By Michael Su 04/16/2009

 Introduction  Fluid characteristics  Navier-Stokes equation  Eulerian vs. Lagrangian approach  Dive into the glory detail (A case study of the 2d fluid simulation)  Advection  Diffusion  Pressure solve  Fluid object couple  One-way and two-way coupling  Real-time fluids

 Broad view of the fluid simulation in graphics community and its potential applications  Basic knowledge about the grid-based fluid simulation  Understanding the challenges of the existing methods  Foundation for the following two fluids- related lectures (smoke & granular material).

 Applications  Games (Half Life, Crysis)  Scientific visualization (Water sewage system, dam construction)  Movie special effects (Finding Nemo, Pirates of Caribbean)  Medical simulation (Blood flow)  What can we achieve so far?  Smoke  Granular flow (Sand)  Newtonian fluid (Water, ocean)  Non-Newtonian fluid (Blood, honey, goop (viscoelaticity flow))  Microscopic phenomena [Zhu, Bridson] Animating Sand as a Fluid, SIGGRAPH 05 [Fedkiw, Stam, Jensen] Visual Simulation of Smoke, SIGGRAPH 01 [Tessendorf] Simulating Ocean Water, SIGGRAPH 01 [Goktekin, Bargteil, O'Brien] A Method for Animating Viscoelastic Fluids, SIGGRAPH 04 [Wang, Mucha, Turk] Water Drops on Surfaces, SIGGRAPH 05

 Basic properties  Pressure  Density  Viscosity (subject to shear stress)  Surface tension  Different types of fluids:  Incompressible (divergence-free) fluids: Fluids doesn’t change volume (very much).  Compressible fluids: Fluids change their volume significantly.  Viscous fluids: Fluids tend to resist a certain degrees of deformation

 Inviscid (Ideal) fluids: Fluids don’t have resistance to the shear stress  Turbulent flow: Flow that appears to have chaotic and random changes  Laminar (streamline) flow: Flow that has smooth behavior  Newtonian fluids: Fluids continue to flow, regardless of the force acting on it

 Non-Newtonian fluids: Fluids that have non- constant viscosity  Phase Transition: Fluids may change physical behavior under different environmental conditions.

 Modeling continuum fluids on discrete systems – It’s all about approximations  Topological variations and different kinds of behaviors with interacting subjects  Numerical stabilities, accuracy and convergence issues  Performance  User control

By Michael Su 04/20/2009

Calculus Review (1)  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) u can be a scalar or a vector u can only be a vector Source, Div(u) > 0 Sink, Div(u) < 0

Calculus Review (2)  Laplacian (∆ or ): Divergence of the gradient  Finite Difference: Derivative approximation u can be a scalar or a vector

 Momentum equation  Incompressibility Claude-Louis Navier (1785~1836) George Gabriel Stokes (1819~1903) u t = k  2 u –(u  )u –  p + f  u=0 Change in velocity Diffusion/Vi scosity AdvectionPressureBody Forces u: the velocity field k: kinematic viscosity

[Mueller, Charypar, Gross] Particle-Based Fluid Simulation for Interactive Applications, SCA03  Borrowed from CFD (Computational Fluid Dynamics)  Common techniques for solving Navier Stoke’s equation:  Eulerian approach (grid-based)  Lagrangian approach (particle-based)  Spectral method  Lattice Boltzmann method [Stam] Stable Fluids, SIGGRAPH 99

 Discretize the domain using finite differences  Define scalar & vector fields on the grid  Use the operator splitting technique to solve each term separately  Evaluation: Derivative approximation Adaptive time step/solver Memory usage & speed Grid artifact/resolution limitation

 Treat the fluid as discrete particles  Apply interaction forces (i.e. pressure/viscosity) according to certain pre-defined smoothing kernels  Evaluations: Mass / Momentum conservation More intuitive Fast, no linear system solving Connectivity information/Surface reconstruction

Case Study: A 2D Fluid Simulator  We focus exclusively on incompressible, viscous fluid  Assuming the gravity is the only external force  No inflow or outflow  Constant viscosity, constant density everywhere in the fluid

