3D render of a branched flow in Poisson disc/blue noise
Channel: Nils Berglund
Date: 2023-06-23

This is a 3D version of the simulation https://youtu.be/mVleSAcvcf0 of a branched flow in a  random potential, based on a Poisson disc process, also known as blue noise. I have used such a process several times on previous videos on wave protections and mangroves, for the placement of obstacles. In a Poisson disc process, points are placed randomly in the plane, but with the constraint that there is a minimal distance between points, as if they were discs of finite radius. 
These simulations were suggested to me by Álvar Daza, one of the people running the web site https://www.blochbusters.com/ , which contains, among other things, simulations and background on branched flows. 
A circular wave is emitted at the center of the displayed region. The wave speed depends on the position in the following way. It is given by a sum of Gaussians, which are centered at the vertices of a Poisson disc process. The amplitude of the Gaussians is also random. The wave speed is thus maximal near the points of the Poisson disc process, and drops nearly to zero in other regions. Due to the interaction with the random wave speed, the wave may be expected to branch when it encounters regions with low propagation speed, though the relation between branching and speed may be more complicated than that.
The video has two parts, showing the same simulation with two different color schemes:
Wave height: 0:00
Energy flux: 1:16
In the first part, the color hue and the z-coordinate show the wave height. In the second part, the color hue and z-coordinate show the direction of the energy flux, while the luminosity shows the intensity of the energy flux, slightly averaged over time. There are absorbing boundary conditions on the outer boundaries of the simulation. 

Reference: Eric J. Heller, Ragnar Fleischmann, Tobias Kramer, Branched flow, Physics Today 74 (12), 44–51 (2021).
https://pubs.aip.org/physicstoday/article/74/12/44/396182/Branched-flowIn-many-kinds-of-irregular-media
Poisson disc sampling: https://bl.ocks.org/mbostock/dbb02448b0f93e4c82c3

Render time: 25 minutes 56 seconds
Color scheme: Part 1 - Cividis by Jamie R. Nuñez, Christopher R. Anderton, Ryan S. Renslow
https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0199239
Part 2 - Magma by Nathaniel J. Smith and Stefan van der Walt
https://github.com/BIDS/colormap

Music: "Saucers Unaware" by Jade Wii@JAdeWii_

See also https://images.math.cnrs.fr/Des-ondes-dans-mon-billard-partie-I.html for more explanations (in French) on  a few previous simulations of wave equations. 

The simulation solves the wave equation by discretization. The algorithm is adapted from the paper https://hplgit.github.io/fdm-book/doc/pub/wave/pdf/wave-4print.pdf 
C code: https://github.com/nilsberglund-orleans/YouTube-simulations
https://www.idpoisson.fr/berglund/software.html
Many thanks to Marco Mancini and Julian Kauth for helping me to accelerate my code!

#branched_flow #wave_equation #waves #refraction