Magnetized triangles in a funnel @NilsBerglund

Nils Berglund | Magnetized triangles in a funnel @NilsBerglund | Uploaded August 2024 | Updated October 2024, 1 minute ago.
Like the video youtu.be/Dsa6xj6tARo , this one shows triangles interacting via contact forces falling through a funnel. There are two differences, however: the funnel is steeper, and the triangles also interact via a torque tending to anti-align their orientations, meaning that the sides of nearby triangles tend to be parallel.
To compute the force and torque of triangle j on triangle i, the code computes the distance of each vertex of triangle j to the faces of triangle 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 triangles have been added, whenever a vertex of triangle j is not on a perpendicular to a face of triangle 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 triangles has been added, as it seems to increase numerical stability.
Unlike in 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.
This simulation has two parts, showing the evolution with two different color gradients:
Orientation: 0:00
Initial x position: 1:27
In the first part, the particles' color depends on their orientation modulo 120 degrees. In the second part, it depends on their initial x position. This allows to get a sense of how particles are mixed while falling through the funnel.
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 21 minutes
Compression: crf 23
Color scheme: Part 1 - HSL/Jet
Part 2 - Turbo, by Anton Mikhailov
gist.github.com/mikhailov-work/6a308c20e494d9e0ccc29036b28faa7a

Music: "Strut Funk" by Dougie Wood

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

Density and direction of speed of more stable vortices on a rotating sphere

Shallow water flowing over an immersed octahedron

Replicants are like any other machine: Trying to model DNA replication

Shallow water flowing over an immersed cube

$Waves escaping a ring of obstacles: Logarithmic spirals This is a simulation of waves originating from a point source crossing a set of circular obstacles placed on 60 logarithmic spirals, having 25 discs each. On each logarithmic spiral, the radial coordinate grows linearly, while the angular coordinate is an affine function of the logarithm of the radius. One particularity of this design is that the energy appears to leave the ring on straight lines that are not radial, but tangent to the spirals at their endpoints. 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: 26 minutes 6 seconds Compression: crf 23 Color scheme: Part 1 - Inferno by Nathaniel J. Smith and Stefan van der Walt Part 2 - Viridis by Nathaniel J. Smith, Stefan van der Walt and Eric Firinghttps://github.com/BIDS/colormap Music: Plaidness by Francis Preve@francispreve 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$ $Waves escaping a ring of obstacles: Poisson disc grid This is a simulation of waves originating from a point source crossing a set of circular obstacles placed randomly in an annular region. The centers of obstacles follow a so-called Poisson disc process, which is the most random distribution that keeps a minimal distance between the points. This video has two parts, showing the same evolution with two different color gradients: Averaged wave energy: 0:00 Wave height: 1:11 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: 32 minutes 37 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: The Great Unknown by Audionautix is licensed under a Creative Commons Attribution 4.0 licence. https://creativecommons.org/licenses/by/4.0/Artist: http://audionautix.com/ 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$

Neutral soap and water with increasing force constant

Weather on the Earth with a random initial state

$What would a superluminal light source look like in a non-relativistic universe? In one word, blurry. In this simulation, a light source moving at about 1.2 times the speed of light is observed through a gradient index lens. This is of course not possible in our relativistic universe, though it can be done with sound waves or water waves. As a result of its high speed, the source creates a shock wave, and that is what is observed in the focal plane shown on the right. The refractive index n(r) 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: 1:31 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: 43 minutes 52 seconds Compression: crf 23 Color scheme: Part 1 - Twilight by Bastian Bechtold https://github.com/bastibe/twilight Part 2 - Plasma by Nathaniel J. Smith and Stefan van der Walt https://github.com/BIDS/colormap Music: I Dont Remmber I Dont Recall by the 129ers 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$