Introduction to Finite Element Method
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- Curtis Bailey
- 5 years ago
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1 p. o C d Eo E. Iodo o E Mod s H L p. o C d Eo E o o s Ass L. o. H L p://s.s..
2 p. o C d Eo E. Cos. Iodo. Appoo o os & o Cs. Eqos O so. Mdso os-es 5. szo 6. wo so Es os 7. os ps o Es 8. Io 9. Co C Isop E. Msos ops. os w H Co q - Bs. Md oo Ass L. o. H L p://s.s..
3 p. o C d Eo E. Cp Appoo o os d o Cs Ass L. o. H L p://s.s..
4 p. o C d Eo E. d Cosdos W s s soo o po? dss o s s soo W s soo? soo sss o qos s w s od odos. How do soos s? O os o I s s o s o soos s o qo w po s -posd po d ds d soo o po W s poss o d soo o os sos? W d ppo soo. W s ppo soo? d Qsos - W s s ppoo? - How w pss s ppoo? A Good o os o W od Appoo od - Yd s ppoo o soo o d o ppoo. - Co o soo s d o ppoo s pod. W s o o Bs Appoo? - M dd s oss soo o soo. - B ow os s oss? - Cos o ps ds w wo sp pos. - W sod d so so o o s ds w wo s o s Ass L. o. H L p://s.s..
5 p. o C d Eo E. szo - pso o oos dsd qs w so s. X X X w ss os o o s - : Coo o so os w s ss - ss os sod dpd o o. X d o - o ss o ss Appoos X X X X w o oo: pd ds do so. os o os: A s o o sp A o sp s sd o od sp o ssod o d o o s w - d o - o. - E od sp dd s sp w ds w wo s sp s sd dd o. os ps o o dd o o sp. Ao oow os po. - L o: L d - L o: - H o: L d / H d / Ass L. o. H L p://s.s..
6 p. o C d Eo E. Ass L. o. H L p://s.s.. G Ids o Bs Appoo L s d o ppo o s oss o o s o o dd o sp. I s s s ss o ppoo od dpd o os o o sd ppoo. Ls q EoLE Mzo Eo: Mz L L d d d d d d d o s Eqo : I ss os ooo os do.
7 p. o C d Eo E. o o o - o o o s poss o o d. o Cs - o o s dd s o do. - : - - d d d d - d d d d - d d d d o o o Ass L. o. H L p://s.s..
8 p. o C d Eo E. Ass L. o. H L p://s.s.. Mzo o LE o Cs M d d d d d M d poss o E Eqo M d o d d d d d d d I s ss os s od o d d d d d d d d d d I s o ss ods d d d d d d. o M
9 p. o C d Eo E. Ep M d s o d d d d d d d / / d / / B pp BC / / Ep M d s o d d d d d d Howo. Appo os o os poos sd zo o s sq os. s poos p o od. Yo s o o s s pso s pzod o. Yo so d so o so sos qos Lp. o o o o ps s do pso o po. Yo sod ps pop dsssos o o ss o w so ps sow o ppo os d o.. E qo o oow zo o d pop od odos. M d. o qo d od odos o oow po. d w M w wq d d Ass L. o. H L p://s.s..
10 p. o C d Eo E. Cp Ep Eqos O so Ass L. o. H L p://s.s..
11 p. o C d Eo E... os w ooos dsp od odos o o d d Appoo szo w sds o : o o o s o so... d Eqo sd : E d o sd : Eo Eso : Ed d d d Ls q Eo d d d d d d d d d d d d d d d d d d d d d d d d d L d d d d d Ls q Eo E o o o E d d d d d d d d d d d d d d d d d d C d d d d d d d d d d d Ass L. o. H L p://s.s..
12 p. o C d Eo E. Mzo os M M L M w... M : -z Mod o p o M o E - s Od ss Codo o Mzo o d d d d d d d d d d d d d d d d d d d o d d d : o pp o p o Wo d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d oo p - { d d } d - : zo pos d soo. - : zo pos d ppo soo. ops o - d d d d : d d d d d d - os ss : d d d d d d d d d d d d d Ass L. o. H L p://s.s..
13 p. o C d Eo E. Aso M op o o o E E oo d d d d d d d d d d d d d d d d d d d d E d d d d d d d d d d d d d E d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d q s ods o o d d d d d d d C d Wd sd Mod d d d Ed d o d - : G Mod p s o s-do ss d d d d d d d d d d d d d d d d d d d d d d d o d d d d - Id s o -z Mod o o p - : o-g Mod po ss o o-s do ss Wd sd Mod s. o p o d d d d d d d d d d d d d d d o poss Ass L. o. H L p://s.s..
