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1、上海交通大学 硕士学位论文 薄板弯曲问题的集中质量非协调有限元模拟 姓名：张文娟 申请学位级别：硕士 专业：计算数学 指导教师：黄建国 20070101 ? ?_?h?; o? v A ? , ? ? U?v ,? h? ? h? ? h? . R qh?%? ! ? , “ $# Kirchhoff ?%:=?%?A B C+.%D 9 ?E a -?FGH?I 7 . J ?K v o$L?H?A ?;M =%N , 9 , O K v QP #?R;Sjr?T ?UWV C9.?D 9 ?E a?F?G?HI 7 ._ vp ,?. o , j Morley 3 ?l $a?# 9 ?%
2、b?y c #? , ?d?GH?e?fp Mw?2$gh 7?8 . . 9 Xij?k W ,?. o Morley 3 ?l ? m? . on n nqp p p X X X Kirchhoff ? ? ? , o o o%, , ,q. . . , Morley 3 3 3 , G G GqH H HrI I I 7 7 7 , 9 9 9rX X Xs s st t t , Matlab. ABSTRACT Plates are the basic components in elastic structures. Their vibration analysis is of gr
3、eat importance in many applied fi elds, such as civil, mechanical, aerospace engineering, etc. The semi-discrete Morley element method for the problem and its error estimates in the energy norm are reviewed fi rst. Then, using the second- order central difference to discretize the time derivative of
4、 the displacement, and applying the quadrature formula to the mass matrix, a fully discrete lumped mass Morley element method is developed for the same problem. Since the technique of mass-lumping is borrowed, the resulting linear system is easy to solve compar- atively. The convergence rate in the
5、energy norm is established in terms of error estimates for the quadrature formula involved combined with error estimates for the static problem. A series of numerical results are given to illustrate the effec- tiveness and effi ciency of the method being proposed in this thesis. KEY WORDS X X X Kirc
6、hhoff plate, Lumped mass, Morley element, Error estimate, Numerical simulation, Matlab D D D E E E F F F G G G H H H I I I I I I J J J K K K L L L M M M N N N O O O P P P Q Q Q R S T?UV;W XZY? ? ?_ R S , .?+,“a?b c?d e?fCg“h % YCij kf;g lCm . n porqZs?t W?uwv rxzy , R C| C? S? ? qs?“ ?C“ k%kl m . R
7、Zfg“% U ? CSCC , q WCpW . RCSC?C r?RrVW ; m R S ? . ? % rZ pz? ,“ ? ? . R?S 3? ? ? p R ?“ ? xy? 9? d e , p v ?k ? “ R ? . ? ?, ? v R 3? . R ? ? | | |“ ? ?. z? ? xz ”X”) ? % o? v A ? , ? ? U v ,? ? h ? h ? ? h ? 13. _ , q s% # T ?% 31.9_ T ? rP? 7,12. 4 oy2 A+ ? 3 , +n_ ?+ BB A = 3: . 19 o v Morley/0
8、q12 3 # T ?W , Y ?lC# 9 ?bp s v 1?2 3 #? . ,%. or_?%Co v 11,27,28, ,%. Q v #? , ? ? 6?8 r Morley 3;:=?%?A ?B =C . D 9 E ap$?F%G%H?I 7 . J K v o?L H A ? M ? A 6 +B G?H I 7 . : ?%*“?,?. o T ?o C9.?D 9 E a)? F?G?H I 7 . % : $ 9?X? , v Matlab w ? ,+_ v ,. o , j Morley 3 ?l-a? # 9 ?b y%c # , ?-g?h 7?8 .
9、.9 Xij?k W ,?. o Morley 3 ?l ? m? . 7 2 I I I 2.1 ?Q?,-?Kirchhoff ? %C ?% f v a?2 +s?/?_14,16,21: utt IJMIJ(u) = f(x,t), (x,t) (0,T, u = nu = 0, (x,t) 0,T, u(x,0) = u0(x), ut(x,0) = u1(x), x , (2.1) 2o MIJ(u) := (1 )KIJ(u) + KLL(u)IJ, KIJ(u) := IJu, 1 I, L, J 2, n _ $%?y. . u0(x) u1(x) 7$?q) , %? )r
10、 ? ?+?. (0,0.5) ? Poisson ? .? ? , , ? ? ? !?.“ #?,$Sobolev=7?A=6BC2. D G R2 ? EF, 8 Wm,p(G) (m 0) %?G H SobolevSUT?L VML XW R kvkHm(G) |v|Hm(G). Hm 0(G) C 0 (G)?L k km,G Y ?Z? (3.3) ? ? =A? H3() H2 0() VM h () CB D ! .?FE “=#)G 7,13 kv hvkL2()+ h2kv hvkh. h3|v|H3(), v H3() H2 0(). (3.7) (+H 356?8 ?
11、! Ph: H2 0() V M h (), - Y v H2 0(), Phv V M h () _? ? ah(Phv,wh) = ah(v,wh), wh VM h ().(3.8) (3.6) Lax-Milgram (+* G PhvI?4.JL2(), ft L2(0,T;L2(), u L(0,T;H3(), utt L(0,T;L2() L2(0,T;H3(), uttt L2(0,T;L2(). max 0tT kuh(t) u(t)kh.ku1h u1kh+ ku0h u0kh +h(|u|L(0,T;H3()+ |utt|L2(0,T;H3() + h2(kuttkL(0
12、,T;L2()+ kutttkL2(0,T;L2() + kfkL(0,T;L2()+ kftkL2(0,T;L2().(3.12) = , ?O ?u0h= Phu0 w hu0, u1h= Phu1 w hu1, max 0tT kuh(t) u(t)kh.h(|u|L(0,T;H3()+ |utt|L2(0,T;H3() + h2(kuttkL(0,T;L2()+ kutttkL2(0,T;L2() + kfkL(0,T;L2()+ kftkL2(0,T;L2().(3.13) P% % % .Q (3.12), (3.7) (3.9) 4 G (3.13),Q$?R?% (3.12) . ? (3.11) ? vh L t (tt,t) + ah(,t) = (tt,t) + (utt,t) + ah(u,t) (f,t) = (tt,t) + d dt (utt,) + ah(u,) (f,) (uttt,) + ah(ut,) (ft,),
