A Material Point Method for Snow Simulation

Copyright of figures and other materials in the paper belongs original authors.

Presented by Ki-hoon Kim

2016.01. 19

Computer Graphics @ Korea University

Alexey Stomakhin et al.SIGGRAPH 2014

Ki-hoon Kim | 2016-01-25 | # 2Computer Graphics @ Korea University

• This handles difficult snow behaviors in a single solver.

• Contributions

We develop a semi-implicit Material Point Method(MPM)

• Designed to efficiently treat the wide range

This is the first time MPM has been used in graphics

Introduction

Ki-hoon Kim | 2016-01-25 | # 3Computer Graphics @ Korea University

• Geometric snow modeling

The bulk of graphics snow is devoted to modeling accumulation

• Granular materials

Traditionally, snow in graphics is thought of as a granular material

• Engineering modeling of snow & Elasto-plastic continuum

Elasto-plastic constitutive relation worked well

Related work

Ki-hoon Kim | 2016-01-25 | # 4Computer Graphics @ Korea University

• Snow dynamics modeling is difficult

Due to snow’s variability, usually stemming from environmental

Our model ignores root causes

• Strength of MPM

Relying on the continuum approximation

Avoiding the need to model every snow grain

Making self-collisions automatic

Better able to handle the dynamics of snow

Paper Overview - Why do they choose MPM?

Ki-hoon Kim | 2016-01-25 | # 5Computer Graphics @ Korea University

• In volume preservation, Grid is more effective than MPM

Snow is compressible

• In stiffness, FEM is more effective than MPM

Remeshing is required with extreme deformation

• MPM is almost ideal for plasticity

The inaccuracies of the deformation gradient accumulate as artificial plasticity

Paper Overview - Comparison with Other methods

Ki-hoon Kim | 2016-01-25 | # 6Computer Graphics @ Korea University

Paper Overview - What is Deformation gradient?

• Deformation Gradient 𝐹 =𝜕𝐱(𝐗)

𝜕𝐗

𝐗 is the undeformed reference configuration

𝐱(𝐗) is the deformed current configuration

• Examples

Referencehttp://www.continuummechanics.org

Ki-hoon Kim | 2016-01-25 | # 7Computer Graphics @ Korea University

• Mass Preservation

𝐷𝜌

𝐷𝑡= 0

• Momentum Preservation

𝜌𝐷𝐯

𝐷𝑡= 𝛁 ∙ 𝜎 + 𝜌𝐠

• Cauchy stress 𝜎 =1

𝐽

𝜕Ψ

𝜕𝐹𝐸𝐹𝐸

𝑇

Ψ is elasto-plastic potential energy density

𝐽 = det(𝐹) & 𝐹 = 𝐹𝐸𝐹𝑃

• If material is isotropic Newtonian fluid 𝜎𝑖𝑗 = −𝑝𝛿𝑖𝑗 + 𝜇(𝜕𝑣𝑖

𝜕𝑥𝑗+

𝜕𝑣𝑗

𝜕𝑥𝑖)

Material point method- Governing equation

Ki-hoon Kim | 2016-01-25 | # 8Computer Graphics @ Korea University

• Particle(material point) 𝑝 holds

Position 𝐱𝑝

Velocity 𝐯𝑝

Mass 𝑚𝑝

Deformation gradient 𝐹𝑝

• The Lagrangian treatment simplifies the discretization of

𝐷𝜌

𝐷𝑡

𝜌𝐷𝑣

𝐷𝑡

Material point method- Particles(Lagrangian)

Ki-hoon Kim | 2016-01-25 | # 9Computer Graphics @ Korea University

• Background grid node 𝑖 holds

Constant Position 𝐱𝑖

Velocity 𝐯𝑖

Mass 𝑚𝑖

• Grid is needed for the computation of stress-based force

𝛁 ∙ 𝜎 terms in the standard FEM

Using cubic B-splines kernel

𝑤𝑖𝑝(𝐱𝑖 , 𝐱𝑝, ℎ) = 𝑁(1

ℎ(𝑥𝑝 − 𝑥𝑖))𝑁(

1

ℎ(𝑦𝑝 − 𝑦𝑖))𝑁(

1

ℎ(𝑧𝑝 − 𝑧𝑖))

