9. 05 L x P(x) P(0) P(x) u(x) u(x) (0 < = x < = L) P(x) E(x) A(x) P(L) f ( d EA du ) = 0 (9.) dx dx u(0) = 0 (9.2) E(L)A(L) du (L) = f (9.3) dx (9.) P
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1 9 (Finite Element Method; FEM) P(0) P(x) u(x) (a) P(L) f P(0) P(x) (b) 9. P(L)
2 9. 05 L x P(x) P(0) P(x) u(x) u(x) (0 < = x < = L) P(x) E(x) A(x) P(L) f ( d EA du ) = 0 (9.) dx dx u(0) = 0 (9.2) E(L)A(L) du (L) = f (9.3) dx (9.) P(x) (boundary condition)(9.2) (9.3) 3 3 (initial value problem) x [ 0, L ] (boundary value problem) 3 x u(x) x, y x, y, z (partial differential equation; PDE) 3
3 06 9. Step Step 2 Step 3 U f W U = L 0 ( ) 2 du 2 EA dx (9.4) dx W = f u(l) (9.5) u(x) (functional) u(0) = 0 I = U W minimize I = U W (9.6) subject to u(0) = 0 [ 0, L ] 6 h = L/6 (nodal point) x 0 = 0, x = h, x 2 = 2h,, x 6 = L [ x i, x j ] u(x) (6.3) u(x) = u i N i,j (x) + u j N j,i (x) (x i < = x < = x j ) (9.7) u i, u j x i, x j u(x) 7 u 0, u, u 6 (9.7) (9.4) 7
4 9. 07 u 0, u, u 6 E A [ 0, L ] U = x x 0 + ( ) 2 du x2 dx + dx ) 2 dx 2 EA x6 x 5 2 EA ( du dx x 2 EA ( ) 2 du dx+ dx [ x i, x j ] u(x) (9.7) 6 5 xj ( ) 2 du x i 2 EA dx = [ dx 2 u i u j ] EA h u i u j U = [ ] EA u 2 0 u h + [ ] EA u 2 u 2 h + [ ] EA u 2 5 u 6 h u u 0 u 2 u + u 5 u 6 U = 2 [ u 0 u u 5 u 6 ] EA h u 0 u. u 5 u 6
5 08 9. u 0 u u N =. u 6 (9.8) K = EA h (9.9) U = 2 ut N K u N (9.0) f W W = f T u N (9.) f = 0. 0 f (9.2) u(0) = 0 a T u N = 0 (9.3)
6 a = (9.4) u(x) (9.6) minimize I(u N ) = 2 ut N K u N f T u N (9.5) subject to a T u N = 0 (9.6) u N u N minimize J(u N, ) = I(u N ) a T u N (9.7) J u N = Ku N f a = 0, K a u N = f a T 0 0 J = at u N = 0 (9.8) u N (9.)(9.2)(9.3) (9.8) x A(x) E [ x i, x j ] du/dx ( u i + u j )/h
7 0 9. P(0) P(L/2) P(L) 9.2 = 2 xj x i 2 EA(x) [ V i,j = u i u j ] E h 2 xj x i ( du dx A(x) dx ) 2 dx = 2 E V i,j V i,j V i,j V i,j ( ) 2 ui + u xj j A(x) dx h x i u i u j [ x i, x j ] [ 0, L ] 6 V 0, V 0, V 0, V 0, + V,2 V,2 K = E V,2 V,2 + V 2,3 V 2,3 h V 4,5 V 4,5 + V 5,6 V 5,6 V 5,6 V 5,6 9.2 f [ 0, L ] 6 u N = [ u 0, u,, u 6 ] T U (9.0) K (9.9) W (9.) f = [ 0, 0, 0, f, 0, 0, 0 ] T u(0) = 0, u(l) = 0 a T 0 u N = 0, a T Lu N = 0
8 9.2 a 0 = [, 0, 0, 0, 0, 0, 0 ] T a L = [ 0, 0, 0, 0, 0, 0, ] T U W J(u N, 0, L ) = 2 ut N K u N f T u N 0 a T 0 u N L a T Lu N u N, 0, L Ku N f 0 a 0 L a L = 0 a T 0 u N = 0 a T Lu N = 0 K A A T u N = f 0 (9.9) A = [ a 0 a L ] = , = 0 L (9.9) 9.2 t P(x) u(x, t) P(x) ρ(x) T = L 0 ( ) 2 u 2 ρa dx = t L 0 2 ρa u2 dx (9.20)
