Introduction to Finite Element Method
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- Clyde Little
- 6 years ago
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1 pt. o C d Eot E. Itodto to t Et Mtod st 5 H L pt. o C d Eot E o to st tt Ass L. o. H L ttp://st.s..
2 pt. o C d Eot E. Cotts. Itodto. Appoto o tos & to Cs. t Eqtos O so. Mtdso os-estt 5. stzto 6. wo so Estt os 7. os ps o Ets 8. Itto 9. Co Ct Isopt Et. Msos ops. os wt H Cott qt - Bs. Md oto tt Ass L. o. H L ttp://st.s..
3 pt. o C d Eot E. Cpt Appoto o tos d to Cs tt Ass L. o. H L ttp://st.s..
4 pt. o C d Eot E. tt Ass L. o. H L ttp://st.s..
5 pt. o C d Eot E. dt Cosdtos Wt s t st soto o po? dss to s t s t t soto Wt s t t soto? soto tt stss t o qtos s w s od odtos. How do t t sotos st? O os o I s s o t s o t t sotos st o qto w t po s -posd po d t dts dt soto o t po Wt t s poss to dt t t soto o os sos? W d ppot soto. Wt s ppot soto? dt Qstos - Wt s t st ppoto? - How w t tt psts t st ppoto? A Good o ost o W otd Appoto od - Yd t st ppoto to t t soto o d o ppoto. - Co to t t soto s d o ppoto s pod. Wt s t to o t Bst Appoto? - M dd s t osst soto to t t soto. - Bt ow os s t osst? - Cos o ps t dst tw two spt pots. - W sod d so sot o to s t dst tw two ts to st tt Ass L. o. H L ttp://st.s..
6 pt. o C d Eot E. stzto - pstto o otos dsttd qtts wt so s. X X X υ w t ss tos o to st υ - t : Coto o so ots wt t s tsts - ss tos sod dpdt to ot. X d o - o ss o ss t Appotos X X X X υ υ w to otto: ptd ds dot sto. os o tos: A s o to st A to st υ s sd to od sp to υ t ssotd ot d t o o s w tt - d o - α α o α. - E od sp dd s t sp w t dst tw two ts t sp s sd t dd o. os tps o o dd o to sp. Ao t t oow os pott. - L o: L - L o: - H o: H L d d / d / tt Ass L. o. H L ttp://st.s..
7 pt. o C d Eot E. tt Ass L. o. H L ttp://st.s.. G Ids o t Bst Appoto Lt s d ot ppot to tt s osst to t to s o o dd t to sp. I ts s t s t tsts o ppoto tod dpd o tos o t o sd t ppoto. Lst q EoLE Mzto Eo: Mz Π L L d d d d d d d o Π st Eqto : I t ss tos otoo os do.
8 pt. o C d Eot E. to o to - to o to s poss t to o t d. to Cs - t to o s dd s. - : - - d d d d - d d d d - d d d d to o to tt Ass L. o. H L ttp://st.s..
9 pt. o C d Eot E. tt Ass L. o. H L ttp://st.s.. Mzto to Cs M Π d d d d d Π Π M d Π Π poss o E Eqto M d o Π Π Π d d d d d d d I s t ss tos s t t od t o Π d d d d d Π d d d d d I s t to ss t t ods t d d d d d d Π Π. o M Π Π
10 pt. o C d Eot E. Ep M Π d st to d d d d d d / d d / / / / /. B pp BC Ep M Π d st to d d d d d d Howo π. Appot os to os poos sd t zto o t st sq os. s poos p to t t od. Yo s tto ot s s pso s t tpzod o t t. Yo so d so to so stos qtos t Lp. o t o o to ps s do pso o po. Yo sod pst pop dsssos o o sts tot wt so ps tt sow o ppot tos d t to.. E qto o t oow zto to d pop od odtos. M Π d. t o qto d t od odtos o t oow po. d w M Π w wq d d tt Ass L. o. H L ttp://st.s..
11 pt. o C d Eot E. Cpt Ept t Eqtos O so tt Ass L. o. H L ttp://st.s..
12 pt. o C d Eot E. tt Ass L. o. H L ttp://st.s..
13 pt. o C d Eot E... os wt ooos dspt od odtos o to d < < d Appoto stzto w sds to : ot t o o st o stto... d Eqto sd : E < < d to sd : < Eo Estto : < Π d d Ed d Lst q Eo Π d d d d d d d d d d d d d d d d d d d d d d d d d Π L d d d d d Lst q Eo E to ot ott E Π { d d d d d d d d d d d d d d d d d C Π d d d d d d d d d d d d} d tt Ass L. o. H L ttp://st.s..
14 pt. o C d Eot E. Mzto os M Π M Π L M Π w..t. υ M Π : -tz Mtod o p o M ott E - st Od ss Codto o Mzto o Π d d d d d d d d d d d d d d d d d o d d d d d d d Π : to pp o p o t Wo Π d d d d d d d d d d d d d d d d d d d d d d d oto p d - υ { d < } d - υ υ - υ υ opts o : zto pos d t t soto. : zto pos d ppot soto. d d d d - t : d d d d d d - ost tss : d d d d d d d d d d d d d d tt Ass L. o. H L ttp://st.s..
