Conclusions

The main objective of this research work was to develop a computational method that would calculate multiple perturbation effects in the eigenvalue of the Boltzmann transport equation for neutrons from a single Monte Carlo simulation. Even though Monte Carlo methods can efficiently estimate the eigenvalue of the transport equation, calculation of small perturbation effects encounters difficulties. It has been shown in this research work by numerical examples, and by other researchers, that subtracting two independent Monte Carlo simulations for estimating perturbation effects in eigenvalue is not efficient and sometimes provides incorrect estimates. This is specifically true for cases in which the difference in eigenvalues is of the same order as the uncertainty of the eigenvalues. Hence, to estimate eigenvalue perturbation effects using Monte Carlo techniques, the unperturbed and perturbed simulations need to be positively correlated. Special Monte Carlo perturbation techniques such as the correlated sampling, derivative operator sampling, or the importance function approach must be employed to obtain an efficient estimate of eigenvalue perturbations. Our work has dealt with the correlated sampling technique.

In chapter II we discussed two Monte Carlo approaches to estimate the eigenvalue of a system, the source iteration method and the fission matrix approach. In the source iteration technique, the eigenvalue of a system is calculated as the ratio of source neutrons of two successive neutron generations. The fission matrix algorithm solves the homogeneous neutron transport equation which holds for every generation. This equation is discretized to generate a matrix equation. Monte Carlo particle tracking is done to estimate contributions of neutrons to each of the matrix elements of the fission matrix, and then the largest eigenvalue of the fission matrix is determined numerically. We have investigated two variations of the fission matrix algorithm. In the cycle fission matrix algorithm the fission matrix is generated from contributions of all the neutrons of a particular batch. In contrast, in the cumulative fission matrix algorithm the fission matrix is formed from contributions of neutrons from all the batches up to and including the last batch. We have tested these two algorithms for problems representing tightly coupled and loosely coupled systems. We have observed from our computational experiments that for tightly coupled systems, both the cycle fission matrix and the cumulative fission matrix algorithm perform well and yield comparable performance. But for loosely coupled systems, the cumulative fission matrix algorithm performs better than the cycle fission matrix algorithm. For loosely coupled systems there is less neutron communication between the different parts of the system, and hence a fission matrix formed with the neutrons of a particular batch provides poor sampling. One characteristic of the cumulative fission matrix algorithm is that it does not provide any uncertainty of the results, since the eigenvalues of different batches are not statistically independent.

The correlated sampling technique was applied to both the source iteration and the cycle fission matrix approach. It was shown, by numerical experiments, that the correlated sampling technique, when applied to the source iteration method, fails to efficiently calculate perturbation effects in the eigenvalue of the transport equation. It is not possible to calculate perturbations in eigenvalues by propagating perturbed weights from one generation to another. Any useful information about the eigenvalue perturbation is lost due to fluctuations in the perturbed weights from one generation to another. However the calculation of the fission matrix constitutes an initial value problem and correlated sampling or derivative operator sampling can be applied directly. The correlated sampling technique, when applied to the fission matrix method, can accurately and efficiently estimate small perturbation effects in the eigenvalue. Next, we introduced the idea of performing the actual Monte Carlo simulation in an artificial reference system different from both the unperturbed and perturbed systems. The choice of a proper reference system is an important issue. The important characteristic of the reference system is that a transition that may occur in any of the unperturbed or perturbed systems could occur in the reference system. Even after satisfying this characteristic there could be many choices for the reference system. In appendix A we have performed computational experiments to address this issue and have shown that varying a reference system does impact the outcome of the simulation and hence this issue should be addressed from theoretical standpoint. For all our simulations, we have taken the reference system to be an average of the unperturbed and all perturbed systems. We have also introduced the concept of adding a forward $\delta$-scatter cross section to the total cross section of all the systems to reduce fluctuations in the adjusting weight factors. All these theories and numerical examples to support them are given in chapter III.

Next, in chapter IV, the ideas of correlated sampling technique, fission matrix approach, and an artificial reference system were combined to develop the multiple perturbation CSFM Monte Carlo technique. In this technique the Monte Carlo simulation is done in an artificial reference system and multiple fission matrices for the multiple perturbed systems and for the unperturbed system are formed by correlating them to the reference system's fission matrix. The correlated sampling technique allows one to form multiple fission matrices from a single Monte Carlo simulation. At the end of the simulation, the dominant eigenvalue of each of the multiple fission matrices of the perturbed systems are evaluated numerically, along with the dominant eigenvalue of the unperturbed fission matrix. We have tested this Monte Carlo technique for various test problems and compared the results to that of the TWODANT code. Satisfactory comparison of results between the multiple perturbation Monte Carlo method and the TWODANT code validates the multiple perturbation method. This method allows significant savings in computational efforts as discussed in chapter IV. This method can be applied to problems in which it is desired to calculate multiple perturbations in the eigenvalue due to small variations in cross sections. Some practical examples of such problems are perturbations in eigenvalue due to changes in soluble boron concentrations, different number of absorber rods in assemblies, and different assembly loading patterns for global core calculations. We have also shown that the CSFM method can be used as a tool for design applications when the eigenvalue of the unperturbed system is known beforehand to a good accuracy by a separate calculation. Even though the CSFM method is meant for small perturbation problems, it has performed well for some large perturbation cases.

We have also implemented parallel Monte Carlo algorithms for particle transport on the KSR-1, BBN Butterfly and IBM-SP2 parallel computers. Both fixed source and eigenvalue type neutron transport algorithms were implemented on these machines. For many applications, Monte Carlo algorithms may be slow compared to deterministic methods, whereas Monte Carlo algorithms are inherently parallel and hence easy to parallelize compared to deterministic methods. We have observed close to linear speedups for some of the example Monte Carlo problems. These speedups are observed for both fixed source and eigenvalue type Monte Carlo algorithms. Theoretical models of speedups for parallel particle transport algorithms were developed and compare well with observed speedup results for all three parallel machines. As the price of MPPs decreases, the inherent parallelism of Monte Carlo will make it a significant computational tool of choice.