Learning mesh-based simulation with Graph Networks

FlagDynamic

cloth dynamics w/ self-collisions

2767 nodes (avg.), adaptively remeshed

250 time steps

ground truth simulator: ArcSim

SphereDynamic

cloth dynamics w/ self+obstacle collisions

1373 nodes (avg.), adaptively remeshed

500 time steps

ground truth simulator: ArcSim

DeformingPlate

structural mechanics
color: von-Mises Stress

1271 nodes (avg.)

400 time steps

ground truth simulator: COMSOL

CylinderFlow

incompressible fluid dynamics
color: velocity in x-direction

1885 nodes (avg.)

600 time steps

ground truth simulator: COMSOL

Airfoil

compressible fluid dynamics
color: velocity in x-direction

5233 nodes

600 time steps

ground truth simulator: SU2

Airfoil: faster & steeper

Model trained on dataset Airfoil

training range:

• angle of attack: [-25, 25]

• mach number: [0.2, 0.7]

extrapolation test range:

1. steeper: angle of attack [-35, 35]

2. faster: Mach number [0.7, 0.9]

TL;DR:
We retain plausible behavior when moderately extrapolating outside the training bracket.

FishFlag (鯉のぼり)

Model trained on dataset FlagDynamic

self-collisions, learned remeshing

TL;DR:
We obtain accurate results when testing on shapes unseen in training; both forward model and sizing field model generalize to new scenes.

HippoFlag 🦛

Model trained on dataset FlagDynamic
(wind speed varied, but constant within trajectories)

self-collisions, learned remeshing

TL;DR:
We can smoothly vary wind direction and speed at test time, even though varying wind speeds/directions were never observed in training.

WindSock

Model trained on dataset FlagDynamic

self-collisions, learned remeshing

• training set: 2.7k nodes,
  simple rectangular flag

• inference: 20k nodes,
  cylinder with tassels

TL;DR:
We can apply a model trained on a simple setup to significantly bigger and more complex scenarios at test time.

Dynamic Remeshing

Dataset FlagDynamic

Comparison: MeshGraphNets with

• ground truth meshing

• learned remeshing

• learned remeshing w/ estimated targets

TL;DR:
We obtain good results with all remeshing variants.

Comparison: GNS

Dataset FlagSimple

• fixed, regular mesh, 1579 nodes

Comparison:

• GNS w/ 1 step of history

• GNS w/ 5 steps of history

• GNS w/ rest positions

• MeshGraphNets

TL;DR:
GNS fails on cloth, GNS+mesh-pos works for regular meshes, but is more prone to show artifacts (and does not work for irregular meshes).

Comparison: GCN-MLP

Dataset Airfoil

• fully dynamic prediction

Comparison:

• GCN-MLP (GCN with with improved architecture, best variant we tried)

• MeshGraphNets

TL;DR:
Even improved GCN variants fail to properly predict the dynamics on harder datasets.

Comparison: UNet

Dataset CylinderFlow
resampled on 128x128 grid

Comparison:

• UNet from [Thuerey et al. 2020]

• MeshGraphNets

TL;DR:
UNet can model overall dynamics well, but are less accurate in the wake regions.

Comparison: UNet

Dataset Airfoil
resampled on 128x128 grid
(here: showing the 107x107 center region)

Comparison:

• UNet from [Thuerey et al. 2020]

• MeshGraphNets

TL;DR:
UNet can model overall dynamics well, but are less accurate around the airfoil, and more prone to show artifacts.

Comparison: GCN (SteadyState)

Dataset AirfoilSteady

• single-step, steady-state prediction

Comparison:

• GCN-baseline from
  [Belbute-Peres et al. 2020]

• MeshGraphNets

TL;DR:
All methods show good results for simple steady-state prediction tasks.

Long rollout

Dataset FlagSimple

• trained on trajectories with 400 steps

• rolled out for 40000 steps

TL;DR:
Our models remain stable even for extremely long rollouts