Biomedical Engineering Reference
In-Depth Information
8.6.4.1 Numerical Schemes
Osher
et al.
[30] have proposed an up-wind method for solving equations of the
form
φ
t
=∇
φ
·
v
, of which
φ
t
=|∇
φ
|
i
e
i
(
x
), from Eq. (8.47), is an example.
The up-wind scheme utilizes one-sided derivatives in the computation of
|∇
φ
|
,
where the direction of the derivative depends, point-by-point, on the sign of
the speed term
i
e
i
(
x
). With strictly regulated time steps, this scheme avoids
overshooting (ringing) and instability.
Under normal circumstances, the curvature term, which is a directional dif-
fusion, does not suffer from overshooting; it can be computed directly from first-
and second-order derivatives of
φ
using central difference schemes. However,
we have found that central differences do introduce instabilities when comput-
ing flows that rely on quantities other than the mean curvature. Therefore, we
use the method of
differences of normals
[101,102] in lieu of central differences.
The strategy is to compute normalized gradients at staggered grid points and
take the difference of these staggered normals to get centrally located approxi-
mations to
N
(as in Fig. 8.20). The normal projection operator
n
⊗
n
is computed
with gradient estimates from central differences. The resulting curvatures are
q-1
n[p,q-1]
N
computed as
difference of normals at
original grid location
n[p-1,q]
n[p,q]
q
n[p,q]
Staggered normals
computed using 6
neighbors (18 in 3D)
q+1
p-1
p
p+1
Figure 8.20:
The shape matrix
B
is computed by using the differences of stag-
gered normals.