9 2 9. L ( ) 2 u U = 2 EA dx (9.2) x 0 P(0) t P(L) f(t) W = f(t) u(l, t) (9.22) u(0, t) 0, t (9.23) (9.7) (9.20) 7 u 0, u, u 6 ρ A [ x i, x j ] u(x, t) u(x, t) = u i (t) N i,j (x) + u j (t) N j,i (x), (x i < = x < = x j ) (9.24) u(x, t) = u i (t) N i,j (x) + u j (t) N j,i (x), (x i < = x < = x j ) (9.25) [ 0, L ] T = x x2 x 0 2 ρa x6 u2 dx + x 2 ρa u2 dx + + x 5 2 ρa u2 dx (9.25) 6 5 xj x i 2 ρa u2 dx = [ 2 u i u j ] ρah u i u j
10 T = [ 2 + [ 2 + [ ] ρah u 0 u 2 u u ] ρah u u 2 2 u u 2 ] ρah u 5 u 6 2 u u 6 T = 2 u N T M u N u N = u 0 u. u 5 u 6, M = ρah M (9.0) (9.) f = [ 0,, 0, f(t) ] T (9.3) L L(u N, u N ) = T U + W + R = 2 u N T M u N 2 u N T Ku N + f T u N + a T u N (9.26) L d L = Ku N + f + a Mü N = 0 (9.27) u N dt u N a a T u N = 0
11 4 9. Ku N f a Mü N R(u N ) = a T u N = 0 R + 2αṘ + α2 R = 0 a T ü N + a T (2α u N + α 2 u N ) = 0 α v N = u N M v N a = Ku N + f a T v N = a T (2αv N + α 2 u N ) (9.28) M a v N a T = Ku N + f a T (2αv N + α 2 u N ) (9.29) u N v N v N u N, v N u N, v N u N, v N u N v N 9.3
12 9.3 5 S h P (x, y) P P(x, y) x u(x, y) y v(x, y) u = [ u, v ] T P(x, y) ε = Lu (9.30) x L = 0 y 0 y x, µ + 2µ 0 D = + 2µ 0 (9.3) 0 0 µ U = 2 εt D ε h ds (9.32) S 9.3 S U
13 6 9. (a) S 9.3 (b) P i P j P k U i,j,k = P i P j P k 2 εt D ε h ds (9.33) P i P j P k u(x, y) = u i N i,j,k (x, y) + u j N j,k,i (x, y) + u k N k,i,j (x, y) v(x, y) = v i N i,j,k (x, y) + v j N j,k,i (x, y) + v k N k,i,j (x, y) γ u = [ u i, u j, u k ] T γ v = [ v i, v j, v k ] T 6 u ξ = at γ u, u η = bt γ u, v ξ = at γ v, v η = bt γ v a = 2 y j y k y k y i, b = 2 x j x k x k x i y i y j x i x j a T γ u ε = b T γ v b T γ u + a T γ v
14 (9.33) U i,j,k = [ ] 2 Huu H uv γ u + 2 µ [ γ T u γ T u γ T v γ T v H vu H vv ] Huu µ Hµ uv Hµ vu Hµ vv γ v γ u γ v (9.34) H uu = aa T h, H uv = ab T h, H vu Hµ uu = 2H uu + H vv, Hµ vv = 2H vv 3 = ba T h, H vv + H uu, H uv µ = H vu = bb T h, Hµ vu = H uv (9.34) [ γ T u, γ T v ] T = [ u i, u j, u k, v i, v j, v k ] T (9.34) [ u T i, u T j, u T k ] T = [ u i, v i, u j, v j, u k, v k ] T H = Huu H vu H uv H vv, H µ = Huu µ Hµ uv Hµ vu Hµ vv, 4, 2, 5, 3, 6, 2, 3, 4, 5, 6, 4, 2, 5, 3, 6, 2, 3, 4, 5, 6 J i,j,k, Jµ i,j,k U i,j,k = 2 [ u T i u T j u T k ] (J i,j,k + µj i,j,k µ ) u i u j u k (9.35) J i,j,k, Jµ i,j,k P i, P j, P k J, J µ
15 8 9. C E D P4 P5 P6 A B F P P2 P3 (a) ABC (b) DEF (c) P P 3 P 6 P (a) ABC a, b a = [,, 0 ] T, b = [, 0, ] T h = H = H µ =
16 J A,B,C = J A,B,C µ = (b) DEF a, b a = [,, 0 ] T, b = [, 0, ] T h = 2 3 ABC 2 2 (A,A) 2 2 (A,B) 2 2 (A,C) (B,A) (C,C) 9.4(c) J, J µ (,) 2 2 (,2) (,3) (6,6) h = 2 P P 2 P 4 J A,B,C, Jµ A,B,C A, B, C P, P 2, P 4 J A,B,C (A,A) J (, ) J A,B,C (A,B) J (, 2) J A,B,C (A,C) J (, 4)
