They Said It Was Impossible… This Simulation Solved It

Jan 25, 2026

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They Said It Was Impossible… This Simulation Solved It

Sand castles, hexapod grains, and a virtual hydraulic press: a new method turns millions of loose grains into a solid elastic material. Can physics really be this fast?

Background

Realistic simulation of granular materials such as sand is computationally demanding. Rigid-body simulation produces realistic results but handles only small object counts. Simulating billions of individual grains is impractical with traditional methods.

A recent paper from the lab of Professor Chris Wojtan addresses this problem. It presents a method that simulates the collective behavior of granular materials built from non-spherical grains. The method reproduces effects such as interlocking, cohesion, and solid-like response, without simulating every grain individually.

Constitutive Models: Drucker-Prager and Mohr-Coulomb

The plastic flow model supports two standard constitutive models for granular materials:

Drucker-Prager: Treats grains as smooth and simple. The model represents sand as a pile of slippery marbles.
Mohr-Coulomb: Accounts for the irregular shape of real grains. It models how grains lock together and where they tend to slip along weak points.

The two models produce similar results when internal friction is low. As friction parameters increase, the models diverge. Each model performs well in some conditions and poorly in others. The authors conclude that traditional techniques cannot reliably produce fast and accurate granular simulations across all conditions.

Effect of Grain Shape

The method simulates grains of different shapes. Each shape produces distinct behavior.

Grain shape	Description	Observed behavior
Spheres (circular grains)	Smooth, round grains	Flow freely; no cohesion
Dolosse	Interlocking, doorknob-like grains; similar to concrete sea-wall armor	Interlock partially; form a steeper heap than spheres
Hexapods (caltrops)	Six-armed, star-shaped grains	Hook onto each other; form cohesive lumps
Dodecafangs	Twelve fang-like hooks	Lock tightly; behave as a solid elastic body

Hourglass Flow

Circular grains flow uniformly through an hourglass.
Star-shaped grains hook together and form a large clump that resists movement.
Dolosse-type grains pass through with more resistance and form clumps.

Sand Bridge

Spheres: The sand bridge collapses immediately. Without interlocking, the grains form a flat pile, held only by friction, which is insufficient to resist gravity.
Dolosse: The bridge collapses. The grains form a much steeper heap at the bottom than spheres do.
Hexapods: Without glue or water, the grains crumble into chunky, cohesive lumps rather than flowing like a fluid.
Dodecafangs: No pile forms. The hooks lock together so tightly that the material behaves as a solid elastic body. The structure jiggles.

Siege Tests

Sand castles built from each grain type were subjected to a virtual sandball impact.

Spheres: The castle collapsed instantly because the grains have no cohesion. The impact produced a splash but no structural resistance.
Dolosse: The castle resisted slightly longer due to higher friction. The impact still exceeded its strength, and the castle collapsed.
Hexapods: The castle was tested under the same impact conditions.
Dodecafangs: A high-speed projectile struck the castle and bounced off. The material absorbed the energy in a manner similar to a block of rubber. The structure then jiggled briefly and returned to rest. This behavior confirms that the loose grains formed a single solid elastic object.

Numerical Homogenization

Simulating every grain in a large castle is computationally infeasible. The method instead uses numerical homogenization.

Procedure

Representative volume: A small box containing a few thousand grains is created.
Compression tests: The box is compressed thousands of times to measure how it resists force.
Rule extraction: The resulting data defines the mathematical behavior of the grain shape.
Periodic extension: The box is treated as a repeating unit that extends endlessly in all three directions, similar to a repeating wallpaper pattern.
Large-scale simulation: The derived rules are applied to a larger, faster simulation. This allows large scenes to be rendered in a fraction of the time.

Homogenized Cauchy Stress Tensor

The method uses integrals that compute the homogenized Cauchy stress tensor. This tensor represents the average stress across the entire box.

The calculation does not measure forces inside each grain, which would be impractical. It calculates the forces acting on the outer walls of the box. The tensor product converts these wall forces into a single average stress value for the material.

