数値計算：有限要素法

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Transcription:

( ) 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 ) = 0 u(0) = 0 E(L)A(L) du dx (L) = f (boundary value problem) ( ) 7 / 61

(finite element method; FEM) 1 2 3 ( ) ( ) 8 / 61

U U = f W L 0 ( ) 2 1 du 2 EA dx, dx W = f u(l) min I = U W, subject to u(0) = 0 ( ) 9 / 61

[0, L] 6 h = L/6 Pi Pj 0 xi xj L x (nodal point) x 0 = 0, x 1 = h, x 2 = 2h,, x 6 = L u i = u(xi ) u(x) 7 u 0, u 1, u 2,, u 6 ( ) 10 / 61

[x i, x j ] u(x) u(x) = u i N i,j (x) + u j N j,i (x), (x i x x j ) u i, u j x i, x j 1 0 0 xi xj L N i,j (x) = x j x { h 1 (x = xi ) = 0 (x = x j ) x 0 0 xi xj L 1 N j,i (x) = x x i { h 1 (x = xj ) = 0 (x = x i ) x ( ) 11 / 61

u(x) = u(x) (0 x L) u 0 N 0,1 (x) + u 1 N 1,0 (x) (x 0 x x 1 ) u 1 N 1,2 (x) + u 2 N 2,1 (x) (x 1 x x 2 ) u 2 N 2,3 (x) + u 3 N 3,2 (x) (x 2 x x 3 ). u 5 N 5,6 (x) + u 6 N 6,5 (x) (x 5 x x 6 ) u(x) 7 u 0, u 1,, u 6 ( ) 12 / 61

u(x) 0 h 2h 3h 4h 5h L P0 P1 P2 P3 P4 P5 P6 x ( ) 13 / 61

u(x) u5 u6 u2 u4 u0 u1 u3 0 h 2h 3h 4h 5h L P0 P1 P2 P3 P4 P5 P6 x ( ) 13 / 61

[x i, x j ] N i,j (x) = x j x, N j,i (x) = x x i h h N i,j(x) = 1 h, N j,i(x) = 1 h u(x) = u i N i,j (x) + u j N j,i (x) du = u i N dx i,j(x) + u j N j,i(x) 1 = u i h + u 1 j h = u i + u j h ( ) 14 / 61

= 7 u 0, u 1,, u 6 L 0 = u N = u 0 u 1 u 2. u 6 x1 x 0 x2 x 1 x3 + + x 2 + + x6 x 5 ( ) 15 / 61

E xj ( ) 2 1 du xj ( ) 2 x i 2 EA 1 dx = dx x i 2 EA ui + u j dx h = 1 E xj 2 h ( u 2 i + u j ) 2 A dx x i = 1 [ [ ] E V ui u i,j j 2 h 2 V i,j V i,j = xj x i A dx = [x i, x j ] V i,j V i,j ] [ ui u j ], ( ) 16 / 61

L ( ) 2 1 du U = 0 2 EA dx dx = 1 [ ] [ [ ] E V u0 u 0,1 V 0,1 u0 1 2 h 2 V 0,1 V 0,1 u 1 [ ] [ 1 [ ] E V u1 u 1,2 V 1,2 u1 2 2 h 2 V 1,2 V 1,2 u 2 [ ] [ 1 [ ] E V u2 u 2,3 V 2,3 u2 3 2 h 2 V 2,3 V 2,3 u 3 [ ] [ 1 [ ] E V u5 u 5,6 V 5,6 u5 6 2 h 2 V 5,6 V 5,6 u 6 ] + ] + ] + + ] ( ) 17 / 61

K = E h 2 U = 1 2 u N T Ku N V 0,1 V 0,1 V 0,1 V 0,1 + V 1,2 V 1,2 V 1,2 V 1,2 + V 2,3 V 2,3 V 2,3......... V 4,5 + V 5,6 V 5,6 V 5,6 V 5,6 ( ) 18 / 61

