Civil Engineering Reference
In-Depth Information
T
k
−
1
{
R
0
}
{
R
}
where
α
k
−
1
=
(3.28)
1
{
R
0
}
T
k
−
{
Q
}
{
R
}
k
−
={
R
}
k
−
1
−
α
k
−
1
{
Q
}
k
−
1
1
2
(b)
{
S
}
k
−
=
[
K
m
]
{
R
}
k
−
1
2
1
2
{
U
}
k
={
U
}
k
−
2
+
ω
k
{
R
}
k
−
1
1
2
T
k
−
{
R
}
2
{
S
}
k
−
1
2
1
where
ω
k
=
(3.29)
T
k
−
{
S
}
2
{
S
}
k
−
1
2
1
{
R
}
k
={
R
}
k
−
2
−
ω
k
{
S
}
k
−
1
1
2
k
{
R
0
}
T
α
k
−
1
{
R
}
β
k
=
T
k
−
1
{
R
0
}
ω
k
{
R
}
{
P
}
k
={
R
}
k
+
β
k
{
P
}
k
−
1
−
β
k
ω
k
{
Q
}
k
−
1
until convergence is achieved. Compared with (3.21) and (3.22), we see a similar, but two
stage process with initialisation followed by stages (a) and (b) in both of which a matrix-
vector multiplication like (3.23) is involved, together with whole array operations and inner
products, readily computed in Fortran 95.
The hybrid BiCGStab(l) version (Sleijpen
et al
., 1994) involves essentially the same
arithmetic. Serial implementations involving it can be found in Chapter 9 and parallel
equivalents in Chapter 12.
3.5.4 Symmetric non-positive definite equations
When Biot's equations for coupled consolidation (2.136, 2.137) are discretised by finite
elements as (3.112) or (3.115) these systems will be seen to be symmetric but non-positive
definite. Although a candidate solution algorithm is the minimum residual method (MIN-
RES), Smith (2000) found that the diagonally preconditioned conjugate gradient method
worked quite effectively and is used herein. However, the whole question of preconditioning
is a developing area (Chan
et al
., 2001).
3.5.5 Symmetric eigenvalue systems
Again in an element-by-element context we seek algorithms which have at their heart
matrix-vector operations like (3.23) and vector or inner product operations which can read-
ily be parallelised. Candidates (Bai
et al
., 2000) are the long-established Lanczos method
and the Jacobi-Davidson method in which interest has recently been revived (Sleijpen and
van der Vorst, 2000). In this topic the Lanczos method is used, which is deceptively simple
(Griffiths and Smith, 1991). Suppose the eigenproblem (3.18) has been reduced to finding
