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YUN (RAYMOND) FEI
QI GUO
RUNDONG WU
LI HUANG
MING GAO
TENCENT GAME AI RESEARCH CENTER
REVISITING INTEGRATION IN
THE MATERIAL POINT METHODA SCHEME FOR EASIER SEPARATION AND LESS DISSIPATION
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MOTIVATION
WHY WE NEED A NEW INTEGRATION SCHEME?
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A CASE IN PRODUCTION
β’ 93,014 strands
β’ 7,179,350 DoFs
β’ Required simulation cost
β ~ 6s per frame (based on 60
frame per simulated second)
β ~ 1h in total (10s simulation)
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Simulate with GPU-MPM?
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PRIOR WORK OF MPM WITH LAGRANGIAN FORCES
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[Jiang et al. 2017][Jiang et al. 2015] [Guo et al. 2018] [Han et al. 2019]
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AN MPM HAIR SIMULATOR
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Init time stepping
Particle to grid mapping
Apply stress on grid
Grid to particle mapping
Plasticity projection
Add Lagrangianimpulses solved from discrete elastic rods
(DER)
Our MPM Hair SimulatorMPM for contact & friction
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HAIRS SIMULATED WITH MLS-MPM + FLIP
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HAIRS SIMULATED WITH MLS-MPM + FLIP
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FLIP INTEGRATOR[BRACKBILL & RUPPEL 1986]
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Positions they should be
Positions they end up with
πππ+1 = ππ
π + Ξπ‘
π
π€ππππβ Interpolated from
current nodal velocity
P2G G2P
Position update during grid-to-particle (G2P) transfer:
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πππ+1 =
π
π€ππππβ
PIC/APIC/POLYPIC INTEGRATORS
PIC grid-to-particle:[Harlow et al. 1955]
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APIC grid-to-particle: (moving-least-square version)[Jiang et al. 2015, Hu et al. 2018]
PolyPIC grid-to-particle:[Fu et al. 2017]
VelocityUpdate
PositionUpdate
πππ+1 =
π
π€ππππβ
πππ+1 =
π
π€ππππβ π₯π
π β π₯ππ π
πππ β1
πππ+1 =
π
π€ππππβ
πππ+1 = argmin
π
π
ππππ ππ
β β
π
πΌ
π π π₯π β π₯ππ ππΌππππΌ
π+1
2
πππ+1 = ππ
π + Ξπ‘
π
π€ππππβ ππ
π+1 = πππ + Ξπ‘
π
π€ππππβ ππ
π+1 = πππ + Ξπ‘
π
π€ππππβ
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THE CONTINUUM ASSUMPTION
β’ FLIP/xPIC assumes material continuous in the
entire domain.
β’ Problem:
β Particle position is always interpolated from
deformed nodal positions
β If any movement is not capturable by grid, it
would be neither capturable by particles.
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Continuous deformation Ξ¨
ππ
ππ‘=πΞ¨ π, π‘
ππ‘β
π
π€ππππβ
πππ+1 =
π
π€ππππβ
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DISCREPANCY BETWEEN RENDERING, EXPECTATION AND SIMULATION
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Sand Hairs Clothes
Expected behavior: separate immediately when pulled apart
Rendering: discrete & separated elements
Simulation: continuum that trap particles until they have no shared weights on a node
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NFLIP
NATURALLY-MODIFIED FLIP
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NFLIP: USE PARTICLE VELOCITY FOR ADVECTION
FLIP grid-to-particle:
πππ+1 =
π
π€ππππβ + πΌ ππ
π β
π
π€πππππ
πππ+1 = ππ
π + Ξπ‘
π
π€ππππβ
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NFLIP grid-to-particle:
πππ+1 =
π
π€ππππβ + πΌ ππ
π β
π
π€πππππ
πππ+1 = ππ
π + Ξπ‘πππ+1
Completely ignore continuity.Assume material continuous everywhere.
