1798 July ) 13) Ota 7) Giacomo 9) 14) 17) 2. Navier-Stokes 2.1 Navier-Stokes 18) 3 Navier-Stokes V t + (V V) = 1 ρ p + ν 2 V (1) t V= (u, v, w)
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1 Vol. 46 No. 7 July 2005 CG Navier-Stokes Computer Animation of Swaying Trees Based on Physical Simulation Yasuhiro Akagi, Syo Sanami and Katsuhiro Kitajima This paper presents a series of techniques for generating animations of trees swaying in the wind, in consideration of the influences that the tree shapes and leaf sizes give to the air current. To do the simulation of the wind around a tree having a complicated shape, it is necessary to consider the influence that some objects obstructing the wind such as leaves or branches give. Generally, the following problem occurs when we use the incompressible Navier-Stokes equations in a physical simulation model of the wind. Computational complexity increases because of considering the details of tree shapes, so it is difficult to generate the animations in real-time. Therefore, this paper proposes a novel method that reduces the computational complexity and realizes an animation in real-time, by means of a boundary condition map expressing space distribution of resistances from tree models automatically. In this case, we make a model as simple resistances decreasing the wind velocity from the parts that have similar shapes like leaves and branches. And also, it has another advantage that the influences between a tree and others can be rapidly calculated by using a hierarchical calculation method. Finally, through many experiments using these methods, it is shown that real-time animations of swaying trees in the wind can be realized. 1. CG Weber 1) Tokyo Agriculture and Technology University 2) 4) 5),6) 1797
2 1798 July ) 13) Ota 7) Giacomo 9) 14) 17) 2. Navier-Stokes 2.1 Navier-Stokes 18) 3 Navier-Stokes V t + (V V) = 1 ρ p + ν 2 V (1) t V= (u, v, w) p ρ ν = (,, x y z) (1) V V (2) V = u x + v y + w =0 (2) z (1) (2) Navier-Stokes (1) (2) 19) SMAC 20) (1) (2) Navier-Stokes SMAC SMAC 1 SMAC
3 Vol. 46 No n (5) p t+1 p t V t (3) ( V t = V t + t 1 ρ pt (V t V t ) ) + ν 2 V t (6) 1 Fig. 1 Staggered cell. (1) t V V t t +1 V t+1 ( V t+1 = V t + t 1 ρ pt+1 (V t V t ) + ν 2 V t ) (3) (3) (V t V t ) ν 2 V t p t+1 (3) divergence V t+1 V t t ( ) p = 2 t+1 ρ + ( (V t V t )+ν 2 V t) (4) V (2) V t 0 p t+1 V t V t+1 =0 (4) V t+1 =0 ( ) p 2 t+1 = 1 ρ t Vt + ( (V t V t )+ν 2 V t) (5) (5) t n V t+1 = V t t φ (7) φ = pt+1 p t ρ (8) (6) Navier-Stokes t +1 p t (7) V t φ Navier-Stokes V t+1 (7) (5) 2 φ = 1 t Vt (9) (9) V t φ (i, j, k) φ i 1,j,k 2φ i,j,k + φ i+1,j,k x 2 + φ i,j 1,k 2φ i,j,k + φ i,j+1,k y 2 + φ i,j,k 1 2φ i,j,k + φ i,j,k+1 z ( 2 = 1 u t + i 1 ut,j,k i ,j,k t x + + v t + i,j 1 vt,k i,j ,k y w t i,j,k 1 + w t i,j,k z ) (10) φ SOR φ (5) p t+1 (8) p t+1 = ρφ + p t p t+1 (7) V t+1 V t+1
4 1800 July Sommerfeld No-Slip Navier- Stokes V t (6) ) 1 2 Fig. 2 Bend of branch CG Terashima 22)
5 Vol. 46 No Table 1 1 Calculation time of the wind with a grid size. grid size calculation time (sec.) 3 Fig. 3 Transmission of a force added to a branch. 4 Fig. 4 Tree model ) m 1 SMAC cm 1 1 1
6 1802 July 2005 M F l a: Ordinary grids and boundary conditions to compute the wind. b: Using virtual resistance to reduce the amount of grids. 5 Fig. 5 Virtual resistance F l = M Vt (11) t F l (6) SMAC CG No-Slip Sc Sb V t+1 t = Vt t Sb Sc (12) (12) V t (1) (2) SMAC (3) (6) (4) (5)
7 Vol. 46 No Level of Detail LOD LOD LOD LOD LOD , Table 2 2 Specification of the experiment environment. CPU Pentium GHz Memory 512 Mbyte Fig. 6 6 Hierarchical calculation. VGA OS Radeon9600 Windows2000
8 1804 July 2005 Wind velocity (a: 0m/s, b: 5m/s, c: 10m/s, d: 20m/s) Fig. 7 7 Variations of the bend of a branch by each wind velocity. Wind velocity (b: 5 m/s, c: 10 m/s, d: 20 m/s) 9 8 Fig. 9 Motion of a branch in each tree of Fig. 8. Wind velocity (a: 0m/s, b: 5m/s, c: 10m/s, d: 20m/s) 8 Fig. 8 Differences of the sway of a tree by each wind velocity. a: Normal size b: Large size 10 Fig. 10 Difference of the sway of a tree at each leaf size b 5m/s 8 c 10 m/s 8 d 20 m/s a b 2 a b a b b (11) 10 a
9 Vol. 46 No a: No wall b: Wall is placed 11 Fig. 11 Motion of a branch with large leaves. 12 Fig. 12 Influence of a wall on the sway of a tree, when it is placed on the upwind side. a 1/ a b b b a a: Upper wind b: Lower wind 13 Fig. 13 Influence from the tree on the upwind side on the difference of the sway of branches SMAC fps 1 SMAC
