snow as a continuumplasticity: von-mises ⌧ 1 ⌧ 2 p q • yield region: cylinder • uncapped /...
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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)
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• 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
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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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• 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
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Thank you!