Civil Engineering Reference
In-Depth Information
Considering next the pressure terms and converting to matrix notation, the
first of Equation 8.53 leads to
[
N
]
T
∂
[
N
]
∂
d
A
{
p
} =
[
k
px
]
{
p
}
(8.61)
x
A
(
e
)
and similarly the second momentum equation contains
[
N
]
T
∂
[
N
]
∂
d
A
{
p
} =
[
k
py
]
{
p
}
(8.62)
y
A
(
e
)
The nodal force components corresponding to the body forces are readily
shown to be given by
[
N
]
T
F
Bx
d
A
{
f
Bx
}=
A
(
e
)
(8.63)
[
N
]
T
F
By
d
A
{
f
By
}=
A
(
e
)
Combining the notation developed in Equations 8.58-8.63, the momentum equa-
tions for the finite element are
[
k
u
]
{
u
}+
[
k
px
]
{
p
}={
f
Bx
}+{
f
x
}
(8.64)
{
}+
{
}={
f
By
}+{
f
y
}
[
k
v
]
v
[
k
py
]
p
where, for completeness, the nodal forces corresponding to the integrals over
element boundaries
S
(
e
)
in Equations 8.57 and 8.59 have been included.
Finally, the continuity equation is expressed in terms of the nodal velocities
in matrix form as
[
N
]
T
∂
[
N
]
∂
[
N
]
T
∂
[
N
]
∂
d
A
{
u
} +
d
A
{
v
}=
[
k
u
]
{
u
}+
[
k
v
]
{
v
}=
0
(8.65)
x
y
A
(
e
)
A
(
e
)
where
[
N
]
T
∂
[
N
]
∂
[
k
u
]
=
[
k
px
]
=
d
A
x
A
(
e
)
(8.66)
[
N
]
T
∂
[
N
]
∂
[
k
v
]
=
[
k
py
]
=
d
A
y
A
(
e
)
As formulated here, Equations 8.64 and 8.65 are a system of 3
M
algebraic equa-
tions governing the 3
M
unknown nodal values
{
u
}
,
{
v
}
,
{
p
}
and can be expressed
