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2 H. Kuninaka and H. Hayakawa Phys. Rev. Lett. 93, 543 (4) M. Y. Louge & E. Adams Phys. Rev. E 65, 33 () Y.Tanaka et. al. Europhys. Lett. 63, 46 (3) K. Okumura et. al. Europhys. Lett. 6, 37 (3) R +X h R +X h (a) (b) J. W. Glasheen and T. A. McMahon, Nature, 38 (996) C. Clanet, F. Hersen and L. Bocquet. Nature, 47 (4)
3 Weight[Cwt.] distance [Yards] Number of ricochet θ c = 8/ σ σ = c
4 G. Birkhoff et. al. (944) W. Johnson & S. R. Reid (975) I. M. Hutchings (976) 円柱衝突の理論 Birkhoffモデル y O y dv m = dt w! pn ds x 4 E. G. Richardson, Proc. Phys. Soc. London, Sect. A 6 (948) surface π sin α pn = ρv 4 + π sin α (Rayleighの式) θc = 7.8/ σ 年月9日木曜日 Flow direction
5 v F = v gl v C. Clanet, F. Hersen and L. Bocquet. Nature, 47 (4)
6 入射角度 θ [deg.] Umin[m/sec] 石の水切りの実験 Skipping Stone 円盤の傾き角度 φ [deg.] v = 3.5[m/sec.] 4 3 Skipping stone 4 円盤の傾き角度 φ [deg.] Magic Angle φm 6 θ φ v C. Clanet, F. Hersen and L. Bocquet. Nature, 47 (4) 年月9日木曜日
7
8
9 SPH( Smoothed Particle Hydrodynamics) MPS( Moving Particle Semi-implicit ) CIP Cubic Interpolated Propagation Level-set
10 ρ (r) = mw (r r j ) j W(r j): Gaussian karnel. v j A (r) = j A (r) = j m A j W (r r j ) ρ j m A j W (r r j ) ρ j ρ j ρ (r j ) ( ) c ρ i ρ ρ i ρ ρ p i = = ρ i < ρ
11 SPH法 () Navier-Stokes方程式 u + (u )u = p + ν u + t ρ! " ζ + ν/3 u ρ! m!a"(r) = Aj W (r r j )!ρ" j j に基づく補間の手続き SPHの運動方程式! mb d!u"i = dt ρi ρj j " uij r ij pi + pj νξ rij # i Wij 斥力相互作用する粒子系の運動方程式 年月9日木曜日
12 Length of smoothing kernel vi Fluid particles fixed on solid wall (Wall particles)
13 r = v z g x Lz Lx
14 円柱と水面の衝突 ー衝突の様子と速度プロファイル 上図a-d; E. G. Richardson, Proc. Phys. Soc. London, Sect. A 6 (948)より引用 年月9日木曜日
15 斜め衝突の間に円柱が受ける力 粒子の初期配置の影響 粒子の配置を正方格子とランダム配置に選んだ二つの結果を比較 force v r.5 force v r Square Lattice Random.6 Square Lattice Random.5.4 fz.3 fz.5.. T=r/cで 時間平均処理 fx -. fx t t r/v r/v 境界(底面)の影響 深さの異なる水槽 用いたシミュレーションの結果.4 force.3 v r fz Lz=7r Lz=4r fz fx. fz. fx -. fx t r/v 年月9日木曜日 6 7 8
16 θ c tank size lx=3r, lz=4r lx=4r, lz=8r θ c = 8/ σ Critical Angle [deg.] -.5. Specific gravity
17 v
18 5cm θ =.5, φ =, ω = 6[rounds/sec.]
19 5cm θ =.5, φ =, ω = [rounds/sec.]
20 v Minimum Velocity [m/sec.] Experiment SPH Incident Angle [deg.] Experiment SPH Tilt Angle [deg.]
21 v θ =8. φ =
22 z φ = const. p ρ(v n). z' f R x = const. f = pnds = disk s face g C D ρ(v n) n/ds disk s face = C Dρ(v n) Sn d v Water x'
23 衝突のODEモデル １ 無次元化した運動方程式 x z = sin φ F = κs(z )z cos φ F! z (x,z) v F = gr f x R = const. CD Rρ κ= πdρ! g z' xz座標系 v d 円盤下側の角の位置 フロード数 #! "! " "!!!! z z π z + S(z! ) = arcsin + + sin φ sin φ sin φ 流体にひたっている面積 年月9日木曜日 x' Water
24 ODEモデルと SPHシミュレーションの比較 定数κの決定 κ=.94, (Fitting parameter) 円板の受ける力 円板の軌道.6.35 fz fx.3.5. z/r fx -.5 年月9日木曜日.5 -. SPH ODE time r/v.5 ODE SPH.5 fz x/r
25 Minimum Velocity vmin [m/sec.] Criterion A Experiment SPH Theory Tilt Angle [deg.] Incident Angle [deg.] Experiment SPH Theory Tilt Angle [deg.]
