湖南大學(xué)機(jī)械與運(yùn)載工程學(xué)院 College of Mechanical & Vehicle Engineering, Hunan University 三維問題的有限元方法 湖南大學(xué)機(jī)械與運(yùn)載工程學(xué)院 College of Mechanical & Vehicle Engineering, Hunan University 2021/1/26 空間實(shí)體單元 特點(diǎn) 思路和平面問題、一維問題完全相同 采用位移法構(gòu)造形函數(shù) 實(shí)體單元的主要類型 四面體，長(zhǎng)方體，直邊六面體，曲邊六面體 湖南大學(xué)機(jī)械與運(yùn)載工程學(xué)院 College of Mechanical & Vehicle Engineering, Hunan University 2021/1/26 四面體常應(yīng)變單元 z=Z x=X z =Z y=Y w 4 v 4 u 4 w 2 u 2 u 2 w 1 u 1 v 1 w 3 u 3 v 3 i j l k 1 = 4 = 2 = 3 = f s y f s z f s x 1 1 1 2 2 2 3 3 3 4 4 4 no de 1 no de 2 no de 3 no de 4 e u v w u v w u v w u v w d 湖南大學(xué)機(jī)械與運(yùn)載工程學(xué)院 College of Mechanical & Vehicle Engineering, Hunan University 2021/1/26 四面體常應(yīng)變單元 形函數(shù) ( , , ) ( , , )h ex y z x y zU N d 1 2 3 4 1 2 3 4 1 2 3 4 no de 1 no de 2 no de 3 no de 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 N N N N N N N N N N N N N 類似于常應(yīng)變?nèi)切螁卧?，采用什么樣的坐?biāo)系呢？ 1 2 3 4 2 3 4 1 V VL P 1= i 2= j 3= k 4= l P y z x 湖南大學(xué)機(jī)械與運(yùn)載工程學(xué)院 College of Mechanical & Vehicle Engineering, Hunan University 2021/1/26 四面體常應(yīng)變單元 形函數(shù) 1 2 3 4 123 4 1 2 3 4 124 3 1 2 3 4 134 2 , , V VL V VL V VL PPP 怎樣確定體積坐標(biāo)？ 2 3 4 1 3 4 1 2 3124 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 , , , P P PPd d ddL L L Ld d d d 1= i 2= j 3= k 4= l P y z x 1 4321 LLLL 是否滿足形函數(shù)的性質(zhì)？ 1 2 3 4123124134234 VVVVV PPPP 湖南大學(xué)機(jī)械與運(yùn)載工程學(xué)院 College of Mechanical & Vehicle Engineering, Hunan University 2021/1/26 四面體常應(yīng)變單元 形函數(shù) j k l iL i n o d e s r e m o t e t h ea t t h e 0 n o d e h o m e a t t h e 1 44332211 44332211 44332211 zLzLzLzLz yLyLyLyLy xLxLxLxLx (Delta 條件 ) 1 4321 LLLL 4 3 2 1 4321 4321 4321 1 1 1 11 L L L L zzzz yyyy xxxx z y x 湖南大學(xué)機(jī)械與運(yùn)載工程學(xué)院 College of Mechanical & Vehicle Engineering, Hunan University 2021/1/26 四面體常應(yīng)變單元 形函數(shù) 因 此 z y x dcba dcba dcba dcba V L L L L 1 6 1 4444 3333 2222 1111 4 3 2 1 1 de t , de t 1 1 11 de t 1 , de t 1 11 j j j j j i k k k i k k l l l l l j j j j i k k i k k l l l l x y z y z a x y z b y z x y z y z y z y z c y z d y z y z y z (Adjoint matrix) 伴隨陣 (Cofactors) 余因子 i j k l i= 1,2 k = 3,4 l = 4,1 湖南大學(xué)機(jī)械與運(yùn)載工程學(xué)院 College of Mechanical & Vehicle Engineering, Hunan University 2021/1/26 四面體常應(yīng)變單元 形函數(shù) l k j i l k j i l k j i z z z z y y y y x x x x V 1 1 1 1 d e t 6 1 四面體體積 )(6 1 zdycxbaVLN iiiiii 湖南大學(xué)機(jī)械與運(yùn)載工程學(xué)院 College of Mechanical & Vehicle Engineering, Hunan University 2021/1/26 四面體常應(yīng)變單元 應(yīng)變矩陣 ( , , ) ( , , )h ex y z x y zU N d ee BdL NdLU NLNB 0 0 0 00 00 00 xy xz yz z y x 44 44 44 4 4 4 33 33 33 3 3 3 22 22 22 2 2 2 11 11 11 1 1 1 0 0 0 00 00 00 0 0 0 00 00 00 0 0 0 00 00 00 0 0 0 00 00 00 2 1 bd cd bc d c b bd cd bc d c b bd cd bc d c b bd cd bc d c b V B 湖南大學(xué)機(jī)械與運(yùn)載工程學(xué)院 College of Mechanical & Vehicle Engineering, Hunan University 2021/1/26 四面體常應(yīng)變單元 剛度和質(zhì)量陣 e TTee V d V Vk B c B B c B 1 1 1 2 1 3 1 4 2 1 2 2 2 3 2 4 3 1 3 2 3 3 3 4 4 1 4 2 4 3 4 4 dd ee T e VV VV N N N N N N N N m N N N N N N N N N N ji ji ji ij NN NN NN 00 00 00 N 湖南大學(xué)機(jī)械與運(yùn)載工程學(xué)院 College of Mechanical & Vehicle Engineering, Hunan University 2021/1/26 四面體常應(yīng)變單元 質(zhì)量陣的積分 1 2 3 4 ! ! ! !d6 ( 3 ) !e m n p q eV m n p qL L L L V V m n p q Eisenberg and Malvern 1973 : 2 0 0 1 0 0 1 0 0 1 0 0 2 0 0 1 0 0 1 0 0 1 0 2 0 0 1 0 0 1 0 0 1 2 0 0 1 0 0 1 0 0 2 0 0 1 0 0 1 0 2 0 0 1 0 0 1 2 0 0 1 0 020 2 0 0 1 0 2 0 0 1 . 2 0 0 20 2 e e V sy m 湖南大學(xué)機(jī)械與運(yùn)載工程學(xué)院 College of Mechanical & Vehicle Engineering, Hunan University 2021/1/26 求單元質(zhì)量陣的另 類方法 特殊的自然坐標(biāo)系 統(tǒng) z x z =Z y i j l k 1 = 4 = 2 = 3 = =0 =1 =1 =const ant P Q 四面體常應(yīng)變單元 z x z =Z y i j l k 1 = 4 = 2 = 3 = =0 =0 =1 =const ant P z x z =Z y i j l k 1 = 4 = 2 = 3 = =1 =1 =1 =0 = constant P Q R 湖南大學(xué)機(jī)械與運(yùn)載工程學(xué)院 College of Mechanical & Vehicle Engineering, Hunan University 2021/1/26 四面體常應(yīng)變單元 0 )( )( 223 223 P P P z yyyy xxxx 0 )()()( )()()( 1122311 1122311 B PB PB z yyyyyyyyy xxxxxxxxx 4 321214444 321214444 )1( )()()()( )()()()( zz yyyyyyyyyyy xxxxxxxxxxx B B 1 2 3 4 (1 ) (1 ) (1 ) N N N N z x z =Z y i j l k 1 = 4 = 2 = 3 = =0 =0 =1 =1 =0 =1 =1 =1 =1 =0 =con st ant P x P ( x 3 x 2 ) +x 2 , y P ( y 3 y 2 ) +y 2 ,0 O B B x B ( x P x 1 ) +x 1 , y B ( y P y 1 ) y 1 ,0 O x =( 1 )( x 4 x B ) x B , y =( 1 )( y 4 y B ) y B , z =( 1 ) z 4 =con st ant =con st ant 湖南大學(xué)機(jī)械與運(yùn)載工程學(xué)院 College of Mechanical & Vehicle Engineering, Hunan University 2021/1/26 四面體常應(yīng)變單元 Jacobian: z y x z y x z y x J 2 4 312141313121 312141313121 6 0 0 d e t V z yyyyyy xxxxxx J 111 000d d e t d d d e TT e V V m N N N N J 1 1 1 2 1 3 1 4 111 2 1 2 2 2 3 2 4 2 000 3 1 3 2 3 3 3 4 4 1 4 2 4 3 4 4 6 d d dee V N N N N N N N N m N N N N N N N N 湖南大學(xué)機(jī)械與運(yùn)載工程學(xué)院 College of Mechanical & Vehicle Engineering, Hunan University 2021/1/26 六面體單元 P P P P 湖南大學(xué)機(jī)械與運(yùn)載工程學(xué)院 College of Mechanical & Vehicle Engineering, Hunan University 2021/1/26 六面體單元 形函數(shù) eNdU 1 2 3 4 5 6 7 8 d ispl a c e m e n t c o m p o n e n ts a t n o d e 1 d ispl a c e m e n t c o m p o n e n ts a t n o d e 2 d ispl a c e m e n t c o m p o n e n ts a t n o d e 3 d ispl a c e m e n t c o m p o n e n ts a t n o d e 4 d ispl a c e m e n t c o e e e e e e e e e d d d d d d d d d m p o n e n ts a t n o d e 5 d ispl a c e m e n t c o m p o n e n ts a t n o d e 6 d ispl a c e m e n t c o m p o n e n ts a t n o d e 7 d ispl a c e m e n t c o m p o n e n ts a t n o d e 8 1 1 1 ( 1 , 2 , , 8 ) ei u vi w d 1 7 5 8 6 4 2 0 z y x 3 0 fsz fsy f sx 87654321 NNNNNNNNN )8,2,1( 00 00 00 i N N N i i i iN 湖南大學(xué)機(jī)械與運(yùn)載工程學(xué)院 College of Mechanical & Vehicle