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 $a?# 9 ?%b?y c #? , ?d?GH?e?fp Mw?2$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 great 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 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 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 Kirchhoff 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 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/0q12 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 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 ? ?+?. (0,0.5) ? Poisson ? .? ? , , ? ? ? !?.“ #?,$Sobolev=7?A=6BC2. D G R2 ? EF, 8 Wm,p(G) (m 0) %?G H SobolevSUT?L VML XW R kvkHm(G) |v|Hm(G). Hm 0(G) C 0 (G)?L k km,G Y ?Z? (3.3) ? ? =A? H3() H2 0() VM h () CB D ! .?FE “=#)G 7,13 kv hvkL2()+ h2kv hvkh. h3|v|H3(), v H3() H2 0(). (3.7) (+H 356?8 ?! Ph: H2 0() V M h (), - Y v H2 0(), Phv V M h () _? ? ah(Phv,wh) = ah(v,wh), wh VM h ().(3.8) (3.6) Lax-Milgram (+* G PhvI?4.JL2(), ft L2(0,T;L2(), u L(0,T;H3(), utt L(0,T;L2() L2(0,T;H3(), uttt L2(0,T;L2(). max 0tT kuh(t) u(t)kh.ku1h u1kh+ ku0h u0kh +h(|u|L(0,T;H3()+ |utt|L2(0,T;H3() + h2(kuttkL(0,T;L2()+ kutttkL2(0,T;L2() + kfkL(0,T;L2()+ kftkL2(0,T;L2().(3.12) = , ?O ?u0h= Phu0 w hu0, u1h= Phu1 w hu1, max 0tT kuh(t) u(t)kh.h(|u|L(0,T;H3()+ |utt|L2(0,T;H3() + h2(kuttkL(0,T;L2()+ kutttkL2(0,T;L2() + kfkL(0,T;L2()+ kftkL2(0,T;L2().(3.13) P% % % .Q (3.12), (3.7) (3.9) 4 G (3.13),Q$?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,),