K = E h 2 U = 1 2 u N T Ku N V 0,1 V 0,1 V 0,1 V 0,1 + V 1,2 V 1,2 V 1,2 V 1,2 + V 2,3 V 2,3 V 2,3......... V 4,5 + V 5,6 V 5,6 V 5,6 V 5,6 ( ) 18 / 61

K = E h 2 U = 1 2 u N T Ku N V 0,1 V 0,1 V 0,1 V 0,1 + V 1,2 V 1,2 V 1,2 V 1,2 + V 2,3 V 2,3 V 2,3......... V 4,5 + V 5,6 V 5,6 V 5,6 V 5,6 ( ) 18 / 61

K = E h 2 U = 1 2 u N T Ku N V 0,1 V 0,1 V 0,1 V 0,1 + V 1,2 V 1,2 V 1,2 V 1,2 + V 2,3 V 2,3 V 2,3......... V 4,5 + V 5,6 V 5,6 V 5,6 V 5,6 ( ) 18 / 61

A V i,j = Ah 1 1 1 2 1 K = EA 1 2 1 h 1 2 1 1 2 1 1 2 1 1 1 ( ) 19 / 61

A V i,j = Ah 1 1 1 1+1 1 K = EA 1 1+1 1 h 1 1+1 1 1 1+1 1 1 1+1 1 1 1 ( ) 20 / 61

A V i,j = Ah 1 1 1 1+1 1 K = EA 1 1+1 1 h 1 1+1 1 1 1+1 1 1 1+1 1 1 1 ( ) 20 / 61

f W = f u(l) = f u 6 W = f T u N f = 0 0 0 0 0 0 f ( ) 21 / 61

u(0) = u 0 = 0 a T u N = 0 a = 1 0 0 0 0 0 0 ( ) 22 / 61

min I = U W, subject to u(0) = 0 min I (u N ) = 1 2 ut N K u N f T u N, subject to a T u N = 0 ( ) 23 / 61

( ) min I (u N ) = 1 2 ut N K u N f T u N, subject to a T u N = 0 min J(u N, λ) = I (u N ) λa T u N = 1 2 ut N K u N f T u N λa T u N ( ) 24 / 61

( ) J u N = Ku N f λa = 0 J λ = at u N = 0 [ K a a T 0 ] [ un λ ] = [ f 0 ] ( ) 25 / 61

( ) J u N = Ku N f λa = 0 J λ = at u N = 0 [ K a a T 0 ] [ un λ ] = [ f 0 ] ( ) 25 / 61

( ) J u N = Ku N f λa = 0 J λ = at u N = 0 [ K a a T 0 ] [ un λ ] = [ f 0 ] = u N λ ( ) 25 / 61

A(x) = a 2bx (a, b a 2bL > 0 ) E V i,j = xj x i (a 2bx) dx = [ ax bx 2] xj x i = {a b(x i + x j )} h V 0,1 = (a bh)h, V 1,2 = (a 3bh)h, V 2,3 = (a 5bh)h, [0, L] 4 a bh K = E h a + bh a + bh 2a 4bh a + 3bh a + 3bh 2a 8bh a + 5bh a + 5bh 2a 12bh a + 7bh a + 7bh a 7bh ( ) 26 / 61

P(x) r(x) = 3(a bx) (a, b a bl > 0 ) A(x) = πr 2 = 3π(a bx) 2 xj ] x=xj V i,j = 3π(a bx) 2 (a bx)3 dx = [π x i b x=x i = π { 3a 2 3ab(x i + x j ) + b 2 (xi 2 + x i x j + xj 2 ) } h [0, L] 4 (x i = ih h = L/4) V 0,1 = π(3a 2 3abh + b 2 h 2 )h V 1,2 = π(3a 2 9abh + 7b 2 h 2 )h V 2,3 = π(3a 2 15abh + 19b 2 h 2 )h V 3,4 = π(3a 2 21abh + 37b 2 h 2 )h ( ) 27 / 61

