interlinked sph pressure solvers for strong fluid-rigid...

Interlinked SPH Pressure Solvers for Strong Fluid-Rigid Coupling

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

Presented by Ki-hoon Kim

2019.10. 10

Computer Graphics @ Korea University

C. Gissler(University of Freiburg) et al.SIGGRAPH 2019

Ki-hoon Kim | 2019-11-12 | # 2Computer Graphics @ Korea University

• Introduction

• Method

• Result & Conclusion

Index

Introduction


• Particle-sampled solids are a popular basis for boundary handling

• In iteration solvers, contact forces are only applied after solver

• This concept causes substantial issues in the two-way coupling

Introduction


• Stabilize the handling of SPH fluid-rigid interfaces

• Fluid-rigid and rigid-rigid contacts are uniformly handled

With SPH Concept

• Our approach can be combined with iterative SPH solver

Contributions


• Particle-Based Fluid Simulation for Interactive ApplicationsM.Muller et al. / SCA 2003

Show the interaction of fluids and deformable objects

Basic of SPH Fluid Simulation on CG

Related Work

SPH


• Predictive-corrective Incompressible SPH(PCI-SPH)B. Solenthaler and R.Pajarola/SIGGRAPH 2009

Predict Pressure Force

Enforce 𝜌𝑛𝑒𝑥𝑡 = 𝜌0

• Implicit Incompressible SPH(IISPH)Markus Ihmsen et al./TVCG 2013

Predict Velocity Field

Enforce 𝛻 ∙ 𝐯𝑛𝑒𝑥𝑡 = 0

• Both methods adjust pressure Force.

Related Work

Iterative Pressure Solver


• Versatile Rigid-Fluid Coupling for Incompressible SPHAkinci et al. / SIGGRAPH 2012

Boundary particles for the solid boundaries

Extends to two-way fluid-solid coupling

Contacts force is based on SPH Pressure

• Direct Forcing for Lagrangian Rigid-Fluid CouplingBecker et al. / TVCG 2009

Predictor-corrector Scheme

Only enforce correction of particle position

Related Work

SPH-Rigid Coupling

Method


• 𝐅𝑟𝑓𝑟

and 𝐅𝑟𝑟𝑟 are only applied after the iterations

By using final pressure field

• 𝐯𝑟∗ is erroneous during the solver iterations.

Generic iterative pressure solver with the boundary handling

𝑙 : Iteration Step𝑝 : Pressure𝐯 : Velocity

Small 𝑓: FluidSmall 𝑟: Rigid Body

Upper *: Predicted

Algorithm 1


• Forces at the fluid-rigid interface are not only applied to fluid.

• Update the velocity 𝐯𝑟∗,𝑙 in each iteration 𝑙.

• Improved predicted velocities 𝐯𝑟∗,𝑙 influence

Pressure refinement 𝑝𝒇𝑙

Fluid velocity 𝐯𝒇∗,𝑙

Iterative pressure solver with interleaved fluid-rigid update

Algorithm 2


• We integrated the forces proposed by Akinci et al[2012]. and Band et al. [2018b]

Fluid-Rigid interface forces

Algorithm 2


• 𝐅𝑟𝑟𝑟 based on artificial density deviations

𝐅𝑟𝑟𝑟 is contact force

• If there is a contact, we calculate a density 𝜌𝑟 > 𝜌𝑟0

𝜌𝑟 is density of rigid particle

𝜌𝑟0 is rest density of rigid particle

• Determine contact forces such that 𝜌𝑟 = 𝜌𝑟0 for all rigid particles

Contact force is artificial pressure force: −𝛻 ∙ 𝑝𝑟 𝑝𝑟 is artificial pressure

Rigid body solver


• Proposed system derived from the continuity equation𝐷𝜌𝑟𝐷𝑡

= −𝜌𝑟𝛻 ∙ 𝐯𝑟

• Discretizing time with

A backward difference

Constraint that the density at 𝑡 + Δ𝑡

• 𝜌𝑟𝑛𝑒𝑥𝑡 = 𝜌𝑟

0

𝜌𝑟0−𝜌𝑟

Δ𝑡= −𝜌𝑟𝛻 ∙ 𝐯𝑟

𝑛𝑒𝑥𝑡 (Eq.1)

