =
, where
a
is the half-width of the rect-
angular piston and
x
represents the
x
coordinate of the observa-
tion point. Likewise, integration with respect to the
′
and with weights
dx
aw
i
i
y
i
variable
is performed with abscissas defined by
222
−
jz
kkk
−
−
e
j
xy
2
2
2
for
kk k
+≤
x
y
2
2
2
kkk
−−
y gy
i
′=+
(6.22)
x
y
i
Hkkz
(,
,)
=
,
(6.26)
ux y
222
−
zk kk
+
−
e
xy
and with weights
dy
′ =
, where
b
is the half-height of the
rectangular piston and
y
represents the
y
coordinate of the
observation point.
bw
2
2
2
for
kk k
+>
i
i
x
y
2
2
2
kkk
+−
x
y
