Civil Engineering Reference
In-Depth Information
(8.6)
^` ^`
^`
RFKu
ª
¬¼
m
0
0
Where
^`
0
is the “residual” or error for a first trial solution of
^
u
namely
^`
0
R
u
. In
elasticity problems with only Neumann boundary conditions, for example,
ª
¬¼
K
would
be the assembled matrix
> @
'
T
and
^
u
a vector of displacements. However, we note that
in an element by element (EBE) iterative solution the system of equations (8.6) need
never be actually assembled.
^` ^` ^` ^`
^`
Set
PR u
;
0 or some intial estimate
0
0
0
^`
^`
ˆ
T
ˆ
choose
R
such that
R
R
z
0
0
0
0
FOR
k
1, 2, 3....
niter
DO
^`
^`
QKP
ª
¬¼
m
k
1
k
1
^`
^`
^`
^`
T
k
ˆ
RR
0
^`
^`
^`
1
u
u
D
P
;
D
k
1
k
1
k
12
k
1
k
1
T
ˆ
QR
0
k
1
^`
^`
^`
R
R
D
Q
k
1
k
12
k
1
k
1
^`
^`
S
ª
¬¼
KR
m
k
1
k
1 2
T
k
^` ^`
^` ^`
RS
^` ^`
^`
12
k
12
u
u
Z
R
;
Z
k
k
1
k
k
12
k
12
T
S
S
k
12
k
12
^` ^`
^`
RR
Z
S
k
k
k
12
k
12
^`
^`
^`
^`
T
ˆ
D
RR
k
1
0
^` ^`
^`
^`
k
PR
E
P
EZ
Q
;
E
k
k
k
k
k
k
k
1
k
1
T
ˆ
Z
RR
k
0
k
1
IF
(converged)
EXIT
END DO
Figure 8.1
Pseudo-code for BiCGStab
For the BEM we could choose any of the GMRES-type class of solution techniques. In
particular, we select the BiCGStab algorithm, which has been shown to be effective in
Finite Element work
1
. It follows the two-stage (“Bi”) procedure shown in Figure 8.1
being dominated by two matrix*vector products such as
^`
^`
QKP
ª
¬¼
(8.7)
m
k
1
k
1
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