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1

2 1 Fig

3 Element-Free Galerkin Method (EFGM) Meshless Local Petrov-Galerkin Method (MLPGM)

4 2 MLS u h (x) 1 p T (x) = [1, x, y]. (1) φ(x) 0.5 φ(x) N Σ (1) J[a] = w i (x) p T (x i )a(x) u i 2 i= x Fig. 2 1 φ i (x) N : w(r) : u i u(x i )

5 w(r) MLS Gauss : : Table 1 MLS 1 c = h R = 4h ( 5 ) Gauss h :

6 3 Element-Free Galerkin Method 2 Poisson (2) (3) (4) 2 Poisson : Fig. 3 Ω Ω (5) λ(s) : Lagrange ( Ω s ) (5) (6) 2 Poisson (6)

7 Table 2 Ω = ( 1, 1) ( 1, 1) Dirichlet u = sinπx cosπy p = 2π 2 sinπx cosπy Fig

8 CPU 10 3 CPU Time [s] Gauss ICCG Total Number of Nodes, N Fig. 5(6) CPU

9 4 Meshless Local Petrov-Galerkin Method 2 Poisson 2 Poisson 2 Poisson : 2 Poisson : Fig. 6 (7) α : (7) Ku = f. (8) 2 Poisson (8)

10 Table 3 Ω = ( 1, 1) ( 1, 1) 1 Dirichlet u = sinπx cosπy p = 2π 2 sinπx cosπy 3200 Fig

11 (a) r s = 0.5h (b) r s = h (c) r s = h Fig

12 10 1 Relative Error, ε Total Number of Nodes, N Fig. 9 (α = 10 6 ) : r s = 0.5h : r s = : r s = 2h : r s = 4h : r s = 10h h : r s = h

13 10 1 Relative Error, ε Total Number of Nodes, N Fig. 10 (r s = 4h) : α = : α = : α = : α = : α = : α =

14 5 EFGM MLPGM Table 4 EFGM MLPGM u δu EFGM Lagrange (Galerkin ) MLPGM (Petrov-Galerkin )

15 EFG MLPG EFG EFG MLPG MLPG Table 5 Lagrange Lagrange

16 Table 6 w σ φ ψ Gauss Shepard (0 MLS ) Gauss MLS MLS Table 7 Galerkin(w) w w Galerkin(σ) σ σ Galerkin(φ) φ φ Petrov-Galerkin(φ, w) φ w Petrov-Galerkin(φ, σ) φ σ Petrov-Galerkin(φ, ψ) φ ψ

17 Lagrange EFG Relative Error, ε Relative Error, ε Total Number of Nodes, N Total Number of Nodes, N (a) Galerkin (b) Petrov-Galerkin Fig. 11 : Galerkin(w) : Galerkin(σ) : Galerkin(φ) : Petrov-Galerkin(φ, w) : Petrov-Galerkin(φ, σ) : Petrov- Galerkin(φ, ψ)

18 EFG Relative Error, ε Relative Error, ε Total Number of Nodes, N Total Number of Nodes, N (a) Galerkin (b) Petrov-Galerkin Fig. 12 : Galerkin(w) : Galerkin(σ) : Galerkin(φ) : Petrov-Galerkin(φ, w) : Petrov-Galerkin(φ, σ) : Petrov- Galerkin(φ, ψ)

19 Lagrange MLPG Relative Error, ε Total Number of Nodes, N Fig. 13 : Galerkin(w) : Galerkin(φ) : Petrov- Galerkin(φ, w)

20 MLPG Relative Error, ε Total Number of Nodes, N Fig. 14 : Galerkin(w) : Galerkin(φ) : Petrov- Galerkin(φ, w)

21 6 EFGM ICCG Gauss MLPGM Lagrange MLPG Petrov-Galerkin(φ, w) Petrov-Galerkin(φ, w)

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#A A A F, F d F P + F P = d P F, F y P F F x A.1 ( α, 0), (α, 0) α > 0) (x, y) (x + α) 2 + y 2, (x α) 2 + y 2 d (x + α)2 + y 2 + (x α) 2 + y 2 = #A A A. F, F d F P + F P = d P F, F P F F A. α, 0, α, 0 α > 0, + α +, α + d + α + + α + = d d F, F 0 < α < d + α + = d α + + α + = d d α + + α + d α + = d 4 4d α + = d 4 8d + 6 http://mth.cs.kitmi-it.c.jp/

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D xy D (x, y) z = f(x, y) f D (2 ) (x, y, z) f R z = 1 x 2 y 2 {(x, y); x 2 +y 2 1} x 2 +y 2 +z 2 = 1 1 z (x, y) R 2 z = x 2 y

