an implicit viscosity formulation for sph...

An Implicit Viscosity Formulation for SPH Fluids

Andreas Peer et al.SIGGRAPH 2015

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

Presented by Min-ju Park

2017.02.22

Computer Graphics @ Korea University

Min-ju Park | 2017. 02. 22 | # 2Computer Graphics @ Korea University

Abstract

• We present a novel implicit formulation for highly viscous fluids

• Compare to explicit method

more efficient

handles a larger range of viscosities

• Compare to existing implicit method

our approach reconstructs the velocity field from a target velocity gradient

• encodes a desired shear-rate damping

• preserves the velocity divergence

• ensures that pressure and viscosity computation do not interfere


1 Introduction

• Explicit method

• [Modeling low reynolds number incompressible flows using SPH.] (Morris et al., Journal of computational physics 136 ‘97)

• Implicit method

• [Accurate viscous free surfaces for buckling, coiling, and rotating liquids] (Batty et al., SIGGRAPH ‘08)

competing constraints might be neglected in explicit approach, while in implicit, this issue has to be addressed.

Explicit Implicit

𝑦 𝑡 + Δ𝑡 = 𝑓 𝑦 𝑡 𝑦 𝑡 + Δ𝑡

= 𝑓 𝑦 𝑡 + Δ𝑡 + 𝑔(𝑦 𝑡 )

Small timestep Large timestep


1 Introduction

contribution

• This paper proposes a novel implicit formulation for highly viscous SPH fluids

• that addresses the interference of pressure and viscosity computation

• target velocity gradient encode the desired viscosity and preserves arbitrary velocity divergences

• SPH pressure solver introduce some divergence to the velocity field to counteract density deviations

• e.g. IISPH in [Implicit incompressible SPH] (Ihmsen et al., TVCG ‘14)

• Our formulation preserves density corrections that have been computed in the preceding pressure projection.

• This also means that only one pressure projection step is required


2 Related work

Lagrangian techniques

• [Animating lava flows] (Stora et al., GI ‘99) and [Modeling and rendering viscous liquids] (Steele et al., CASA ‘04)

• these approximation improve the stability of low viscous SPH, but are less efficient for highly viscous fluids

• prone to overshooting

• Use an implicit formulation

• it allows for large timestep, so overshooting cannot occur

• Correctly account for desired velocity divergence

• existing formulation assume a divergence-free velocity field

• Our novel formulation bases on target velocity gradient

• does not interfere with the pressure solver

• dose not require a final pressure solver because it possibly perturbs the result of the viscosity solver


2 Related work

Eulerian techniques

• Many Eulerian approaches require some special treatment of free surface to allow for rotational fluid movement or only work on divergence-free velocity field

• [Accurate viscous free surfaces for buckling, coiling, and rotating liquids] (Batty et al., SIGGRAPH ‘08)

• [Multiple interacting liquids] (Losasso et al., ACM TOG ‘06)

• Our Lagragian approach does not require any special treatment and only work on divergence-free

• compute a desired velocity gradient

• does not affect rotation rate

• preserves divergence


2 Related work

Viscosity in multiphase fluids

• By averaging the viscosity constants, handling the interfaces is done in Lagrangian approaches

• [Particle-based fluid-fluid interaction] (Muller et al., SIGGRAPH ‘05)

• In Eulerian, an method is proposed to extrapolate ghost velocities across the interface to counteract distortions in the velocity gradient caused by variable viscosities

• [Discontinuous fluids] (Hong et al., ACM TOG ‘05)

• Our viscosity formulation is incorporated in [Density contrast SPH interfaces] (Solenthaler et al., SIGGRAPH ‘08)

• It does not impose additional costs for multiple phases with different densities and viscosities.


2 Related work

Fluid-solid blending and viscoelastic fluids

• The Navier-Stokes equations are augmented with a term accounting for elastic forces for SPH

• [A point-based method for animating elastoplastic solids] (Gerszewski et al., SIGGRAPH ‘09)

• Our method relies on the plain Navier-Stokes equations for fluids

• does not account for rigid motion and elasticity


2 Related work

Position-based methods

• Our method is implicit

as we solve a linear system for unknown velocities at next timestep

• Similarity to PBD

[A survey on position-based simulation methods in computer graphics] (Bender et al., CG Forum ’14)

• formulate a set of constraints that should be fulfilled at the next timestep

• employ a non-standard parameter between zero to one

our approach could also be seen as a contribution to the development of PBD from geometric towards physical motivations

• Our formulation imposes velocity constraints to damp shear rates which is the physical effect of shear viscosity.


