Biomedical Engineering Reference
In-Depth Information
approximation. The linear approximation allows the solutions in the theorem
to apply, where it is observed that the characteristics travel in circles with vary-
ing speed
F
, and the linear approximation of
F
designates a center of rotation,
(
A
,
B
), depending on where
F
crosses zero. This leads to a generalized form of
Eq. 4.28:
∇
φ
(
y
)
×
(
x
−
y
)
=
x
−
y
2
(
k
·
(
∇
F
(
y
)
×∇
φ
(
y
)))
2
F
(
y
)
(4.52)
.
Note that Eq. 4.28 is recovered if
∇
F
=
0 is assumed. Equations 4.27 and 4.52
are solved in the same manner as described in Section 4.2.3.
Using the modified velocity extension method on the earlier two-circle ex-
ample produces the correct results as shown in Fig. 4.15.
Another example that illustrates the difference between the two velocity
extension methods is given by an initial circle, with
F
varying linearly with
respect to
x
, and near zero on the left side. The largest difference between the
two methods can be seen on the side where
F
is small. In the old method,
1
0.8
0.6
0.4
0.2
0
−
0.2
−
0.4
−
0.6
−
0.8
−
1
−
1
−
0.8
−
0.6
−
0.4
−
0.2
0
0.2
0.4
0.6
0.8
1
Figure 4.15:
Two-circle example with the modified velocity extension method.