An analogy illustrates the principle. To determine the pressure within a crowd, one measures how hard the crowd pushes against the walls of the room. Individual forces inside the crowd do not need to be measured.

Limitations

Computation time: Deriving the rules for the hexapod grain shape required 705 hours of computation.
Rigid grains: The method assumes that grains are rigid. It does not model deformable grains, such as soft particles.

Conclusion

The method reproduces the behavior of granular materials with non-spherical grains. It captures interlocking, cohesion, and solid-like response, which traditional methods do not reliably model. The approach reduces the simulation of millions of grains to the analysis of a small representative volume. The authors note that the method establishes feasibility, and that optimization can follow.

Introduction and the Sand Castle Challenge

The presenter introduces a granular simulation paper from Chris Wojtan's lab that simulates sand made of billions of grains, and compares it to earlier rigid body methods.

The paper is from Chris Wojtan's lab.
The presenter worked at the Disney research lab in Switzerland around the time of an earlier sand castle paper.
The earlier sand castle simulation used up to billions of grains and used no AI.
Rigid body simulation is realistic but only works for small object counts.

Plastic Flow Models: Drucker-Prager vs Mohr-Coulomb

The paper's plastic flow handles two sand models, Drucker-Prager and Mohr-Coulomb, which agree in simple cases but diverge as friction increases.

Drucker-Prager models sand as smooth, slippery marbles.
Mohr-Coulomb models sand as jagged grains that lock together and have weak slip planes.
The two models look similar with low internal friction but behave differently as friction parameters increase.
Traditional techniques struggle to give quick and accurate granular simulations.

Grain Shapes and Their Behavior

Grain shape determines clumping and flow. Hexapod grains hook together and clump, and interlocking shapes resist falling apart.

Circular grains flow freely through an hourglass.
Hexapod star-shaped grains hook onto each other and clump together.
Interlocking grains barely fall apart and create a big clump refusing to move in an hourglass.
Another object type passes through an hourglass with more resistance and clumps like caramel.

Sand Bridges and Shape Tests

Sand bridge tests show that spherical grains collapse immediately, while dolosse and hexapods form steeper or cohesive heaps.

Spherical grains have no interlocking, so the sand bridge collapses immediately and forms a flat pile.
Dolosse-shaped grains interlock somewhat and form a much steeper heap than spheres.
Hexapods with six arms form chunky, cohesive lumps without glue or water.
Dodecafangs with 12 fang-like hooks lock together so tightly that the material acts like a solid elastic body and jiggles.

Siege Test on Castles

A virtual sandball is fired at castles made of different grains. Spheres collapse instantly, dolosse hold slightly better, and dodecafang castles bounce back like rubber.

Spherical grains have zero cohesion and the castle collapses instantly, rated zero out of ten.
Dolosse-shaped grains put up a slightly better fight and receive one out of ten.
Hexapod castles behave differently under impact, with the presenter describing them as caltrops.
Dodecafang-based piles absorb a high-speed projectile and bounce back like a block of rubber.

Numerical Homogenization and the Mathematics

Numerical homogenization makes large scenes feasible by simulating a small periodic box of grains, measuring its average stress, and transferring the rules to larger simulations.

The team packed a tiny box with just a few thousand grains and squashed it thousands of times.
The box is treated as repeating 3D wallpaper in all directions.
The integrals compute the homogenized Cauchy stress tensor, which is the average pressure for the box.
Forces are measured only on the outer walls of the box, not inside every grain.
The tensor product translates wall forces into a single pressure value, like measuring crowd pressure against a room's walls.

Limitations and Closing

The method's main weakness is computational cost, and it assumes the grains are hard. The presenter argues optimization can come later, citing the First Law of Papers.

Simulating the pink hexapods took 705 hours of computation, almost a month, to learn the rules for one grain shape.
The method assumes grains are hard as rock and does not handle squishy grains.
The presenter cites the First Law of Papers: research is a process, and results should be judged by where they will be later.