• ℎ is Kernel radius

Material point method- Background grid(Eulerian)

Ki-hoon Kim | 2016-01-25 | # 10Computer Graphics @ Korea University

Material point method- Full method overview

Ki-hoon Kim | 2016-01-25 | # 11Computer Graphics @ Korea University

• Rasterize particle data to the grid

Mass 𝑚𝑖𝑛 = Σ𝑝𝑚𝑝𝑤𝑖𝑝

𝑛

Velocity 𝐯𝑖𝑛 = Σ𝑝𝐯𝑝

𝑛𝑚𝑝𝑤𝑖𝑝𝑛 /𝑚𝑖

𝑛

• Compute particle volumes and densities

Only once

• Compute grid forces

Stress-based forces

• Update velocities on grid

• Grid-based body collisions

• Solve the linear system

• Update deformation gradient

• Update particle velocities and positions

Material point method- Full method overview

Ki-hoon Kim | 2016-01-25 | # 12Computer Graphics @ Korea University

• Rasterize particle data to the grid

Make velocity field

• Compute grid forces

MPM: Stress-based forces

FLIP: Pressure forces

• Update velocity field

𝐯𝑛+1 = 𝐯𝑛 + Δ𝑡𝑚−1𝐟

• Update particle velocity

Material point method- Method flow comparison with FLIP

Ki-hoon Kim | 2016-01-25 | # 13Computer Graphics @ Korea University

• Energy density function is

Ψ 𝐹𝐸 , 𝐹𝑃 = 𝜇 𝐹𝑃 ||𝐹𝐸 − 𝑅𝐸||𝐹2 +

𝜆 𝐹𝑃

2𝐽𝐸 − 1 2

𝐹𝐸 = 𝑅𝐸𝑆𝐸 by the polar decomposition

1st term of the right side : Shape based

2nd term of the right side : Volume based

• Functions of the 𝐹𝑃

𝜇 𝐹𝑃 = 𝜇0𝑒𝜉(1−𝐽𝑝)

𝜆 𝐹𝑃 = 𝜆0𝑒𝜉(1−𝐽𝑝)

𝜇0 and 𝜆0 are initial parameters

𝜉 is plastic hardening parameter

Constitutive model- Elasto–plastic energy density function

Ki-hoon Kim | 2016-01-25 | # 14Computer Graphics @ Korea University

• The total elastic potential energy can be expressed as

Φ = 𝛀𝟎 Ψ 𝐹𝐸 𝐗 , 𝐹𝑃 𝐗 𝑑𝐗 = Σ𝑝 𝑉𝑝0 Ψ𝑝 𝐹𝐸𝑝, 𝐹𝑃𝑝

• 𝛀𝟎 is the reference configuration

• Spatial discretization of the stress-based forces

−𝐟𝑖=𝜕Φ(𝐱)

𝜕𝐱𝑖= Σ𝑝𝑉𝑝

0 𝜕Ψ𝑝

𝜕𝐹𝐸𝑝𝐹𝐸𝑝

𝑇𝛻𝑤𝑖𝑝

• Expression in terms of the Cauchy stress and sequence index

𝐟𝑖𝑛 = −Σ𝑝𝑉𝑝

𝑛 𝜎𝑝𝛻𝑤𝑖𝑝

Cauchy stress 𝜎𝑝 =1

𝐽𝑝𝑛

𝜕Ψ𝑝

𝜕𝐹𝐸𝑝𝐹𝐸𝑝

𝑇

Occupied volume 𝑉𝑝𝑛 = 𝐽𝑝

𝑛𝑉𝑝0

Stress-based forces- Forces derivation from Total potential energy

Ki-hoon Kim | 2016-01-25 | # 15Computer Graphics @ Korea University

• Predicted grid node position

𝐱𝑖(𝐯𝑖) = 𝐱𝑖 + Δ𝑡𝐯𝑖

• Predicted Deformation Gradient

𝐹𝐸𝑝𝑛+1 𝐱 = 𝐼 + Σ𝑖 𝐱𝑖 − 𝐱𝑖 𝛻𝑤𝑖𝑝

𝑛 𝑇𝐹𝐸𝑝

𝑛

• Predicted grid forces

𝐟𝑖𝑛 = 𝐟𝑖 𝐱𝑖(𝟎)

