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2 9. (FEM, nite element method) (element) (mesh discretization) DT LOK DT M([ ],[ ] )=[ ] DIMENSION M(,NF) DT M/,4,5,,5,,,5,6,,6,,,6,7, 4,8,9, 4,9,5,.../ DIMENSION M(4,NF) DO I=,IF- DO J=,JF K=(I-)*JF+J N=(I-)*(JF-)+J M(,N)=K M(,N)=K+JF M(,N)=K+JF+ M(4,N)=K+ ONTINUE nodal pt number element number 9.: ( ) 9. (a) (b) 7% (b)

3 5 (a) 67.7 (b) (a) / 9.: (a) (b) 9.: 9.. u(x) u(x) (interpolation function) u(x) u (x) i (x)u i = (x)u (9.) (shape function base function) U u ( u j u jk ::: ) FEM u(x) U u(x) u j u j = i ju i u jk i jk U i u(x) ud X n u k u l d X e u d = X i (x) d U i n e i k j l d U i U j n e u j = u=xj u jk = u=xj xk u i k = ui=xk

4 4 R d R e d P n u (x y) u (x y) = nx = c (x y) c + c x+c y + c 4 x +c 5 xy+c 6 y + c 7 x +c 8 x y+c 9 xy +c y + (9.) c (9.) U i = u (x i y i )= nx = c (x i y i ) (i = ::: n) (9.) (9.) (9.) 6 (9.) (9.) (9.) c (x) = (x ) ++ (x n ) n... n(x) = n (x ) ++ n (x n ) n ::: n T T = b T ; = b (9.4) b x y x y x y x y x x y y x y x x y y x y x x y y x 4 y 4 x4 x 4y 4 y4 x 5 y 5 x5 x 5y 5 y5 x 6 y 6 x6 x 6y 6 y6 b x y x y x xy y (9.4) ramer = x y x y x y x y x y x y x y x y (9.5) x y

5 5 = jj = x y x y x y = (x ; x )(x ; x ) (9.5) 4 4P 4P 4P (area coordinates) cc cc c c : c c ; HH c P c x = y ;y y ;y y ;y y x ;x x ;x x ;x (9.6) e l m n d = l m n (l+m+n+) (9.7) 6 - P t dt ds X X? 6 P PPPPPPP s?? a - h 9.4: (9.7) = as= = fa(h;s);htg= = ht= (9.7) I = l m n d = al h m+n h l+m+n s l ds e a(;s=h) fa(;s=h) ; tg m t n dt

6 6 b (b;t) m t n dt = n m+ b (b;t) m+ t n; dt = = m n (m+n) t s I = m n a l+m+n+ h ; h (m+n+) l+m+n (h;s) m+n+ s l l m n ds = (l+m+n+) b (b;t) m+n dt = m n bm+n+ (m+n+) [ ] i i = jj i b b u u x u y Hermite (9.) u u x u y u U = u u x u y u u x u y u u x u y u 4 = jj b Hermite x y x x y y x x y x y y x y x x y y x y x x y y x y x x y y x x y x y y x y x x y y x y x x y y x y x x y y x x y x y y x y x x y y x y x x y y x 4 y 4 x4 x 4y 4 y4 x4 x4 y 4 x 4y4 y4 ( x y x xy y b x x y xy y ) xy u u = u(x y) u u n

7 7 s u(s) =d +d s+d s d d d u u u n u n (s) =d +d s Hermite u u(s) =d +d s+d s +d s 4 u u s u n u n (s) =e +e s+e s u n 5 u u x u y u xx u xy u yy u n 5 Hermite u n u(x y) = c + c x + c 5 x + c y + c 4 xy + c 7 x y + c 6 y + c 8 xy + c 9 x y (9.8) 4 xy Lagrange 8 Lagrange Lagrange (9.4) b Lagrange x y x y x y x y x y x y x 4 y 4 x 4y 4 x y x x y y x y x y x y x y x x y y x y x y x y :::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::: x 9 y 9 x9 x 9y 9 y9 x9 y 9 x 9y9 x9 y9 b x y xy ; x y x xy y x y xy x y ; ( i ) i = i = 4

