introduction to modeling fluid dynamics
DESCRIPTION
Introduction to Modeling Fluid Dynamics. Different Kind of Problem. Can be particles, but lots of them Solve instead on a uniform grid. Particle Mass Velocity Position. Fluid Density Velocity Field Pressure Viscosity. No Particles => New State. No Particles => New Equations. - PowerPoint PPT PresentationTRANSCRIPT
The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Introduction to Modeling Fluid Dynamics
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Different Kind of Problem• Can be particles, but lots of
them• Solve instead on a uniform
grid
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No Particles => New StateParticle• Mass• Velocity• Position
Fluid• Density• Velocity Field• Pressure• Viscosity
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No Particles => New Equations
Navier-Stokes equations for viscous, incompressible liquids.
fuuuu
u
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What goes in must come out
Gradient of the velocity field= 0 Conservation of Mass
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Time derivative
Time derivative of velocity field Think acceleration
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u
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au
t
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Advection term
Field is advected through itself Velocity goes with the flow
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u
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Diffusion term
Kinematic Viscosity times Laplacian of uDifferences in Velocity damp out
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u
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Pressure term
Fluid moves from high pressure to low pressureInversely proportional to fluid density,
ρ
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External Force Term
Can be or represent anythyingUsed for gravity or to let animator “stir”
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Navier-Stokes
How do we solve these equations?
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Discretizing in space and time
• We have differential equations
• We need to put them in a form we can compute
• Discetization – Finite Difference Method
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Discretize in Space
X Velocity
Y Velocity
Pressure
Staggered Grid vs Regular
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Discretize the operators
• Just look them up or derive them with multidimensional Taylor Expansion
• Be careful if you used a staggered grid
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Example 2D Discetizations
-1 0 1
1
-1
1 -4 1
1
1
Divergence Operator
Laplacian Operator
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Make a linear system
It all boils down to Ax=b.
dddd nnxnnx
b
b
x
x
x
2
1
2
1
??
??
???
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Simple Linear System
• Exact solution takes O(n3) time where n is number of cells
• In 3D k3 cells where k is discretization on each axis
• Way too slow O(n9)
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Need faster solver
• Our matrix is symmetric and positive definite….This means we can use♦ Conjugate Gradient
• Multigrid also an option – better asymptotic, but slower in practice.
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Time Integration
• Solver gives us time derivative
• Use it to update the system state
U(t+Δt)
U t
U(t)
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Discetize in Time
• Use some system such as forward Euler.
• RK methods are bad because derivatives are expensive
• Be careful of timestep
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Time/Space relation?
• Courant-Friedrichs-Lewy (CFL) condition
• Comes from the advection term
u
xt
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Now we have a CFD simulator
• We can simulate fluid using only the aforementioned parts so far
• This would be like Foster & Metaxas first full 3D simulator
• What if we want it real-time?
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Time for Graphics Hacks
• Unconditionally stable advection♦ Kills the CFL condition
• Split the operators♦ Lets us run simpler solvers
• Impose divergence free field♦ Do as post process
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Semi-lagrangian Advection
CFL Condition limits speed of information travel forward in time
Like backward Euler, what if instead we trace back in time?
p(x,t) back-trace
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Divergence Free Field
• Helmholtz-Hodge Decomposition♦ Every field can be written as
• w is any vector field• u is a divergence free field• q is a scalar field
quw
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Helmholtz-Hodge
STAM 2003
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Divergence Free Field
• We have w and we want u
• Projection step solves this equation
q
q
q
2
2
w
uw
uw
qwu
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Ensures Mass Conservation• Applied to field before
advection• Applied at the end of a step
• Takes the place of first equation in Navier-Stokes
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Operator Splitting
• We can’t use semi-lagrangian advection with a Poisson solver
• We have to solve the problem in phases
• Introduces another source of error, first order approximation
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Operator Splitting
0 u
uu u2 p1 ftu
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Operator Splitting
1. Add External Forces
2. Semi-lagrangian advection
3. Diffusion solve
4. Project field
f
uu
u20 u
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Operator Splitting
u2f
uu
W0 W1 W2 W3 W4
u(x,t)
u(x,t+Δt)
0 u
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Various Extensions
• Free surface tracking• Inviscid Navier-Stokes• Solid Fluid interaction
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Free Surfaces
• Level sets ♦ Loses volume♦ Poor surface detail
• Particle-level sets♦ Still loses volume♦ Osher, Stanley, & Fedkiw, 2002
• MAC grid♦ Harlow, F.H. and Welch, J.E., "Numerical
Calculation of Time-Dependent Viscous Incompressible Flow of Fluid with a Free Surface", The Physics of Fluids 8, 2182-2189 (1965).
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Free Surfaces
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MAC Grid Level Set
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Inviscid Navier-Stokes
• Can be run faster• Only 1 Poisson Solve needed• Useful to model smoke and
fire♦ Fedkiw, Stam, Jensen 2001
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Solid Fluid Interaction
• Long history in CFD• Graphics has many papers on
1 way coupling♦ Way back to Foster & Metaxas, 1996
• Two way coupling is a new area in past 3-4 years♦ Carlson 2004
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Where to get more info
• Simplest way to working fluid simulator (Even has code)♦ STAM 2003
• Best way to learn enough to be dangerous♦ CARLSON 2004
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ReferencesCARLSON, M., “Rigid, Melting, and Flowing Fluid,” PhD Thesis, Georgia Institute of Technology, Jul. 2004.
FEDKIW, R., STAM, J., and JENSEN, H. W., “Visual simulation of smoke,” in Proceedings of ACM SIGGRAPH 2001, Computer Graphics Proceedings, Annual Conference Series, pp. 15–22, Aug. 2001.
FOSTER, N. and METAXAS, D., “Realistic animation of liquids,” Graphical Models and Image Processing, vol. 58, no. 5, pp. 471–483, 1996.
HARLOW, F.H. and WELCH, J.E., "Numerical Calculation of Time-Dependent Viscous Incompressible Flow of Fluid with a Free Surface", The Physics of Fluids 8, 2182-2189 (1965).
LOSASSO, F., GIBOU, F., and FEDKIW, R., “Simulating water and smoke with an octree data structure,” ACM Transactions on Graphics, vol. 23, pp. 457–462, Aug. 2004.
OSHER, STANLEY J. & FEDKIW, R. (2002). Level Set Methods and Dynamic Implicit Surfaces. Springer-Verlag.
STAM, J., “Real-time fluid dynamics for games,” in Proceedings of the Game Developer Conference, Mar. 2003.