Engineering, Hunan University 2021/1/26 六面體單元 坐標(biāo)轉(zhuǎn)換 4( - 1, 1, - 1) (1, - 1, 1)6 (1, - 1, - 1)2 1 7 5 8 6 4 2 0 z y x 3 0 f sz f sy f sx 8( - 1, 1, 1) 7 (1, 1, 1) ( - 1, - 1, 1)5 ( - 1, - 1, - 1)1 3(1, 1, - 1) ii i ii i ii i zNz yNy xNx ),( ),( ),( 8 1 8 1 8 1 )1)(1)(1(81 iiiiN (Tri-linear functions) 湖南大學(xué)機(jī)械與運(yùn)載工程學(xué)院 College of Mechanical & Vehicle Engineering, Hunan University 2021/1/26 六面體單元 87654321 BBBBBBBBB 0 0 0 00 00 00 xNyN xNzN yNzN zN yN xN ii ii ii i i i ii LNB ee BdL NdLU 如何推導(dǎo) B矩陣？（形函數(shù)在自然坐標(biāo)系下得到） 湖南大學(xué)機(jī)械與運(yùn)載工程學(xué)院 College of Mechanical & Vehicle Engineering, Hunan University 2021/1/26 六面體單元 z z Ny y Nx x NN z z Ny y Nx x NN z z Ny y Nx x NN iiii iiii iiii 鏈?zhǔn)椒▌t z N y N x N N N N i i i i i i J where z z z y y y x x x J 湖南大學(xué)機(jī)械與運(yùn)載工程學(xué)院 College of Mechanical & Vehicle Engineering, Hunan University 2021/1/26 六面體單元 8 8 8 1 1 1 ( , , ) , ( , , ) , ( , , )i i i i i i i i i x N x y N y z N z or 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 i i i i i i i i i i i i i i i i i i i i i i i i i i i N z N z N z N y N y N y N x N x N x J 1 1 1 2 2 2 3 5 6 7 81 2 4 3 3 3 4 4 43 5 6 7 81 2 4 5 5 5 6 6 61 2 3 4 5 6 7 8 7 7 7 8 8 8 x y z x y zN N N N NN N N x y z x y zN N N N NN N N x y z x y zN N N N N N N N x y z x y z J 湖南大學(xué)機(jī)械與運(yùn)載工程學(xué)院 College of Mechanical & Vehicle Engineering, Hunan University 2021/1/26 六面體單元 i i i i i i N N N z N y N x N 1 J 0 0 0 00 00 00 xNyN xNzN yNzN zN yN xN ii ii ii i i i ii LNB 111TT 111d d e t d d d e e V A k B c B B c B J ),(d)d,( 1 1 1 1 1 1 1 1 1 jjikji n i m j l k fwwwfI 111 111d d e t d d d e TT e V V m N N N N J 湖南大學(xué)機(jī)械與運(yùn)載工程學(xué)院 College of Mechanical & Vehicle Engineering, Hunan University 2021/1/26 六面體單元 d e t ea b c V J 88 7877 686766 58575655 4847464544 282726252433 28272625242322 1817161514131211 . m mm mmm mmmm mmmmm mmmmmm mmmmmmm mmmmmmmm m sy e 對(duì)于長(zhǎng)方體單元 湖南大學(xué)機(jī)械與運(yùn)載工程學(xué)院 College of Mechanical & Vehicle Engineering, Hunan University 2021/1/26 六面體單元 ddd 00 00 00 ddd 00 00 00 00 00 00 ddd 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ji ji ji j j j i i i jiij NN NN NN abc N N N N N N abc abc NNm ij ij ij ij m m m 00 00 00 m )1)(1)(1( 8 d)1)(1(d)1)(1(d)1)(1( 64 ddd 3 1 3 1 3 1 1 1 1 1 1 1 1 1 1 1 jijiji jijiji jiij hab abc NNabcm 湖南大學(xué)機(jī)械與運(yùn)載工程學(xué)院 College of Mechanical & Vehicle Engineering, Hunan University 2021/1/26 六面體單元 E.g. 2168)111)(111)(111(8 3 1 3 1 3 133 abcabcm 216 1 216 2 216 4 216 8 46352817 184538276857473625162413 483726155814786756342312 8877665544332211 abc mmmm abc mmmmmmmmmmmm abc mmmmmmmmmmmm abc mmmmmmmm 湖南大學(xué)機(jī)械與運(yùn)載工程學(xué)院 College of Mechanical & Vehicle Engineering, Hunan University 2021/1/26 六面體單元 (Contd) 8 48. 248 4248 42128 242148 1242248 21244248 216 sy abc ex m 對(duì)于 X方向 (Rectangular hexahedron)