IPSJ SIG Technical Report Vol.2014-CG-155 No /6/28 1,a) 1,2,3 1 3,4 CG An Interpolation Method of Different Flow Fields using Polar Inter

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1 ,a),2,3 3,4 CG An Interpolation Method of Different Flow Fields using Polar Interpolation Syuhei Sato,a) Yoshinori Dobashi,2,3 Tsuyoshi Yamamoto Tomoyuki Nishita 3,4 Abstract: Recently, realistic fluid animations are often used in movies or video games. These fluid animations are usually generated by using fluid simulation, but computational costs are too expensive for creating realistic animations. Therefore, if the user would like to obtain results different from simulated results, the user must execute fluid simulation repeatedly and must take a long time. To address this problem, this paper proposes a method for interpolating between two different velocity fields of fluid. By using our system, the user can create various fluid animations from the two input velocity fields, without executing fluid simulation. In our method, control points are placed at the positions where features of these two velocity fields are similar, we interpolate between these control points by using polar interpolation. Then, by deforming the two velocity fields following the interpolated control points, plausible velocity fields are interpolated. Keywords: fluid simulation, interpolation of flow fields, polar decomposition interpolation. (CG) Hokkaido University, Sapporo, Hokkaido , Japan 2 JST CREST 3 UEI Research 4 Hiroshima shudo University, Hiroshima, Hiroshima , Japan a) [][2][4][5] c 204 Information Processing Society of Japan

2 [6][7] [3] Stam Navier-Stokes CG [5] Stam [] [6][7] Navier-Stokes [3] 3. Navier-Stokes NS u t = (u )u ρ p + ν 2 u + f () u = 0 (2) u ρ p ν f () (2) Fedkiw [2] NS D t = (u )D + D s (3) D D s () (2) 2 c 204 Information Processing Society of Japan 2

3 ~ (-t) v t x 0 v v x t ~ v x t t(0 < t < ) 2 t (n) v (n) v n n = 0,, 2,..., N N v x 0 x p p = 0,, 2,, P P 2 6. x 0 x 2 x t x 0 x x 0 = (r 0 cos(θ 0 ), r 0 sin(θ 0 )) x = (r cos(θ ), r sin(θ )) r 0 r θ 0 θ (, 0) θ = 0 r 0 r θ 0 θ x t r t θ t r t = ( t)r 0 + tr θ t = ( t)θ 0 + tθ x 0 x 2 c 204 Information Processing Society of Japan 3

4 φ 0 x 0 x φ 2 ϕ 2 x 0 x ϕ 0 ϕ ( 2 ) ϕ t (a) ϕ t = ( t)ϕ 0 + tϕ (n) v (n) v t (n) 7. r t θ t ϕ t v t (n) (n) ṽ 0 (n) v (n) ṽ (n) v t (n) v t (n) = ( t)ṽ 0 (n) + tṽ (n) (4) 3 (b) ṽ 0 (n) ṽ (n) r t θ t x x 0 x x 0 = ((r + r x,0 ) cos(θ + θ x,0 ), (r + r x,0 ) sin(θ + θ x,0 )) x = ((r + r x, ) cos(θ + θ x, ), (r + r x, ) sin(θ + θ x, )) r θ x = (r cos θ, r sin θ) r x,a θ x,a (a = 0or) r x,a = p θ x,a = p λ x,a r a λ x,a θ a λ x,a = w x,a q w x,a(q) w x,a = x a x q q = 0,, 2,, P x x ϕ x,a λ x,a c 204 Information Processing Society of Japan 4

5 ϕ x,a = p λ x,a ϕ a ϕ x,a x a ṽ 0 (n) ṽ (n) v t (n) 8. 3 CPU Intel Core i7 2600K 6GB GPU NVIDIA GeForce GTX 680 3(a) (b) (a) (b) 2 t = 0.5 3(b) r t r t CREST [] R. Bridson : Fluid Simulation for Computer Graphics, AK Peters (2008). [2] R. Fedkiw, J. Stam, and H.W. Jensen : Visual simulation of smoke, In Proc. SIGGRAPH 200, 5-22 (200). [3] T. Kim, and J. Delaney : Subspace fluid re-simulation, ACM Transactions on Graphics 32, 4, Article 62 (203). [4] D.Q. Nguyen, R. Fedkiw, H.W. Jensen : Physically Based Modeling and Animation of Fire, In Proceeding of ACM SIGGRAPH 2002, , (2002) [5] J. Stam : Stable fluids, In Proceedings of ACM SIG- GRAPH 999, Annual Conference Series, 2-28 (999). [6] A. Treuille and A. Lewis and Z. Popovic : Model reduction for real-time fluids, ACM Transactions on Graphics 25, 3, (2006). [7] M. Wicke and M. Stanton and A. Treuille : Modular bases for fluid dynamics, ACM Transactions on Graphics 28, 3, Article 39 (2009) () (2) c 204 Information Processing Society of Japan 5

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