Graph Equilibrium Radiative Transfer

Radiative heat transfer as an equilibrium on a graph

Last updated

30 September 2026

Graph Equilibrium Radiative Transfer (GERT) treats a radiating domain as a graph. Surfaces and volumes of participating medium are the nodes, and the exchange factors between them are the edges: the fraction of the radiation leaving one element that first interacts with another. Every row of this matrix sums to one, because all radiation leaving an element arrives somewhere.

Ray tracing gives the exchange factors, following each ray only to its first interaction. Everything after that, every reflection, scattering event, absorption and re-emission, is resolved by a single linear system. Its solution is non-negative, and it conserves energy to machine precision as an algebraic identity, however well or badly the exchange factors were sampled. Monte Carlo sampling leaves the factors slightly out of balance with each other, so they are first projected onto the set of matrices that satisfy reciprocity and energy conservation exactly, which removes much of the sampling noise as well.

Centreline source function against optical depth: a wide blue band of raw results around a straight black line, with the orange smoothed results lying on the line. — A million elements: the centreline of a 1000 m × 1 m domain with extinction coefficient 100 m⁻¹, meshed into 1001 × 1001 cells and traced with a billion rays. Raw Monte Carlo exchange factors (blue) scatter around the diffusion approximation (black); smoothed factors (orange) fall onto it, with 97.5 % of the deviation removed. Tracing, smoothing and solving took 1.8 minutes on a desktop PC.

The methods are implemented in RayTraceHeatTransfer.jl, a registered Julia package. It handles grey, spectral and directional problems in two-dimensional participating media, and surface exchange in three-dimensional enclosures.

Where to go next

Papers: the theory, with proofs, published as preprints on arXiv.
Examples: worked problems with the package, each validated against an independent reference.
About: the author and the software.

---
title: "Graph Equilibrium Radiative Transfer"
subtitle: "Radiative heat transfer as an equilibrium on a graph"
---

Graph Equilibrium Radiative Transfer (GERT) treats a radiating domain as a graph. Surfaces and volumes of participating medium are the nodes, and the exchange factors between them are the edges: the fraction of the radiation leaving one element that first interacts with another. Every row of this matrix sums to one, because all radiation leaving an element arrives somewhere.

Ray tracing gives the exchange factors, following each ray only to its first interaction. Everything after that, every reflection, scattering event, absorption and re-emission, is resolved by a single linear system. Its solution is non-negative, and it conserves energy to machine precision as an algebraic identity, however well or badly the exchange factors were sampled. Monte Carlo sampling leaves the factors slightly out of balance with each other, so they are first projected onto the set of matrices that satisfy reciprocity and energy conservation exactly, which removes much of the sampling noise as well.

![A million elements: the centreline of a 1000 m × 1 m domain with extinction coefficient 100 m⁻¹, meshed into 1001 × 1001 cells and traced with a billion rays. Raw Monte Carlo exchange factors (blue) scatter around the diffusion approximation (black); smoothed factors (orange) fall onto it, with 97.5 % of the deviation removed. Tracing, smoothing and solving took 1.8 minutes on a desktop PC.](assets/large_scale_grey_diffusion.png){fig-alt="Centreline source function against optical depth: a wide blue band of raw results around a straight black line, with the orange smoothed results lying on the line."}

The methods are implemented in [RayTraceHeatTransfer.jl](https://github.com/NikoBiele/RayTraceHeatTransfer.jl), a registered Julia package. It handles grey, spectral and directional problems in two-dimensional participating media, and surface exchange in three-dimensional enclosures.

## Where to go next

- [Papers](papers.qmd): the theory, with proofs, published as preprints on arXiv.
- [Examples](examples/index.qmd): worked problems with the package, each validated against an independent reference.
- [About](about.qmd): the author and the software.