Fluid Dynamics in Graphics [pi] Overview Fields Math Background - - PowerPoint PPT Presentation

Fluid Dynamics in Graphics

[pi]

Overview

• Math Background
• Physics
• Simulation
• Related Phenomena
• Frontiers in Graphics

Fields

• Domain
• Scalar Field
• Vector Field

Types of Derivatives

• Derivative measures how something

changes.

• Temporal Derivatives
• Spatial Derivatives

Overview

• Math Background
• Physics
• Simulation
• Related Phenomena
• Frontiers in Graphics

Representation

(density) (velocity) (domain)

“Coffee Cup” Equations

• Density
• Velocity

s.t.

Navier-Stokes

Density Advection Velocity Advection

s.t.

Projection

s.t.

(divergence)

Div > 0 Div < 0 Div = 0

?

Projection

Div 0 Div = 0

Projection

s.t.

Diffusion

s.t.

External Forces

• Gravity
• Heat
• Surface Tension
• User-Created Forces (stirring coffee)

s.t.

Physics Recap

• Physical quantities represented as fields.
• PDE describes the dynamics.

– explains what we see in here…

Overview

• Math Background
• Physics
• Simulation
• Related Phenomena
• Frontiers in Graphics

Simulation Challenges

• How to represent physical quantities?
• How to evolve this representation?

Simulation Representation

• Particle system?
• Very Low Resolution

Simulation Representation

• Recall we’re dealing with fields:
• Grid Representation

– Each grid cell represents integral over underlying quantities – Derivatives Easy to Implement

(density) (velocity)

Simulation Challenges

• How to represent physical quantities?
• How to evolve this representation?

Explicit Integration

• Very simple method to “implement” physics

Explicit Integration Stable Fluids

• Invented by Jos Stam
• Simple + Fast + Unconditionally Stable fluid solver

Splitting Methods

• Suppose we had a system:
• …and we define a simulation Sf.
• Then we could define:

Splitting Methods

Advect Project Diffuse Add Forces

Advection Projection

Diffusion

• Solved implicitly (like projection)
• I don’t have a picture of this.

Add Forces (e.g. heat) Simulation Recap

• Decided Upon grid-based represenation.
• Explicit Methods will not work.
• Stable Fluids solves all our problems…

– …maybe.

Overview

• Math Background
• Physics
• Simulation
• Related Phenomena
• Frontiers in Graphics

Smoke

[Stam et. al. 2001]

Water

low pressure high pressure discontinuous change in pressure

Water

[Foster and Fedkiw 2001]

Particle/Level-Set Method

Water (Revisited)

[Enright et. al. 2002]

Water-Solid Interaction

[Carlson et. al. 2004]

Water-Solid Interaction

[Carlson et. al. 2004]

Melting

[Carlson et. al. 2002]

Explosions

• Not Incompressible

– Implies div > 0

Fire

fuel source burning gass region heat

Fire

[Nguyen et. al. 2002]

Overview

• Math Background
• Physics
• Simulation
• Related Phenomena
• Frontiers in Graphics

– Other phenomena – New Directions!

Breaking Waves Splashing Small Scale Water Phenomena Overview

• Math Background
• Physics
• Simulation
• Related Phenomena
• Frontiers in Graphics

– Other phenomena – New Directions!

Control

<stanford bunny>

Other Directions

• Realtime…
• Model Reduction
• Automatic Discovery of Dynamics