Fluid Simulation A Practical Approach: Part I

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Fluid Simulation A Practical Approach Part I

• By Michael Su

04/16/2009
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Agenda

• 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

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Objectives

• 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).

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Introduction

• 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

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Fluid Characteristics (1)

• Basic properties
• Pressure
• Density
• Viscosity (subject to shear stress)
• Surface tension
• Different types of fluids
• Incompressible (divergence-free) fluids Fluids
  doesnt change volume (very much).
• Compressible fluids Fluids change their
• volume significantly.
• Viscous fluids Fluids tend to resist a
• certain degrees of deformation

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Fluid Characteristics (2)

• Inviscid (Ideal) fluids Fluids dont 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

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Fluid Characteristics (3)

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

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Challenges

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

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Fluid Simulation A Practical Approach Part II

• By Michael Su

04/20/2009
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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
Source, Div(u) 0
Sink, Div(u)
u can only be a vector
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Calculus Review (2)

• Laplacian (? or ) Divergence of the
  gradient
• Finite Difference Derivative approximation

u can be a scalar or a vector
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Navier-Stokes Equation

• Momentum equation
• Incompressibility

ut k?2u (u??)u ?p f
George Gabriel Stokes (18191903)
Claude-Louis Navier (17851836)
??u0
u the velocity field
k kinematic viscosity
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Techniques Overview

• Borrowed from CFD (Computational Fluid Dynamics)
• Common techniques for solving Navier Stokes
  equation
• Eulerian approach (grid-based)
• Lagrangian approach (particle-based)
• Spectral method
• Lattice Boltzmann method

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Eulerian Approach

• 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

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Lagrangian Approach

• 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

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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

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Simulation Loop
Advection
Body Force
Diffusion
Scalar/Vector fields defined on the grid
Pressure Solve
ut k?2u (u??)u ?p f
??u0
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The Power of Operator Splitting

• 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

Un A B D P
U
U
U
Un1
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Advection (1)

• 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)

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Advection (2)
Forward Euler Advection
Semi-Lagragian Advection
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Diffusion (1)

• 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 Viscosity
High Viscosity
Figures from Carlson, Mucha, Turk Melting and
Flowing, SCA 02
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Diffusion (2)

• Diffusion equation
• Approaches
• Explicit formulation
• Implicit formulation
• (for high viscosity)

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Diffusion (3)
0
0
0
0
2.5
5
0
0
0
0
-10
2.5
2.5
0
0
0
0
0
0
2.5
0
Before the diffusion
After the diffusion (k 0.5, time step size 1)
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Pressure Solve (1)

• Its 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

Compressible
Incompressible
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Pressure Solve (2)

• 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)
(Poisson Equation)
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Pressure Solve (3)

• What about the pressure on boundary nodes?
• Free surface The fluid can
• evolve freely (p 0)
• Solid wall The fluid cant
• penetrate the wall but can
• flow freely in tangential
• directions (Neumann BC)

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Issues

• Possible reasons why your simulation doesnt 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

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Fluid-Object Coupling

• 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

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Real-time Fluids (1)

• 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

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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

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References (1)

• 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

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References (2)

• 9 D. Hinsinger, F. Neyret, M. Cani. Interactive
  Animation of Ocean Waves. SCA 02