D xy D (x, y) z = f(x, y) f D (2 ) (x, y, z) f R z = 1 x 2 y 2 {(x, y); x 2 +y 2 1} x 2 +y 2 +z 2 = 1 1 z (x, y) R 2 z = x 2 y 5 5. 2 D xy D (x, y z = f(x, y f D (2 (x, y, z f R 2 5.. z = x 2 y 2 {(x, y; x 2 +y 2 } x 2 +y 2 +z 2 = z 5.2. (x, y R 2 z = x 2 y + 3 (2,,, (, 3,, 3 (,, 5.3 (. (3 ( (a, b, c A : (x, y, z P : (x, y, x

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C : q i (t) C : q i (t) q i (t) q i(t) q i(t) q i (t)+δq i (t) (2) δq i (t) δq i (t) C, C δq i (t 0 )0, δq i (t 1 ) 0 (3) δs S[C ] S[C] t1 t 0 t1 t 0

C : q i (t) C : q i (t) q i (t) q i(t) q i(t) q i (t)+δq i (t) (2) δq i (t) δq i (t) C, C δq i (t 0 )0, δq i (t 1 ) 0 (3) δs S[C ] S[C] t1 t 0 t1 t 0 1 2003 4 24 ( ) 1 1.1 q i (i 1,,N) N [ ] t t 0 q i (t 0 )q 0 i t 1 q i (t 1 )q 1 i t 0 t t 1 t t 0 q 0 i t 1 q 1 i S[q(t)] t1 t 0 L(q(t), q(t),t)dt (1) S[q(t)] L(q(t), q(t),t) q 1.,q N q 1,, q N t C :

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4 2016 3 8 2.,. 2. Arakawa Jacobin., 2 Adams-Bashforth. Re = 80, 90, 100.. h l, h/l, Kármán, h/l 0.28,, h/l.., (2010), 46.2., t = 100 t = 2000 46.2 < Re 46.5. 1 1 4 2 6 2.1............................

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SO(3) 7 = = 1 ( r ) + 1 r r r r ( l ) (5.17) l = 1 ( sin θ ) + sin θ θ θ ϕ (5.18) χ(r)ψ(θ, ϕ) l ψ = αψ (5.19) l 1 = i(sin ϕ θ l = i( cos ϕ θ l 3 = i ϕ

SO(3) 7 = = 1 ( r ) + 1 r r r r ( l ) (5.17) l = 1 ( sin θ ) + sin θ θ θ ϕ (5.18) χ(r)ψ(θ, ϕ) l ψ = αψ (5.19) l 1 = i(sin ϕ θ l = i( cos ϕ θ l 3 = i ϕ SO(3) 71 5.7 5.7.1 1 ħ L k l k l k = iϵ kij x i j (5.117) l k SO(3) l z l ± = l 1 ± il = i(y z z y ) ± (z x x z ) = ( x iy) z ± z( x ± i y ) = X ± z ± z (5.118) l z = i(x y y x ) = 1 [(x + iy)( x i y )

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/ n (M1) M (M2) n Λ A = {ϕ λ : U λ R n } λ Λ M (atlas) A (a) {U λ } λ Λ M (open covering) U λ M λ Λ U λ = M (b) λ Λ ϕ λ : U λ ϕ λ (U λ ) R n ϕ

/ n (M1) M (M2) n Λ A = {ϕ λ : U λ R n } λ Λ M (atlas) A (a) {U λ } λ Λ M (open covering) U λ M λ Λ U λ = M (b) λ Λ ϕ λ : U λ ϕ λ (U λ ) R n ϕ 4 4.1 1 2 1 4 2 1 / 2 4.1.1 n (M1) M (M2) n Λ A = {ϕ λ : U λ R n } λ Λ M (atlas) A (a) {U λ } λ Λ M (open covering) U λ M λ Λ U λ = M (b) λ Λ ϕ λ : U λ ϕ λ (U λ ) R n ϕ λ U λ (local chart, local coordinate)

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H 0 H = H 0 + V (t), V (t) = gµ B S α qb e e iωt i t Ψ(t) = [H 0 + V (t)]ψ(t) Φ(t) Ψ(t) = e ih0t Φ(t) H 0 e ih0t Φ(t) + ie ih0t t Φ(t) = [

H 0 H = H 0 + V (t), V (t) = gµ B S α qb e e iωt i t Ψ(t) = [H 0 + V (t)]ψ(t) Φ(t) Ψ(t) = e ih0t Φ(t) H 0 e ih0t Φ(t) + ie ih0t t Φ(t) = [ 3 3. 3.. H H = H + V (t), V (t) = gµ B α B e e iωt i t Ψ(t) = [H + V (t)]ψ(t) Φ(t) Ψ(t) = e iht Φ(t) H e iht Φ(t) + ie iht t Φ(t) = [H + V (t)]e iht Φ(t) Φ(t) i t Φ(t) = V H(t)Φ(t), V H (t) = e iht V (t)e

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