( ) 28 / 61

P(0) P(x) u(x,t) 0 P(L) f(t) P(0) P(x) t P(L) ( ) 29 / 61

T = L 0 ( ) 2 1 u L 2 ρa 1 dx = t 0 2 ρa u2 dx 7 u 0, u 1,, u 6 [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 ) u(x, t) = u i (t) N i,j (x) + u j (t) N j,i (x), (x i x x j ) ( ) 30 / 61

xj x i 1 2 ρa u2 dx = 1 2 m i,j; i,j = xj = 1 2 m i,j; j,i = m j,i; i,j = xj ρa { u i (t) N i,j (x) + u j (t) N j,i (x)} 2 dx x i [ ] [ ] [ ] m ui u i,j; i,j m i,j; j,i ui j m j,i; i,j m j,i; j,i u j x i ρa (N i,j ) 2 dx, m j,i; j,i = xj x i ρa N i,j N j,i dx xj x i ρa (N j,i ) 2 dx, ( ) 31 / 61

L T = 1 ρa u 2 dx 2 0 = 1 [ ] [ ] [ m u0 u 0,1; 0,1 m 0,1; 1,0 u0 1 2 m 1,0; 0,1 m 1,0; 1,0 u 1 1 [ ] [ ] [ m u1 u 1,2; 1,2 m 1,2; 2,1 u1 2 2 m 2,1; 1,2 m 2,1; 2,1 u 2 1 [ ] [ ] [ m u2 u 2,3; 2,3 m 2,3; 3,2 u2 3 2 m 3,2; 2,3 m 3,2; 3,2 u 3 1 [ ] [ ] [ m u5 u 5,6; 5,6 m 5,6; 6,5 u5 6 2 m 6,5; 5,6 m 6,5; 6,5 u 6 ] + ] + ] + + ] = 1 2 u N T M u N ( ) 32 / 61

M = m 0,1; 0,1 m 0,1; 1,0 m 1,0; 0,1 m 1,0; 1,0 + m 1,2; 1,2 m 1,2; 2,1. m 2,1; 1,2.. m4,5; 5,4 m 5,4; 4,5 m 5,4; 5,4 + m 5,6; 5,6 m 5,6; 6,5 m 6,5; 5,6 m 6,5; 6,5 ( ) 33 / 61

A ρ xj x i (N i,j ) 2 dx = xj x i N i,j N j,i dx = M = ρah 6 xj x i (N j,i ) 2 dx = h 3 xj x i N j,i N i,j dx = h 6 2 1 1 4 1 1 4 1 1 4 1 1 4 1 1 4 1 1 2 ( ) 34 / 61

A ρ M = ρah 6 2 1 1 4 1 1 4 1 1 4 1 1 4 1 1 4 1 1 2 ( ) 35 / 61

A ρ M = ρah 6 2 1 1 2+2 1 1 2+2 1 1 2+2 1 1 2+2 1 1 2+2 1 1 2 ( ) 36 / 61

A ρ M = ρah 6 2 1 1 2+2 1 1 2+2 1 1 2+2 1 1 2+2 1 1 2+2 1 1 2 ( ) 36 / 61

A(x) = a 2bx (a, b a 2bL > 0 ) ρ xj x (N i,j ) 2 dx = h x i 12 (3x i + x j ), xj x (N j,i ) 2 dx = h x i 12 (x i + 3x j ), xj x i x N i,j N j,i dx = xj x i x N j,i N i,j dx = h 12 (x i + x j ) m i,j; i,j = ρh 6 {2a b(3x i + x j )}, m j,i; j,i = ρh 6 {2a b(x i + 3x j )}, m i,j; j,i = m j,i; i,j = ρh 6 {a b(x i + x j )} ( ) 37 / 61

[0, L] 4 2a bh M = ρh 6 a bh a bh 4a 8bh a 3bh a 3bh 4a 16bh a 5bh a 5bh 4a 24bh a 7bh a 7bh 2a 15bh ( ) 38 / 61

