1. Stable, Circulation- Preserving, Simplicial Fluids Sharif Elcott, Yiying Tong, Eva Kanso, Peter Schröder, and Mathieu Desbrun

2. “Simplicial”  A k-simplex is a convex combination of k+1 points  0-simplex -> vertex  1-simplex -> edge  2-simplex -> triangle  A simplicial complex is just a space constructed from a combination of k-simplices  A k-form is an object that is integrated over a k-simplex  0-forms live on vertices  1-forms lives on edges  etc…

3. “Circulation-Preserving”  Circulation is just the flow around the boundary of some loop  The line integral around a closed curve of a velocity field  Kelvin’s Theorem  The circulation around a loop is conserved as the loop gets advected through the fluid.

4. “Stable”  Want numerical stability even for large timesteps  “Stable Fluids” algorithm by Stam 1999  Set up a grid and velocity field  Update velocity field by backtracking – “where did this particle come from?”  Unconditionally stable, since the maximum velocity value never larger than the maximum of the previous field.  Problem: Using a velocity field disregards divergence free constraint  Incompressible Fluids: “What goes in must come out”  Result: significant energy loss and excessive diffusion of vorticity

5. Stable, Circulation-Preserving, Simplicial Fluids  This method will conserve defining physical properties  No previous efforts to conserve fundamental invariants of fluid motion  Exactly preserves circulation along discrete loops  Avoid visually disturbing diffusion of vorticity  Intrinsic representation yields easier implementation for arbitrary topologies and boundary conditions  Similar computational cost to Stable Fluids

6. Euler Fluids  Vorticity  Physical intuition: Local spinning motion of a parcel of fluid  Euler’s Equation for fluid motion:  u = velocity  p = pressure  (Assuming unit density)

7. Euler Fluids  Vorticity is the curl of velocity  Taking the curl, we obtain:  Where L u is the Lie derivative  Implies that vorticity is simply advected along the flow  ie, local spin moves with the fluid flow

8. Euler Fluids  Stokes’ Theorem:  Integrated vorticity on a bounded surface = circulation along the boundary of that surface  If vorticity remains constant as it moves with the flow, then the circulation of the loop remains constant  Agrees with Kelvin’s Theorem!

9. Discrete Setup  Let M be a 2D triangle mesh of the space of the flow  We will also need a dual mesh  For 2D case:  Primal vertex (0-form) -> dual face (2-form)  Primal edge (1-form) -> dual edge (1-form)  Primal face (2-form) -> dual vertex (0-form)  We will store physical quantities (velocity, vorticity) intrinsically on the mesh at any simplex

10. What goes where?  Velocity as discrete flux (on edges)  Let  ij be the flux across the edge from vertex i to vertex j  Divergence as net flux (on triangles)  What goes in, must come out -> divergence-free  Due to incompressibility of the fluid  Sum of fluxes of a triangle must be 0  Vorticity as flux spin (on primal vertices/dual faces)  If we know the circulation around the loop, we know how much it will spin Φ

11. Operators  d – discrete exterior derivative  Implemented via incidence matrices  Key: Stokes’ Theorem  For any k-form , and k+1 dimensional  domain  Note: then the boundary of the domain is of dimension k  We can define d if we know how to evaluate the integral along the boundary!  Ex. Let f be a function with values at vertices (a 0-form)  Then, df is a 1-form that can be integrated along the edge

12. Operators  d goes from a k-form to a k+1 form  d 0 : maps 0-forms to 1-forms, corresponding to Gradient  d 1 : maps 1-forms to 2-forms, corresponding to Curl  We will also need to consider the codifferential operator:  Adjoint of the exterior derivative  Define the Laplacian:

13. Operators  * – Hodge Star  Takes us from the primal -> dual mesh (and vice versa)  Primal vertex (0-form) dual face (2-form)  Primal edge (1-form) dual edge (1-form)  Primal face (2-form) dual vertex (0-form)  diagonal Hodge Star  Matrix with only entries along the diagonal  is the volume of a dual cell (1 for vertices, length for edges, area for faces)  is the volume of a primal cell (1 for vertices, length for edges, area for faces)

14. Geometric Integration of Fluid Motion  Goal: ensure discrete version of Kelvin’s Circulation Theorem  Circulation (and therefore vorticity by Stokes’ Theorem!) is preserved along every loop  Backtrack loop of dual vertices in the current interpolated velocity field  We know where each dual face came from, so we can calculate the circulation.  Assign this value as circulation around original loop at current time. Preserve circulation by construction!

15. The Algorithm

16.  But now what?  The goal is to update the velocity field. We’ve calculated circulation/vorticity for the backtracked loop, and assigned it to the current loop.  We know how to convert from velocity to vorticity:  But how do we convert back our updated vorticites to a new velocity field (ie new fluxes across edges)?

17. Helmholtz- Hodge Decomposition  Any vector field can be decomposed into 3 components: a scalar potential, a vector potential, and a harmonic part.  In terms of the exterior derivative and codifferential:  Drop h, since the harmonic part only matters when boundary is nontrivial  Taking derivative:  Laplacian:

18. Helmholtz- Hodge Decomposition  If the Laplacian is 0:  Now, the decomposition is just:  Recall that vorticity is:  Given vorticity, we have a classic Poisson problem:  Result? We can recover flux from vorticity and thus update the velocity field after each timestep!

19. Results  New method that exactly preserves discrete vorticity  High degree of visual realism  Works on arbitrary simplicial meshes  Everything is stored intrinsically!