Steady State Flow - Programming the Finite Element Method - page 347

commented “pcg equation solution” carries out the vector operations described in equa-
tions (3.22) within the iterative loop labelled pcg . The matrix-vector multiply needed
in the first of (3.22) is done using (3.23). The node number vector num “gathers” the
appropriate components of p , to be multiplied by the element conductivity matrix kc
retrieved from storkc . Similarly, the vector num “scatters” the result of the matrix-
vector multiply to appropriate locations in u . A tolerance cg tol enables the iterations to
be stopped when successive solutions are “close enough”, but since pcg is a loop which
might carry on “forever”, an iteration ceiling, cg limit , is specified also. The nodal net
flow rates are accumulated from element-by-element matrix-vector products in the loop
called elements 3 .
The problem given in Figure 7.20 is solved once more using Program 7.5 with the
data given in Figure 7.22. The only additional data are cg tol and cg limit ,the
pcg convergence tolerance, and iteration ceiling respectively. The output in Figure 7.23 is
essentially identical to that obtained previously using an assembly strategy in Figure 7.21,
apart from the additional comment indicating that the pcg algorithm took 18 iterations to
converge.

element nod nels nn nip ndim cg_tol cg_limit np_types
'hexahedron' 8 64 125 8 3 1.0e-7 200 2
prop(kx,ky,kz)
2.0 2.0 2.0
1.0 1.0 5.0
etype
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
g_coord
0.0 0.0 0.0 1.0 0.0 0.0 2.0 0.0 0.0 3.0 0.0 0.0
.
g_coord data for nodes 5-120 omitted here
.
0.0 4.0 -4.0 1.0 4.0 -4.0 2.0 4.0 -4.0 3.0 4.0 -4.0
4.0 4.0 -4.0
g_num
6 1 2 7 31 26 27 32 7 2 3 8 32 27 28 33

.
g_num data for elements 3-62 omitted here
.

98 93 94 99 123 118 119 124 99 94 95 100 124 119 120 125
loaded_nodes,(k,loads(k),i=1,loaded_nodes)
1
21 100.0
fixed_freedoms,(node(i),value(i),i=1,fixed_freedoms)
61
1 0.0 2 0.0 3 0.0 4 0.0 5 0.0 10 0.0 15 0.0 20 0.0

.
fixed freedom data for 48 nodes omitted here
.
121 0.0 122 0.0 123 0.0 124 0.0 125 0.0

Figure 7.22 Data for Program 7.5 example