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 Yonghao Yue 1 2 3 1 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 1960 2 1 The University of Tokyo 2 Wakayama University 3 Hokkaido University 1 c 2011 Information Processing Society of Japan

2. 1 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 10 25 [13] 3. SPH 2 c 2011 Information Processing Society of Japan

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

(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 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

10 (a) (a) SPH (b) 1 dt time step 0.00075 ν 0.12 m 0.016 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) 11 11 12 6. POV-Ray 5. SPH CUDA 11 12 40,960 Intel(R) Core(TM)2 Duo 3.33GHz CPU 3.25GB GPU NVIDIA GeForce GTX 480 PC 1 6 1 1 2 10(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, 2003. 6 c 2011 Information Processing Society of Japan

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, 2005. 5) T. G. Goktekin, A. W. Bargteil, and J. F. O Brien. A method for animating viscoelastic fluids. In Proceedings of SIGGRAPH 2004, pages 463 468, 2004. 3) M.D. Cowley and R.E. Rosensweig. The interfacial stability of a ferromagnetic fluid. Journal of Fluid Mechanics, 30(4):671 688, 1967. 6) 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):1273 1302, 2010. 4) R. Fedkiw, J. Stam, and H. W. Jensen. Visual simulation of smoke. In Eugene Fiume, editor, Proceedings of SIGGRAPH 2001, pages 15 22, 2001. 7) T.Ishikawa, Y.Yue, Y. Dobashi, and T. Nishita. Visual simulation of solar photosphere based on magnetohydrodynamics. In Proceedings of IEVC 2010. IIEEJ, 7 c 2011 Information Processing Society of Japan

2010. 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):2215 2223, 2010. 9) R.E. Rosensweig. Magnetic fluids. Annual Review of Fluid Mechanics, 19:437 461, 1987. 10) J.Stam. Stable fluids. In Proceedings of SIGGRAPH 1999, pages 121 128, 1999. 11) S.Sudo, H.Hashimoto, A.Ikeda, and K.Katagiri. Some studies of magnetic liquid sloshing. Journal of Magnetism and Magnetic Materials, 65(2):219 222, 1987. 12) B.Thomaszewski, A.Gumann, S.Pabst, and W.Strasser. Magnets in motion. In Proceedings of SIGGRAPH Asia 2008, pages 162:1 162:9, 2008. 13) 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):1370 1373, 2011. 14) 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 217 225, 2010. 15). 23., 1989. 16).., 1988. 8 c 2011 Information Processing Society of Japan

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