3 Method

• Based on a desired velocity gradient at the next timestep, we compute momentum-preserving velocities that correspond to the predicted gradient.

predict a velocity gradient

preserves the corrections of density deviations

pressure & viscosity solvers do not interfere

Compute velocity changes due to pressure forces

↓

Viscosity solver computes velocities that account for viscosity

(but do not influence the rate of change of the volume)


3 Method

3.1 Decomposition of the velocity gradient

𝛻𝐯 =1

2𝛻𝐯 − 𝛻𝐯 𝑇 +

1

3𝛻 ∙ 𝐯 𝐈 +

1

2𝛻𝐯 + 𝛻𝐯 𝑇 −

1

3𝛻 ∙ 𝐯 𝐈

= R + V + S

• Spin tensor R

rate of rotation, i.e. vorticity

• Expansion-rate tensor V

density change (that computed by pressure solver)

preserved by viscous solver

• Shear-rate tensor S

rate of shear strain

(1)


3 Method

3.2 Prediction of the velocity gradient

• Estimates 𝛻𝐯 at time 𝑡 & its decomposition, then computes predicted velocity gradient 𝛻𝜏𝐯 at time 𝑡 + ∆𝑡

⑴ Particle density > rest density

𝛻τ𝐯 = 𝐑 + 𝐕 + ξ𝐒

with 0 ≤ ξ ≤ 1

⑵ Particle density < rest density

① divergence of velocity field is negative

𝛻τ𝐯 = 𝐑 + 𝐕 + ξ𝐒

② divergence of velocity field is positive

𝛻τ𝐯 = 𝐑 + ξ𝐒

(2)

(3)


3 Method

3.2 Prediction of the velocity gradient

[Figure 2]

• Estimates 𝛻𝐯 at time 𝑡 & its decomposition, then computes predicted velocity gradient 𝛻𝜏𝐯 at time 𝑡 + ∆𝑡

⑴ Particle density > rest density

𝛻τ𝐯 = 𝐑 + 𝐕 + ξ𝐒

with 0 ≤ ξ ≤ 1

⑵ Particle density < rest density

① divergence of velocity field is negative

𝛻τ𝐯 = 𝐑 + 𝐕 + ξ𝐒

② divergence of velocity field is positive

𝛻τ𝐯 = 𝐑 + ξ𝐒

(2)

(3)


3 Method

3.3 Reconstruction of the velocity field

• Final velocities 𝐯𝑖(𝑡 + ∆𝑡) are reconstructed from𝛻𝜏𝐯𝑖 𝑡 + ∆𝑡

use first-order Taylor approximation

𝐯𝑖 𝑡 + ∆𝑡 = 𝐯𝑗 𝑡 + ∆𝑡 +𝛻𝜏𝐯𝑖 + 𝛻

𝜏𝐯𝑗

2𝐱𝑖𝑗

𝛻𝜏𝐯𝑖+𝛻𝜏𝐯𝑗

2is used to guarantee momentum-preserving velocity

changes at the particles and accounts for the diffusion of the rotation rate

(4)


3 Method

3.4 Viscous SPH fluid solver

Compute 𝐯∗ without pressure and viscosity

Compute pressure and pressure force, then update 𝐯∗∗ from 𝐯∗

(use the pressure solver of one of IISPH, WSCSPH, PCISPH or PBF)

Solve viscosity, then update 𝐯 𝑡 + ∆𝑡 from 𝐯∗∗


4 Implementation

Velocity gradient

• Following Monaghan’s second golden rule of SPH

∇𝐯𝑖∗∗=1

𝜌𝑖

𝑗

𝑚𝑗(𝐯𝑗∗∗ − 𝐯𝑖

∗∗) × 𝛻𝑊𝑖𝑗

with 𝛻𝑊𝑖𝑗 = 𝛻𝑊(𝐱𝑖 − 𝐱𝑗)

this equation reliably computes the gradient even at free surfaces and inter faces between different phases

(5)


4 Implementation

Linear system

• Existing SPH formulation is deteriorated at free surfaces due to the incomplete neighborhood, so we consider the normalized SPH interpolation