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NFLIP: USE PARTICLE VELOCITY FOR ADVECTION
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dissipated, difficult to separate
slow to separate, then accelerated
dissipated, easy to separate
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IGNORING CONTINUITY LEADS TO ERROR
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Ignoring continuity
Wrong deformation gradient
Wrong stress
Wrong contacts and frictions
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SFLIP
A SEPARABLE FLIP
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A SEPARABLE FLIP
FLIP grid-to-particle:
πππ+1 =
π
π€ππππβ + πΌ ππ
π β
π
π€πππππ
πππ+1 = ππ
π + Ξπ‘
π
π€ππππβ
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SFLIP grid-to-particle:
πππ+1 =
π
π€ππππβ + πΌ ππ
π β
π
π€πππππ
πππ+1 = ππ
π + Ξπ‘
π
π€ππππβ + π½ππΌ ππ
π β
π
π€πππππ
π½π: the trap-breaking ratio
FLIP: βπ, π½π = 0
NFLIP: βπ, π½π = 1
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THE TRAP-BREAKING RATIO π·p
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π½p = π½min, mostly use interpolated nodal velocity to advect this particle
Un
se
pa
rate
d
ApplyForce
Se
pa
rate
d
π½p = π½max, mostly use particle velocity to advect this particle
P2G
ApplyForce
(zero when separated)
P2G
G2P
G2P
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DETERMINE SEPARATION THROUGH VOLUME RATIO
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π1, π2: stress
π½π: volume ratio
ππ: pressure
π : bulk modulus
π½ππ: critical volume ratio
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DETERMINE SEPARATION THROUGH VOLUME RATIO
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π1, π2: stress
π½π: volume ratio
ππ: pressure
π : bulk modulus
π½ππ: critical volume ratio
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A SEPARABLE FLIP - SUMMARIZED
SFLIP grid-to-particle:
πππ+1 =
π
π€ππππβ + πΌ ππ
π β
π
π€πππππ
πππ+1 = ππ
π + Ξπ‘
π
π€ππππβ + π½ππΌ ππ
π β
π
π€πππππ
π½π = ΰ΅
0, in boundary
π½min, π½π < π½ππ
π½max, π½π β₯ π½ππ
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SFLIP
DIDACTIC EXAMPLES
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TWO PERFECTLY SMOOTH FINITE ELEMENT SQUARESNO FORCE APPLIED FROM ONE SQUARE TO ANOTHER
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Ground Truth
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2D FIBERS
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2D HOURGLASS
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2D CLAY FRACTURE
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ASFLIP
AFFINE-AUGMENTED SEPARABLE FLIP
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NAΓVE MODIFICATION TO IMPROVE APIC
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APIC grid-to-particle transfer:
πππ+1 =
π
π€ππππβ
πππ+1 =
π
π€ππππβ π₯π
π β π₯ππ π
πππ β1
πππ+1 = ππ
π + Ξπ‘
π
π€ππππβ
Particle-to-grid transfer:
πππππ
π =
π
π€ππππ πππ + ππ
π πππ β ππ
π
ASPIC grid-to-particle transfer:
πππ+1 =
π
π€ππππβ
πππ+1 =
π
π€ππππβ π₯π
π β π₯ππ π
πππ β1
πππ+1 = ππ
π + Ξπ‘
π
π€ππππβ + π½ππΌ ππ
π β
π
π€πππππ
π½π = ΰ΅
0, in boundary or source
π½min, π½π < π½ππ
π½max, π½π β₯ π½ππ
Add positional adjustment
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Assume zero force applied (i.e., ππβ = ππ
π):
πππ+1 = ππ
π + Ξπ‘
π
π€πππ ππ
β + π½ππΌ πππ β
π
π€πππ ππ
π
πππ+1 =
π
π€πππ ππ
β
ASPICAPIC + POSITIONAL ADJUSTMENT
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ASPIC doesnβt work.
πππ+1 = ππ
π + Ξπ‘
π
1 β π½ππΌ π€πππ ππ
π + π½ππΌπ€πππβ1ππ
πβ1
Nodal velocity at π‘πβ1Nodal velocity at π‘π
Only depends on nodal velocity
Root cause: PIC-style velocity update never preserves particle velocity from previous steps
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AFFINE-AUGMENTED FLIP (AFLIP)
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AFLIP grid-to-particle transfer (G2P):
πππ+1 =
π
π€ππππβ + πΌ ππ
π β
π
π€πππππ
πππ+1 =
π
π€ππππβ π₯π
π β π₯ππ π
πππ β1
πππ+1 = ππ
π + Ξπ‘
π
π€ππππβ
AFLIP particle-to-grid transfer (P2G):
πππππ
π =
π
π€ππππ πππ + ππ
π πππ β ππ
π Preserves affine momentum during P2G
Preserves high-frequency momentum during G2P(damped through Ξ± < 1)
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2D WEAKLY-COMPRESSIBLE LIQUID
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ASFLIPAFFINE-AUGMENTED SEPARABLE FLUID-IMPLICIT PARTICLE
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ASFLIP grid-to-particle transfer (G2P):
πππ+1 =
π
π€ππππβ + πΌ ππ
π β
π
π€πππππ
πππ+1 =
π
π€ππππβ π₯π
π β π₯ππ π
πππ β1
πππ+1 = ππ
π + Ξπ‘
π
π€ππππβ + π½ππΌ ππ
π β
π
π€πππππ
π½π = ΰ΅
0, in boundary or source
π½min, π½π < π½ππ
π½max, π½π β₯ π½ππ
ASFLIP particle-to-grid transfer (P2G):
πππππ
π =
π
π€ππππ πππ + ππ
π πππ β ππ
π Preserves affine momentum during P2G
Preserves high-frequency momentum during G2P(damped through Ξ± < 1)
Improves upon cases where continuum assumption no longer applies, preserving sub-grid movements
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2D FINITE ELEMENT SQUARES
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Ground Truth
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MORE EXAMPLES
WATER, SAND, HAIRS AND CLOTHES
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Grid-resolution dependent wrinkling[Guo et al. 2018]
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SUMMARY
IntegratorsHigh-
frequency motion
Undamped affine motion
Handle Boundary Condition
Easy Separation
Energetic Level
PIC β β β β β
APIC β β β β ββ
FLIP β β β β ββ
NFLIP β β β β ββ
SFLIP β β β β ββ
AFLIP β β β β βββ
ASFLIP β β β β βββ
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FUTURE WORK
β’ Numerical volume gain
β’ Simulators based on Chorinβs projection
β’ Determine separation through fracture mechanics
β’ More accurate (self-) contact resolution
β’ Coupling with principled, discrete physics
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TENCENT GAME AI RESEARCH CENTER
PRE-PRINT, VIDEO, SOURCE CODE
http://yunfei.work/asflip/