Motivation Movie Game Engineering Introduction Ideally Looks good Fast simulation Looks good? Look plausible Doesn’t need to be exactly

Motivation

Movie Game Engineering

Introduction

Ideally Looks good Fast simulation

Looks good? Look plausible Doesn’t need to be exactly correct doesn’t suffer from excessive numerical

dissipation

Overview

Simulation Compressible/Imcompressible Equation Vortex particle method Using octree data structure

Rendering Thick smoke: plain particles Thin smoke: adaptive particles Using Compensated Ray Marching

Compressible Fluids Yngve, G. D., O'Brien, J. F., Hodgins, J. K., 2000,

Animating Explosions. The proceedings of ACM SIGGRAPH 2000, New Orleans, July 23-28, pp. 29-36.

Operator Splitting

Stam used the technique in his Stable Fluids paper, 1999.

Allows us to solve the Navier-Stokes equation in Parts.

As long as each part is stable the technique will be stable.

Semi-Lagrangian Convection

Trace each point p in the field backwards in time. The new velocity at x is therefore the velocity that the particle had a time step ago at the old location.

Smoke Viscous Diffusion

In the simulation of smoke, it is reasonable to consider the fluid inviscid. Therefore this term is zero and need not be solved for.

In fact, the implicit solver that is used will take energy away from the system anyway, so there is a numerical dampening that can look like a non-zero viscosity.

Body Forces

Gravity, user forces, and object interaction.

The visible smoke is from a density field.

Areas of high density fall in the direction of gravity.

Heat moves against gravity (hot air rises).

Incompressibility

At each time step we start with a velocity field that is divergence free. We need this to be true at the end of the time step because the fluid is incompressible.

After solving for convection, and body forces the velocity field is not divergence free.

Incompressibility

Fedkiw, R., Stam, J. and Jensen, H.W.,"Visual Simulation of Smoke",SIGGRAPH 2001, 23-30 (2001).

Incompressible Euler Equations

self-advection Forces

incompressible

(Navier-Stokes without viscosity)

Additional Equations

smoke’sdensity

temperature

We now know all the finite differences we need to add the forces for heat and density…

Vorticity Confinement

The numerical dissipation, and the coarse grid size, cause the fine scale detail of turbulent swirling smoke to vanish.

Identify where the curl is highest, and add back in a rotational force there.

“Vorticity Confinement” force preserves

swirling nature of fluids.

What is vorticity?

A measure of the local rotation in a fluid flow.

Pressure Boundary Conditions A Neumann boundary condition is a

restriction on the derivative of a function.

Velocity Boundary Condition A Dirichlet boundary condition is a

restriction on the value of a function.




Solving the System

Need to calculate:

Start with initial state

Calculate new velocity fields

New state:

Smoke Simulation

While (simulating) Get external forces (if any) from UI Get density/heat sources (from UI or init

grid cells) Update velocity Update density Update temperature Display density

Smoke Simulation



Update Velocity

Equation: First term:

Advection Move the fluid through its velocity field

(Du/Dt=0) Second term:

external forces Final term:

pressure update

Update Velocity

Add external forces (fbuoy + forces from UI + fconf)

Advection (semi-lagrangian step – trace particles back)

Pressure update (solve linear system)

Smoke Simulation



Update density

Equation First term:

Advection Move the temperature through velocity field

Second term: Diffusion

Can skip this term

Second term: external sources

Update density

Add Sources (pick grid cells or from UI)


Diffusion (solve linear system – can skip this step)

Smoke Simulation



Update temperature

Equation First term:

Advection Move the temperature through velocity

field

Second term: Diffusion

Can skip this term

Update Temperature

Add Sources (grid cells or objects or UI)


Diffusion (solve linear system – can skip this step)

Smoke Simulation



Display density

Use any approach you want Visualize the density field: just opengl render a bunch of cubes

(corresponding to grid cells) that have alpha values based on how dense the smoke is.

Numerical Dissipation

‘ Stable Fluids’ method dampens the flow Typical with semi- Lagrangian methods

Improve using “Vorticity Confinement” force

Total Forces

User supplied fields

Buoyancy force

New confinement force

Results

Results

Other Methods

Vortex particle method Using octree data structure

Vortex particle method-Lagrangian primitives

Curves carry the vorticity

Each local vortex induces a weighted rotation

dx

xp

xp3

4

)(

ω

v dlxp

xp

3

4

)(

ω

v

Vortex particle method-Lagrangian primitives

Curves carry the vorticity

Each local vortex induces a weighted rotation

Method of simulation

Vortex particles (for motion) organized as curves. = tangent

Smoke particles (for visualisation)

Curves carry vortices Vortices induce a velocity field velocity field deforms curves & smoke

At every step: Advect the curves Stretch Advect the smoke











Video

Smoke Rendering Thick smoke: plain particles Thin smoke: adaptive particles [AN05]

accumulate stretching

01nTOT JJJJ ... 01nTOT JJJJ ...

Smoke Rendering

Thin smoke behaves like a surface

[ William Brennan ][ William Brennan ]

ee

nn

ll

eJJn 1T eJJn 1T

Smoke Rendering

Using Compensated Ray Marching

References

Documents

Motivation Movie Game Engineering Introduction Ideally Looks good Fast simulation Looks good? Look plausible Doesn’t need to be exactly