Graph Equilibrium Radiative Transfer

Analytic Solutions and Computational Methods

Author

Nikolaj Maack Bielefeld

Published

October 10, 2025

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Overview

This book presents the Graph Equilibrium Radiative Transfer (GERT) framework, a suite of analytical algorithms for computing equilibrium states on directed graphs. Radiative transfer is an important application in its own right, and also offers detailed validation opportunities. The GERT algorithms are general and extend beyond radiative transfer to equilibrium problems on directed graphs.

Note🚧 Book Under Development

This book is currently being written and expanded. New chapters are added regularly as the content is developed and refined. The complete version will be available at gert.net upon completion.

Mathematical Framework

The GERT methods treat radiative systems as stochastic graphs, where nodes represent surfaces or volumes, and edge weights represent energy transfer probabilities. Energy conservation is enforced through the requirement that outgoing probabilities from each node sum to 1 (row-stochastic matrices).

Unlike Monte Carlo approaches, this linear algebraic structure allows results to be proven with mathematical certainty, establishing rigorous guarantees on accuracy, convergence, and physical correctness.

Analytical Solutions

This book presents analytic methods for solving multiple scattering and absorption-reemission in radiative transfer systems. Given perfect optical knowledge of the domain, in the form of perfect row-stochastic exchange factors, these methods provide exact solutions to the energy redistribution problem. In reality, exchange factors do not perfectly match the domain in question. Exchange factors can be estimated numerically using Monte Carlo sampling for complex enclosures containing media which emit, absorb and scatter radiation (participating media). For convex transparent enclosures composed of polygon surfaces, the set of exchange factors can be computed analytically. In both cases, the exchange factors must satisfy both The First and The Second Law of Thermodynamics. This book presents a simple and effective algorithm to enforce these physical constraints (exchange factor smoothing, Part IV), with minimal distortion of the initial factors.

Advantages of the methods presented here:

  • Part III: Exchange Factor Transformation
    • Proven machine precision global energy conservation under specified conditions
    • Proven non-negative radiation results
    • Proven convergence conditions
    • Faster than traditional multiple scattering Monte Carlo ray tracing
      • Especially for high extinction media
      • And for cases with several non-zero specified source fluxes (direct solve, no iteration)
  • Part IV: Exchange Factor Smoothing
    • Simultaneously enforces The First and The Second Law of Thermodynamics to machine precision
    • The convergence rate of the algorithm improves as the number of exchange factors scales
    • Remains feasible and effective even for billions of exchange factors
    • Guaranteed convergence rate via continued fractions proof
  • Part V: Anisotropic Extension
    • Allowance for dependencies on the incident ray direction
    • Relaxes the requirements to obtain machine precision energy conservation
  • Part VI: Spectral Extension
    • Extension to spectral problems via spectral bins
    • The convergence rate of the algorithm improves as the number of spectral bins scales
    • An empirical expression for the approximate convergence rate

These methods serve as a reference standard for validating more complex computational approaches. The author has successfully used them to build and validate a full two-dimensional multiple scattering Monte Carlo code.

Limitations of the methods presented here:

  • The outgoing radiation distribution from each element is fixed by the exchange factors
    • For this reason specular reflections are not possible
    • Both emission and reflection-scattering share the same distribution
      • For surfaces a diffuse Lambertian distribution is used
      • For volumes a uniform spherical distribution is used
  • The methods presented here split the domain into discrete elements and spectral bins
    • If continuous results are needed use of interpolation or other methods is necessary
  • Matrix methods suffer from computational memory limitations due to quadratic scaling
    • The methods presented here have been tested on a 64 GB RAM workstation

Resources

Book Source Code: Complete source for this book is available at github.com/NikoBiele/graph-equilibrium-radiative-transfer

Julia Package: All methods are implemented in the registered package RayTraceHeatTransfer.jl

Contents

Navigate through the book using the sidebars, or jump directly to any part:

Part I: Introduction and Overview

Part II: Domain and Geometry Setup

Part III: Exchange Factor Transformation

Part IV: Exchange Factor Smoothing

Part V: Anisotropic Extension

Part VI: Spectral Extension

Part VII: Beyond Radiative Transfer

Your journey into analytical radiative transfer starts here.