14 p. o C d Eo E. d Ep : d - E soo : - s : App BCs : d d d d. o 6 - od : App BCs : d d d d d d d 6 5 d d 6 s Eqo : 6 5 d Ep : s d - E oo : s - s : d App BCs : d d s d. o Eo =.5 % Ass L. o. H L p://s.s..
15 p. o C d Eo E. Ass L. o. H L p://s.s.. - od : App BCs : d d d d d 5 d d s d 6 s d s Eqo : 6 5???? - I o Eo= / s s d d Eo= / os os d d o o o pssos dpo w ss s pod. - w Op ***** -od oo ***** = E+ Eos o -od poo o o =.9E+ % o =.66E+ %
16 p. o C d Eo E. ***** -od oo ***** = E+ =.E+ Eos o -od poo o o =.9E+ % o =.66E+ % ***** -od oo ***** =.98E+ = E+ =.5585E+ Eos o -od poo o o =.888E- % o =.9568E+ % ***** 5-od oo ***** =.98E+ = E+ =.5585E+ 5 =.787E- Eos o 5-od poo o o =.888E- % o =.9568E+ % ***** 6-od oo ***** = -.577E- = -.58E+ = E+ 5 =.69E+ 6 = -.6E+ Eos o 6-od poo o o =.E- % o =.968E- % ***** 7-od oo ***** = -.577E- = -.58E+ = E+ 5 =.69E+ 6 = -.6E+ 7 =.59577E-9 Eos o 7-od poo o o =.E- % o =.968E- % ***** 8-od oo ***** =.E- = E+ =.6566E- 5 =.6766E+ 6 =.7E+ 7 = E+ 8 =.8796E+ Ass L. o. H L p://s.s..
17 p. o C d Eo E. Eos o 8-od poo o o =.576E-5 % o =.5895E- % ***** 9-od oo ***** =.87E- = E+ =.65578E- 5 =.6767E+ 6 =.59E+ 7 = E+ 8 =.87955E+ 9 =.69885E-6 Eos o 9-od poo o o =.576E-5 % o =.5895E- % ***** -od oo ***** = -.69E-5 = E+ = -.958E- 5 =.557E+ 6 = -.586E- 7 = -.578E+ 8 = -.999E- 9 =.79E+ = E- Eos o -od poo o o =.8956E-7 % o =.55E-6 % ***** -od oo ***** = -.8E-5 = E+ = E- 5 =.555E+ 6 = -.56E- 7 = E+ 8 = E- 9 =.968E+ = E- =.6686E- Eos o -od poo o o =.69E-7 % o =.5E-6 % Ass L. o. H L p://s.s..
18 p. o C d Eo E. - s pos - Eo % - -6 o Od o poo o o os.8.6 X.. E d od od X-Cood o o ppo o o d ods o poo Ass L. o. H L p://s.s..
19 p. o C d Eo E. 'X - - E d od od X-Cood o o d ppo o o d ods o poo Howo. o oow OE o s o o pos o opsso s w s s od q. Ass ooos dsp BCs. d w d w EI q d d. O w odo dos pp o po od o o.? sss ps d s o odo.. Appo soos o d-d d w o od poos w o ow d wo ows. o o s ppd. Cop o soos d dsp oo o d s o o soo d dsss. Ass L. o. H L p://s.s..
20 p. o C d Eo E... os w o Bod Codos o o - Eqo d d - Bod Codos d d o d o d d Eo Mzo: Eo Eso d d d d d d d d d d d d d d d d d d d d L d d d d d d d d d d d d d E o o o E C d d d d d d d d d d d d { d d d d d d d d d d d d d d d d d d d d d d d d d d d d d o d d d d d} Mzo os M M L M w... Ass L. o. H L p://s.s..
21 p. o C d Eo E. M : -z Mod o p o M o E - s Od ss Codo o Mzo o d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d o : o p o p o Wo d d d d d d d d d d d d d d d d d d Aso M op o o o E E oo d d d d d d d d d d d d d d d d d d d d d d E d d d d d d d d d d d d d d d d E d d d d d d d d d d d d d d d d d d q s ods o o d d d d d d d d d d d d d d d d C d d d d Ass L. o. H L p://s.s..
22 p. o C d Eo E. Ass L. o. H L p://s.s.. Wd sd Mod o d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d E Wd sd s. o p poss o o d d d d d d d d d d d d d d d d d d
23 p. o C d Eo E... I Codo - o q - Io o os w dso d?? O o o w so zo d d w o o o - C w d o w odo? d d d d d d d d d d d d s ss s s - d + d os d d d Ass L. o. H L p://s.s..
