考試大禮包有限元版第三章

1、 ! #$%&()*+,- !#$%&()*+,-./01256789:;8=AB !#$%2CDEF9G8HIJK2LMNG2OP0QRS?7T2UVWRitz 01:./52EFWXM YF2)*ZQLagrange & Hermite _2ab./2 cdef.3-1 #$%&(8ghiJK9jklmFknoMNG20Qpkqrs ihi0tiiuab$%iiiv#wx*yz2*JK|,- V! #VF(x,y)=0$ 3-1(a)%Xg&% PA,PB 2!(9 N)&*9+,Vz% k 2+-./901:23245 F(x,y)=0 67V!2hiJKjkn8M019qrs|hi0t

2、 :hia1qrhi w !%:;?8ABCV9ABD$ 3-1f%s(b)EF !X(GH2V!2hiea X 0Y 0-HkJH%K9LsT k l0 AB X 0Y 0 s!sATTBINAPANByMuab$% (:hia1NOqrhi)PBV!2hia1jqr%x!y!D$ 3-1 (c) EF X#2uab%srA9rBPx y #xo$r , $r!P $x ,$y ,$x!,$y!uab$%p ArA pBrB T AB 06QRS%/*01*(T8UVW%bATTByPArA p y p y k l AB AB 0AABB0PBr 2 B 1 p A y A pB yB2 k

3、 AB l00oxc$XY|+ 1 ,- 17,(3) v#wx* :Z5qrhihia1_v#yz 1 k AB l 2 py p yp0A AB B2v#yzwx*9Ls 0phia1k68MqrV!hi2019a,- V!bc+de2fb98M012gh,+shi0t+:hia16qrhiJKWuab$%+:hia1N OqrhiJKWb !#$%+:Zi6qrhiJK !#$%jkOr%s23456789:;)?=A9:BC56DE=A9:FG%&HI-CJI-K#$-CLDEKM8Aj%8ekl$%9S2Km,Enopkpq9Nrs%t6ukvwx%yza%|abef|&|_ef|2

4、 9n5!8g#$ !#$%&hi 0t2de%VpkqvM&- V! +,D| ()*2+,+-%9.5/% 9(012 f(x)x=0E3459x=L 3)3#6=HP& ,ab u(x)78s iu(0)=0 (9:;)iiu(x) :O, L6/9?=Cx, u$ XYM(iii) :1 LLp (u) EA(u) dx f udx P u(L)2(3-1-1)2 00w !*; u(x)+$u(x)%NGa u(x) 2?(3-1-2)u(0) 01 LL (u u) EA(u u)2 dx f (u u)dx P u(L) u(L)2 0bp0NAQsLLLP (uu) P (u)

5、 EAu udxf udx0001 L P EA(u ) dx22 0w !*9LBP(u)P (u u) P (u) 0+ 2 ,- 17 ,f(x)D2FrGef+sHI78(3-1-2)2$u(x) ,LL P EAu udxf udPxu L0(3-1-3)0Q(3-1-3)L#x*ef0;JFK5 u(x) L:/99M3-1-3N/8O9PQF3-1-2,RL EAu f udx EAu (L) Pu(L) 0 (3-1-4)0(3-1-4) QHI$u(x) SH?2FrGef+sEAu f 0EAu(L) P 06rtNT?shi0tHefU N%&HI-OPQRDESTV7R

6、9WwXMqr01:HI6deX UVWXSYZ;AFY0tqrZ9u r78sEAu u(0) 0EAu(L) P 0f 0hi0tabefHefF !#$%qrZ9GAyz78abefu (0) 0 b_;ab% &HI-Acdef u(x) :P mwx*Z9hi0t&HefmQR7 8W u(x) :P O_wx*Z9hi0t&Hef#O_2QR78:;MI n9_;ghbcd;(Natural Boundary Condition )9babefS%ij;kEssential Boundary ConditionlaUVWXSYZmnAopTqkrO/slFY0tqrZGA u(x)

