Optimization and Computational Fluid Dynamics - Department of ...

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Table 2.3 Speed-up obtained using the parallel optimization method (Case A)
Number of processors Wall-clock time Speed-up
1 1280 min 1.00
2 661 min 1.94
5 294 min 4.35
10 153 min 8.37
Table 2.3 shows the resulting CPU times needed for the evaluation of 20 generations
consisting of 40 individuals, performed with an increasing number
of processors on the PC-cluster. The speed-up is defined here as the ratio
between the wall-clock time obtained when using Nproc processors compared
to the one needed with a single processor. The theoretical optimal value of
the speed-up using Nproc processors is Nproc. In practical cases, the communication
times between processors and load-imbalance reduce the speed-up
valuebelowthetheoreticalmaximum.
In the present case, the obtained parallel speed-up is nearly optimal. This
is not really a surprise since the quantity of information transferred between
the so-called farmer and workers is very small: four real parameters in one
direction, two in the reverse direction. The communication times are therefore
almost negligible compared to the CPU times required for the evaluation of
the objectives on each processor. Deviation from the optimal speed-up is
thus mainly due to boundary effects at the end of each optimization iteration
when some worker PCs become inactive for a short time waiting for the next
iteration to start.
2.4 Multi-objective Optimization of a Laminar Burner
(Case B)
2.4.1 Governing Equations
Laminar flows involving chemical reactions are considered in this section. In
the presented application Mach numbers M are very low. It is observed that
pressure variations through laminar flames at low Mach numbers are always of
the order of magnitude of a few Pascal and stem mainly from hydrodynamical
and not from compressibility effects. Stated differently, density variations only
result from heat release due to chemical reactions and from changes in the
mixture composition, but not from local fluid compression. Temperature and
density vary in opposite directions, such that their effects within the ideal
gas law compensate. These physical observations motivate the decomposition
of pressure into a bulk background uniform thermodynamic pressure pu and