Civil Engineering Reference
In-Depth Information
In non-linear problems which are “ill-conditioned”, pcg iterations could become signif-
icant despite the unconditional stability exhibited by Program 8.3. The oldest and simplest
mesh-free process using lumped mass, is the “explicit” method, based on equation (3.98),
where
]
−
1
{
}
1
=
[
M
m
(
[
M
m
]
−
t
[
K
c
]
)
{
}
0
(8.2)
The element integration loop follows the now standard course but the element matrix
(
is stored in
store mm
for each element rather than assembled as would
be the case for a traditional implicit technique. Then in the time-stepping loop, element
matrices are recovered from
store mm
and the product
[
m
m
]
−
t
[
k
c
]
)
[
M
m
)
{
}
0
(
]
−
t
[
K
c
]
formed
“element-by-element” using the summation,
nels
1
(
[
m
]
−
t
[
k
]
)
i
{
φ
}
0
i
m
c
i
=
. The result of this element-by-
element product is called
newlo
in programming terminology.
This having been done, the global
where
{
φ
}
0
i
is the appropriate part of
{
}
0
for element
i
{
}
1
called
loads
is computed by multiplying
newlo
by the inverse of the global mass matrix
globma
. The process is illustrated by
the structure chart in Figure 8.14.
The problem shown on Figure 8.5 has been analysed again with the data shown in
Figure 8.15 with output as Figure 8.16. The only difference from Figure 8.5, is that being
type_2d dir
'plane' 'x'
nxe nye np_types
5 5 1
prop(cx,cy)
1.0 1.0
etype(not needed)
x_coords, y_coords
0.0 0.2 0.4 0.6 0.8 1.0
0.0 -0.2 -0.4 -0.6 -0.8 -1.0
dtim nstep npri nres ntime
0.01 150 10 31 100
loads(i),i=1,neq
0.0 0.0 0.0 0.0 0.0 0.0
100.0 100.0 100.0 100.0 100.0 0.0
100.0 100.0 100.0 100.0 100.0 0.0
100.0 100.0 100.0 100.0 100.0 0.0
100.0 100.0 100.0 100.0 100.0 0.0
100.0 100.0 100.0 100.0 100.0 0.0
fixed_freedoms,(node(i),value(i),i=1,fixed_freedoms)
11
1 0.0 2 0.0 3 0.0 4 0.0 5 0.0 6 0.0
12 0.0 18 0.0 24 0.0 30 0.0 36 0.0
nci
10
Figure 8.15
Data for Program 8.4 example
