Mesh optimisation - Finite Element Mesh Generation - page 350

Table 6.3 Convergence characteristics of QL, LO and GETMe

QL LO GETMe

Cycle NM NM NM α min α mean α min α mean α min α mean

0 0.000468 0.581 0.000468 0.581 0.000468 0.581

1 0.01877 0.7321 37,654 0.02629 0.6769 59,343 0.01909 0.6838 51,539

2 0.02025 0.769 27,158 0.02513 0.723 59,246 0.01908 0.7345 50,287

3 0.02025 0.7781 16,117 0.02678 0.7456 59,086 0.03237 0.7591 48,200

4 0.02025 0.7806 7837 0.06244 0.757 58,935 0.03237 0.7722 46,067

5 0.02025 0.7812 3124 0.08931 0.7631 58,820 0.03237 0.7797 44,022

6 0.02025 0.7814 1045 0.1015 0.7666 58,683 0.03237 0.7845 42,017

7 0.02025 0.7815 317 0.107 0.7687 58,730 0.03237 0.7876 40,317

8 0.02025 0.7815 98 0.107 0.7698 58,731 0.03237 0.7898 38,670

9 0.02025 0.7815 38 0.107 0.7706 58,765 0.03237 0.7913 37,315

10 0.02025 0.7815 12 0.107 0.7709 58,846 0.03237 0.7924 36,150

first three cycles. As for the other two smoothing schemes, there is still consistent improve-
ment in each cycle of iteration, even though the rate is diminishing to a very little progress
after, say, ten cycles. From the convergence characteristics of the three smoothing methods,
a combined scheme might be the most effective. As a numerical experiment, a combined
scheme (4 + 2 + 4) consisting of four cycles of QL, followed by two rounds of LO and further
optimised with four cycles of GETMe is tested. The average results on a number of meshes
are given by α min = 0.08448, α mean = 0.8033 and the CPU time taken = 0.766 s. Relative to
the three smoothing methods, the combined scheme gives the best performance on both α min
and α mean , and in terms of CPU time, it is similar to GETMe.
The performance of the three schemes is plotted in Figure 6.16. We can see that the QL
smoothing reaches the converged value basically in the first four iterations; as for the other
two schemes, improvement could still be made in the tenth cycle, though it is already pretty
small. QL smoothing is the most efficient in the first five cycles of iteration, whereas GETMe
is the most consistent, and all three schemes can be used, in general, to improve triangular
meshes of various characteristics.

0.8

0.75

0.7

Quality Laplace
Local optimisation
GETMe

0.65

0.6

0.55

0123456789 0

Number of iterations

Figure 6.16 Convergence characteristics.