Discussion of Numerical Results

Various eigenvalue perturbation problems have been solved in the previous section using the multiple perturbation CSFM approach. The methodology has worked well for both small and large perturbations, even though the method is specifically meant for problems where the perturbation is small and hence direct Monte Carlo subtraction encounters difficulty. Results of multiple $\Delta$Ks, from a single Monte Carlo simulation, have compared well with the corresponding $\Delta$Ks computed by the TWODANT code. Comparing the relative errors of $\Delta$Ks between the TWODANT code and the multiple perturbation CSFM Monte Carlo approach, we observe that the results always agree within less than 4% for all test cases shown here. Since the relative errors represent comparison of $\Delta$K results between the CSFM method and the TWODANT code, an agreement within 4% can be considered to be quite accurate. Further observation of all the tabulated results in section 4.3 shows that in general, relative errors of $\Delta$Ks are slightly larger when perturbations in $\Sigma_t$were made, compared to perturbations in cross sections other than $\Sigma_t$. This is due to the fact that a perturbation in $\Sigma_t$,in addition to biasing the fission reactions, also forces the CSFM method to bias the weight adjusting factors (WF) for every distance to collision sampled. On the other hand, a perturbation in the $\nu\Sigma_f$ alone requires biasing in the fission reactions only. We have not noticed any general trend in the deterioration of results in going from homogeneous to heterogeneous problems or from one energy group to two energy group problems. This suggests that the two energy group CSFM method can be easily extended to multigroup problems. The reference system for each test problem was chosen using equation (4.2) and the $\delta$-scatter cross sections were chosen using the conditions given in equations (4.5), (4.6), (4.7) and (4.8). These equations provided reasonable choices for both the reference system and $\delta$-scatter cross sections, and can easily be extended to multiple groups.

Since multiple $\Delta$Ks due to multiple perturbations are computed from a single Monte Carlo simulation, a significant reduction in computational effort is achieved. We provide two actual wall clock timing results in table 4.8 to show this. These timing results were obtained on a dedicated HP700 series machine. From the timing results of table 4.8 and various other timing results obtained on non-dedicated machines (and hence not shown here), we can conclude that it requires less than 10% extra computational effort for each additional $\Delta$K calculation compared to the first $\Delta$K calculation. Most of this extra computational effort is due to the number of times the matrix iterative algorithm is invoked. However, the computational time spent in the matrix iterative algorithm is relatively modest compared to the time spent in Monte Carlo particle tracking.

 

$$\Delta$$ 1|c|Problem 1|c|Wall Clock

1|c|Type 1|c|calculated 1|c|time (sec)

First problem one 1185.5

of table 4.3 three 1419.4

Second problem one 1117.1

of table 4.3 three 1359.3