L(u N, u N ) = T U + W + λr = 1 2 u N T M u N 1 2 u N T Ku N + f T u N + λa T u N L d L = Ku N + f + λa Mü N = 0 u N dt u N ( ) 39 / 61

R = a T u N Ṙ = a T u N R = a T ü N a T ü N + 2νa T u N + ν 2 a T u N = 0 ( ) 40 / 61

Mü N λa = Ku N + f a T ü N = a T (2ν u N + ν 2 u N ) u N = v N [ M a a T ] [ vn λ ] [ = Ku N + f a T (2νv N + ν 2 u N ) ] ( ) 41 / 61

λ, µ ( ) 42 / 61

P i P i P i Pk Pi Pj ( ) 43 / 61

P i P i P i Pk Pi Pj ( ) 43 / 61

P i P i P i Pk Pi uk ui Pj uj P i P i P i u i = [ ui v i ] [ uj, u j = v j ] [ uk, u k = v k ] ( ) 43 / 61

U i,j,k = 1 [ u T 2 i uj T uk T K i,j,k (6 6) ] Ki,j,k u i u j u k K i,j,k = λj i,j,k λ J i,j,k λ, Jµ i,j,k (6 6) + µj i,j,k µ ( ) 44 / 61

1 P2 P2 1 1 P3 P0 1 P1 P1 ( ) 45 / 61

J 0,1,2 λ = J 3,2,1 λ = 1 2 J 0,1,2 µ = J 3,2,1 µ = 1 2 1 1 1 0 0 1 1 1 1 0 0 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 3 1 2 1 1 0 1 3 0 1 1 2 2 0 2 0 0 0 1 1 0 1 1 0 1 1 0 1 1 0 0 2 0 0 0 2 ( ) 46 / 61

P2 P3 P2 P2 P3 1 = 1 + 1 1 P0 1 P1 P0 1 P1 P1 ( ) 47 / 61

U = U 0,1,2 + U 3,2,1 = 1 2 [ u T 0 u T 1 u T 2 u T 3 ] K u 0 u 1 u 2 u 3 K (8 8) J λ, J µ (8 8) K = λj λ + µj µ ( ) 48 / 61

J λ (0, 0) J 0,1,2 λ = 1 2 1 1 1 0 0 1 1 1 1 0 0 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 ( ) 49 / 61

J λ (0, 1) J 0,1,2 λ = 1 2 1 1 1 0 0 1 1 1 1 0 0 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 ( ) 49 / 61

J λ (0, 2) J 0,1,2 λ = 1 2 1 1 1 0 0 1 1 1 1 0 0 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 ( ) 49 / 61

J λ (1, 0) J 0,1,2 λ = 1 2 1 1 1 0 0 1 1 1 1 0 0 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 ( ) 49 / 61

J λ (1, 1) J 0,1,2 λ = 1 2 1 1 1 0 0 1 1 1 1 0 0 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 ( ) 49 / 61

J λ (1, 2) J 0,1,2 λ = 1 2 1 1 1 0 0 1 1 1 1 0 0 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 ( ) 49 / 61

J λ (2, 0) J 0,1,2 λ = 1 2 1 1 1 0 0 1 1 1 1 0 0 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 ( ) 49 / 61

J λ (2, 1) J 0,1,2 λ = 1 2 1 1 1 0 0 1 1 1 1 0 0 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 ( ) 49 / 61

J λ (2, 2) J 0,1,2 λ = 1 2 1 1 1 0 0 1 1 1 1 0 0 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 ( ) 49 / 61

J 0,1,2 λ J λ J λ = 1 2 1 1 1 0 0 1 1 1 1 0 0 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 ( ) 50 / 61

J λ (3, 3) J 3,2,1 λ = 1 2 1 1 1 0 0 1 1 1 1 0 0 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 ( ) 51 / 61

J λ (3, 2) J 3,2,1 λ = 1 2 1 1 1 0 0 1 1 1 1 0 0 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 ( ) 51 / 61

