cs 775 - advanced computer graphics · cs775: lecture 10 parag chaudhuri fluid simulation eulerian...

CS 775: Advanced Computer Graphics

Lecture 10: Grid Fluids

CS775: Lecture 10 Parag Chaudhuri

Fluid Simulation

● Calculus Review

— Gradient ( ): A vector pointing in the direction of the greatest rate of increment

— Divergence ( ): Measure how the vectors are converging or diverging at a given location (volume density of the outward flux)

∇

can be a scalar or a vectoru∇ u=( ∂u∂ x,∂u∂ y,∂u∂ z )

http://en.wikipedia.org/wiki/Gradient

∇⋅

http://www.cs.unc.edu/~lin/COMP768S09/LEC/fluid.ppt

∇⋅u=∂ u∂ x

+∂ u∂ y

+∂ u∂ z

can only be a vectoru


Fluid Simulation

● Calculus Review

— Laplacian ( ): Divergence of the gradient

— Finite Difference: Approximating a derivative

∇2

can be a scalar or a vectoru∇ 2u=( ∂

2u

∂ x2 +∂

2u

∂ y2 +∂

2u

∂ z 2 )

∂ u∂ x

=ui+1−u ix i+1−x i

∂ u∂ t

=u i+1−u it i+1−t i

over space

over time


Fluid Simulation

● Calculus Review

then, ∇ F=[∂ f∂ x

∂ f∂ y

∂ f∂ z

∂ g∂ x

∂ g∂ y

∂ g∂ z

∂ h∂ x

∂ h∂ y

∂ h∂ z

] ∇⋅F=∂ f∂ x

+∂ g∂ y

+∂h∂ z

If F (x , y , z)=f (x , y , z) i+g(x , y , z) j+h(x , y , z) k

and, ∇ 2 F=∇⋅(∇ F )=[∂

∂ x(∂ f∂ x

)+ ∂∂ y

(∂ f∂ y

)+ ∂∂ z

(∂ f∂ z

)

∂∂ x

(∂ g∂ x

)+ ∂∂ y

(∂ g∂ y

)+ ∂∂ z

(∂ g∂ z

)

∂∂ x

(∂h∂ x

)+ ∂∂ y

(∂h∂ y

)+ ∂∂ z

(∂h∂ z

) ]


Fluid Simulation

● NavierStokes Equation

● Derivation Sketch

∂ u∂ t

=−u⋅∇ u−1ρ ∇ p+ν∇

2 u+ g Momentum Equation

mD uD t

=F where Dϕ( x , t )Dt

=∂ϕ( x , t )

∂ t+

∂ϕ( x , t )∂ x

⋅∂ x∂ t

=∂ϕ( x , t )

∂ t+∇ ϕ( x , t )⋅u

mD uD t

=m g−V ∇ p+V μ ∇2 u

D uD t

= g−1ρ ∇ p+

μρ ∇

2 u

∂ u∂ t

=−u⋅∇ u−1ρ ∇ p+ν∇

2 u+ g

u is fluid velocityμ is dynamic viscosityg is acceleration due to gravity


Fluid Simulation

● NavierStokes Equation

● Derivation Sketch

∇⋅u=0 Incompressibility Equation

u is fluid velocity

Ω is an arbitrary chunk of fluid

n is normal at surface

ddt

(VolumeΩ)=∫∫Su . n . dS

=∫∫∫Ω∇⋅u .dΩ

=0 for incompressibility

⇒∇⋅u=0

VolumeΩ is its volume

S is its surface boundary


Fluid Simulation● Eulerian Viewpoint

— Discretize the domain using finite differences

— Define scalar & vector fields on the grid

— Use the operator splitting technique to solve each term separately

— Advantages

› Derivative approximation

› Adaptive time step/solver

— Disadvantages

› Memory usage & speed

› Grid artifact/resolution limitation

http://www.eng.nus.edu.sg/EResnews/022014/images/rd02fig1.jpg


Fluid Simulation● Lagrangian Viewpoint

— Discretize the domain using particles

— Define interaction forces between neighbouring particles using smoothing kernels.

— Advantages

› Mass Conservation

› Intuitive— Disadvantages

› Surface tracking

http://mecmail.eng.monash.edu.au/~mct/mct/docs/sph/sph.html


Fluid Simulation● Solving the NavierStokes Equation

— Eulerian Viewpoint

— Operator Splitting

› One complicated Multidimensional operator solved as a series of simple, lower dimensional operators

› Each operator can have its own integration scheme

› High Modularity and Easy to debug

un=A+B+D+P

u2

u1

u3

un+1


Fluid Simulation● Advection

● Euler

● SemiLagrangian (look back in time)

∂ u∂ t

=−u⋅∇ u

uin+1−ui

n

Δ t=−u⋅

ui+1n − ui−1

n

2 Δ xUnstable!

u?n+1−ui

n

Δ t=0

Unconditionally stable!but hasNumerical Dissipation


Fluid Simulation● Incompressibility and Pressure Solve

— Advection may introduce compression/expansion in the field

Discrete Multiscale Vector Field Decomposition, Tong et al, 2003


Fluid Simulation● HemlholtzHodge Decomposition

Discrete Multiscale Vector Field Decomposition, Tong et al, 2003

=

+u ∇ p udiv_free

Input Velocity FieldCurl Free (irrotational)

Divergence Free (incompressible)

=



udiv_free=u−∇ p −(1)

∇⋅udiv_free=0 −(2)

∇⋅∇ p=∇⋅u

Solve (3), then plug into (1) to find new incompressible velocity field.



Staggered Grids



— Divergence

— Gradient

∇⋅u=∂ u∂ x

+∂ u∂ y

≈u2−u1+v2−v1

Δx

∇ x p=∂ p∂ x

≈p2− p1

Δx



— Laplacian (Divergence of Gradient)

∇⋅∇ p≈

pi+1, j− pi , jΔ x

−pi , j− pi−1, j

Δ x+pi , j+1− p i , j

Δ x−pi , j− pi , j−1

Δ xΔ x



— Boundary Conditions

u⋅n=0



— Solving

› A Poisson Equation

› Sparse, positive definite linear system of equations

› One equation per cell, cells globally coupled› Conjugate Gradients solver

∇⋅∇ p=∇⋅u


Fluid Simulation● Viscosity

— Solving

— Dicretize and solve

› If is , explicit integration● No need to solve linear system

› If is , implicit integration● Stable for high viscosities.

∂ u∂ t

=ν∇2 u

unew= uold+Δ t ν∇2 u*

u*

u*unew

uold


Fluid Simulation● Complete Solver

Velocity Solver

Advect Velocities

Add Viscosity

Add Gravity

Pressure Projection



Velocity Solver

Advect Velocities

Add Viscosity

Add Gravity

Pressure Projection

Done Finally?



Velocity Solver

Advect Velocities

Add Viscosity

Add Gravity

Pressure Projection

Not Quite!



Velocity Solver

Advect Velocities

Add Viscosity

Add Gravity

Pressure Projection

Surface Tracking and Reconstruction

cs 775 - advanced computer graphics · cs775: lecture 10 parag chaudhuri fluid simulation eulerian...

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