Scalar/Vector fields defined on the grid AdvectionBody Force Diffusion Pressure Solve u t = k  2 u –(u  )u –  p + f  u=0

The Power of Operator Splitting U n + A + B + D + P U*U* U ** U *** U n+1  One complicated Multi- dimensional operator => A series of simple, lower dimensional operators  Each operator can have its own integration scheme and different time step sizes  High modularity and easy to debug

 Sometimes called “Convection” or “Transport”  Define how a quantity moves with the underlying velocity field  This term ensures the conservation of momentum  Advection equation:  Approaches:  Forward Euler (unstable)  Semi-Lagragian advection (stable for large time steps, but suffers from the dissipation issue)

Forward Euler AdvectionSemi-Lagragian Advection

 Define how a quantity in a cell inter-changes with its neighbors  Diffusion = Blurring  The viscous fluid can be achieved by applying diffusion to the velocity field Low ViscosityHigh Viscosity Figures from [Carlson, Mucha, Turk] Melting and Flowing, SCA 02

 Diffusion equation:  Approaches:  Explicit formulation  Implicit formulation (for high viscosity) Unknowns

Before the diffusionAfter the diffusion (k = 0.5, time step size =1)

 It’s sometimes called “Pressure Projection”  What does the pressure do?  Keep the fluid at constant volume (incompressible, conservation of mass).  Make sure the velocity field stays divergence-free CompressibleIncompressible

 Equation to solve:  How to solve for pressure:  Taking divergence of both sides of (1), we will have  Build a system of equations and solve Ap = d using an iterative method such as Conjugate Gradient  Update the velocity field from the pressure gradient s.t. (1) Unknowns (Poisson Equation)

 What about the pressure on boundary nodes?  Free surface: The fluid can evolve freely (p = 0)  Solid wall: The fluid can’t penetrate the wall but can flow freely in tangential directions (Neumann BC) Free surface Solid wall

 Possible reasons why your simulation doesn’t look right:  CFL condition violation => Smaller time steps / Implicit solver  Flux conservation => BCs may not be set correctly  Grid resolution/ Memory => Adaptive grids  Numerical dissipation => Back and Forth Error Compensation and Correction [4] / Vorticity confinement [5]  Handle the interface and complex topological changes => Level set method [6]  Volume loss => Particle level set [7]

 One-way coupling:  Solid-Fluid interaction: The fluid has no influence on the solid  Fluid-Solid interaction: The solid has no influence on the fluid  Two-way coupling:  Manipulate the boundary conditions  Finite Element techniques: ALE & DLM  Rigid Fluid: Treat the solid as fluids and enforce the rigidity constraint [8]

 Principles:  Cheap to compute  Low memory consumption  Stability  Plausibility  Interactivity  Common techniques:  Procedural water: Superimpose sine waves of a variety of amplitudes and directions. [9]

Real-time Fluids (2)  Heightfield approximations: If the surface is the only interest, it can be represented using a 2d heightfield and animated by 2d wave equations with interaction forces.  Particle systems: This approach is good at simulating a small amount of water such as a puddle, a bubble, or splashing fluids H(x, y)

 [1] R. Bridson and M. Müller-Fischer. Fluid Simulation. SIGGRAPH 07 Course Notes  [2] R. Bridson. Fluid Simulation for Computer Graphics. A K Peters, 2008  [3] J. Stam. Real-Time Fluid Dynamics for Games. GDC 2003  [4] B. Kim, Y. Liu, I. Llamas, and J. Rossignac. FlowFixer: Using BFECC for Fluid Simulation. EGWNP 05  [5] R. Fedkiw, J. Stam, and H.W. Jenson. Visual Simulation of Smoke. SIGGRAPH 01  [6] N. Foster, R. Fedkiw, Practical Animation of Liquids. SIGGRAPH 01  [7] D. Enright, S. Marschner, R. Fedkiw. Animation and Rendering of Complex Water Surfaces. SIGGRAPH 02  [8] M. Carlson, P. J. Mucha, G. Turk. Rigid Fluid: Animating the Interplay Between Rigid Bodies and Fluid. SIGGRAPH 04

References (2)  [9] D. Hinsinger, F. Neyret, M. Cani. Interactive Animation of Ocean Waves. SCA 02