Papers
The methods behind GERT and RayTraceHeatTransfer.jl are published as preprints on arXiv. Each paper is self-contained, and its results can be reproduced with the package.
Non-Negativity, Non-Singularity and Energy Conservation of the Zonal Method with Non-Reciprocal Exchange Factors
Nikolaj Maack Bielefeld · arXiv:2512.22157
The foundational paper. It writes the zonal balances of Hottel’s method in power variables, for a first-interaction exchange factor matrix, as one linear system for any mix of prescribed temperatures and prescribed sources, on general domains with media ranging from transparent to absorbing, emitting and scattering, and boundaries ranging from absorbing to reflecting. The paper proves that this system has a unique, non-negative solution, and that energy is conserved to machine precision as an algebraic identity, with no assumption of reciprocity: the guarantees hold for Monte Carlo exchange factors as traced, without smoothing. The proofs rest on a single minimum principle and are machine-checked in Lean 4. It reproduces the textbook results for parallel plates and concentric cylinders exactly, agrees with the diffusion approximation and with Crosbie and Schrenker, and agrees exactly with Noble’s formulation of Hottel’s zonal method.
@misc{bielefeld2026zonal,
title = {Non-Negativity, Non-Singularity and Energy Conservation of the
Zonal Method with Non-Reciprocal Exchange Factors},
author = {Bielefeld, Nikolaj Maack},
year = {2026},
eprint = {2512.22157},
archivePrefix = {arXiv},
primaryClass = {physics.comp-ph}
}In preparation
Geometric Projection Smoothing for Radiation Exchange Factors. Monte Carlo exchange factors violate reciprocity and energy conservation through sampling noise. This paper projects them onto the set of matrices that satisfy both, with a proven convergence rate, and shows that the projection removes much of the noise while it enforces the physics.