𝐯𝑟𝑛𝑒𝑥𝑡 is desired(Unknown) velocity at 𝑡 + Δ𝑡to obtain desired density 𝜌𝑟

𝑛𝑒𝑥𝑡 = 𝜌𝑟0

System of Rigid body Solver

Continuity Equation


• 𝐯𝑟𝑛𝑒𝑥𝑡 can be written as

𝐯𝑟𝑛𝑒𝑥𝑡 = 𝐯𝑹

𝑛𝑒𝑥𝑡 +𝛚𝑅𝑛𝑒𝑥𝑡 × 𝐫𝑟

𝑛𝑒𝑥𝑡 (Eq. 2)

𝑅 is respective rigid body

𝐯𝑅 is linear velocity of 𝑅

𝛚𝑅 is angular velocity of 𝑅

𝐫𝑟 is vector from center of mass of 𝑅 to rigid particle


Velocity Decomposition of Rigid particles


• Using integration and Newton’s 2nd law, write 𝐯𝑹𝑛𝑒𝑥𝑡

𝐯𝑹𝑛𝑒𝑥𝑡 = 𝐯𝑅 + Δ𝑡

1

𝑀𝑅(𝐅𝑅 + Σ𝑘𝐅𝑘

𝑟𝑟) (Eq. 3)

𝑘 is rigid particle of 𝑅

𝑀𝑅 is mass of 𝑅

𝐅𝑅 : Comprises all momentum-changing except 𝐅𝒓𝑟𝑟

• Fluid-Rigid Interface force

• Gravity

• Etc.

• The angular velocity 𝝎𝑹𝑛𝑒𝑥𝑡 in (Eq. 2) can be written as

𝝎𝑹𝑛𝑒𝑥𝑡 = 𝝎𝑅 + Δ𝑡𝐈𝑅

−1(𝛕𝑅 + 𝐈𝑅𝛚𝑅 ×𝛚𝑅 + Σ𝑘𝐫𝑘 × 𝐅𝑘𝑟𝑟) (Eq. 4)

𝐈𝑅 is inertia tensor of 𝑅

𝛕𝑅 is all source torques except contact force


Velocity Integration of Rigid body


• Using (Eq. 2) to (Eq. 4), we write (Eq. 1) as

Unknown velocities are replaced by unknown rigid-rigid contact forces

• 𝐯𝑟𝑛𝑒𝑥𝑡 → 𝐅𝑘

𝑟𝑟


Rewrite Continuity Equation

(Eq. 5)


• Approximation 𝐫𝒓𝑛𝑒𝑥𝑡 = 𝐫𝑟

• The source terms 𝑠𝑟 =𝜌𝑟0−𝜌𝑟

Δ𝑡+ 𝜌𝑟𝛻 ∙ 𝐯𝑟

𝑠

𝐯𝑟𝑠 =

𝐅𝑅 and 𝛕𝑅 include the fluid-rigid interface forces and all forces

They do not include the rigid-rigid contact forces.

We assume to be constant.


Assumption to Linearize


• We move 𝑠𝑟 to left-hand side and leave the rigid-rigid contact force terms on the right-hand side.

• (Eq. 6) can be written as

𝐊𝑟𝑘 =1

𝑀𝑅𝕀 − 𝐫𝑟𝐼𝑅

−1 𝐫𝑘

𝕀 is the identity matrix

𝐫𝑟 is the skew-symmetric cross-product matrix of 𝐫𝑟• Ex) 𝐱 × 𝐲 = �ු�𝐲 [𝐱, 𝐲 are column vector, �ු� is matrix]

𝐊 is referred to as collision matrix and has been proposed before


Linearize System(b=Ax)

(Eq. 6)

(Eq. 7)


• We model the contact forces as pressure forces.𝐅𝑘𝑟𝑟 = −𝑉𝑘𝛻𝑝𝑘

𝑉𝑘 is artificial volume

𝑝𝑘 is unknown artificial pressure

• This pressure is only used for rigid-rigid contact

• (Eq. 7) can be written as


Artificial Pressure

(Eq. 8)


