Introduction to Finite Element Method

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p. o C d Eo E. Cp Appoo o os d o Cs Ass L. o. H L p://s.s..

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..

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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.

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..

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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..

p. o C d Eo E. Cp Ep Eqos O so Ass L. o. H L p://s.s..

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..

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..

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..

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.5.955 Eo =.5 % Ass L. o. H L p://s.s..

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 ***** = -.8979E+ Eos o -od poo o o =.9E+ % o =.66E+ %

p. o C d Eo E. ***** -od oo ***** = -.8979E+ =.E+ Eos o -od poo o o =.9E+ % o =.66E+ % ***** -od oo ***** =.98E+ = -.7657E+ =.5585E+ Eos o -od poo o o =.888E- % o =.9568E+ % ***** 5-od oo ***** =.98E+ = -.7657E+ =.5585E+ 5 =.787E- Eos o 5-od poo o o =.888E- % o =.9568E+ % ***** 6-od oo ***** = -.577E- = -.58E+ = -.5859E+ 5 =.69E+ 6 = -.6E+ Eos o 6-od poo o o =.E- % o =.968E- % ***** 7-od oo ***** = -.577E- = -.58E+ = -.5859E+ 5 =.69E+ 6 = -.6E+ 7 =.59577E-9 Eos o 7-od poo o o =.E- % o =.968E- % ***** 8-od oo ***** =.E- = -.5797E+ =.6566E- 5 =.6766E+ 6 =.7E+ 7 = -.87585E+ 8 =.8796E+ Ass L. o. H L p://s.s..

p. o C d Eo E. Eos o 8-od poo o o =.576E-5 % o =.5895E- % ***** 9-od oo ***** =.87E- = -.5797E+ =.65578E- 5 =.6767E+ 6 =.59E+ 7 = -.87587E+ 8 =.87955E+ 9 =.69885E-6 Eos o 9-od poo o o =.576E-5 % o =.5895E- % ***** -od oo ***** = -.69E-5 = -.567665E+ = -.958E- 5 =.557E+ 6 = -.586E- 7 = -.578E+ 8 = -.999E- 9 =.79E+ = -.5856E- Eos o -od poo o o =.8956E-7 % o =.55E-6 % ***** -od oo ***** = -.8E-5 = -.56769E+ = -.7975E- 5 =.555E+ 6 = -.56E- 7 = -.56569E+ 8 = -.95567E- 9 =.968E+ = -.7669E- =.6686E- Eos o -od poo o o =.69E-7 % o =.5E-6 % Ass L. o. H L p://s.s..

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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

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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

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 + -

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

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

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.5 9 8 9 8 8 8 5 sp- o ss

p. o C d Eo E. Ass L. o. H L p://s.s.. - Cs : 9 9 9.6. szo Eqo d d w s d -od d opo. sp- o ss

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.

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.

p. o C d Eo E. Cp Mdso os Es Ass L. o. H L p://s.s..

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..

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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..

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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 6 6 6 E B ss M 5 5 5 5 5 5 5 5 E dzdd E dzdd E

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p. o C d Eo E. 5 Cp6 wo-so Es os z 6.. ss 6.. 6.. As Ass L. o. H L p://s.s..

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p. o C d Eo E. Ass L. o. H L p://s.s.. 55 6.. : 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

p. o C d Eo E. Ass L. o. H L p://s.s.. 56 6.. 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

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..

p. o C d Eo E. Ass L. o. H L p://s.s.. 58 7.. Cos C E sp Ipoo 6 5 w + + Copos B ss M B B B B A da

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..

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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..

p. o C d Eo E. 65 7.5. H Od E - dp Q8 E 5 8 ˆ ˆ 5 ˆ 5 8 Ass L. o. H L p://s.s..

p. o C d Eo E. 66 7.6. 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..

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.

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..

p. o C d Eo E. Ass L. o. H L p://s.s.. 69 8.. 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 = / 69 8966. - 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

p. o C d Eo E. Ass L. o. H L p://s.s.. 7 W W W W W W W 5 5 9 7759 55555 55555 55555 = 6669 8 /.. W W W W W W W 8 9 =.88888 88888 88888 - 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

p. o C d Eo E. 7 8.. dd Io Q8 5 5 6 6 7 7 6 7 6 7 5 6 7 5 6 7 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..

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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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p. o C d Eo E. 75 Cp 9 Co C Isop E Ass L. o. H L p://s.s..

p. o C d Eo E. 76 Co odo s odo s ssd s o d dsp d dd q o od w oods d dsp d o od sp. Copss Codo H H H H A H w A A A H.O.. A A Ass L. o. H L p://s.s..

p. o C d Eo E. 77 Cp Msos ops. ss Eo oo d Lo Io.. Iop E - Q6 d QM 6.. Codso & s.. o.5. Coss s os.6. Coss Coos os Ass L. o. H L p://s.s..

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p. o C d Eo E. 85 A- C L I p dsp O o p dsp Ass L. o. H L p://s.s..

p. o C d Eo E. Ass L. o. H L p://s.s.. 86 ss w o- od - G od - od - As od A A A / / / /

p. o C d Eo E. 87 C Ass L. o. H L p://s.s..

p. o C d Eo E. Ass L. o. H L p://s.s.. 88 - ss o s - Cop I I - Eq I I I I I I - Eqo I I

p. o C d Eo E. Ass L. o. H L p://s.s.. 89.5. Coss s os Mzo o M s o A s od Op odo s ooos sppo odos M o-ooos sppo odos sppo M M o G o-ooos L Coss do A = o C C C C I

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p. o C d Eo E. 95 os os p Io EI GA d d d d d GA dw dw d EI d d EI As 6 o. dw dw dw d o d d d wq EI d o p o po. s odo ssd s o sps o w d sd. o od psso s pssd s d d 6 dw dw wq d d d EI d d d d EI o p psso s wo s: dw. s odo qs s o s d sz sod os s s ss o. sod odo qs d o dsp d o s s oo d w s d o ss. o p o dsp d oo d pod sp os w d. sod odo os d o w s q o d d. o oo d sod os w. s odo ssd o w oo d s os o s o o q o C w ows o d od oo o d ds ss soo. I s o o z s o ps do o s sp os w ss ss soos. ood d odd s od sp os. I s sp os o s od o o ds pod s od o oo d sod ws s d o sp o oo ds s ws ssp o dsp d. ow oo d dsp d o dpd ds s sow so soo sp o oo d s o dd dpd d s ood o sp os ds s-op o d o ss. B ws w o s C -oo o ss o s o ps. Ass L. o. H L p://s.s..

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p. o C d Eo E. 97 Cp Md oo Ass L. o. H L p://s.s..

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