𝐯𝑖(𝑡 + ∆𝑡) =1

𝜌𝑖

𝑗

𝑚𝑗 𝐯𝑗 𝑡 + ∆𝑡 +𝛻τ𝐯𝑖∗∗ + 𝛻𝜏𝐯𝑗

∗∗

2𝐱𝑖𝑗 𝑊𝑖𝑗

𝜌𝑖 −𝑚𝑖𝑊𝑖𝑖 𝐯𝑖 𝑡 + ∆𝑡 − 𝑗≠𝑖𝑚𝑗𝐯𝑗 𝑡 + ∆𝑡 𝑊𝑖𝑗

= 𝑗𝑚𝑗 𝐯𝑗 𝑡 + ∆𝑡 +𝛻τ𝐯𝑖∗∗+𝛻𝜏𝐯𝑗

∗∗


(6)

(11)


4 Implementation

Linear system

• Existing SPH formulation is deteriorated at free surfaces due to the incomplete neighborhood, so we consider the normalized SPH interpolation

𝜌𝑖 −𝑚𝑖𝑊𝑖𝑖 𝐯𝑖 𝑡 + ∆𝑡 − 𝑗≠𝑖𝑚𝑗𝐯𝑗 𝑡 + ∆𝑡 𝑊𝑖𝑗

= 𝑗𝑚𝑗 𝐯𝑗 𝑡 + ∆𝑡 +𝛻τ𝐯𝑖∗∗+𝛻𝜏𝐯𝑗

∗∗


solve the linear system A𝐯 = b,

𝑎𝑖𝑖 = 𝜌𝑖 −𝑚𝑖𝑊𝑖𝑖 ,

𝑎𝑖𝑗 = −𝑚𝑗𝑊𝑖𝑗

(11)


4 Implementation

Boundary handling

• Sticky boundary

𝐯𝑖 𝑡 + ∆𝑡 =1

𝜌𝑖 𝑗∉𝑏𝑚𝑗 𝐯𝑗 𝑡 + ∆𝑡 +

𝛻τ𝐯𝑖∗∗+𝛻𝜏𝐯𝑗

∗∗


+1

𝜌𝑖 𝑏𝑚𝑏 𝐯𝑏

∗∗𝑊𝑖𝑏

• Separating boundary

𝛻 ∙ 𝐯𝑖𝑙 =

𝑏

𝑚𝑏( 𝐯𝑖𝑙 − 𝐯𝑏

∗∗)𝛻𝑊𝑖𝑏

if this divergence is negative,

𝐯𝑖𝑙 =𝐧𝑖𝐧𝑖 ∙ 𝐧𝑖

𝑏

𝑚𝑏 𝐯𝑏∗∗𝛻𝑊𝑖𝑏

with 𝐧𝑖 = 𝑏𝑚𝑏 𝛻𝑊𝑖𝑏

(12)

(13)

(14)


4 Implementation

Boundary handling

• Sticky boundary use Conjugate gradient method

𝐯𝑖 𝑡 + ∆𝑡 =1

𝜌𝑖 𝑗∉𝑏𝑚𝑗 𝐯𝑗 𝑡 + ∆𝑡 +


∗∗


+1

𝜌𝑖 𝑏𝑚𝑏 𝐯𝑏

∗∗𝑊𝑖𝑏

• Separating boundary use Jacobi method

𝛻 ∙ 𝐯𝑖𝑙 =

𝑏

𝑚𝑏( 𝐯𝑖𝑙 − 𝐯𝑏

∗∗)𝛻𝑊𝑖𝑏

if this divergence is negative,

𝐯𝑖𝑙 =𝐧𝑖𝐧𝑖 ∙ 𝐧𝑖

𝑏

𝑚𝑏 𝐯𝑏∗∗𝛻𝑊𝑖𝑏

with 𝐧𝑖 = 𝑏𝑚𝑏 𝛻𝑊𝑖𝑏

(12)

(13)

(14)


4 Implementation

Multiple phase

• Incorporate the approach of [Density contrast SPH interfaces] (Solenthaler et al., SIGGRAPH ‘08) into IISPH

viscosity system is solved independently for each phase

• Interaction between the phases is accomplished

via pressure and cohesion forces in [Versatile rigid-fluid coupling for incompressible SPH] (Akinci et al., ACM TOG ‘12)


5 Discussion

• Interpretation of the system

1 −𝑚𝑖

𝜌𝑖𝑊𝑖𝑖 𝐯𝑖 𝑡 + ∆𝑡 −

1

𝜌𝑖 𝑗≠𝑖𝑚𝑗𝐯𝑗 𝑡 + ∆𝑡 𝑊𝑖𝑗

=1

𝜌𝑖 𝑗𝑚𝑗 𝐯𝑗 𝑡 + ∆𝑡 +


∗∗


the equation actually estimates velocities that meet an approximated target Laplacian