7、,/92bcdzej ufC2(0, L)bF !#$%qrZ9%ygvhZ#L:9GA u(x)h0j/ej ufH|(0, L)1M h5EHJKK0Zij-% a9k % t9)RD/H (fx, fy), AB 45y, vnq(x)P(x) BCCD AD 21(EH% q(x)9l(EH% p(x) #P &mnef%s12 E nTnu, vtdxdyPuT f x tdxdy f y v x, uuT p v q tdxAD $ XYX1H1 %8cdzh0j/2yzGH2yzoT9C2 %bcdz/92yzGH2yzoT+ 3 ,- 17 ,AD(fx, fy)BOCnupA

8、B=vpAB=0QFT x0 u T E T E T E 0 x 112E T y v y 2 xy yx kRsQdxdy d xL dsxdyxXdxdy d yL dsxdyYyI= LX#LY %ijt7(2=1u n 20(vw#P 2x*ef%suT f xuT p1 T E P ( 2 tdxdy) v v tdxdy q tdx fAD y T0 TuT x x LX0 u x f 0 x tdxdy 0Ly tdsy fY v y vy LL XY yx yx x yuT pq tds 0(3-1-5)vAD xIRy(9=1u n 20(vw%kRyu ABs$u=$v=0

9、M(3-1-5)HI$u,$v SH?2FrGef%T x0 x f 0 x 0 y y f y yx yx + 4 ,- 17 ,ABBCCDDALX-1010LY0-101y BCs)y=*xy=0 y CDs)x=*xy=0 y ADs)y=q,*xy=p%aef2kR u, v jY,26#._9;|sX 2h54a$ 3-4 EF24JK9v#&mnef%vq(x)Q1 LP (v) EI (v)22 0CL q vdx Q v(L) M v(L)0v (0) v(0) 0#x*ef$ XY4L P EIv vLQq vdx vL M v0dx(3-1-6)L006QN/XO9PxI

10、Rar78mnef $v(0)=$v/(0)=0M,L EIv q vdx ) M vL EIv(LP0 EIv Q (v(3-1-7) L0 x L:(3-1-7)HI$v(x) SH?2FrGef+sEIv q 0EIv(L) M 0hi0tEIvfef Q 0 x LY0t2cz% %a vv 2ef%fef9%a vv 2e f%mneftu/*STWXv v(x)4wxAyznv|C4(0,L)u#$%&(WXvCb1& 2C! v(x)DSOv|H2(0,L)l!28g#K9jj%zNG9czNG2#O$%8&Gt()DnOI=9|Y0tcz*+kab%kl+,*9)-9|jY,G

11、A*+1 (0) 1 (L) 2 (0) 2 (L) 0v 11 (x) 2 2 (x)%23yz91 ,1 %5Yz9Mqr;8JK2#*w6yjWH1 LL33 EI (v ) dx P v() 2EIL(1 1 2 L 2 L ) 242L L22223P1216640#*6y2x*efP 0P 0, 21R8#z0tG3 PL4 2L 1216 EI3 PL2 4L 1264 EI#z0tG23yz=25Yz 7 PL ,128 EI 1 P 64 EI12=#ab2O_*%PL3Lv( ) 0.00952 4EI+ 6 ,- 17 ,1MsH,-mnb%mn (x) sin 2 x

12、(x) sin x ,21LL;?02kyzG?78_,jY,kRmnef2GAj)-?s 44 2122EIL 21 2 P PP2124 L P1LA 0 0,P 2jAQ2 PL31PL31 , 28EI4EI4=#abPL3Lv( ) 0.011549 4EI7JK2m%3PL3PL3Lv( ) 0.011719 4256EIEIXM01AQ2=#abB*!am*Kw2kyz9:8CDE%F2efNGamnefRQR78HI Ritz 1lJPNGA; 8#9x+KbgO_2 :fLK8gGMQ NNTR90wNOQP8Ayz%kyzZ9a;&yzjkAdHIO9_ef&jY,G132

13、4x/.0 11x0 21xAP6QQR789GR2nS%,178mn0,ef;:8TUnP6QRR9x:b 8TUn9V8&ZWXM2ij#DEZYZZ-0Dd=Po$Bb2 Ritz 12EF)Ry-n 31x0 41x.3-3 *+,-+A%mn)./0uu4xmnk0A12s3,-mnA4H5678luu3u1u209:(; Lagrange 0,- $ 3-6 %8()*2+9.5/&(012 f jk+ x 2yzBb?mniT0, LH;_;|+8gx0$iT./9%Ub aw3# i 2ab*9.i7T+j?2u(xi ) uiVz ui hM#O#A*Bk+l8CA./mnA

