Future Work

Finally, we would like to suggest some future research directions based on this work that would either provide more insight into the multiple perturbation approach or make it applicable to other problems of interest. In appendix A we perform a computational study on the choice of the reference system. Results of this study suggest that an optimized choice of a reference system for a given problem is desirable. An optimized reference system could be determined by requiring minimum variance in the result. This requires further theoretical study and we will leave this as a future research topic.

The multiple perturbation method developed here is for an isotropic scattering medium. A possible next step would be to extend this approach to anisotropic scattering. Another possibility is to extend this approach to continuous energy Monte Carlo.

Calculation of multiple perturbations in eigenvalue involves solution of fixed source Monte Carlo problems over a number of fission generations. We were able to calculate perturbations in the eigenvalue because this method was able to successfully solve fixed source problem for each generation. This suggests that this method could be applied for solving perturbation problems in reaction rates. This would lead to solution of perturbation problems in shielding calculations where different shielding materials are tested.

In this research work we applied the fission matrix approach of eigenvalue calculation to the correlated sampling perturbation technique. We have mentioned that another Monte Carlo perturbation technique is the derivative operator sampling approach. Another extension to this work might be to combine the fission matrix and derivative operator sampling methods to calculate multiple eigenvalue perturbations.

CHOICE OF A REFERENCE SYSTEM

The multiple perturbation method developed in this dissertation requires that the Monte Carlo simulations be performed in a reference system different from the unperturbed and all perturbed systems. We have chosen the reference system cross sections to be the arithmetic average of the cross sections of the unperturbed and all perturbed systems. It may be possible to choose another reference system that is more appropriate for a given problem. An optimized reference system for a given problem would be the one that produces the most accurate $\Delta$K along with the minimum variance. Our choice of reference system is based on intuition rather than any theoretical study. We believe that it might be possible to study the choice of a reference system as an optimization problem. We suggested this optimization problem as a future research topic in chapter VI. In this appendix we perform numerical experiments to investigate the effect of the choice of a reference system on the result of $\Delta$K. We will perform these numerical experiments on a few problems that have been used as example problems in this dissertation.

For the numerical experiment, we choose a problem in which there is only one perturbed system corresponding to an unperturbed system. It is understood that similar observations can be made for problems with multiple perturbed systems. The reference system's cross sections will be varied according to the following formula,

$$\Sigma_x^{ref} = \alpha\Sigma_x^{up} + (1-\alpha)\Sigma_x^{p}\;\;\;,\;\;\; 0 \le \alpha \le 1,$$ (129)

where $\Sigma_x$ refers to some cross section. We would look at the $\Delta$K results as a function of $\alpha$.

The first example is for a one energy group homogeneous slab problem with a thickness of 16 cm and vacuum boundary conditions on both ends. Cross section perturbation was done over the entire slab. Unperturbed cross sections of the slab are as follows,

$$\Sigma_t^{up}=1.0, \Sigma_s^{up}=0.9, \nu\Sigma_f^{up}=0.11, \Sigma_a^{up}=0.1.$$

All the cross sections have units of cm^-1. Perturbation was done in the fission cross section of the slab, and the perturbed cross sections are as follows,

$$\Sigma_t^{p}=1.0, \Sigma_s^{p}=0.9, \nu\Sigma_f^{p}=0.111, \Sigma_a^{p}=0.1.$$

Table A.1 shows the $\Delta$K results as a function of $\alpha$. The Monte Carlo results were generated from 30 inactive batches, 70 active batches, and 2000 neutrons per batch.

It appears from table A.1 that for this problem, when the reference system is chosen as the unperturbed system, the most accurate result for $\Delta$K is produced. It should be noted that the errors are not linearly dependent on $\alpha$.This analysis assumes the TWODANT result as exact.

Next, we perturb the absorption cross section for the same homogeneous one energy group slab problem as in table A.1. The perturbed cross sections are as follows,

$$\Sigma_t^{p}=1.001\;\;\;, \Sigma_s^{p}=0.9\;\;\;, \nu\Sigma_f^{p}=0.11\;\;\;, \Sigma_a^{p}=0.101.$$

 

$$\alpha \Delta \Delta$$ 1c|% error

$${\pm}$$ 1.0 0.00903 0.009

$$\delta^{ref}$$

$${\pm}$$ 0.75 0.00903 0.05

$$\delta^{ref}$$

$${\pm}$$ 0.50 0.00903 0.22

$$\delta^{ref}$$

$${\pm}$$ 0.25 0.00903 0.1

$$\delta^{ref}$$

$${\pm}$$ 0.0 0.00903 0.12

$$\delta^{ref}$$

Perturbation results as a function of $\alpha$ are shown in table A.2. The Monte Carlo results were generated using 30 inactive batches, 70 active batches and 2000 neutrons per batch.

From table A.2 we observe that the most accurate result is achieved when $\alpha$ equals 0.25. We observe for this case also that the errors are not a linear function of $\alpha$. Again these conclusions are made based on the fact that the TWODANT result is the correct one.

The above two numerical experiments suggest that even though the results are a function of the chosen reference system, it is difficult to determine the relation. For complicated systems (heterogeneous, multigroup, etc.), it will be more difficult to

 

$$\alpha \Delta \Delta$$ 1c|% error

$$\mp$$ 1.0 -0.008807 0.96

$$\delta^{ref}$$

$$\mp$$ 0.75 -0.008807 0.94

$$\delta^{ref}$$

$$\mp$$ 0.50 -0.008807 1.13

$$\delta^{ref}$$

$$\mp$$ 0.25 -0.008807 0.82

$$\delta^{ref}$$

$$\mp$$ 0.0 -0.008807 0.87

$$\delta^{ref}$$

determine this relationship. This is an optimization problem, and a theoretical investigation is required to gain more insight into it.

TWODANT ACCURACY

In this appendix we perform accuracy test for the TWODANT code. For a given problem we vary the mesh size as well as the quadrature sets to observe the effect of these variations on TWODANT calculated eigenvalue results. These results guide us to determine up to how many digits, after the decimal point, the TWODANT calculated $\Delta$Ks are accurate.

The test problem is a homogeneous, one energy group, slab, with thickness of 16 cm and vacuum boundary conditions on both ends. Tables B.1 and B.2 show the different eigenvalue results due to different mesh sizes and quadrature sets.

In most of our TWODANT calculations, for chapter 2, 3 and 4 results, we have used mesh sizes and quadrature sets which are equivalent to that of the first row of table B.1. This implies that the TWODANT $\Delta$Ks are accurate only up to five digits after the decimal point. We observe that even though the inner and outer iteration convergence criteria are set to 10^-12 the TWODANT eigenvalues are not accurate up to that many digits after the decimal point.

 

1|cMesh Size (mfp) 1|c|quadrature sets 1|c|convergence criteria 1c|eigenvalue

0.125 32 10^-12 0.993070

0.0625 32 10^-12 0.993075

0.03125 32 10^-12 0.993077

0.015625 32 10^-12 0.993077

 

1|cMesh Size (mfp) 1|c|quadrature sets 1|c|convergence criterion 1c|eigenvalue

0.125 16 10^-12 0.993056

0.0625 16 10^-12 0.993061

0.03125 16 10^-12 0.993063

0.015625 16 10^-12 0.993063