Civil Engineering Reference
In-Depth Information
Input and initialisation
For all elements
For all integrating points
Form and sum [k
c
] contribution
Form and sum [m
m
] contribution
Store element [k
c
]
Assemble global lumped mass
For all elements
Retrieve [k
c
]
Invert [[m
m
] +
q∆
t[k
c
]/2]
-1
=[b]
Form [b][[m
m
]
−
(1
− q
)
∆
t[k
c
]/2]=[a]
Store [a]
For all time steps
For both passes
Gather appropriate part of {
f
}
0
Compute {
f
}
1
=[a]{
f
}
0
Print solution.
Figure 3.20 Structure chart for the element-by-element product algorithm (two pass)
and carrying out the product (3.101) by sweeping twice through the elements. They suggest
from first to last and back again, but clearly various choices of sweeps could be employed.
It can be shown that as
t
→
0, any of these processes converges to the true solution of
the global problem. A structure chart for the process is shown in Figure 3.20 and examples
of all the methods described in this section are implemented in Chapter 8. Consistent mass
versions are described by Gladwell
et al
. (1989).
3.11 Solution of coupled Navier-Stokes problems
For steady state conditions, it was shown in Section 2.16 that a non-linear system of alge-
braic equations had to be solved, involving, at the element level, submatrices [
c
11
], [
c
12
],
etc. These element matrices contained velocities
u
and
v
(called
ubar
and
vbar
in the
programs) together with shape functions and their derivatives for the velocity and pressure
variables. It was mentioned that it would be possible to use different shape functions for
