Civil Engineering Reference
In-Depth Information
formally as the system
=
{
u
}
{
f
Bx
}
[
k
u
]
[0]
[
k
px
]
⇒
k
(
e
)
(
e
)
=
f
(
e
)
{
v
}
{
f
By
}
{
(8.67)
[0]
[
k
v
]
k
py
]
{
p
}
0
}
[
k
u
]
[
k
v
]
[0]
where
[
k
(
e
)
]
represents the complete element stiffness matrix. Note that the ele-
ment stiffness matrix is composed of nine
M
×
M
submatrices, and although the
individual submatrices are symmetric, the stiffness matrix is
not
symmetric.
The development leading to Equation 8.67 is based on evaluation of both
the velocity components and pressure at the same number of nodes. This is
not necessarily the case for a fluid element. Computational research [7] shows
that better accuracy is obtained if the velocity components are evaluated at
a larger number of nodes than pressures. In other words, the velocity compo-
nents are discretized using higher-order interpolation functions than the pres-
sure variable. For example, a six-node quadratic triangular element could be
used for velocities, while the pressure variable is interpolated only at the cor-
ner nodes, using linear interpolation functions. In such a case, Equation 8.66
does not hold.
The arrangement of the equations and associated definition of the element
stiffness matrix in Equation 8.67 is based on ordering the nodal variables as
T
{}
=
[
u
1
u
2
u
3
v
1
v
2
v
3
p
1
p
2
p
3
]
(using a three-node element, for example). Such ordering is well-suited to illus-
trate development of the element equations. However, if the global equations for
a multielement model are assembled and the global nodal variables are similarly
ordered, that is,
T
{
}
=
[
U
1
U
2
···
V
1
V
2
···
P
1
P
2
···
P
N
]
the computational requirements are prohibitively inefficient, because the global
stiffness has a large
bandwidth
. On the other hand, if the nodal variables are
ordered as
T
U
N
V
N
P
N
]
computational efficiency is greatly improved, as the matrix bandwidth is signifi-
cantly reduced. For a more detailed discussion of banded matrices and associated
computational techniques, see [8].
{
}
=
[
U
1
V
1
P
1
U
2
V
2
P
2
···
EXAMPLE 8.3
Consider the flow between the plates of Figure 8.4 to be a viscous, creeping flow and
determine the boundary conditions for a finite element model. Assume that the flow is
fully developed at sections
a-b
and
c-d
and the constant volume flow rate per unit thick-
ness is
Q
.
