Graphics Reference
In-Depth Information
In three dimensions,
(
A
(
i−
1
,j,k
)
,
(
i,j,k
)
/E
(
i−
1
,j,k
)
)
2
A
(
i,j,k
)
,
(
i,j,k
)
−
(
A
(
i,j−
1
,k
)
,
(
i,j,k
)
/E
(
i,j−
1
,k
)
)
2
(
A
(
i,j,k−
1)
,
(
i,j,k
)
/E
(
i,j,k−
1)
)
2
−
−
−
A
(
i−
1
,j,k
)
,
(
i,j,k
)
×
(
A
(
i−
1
,j,k
)
,
(
i−
1
,j
+1
,k
)
+
A
(
i−
1
,j,k
)
,
(
i−
1
,j,k
+1)
)
/E
(
i−
1
,j,k
)
E
(
i,j,k
)
=
−
A
(
i,j−
1
,k
)
,
(
i,j,k
)
×
(
A
(
i,j−
1
,k
)
,
(
i
+1
,j−
1
,k
)
+
A
(
i,j−
1
,k
)
,
(
i,j−
1
,k
+1)
)
/E
(
i,j−
1
,k
)
−
A
(
i,j,k−
1)
,
(
i,j,k
)
×
(
A
(
i,j,k−
1)
,
(
i
+1
,j,k−
1)
+
A
(
i,j,k−
1)
,
(
i,j
+1
,k−
1)
)
/E
(
i,j,k−
1)
If you're curious, the intuition behind MIC (and why it outperforms IC)
lies in a Fourier analysis of the problem. If you decompose the error as
a superposition of Fourier modes, some low frequency (smooth) and some
•
Set tuning constant
τ
=0
.
97 and safety constant
σ
=0
.
25
•
For
i=1
to
nx
,
j=1
to
ny
,
k=1
to
nz
:
•
If cell (
i, j, k
) is fluid:
•
(
Aplusi
i−
1
,j,k
∗
precon
i−
1
,j,k
)
2
Set
e
=
Adiag
i,j,k
−
precon
i,j−
1
,k
)
2
−
(
Aplusj
i,j−
1
,k
∗
(
Aplusk
i,j,k−
1
∗
precon
i,j,k−
1
)
2
−
τ
Aplusi
i−
1
,j,k
∗
−
(
Aplusj
i−
1
,j,k
+
Aplusk
i−
1
,j,k
)
∗
precon
2
i−
1
,j,k
+
Aplusj
i,j−
1
,k
∗
(
Aplusi
i,j−
1
,k
+
Aplusk
i,j−
1
,k
)
∗
precon
i,j−
1
,k
+
Aplusk
i,j,k−
1
∗
(
Aplusi
i,j,k−
1
+
Aplusj
i,j,k−
1
)
∗
precon
2
1
i,j,k−
•
If
e<σ
Adiag
i,j,k
,set
e
=
Adiag
i,j,k
precon
i,j,k
=1
/
√
e
•
Figure 4.6.
The calculation of the MIC(0) preconditioner in three dimensions.
