CG [7] Thomaszewski [12] Baranoski [1] [2] (a) (b) (c) 3 a b c 3(a) E g 3(b) E mag 3(c) E s 3 2 [16] SPH SPH 1960 Rosenswig 4 [9] Sudo [11] Han

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1 1 Yonghao Yue SPH smoothed particle hydrodynamics Visual Simulation of the Magnetic Fluid to Represent the Spike Phenomenon Tomokazu Ishikawa, 1 Yonghao Yue, 1 Kei Iwasaki, 2 Yoshinori Dobashi 3 and Tomoyuki Nishita 1 In this paper, we focus on magnetic fluids. Magnetic fluids behave as both fluids and as magnetic bodies, and these characteristics allow them to generate spike-like shapes along a magnetic field. Magnetic fluids are popular materials for use in works of art. Our goal is to simulate such works of art. It is known, however, that the spikes are difficult to simulate using fully physical-based methods. Therefore, we propose a visually plausible method that combines a procedural approach together with the SPH (smoothed particle hydrodynamics) method. We demonstrate that the spike shapes can be simulated when a magnetic force is applied. 1. CG CG, Navier-Stokes,.,,,,. CG 1 NASA The University of Tokyo 2 Wakayama University 3 Hokkaido University 1 c 2011 Information Processing Society of Japan

2 CG [7] Thomaszewski [12] Baranoski [1] [2] (a) (b) (c) 3 a b c 3(a) E g 3(b) E mag 3(c) E s 3 2 [16] SPH SPH 1960 Rosenswig 4 [9] Sudo [11] Han [6] MPS Moving Particle Semi-implicit FEM Finite Element Method [13] 3. SPH 2 c 2011 Information Processing Society of Japan

3 3.1 Navier-Stokes m u = 0 (1) u t = (u )u 1 ρ p + ν 2 u + F (2) (1) (2) u t ρ p ν F Navier-Stokes SPH i x i F sur(x i) [8] F sur (x i ) = k j A i x ij x ij 2 (3) k A i i x ij = x j x i x ij = x j x i 3.2 SPH N S N S m = q m d (4) q m d p N p S d = p N p S x H dipole (x) H dipole (x) = 1 4πµ m x x 3 (5) µ x = x j H x j H(x j ) = H dipole (x j ) V 4πµ N i=1 i j χh(xi) xij V SPH x i i SPH N SPH (6) SPH χh(xi) xij χh(x i ) = χh(x i) χh(x i) ( x ij x ( x ij y ( x ij z ) + x ij ) + x ij ) + x ij H(x i) (χh(x x i)) (χh(xi)) y (χh(xi)) z (6) (7) 3 c 2011 Information Processing Society of Japan

4 (χh(x i )) = N j=1 j i m j ρ j χh(x j ) w(x ij ) (8) w(x ij ) { 315 (h 2 r 2 ) 3 0 r h 64πh w(r) = 9 0 h < r, (9) r h SPH SPH SPH 4 (6) RMSE SPH F mag(x i) F mag (x i ) = µ H(x i) 2 ϕ i = µ H(x i) 2 2 ϕ i 3.3 j 2 (10) (11) ϕ j ϕ i = m j w(x ij ) (12) ρ j 4 (6) [9] SPH z(x, y) [15] z(x, y) = (k 1,k 2 ) Ω C 0 (sin k 1 x + C 1 cos k 1 x)(sin k 2 y + C 2 cos k 2 y) (13) C 0 C 1 C 2 Ω k 2 1+k 2 2=k 2 x y xy 5 (13) z(x, y) = C 0(cos k 2 ( 3x + y) + cos k 2 ( 3x y) + cos ky) (14) (14) 6 4 c 2011 Information Processing Society of Japan

