Fluid Simulation in Computer Graphics
Tomáš Skřivan
Fluid Simulation in Computer Graphics
Tomáš Skřivan
About me
- I like math and pretty pictures
- fluid simulation has both
- Work on Houdini at SideFX
- mainly working on FLIP solver
- Did internship at Weta Digital
- wave curve solver for Avatar 2
What are fluids?
- liquids like water, honey, gel, ...
- smoke, fire, ...
- almost anything
Two main computational approaches
Particle Based
Grid Based
- each particle has
- position
- velocity
- force
- each pixel/voxel has
- density
- velocity
- pressure
Particle Based
m_i a_i = F_i
F_i = \sum_j - \nabla_{p_i} E (\| p_i - p_j \|)
E(x) = e^{-x^2/r^2}
- Based on Newton's second law
- Force on each particle
p_i
p_j
\begin{align*}
v_i^{n+1} &= v_i^{n} + \Delta t \, a_i \\
p_i^{n+1} &= p_i^{n} + \Delta t \, v_i^{n}
\end{align*}
Particle Based
m_i a_i = F_i
F_i = \sum_j \nabla_{p_i} E (\| p_i - p_j \|)
E(x) = e^{-x^2/r^2}
- Based on Newton's second law
- Force on each particle
p_i
p_j
\begin{align*}
v_i^{n+1} &= v_i^{n} + \Delta t \, a_i \\
p_i^{n+1} &= p_i^{n} + \Delta t \, v_i^{n}
\end{align*}
Let's implement it!
Mathematical description "particle based"
Mathematical description
Mathematical description "particle based"
Mathematical description "particle based"
m \frac{d}{dt}
velocity
F
o
r
c
e
s
=
m a = F
Mathematical description "particle based"
Mathematical description "grid based"
Mathematical description "grid based"
Mathematical description "grid based"
Mathematical description
\frac{d}{dt} v = \frac{\partial v}{\partial t} + v \cdot \nabla v
F
o
r
c
e
s
=
\frac1m
Mathematical description
\frac{\partial v}{\partial t} + v \cdot \nabla v
\int_{\partial \Omega} N \cdot \sigma \, dA = \int_{\Omega} \text{div}\, \sigma \, dV
F = N \cdot \sigma
normal vector
stress tensor
(just 3x3 matrix)
F
o
r
c
e
s
=
\frac1m
=
= \frac1m \text{div} \, \sigma
F
o
r
c
e
s
Mathematical description
\frac{\partial v}{\partial t} + v \cdot \nabla v = \frac1\rho \text{div} \, \sigma
\sigma = - p I + \frac{\mu}{2} (\nabla v + \nabla v ^T)
\frac{\partial v}{\partial t} + v \cdot \nabla v = - \frac1\rho \nabla p + \nu \Delta v
Navier-Stokes equation:
\frac{\partial v}{\partial t} + v \cdot \nabla v = - \frac1\rho \nabla p + \nu \Delta v
Particle Based
Lagrangian methods
Grid Based
Eulerian methids
m \frac{d^2 p_i}{dt^2} = \sum_j - \nabla_{p_i} E (\| p_i - p_j \|)
Two main computational approaches
\frac{\partial v}{\partial t} + v \cdot \nabla v = - \frac1\rho p + \nu \Delta v
Grid Based Approach
\frac{\partial c}{\partial t} + v \cdot \nabla c = 0
Hybrid Approach
- particles for advection
- pressure solved on grid
Houdini Fluid Solver
Houdini Fluid Solver
particle to grid
external forces
pressure solve on the grid
grid to particles
particle advection
Thank You
For Your Attention