• it can be interpreted as 𝛻2𝐯𝑖 𝑡 + ∆𝑡 = 𝛻2τ𝐯𝑖∗∗

• Parameter 𝜉

non-standard parameter

dose not correspond to dynamic or kinematic viscosity

governs damping of the shear rate, so governs viscosity

depend on the timestep and resolution

• the time dependency of parameter is not handled as timesteps

(10)


5 Discussion

• Limitations

use various numerical approximation, e.g. SPH approximation and Taylor approximation → results in numerical errors

• But, these errors are small as negligible

should not be used for low viscous fluids such as water

rather expensive for rigid-like materials

• it requires many solver iterations

• viscoelastic model should be preferred


6 Results

• Our viscosity solver is combined with an IISPH pressure solver

cubic spline kernel

• [Smoothed particle hydrodynamics] (Monaghan, REPORTS ON PROGRESS IN PHYSICS 2005)

surface tension

• [Weakly compressible SPH for free surface flows] (Becker et al., ACM SIGGRAPH 2007)

one- or two-way coupled boundary handling

• [Versatile rigid-fluid coupling for incompressible SPH] (Akinci et al.,

CM TOG 2012) and [Boundary handling and adaptive time-stepping for PCISPH] (Ihmsen et al., VRIPHYS 2010)

maximum volume deviation is below 0.1%

• guaranteed by IISPH pressure solver


6 Results

Buckling and coiling

∆t = 0.002s

d = 0.04m //particles space

iter. = 50

• neighborhood & pressure requires 25ms per step

• viscosity requires 195ms


6 Results

Viscosity range

∆t = 0.001s

d = 0.012m

viscous iter. = 25 for 𝜉𝑙𝑒𝑓𝑡 = 0.8

viscous iter. = 87 for 𝜉𝑟𝑖𝑔ℎ𝑡 = 0.2

• both pressure solver requires 5 iterations


6 Results

Rotation rate and density error

∆t = 0.001s

d = 0.02m

𝜉 = 0.5

viscous iter. = 17

pressure iter. = 3

[Figure 6] densities before and after the viscous solve


6 Results

Time-varying viscosity

∆t = 0.001s

d = 0.02m

𝜉𝑏 = 0, 𝜉𝑦 and 𝜉𝑟 = raise form 0 to 0.99 within 3s and 15s


6 Results

Boundary handling

∆t = 0.002s

d = 0.025m

𝜉 = 0.5

viscous iter. = 17 for sticky boundary with Conjugate Gradient

viscous iter. = 230 for separating boundary with Jacobi


6 Results

Complex boundaries

∆t = 0.0004s

d = 0.02m

𝜉 = 0.5

viscous iter. = 12


6 Results

Two-way coupling

∆t = 0.0005s

d = 0.014m

𝜉 = 0.8

viscous iter. = 9


6 Results

Multiple phases

∆t = 0.001s

d = 0.05m

𝜉 = 1 to 0

viscous iter. = 12 to 140


6 Results

Large-scale scenarios

∆t = 0.04s

d = 0.25m

viscous iter. = up to 30

pressure iter. = up to 10


6 Results

Explicit and implicit viscosity solvers

• Comparison of explicit and implicit

(left) use explicit method in [On the problem of penetration in particle methods] (Monaghan, Journal of Computational physics 82 ‘89)

(right) our implicit method

• explicit required 20min, while implicit required only 11sec

• explicit use the timestep of 0.00002s, but implicit 0.01s

[Figure 12]


6 Results


• Implicit approaches have the capability to obtain the equilibrium faster than explicit approaches

as the shear rates are propagated within solver iterations instead of per simulation step.

[Figure 13] (b) the fluid is initialized in a static cylinder / stopping the cylinder(c) cylinder starts rotation with constant velocity


6 Results


• Viscosity varies in all setting

(a) and (b) : artificial viscosity (explicit)

(c),(d) and (e) : our method (implicit)

[Figure 14]


7 Conclusion and future work

• Our viscosity formulation preserves the density correction

this is achieved by decomposing the gradient

• By integrating our formulation into IISPH

a maximum volume deviation can be guaranteed independent from the grade of viscosity

• More efficient than explicit method

work well with large timestep

versatile in wide range of viscous material

work well in one-/two-way coupling and multiphase simulation


7 Conclusion and future work

• Our method heavily damps shear rates

particle movement is strongly restricted

after simulation, the pattern of initial particle sampling is visible

• it is problematic for regular sampling patterns

• Conjugate gradient method does not work in separating boundary

in separating boundary, intermediate results have to be adjusted in each solver iteration

an implicit viscosity formulation for sph...

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