Civil Engineering Reference
In-Depth Information
Application of the Green-Gauss theorem to the first integral in expression (8.55)
yields
y
n
y
d
S
i
=
1,
M
(8.56)
where
S
(
e
)
is the element boundary and (
n
x
,
n
y
) are the components of the unit
outward normal vector to the boundary. Hence, the integral in expression (8.54)
becomes
∂
∂
N
i
∂
N
i
∂
d
A
N
i
∂
A
(
e
)
S
(
e
)
u
∂
∂
u
u
x
n
x
+
∂
u
−
+
=−
x
∂
x
y
∂
y
∂
∂
N
i
∂
d
A
N
i
∂
y
n
y
d
S
A
(
e
)
S
(
e
)
2
u
2
u
+
∂
u
x
n
x
+
∂
u
−
=−
x
2
y
2
∂
∂
∂
∂
∂
d
A
A
(
e
)
N
i
∂
∂
u
x
+
∂
N
i
∂
∂
u
+
(8.57)
x
∂
y
∂
y
Note that the first term on the right-hand side of Equation 8.57 represents a nodal
boundary force term for the element. Such terms arise from shearing stress. As
we observed many times, these terms cancel on interelement boundaries and
must be considered only on the global boundaries of a finite element model.
Hence, these terms are considered only in the assembly step. The second integral
in Equation 8.57 is a portion of the “stiffness” matrix for the fluid problem, and
as this term is related to the
x
velocity and the viscosity, we denote this portion
of the matrix
[
k
u
]
. Recalling that Equation 8.57 represents
M
equations, the
integral is converted to matrix form using the first of Equation 8.52 to obtain
A
(
e
)
∂
d
A
[
N
]
T
∂
[
N
]
T
∂
∂
[
N
]
∂
+
∂
∂
[
N
]
∂
{
u
}=
[
k
u
]
{
u
}
(8.58)
x
x
y
y
Using the same approach with the second of Equation 8.53, the results are
similar. We obtain the analogous result
N
i
∂
d
A
N
i
∂
y
n
y
d
S
A
(
e
)
S
(
e
)
2
v
2
v
∂
v
∂
v
−
+
=−
x
n
x
+
x
2
y
2
∂
∂
∂
∂
∂
d
A
A
(
e
)
N
i
∂
∂
v
x
+
∂
N
i
∂
∂
v
+
(8.59)
x
∂
y
∂
y
Proceeding as before, we can write the area integrals on the right as
A
(
e
)
∂
d
A
[
N
]
T
∂
[
N
]
T
∂
∂
[
N
]
∂
+
∂
∂
[
N
]
∂
{
v
}=
[
k
v
]
{
v
}
(8.60)
x
x
y
y
