Civil Engineering Reference
In-Depth Information
The
M
×
M
element stiffness matrix is
d
x
d
y
k
(
e
)
=
[
N
]
T
∂
[
N
]
T
∂
∂
∂
[
N
]
∂
+
∂
∂
[
N
]
∂
(8.20)
x
x
y
y
A
(
e
)
and the nodal forces are represented by the
M
×
1
column matrix
f
(
e
)
=
[
N
]
T
(
un
y
−
vn
x
)d
S
(8.21)
S
(
e
)
Since the nodal forces are obtained via integration along element boundaries and
the unit normals for adjacent elements are equal and opposite, the forces on
interelement boundaries cancel during the assembly process. Consequently, the
forces defined by Equation 8.21 need be computed only for element boundaries
that lie on global boundaries. This observation is in keeping with similar obser-
vations made previously in context of other problem types.
8.3.2 Boundary Conditions
As the governing equation for the stream function is a second-order, partial dif-
ferential equation in two independent variables, four boundary conditions must
be specified and satisfied to obtain the solution to a physical problem. The man-
ner in which the boundary conditions are applied to a finite element model is
discussed in relation to Figure 8.4a. The figure depicts a flow field between two
parallel plates that form a smoothly converging channel. The plates are assumed
sufficiently long in the
z
direction that the flow can be adequately modeled as
two-dimensional. Owing to symmetry, we consider only the upper half of the
flow field, as in Figure 8.4b. Section
a-b
is assumed to be far enough from the
convergent section that the fluid velocity has an
x
component only. Since we ex-
amine only steady flow, the velocity at
a-b
is
U
ab
=
constant
.
A similar argument
applies at section
c-d
, far downstream, and we denote the
x
-velocity component
at that section as
U
cd
=
constant. How far upstream or downstream is enough to
make these assumptions? The answer is a question of solution convergence. The
distances involved should increase until there is no discernible difference in the
flow solution. As a rule of thumb, the distances should be 10-15 times the width
of the flow channel.
As a result of the symmetry and irrotationality of the flow, there can be no
velocity component in the
y
direction along the line
y
=
0
(i.e., the
x
axis). The
velocity along this line is tangent to the line at all values of
x
. Given these obser-
vations, the
x
axis is a streamline; hence,
=
1
=
constant along the axis.
Similarly, along the surface of the upper plate, there is no velocity component
normal to the plate (imprenetrability), so this too must be a streamline along
which
=
2
=
constant. The values of
1
and
2
are two of the required
boundary conditions. Recalling that the velocity components are defined as
first partial derivatives of the stream function, the stream function must be
known only within a constant. For example, a stream function of the form
