Fluid Simulation for Computer Graphics Robert Bridson, A K Peters, 2008: link to publisher's site.

From the preface:
This book is designed to give the reader a practical introduction to fluid simulation for graphics. The field of fluid dynamics, even just in animation, is vast and so not every topic will be covered, and many wonderful papers will sadly be passed over in the hope of distilling the essentials; this is far from a thorough survey. The focus of this book is animating fully three-dimensional incompressible flow---from understanding the math and the algorithms to actual implementation. However, there is also a small amount of material on height field simplifications which are important for efficiently modeling large bodies of water.

From the back cover:
"My colleagues and I have an alternate name for Bridson's 'Fluid Simulation for Computer Graphics' book; we like to call it 'How to Write a Production Quality Fluid Simulator'. I've been a huge fan of this text since it was SIGGRAPH course notes a couple of years ago. This book has amplified and improved on the work started back then. For a practitioner in this field, it provides everything needed: a good foundation in the theory of finite difference fluid simulation, practical working advice on how to deal with the mathematics of simulation, and clear explanations of advanced techniques (like solid/fluid coupling). All of this and it's very well written and a pleasure to read."
-- Doug Roble, Creative Director of Software, Digital Domain, Scientific and Technical Academy Award 2007

From the back cover:
"This book pulls together the essence of fluid simulation into a single volume, presenting it in a fashion that is both useful to researchers and accessible to implementers. I highly recommend it for anyone interested in fluid simulation for computer graphics."
-- Ron Fedkiw, Stanford University, Scientific and Technical Academy Award 2007

Errata and Extra Comments

Here are a few issues I didn't correct in time; please email me at rbridson@cs.ubc.ca if you find more problems or sources of confusion!

• 4.5.3 (p. 70): Note that in the right-hand-side of the equation, the coefficients in front of the solid velocities (differences of volume fractions) drop to zero away from the solid boundaries---thus in practice, you never need to worry about extrapolating solid velocities more than a grid cell width from the solid boundary.
• 6.1 (p. 85): The Zhu and Bridson formula is only well-defined near particles (where the denominator in the fraction is nonzero), and should default to some reasonable positive value such as the maximum particle radius elsewhere. Note this region should always be well outside the fluid (you don't want the particle radius to be larger than the support of the kernel function!), so the default value chosen doesn't matter too much as long as it's greater than zero, particularly if you subsequently reinitialize to signed distance on a grid. However, for a higher quality approach see Further Developments below.
• 6.2.4 (p. 93): For more details on allowing water drops to separate reasonably gracefully from solid surfaces, refer to [Rasmussen et al. 04]. For example, in semi-Lagrangian advection if phi is being interpolated inside a solid, if the fluid velocity dotted with the solid normal is greater than some positive tolerance---i.e. the fluid looks like it's separating from the wall---then phi should be set to the negative of the solid's level set function (a positive quantity); otherwise it's interpolated as usual, from the typically negative extrapolated liquid phi.
• 6.3 (p. 94): Missing from the fluid simulation loop are the details on moving solids and the adjustments to the liquid phi inside those solids. Conceptually the solids should be moved simultaneously in step 4 as the fluid variables are advected, so the semi-Lagrangian boundary condition handling will use the time n solid positions but the pressure solve will use the time n+1 solid positions. If you adjust the liquid phi from the solid level sets, do it between steps 4 and 5: after advection but before pressure projection.
• 6.3 (p. 94): The technique of using the normal component of the solid velocity but the tangential component extrapolated from the fluid should cite the technical sketch [Houston et al. 03]. However, as long as the linear system for pressure is solved reasonably accurately, this generally proves unnecessary.
• 6.4 (p. 97): The phrase "we can set p=0 in the grid cells that have no water at all" is ambiguous; we've seen fairly good results by specifying p=0 whenever one or more of the adjoining u-, v- or w-cells has zero liquid but isn't entirely occupied by solid. This is in fact consistent with the ghost fluid approach, which sets p=0 where the liquid level set function phi is postive.
• 7.2 (p. 104): In the definition of alpha, all the occurences of i+1,j,k should be subscripts of phi.
• 11.1 (p. 156): Note that the force of gravity should also be applied to particles with mass; the book only discusses the forces due to the fluid.
• 11.5 (p. 167): In the second paragraph, discussing the additional matrix term, the word density is referring to the large number of nonzeros in the matrix, not a physical ratio of mass to volume.
• 12.1.4 (p. 174): At the end of the section, the statement "the depth is zero (h=b)" should instead be "the height is constant". See section 12.3 where in the discretization we extrapolate height h, not depth. For an upward sloping beach, the depth will be considered negative in the dry areas.
• 13.1 (p. 180): The first momentum equation has the term grad phi instead of grad p: phi should only substitute in the velocity terms, not the pressure term. This is fixed in subsequent equations.
• 13.3 (p. 189): The second equation for h_pq is missing a factor of A_ij in the second term. This is corrected in the line below.
• A.1.3 (p. 197): The last formula on the page for curl should have the partial derivatives dz/du and dx/dv in place of "pdzu" and "pdxv" respectively.
• A.1.7 (p. 201): The free index in the matrix-vector product A_ij x_j is i, not j.
• A.1.7 (p. 202): The third expression in the list of examples is a slight abuse of notation: it would be better to write it as f(x_i) + df/dx_j dx_j, i.e. change the repeated i in the second term to a j.

Further Developments

This book covers much of the state-of-the-art in graphics, concentrating on grid-based 3D incompressible flow solvers, up to about January 2008. However, there were a few extra developments since then I'd ideally want included in the book:

• Fluid surface reconstruction from particles, MSc thesis by my past student Brent Williams, based on research also with Marcus Nordenstam: this covers a new method for generating high quality smooth surfaces from volumetric particle clouds. This can give significantly better results than the methods discussed in section 6.1 in the book: it's worked well on the big screen in a number of recent feature films such as Hell Boy II: The Golden Army.
• Accurate viscous free surfaces for buckling, coiling, and rotating liquids, C. Batty and R. Bridson, SCA 2008. This finally treats the free surface condition in viscous flow correctly, resolving phenomena such as honey coiling as it pours, slime properly oozing, etc. This mostly resolves the open questions in section 8.5.
• Evolving sub-grid turbulence for smoke animation, H. Schechter and R. Bridson, SCA 2008. This develops further quasi-procedural methods for adding turbulent detail to smoke, but perhaps even more importantly takes a first step at reducing the first order time splitting error that plagues virtually all grid-based fluid solvers in graphics, left as an open question in section 9.1.

Please note that, just like the book, this is not an attempt to reference all the great work being done in the area: it's again my personal view of some particularly useful starting points for advanced development in one approach to a practical high quality grid-based fluid solver. In general, it's worth checking out papers at graphics conferences for continuing advances---try the list maintained by Ke-Sen Huang at kesen.huang.googlepages.com.

Course Notes

This book grew out of my part of a set of notes for SIGGRAPH courses held in 2006 and 2007, available at www.cs.ubc.ca/~rbridson/fluidsimulation. The book has well more than double the material I provided and has corrected several small errors in the notes, but if you want a preview of the style of the text or a quick introduction to some of the basics of fluid simulation, you should take a look. The notes also have coverage of a few different algorithms, notably SPH, and on the web site there are are also some slides and a bit of code.

Robert Bridson - August 19, 2008