26 衝突のODEモデル 解析解 反発 の定義 通常は 水面の高さを基準 にとる 位置条件 ここでは 重力方向の速度を基準にとる 速度条件 円盤の重力方向の速度が反転したら 反発が 起こったものと 見なす x (a) z (b) (c) (d) 年月9日木曜日 = sin φ F = κs(z )z cos φ F! 変曲点の存在条件をしらべる 球と水面の衝突の実験で得られた 衝突後の球の軌道 E. G. Richardson, Proc. Phys. Soc. London, Sect. A 6 (948)
27 v min = θ max = arccos gr cos(θ + φ) F { x sin φ + ( x sin φ + σd cos φ C D R sin φ } ) σd cos φ C D R sin φ φ Minimum Velocity vmin [m/sec.] Experiment SPH Theory Tilt Angle [deg.] Incident Angle [deg.] Experiment SPH Theory Tilt Angle [deg.]
28 35 3 σ. Angle [deg.] Experiment SPH Theory Incident Angle [deg.]
29 Incident speed [m/sec.] n > 38 n > 3 n > n > n > 5 θ Angle of incidence [deg.] Angle of incidence [deg.] 4 3 n > 3 n > 38 v 5[m/sec.] n > n > n > Tilt angle [deg.]
30
31 シミュレーション法の応用分野と 今後の課題 濡れ を伴う粉体系へのシミュレーション法の開発 地盤の液状化現象への応用 R+X 非常に大きな変形をともなう粘弾性体 のシミュレーション ゲルの衝突 水滴の衝突 R+X h h (a) (b) K. Okumura et. al. Europhys. Lett. 6, 37 (3) 高圧の物理学の実験 自由表面を持つ様々な現象への 数値的なアプローチ 円形跳水の問題 Clive Ellegaard et. al., Nonlinearity,, (999) Nature, 39, 3 (998) 年月9日木曜日
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II 29 7 29-7-27 ( ) (7/31) II (http://www.damp.tottori-u.ac.jp/~ooshida/edu/fluid/) [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Reynolds [ (4.6), (45.8)] [ p.186] Navier Stokes I Euler Navier
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II 14 14-7-8 8/4 II (http://www.damp.tottori-u.ac.jp/~ooshida/edu/fluid/) [ (3.4)] Navier Stokes [ 6/ ] Navier Stokes 3 [ ] Reynolds [ (4.6), (45.8)] [ p.186] Navier Stokes I 1 balance law t (ρv i )+ j
A Precise Calculation Method of the Gradient Operator in Numerical Computation with the MPS Tsunakiyo IRIBE and Eizo NAKAZA A highly precise numerical
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1 (Berry,1975) 2-6 p (S πr 2 )p πr 2 p 2πRγ p p = 2γ R (2.5).1-1 : : : : ( ).2 α, β α, β () X S = X X α X β (.1) 1 2
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2 3 4 (Neural Network) (Deep Learning) (Deep Learning) ( x x = ax + b x x x ? x x x w σ b = σ(wx + b) x w b w b .2.8.6 σ(x) = + e x.4.2 -.2 - -5 5 x w x2 w2 σ x3 w3 b = σ(w x + w 2 x 2 + w 3 x 3 + b) x,
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医系の統計入門第 2 版 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/009192 このサンプルページの内容は, 第 2 版 1 刷発行時のものです. i 2 t 1. 2. 3 2 3. 6 4. 7 5. n 2 ν 6. 2 7. 2003 ii 2 2013 10 iii 1987
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米田 戸倉川月 7 限 193~21 西 5-19 応用数学 A 積分定理 Gaussの定理 divbd = B nds Stokesの定理 E bds = E dr Green の定理 g x f y dxdy = fdx + gdy = f e i + ge j dr Gauss の発散定理 S n FdS = Fd 1777-1855 ドイツ Johann arl Friedrich Gauss
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1. ( ) 1.1 t + t [m]{ü(t + t)} + [c]{ u(t + t)} + [k]{u(t + t)} = {f(t + t)} (1) m ü f c u k u 1.2 Newmark β (1) (2) ( [m] + t ) 2 [c] + β( t)2
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212 1 6 1. (212.8.14) 1 1.1............................................. 1 1.2 Newmark β....................... 1 1.3.................................... 2 1.4 (212.8.19)..................................
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99 3 (Coriolis) cm m (free surface wave) 3.1 Φ 2.5 (2.25) Φ 100 3 r =(x, y, z) x y z F (x, y, z, t) =0 ( DF ) Dt = t + Φ F =0 onf =0. (3.1) n = F/ F (3.1) F n Φ = Φ n = 1 F F t Vn on F = 0 (3.2) Φ (3.1)
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微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 初版 1 刷発行時のものです.
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( ; ) C. H. Scholz, The Mechanics of Earthquakes and Faulting : - ( ) σ = σ t sin 2π(r a) λ dσ d(r a) =
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