8 8 u () =() u = u + u + u + 4 u 4 i = 4 (+ i)(+ i ) (i = 4) (9.9) i i i 4 i ; ; i ; ; (9.9) u x y u x y x = x ; (x y) (x y) y = y ; (x y) (x y) (9.) x y = x x + x x = x y + x y = y x + y x = y y + y y (metrics) x = y =J x = ;y =J y = ;x =J y = x =J (9.) J Jacobian (x y) ( ) = x y ;x y (9.) (9.) x = x x y y y = J y ;y ;x x (9.) x = y ;y = y J ;x x J y ; y ; x x (9.4) ;+ ; + ;; = 4 ;+ ;; + ; (x y) = x y x y x y x 4 y 4 (9.5)

9 9 (9.9) x x() =() x x = x ( i ) = i (+ i )=4 ( i ) = i (+ i )=4 e k (x) j (x)d = ; ; k () j ()J()dd (9.6) 6.. auss (auss-legendre ) s = r = serendipity element Lagrange element 9.5: 8 (8-node quadratic serendipity element) i = 8 >< >: 4 (+ i)(+ i )( i + i ;) ( i = i = ) (; )(+ i ) ( i = i = ) (+ i)(; ) ( i = i = ) 9 (quadratic Lagrangian element) u( ) = 9X i= i ( )u i = X X r= s= L r Lagrange (9.7) L r ()L s ()u i (i = r+(s;)) (9.8) L () =; (;) L () =; L () = (+) ( ) =L ()L () =;(; )(;)= serendipity (;)(+) ;4(;) ;(; ) = 4 (;)(+) ;(; ) ;4(;) Horace Walpole \The Three Princes of Serendip"( ) () i i( ) i = a i(a ) = a +i (a b) = ( ;a )+( ;b ) =

10 Lagrange (;)(;) 4(;) (+)(+) = 4 (;)(;) ;(; )(;) (+)(+) x u x() = i ()x i u() = i ()u i i () i () i () i () u x HH sub-parametric element isoparametric element super-parametric element serendipity Lagrange Lagrange serendipity 5 (9.4) b x y z x y z x y z x 4 y 4 z 4 x y z y z z x x y x y z x y z y z z x x y x y z ::::::::::::::::::::::::::: ::::::::::::::::::::::::::: x 8 y 8 z 8 y 8z 8 z 8x 8 x 8y 8 x 8y 8z 8 b x y z ; x y z yz zx xy xyz

11 9.. (calculus of variations) FEM Fermat astigliano FEM x x u(x ) u(x ) (functional) x J[u] = F (x u u )dx =min (9.9) x 4 F x u u (= du=dx) u x J[u] u(x) u J u(x) x u u u+u u+(x) (9.9) x J[u+] = F (x u+ u + )dx x d d J[u+] = x x x (F u + F u )dx = (x )=(x )= x F u ; d dx F u dx = d dx F u ; F u = (9.) u F u u + u F u u + F u x ; F u = (9.) Euler (extremal) df u =dx;f u F u (variational derivative) Legendre F u u 6= (9.) 4 R. ourant and D. Hilbert, Methods of Mathematical Physics, Vol.I, 95, Interscience Publishers.

12 (extremum) u(x) u = u 6= u = u (variation) x x (F u u+f u u )dx = F u ; d x dx F u u dx +(F u u) x=x ; (F u u) x=x x J (rst derivative) J u+u Euler u(x) v(x) ::: x x F (x u v : : : u v :::)dx = min u+u v+v ::: x x (F u u+f u u + F v v+f v v +)dx = F u u x + F v v x + x x Euler x x F u ; d dx F u u dx + x x F v ; d dx F v v dx + = d dx F u ; F u = d dx F v ; F v = u F u u + u F u u + F u x ; F u = v F v v + v F v v + F v x ; F v = u ::: u (n) x F (x u u u ::: u (n) )dx = min x u(x ) u (x ) ::: u (n) (x ) u(x ) u (x ) ::: u (n) (x ) x x (F u u+f u u ++F u(n) u (n) )dx = x x F u ; d dx F u + d dx F u ;+(;) n dn dx n F u (n) u dx =

