A particle sorter with linearly increasing gaps @NilsBerglund

Nils Berglund | A particle sorter with linearly increasing gaps @NilsBerglund | Uploaded September 2024 | Updated October 2024, 1 hour ago.
The particle sorter in this simulation is inspired by a comment to a previous video. Instead of using several sieves one above the other, it uses a single sieve, with rotating obstacles of decreasing radius and increasing gaps between them.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. In addition, the obstacles have a circular motion of small amplitude, which reduces the chances of particles getting stuck even more.
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 9 minutes
Compression: crf 23
Color scheme: Turbo, by Anton Mikhailov
gist.github.com/mikhailov-work/6a308c20e494d9e0ccc29036b28faa7a

Music: "Fairy Meeting" by Emily A. Sprague@emilysprague6983

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

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$Waves escaping a ring of obstacles: Polar grid This is a simulation of waves originating from a point source crossing a set of circular obstacles placed on a polar grid, meaning that the locations form a regular grid in polar coordinates. Here there are 60 radial lines with 25 discs each. As one would expect, waves pass this lattice easier than for other arrangements seen on this channel. Still, diffraction has a large impact on the outgoing waves. The fact that the 60-fold symmetry is not preserved is probably due to round-off errors coming from the discretization. This video has two parts, showing the same evolution with two different color gradients: Averaged wave energy: 0:00 Wave height: 1:14 In the first part, the color hue depends on the energy of the wave, averaged over a sliding time window. In the second part, it depends on the height of the wave. 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. The color in the central region has been brightened to white, because the waves tend to make large-amplitude oscillations there, which would not be very pleasant to watch. 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: 30 minutes 38 seconds Compression: crf 23 Color scheme: Part 1 - Magma by Nathaniel J. Smith and Stefan van der Walt https://github.com/BIDS/colormap Part 2 - Twilight by Bastian Bechtold https://github.com/bastibe/twilight Music: Desert Planet by Quincas Moreira@QuincasMoreira 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$ $The double peak disappears when the source filmed with a gradient index lens is slower The recent video https://youtu.be/idB6qn802lM showed a source of light moving at about 33% of the speed of light being filmed by a gradient index lens. It showed a somewhat strange double peak, or echo being formed by the lens, making the source appear doubled. The question arose in the comments whether this was due to the very fast speed of the source. This simulation shows a similar situation, with a source moving at a slower speed, about 17% of the speed of light. It turns out that there is no double peak here, confirming that in the other simulation it must have been an effect of the higher speed. 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: 2:16 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: 1 hour 6 minutes 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: Lazy Boy Blues by the Unicorn Heads@UnicornHeads 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$

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$Waves branching through Poisson disc obstacles This is a simulation of waves originating from a point source crossing a set of obstacles forming a Poisson disc process, also known as blue noise. This is obtained by placing the obstacles randomly, but with an imposed minimal distance. Note the nice standing wave patterns in the wave height part, and the filamentation in the energy representation, similar to 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: 37 minutes 6 seconds Compression: crf 23 Color scheme: Part 1 - Parula, originally from Matlab https://www.mathworks.com/help/matlab/ref/colormap.html Part 2 - Turbo, by Anton Mikhailov https://gist.github.com/mikhailov-work/6a308c20e494d9e0ccc29036b28faa7a Music: Desert Catharsis 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$

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