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

Aug 27, 2023 •46 likes •186 views

Fluid Dynamics in Graphics [pi] Overview Fields Math Background Domain Physics Simulation Scalar Field Related Phenomena Frontiers in Graphics Vector Field Types of Derivatives Overview Derivative

  1. Fluid Dynamics in Graphics [pi] Overview Fields • Math Background • Domain • Physics • Simulation • Scalar Field • Related Phenomena • Frontiers in Graphics • Vector Field

  2. Types of Derivatives Overview • Derivative measures how something • Math Background changes . • Physics • Temporal Derivatives • Simulation • Spatial Derivatives • Related Phenomena • Frontiers in Graphics Representation “Coffee Cup” Equations Navier-Stokes • Density (density) • Velocity (domain) (velocity) s.t.

  3. Density Advection Velocity Advection s.t. Projection Projection s.t. (divergence) ? Div > 0 Div < 0 Div = 0 Div � 0 Div = 0

  4. Projection Diffusion s.t. s.t. External Forces Physics Recap • Physical quantities represented as fields. • PDE describes the dynamics. – explains what we see in here… s.t. • Gravity • Heat • Surface Tension • User-Created Forces (stirring coffee)

  5. Overview Simulation Challenges • Math Background • How to represent physical quantities? • Physics • How to evolve this representation? • Simulation • Related Phenomena • Frontiers in Graphics Simulation Representation Simulation Representation • Recall we’re dealing with fields : • Particle system? (density) (velocity) • Grid Representation • Very Low Resolution – Each grid cell represents integral over underlying quantities – Derivatives Easy to Implement

  6. Simulation Challenges Explicit Integration • How to represent physical quantities? • Very simple method to “implement” physics • How to evolve this representation? Explicit Integration Stable Fluids • Invented by Jos Stam • Simple + Fast + Unconditionally Stable fluid solver

  7. Splitting Methods Splitting Methods • Suppose we had a system: • …and we define a simulation S f . Advect Project Diffuse Add Forces • Then we could define: Advection Projection

  8. Add Forces (e.g. heat) Diffusion • Solved implicitly (like projection) • I don’t have a picture of this. Simulation Recap Overview • Decided Upon grid-based represenation. • Math Background • Explicit Methods will not work. • Physics • Stable Fluids solves all our problems… • Simulation – …maybe. • Related Phenomena • Frontiers in Graphics

  9. Smoke Water low pressure discontinuous change in pressure high pressure [Stam et. al. 2001] Water Particle/Level-Set Method [Foster and Fedkiw 2001]

  10. Water (Revisited) Water-Solid Interaction [Enright et. al. 2002] [Carlson et. al. 2004] Water-Solid Interaction Melting [Carlson et. al. 2004] [Carlson et. al. 2002]

  11. Explosions Fire • Not Incompressible – Implies div > 0 burning gass region heat fuel source Fire Overview • Math Background • Physics • Simulation • Related Phenomena • Frontiers in Graphics – Other phenomena – New Directions! [Nguyen et. al. 2002]

  12. Breaking Waves Splashing Small Scale Water Phenomena Overview • Math Background • Physics • Simulation • Related Phenomena • Frontiers in Graphics – Other phenomena – New Directions!

  13. Control <stanford bunny>

  14. Other Directions • Realtime… • Model Reduction • Automatic Discovery of Dynamics

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