# Realistic Animation of Fluids

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1 1 Realistic Animation of Fluids Nick Foster and Dimitris Metaxas Presented by Alex Liberman April 19, 2005

2 2 Previous Work Used non physics-based methods (mostly in 2D) Hard to simulate effects that rely on physical phenomena and volume; rotational and pressurebased effects Hard to make fluid interact with dynamic objects and to simulate the buoyancy force on such objects Quintessential Player

3 3 Inspiration from CFD Computational Fluid Dynamics has been used in engineering problems since the mid '60s Very detailed fluid models exist Unfortunately most methods are needlessly-complex (for graphics) and scale poorly End-user must know what he/she is doing (more than the average animator is interested in)

4 4 Main contributions from this paper Simulation Solving the Navier-Stokes equations Boundary Conditions Initializing fluid boundaries and setting up global behavior Surface Tracking Tracking the evolution of the fluid surface for rendering

5 5 Simulation: Navier-Stokes (N-S) N-S are a set of equations that describe the behavior of various fluids (including gases) For graphics the incompressible Navier-Stokes equations are sufficient Next Level Inc. ets/nuclear_bomb_test.jpg

6 6 Simulation: Equations Foster & Metaxas formulation: A vector-based formulation: u fluid velocity (vector field) (u, v, w) u t fluid acceleration (vector field) ( u/ t, v/ t, w/ t) p fluid pressure (scalar field) ρ fluid density (~1.0 for water assumed in F&M) ν viscosity or fluid thickness (assumed constant)

7 7 Simulation: More N-S detail a= F net m Mass Conservation: net fluid flow is 0 u u Advection: Describes in what direction a neighboring region of water pushes water at u v 2 u Momentum Diffusion: Describes how quickly variations in velocity are damped-out; depends on fluid viscosity 1 ρ p Pressure Gradient: Describes in which direction fluid at u is pushed to reach a lower pressure area

8 8 Simulation: Discretization Foster & Metaxas '00

9 9 Simulation: Velocity field update Intimidating, but really means... CFL Condition: traverse at most one cell in a single time-step

10 10 Simulation: Velocity field update (ctd) Intimidating, but really means... CFL Condition: traverse at most one cell in a single time-step u t 1 =u t h u t t u=u Is an Euler step that uses a finite-difference approximation of u t (t) Maximum value of h is dictated by the CFL

11 11 Simulation: Finite differences

12 12 Simulation: Problem

13 13 Simulation: Problem compressible fluids? However: u 0

14 14 Simulation: Enforcing incompressibility Compute divergence D= u i, j, k Computer pressure correction p i, j, k = D i, j, k Update cell-face velocities u i 1/2, j, k =u i 1/2, j, k t / x p i, j, k u i 1/2, j, k =u i 1/2, j, k t / x p i, j, k... Iterate the above steps until D 0 Update pressure field: p i, j, k = p i, j, k p i, j, k

15 15 Boundary Conditions free-slip boundary Fluid is free to move parallel to the boundary non-slip boundary Fluid is fixed to boundary

16 16 Inflow/Outflow conditions (sources and sinks) inflow condition u c Set a constant fluid velocity u c on the face of a source cell outflow condition u f Set sink cell's face velocity to adjacent fluid cell's velocity, and allow it to relax

17 17 Boundary condition for free surface cells Surface cell's interface's velocity = sum of the remaining velocities This maintains the condition u 0

18 18 Surface Tracking: Marker Particles Generate marker particles at sources (and render them only in cells which contain fluid) 999/images/art5fig7.gif Use weighted (bilinear) interpolation to compute a marker's velocity from four nearest cell velocities Propagate particles using standard integration techniques Easy to simulate violent effects like splashing and overturning McFluid

19 19 Surface Tracking: Free Surface Particles Start at known Full or Empty regions and grow particles via 4-connected search Insert particles when too sparse, remove when too dense Problematic: how to efficiently detect sparse/dense areas?

20 20 Surface Tracking: Height Field 1.gif For smooth fluids (ocean, lake, puddles) w/o splashes H t 1 =H t t w t t H un t 1 un t+1 velocity field values of neighbor cells

21 21 Miscellaneous & Questions? Better Explanation and Implementation Guide Available as part of Mark Carlson's PhD thesis: Other Simulation Techniques SPH; simulate motion of marker particles directly CFD Math & Derivations Pain available at:

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