Civil Engineering Reference
In-Depth Information
The following sequence completes the integration:
CALL cross_product(vol,funf,volf)
nip
c
=
(3.111)
det
i
*weights(i)*volf
i
i
=
1
The volumetric strain (
v
) at any point within a displacement element is easily retrieved
from the product,
DOT_PRODUCT(vol,eld)
where
eld
holds the element nodal dis-
placements.
To integrate equations (3.107) with respect to time there are again many methods avail-
able, but we consider only the simplest linear interpolation in time using finite differences,
similar to that used for first order uncoupled problems in equation (3.94).
3.12.1 Absolute load version
This approach applies the full external loading at each time step and is suitable for linear
elastic problems. Interpolation in time using
θ
, and elimination of the derivative terms as
was done for first order problems in (3.94), leads to the following recurrence equations at
the element level:
θ
[
k
m
]
{
−
(
1
{
θ
[
c
]
u
}
−
θ)
[
k
m
]
−
(
1
−
θ)
[
c
]
u
}
1
=
θ
[
c
]
T
−
θ
2
t
[
k
c
]
θ
[
c
]
T
{
u
w
}
θ(
1
−
θ)t
[
k
c
]
{
u
w
}
0
(
1
θ
{
−
θ)
{
f
}
f
}
+
0
+
(3.112)
{
0
}
{
0
}
1
represents the external loading vector which may itself be time dependent.
The left and right hand side element matrices, called [
k
e
] and [
k
d
] respectively, are
formed from their constituent matrices by subroutine
fmkdke
. The second equation has
been multiplied through by
θ
to preserve symmetry of the [
k
e
] matrix; however, the right
hand side matrix [
k
d
] is unsymmetric.
A Crank-Nicolson type of approximation,
θ
=
where
{
f
}
1
/
2 would be a popular choice in
equation (3.112); however, it will be shown in Chapter 8 that this approximation can
lead to oscillatory results. The oscillations can be smoothed out either by using the fully
implicit version with
θ
=
1, or by writing the first of (3.112) with
θ
=
1 and the second
1
/
2. Examples of this algorithm with constant
θ
are presented in Chapter 9. In all
cases, a right hand side matrix-by-vector multiplication is followed by an equation solution
for each timestep. As before a saving in computer time can be achieved if [
k
m
], [
c
] and
[
k
c
] are independent of time, and constant
t
is used, because the left hand side matrix
needs to be factorised only once.
Note that the left hand side matrix is always symmetrical whereas the right hand side
is not. Therefore, if using an assembly approach
fsparv
can be used to assemble the left
hand side system equations from (3.112), where
formtb
must be used to assemble the right
hand side system followed by
bantmul
to complete the matrix-vector multiply. Element-
by-element summation is most effective in the right-hand side operations in (3.112) due
with
θ
=
