Civil Engineering Reference
In-Depth Information
nip
c21
=
(3.106)
det
i
*weights(i)*nfd1
i
i
=
1
nip
c23
=
det
i
*weights(i)*nfd2
i
i
=
1
where
rho
is the mass density. The other submatrices
c13
,
c22
,and
c31
are set to zero.
The (unsymmetrical) matrix built up from these submatrices is called [
k
e
], and is formed
by a special subroutine called
formupv
(
formupvw
in 3D) and the global, unsymmetrical,
band matrix is assembled using
formtb
. The appropriate equation solution routines are
gauss_band
and
solve_band
as shown in Table 3.8. In an element-by-element context,
the iterative solution method is BiCGStab(l) following the algorithm described in (3.27)
to (3.29).
3.12 Solution of coupled transient problems
The element equations for Biot consolidation were shown in equation (2.139) to be given by
[
k
m
]
{
u
} +
[
c
]
{
u
w
} = {
f
}
[
c
]
T
d
u
d
t
−
[
k
c
]
{
u
w
} = {
}
0
(3.107)
where [
k
m
] and [
k
c
] are the now familiar solid stiffness and fluid conductivity matrices.
The matrix [
c
] is the connectivity matrix which is formed from integrals of the form,
∂N
j
∂x
N
i
d
x
d
y
(3.108)
where the first derivative term comes from the displacement field, and the second term
comes from the excess pore pressure field. The programs described in Chapter 9 use differ-
ent element types for the displacements (8-node) and the excess pore pressures (4-node).
In Chapter 12, the 3D elements have 20 displacement and 8 pore pressure nodes.
The integrals that generate the [
c
] matrix involve the product of the vectors
vol
and
funf
,where
vol
is derived from the familiar
deriv
array for the solid elements, and
takes the form, for 2D,
∂N
1
d
x
T
∂N
1
d
y
∂N
2
d
x
∂N
2
d
y
··· ···
∂N
8
d
x
∂N
8
d
y
vol
=
(3.109)
and
funf
holds the fluid component shape functions as
[
N
1
N
2
N
3
N
4
]
T
funf
=
(3.110)
