Information Technology Reference
In-Depth Information
2 Analysis Methods
The FEM model is presented as follows:
∂
φ
[
M
]
+
[
N
]
φ
=
{
Q
}
(1)
∂
t
[
M
]
=
[
S
]
+
n
3
A
[
]
(2)
*
1
*
2
[
N
]
=
[
D
]
−
[
B
]
−
[
B
]
(3)
{
Q
}
=
{
Q
}
+
{
Q
}
+
{
Q
}
(4)
1
2
3
φ
is potential function;
t
is time;
[
M
and
]
[
N
is coefficient
]
While, in eq.(1),
{
Q
is column vector; the calculation methods of
}
[
M
,
]
matrixes respectively;
[
N
and
]
{
Q
are shown in eq.(2)~(4). In eq.(2),
}
[
S
is the superposition of
]
e
,
∫
e
[
S
]
[
S
]
=
CN
N
d
Ω
,
e
means element,
C
is water capacity of
e
,
N
i
j
is shape function of
e
,
i
,
j
∈
[
nnode
]
,
nnode
is element knots,
Ω
is region of
e
,
∫
e
][
,
S
is slope surface bor-
der of
e
;
n
is the cosine of the angle between the outer normal vector and Z
axis. In eq.(3),
e
;
[
A
is the superposition of
]
[
A
]
A
=
N
N
d
S
i
j
e
,
∫
e
T
[
D
is the superposition of
]
[
D
]
[
D
]
=
[
B
]
[
k
][
B
]
Ω
,
i
j
∂
N
∂
N
∂
N
T
i
i
i
*
[
B
]
=
[
,
,
]
[
B
is the superposition
]
,
[
k
is conductivity matrix;
∂
x
∂
y
∂
z
∂
N
1
1
∫
j
e
e
e
e
5
/
3
[
1
B
]
[
λ
]
[
B
]
=
N
d
S
[
λ
]
=
diag
(
λ
)
λ
=
h
of
,
x
i
x
xi
xi
i
∂
x
n
f
xi
n
is Manning roughness,
h
is water depth of node
(
f
2
+
f
2
)
1
/
4
(
z
+
h
/
n
)
xi
yi
i
i
3
i
,
z
is z coordinate of node
i
,
f
f
are slope gradients along
x
and
y
and
xi
yi
*
e
e
e
directions of node
i
;
[
B
is the superposition of
]
[
B
]
[
λ
]
[
B
]
=
,
2
y
2
∂
N
1
f
∫
j
e
5
/
3
N
d
S
[
λ
]
=
diag
(
λ
)
λ
=
h
xi
,
.
i
y
yi
yi
i
∂
y
n
2
2
1
/
4
(
f
+
f
)
(
z
+
h
/
n
)
xi
yi
i
i
3
e
,
∫
e
{
Q
is the superposition of
}
{
1
Q
}
{
Q
}
=
N
q
d
S
q
is the normal
In eq.(4),
,
1
i
1
1
S
of element which does not contain slope surface;
{
Q
is the
}
boundary flux on
Search WWH ::
Custom Search