13 Euler F u ; d dx F u + d dx F u ;+(;) n dn dx n F (n) u = n = F u ; (u F u u + u F u u + u F u u + F u x) + u (4) F u u + u (u F u u u + u F u u u + u F u uu + F u xu ) + u F u u + u (u F u u u + u F u u u + u F u uu + F u xu ) + u F u u + u (u F u u u + u F u u u + u F u uu + F u xu) +(u F u u x + u F u u x + u F u ux + F u xx) = u(x y) 5 I F (x y u u x u y )dxdy = min (F u u + F ux u x + F uy u y )dxdy = (F ux dy ; F uy dx)u + ; u Euler (F u ; x F u x (F u ; x F u x ; y F u y )u dxdy = x F u x + y F u y ; F u = ; y F u y )u dxdy = u xx F uxu x +u xy F uxu y + u yy F uyu y + u x F uxu + u y F uyu + F uxx + F uyy ; F u = Euler F = ruru = ux + u y (u x + u y )dxdy =min (u xu x +u y u y )dxdy = 5 RR a ru dxdy a ru = r(au);r au auss RR r (au)dxdy = H anu ds n ds nds =(dy ;dx)

14 4 u x u x +u y u y = ruru = r(uru);ur u auss nru = u n I u n u ds ; ; (u xx +u yy )u dxdy = Euler Laplace r u u xx +u yy = Poisson u xx +u yy = g n (u x + u y )+gu o dxdy F =(r u) = u xx +u xx u yy +u yy (u xx +u xx u yy +u yy )dxdy = min (u xx u xx +u yy u xx +u xx u yy +u yy u yy )dxdy = r ur u dxdy = r ur u = r(r uru) ;rr uru = r(r uru) ;r(rr uu) + r r uu auss I (r uu n ;r u n u)ds + ; r r u u dxdy = Euler (biharmonic equation) r r u u xxxx +u xxyy +u yyyy = u x x F (x u u )dx = min (F u u) x=x ; (F u u) x=x + x x F u ; d dx F u u dx = Euler F u = (x = x x = x ) (natural boundary condition) x F (x u u )dx+' (u ) ; ' (u ) = min x (F u +' )u x=x ; (F u +' )u x=x + x x F u ; d dx F u u dx =

15 5 F u +' = (x = x ) F u +' = (x = x ) Poisson n I (u x +u y odxdy )+gu + ;u u ds = min ; (u xx +u yy ;g)u dxdy + (u n +u;)u ds = ; ; I u n +u = on ; 9..4 FEM Laplace r u = FEM alerkin 6X k= (u k ;u )= x = x x = x k xy Taylor u k = u +ha k ru + a (a k r) u + a (a k r) u + 4 a4 (a k r) 4 u + 5 a5 (a k r) 5 u + a a = ( ) a = (= p =) a = (;= p =) a 4 = (; ) a 5 =(;= ; p =) a 6 =(= ; p =) u k Taylor u u x u y u xx u xy u yy u xxx u xxy u xyy u yyy p p p p u 4 8 u u ; ; ; ; u 4 4 ;8 u 5 ; ; ; ; ; ; u 6 ; ; ; ; sum 6

16 6 u xxxx u xxxy u xxyy u xyyy u yyyy u xxxxx u xxxxy u xxxyy u xxyyy u xyyyy u yyyyy 6 p 6 6 p p p 9 9 p ;4 6 ;4 ; 5 ; ;5 6 ; ; ;5 ; ; ;5 ; ;4 6 ;4 ;5 ; 5 ; 9/4 8/4 9/4 sum 6X k= (u k ;u )= a r u a4 r r u +O(a 6 ) Laplace r r u = r u = a 6X k= (u k ;u )+O(a 4 ) (9.) Laplace Poisson r u = ; = c + c x + c y c c c Laplace I K S SSS a b S SS g S S ((((((((((((((( h O E hhhhh EE h c J 9.6: 9.4

Microsoft Word - 計算力学2007有限要素法.doc

Microsoft Word - 計算力学2007有限要素法.doc 95 2 x y yz = zx = yz = zx = { } T = { x y z xy } () {} T { } T = { x y z xy } = u u x y u z u x x y z y + u y (2) x u x u y x y x y z xy E( ) = ( + )( 2) 2 2( ) x y z xy (3) E x y z z = z = (3) z x y