FLOW SIMULATIONS USING PARTICLES Bridging Computer Graphics by Petros Koumoutsakos,Georges-Henri Cottet,Diego Rossinelli

By Petros Koumoutsakos,Georges-Henri Cottet,Diego Rossinelli

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As we are solving the conservation law formulation (76) to extend both the concentrations c and the velocities u off the interface Γ, while we require this extension to satisfy the requirements (75). As we are only interested in the concentrations on Γ it suffices to extend the quantities into a narrow band around the level set, which we define as Γe = x |ϕ(x)| ≤ γ . (80) All calculations are restricted to this narrow band. We periodically extend the concentrations by using solving the following PDE: ∂cs + sign(ϕ)∇ϕ · ∇cs = 0 , ∂τ (81) As particles follow the flow map u their locations eventually become distorted and need to be regularized to ascertain convergence [48].

37 Figure 19: Zalesak’s Slotted Sphere. Top: Eulerian Level Sets using 100x100x100 cells (from Enright et al. [27]). Middle: Hybrid Eulerian-Particle Level Sets of Enright et al. [27] using 100x100x100 cells and subscale particles. Bottom: Lagrangian Particle Level Sets [37]. The simulation uses 24351 Lagrangian particles in a narrow band, with an effective resolution of 64x64x64 cells. 38 The equations for the evolution of level sets can be cast, however in a lagrangian form enabling the use of all the tools, including remeshing, that we have developed so far in particle methods.

1 involves maps for which the3. Jacobian is readily avail- to Ω with adapted map able. fIf the mapping is invertible, it can be changed at any time during a simulation. and differentiable. a map asinverse described ˜ in transfer particles to Ω using an adapted map g ; (45) makes it impossible ˆ and perform diffusion; 2. remesh in Ω to leap back and forth particles from physical to reference˜ space. However its differentiability 3. transfer particles to Ω with adapted map f . enables us to cast the governing equations into reference space and solve the problem Like that, analytic, invertible maps could be dynamically adapted to the flow field.

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