10 1806 July Table 3 Creation time of the animation at each grid size. grid size BCM (sec.) SMAC (sec.) BCM: Boundary Condition Map Grid size (a: 5 3,b:10 3,c: 20 3 ) Fig Difference of the sway of a tree at each grid size. 14 Fig. 14 Experimental verification of the acceleration technique. 4 Table 4 Calculation time in each case. Method Calculation time (s) BCM 3.43 BCM+HC BCM+HC+LOD BCM: Boundary Condition Map HC: Hierarchical Calculation LOD: Level of Detail LOD LOD 14 4 LOD LOD LOD m 80 cm 5cm
11 Vol. 46 No SMAC LOD 5. (1) Navier-Stokes SMAC (2) (3) LOD 2 (4) ) Weber, J. and Penn, J.: Creation and rendering of realistic trees, Proc. SIGGRAPH 95 Conference, pp (1995). 2) Soler, C., Sillion, F., Blaise, F. and Dereffye: A physiological plant growth simulation engine based on accurate radiant energy transfer, Tech. Rep., Vol.4116, INRIA (2001). 3) CG D-II Vol.76, No.8, pp (1993). 4) D-II Vol.79, No.8, pp (1996). 5) MVE pp (1997). 6) Shlyakhter, I., Rozenoer, M., Dorsey, J. and Teller, S.: Reconstructing 3D Tree Models from Instrumented Photographs, IEEE Computer Graphics and Applications, Vol.21, No.3, pp (2001). 7) Ota, S., Tamura, M., Fujita, K., Muraoka, K., Fujimoto, T. and Chiba, N.: 1/f β Noise- Based Real-Time Animation of Trees Swaying in Wind Fields, Proc. Computer Graphics International, pp (2003). 8) Shinya, M. and Fournier, A.: Stochastic motion-motion under the influence of wind, Computer Graphics Forum, pp.c119 C128 (1992). 9) Giacomo, T.D., Capo, S. and Faure, F.: An Interactive Forest, Eurographics Workshop on
12 1808 July 2005 Computer Animation and Simulation, pp (2001). 10) Sakaguchi, T. and Ohya, J.: Modeling and Animation of Botanical Trees for Interactive Virtual Environments, Symposium on Virtual Reality Software and Technology, pp (1999). 11) D-II Vol.80, No.7, pp (1997). 12) Perbet, F. and Cani, M.: Animating Prairies in Real-Time, Proc. Conference on the 2001 Symposium on Interactive 3D Graphics, pp (2001). 13) I Vol.22, No.5, pp (1993). 14) Stam, J.: Stable Fluids, SIGGRAPH 99, pp (1999). 15) Foster, N.: Animation and Rendering of Complex Water Surfaces, SIGGRAPH 02, pp (2002). 16) Witting, P.: Computational Fluid Dynamics in a Traditional Animation Environment, SIGGRAPH 99, pp (1999). 17) Rudolf, M.J. and JacekRaczkowski: Modeling the Motion of Dense Smoke in the Wind Field, Computer Graphics Forum 2000, Vol.19, No.3, pp (2000). 18) (1995). 19) (1999). 20) Amsden, A.A., Harlow and F.H: The SMAC Method: A Numerical Technique for Calculating Incompressible Fluid Flows, LA-4370 (1970). 21) Perttunen, J., Sievanen, R. and Nikinmaa, E.: LIGNUM: A tree model based on simple structural units, Annals of Botany, Vol.77, pp (1996). 22) Terashima, I., Kimura, K., Sone, K., Noguchi, K., Ishida, A., Uemura, A. and Matsumoto, Y.: Differential analyses of the effects of the light environment on development of deciduous trees: Basic studies for tree growth modeling, Diversity and Interaction in a Temperate Forest Community, Vol.158, pp (2002). 23) Akagi, Y., Sanami, S. and Kitajima, K.: Study on generation of tree shapes with analysis of common features of species, NICOGRAPH International 2005, pp (2005). 16 Fig. 16 Trees like a Japanese cedar. 17 Fig. 17 Trees like a camphor tree. A.1 4 A A ( ) ( ) CG CAD
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82016 pp. 45~58 STUDIES IN ART 8 Bulletin of Tamagawa University, College of Arts 2016 This study aimed to identify the contemporary issues considered necessary in today s keyboard harmonica instruction,
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See Boundary space and human consciousness from holy place and monster s distribution MATSUI Koichi TAKAHASHI Seiichi Amami Oshima has created own unique culture, which is based on Ryukyu culture and also
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THE INSTITUTE OF ELECTRONICS, INFORMATION AND COMMUNICATION ENGINEERS TECHNICAL REPORT OF IEICE.,, 464 8601 470 0393 101 464 8601 E-mail: matsunagah@murase.m.is.nagoya-u.ac.jp, {ide,murase,hirayama}@is.nagoya-u.ac.jp,
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Kanji Vocabulary Development for Intermediate and Advanced Learners Kano, Chieko In recent Japanese language teaching, a communicative approach based on the functions and needs of actual situations is
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