Nils Berglund | Hexagons make nice sand piles @NilsBerglund | Uploaded August 2024 | Updated October 2024, 3 minutes ago.
This is a variant of the video youtu.be/zvhWklW_XlQ showing magnerized squares thrown into a funnel, but with hexagons instead of squares. The behavior turns out to be quite different, and much closer to that of a sand pile, including avalanches.
To compute the force and torque of hexagon j on hexagon i, the code computes the distance of each vertex of hexagon j to the faces of hexagon 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 hexagons have been added, whenever a vertex of hexagon j is not on a perpendicular to a face of hexagon i. This is important, because otherwise hexagons can approach each other from the vertices, and when one vertex moves sideways, it is suddenly strongly accelerated, causing numerical instability. In addition, a torque has been added between hexagons, that tends to align them, as if they were magnets. A weak Lennard-Jones interaction between squares 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:
Initial x position: 0:00
Velocity: 1:58
In the first part, the particles' color depends on their initial x position. This allows to get a sense of how particles are mixed while falling through the funnel. In the second part, the due depends on the direction in which the particles are moving, while the luminosity depends on their speed. Both quantities are averaged over a time interval.
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 36 minutes
Compression: crf 23
Color scheme: Part 1 - Turbo, by Anton Mikhailov
gist.github.com/mikhailov-work/6a308c20e494d9e0ccc29036b28faa7a
Part 2 - Twilight by Bastian Bechtold
github.com/bastibe/twilight
Music: "Thinking Machines" 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
This is a variant of the video youtu.be/zvhWklW_XlQ showing magnerized squares thrown into a funnel, but with hexagons instead of squares. The behavior turns out to be quite different, and much closer to that of a sand pile, including avalanches.
To compute the force and torque of hexagon j on hexagon i, the code computes the distance of each vertex of hexagon j to the faces of hexagon 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 hexagons have been added, whenever a vertex of hexagon j is not on a perpendicular to a face of hexagon i. This is important, because otherwise hexagons can approach each other from the vertices, and when one vertex moves sideways, it is suddenly strongly accelerated, causing numerical instability. In addition, a torque has been added between hexagons, that tends to align them, as if they were magnets. A weak Lennard-Jones interaction between squares 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:
Initial x position: 0:00
Velocity: 1:58
In the first part, the particles' color depends on their initial x position. This allows to get a sense of how particles are mixed while falling through the funnel. In the second part, the due depends on the direction in which the particles are moving, while the luminosity depends on their speed. Both quantities are averaged over a time interval.
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 36 minutes
Compression: crf 23
Color scheme: Part 1 - Turbo, by Anton Mikhailov
gist.github.com/mikhailov-work/6a308c20e494d9e0ccc29036b28faa7a
Part 2 - Twilight by Bastian Bechtold
github.com/bastibe/twilight
Music: "Thinking Machines" 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
