On the Use of Graph Theory to Interpret the Output Results from a Monte-Carlo Depletion Code

Abstract

The analysis of the results of a depletion code is often considered a tedious and delicate task, for it requires both the processing of large volumes of information (the time-dependent composition of up to thousands of isotopes) and an extensive knowledge of nuclear reactions and associated nuclear data. From these observations, dedicated developments have been integrated to the upcoming version of the Monte Carlo depletion code VESTA 2.2 in order to implement an innovative representation of depletion problems. The aim is to provide users with an adaptable and efficient framework to ease the analysis of the results of the code and facilitate their interpretation. This effort ultimately culminates in the development of the representation of the isotopic evolution of a given system as a directed graph.

In this paper, it is shown that the Bateman equation encoded in the VESTA code indeed possesses a natural interpretation in terms of a directed cyclic graph, and it is proposed to explore some of the insight one can gain from the graph representation of a depletion problem. Starting from the new capabilities of the code, it is shown how one can build on the wealth of existing methods of graph theory in order to gain useful information about the nuclear reactions taking place in a material under irradiation. The graph representation of a depletion problem being especially simple in activation problems—for then only a limited number of nuclides and reactions are involved—the graph representation and its associated tools will be used to study the evolution of the structure materials of a simplified model of the ITER fusion reactor.

Keywords:

• Graph theory
• Monte Carlo
• depletion
• VESTA

Acknowledgments

The author would like to thank Mariya Brovchenko for helpful discussions and comments.

Notes

^a The concept of bridges and its generalization to cut-sets of edges allows one to search for sets of edges whose removal disconnects the graph.