Transport on Adaptive Random Lattices

TL;DR

This paper introduces a novel Markov process-based algorithm on a Voronoi-Delaunay lattice for solving linear transport equations, offering improved efficiency and representing medium properties through a random lattice structure.

Contribution

The paper presents a new transport simulation method using a Voronoi-Delaunay grid derived from a random point process, enhancing efficiency and medium representation.

Findings

The method is significantly faster than existing approaches.

The lattice accurately models the medium's free path space.

Transport operations are simplified to particle movement between nodes.

Abstract

In this paper, we present a new method for the solution of those linear transport processes that may be described by a Master Equation, such as electron, neutron and photon transport, and more exotic variants thereof. We base our algorithm on a Markov process on a Voronoi-Delaunay grid, a nonperiodic lattice which is derived from a random point process that is chosen to optimally represent certain properties of the medium through which the transport occurs. Our grid is locally translation and rotation invariant in the mean. We illustrate our approach by means of a particular example, in which the expectation value of the length of a grid line corresponds to the local mean free path. In this example, the lattice is a direct representation of the `free path space' of the medium. Subsequently, transport is defined as simply moving particles from one node to the next, interactions taking place at each point. We derive the statistical properties of such lattices, describe the limiting behavior, and show how interactions are incorporated as global coefficients. Two elementary linear transport problems are discussed: that of free ballistic transport, and the transport of particles through a scattering medium. We also mention a combination of these two. We discuss the efficiency of our method, showing that it is much faster than most other methods, because the operation count does not scale with the number of sources. (ABRIDGED)