24 p. o C d Eo E. - C w d o w odo? d d d d d d 6 d d d d d d 6 s ss s s - d + d os d - C w d?? d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d o d s d o s oos. o ps po o w ood o odo pss op odo w ss dsp d o sod q dd dd s d os. Ass L. o. H L p://s.s..
25 p. o C d Eo E. d - C w d d o w odo? d d d d d d d d d d d d d d o d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d [ ] d d d d d d o d s q & s dsoos. Howo. L o oow OE w ooos dsp BCs. d w q d. o so pop o dd o. soo.. Appo soos o s o o od q o sp d o od ppd d poos w o ow wo ows d ows. Cop o soos d dsp oo o d s o o soo d dsss.. Id odo o oow s d s o dd odo. d w dw d w d d d d d d Ass L. o. H L p://s.s..
26 p. o C d Eo E... O d o p o Wo s wpo I do od s q d Q-o ss d s q w s sd s doo wo do Q os o od s q o wo o doo do Q-ssss. Qd d w M wpo - Coos o I A sod od o ss oow s sod od. H d od o A sod s s. A d - Ep : B po o qo o po : psso o wo : : A pop o sp d w EI q d d w d w d w w EI q d EI d wqd d d d I w s dsp dd od ppd od psso o pp o wo os s oows. Q M M d wqd w d w EI w M s o dd od ppd d s d d o ppd. Ass L. o. H L p://s.s..
27 p. o C d Eo E. Ass L. o. H L p://s.s.. M wpo - s o A A : A pop o sp - Ep : ss po Eq Eqos os Y X H o Wo Epsso Y X H s os Y X Y X s os Y X Y X s os
28 p. o C d Eo E. os os EA s s EA X Y X Y I o ss osss o od ppd - o do X Y X os os EA.5. E szo o szo... od d d d d d d d d d d d d d d d d d d o dss Ipoo o sp d E L L = + L Ass L. o. H L p://s.s..
29 p. o C d Eo E. Ass L. o. H L p://s.s.. L L L L L L L L L = = szd o o o L L L L L L L L d d d d d d d d d d d d d d d d d d d d B B B B L d d d d d d d d d d d d d d Cop Codo Co q L L C C L Go s Eqo Go ss Eqo. C C C C C C C C dss o
30 p. o C d Eo E. Ass L. o. H L p://s.s.. -z p Ipo o EM d d d d d d dss o d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d B B + -
31 p. o C d Eo E. Ass L. o. H L p://s.s.. E od. Go qos do od odos o od.. w o o G.E. d B.C. o pp o q.. sz do d od w s.. Ass dsp d sp os d od s w. 5. C sss d ss od o op. d B B 6. C q od o d ss od o op. d 7. App dsp od odos d so sss qo. 8. C s sss d o o. Ep w Es d o ods - p o M L L L L L = = ] [ L d d d d B - E ss M ] [ ] [ d
32 p. o C d Eo E. Ass L. o. H L p://s.s.. - Cop M C L C L C L - Go ss M C C C C C C C C
33 p. o C d Eo E. Ass L. o. H L p://s.s.. - o d C C C C - s Eqo - Cs : d d sp- o ss
34 p. o C d Eo E. Ass L. o. H L p://s.s.. - Cs : szo Eqo d d w s d -od d opo. sp- o ss
35 p. o C d Eo E. Ass L. o. H L p://s.s.. Opo C - ppos s ppod d -od po d d Eqos o o ods Eqos o Bod ods w sp BCs I s dsp BC s spd od ods d qos d o s p o o o ods. BC ppd o d qo o od d o od ods. - Ep Cs Eqos o Bod ods w o BCs I s o BC s spd od ods so sp o od odo s s os od s dd.
36 p. o C d Eo E. Ass L. o. H L p://s.s.. - d qo od + : - Appoo o o BC d opo. d d - so o o BC o qos o od od. - oo o os od s ssd wo oss o. qo o od od os - Ep Cs Howo. s o qs o pos -od OE osdd owo popos s poo sp os o d d sss o.. M qo o s o od od d. opos M qos o od odos o pos s pop os ods. sz w 5 ods d wo od ods. o o so ss qos.
37 p. o C d Eo E. Cp Mdso os Es Ass L. o. H L p://s.s..
38 p. o C d Eo E... o o E Go Eqos d Bod Codos Eq Eqo : Cos Lw : : -sp osp : sp Bod odo : o o Bod Codo : o C s o o Bod : o E o pp. - 6 o o o Es oso d d d o o E d d d Ass L. o. H L p://s.s..