J λ (3, 1) J 3,2,1 λ = 1 2 1 1 1 0 0 1 1 1 1 0 0 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 ( ) 51 / 61

J λ (2, 3) J 3,2,1 λ = 1 2 1 1 1 0 0 1 1 1 1 0 0 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 ( ) 51 / 61

J λ (2, 2) J 3,2,1 λ = 1 2 1 1 1 0 0 1 1 1 1 0 0 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 ( ) 51 / 61

J λ (2, 1) J 3,2,1 λ = 1 2 1 1 1 0 0 1 1 1 1 0 0 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 ( ) 51 / 61

J λ (1, 3) J 3,2,1 λ = 1 2 1 1 1 0 0 1 1 1 1 0 0 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 ( ) 51 / 61

J λ (1, 2) J 3,2,1 λ = 1 2 1 1 1 0 0 1 1 1 1 0 0 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 ( ) 51 / 61

J λ (1, 1) J 3,2,1 λ = 1 2 1 1 1 0 0 1 1 1 1 0 0 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 ( ) 51 / 61

J 3,2,1 λ J λ J λ = 1 2 0 0 0 0 0 0 0 1 1 0 1 1 0 1 1 0 1 1 0 0 0 0 0 0 0 1 1 0 1 1 0 1 1 0 1 1 ( ) 52 / 61

J 0,1,2 λ J 3,2,1 λ 1 1 1 0 0 1 1 1 1 0 0 1 J λ = 1 1 1 1 0 0 1 0 0 0 0 0 1 1 0 1 1 2 0 0 0 1 1 0 1 1 1 1 1 0 0 1 0 0 0 1 1 0 1 1 0 1 1 0 1 1 ( ) 53 / 61

J 0,1,2 µ J µ J µ = 1 2 3 1 2 1 1 0 1 3 0 1 1 2 2 0 2 0 0 0 1 1 0 1 1 0 1 1 0 1 1 0 0 2 0 0 0 2 ( ) 54 / 61

J 3,2,1 µ J µ J µ = 1 2 1 0 0 1 1 1 0 2 0 0 0 2 0 0 2 0 2 0 1 0 0 1 1 1 1 0 2 1 3 1 1 2 0 1 1 3 ( ) 55 / 61

J 0,1,2 µ J 1,3,2 µ J µ = 1 2 3 1 2 1 1 0 1 3 0 1 1 2 2 0 3 0 0 1 1 1 1 1 0 3 1 0 0 2 1 1 0 1 3 0 2 0 0 2 1 0 0 3 1 1 1 0 2 1 3 1 1 2 0 1 1 3 ( ) 56 / 61

U = 1 2 u N T Ku N, T = 1 2 u N T M u N = = ( ) 57 / 61

x = x 1 x 2 x 3 y x y x y 1 = y x x 2 y x 3 ( ) 58 / 61

b = b 1 b 2 b 3 y = b T x = b 1 x 1 + b 2 x 2 + b 3 x 3 y x 1 = b 1, y x 2 = b 2, y x 3 = b 3 ( b T x ) = b x ( ) 59 / 61

A = a 11 a 12 a 13 a 12 a 22 a 23 a 13 a 23 a 33 y = 1 2 x T Ax = 1 2 { a11 x 2 1 + a 22 x 2 2 + a 33 x 2 3 + 2a 12 x 1 x 2 + 2a 13 x 1 x 3 + 2a 23 x 2 x 3 } y x 1 = 1 2 {a 11 2x 1 + 2a 12 x 2 + 2a 13 x 3 } = a 11 x 1 + a 12 x 2 + a 13 x 3 ( ) 60 / 61

y x 1 = a 11 x 1 + a 12 x 2 + a 13 x 3 y x 2 = a 12 x 1 + a 22 x 2 + a 23 x 3 y x 3 = a 13 x 1 + a 23 x 2 + a 33 x 3 ( ) 1 x 2 x T Ax = Ax ( ) 61 / 61

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