• We use SPH Interpolation

𝐴𝑖 = Σ𝑗𝑚𝑗

𝜌𝑗𝐴𝑗𝑊𝑖𝑗

𝐴 is arbitrary scalar quantity

𝑚 is mass

𝜌 is density

𝑊 is Kernel function

• Spatial derivatives

Gradient: 𝛻𝐴𝑖 = 𝜌𝑖Σ𝑗𝑚𝑗(𝐴𝑖

𝜌𝑖2 +

𝐴𝑗

𝜌𝑗2)𝛻𝑊𝑖𝑗

Divergence: 𝛁 ∙ 𝐀𝑖 =1

𝜌𝑖Σ𝑗𝑚𝑗(𝐀𝑖 − 𝐀𝑗)𝛻𝑊𝑖𝑗

Introduced in “Smoothed particle hydrodynamics and magnetohydrodynamics”Daniel J. Price /Journal of Computational Physics 2012

Implementation

SPH


• Artificial Rest Volume

𝑉0𝑟 =

𝛾

Σ𝑘𝑊𝑟𝑘

𝛾 is user parameter. We use 0.7

• Artificial Rest Density

Artificial Rest Density is irrelevant for the simulation.

Implementation

Artificial Rest(Initial) Values


• Artificial Density

𝜌𝑟 = 𝜌𝑟0𝑉𝑟

0Σ𝑘𝑊𝑟𝑘 or Σ𝑘𝜌𝑘0𝑉𝑘

0𝑊𝑟𝑘

If other objects are in contact, we get a density deviation

• 𝜌𝑟 > 𝜌𝑟0

• Artificial Volume

𝑉𝑟 =𝜌𝑟0𝑉𝑟

0

𝜌𝑟

Implementation

Compute Artificial Values


• 𝑠𝑟 =𝜌𝑟0−𝜌𝑟

Δ𝑡+ 𝜌𝑟𝛻 ∙ 𝐯𝑟

𝑠

𝜌𝑟 has been computed.

• 𝐯𝑟𝑠 =

We can compute this term directly.

• Divergence term

Implementation

Compute 𝒔𝒓(Left-hand Side of (Eq. 8))

(Eq. 9)


• Compute 𝜌𝑟𝛁 ∙ (Δ𝑡Σ𝑘𝑉𝑘𝐊𝑟𝑘𝛁𝑝𝑘)

• Step 1-1. Compute Gradient Pressure(Pressure Force)

• Step 2: Accumulates Velocity change per Rigid body

• Step 3: Compute Velocity change per Rigid Particle𝐯𝑟𝑟𝑟 = 𝐯𝑅

𝑟𝑟 +𝝎𝑟𝑟𝑟 × 𝐫𝑟

= −Δ𝑡Σ𝑘𝑉𝑘𝐊𝑟𝑘𝛁𝑝𝑘

• Step 4: Compute Divergence Using 𝐯𝑟𝑟𝑟 = −Δ𝑡Σ𝑘𝑉𝑘𝐊𝑟𝑘𝛁𝑝𝑘

Implementation

Compute (Right-hand Side of (Eq. 8))


• We use a relaxed Jacobi solver

𝛽𝑟𝑅𝐽

is relaxation coefficient

• 𝛽𝑟𝑅𝐽

=0.5

𝑛𝑢𝑚_𝑐𝑜𝑛𝑡𝑎𝑐𝑡𝑠

𝑏𝑟 is diagonal elements of linear system

Implementation

Solver step

(Eq. 10)


Combining fluid and rigid body solver

Algorithm 3

Result & Conclusion


Result


Performance Comparison(Akinci)

Part of Table2


Performance Comparison(with bullet)

Table 3


Stability Comparison(with bullet)

Table 3


• The accuracy of the contact handling is related to the size of a rigid body and a fluid particle.

• The accuracy of the contact handling is coupled to the accuracy of the fluid simulation.

• Performance gain is reduced in

Scenarios with smaller density ratios

Calm scenes with slowly moving particles

Limitations


• We have introduced a strong two-way coupling method

• The technique shares characteristics of a unified approach by solving pressure

• We solve a monolithic system, we show that it is a flexible and useful extension.

• In Future

Investigate the applicability of our concept to alternative Lagrangian fluid formulations.

Conclusion

interlinked sph pressure solvers for strong fluid-rigid...

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