39 p. o C d Eo E. Ass L. o. H L p://s.s.... Eo Mzo Eo Eso : d d Ls q Eo L d d d d d d d d d d d E o o o E L C d d d d d d d d d d d d d d d d d d d d d d d d
40 p. o C d Eo E. Ass L. o. H L p://s.s.. Mzo os L M M M w... M : -z Mod o p o M o E d s z w } o { d d d d d - : soo. - : A ppo soo. : o p o p o Wo d d d d d d d d d d d d d d d Aso M op o o o E d d d d d d d d d d d d d d d d d d d
41 p. o C d Eo E. Ass L. o. H L p://s.s.. d d d d d d d d d d d d d d d d d d d d E E E E E E s pos d d d d o. o??. = q s ods o o E.. p o Wo I oow q s d o pp o wo ods. dss o d d d d d d d d d d d d d d d
42 p. o C d Eo E. Ass L. o. H L p://s.s.. I pp o wo ods pp o po ods s od qo o o po ss d. ppo so o pp o wo s d d d dss o Eq o E - E o d d - Appo o d d. d d.. Eq o Wd sd Mod - szo : d d d d o poss o.. qss o soo I wo soos ss pp o wo d d d d d d dss o dss o B s wo qos dss o d
43 p. o C d Eo E. Cp 5 szo 5.. -z p szo 5.. E szo 5.. E o L s s Ass L. o. H L p://s.s..
44 p. o C d Eo E. Ass L. o. H L p://s.s z p szo Appoo p p p p p p p p p o M o E d d d p p p p q q p p M o o d d d d d d d d d d d d d d p p p p p p p p q q q q p p q q p p d o p o Wo d d d p p p q q p p p p q q p p p p p p q q p p d d d d d d d d d o d o p p q pq
45 p. o C d Eo E. Ass L. o. H L p://s.s.. M o Wo Epsso d d d d d d d d sp
46 p. o C d Eo E. Ass L. o. H L p://s.s.. - M o B ss-s dsp o B E s Eqo B B d d d d d d d d d B B o dss o d d d
47 p. o C d Eo E. Ass L. o. H L p://s.s.. Ep sp d B soo dsp d o s s ssd s E B ss M E dzdd E dzdd E
48 p. o C d Eo E. Ass L. o. H L p://s.s.. Lod d s Eqo E EI EI E 5 sp d ss 6 5 EI EI I I M I M I 6 5
49 p. o C d Eo E. Ho Wo 5. dsp d o p s pssd s oows. w w z z z z w w s opo s dsp d. Ass d p s d p sss odo d sss opos. pssos o o po d wo s p s s o s od o pp s. : po o do o ss z o qo d poss od odos o o ds.. Az s sow oow d p sss odo -z od q Ass L. o. H L p://s.s..
50 p. o C d Eo E. Ass L. o. H L p://s.s E szo o szo Τ Τ d d d d d d dsp d w X.
51 p. o C d Eo E. Ass L. o. H L p://s.s.. s d B sss d E B
52 p. o C d Eo E. Ass L. o. H L p://s.s.. ss Eqo d d d B B Cop Codos dss o E od. Go qos do od odos o od.. w o o G.E. d B.C. o pp o q.. sz do d od w s.. Ass dsp d sp os d od s w. 5. C sss d ss od o op. d B B 6. C q od o d ss od o op. d d 7. App dsp od odos d so sss qo. 8. C s sss d o o.
53 p. o C d Eo E. 5.. E o L s s o I - Ass od Lod o - C. o o A - C. o Bd wd poss Co o E sss d Lod o Loop o s Go ss M d Lod o Ass E..M d E.L.. o Go M Gss Eo Bd so oposo osposs C. & ss C o o o sp - Coo : # o ods # o s # o sppo # o os ppd ods - Go : od Coods E oo p M ops Ids - M ops - Bod Codo o od odo d os ppd ods sp od odo - Msos opos Ass L. o. H L p://s.s..
54 p. o C d Eo E. 5 o o W o p o M o E d d d o dss Eqo & Bod Codos d p o Wo d d o dss o E oo Mzo o Eo d o dss Wd sd Mod G Mod d d o dss Appo oo -z p szo p o M o E d d d o dss d p o Wo d d o dss p o M o E o dss d d d - o dss - E szo o. H L p://s.s.. d p o Wo d d o dss o dss Ass L.
55 p. o C d Eo E. 5 Cp6 wo-so Es os z 6.. ss As Ass L. o. H L p://s.s..