𝐟𝑖𝑛+1=𝐟𝑖 𝐱𝑖(𝐯

𝑛+1 )

Stress-based forces- Predicted quantities

Ki-hoon Kim | 2016-01-25 | # 16Computer Graphics @ Korea University

• Grid velocity Update

𝐯𝑖𝑛+1 = 𝐯𝑖

𝑛 + Δ𝑡𝑚𝑖−1( 1 − 𝛽 𝐟𝑖

𝑛 + 𝛽𝐟𝑖𝑛+1)

• 𝛽 is choose between

0 : Explicit

0~1: Semi implicit

1 : Backward Euler

𝐯𝑖𝑛+1 ≈ 𝐯𝑖

𝑛 + Δ𝑡𝑚𝑖−1(𝐟𝑖

𝑛 + 𝛽Δ𝑡Σ𝑗𝜕𝐟𝑗

𝑛

𝜕 𝐱𝑗𝐯𝑗𝑛+1)

• 𝐟𝑖𝑛+1 = 𝐟𝑖 𝐱𝑖(𝐯

𝑛+1 ) = 𝐟𝑖 𝐱𝑖(𝟎) + 𝛿𝐟𝑖(𝐯𝑛+1)

• 𝛿𝐟𝑖(𝐯𝑛+1) ≈ Δ𝑡Σ𝑗

𝜕𝐟𝑗𝑛

𝜕 𝐱𝑗𝐯𝑗𝑛+1

Stress-based forces- Semi-implicit update(1)

Ki-hoon Kim | 2016-01-25 | # 17Computer Graphics @ Korea University

• The result of the previous page leads to linear system

Σ𝑗 𝐼𝛿𝑖𝑗 + 𝛽Δ𝑡2𝑚𝑖−1 𝜕2Φ𝑛

𝜕 𝐱𝑖𝜕 𝐱𝑗𝐯𝑗𝑛+1 = 𝐯𝑖

∗

𝐯𝑖∗ = 𝐯𝑖

𝑛 + Δ𝑡𝑚𝑖−1𝐟𝑖

𝑛

Using the conjugate residual method

Stress-based forces- Semi-implicit update(2)

𝐴 𝐱 = 𝐛

Ki-hoon Kim | 2016-01-25 | # 18Computer Graphics @ Korea University

• Step 1. From page 15(PPT 26) Temporarily defining

𝐹𝐸𝑝𝑛+1 = 𝐼 + Δ𝑡𝛻𝐯𝑝

𝑛+1 𝐹𝐸𝑝𝑛 , 𝐹𝑃𝑝

𝑛+1 = 𝐹𝑃𝑝𝑛

𝐹𝑝𝑛+1 = 𝐹𝐸𝑝

𝑛+1 𝐹𝑃𝑝𝑛+1

• All the changes get attributed to the elastic part

• Step 2. Clamping singular values of 𝐹𝐸𝑝𝑛+1 to the permitted range

𝐹𝐸𝑝𝑛+1 = 𝑈𝑝

Σ𝑝𝑉𝑝𝑇 , then Σp = clamp( Σ𝑝, [1 − 𝜃𝑐 , 1 + 𝜃𝑠])

• 𝜃𝑐 : Critical compression

• 𝜃𝑠 : Critical stretch

• Step 3. Push changes of the elastic part into the plastic part

𝐹𝐸𝑝𝑛+1 = 𝑈𝑝Σ𝑝𝑉𝑝

𝑇 and 𝐹𝐸𝑝𝑛+1 = 𝐹𝐸𝑝

𝑛+1 −1𝐹𝑝

𝑛+1

Deformation gradient update

Ki-hoon Kim | 2016-01-25 | # 19Computer Graphics @ Korea University

Results

Ki-hoon Kim | 2016-01-25 | # 20Computer Graphics @ Korea University

• Limitations

A lot of our parameters were tuned by hand

This neglects interactions with the air

Using adaptive sparse grids would save memory

• Conclusion

Presenting a constitutive model and simulation technique

Demonstrating a wide variety of snow behaviors

Discussion and Conclusion