## Nuclear Engineering ETDs

#### Title

Diagnostics of source distribution and particle population in Monte Carlo source iteration methods

#### Publication Date

6-24-2010

#### Abstract

This thesis is concerned with the development of mesh-input-free diagnostics for the determination of the iteration at which the source distribution of a Monte Carlo simulation has reached a stationary state, as well as the sufficiency of particle population size for a given tally cell volume, so as to reduce bias and increase accuracy of estimations of physical properties. Such physical properties can include, but are not limited to, neutron effective multiplication, power distribution, neutron flux and various interaction rates. When the physical properties of a Monte Carlo simulation are accurately estimated, they can be used to predict the actual behavior of a nuclear system, only being limited to the assumptions used to create the model. Five methods were used to describe the state of the source distribution. Four of the methods were established indicators of the source distributions state that required the input of a mesh, which divided the geometry into bins. These indicators are the Shannon entropy, Jensen measure, the progressive relative entropy and the posterior relative entropy. The fifth indicator of the source distribution's state was developed as to eliminate the need for the input of a mesh upon the geometry. This method will be identified as the regionwise average position indicator or RAPI and is calculated by taking the sum of the distances of the regionwise average particle positions in the model at each cycle from the corresponding regionwise average particle positions at the first cycle. In conjunction with the Shannon entropy, Jensen measure, progressive relative entropy and the RAPI, an on-the-fly step-refined judgment of the indicators of the source distribution's state will be employed to determine at which cycle or iteration the indicators have reached convergence, signifying that the simulated source distribution has begun to fluctuate around the true source distribution. This step-refined on-the-fly diagnostic of the source distribution was developed from the Wilcoxon rank sum in non-parametric statistics. The posterior relative entropy cycle of convergence is determined to be the cycle at which the posterior relative entropy becomes less than the average value of the posterior relative entropy over the second half of active cycles. The cycle of convergence was determined for three different models by the use of the above described methods. The resulting cycle of convergence obtained by the use of the indicators requiring a mesh input was compared against that obtained from the mesh-input-free indicator, RAPI, for each of the models. It was found that the RAPI was an excellent representation of the source distribution's state and more conservative than the posterior relative entropy diagnosis. The RAPI can be used to determine the cycle at which the source distribution converged to an equilibrium fluctuation range of stationary state, thus eliminating the need for mesh-input for physical property estimation. Applications of graph theory techniques to Monte Carlo methods were also investigated as a means of meshless convergence indication, but drawbacks for such an application led to a particle population diagnostic investigation. This was done because meshless particle population diagnosis for the power distribution has yet to be done in Monte Carlo source iteration methods. In power distribution calculations, tally cells are used to estimate the power distribution in a model. To approach this problem, the concept of Euclidian minimum spanning trees (EMST) was applied to the source distribution to develop a meshless diagnosis of the particle population. One source particle effect is the characteristic volume of one particle and is defined to be the cubic of the average edge length of an EMST. Then using this characteristic volume, weak and strong requirements of the particle population size were defined for minimum tally cell volume. This diagnostic was compared against a verified population diagnostic, which requires a mesh input, termed as PD-MESH in this thesis. These diagnostic methods were used in the analysis of a pressurized water reactor initial full core simulation. The comparison of the EMST-based population diagnosis to PD-MESH showed that it can be used to determine if a population size is of sufficient size for power distribution calculations, eliminating the need for mesh-based diagnosis.

#### Document Type

Thesis

#### Language

English

#### Degree Name

Nuclear Engineering

#### Level of Degree

Masters

#### Department Name

Nuclear Engineering

#### First Advisor

Ueki, Taro

#### First Committee Member (Chair)

Ueki, Taro

#### Second Committee Member

Busch, Robert

#### Third Committee Member

Prinja, Anil

#### Recommended Citation

Chapman, Bryan Scott. "Diagnostics of source distribution and particle population in Monte Carlo source iteration methods." (2010). http://digitalrepository.unm.edu/ne_etds/23