Falling triangles @NilsBerglund

Nils Berglund | Falling triangles @NilsBerglund | Uploaded July 2024 | Updated October 2024, 2 hours ago.
Having modeled interacting falling sticks in the video youtu.be/BmLfzkOkJ6o , the next step is to simulate regular polygons, starting with triangles in this video. This turned out to be somewhat tricky to code, because the interactions have to be designed carefully, avoiding any discontinuity in the force.
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.
The temperature is controlled by a thermostat with constant temperature. There is a constant gravitational force directed downward.
This simulation has two parts, showing the evolution with two different color gradients:
Orientation: 0:00
Kinetic energy: 1:03
In the first part, the particles' color depends on their orientation modulo 120 degrees. In the second part, it depends on their kinetic energy, averaged over a sliding time window.
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 temperature is controlled by a thermostat, implemented here with the "Nosé-Hoover-Langevin" algorithm introduced by Ben Leimkuhler, Emad Noorizadeh and Florian Theil, see reference below. The idea of the algorithm is to couple the momenta of the system to a single random process, which fluctuates around a temperature-dependent mean value. Lower temperatures lead to lower mean values.
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: 40 minutes 2 seconds
Compression: crf 23
Color scheme: Part 1 - HSL/Jet
Part 2 - Turbo, by Anton Mikhailov
gist.github.com/mikhailov-work/6a308c20e494d9e0ccc29036b28faa7a

Music: "Meet & Fun!" by Ofshane@Ofshane

Reference: Leimkuhler, B., Noorizadeh, E. & Theil, F. A Gentle Stochastic Thermostat for Molecular Dynamics. J Stat Phys 135, 261–277 (2009). doi.org/10.1007/s10955-009-9734-0
maths.warwick.ac.uk/~theil/HL12-3-2009.pdf

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 #ions #quasicrystal

$Filming a moving source with a gradient index lens Like the video https://youtu.be/KB_Roqs38F8 , this one shows a simulation of light from a nearby source crossing a gradient index lens. The difference is that here, the source is moving on a path perpendicular to the axis of the lens. Note that the speed of the source is very large, about 33% of the speed of light, but the simulation does not take any relativistic effects into account. The refractive index along the central axis of the lens is quite large, with a value of 1.82, in order to obtain the small focal distance required by the simulation. Im not quite sure what causes the echo of the transmitted waves, which was not present in other simulations with a static source. Is might be due to light reflecting on the horizontal boundaries of the lens. If someone has a better explanation, please feel free to leave a comment. 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 sqrt(n0² - a*r²), where r is the distance to the axis of symmetry. This results in the incoming waves being focused on 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: 1:39 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: 51 minutes 32 seconds Compression: crf 23 Color scheme: Part 1 - Twilight by Bastian Bechtold https://github.com/bastibe/twilight Part 2 - Inferno by Nathaniel J. Smith and Stefan van der Walt https://github.com/BIDS/colormap Music: Runner by Silent Partner 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$ $A gradient index lens with quadratic refractive index exposed to two light sources$

Winnowing: Separating particles by size using wind

$3D representation of a gradient index lens with quadratic refractive index$ $Waves of two different frequencies crossing a diffraction grating$

A shallower laminar flow over an immersed dodecahedron

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

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$