17 20 9. J A,B,C (C,C) J (4, 4) J A,B,C µ J µ P 5 P 4 P 2 J A,B,C, Jµ A,B,C A, B, C P 5, P 4, P 2 J A,B,C J A,B,C J A,B,C J A,B,C (A,A) J (5, 5) (A,B) J (5, 4) (A,C) J (4, 2) (C,C) J (2, 2) J A,B,C µ J µ J, J µ J =
18 9.3 2 P4 P5 P P2 P6 P3 p J µ = K = J + µj µ 9.5 P P 3 P 6 P 4 P 3 P 6 p = [ p x, p y ] T P 3 P 6 p P 3 P 6 h (u 3 + u 6 )/2 p W = (p P 3 P 6 h) T ( u3 + u 6 2 f T = [ ) = f T u N ] p x /2 p y / p x /2 p y /2
19 22 9. P P 4 u = 0, v = 0, u 4 = 0, v 4 = 0 A T u N = A T = K A u N = f (9.36) A T (9.8) u N = [,x,,y, 4,x, 4,y ] T P, P 4 x, y 9.4 P i P j P j u(x, y, t) = u i (t)n i,j,k (x, y)+u j (t)n j,k,i (x, y)+u k (t)n k,i,j (x, y)(9.37) t u i, u j, u k u = u i N i,j,k + u j N j,k,i + u k N k,i,j (9.38)
20 P i P j P j T i,j,k = P i P j P j 2 ρ ut u h ds (9.39) ρ ρ (9.38) 6 6 u T i,j,k = [ ] i u 2 i u j u k M i,j,k u j u k (9.40) M i,j,k = ρh 2 2I I I I 2I I I I 2I I 2 T = 2 ut N M u N (9.4) M (c) P P 3 P 6 P 4 M = ρh 24 2I I I I 6I I 2I 2I I 4I 2I I I 2I 4I I 2I 2I I 6I I I I 2I
21 24 9. E c E (a) E (b) c E c c (c) (d) (e) 9.6 T U W 9.5 Mü N Ku N + f + A = 0 (9.42) A T ü N + A T (2α u N + α 2 u N ) = 0 (9.43) v N = u N M A v N = A T Ku N + f A T (2αv N + α 2 u N ) (9.44) M v N (a)) 9.6(b))
22 (c)) 9.6(e)) 9.6(e)) σ ε σ = Eε σ = c ε σ = Eε + c ε σ = E c ε + E ε σ = E c + c 2 ε + Ec 2 c + c 2 ε + c c 2 c + c 2 ε 9.3 (J +µj µ )u N E, µ J + µj µ ε u N J, J µ, µ E ν = νe ( + ν)( 2ν), µ = E 2( + ν) ε u N ( v J + µ v J µ ) u N v, µ v c ν v = νc ( + ν)( 2ν), µv = c 2( + ν)
23 26 9. f f µ f = J u N + v J u N f µ = µj µ u N + µ v J µ u N σ = Eε + c ε σ f E c v ε J u N f σ f µ E µ c µ v ε J µ u N f µ f f µ f, f µ f f µ = v f + J u N = µ µ v f µ + µj µ u N (J + µj µ )u N ( v J + µ v J µ ) u N (J + µj µ )u N ( v J + µ v J µ ) u N f f µ f = v f + J u N, f µ = µ µ v f µ + µj µ u N f f µ f = f + v 2 v + J u N + v v 2 v 2 v + J ü N v 2 v + v 2 f µ = µ µ v + f µ + µµv 2 µv 2 µ v + J µ u N + µv µ v 2 µv 2 µ v + J µ ü N µv 2 (9.44) Ku N f, f µ
24 M A A T v N = Ku N Bv N + f (9.45) A T (2αv N + α 2 u N ) B = v J + µ v J µ M A A T v N = f f µ A T (2αv N + α 2 u N ) (9.46) f f µ = v f + J v N = µ µ v f µ + µj µ v N M A v N f = f µ (9.47) A T A T (2αv N + α 2 u N ) f = v + f + v 2 v 2 v + J v N + v v 2 v 2 v + J v N v 2 f µ = µ µ v + f µ + µµv 2 µv 2 µ v + J µ v N + µv µ v 2 µv 2 µ v + J µ v N µv 2 v N f, f µ 20 ( ) ( )
25 28 9. x P(0) P(L) P(x) ρ(x) U grav = L 0 ρag u(x) dx ρ A [ 0, L ] n h = L/n [ x i, x j ] 2 () x i, y i, x j, y j, x k, y k h J i,j,k, Jµ i,j,k (2) J, J µ 9.4(c) , h 3 (9.34) 4 () (9.9) K e = [,,, ] T (2) 9.4(c) P P 3 P 6 P 4 K = J + µj µ
