How does the particle size sorter work at very low friction? @NilsBerglund

Nils Berglund | How does the particle size sorter work at very low friction? @NilsBerglund | Uploaded September 2024 | Updated October 2024, 1 minute ago.
Not too well. This simulation uses the same sorting set-up as in the video youtu.be/ZA3gcbNm1l8 , but the friction here is four times weaker. As a result, the particles bounce more, and many of them end up in the wrong bin.
The sorting set-up consists in three grids of obstacles with decreasing space between them, acting as particle sieves. The obstacles rotate and exert a tangential force on the particles, in order to decrease clogging of the sieves. The angular speed of the obstacles varies periodically in time, in order to reduce the chance of particles getting stuck.
The conveyor belt effect results from the segments forming the belt exerting a tangential force on the polygons, in addition to the normal force. The tangential force is proportional to the difference between the tangential speed of the polygon and the speed of the belt.
To compute the force and torque of polygon j on polygon i, the code computes the distance of each vertex of polygon j to the faces of polygon i. If this distance is smaller than a threshold, the force increases linearly with a large spring constant. In addition, radial forces between the vertices of the polygons have been added, whenever a vertex of polygon j is not on a perpendicular to a face of polygon i. This is important, because otherwise triangles can approach each other from the vertices, and when one vertex moves sideways, it is suddenly strongly accelerated, causing numerical instability. A weak Lennard-Jones interaction between polygons has been added, as it seems to increase numerical stability.
Unlike in some previous videos involving interacting polygons, there is no thermostat in this simulation. Instead, friction forces (both linear and angular) have been added for numerical stability. In addition, the particles are subject to a gravitational force directed downwards.
The color of the polygons depends on their size.
To save on computation time, particles are placed into a "hash grid", each cell of which contains between 3 and 10 particles. Then only the influence of other particles in the same or neighboring cells is taken into account for each particle.
The Lennard-Jones potential is strongly repulsive at short distance, and mildly attracting at long distance. It is widely used as a simple yet realistic model for the motion of electrically neutral molecules. The force results from the repulsion between electrons due to Pauli's exclusion principle, while the attractive part is a more subtle effect appearing in a multipole expansion. For more details, see en.wikipedia.org/wiki/Lennard-Jones_potential

Render time: 37 minutes 43 seconds
Compression: crf 23
Color scheme: Turbo, by Anton Mikhailov
gist.github.com/mikhailov-work/6a308c20e494d9e0ccc29036b28faa7a

Music: "Fanta Mankane" by the Mini Vandals@theminivandals1840
featuring Mamadou Kolta and Lasso

Current version of the C code used to make these animations:
github.com/nilsberglund-orleans/YouTube-simulations
https://www.idpoisson.fr/berglund/software.html
Some outreach articles on mathematics:
https://images.math.cnrs.fr/auteurs/nils-berglund/
(in French, some with a Spanish translation)

#molecular_dynamics #polygons #conveyor

A more symmetric version of resonances in a circle excited by five out of phase sources

$Almost a branched flow: Waves crossing a finer percolation-style arrangement Like the video https://youtu.be/QhRmQ_ZX31Y , this one shows waves originating from a point interacting with obstacles obtained by randomly deleting circles from a regular square lattice. 15% of the points of the regular lattice have been kept. The arrangement is more random than a square lattice, but not as random as a Poisson disc process. The lattice is finer as in the previous simulation, as it contains four times as many points. I also chose different color gradients. In particular the energy part shows a filamentation of the energy distribution not unlike that of a branched flow, illustrated for instance in the video https://youtu.be/kOR6OiRlL8Y This video has two parts, showing the same evolution with two different color gradients: Wave height: 0:00 Averaged wave energy: 1:27 In the first part, the color hue depends on the height of the wave. In the second part, it depends on the energy of the wave, averaged over a sliding time window. The contrast has been enhanced by a shading procedure, similar to the one I have used on videos of reaction-diffusion equations. The process is to compute the normal vector to a surface in 3D that would be obtained by using the third dimension to represent the field, and then to make the luminosity depend on the angle between the normal vector and a fixed direction. There are absorbing boundary conditions on the borders of the simulated rectangle. The display at the right shows a time-averaged version of the signal near the right boundary of the simulated rectangular area. More precisely, it shows the square root of an average of squares of the respective field value (wave height or energy). Render time: 36 minutes 12 seconds Compression: crf 23 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 - Turbo, by Anton Mikhailov https://gist.github.com/mikhailov-work/6a308c20e494d9e0ccc29036b28faa7a Music: Magical Gravity by Asher Fulero@AsherFulero See also https://images.math.cnrs.fr/des-ondes-dans-mon-billard-partie-i/ 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! #wave #diffraction$

More stable vortices in Eulers equations on a rotating sphere

Vortices on a sphere like octahedral symmetry

Simulation of a meteor impact in the Atlantic Ocean

$When waves crossing a square lattice have a larger phase than group velocity The simulation https://youtu.be/15fpsopRyn8 , that showed waves from two different sources having different frequencies cross a regular square grid of obstacles, revealed another interesting feature: After a while, the waves from the source with a lower frequency seemed to propagate through the lattice at a larger speed than the wave speed. This has to be an instance of the situation where the phase velocity (measuring how fast wave fronts move) is larger than the group velocity (measuring the speed of energy transfer). This video focuses on this phenomenon, by using only one source, having the lower frequency, and an additional plot of the signal in the x-direction. This video has two parts, showing the same evolution with two different color gradients: Wave height: 0:00 Averaged wave energy: 1:16 In the first part, the color hue depends on the height of the wave. In the second part, it depends on the energy of the wave, averaged over a sliding time window. The contrast has been enhanced by a shading procedure, similar to the one I have used on videos of reaction-diffusion equations. The process is to compute the normal vector to a surface in 3D that would be obtained by using the third dimension to represent the field, and then to make the luminosity depend on the angle between the normal vector and a fixed direction. There are absorbing boundary conditions on the borders of the simulated rectangle. The displays at the right and bottom show a time-averaged version of the signal near the right boundary of the simulated rectangular area. More precisely, it shows the square root of an average of squares of the respective field value (wave height or energy). Render time: 30 minutes 28 seconds Compression: crf 23 Color scheme: Part 1 - Viridis by Nathaniel J. Smith, Stefan van der Walt and Eric Firing Part 2 - Inferno by Nathaniel J. Smith and Stefan van der Walt https://github.com/BIDS/colormap Music: Future Girl by RKVC@rkvc See also https://images.math.cnrs.fr/des-ondes-dans-mon-billard-partie-i/ 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! #wave #diffraction #phase_velocity$

Mega-hextris: Coagulating falling hexagons

Looking for quasicrystals: Interacting kites-and-darts-type molecules