Civil Engineering Reference
In-Depth Information
iy,ptot,totd(nf(2,node(1))),iters,REAL(cg_tot)/REAL(iters)
IF(iters==limit)THEN
CALL dismsh(loads,nf,0.05_iwp,g_coord,g_num,13)
CALL vecmsh(loads,nf,0.05_iwp,0.1_iwp,g_coord,g_num,14)
STOP
END IF
END DO load_incs
STOP
END PROGRAM p62
New scalar integers:
cg iters
conjugate gradient iteration counter
conjugate gradient iteration ceiling
cg limit
keeps running total of
cg iters
cg tot
New scalar reals:
alpha
α
from equations (3.22)
β
from equations (3.22)
beta
pcg convergence tolerance
cg tol
T
holds dot product
{
R
}
k
{
R
}
k
from equations (3.22)
up
New Scalar logical:
cg converged
set to
.TRUE.
if pcg process has converged
New dynamic real arrays:
d
preconditioned rhs vector
diagonal preconditioner vector
diag precon
“descent” vector used in equations (3.22)
p
holds element stiffness matrices
storkm
vector used in equations (3.22)
u
“old” solution vector
x
“new” solution vector
xnew
For non-linear problems, computation times are now more demanding. For example
Program 6.1 required some 765 linear elastic solutions to reach the failure load. Clearly
vectorised and parallelised algorithms will become essential for much larger problems.
The mesh-free approach first demonstrated in Program 5.5 will be attractive for large non-
linear problems and is implemented in Program 6.2. The data in Figure 6.13 are identical to
those used in Program 6.1 with the addition of a conjugate gradient convergence tolerance
cg tol
set to 0.0001, and conjugate gradient iteration ceiling
cg limit
set to 100. The
results are shown in Figure 6.14, and are nearly identical to those in Figure 6.10. The results
table in Figure 6.14 also indicates that approximately 50 conjugate gradient iterations were
needed on average for each plastic iteration. In scalar computations, therefore, this algorithm
will not be attractive, but it does have some strong attractions for parallel computation.
Because a “constant stiffness” approach is being used, groups of elements in Figure 6.9 have
