Snow as a continuum

Chenfanfu Jiang

⌦0 ⌦t

x = �(X, t)

X = ��1(x, t)

Johan Gaume

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Talk overview - coming from a graphics perspective

The start of snow: a purely computer graphics effort

Second law of thermodynamics teaches a lesson

Constitutive modeling: learn and improve

Continuum damage: from snow to glacier

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Talk overview - coming from a graphics perspective

The start of snow: a purely computer graphics effort

Second law of thermodynamics teaches a lesson

Constitutive modeling: learn and improve

Continuum damage: from snow to glacier

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The motivation

``Making a new movie… with a lot of snow in it.. need to look real’’

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The question

The first question: how do we represent snow (discretely)?

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Height field: snow is a surfaceRobert W. Sumner, James F. O'Brien, and Jessica K. Hodgins. "Animating Sand, Mud, and Snow". In Proceedings of Graphics Interface 98, pages 125–132, June 1998.

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Rigid bodies

Height field + Rigid bodies + Passive particles

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Finite elements and discrete elements: snow is a solid

• Challenges • Contact • Friction • Scalability

DEM snow: seems to be popular in computational mechanics

Polyhedral FEM

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Eulerian view: snow is a fluid flow

Challenges: small scale dynamics and fracture propagation, and rendering

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Fluid-like stuff without meshing: particles“Molecular” add-on for blender: springs that can break when too stretched

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Particles with real strain-stress relationship

"A material point method for snow simulation." Stomakhin, Alexey, Craig Schroeder, Lawrence Chai, Joseph Teran, and Andrew Selle. SIGGRAPH 2013

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MPM is hybrid Lagrangian/Eulerian

• Grid handles: the Galerkin DOFs, discretization, function space, …. • The transfer/embedding handles: coupling, quadrature rule, non-

penetration, topology change, … • Each individual particle handles: constitutive modeling - the physics

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Developments

A Temporally Adaptive Material Point Method with Regional Time Stepping Yu Fang, Yuanming Hu, Shi-Min Hu, Chenfanfu Jiang, ACM SIGGRAPH / Eurographics Symposium on Computer Animation (SCA 2018)

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More developments• More accurate advection

• The Affine Particle-In-Cell Method, Chenfanfu Jiang, Craig Schroeder, Andrew Selle, Joseph Teran, Alexey Stomakhin, ACM Transactions on Graphics (SIGGRAPH 2015)

• An Angular Momentum Conserving Affine-Particle-In-Cell Method, Chenfanfu Jiang, Craig Schroeder, Joseph Teran, Journal of Computational Physics, 338(1), pp. 137-164, 2017 (JCP 2017)

• High performance integrator • Optimization Integrator for Large Time Steps ,Theodore Gast, Craig Schroeder, Alexey

Stomakhin, Chenfanfu Jiang, Joseph Teran IEEE Transactions on Visualization and Computer Graphics (TVCG 2015)

• Spatial adaptivity • An Adaptive Generalized Interpolation Material Point Method for Simulating Elastoplastic

Materials, Ming Gao, Andre Pradhana, Chenfanfu Jiang, Eftychios Sifakis, ACM Trans. Graph. 36, 6, Article 223, (SIGGRAPH Asia 2017)

• Coupling with rigid bodies • A Moving Least Squares Material Point Method with Displacement Discontinuity and Two-

Way Rigid Body Coupling ,Yuanming Hu, Yu Fang, Ziheng Ge, Ziyin Qu, Yixin Zhu, Andre Pradhana, Chenfanfu Jiang, (SIGGRAPH 2018)

• …

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• Mathematical models, numerical formulations, and performance…

MPM resources

SIGGRAPH 19 Course http://mpm.graphics

SIGGRAPH 16 Course http://mpm.graphics

The material point method for simulating continuum materials

Chenfanfu Jiang, Craig Schroeder, Joseph Teran, Alexey Stomakhin, and Andrew Selle

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Finite strain hyperelasticity

Deformation gradient

F(X, t) =@�

@X

LARGE deformation

⌦0 ⌦t

x = �(X, t)

X = ��1(x, t)

• The existence of an energy density function is convenient! • The advantages of hyperelasticity: good models, and good solvers

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Graphics has favored hyperelasticity (F) =