26 29 e x e y e θ = [, 0,, 0,, 0,, 0,, 0,, 0 ] T = [ 0,, 0,, 0,, 0,, 0,, 0, ] T = [ /2,, /2, 0, /2,, /2,, /2, 0, /2, ] T P P 4 P 4P 5 [ 0, t p ] v p t h P3 P4 P5 P6 P9 P0 P P2 P5 P6 P7 P8 P P2 P3 P θ(s) (0 < = s < = L) 7 (9.6) (9.) () u(x) U(u) = L 0 ( ) 2 du 2 EA dx dx u(x) δu(x) u(0) = 0 δu(0) = 0 U(u + δu) = L 0 ( du 2 EA dx + dδu ) 2 dx dx U(u)
27 30 9. δu = E(L)A(L) du L ( d (L) δu(l) EA du ) δu dx dx 0 dx dx (2) W (u) = f u(l) δw = W (u + δu) W (u) = f δu(l) (3) I δi = δu δw L ( d δi = EA du ) { δu dx + E(L)A(L) du } 0 dx dx dx (L) f δu(l) (9.6) δu(x) δi = 0 δu(x) δi = 0 ( d EA du ) dx dx = 0, E(L)A(L) du dx (L) f = 0 (9.6) (9.) (9.3) (variation) 8 E xx, E yy, 2E xy E xx = u x + 2 (u2 x + v 2 x), E yy = v y + 2 (u2 y + v 2 y) 2E xy = u y + v x + u x u y + v x v y E xx = 0 E yy = 0 2E xy = 0 (9.30) 0 ε = [ ε xx, ε yy, 2ε xy ] T E = [ E xx, E yy, 2E xy ] T
数値計算：有限要素法
( ) 1 / 61 1 2 3 4 ( ) 2 / 61 ( ) 3 / 61 P(0) P(x) u(x) P(L) f P(0) P(x) P(L) ( ) 4 / 61 L P(x) E(x) A(x) x P(x) P(x) u(x) P(x) u(x) (0 x L) ( ) 5 / 61 u(x) 0 L x ( ) 6 / 61 P(0) P(L) f d dx ( EA du dx
変 位 変位とは 物体中のある点が変形後に 別の点に異動したときの位置の変化で あり ベクトル量である 変位には 物体の変形の他に剛体運動 剛体変位 が含まれている 剛体変位 P(x, y, z) 平行移動と回転 P! (x + u, y + v, z + w) Q(x + d x, y + dy,
変 位 変位とは 物体中のある点が変形後に 別の点に異動したときの位置の変化で あり ベクトル量である 変位には 物体の変形の他に剛体運動 剛体変位 が含まれている 剛体変位 P(x, y, z) 平行移動と回転 P! (x + u, y + v, z + w) Q(x + d x, y + dy, z + dz) Q! (x + d x + u + du, y + dy + v + dv, z +
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W u = u(x, t) u tt = a 2 u xx, a > 0 (1) D := {(x, t) : 0 x l, t 0} u (0, t) = 0, u (l, t) = 0, t 0 (2)
3 215 4 27 1 1 u u(x, t) u tt a 2 u xx, a > (1) D : {(x, t) : x, t } u (, t), u (, t), t (2) u(x, ) f(x), u(x, ) t 2, x (3) u(x, t) X(x)T (t) u (1) 1 T (t) a 2 T (t) X (x) X(x) α (2) T (t) αa 2 T (t) (4)
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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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No δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i x j δx j (5) δs 2
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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
統計的データ解析
ds45 xspec qdp guplot oocalc (Error) gg (Radom Error)(Systematc Error) x, x,, x ( x, x,..., x x = s x x µ = lm = σ µ x x = lm ( x ) = σ ( ) = - x = js j ( ) = j= ( j) x x + xj x + xj j x + xj = ( x x
(3) (2),,. ( 20) ( s200103) 0.7 x C,, x 2 + y 2 + ax = 0 a.. D,. D, y C, C (x, y) (y 0) C m. (2) D y = y(x) (x ± y 0), (x, y) D, m, m = 1., D. (x 2 y
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II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2
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