An improved device sorting particles by size @NilsBerglund

Nils Berglund | An improved device sorting particles by size @NilsBerglund | Uploaded September 2024 | Updated October 2024, 1 hour ago.
This variant of the videos youtu.be/qozGp_KR_DA and youtu.be/feEDk9e4HPM of a device sorting particles by size contains two improvements. The first one, suggested in a comment, is to have a conveyor belt bringing the particles from the opposite side, and dropping them more to the left, in order to use a longer stretch of the sieves. The second improvement is that the circular obstacles forming the sieves do not rotate at constant speed, but rotate sometimes faster and sometimes slower. This reduces the number of particles getting stuck in the sieves, in a similar way as "rattling" the sieves now and then would.
The sorting set-up consists in three grids of obstacles with decreasing space between them, acting as particle sieves. 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: 1 hour 17 minutes
Compression: crf 23
Color scheme: Turbo, by Anton Mikhailov
gist.github.com/mikhailov-work/6a308c20e494d9e0ccc29036b28faa7a

Music: "Running Errands" by TrackTribe@TrackTribe

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

Four waves in shallow water on the sphere

$A light source moving in and out of focus of a gradient index lens This video was requested in a comment to a previous video featuring a gradient index lens. It shows a source of light moving towards the lens on its axis of symmetry. The lens is designed in such a way that the focal plane shown on the right corresponds to a symmetrical position of the source on the left. As the source moves with respect to the optimal position, one can see the light reaching the focal plane become first slightly more focused, and then again unfocused. The refractive index n(r) of the gradient index lens used here depends on the distance r to the horizontal symmetry axis of the lens in a quadratic way, like n(r) = n0 - a*r², with n0 = 1.25 and a = 0.4375, making the refractive index smaller than 1 at the outer boundary of the lens, where r = 1. In order to visualize the refractive index as well, the luminosity of the background depends on the refractive index. The videos are inspired by Huygens Optics recent short https://youtube.com/shorts/VGd3Ajnp6e0 showing the principle of a gradient index lens. Lenses focus incoming rays of light by delaying them more near the center of the lens than at its periphery. This is often done with a material of constant index of refraction, by making the lens thicker near the center, as shown for instance in the simulation https://youtu.be/rrJJBh9ubUE . However, one can also build lenses of constant thickness, by making the index of refraction of their material depend on the location in the lens. In this simulation, the index decreases like the square of the distance to the center (that is, it is of the form n0 - a*r², where r is the distance to the axis of symmetry). This results in the incoming waves being focused at two points in the (estimated) focal plane, marked by a vertical line. The plot to the right shows a time-averaged value of the field along that plane. 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. There are absorbing boundary conditions on the borders of the simulated rectangle. The display at the right shows the signal along the focal plane, which is indicated by a vertical line. Render time: 38 minutes 3 seconds Compression: crf 23 Color scheme: Part 1 - Viridis, by Nathaniel J. Smith, Stefan van der Walt and Eric Firing Part 2 - Plasma by Nathaniel J. Smith and Stefan van der Walt https://github.com/BIDS/colormap Music: Finding Light by Dan Lebowitz@lebo_tone 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! #wave #lens #gradient_index$

Four interacting pressure systems on a rotating sphere

$Waves of two different frequencies crossing a larger regular square lattice This is a variant of the video https://youtu.be/pObtx2xmZDA , showing waves from sources with two different frequencies, crossing a regular square lattice of scatterers. The lattice is here broader than in the previous simulation, and the obstacles have a larger radius, in order to provide a better separation between wavelengths. One feature I found quite striking is that waves from the top source seem to accelerate when they reach the lattice. The speed of waves being fixed by the simulation, this has to be a situation where the phase velocity exceeds the group velocity. The frequency of the lower source is three times the frequency of the upper one. The resulting wavelengths are such that the open intervals in the grating are roughly between both wavelengths. As a result, waves of the lower source pass the grating a bit more easily, as can be best seen on the energy plot. This video has two parts, showing the same evolution with two different color gradients: Wave height: 0:00 Averaged wave energy: 1:20 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: 37 minutes 23 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: Alone by Emmit Fenn@emmitfenn 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 #grating$ $Refraction and reflection of a shock wave - Source in the lower speed medium$

Bloopers 17: You look lonely. I can fix that. Replicated DNA getting pinned

A conveyor belt with shovels at lower friction

Increasing the wavelength even more in a magnetron-shaped resonator

Using seven sources to excite resonant modes in a circle