µ

2

�tr(FFT )� 3

�� µ log(J) +

�

2log2(J)

P =@

@F= µ(F� FT ) + � log(J)F�T � =

1

JPFT

[Stomakhin et al. 12] Energetically Consistent Invertible Elasticity [Xu et al. 15] Nonlinear Material Design Using Principal Stretches [Smith et al. 18] Stable Neo-hookean Flesh Simulation …… [Bouaziz et al. 14] Projective Dynamics: Fusing Constraint Projections for Fast Simulation [Overby et al. 17] ADMM ⊇ Projective Dynamics: Fast Simulation of Hyperelastic Models with Dynamic Constraints [Liu et al. 17] Quasi-Newton Methods for Real-time Simulation of Hyperelastic Materials

⇢Dv

Dt= r · ��� + ⇢g.

Conservation of momentum

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Collision is automatic in MPM

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[Gao18] GPU Optimization of Material Point Methods, Ming Gao*, Xinlei Wang*, Kui Wu* (*joint), Andre Pradhana, Eftychios Sifakis, Cem Yuksel, Chenfanfu Jiang, SIGGRAPH Asia 2018

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Need for permanent deformation

FE = U⌃VT �i 2 [1� 0.01, 1 + 0.01]

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Need for permanent deformation

FE = U⌃VT �i 2 [1� 0.01, 1 + 0.01]

elasticity elastoplastic

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Hardening and softening

E = E0e�⇠Jp

elastoplastic withhardening

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Comparision

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The weirdness

Popping particles, and strange instabilities

video from shutterstock

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Energy rate and second law of thermodynamics

The hardening variable depends on the history of snow.

By looking at the rate of plastic deformation, we can prove that if Jp>1 and rate of Jp is negative, then total energy may increase.

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Elastoplasticity and flow rule

Principle of Maximum Dissipation

Plastic volume conservation

Associative flow rule

Non-associative flow rule

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Plasticity: von-Mises

⌧1

⌧2

p

q

• Yield region: cylinder • Uncapped / capped • Hardening: radius (yield stress)

• Commonly describing metal bending

[Fang19] Silly Rubber: An Implicit Material Point Method for Simulating Non-equilibrated Viscoelastic and Elastoplastic Solids ,Yu Fang, Minchen Li, Ming Gao, Chenfanfu Jiang, SIGGRAPH 2019

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Plasticity: Drucker-Prager

⌧1

⌧2

p

q

• Yield region: cone • Uncapped / capped • Hardening: internal friction angle

• Commonly describing granular media (dry, wet): sand, soil …

[Hu18]

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Soil: Modified Cam-Clay (MCC)

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Parametrization

MCC

Ours

Beta: cohesion M: friction p0: pre-consolidation

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Cohesion and friction

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Hardening and packing

video from shutterstock

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Hardening and packing

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Character

Snowball—big!

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17 million particles, 100s/frame on Gtx1080

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Plasticity: Modified Cam-Clay (MCC)

Johan

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Plasticity: Modified Cam-Clay (MCC)

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Animation and VFX industry

[Gaume18]

http://www.jixiefx.com/

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• Sharp interface • Solid-fluid coupling • Porous media

Glacier calving and more general fracture video from shutterstock

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Continuum damage mechanics

[Wolper19] CD-MPM: Continuum Damage Material Point Methods for Dynamic Fracture Animation, Joshuah Wolper, Yu Fang, Minchen Li, Jiecong Lu, Ming Gao, Chenfanfu Jiang, SIGGRAPH 2019

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Acknowledgements

• Students and postdocs • Joshuah Wolper, Yu Fang, Minchen Li, Ziyin Qu, Xinlei Wang, Yuxing

Qiu, Jiecong Lu, Yue Li, Ming Gao, Andre Pradhana, …

• Collaborators • Xinxin Zhang, Danny Kaufman, Timothy Langlois, Eftychios Sifakis,

Cem Yuksel, Baoquan Chen, Shi-Min Hu, Min Tang, Song-Chun Zhu, Alexey Stomakhin, Ken Museth, Yixin Zhu, Kun Wu, Grant Kot, Johan Gaume, Stuart Slattery, Doug Kothe, …

• Funding and support • NSF, DOE, Adobe, NVidia, SideFX

Thank you!