# US 10,503,845 B2 · Particle-in-cell methods preserving shearing and rotation

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- Number: US 10,503,845 B2
- Document ID: US10503845B2
- Status: Granted
- Kind: B2
- Application number: 14/981,309
- Filed: 2015-12-28
- Published: 2017-06-29
- Granted: 2019-12-10
- Earliest claimed: 2015-12-28
- Assignees: Disney Enterprises, Inc.
- Applicants: Disney Enterprises, Inc.
- Inventors: Alexey Stomakhin; Joseph Teran
- Web page: https://patentgrep.com/patent/US10503845B2

## CPC

- G06F 30/20: Design optimisation, verification or simulation (optimisation, verification or simulation of circuit designs G06F30/30)
- G06F 17/10: Complex mathematical operations {(function generation by table look-up G06F1/03; evaluation of elementary functions by calculation G06F7/544)}
- G06F 2111/10: Numerical modelling
- G06F: ELECTRIC DIGITAL DATA PROCESSING (computer systems based on specific computational models G06N)

## Abstract

The disclosure provides an approach for reducing information loss when transferring from particles to grid and vice versa in a hybrid Lagrangian/Eulerian simulation, while also preserving stability. In one aspect, referred to as the rigid particle-in-cell (RPIC) method, a simulation application stores a sampling of local angular momentum for each particle, helping to preserve the angular momentum that would otherwise be lost in the transfer from grid to particles. In another aspect that is more generally applicable and referred to herein as the rigid and shearing particle-in-cell (RSPIC) method, the simulation application stores, for each particle, a matrix with rotational and shearing information. The RSPIC method effectively reduces dissipation, preserves angular momentum exactly, and prevents instabilities. Further, the RSPIC method is applicable to both incompressible liquids and material point method (MPM) simulations of materials such as solids.

## Claims

### Claim 1

A computer-implemented method of simulating and rendering a visual depiction of a system, comprising:
simulating the system, wherein the system includes at least one material, and wherein simulating the system comprises, for each of one or more iterations:
transferring system information including angular momentum information from particles to a grid on which dynamics are computed, and
transferring the system information including the angular momentum information and velocity information back from the grid to the particles on which at least velocity changes are computed,
wherein the transfer from the particles to the grid and the transfer back from the grid to the particles act as filters and conserve angular momentum; and
rendering for display one or more images depicting the system including the at least one material based, at least in part, on the simulation of the system.

### Claim 2 (depends on 1)

The method of claim 1, wherein:
the system information that is transferred from the particles to the grid and back from the grid to the particles further includes shearing information; and
the transfer from the particles to the grid and the transfer back from the grid to the particles also preserve shearing modes.

### Claim 3 (depends on 2)

The method of claim 2, wherein:
the dynamics computed on the grid include at least one of forces, collision responses, and incompressible projections;
at least the velocity changes and angular momentum changes are computed on the particles; and
the particles are advected.

### Claim 4 (depends on 2)

The method of claim 2, wherein the angular momentum and the shearing information are stored in a matrix.

### Claim 5 (depends on 2)

The method of claim 2, wherein:
the system includes a fluid; and
the system information is transferred from the particles to marker-and-cell (MAC) faces of the grid.

### Claim 6 (depends on 2)

The method of claim 2, wherein:
the system includes a solid; and
the one or more iterations are iterations in a material-point-method (MPM) loop.

### Claim 7 (depends on 2)

The method of claim 2, further comprising, blending the simulation of the system with a particle-in-cell (PIC) simulation technique.

### Claim 8 (depends on 2)

The method of claim 2, wherein:
the system includes a plurality of materials; and
the simulation is a coupled simulation of the plurality of materials.

### Claim 9 (depends on 2)

The method of claim 2, wherein the dynamics computed on the grid include mesh-based Lagrangian forces.

### Claim 10

A non-transitory computer-readable storage medium storing a program, which, when executed by a processor performs operations for simulating and rendering a visual depiction of a system, the operations comprising:
simulating the system, wherein the system includes at least one material, and wherein simulating the system comprises, for each of one or more iterations:
transferring system information including angular momentum information from particles to a grid on which dynamics are computed, and
transferring the system information including the angular momentum information and velocity information back from the grid to the particles on which at least velocity changes are computed,
wherein the transfer from the particles to the grid and the transfer back from the grid to the particles act as filters and conserve angular momentum; and
rendering for display one or more images depicting the system including the at least one material based, at least in part, on the simulation of the system.

### Claim 11 (depends on 10)

The computer-readable storage medium of claim 10, wherein:
the system information that is transferred from the particles to the grid and back from the grid to the particles further includes shearing information; and
the transfer from the particles to the grid and the transfer back from the grid to the particles also preserve shearing modes.

### Claim 12 (depends on 11)

The computer-readable storage medium of claim 11, wherein:
the dynamics computed on the grid include at least one of forces, collision responses, and incompressible projections;
at least the velocity changes and angular momentum changes are computed on the particles; and
the particles are advected.

### Claim 13 (depends on 11)

The computer-readable storage medium of claim 11, wherein the angular momentum and the shearing information are stored in a matrix.

### Claim 14 (depends on 11)

The computer-readable storage medium of claim 11, wherein:
the system includes a fluid; and
the system information is transferred from the particles to marker-and-cell (MAC) faces of the grid.

### Claim 15 (depends on 11)

The computer-readable storage medium of claim 11, wherein:
the system includes a solid; and
the one or more iterations are iterations in a material-point-method (MPM) loop.

### Claim 16 (depends on 11)

The computer-readable storage medium of claim 11, the operations further comprising, blending the simulation of the system with a particle-in-cell (PIC) simulation technique.

### Claim 17 (depends on 11)

The computer-readable storage medium of claim 11, wherein:
the system includes a plurality of materials; and
the simulation is a coupled simulation of the plurality of materials.

### Claim 18 (depends on 11)

The computer-readable storage medium of claim 11, wherein the dynamics computed on the grid include mesh-based Lagrangian forces.

### Claim 19

A system, comprising:
a processor; and
a memory, wherein the memory includes a program configured to perform operations for simulating and rendering a visual depiction of a system, the operations comprising:
simulating the system, wherein the system includes at least one material, and wherein simulating the system comprises, for each of one or more iterations:
transferring system information including angular momentum information from particles to a grid on which dynamics are computed; and
transferring the system information including the angular momentum information and velocity information back from the grid to the particles on which at least velocity changes are computed,
wherein the transfer from the particles to the grid and the transfer back from the grid to the particles act as filters and conserve angular momentum, and
rendering for display one or more images depicting the system including the at least one material based, at least in part, on the simulation of the system.

### Claim 20 (depends on 19)

The system of claim 19, wherein:
the system information that is transferred from the particles to the grid and back from the grid to the particles further includes shearing information; and
the transfer from the particles to the grid and the transfer back from the grid to the particles also preserve shearing modes.