56 p. o C d Eo E. Ass L. o. H L p://s.s ss ss : : Modd ss-s o E E E E E Ipoo o sp -sp o B p o wo A A A A d da da d da da B B
57 p. o C d Eo E. Ass L. o. H L p://s.s : ss : ss-s o E Ipoo o sp -sp o B p o wo A A A A d da da d da da B B
58 p. o C d Eo E. Ass L. o. H L p://s.s As : z z zz z z ss : z ss-s o z zz z zz E Ipoo o sp -sp o z zz z z z z z z z z B p o wo A zz zz A z z A A da da d da da d B B
59 p. o C d Eo E. 57 Cp 7 os ps o Es 7.. Cos C E 7.. Isop oo 7.. B Isop E 7.. H Od E - L 7.5. H Od E - dp 7.6. Isop E Ass L. o. H L p://s.s..
60 p. o C d Eo E. Ass L. o. H L p://s.s Cos C E sp Ipoo 6 5 w + + Copos B ss M B B B B A da
61 p. o C d Eo E. 59 Howo 6 - C B w od ss - Ip o ow po o - s pos s C. W o d o po osd pd so o s dd o ps o s o o po o owos. sss ow o s wo od odos osos s oo qo d p o d s od s so po o o od odos w C s. o o od ss s o od odo o o o w C s d o o s w s 5 d s os o d s od s. s od odo o o o. sss o s - osso sss oo. s s pos d s o dsp d sss o s o o dsssos. Copso o o ss w o soos s s soos odso soos s so odd o o dssso. Ass E =. =.. Ass L. o. H L p://s.s..
62 p. o C d Eo E. Ass L. o. H L p://s.s Isop oo Ipoo o Go X
63 p. o C d Eo E. Ass L. o. H L p://s.s.. 6 Ipoo o sp E s o sp p os o J J X J > : pp = : Isop < : p d d J da A B B B B
64 p. o C d Eo E B Isop E p os p ood ss Ass L. o. H L p://s.s..
65 p. o C d Eo E H Od E - L p o o - Od o - od O so IJ I J Ass L. o. H L p://s.s..
66 p. o C d Eo E. 6 Q9 E o o ods : ++ o o s - od poos : o o ps s : s oos Cop oo s Ass L. o. H L p://s.s..
67 p. o C d Eo E H Od E - dp Q8 E 5 8 ˆ ˆ 5 ˆ 5 8 Ass L. o. H L p://s.s..
68 p. o C d Eo E Isop E o o ods o sds o o - od..: o o s - od poos : o o ods : - = s oos A Cood s os A A os A os A A A A A A p os - C E - L E 5 6 Ass L. o. H L p://s.s..
69 p. o C d Eo E. Ass L. o. H L p://s.s.. 67 Ipoo o Go ~ ~ ] ~ [ ~ ~ ~ ~ ~ ~ X X Ipoo o sp E [ ] [ ~ ] s o sp p os ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ J ~ ~ o J X J ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ Howo 7 sp os q sdp d p ood.
70 p. o C d Eo E. 68 Cp8 Io 8. Gss Qd 8.. dd Io 8.. pos Zo E od 8.. Io Ass L. o. H L p://s.s..
71 p. o C d Eo E. Ass L. o. H L p://s.s Gss Qd O so d W I o s poo s poss o os qd ds o. - s os: W = W d - s s od: O po s ood o. - s sod od: W W W W W W W W W W W W W W W W W W W 5775 = / s d od: wo po s o. - s o od: 5 5 W W W W W W W W W W d W W W W W W W W W 5 5
72 p. o C d Eo E. Ass L. o. H L p://s.s.. 7 W W W W W W W = /.. W W W W W W W 8 9 = Bs o s odo w d o dd o ows o -pos G.Q.. - W -- poos w -pos G.Q. o - od poos w odos o G.Q. -w s w d +-po G.Q.. - ss Eqo B B B B B B B B d Ad A J d W A J d Ad A J d W A J wo-dso Cs Es - Qd d d W d W W WW - ss qo A J W W d d J da B B B B B B p p A W d d J W W d d J da wo-dso Cs Es A J W d d J da B B B B B B
73 p. o C d Eo E dd Io Q s sss - o op poos: - o ps s: dd Io d o od o o o ps s sss E s sss od s pod o Gss po Gss po dd o Ass L. o. H L p://s.s..
74 p. o C d Eo E pos Zo E od Idpd sp Mods o B E d Bod oo zo od pos Zo E Mod Ho ss od Ass L. o. H L p://s.s..
75 p. o C d Eo E. 7 Zo E od o Q8 E Zo E Mods o Q9 E Zo E Mods Ass L. o. H L p://s.s..
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