14、(DEFGHKIij;7JKLM56#D u(0)=0 2efG7.A#|2ab u1=0 Ljko$b;kl:Hm0z2Zn#DGovDp EqZ_;&efamnefJKrs,ytu2vw192xIHx(_ef&jY,JKX_,&jY,655 2 i(x)& u(x)SL:y|z#|923t Nx+si i(x)& u(x):./5/99:#DR/9W) u(x)2N$OyzC2eHPO&yzQR!44#ASTUQii HI：2VW0, LXDSOQ ：(x) u(x)YZ(0AmnQZZ?mn24+cJK923yz v R v E:0, L6/90, L%;_;|,+8g$iv1324x/

15、.0 11xT./9%Ubaw3#2ijtqq%5g././&! Nh|kyz0685 kyz 1(x,y)b 9(x,y)78i i(xj,yj)=(ij1 j=iM0 ji(ii) i(x,y):&g./5+xy 2u,yzM23yz9u( x, y) uiii1v(x, y) viii 1ef i(xj,yj) 25 9:#6,X7=u(xi , yi ) uiv(xi , yi ) viVz uivi h%#62ab*:&g./59uv + xy 2?Z8ONOQ9jk8g#62yz*8m59 jKb22-3-2OF:f:./5 uv /99;Idzuxuyvxvy%Yzy./#D i

16、j 9uv Au,h9?Z;e6Xg #62yz* uiviujvj Ev 59BC./Z uv /9D$ 3-11 EFxC./ZIu,vdzuxuyvxvy8N/99,-2DE*9x:t5S?h0j/ E0223yz78h5EHJKbcJK_,&jY,2GAjF Ritz 1AO_Xv#K52abFWHix$3-11u U v u9 v 9 F8g./, 6 gI=)%8g 2G18 2-0Eq9):8g./5,|M# 9b6Q+ 18 g 7JK2v#yzjkOP%uT 01 T E tdxdy 2 tdx Pv q ADuT 0128 e vT tdxd qtdxAD i1i+ 12

17、,- 17 ,jyT 0 0 xx E 0U T 018 0U tdxdy U ADTT qtdxy y 2 i1 ei xy xy 0128 TTTUkUU qtdxi i17441U 1 KT U T U (3-4-1)F24#wx*U T K U U T F 0LK U F v7 u8 v8 u9 v9 0mnefu7jk!KFI-IHP:GoKF2t=pko$5./p Eq ):8g./5,|MF9Bk1 ,|M-|MFki #+Kb?A-./Cij;w7JKLM56QR34mA./mn0156?s)O/sQ;+ O?$2%a,-/012%ttv3-5 12=8W99F()*ZQ5 2

18、23yzPRRJKlJ2Pom?ZQd9Bb,-/8S+O_RS:TqUUpV./W &-h!Z,-/efcdRPo%ygvcd,9(./5K52abF23yzE78kn|28A9:32N$Oyz;+ !#$%EGA29R+P6QRR2M&4WX56(Vty1*7kt89l()?;8ef2%+s./&-XZ9Po:&g./5SYOZMYEhWD%,-/&-YOv#2Po !*97R# OyPo78;8ef28gMD+9D%Po:/gij6%8YEhWP_EHhLha9bcdW 2./Lu?mPM2O_X:;64A:+:K5./abFZN34abefefmnef9;&ef:GovDEqZgpk34D%./abFhmZMpbiQ9ojIkyp6yl=2efVHo$efM&X+N2a()*+9Gn?Z8ONOQuxu 1 2 xLjI=51 %pDhb952x %YEhO 2ah5EHJK9on xy 2?Z8ONOQLj9L 5 3x 3 5u x y y y 123122 5 32 v x y y x x 354564622+ 14 ,- 17 ,I=K8gpa5%8gpDabsbghb8gqxKbgpa5%YEhOW u v u v