5 5 6 (14) 9 SPH (14) M c M c [15] 7 (14) 8 M 2 c = 2 µ (1 + 1 γ ) (ρ 1 ρ 2)gα, (16) (14) ( 5) [9] 7 (14) 7 l c 8 C 0 C 0 ρ 1 ρ 2 α x y x y x y 9 (14) 4. C 0 = βh(x) (15) β H(x) x Yu [14] SPH 5 c 2011 Information Processing Society of Japan

6 10 (a) (a) SPH (b) 1 dt time step ν 0.12 m R 0.5 h 1.3 g 9.8 k 7.5 q m 5.0 µ 4 π 10 7 χ 0.01 (b) POV-Ray 5. SPH CUDA ,960 Intel(R) Core(TM)2 Duo 3.33GHz CPU 3.25GB GPU NVIDIA GeForce GTX 480 PC (a) 10(b) (13). 1) G.V.G. Baranoski, J.G. Rokne, P.Shirley, T.S. Trondsen, and R.Bastos. Simulation the aurora. Visualization and Computer Animation, 14(1):43 59, c 2011 Information Processing Society of Japan

7 情報処理学会研究報告 (a) t = 0.0 sec (b) t = 1.6 sec 図 11 (a) t = 5.2 sec (c) t = 3.2 sec (d) t = 4.8 sec 磁性流体のスパイクの形成 スパイク形状は磁性流体の下部に磁石を近づけていくと成長する (b) t = 6.8 sec (c) t = 8.4 sec (d) t = 10.0 sec 図 12 磁場を除いた場合の磁性流体の動き 2) G.V.G. Baranoski, J.Wan, J.G. Rokne, and I.Bell. Simulating the dynamics of auroral phenomena. ACM Transactions on Graphics (TOG), 24(1):37 59, ) T. G. Goktekin, A. W. Bargteil, and J. F. O Brien. A method for animating viscoelastic fluids. In Proceedings of SIGGRAPH 2004, pages , ) M.D. Cowley and R.E. Rosensweig. The interfacial stability of a ferromagnetic fluid. Journal of Fluid Mechanics, 30(4): , ) K.Han, Y.T. Feng, and D.R.J. Owen. Three-dimensional modelling and simulation of magnetorheological fluids. International Journal for Numerical Methods in Engineering, 84(11): , ) R. Fedkiw, J. Stam, and H. W. Jensen. Visual simulation of smoke. In Eugene Fiume, editor, Proceedings of SIGGRAPH 2001, pages 15 22, ) T.Ishikawa, Y.Yue, Y. Dobashi, and T. Nishita. Visual simulation of solar photosphere based on magnetohydrodynamics. In Proceedings of IEVC IIEEJ, 7 c 2011 Information Processing Society of Japan

8 ) K.Iwasaki, H.Uchida, Y.Dobashi, and T.Nishita. Fast particle-based visual simulation of ice melting. Computer Graphics Forum (Pacific Graphics 2010), 29(7): , ) R.E. Rosensweig. Magnetic fluids. Annual Review of Fluid Mechanics, 19: , ) J.Stam. Stable fluids. In Proceedings of SIGGRAPH 1999, pages , ) S.Sudo, H.Hashimoto, A.Ikeda, and K.Katagiri. Some studies of magnetic liquid sloshing. Journal of Magnetism and Magnetic Materials, 65(2): , ) B.Thomaszewski, A.Gumann, S.Pabst, and W.Strasser. Magnets in motion. In Proceedings of SIGGRAPH Asia 2008, pages 162:1 162:9, ) G.Yoshikawa, K.Hirata, F.Miyasaka, and Y.Okaue. Numerical analysis of transitional behavior of ferrofluid employing mps method and fem. IEEE Transactions on Magnetics, 47(5): , ) J.Yu and G.Turk. Reconstructing surfaces of particle-based fluids using anisotropic kernels. In Proceedings of the 2010 ACM SIGGRAPH/Eurographics Symposium on Computer Animation, pages , ). 23., ).., c 2011 Information Processing Society of Japan

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