FLIP fluid simulation with sample code

I've been continuing to play around with incompressible fluid simulation and have implemented the FLIP method (FLuid Implicit Particle). The FLIP algorithm represents the fluid concentration and velocity field using a large set of particles.  At every iteration these particles are transferred to an auxiliary grid using kernel density estimation which defines a grid-based velocity field which is made incompressible by pressure projection and then used to advect the particles. Finally the velocity of each particle is corrected using the difference between the projected velocity field and the initial velocity field obtained by kernel density estimation.

The primary benefit of this approach is that it has extremely low numerical viscosity, making simulating fluids such as smoke and water possible at sane grid resolutions.  The downside is considerably more complexity than a purely grid-based solver; you need to track particles, perform the mapping between particles and grid (and vice versa) and have some sensible scheme to reseed particles when either there are too few or too many in a region.

To play with the method I implemented it in C++, largely following Robert Bridson's Fluid Simulation for Computer Graphics.  It is a canned example demonstrating variable density flow under the influence of gravity using an body force proportional to the fluid 'concentration', a sort of poor-man's Boussinesq approximation.  The simulation below was performed on a 100x100 auxiliary grid with 16 particles per grid-cell. This is more than the 2x2 typically used, but helps get around reseeding.  As this is not free-surface flow, there are particles everywhere which allows causes the smoke to form nice instabilities and finally turbulent mixing (although I did not run it particularly far).

In order to have an unmangled version of the code available to myself, I am posting it here.  It is uncommented and unstructured, but (hopefully) fairly understandable once you understand the basic algorithm.  The code is bundled with the GMM++ sparse matrix library for the pressure solves and required VTK and CMake for output and as a build-system respectively.

You can download the code here: FLIP2.zip

Latex formulas as images using Python

I find it endlessly frustrating to incorporate math into vector-graphics documents and Powerpoint presentations. While Powerpoint does allow importing equations from the equation editor, they usually look terrible.  However there is an online Latex equation editor run by codecogs.com that allows you to generate images of expressions.  This works great, but it's inconvenient to save and regenerate images after using it.

To that end, I've written a python function that will generate the URL for the Latex image-generation script, get the image data using HTTP and write it to disk as a file of your choosing.  It uses the python requests module (available via pip) and ImageMagick:

import os, requests

def formula_as_file( formula, file, negate=False ):
    tfile = file
    if negate:
        tfile = 'tmp.png'
    r = requests.get( 'http://latex.codecogs.com/png.latex?\dpi{300} \huge %s' % formula )
    f = open( tfile, 'wb' )
    f.write( r.content )
    f.close()
    if negate:
        os.system( 'convert tmp.png -channel RGB -negate -colorspace rgb %s' %file )

The code is appallingly simple for how much time it has saved me; it allows me to regenerate formulas for conference posters by simply running a python script and since most vector-graphics packages simply link to bitmap images allows the resulting layouts to update automatically.  Even better is that CodeCogs image generation script produces transparent PNGs so you can drop them on top of lightly colored backgrounds.  By setting the negate flag you will get a white on transparent PNG that can be used over dark backgrounds.

Here are some examples:

formula_as_file( r'\Gamma_{Levin}(x) = \| \nabla p(x) \|_2^{0.8} + \sum_i |\frac{\partial^2 p(x)}{\partial x_i^2}|^{0.8}', 'reg_levin.png', True )

formula_as_file( r'\Gamma_{MTV}(x) = \| \sum_{i=1}^3 \left( \| \nabla p^{(i)}(x)\|_2^2 \right) \|_2', 'reg_MTV.png' )

Finally no more ugly power-point presentations!