15 pt. o C d Eot E. Asot M opt o ot ott E Π Π 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 Π 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 qt s ods o o d d d d d d d C d Wtd sd Mtod d π φ Ed φ d o d - φ : G Mtod pt st o s-dot sst π φ 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 - Idt st to t -tz Mtod o to p - φ : to-g Mtod po sst o o-s dot sst Wtd sd Mtod s. to p π π o d d d d d d d π d d d d d d d d o poss tt Ass L. o. H L ttp://st.s..
16 pt. o C d Eot E. d Ep : d - Et soto : - st t : App BCs : d d d d. o 6 - od : App BCs : d d d d d d d 6 d 6 5 d st Eqto : 6 5 d Ep : π s π d - Et oto : s π - st t : App BCs : d d d π s π d π. o π π π Eo.5 % π tt Ass L. o. H L ttp://st.s..
17 pt. o C d Eot E. tt Ass L. o. H L ttp://st.s.. - od t : App BCs : d d d d d 5 d d π π π s d π π π 6 s d st Eqto : 6 5 π π π???? - I to Eo / s s π π d d t Eo / os os π π π π d d o t to t o pssos t dpot wt sts s pod. - w Otpt ***** t-od oo ***** E Eos o t-od poo to o.9e % t o.66e %
18 pt. o C d Eot E. ***** t-od oo ***** E.E Eos o t-od poo to o.9e % t o.66e % ***** t-od oo *****.98E E.5585E Eos o t-od poo to o.888e- % t o.9568e % ***** 5t-od oo *****.98E E.5585E 5.787E- Eos o 5t-od poo to o.888e- % t o.9568e % ***** 6t-od oo ***** -.577E- -.58E E 5.69E 6 -.6E Eos o 6t-od poo to o.e- % t o.968e- % ***** 7t-od oo ***** -.577E- -.58E E 5.69E 6 -.6E E-9 Eos o 7t-od poo to o.e- % t o.968e- % ***** 8t-od oo *****.E E.6566E E 6.7E E E tt Ass L. o. H L ttp://st.s..
19 pt. o C d Eot E. Eos o 8t-od poo to o.576e-5 % t o.5895e- % ***** 9t-od oo *****.87E E.65578E E 6.59E E E E-6 Eos o 9t-od poo to o.576e-5 % t o.5895e- % ***** t-od oo ***** -.69E E -.958E E E E E- 9.79E E- Eos o t-od poo to o.8956e-7 % t o.55e-6 % ***** t-od oo ***** -.8E E E E E E E E E-.6686E- Eos o t-od poo to o.69e-7 % t o.5e-6 % tt Ass L. o. H L ttp://st.s..
20 pt. o C d Eot E. - st pots - Eo % - -6 to t Od o poo to o os.8.6 X.. Et d od t od X-Coodt ot o ppot to o dt ods o poo tt Ass L. o. H L ttp://st.s..
21 pt. o C d Eot E. 'X - - Et d od t od X-Coodt ot o dt ppot to o dt ods o poo Howo. Π o t oow OE o st to o post o opsso s w s ts od q. Ass ooos dspt BCs. d w d w EI q d d. O wt odto dos t pp o pott od o o.? sss t ps d t tt s o t odto.. Appot sotos o d-d d wt o od poos wt o ow d two ows. o o s ppd. Cop o sotos d dspt otto ot d s o to t soto d dsss. tt Ass L. o. H L ttp://st.s..
22 pt. o C d Eot E... os wt to Bod Codtos o to - t Eqto d < < d - Bod Codtos d d t o d t o d d Eo Mzto: Eo Estto Π d d d d d d d d d d d d d d d d d d d d Π L d d d d d d d d d d d d d E to ot ott E Π C Π d d d d d d d d d d d d { d d d d d 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} Mzto os M Π M Π L M Π w..t. υ tt Ass L. o. H L ttp://st.s..
23 pt. o C d Eot E. M Π : -tz Mtod o p o M ott E - st Od ss Codto o Mzto o Π d d d d d d 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 Π : to p o p o t Wo Π d d d d d d d d d d d d d d d d d d Asot M opt o ot ott E t Et oto Π Π 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 Π 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 qt 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 tt Ass L. o. H L ttp://st.s..
24 pt. o C d Eot E. Wtd sd Mtod π d d Ed d d d d d d d d d 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 Wtd sd s. to p π o d d d d d d d d d d d d d d d π d d d o poss Π tt Ass L. o. H L ttp://st.s..
25 pt. o C d Eot E... Itt Codto - t ott qt - Itto o tos wt dsott d?? - - O to to wt tsto zo d d w o o o - C w tt d o wt odto? d d d d d d d d d d d d st t ss s s d t d t t os d d d tt Ass L. o. H L ttp://st.s..
26 pt. o C d Eot E. - C w tt d o wt odto? d d d d d d 6 d d d d d d 6 st t ss s s d t d t t os d d d - C w tt d?? d d d d d d d d d d d d d d d 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 t dt t s t d o s otos. o t ps pot o w t otod ott odto psts t optt odto w stts tt t dspt d ot sod q dtd dd s d tos. tt Ass L. o. H L ttp://st.s..
27 pt. o C d Eot E. d - C w tt d d o wt odto? 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 t dt t s q & t s dsotos. Howo. Π L Π. o t sot popt o Π o t oow OE wt t ooos dspt BCs. d w q d Π dd o. t t soto.. Appot sotos o t st to o od q o t sp d ott od ppd t t d poos wt o ow two ows d t ows. Cop o sotos d dspt otto ot d s o to t soto d dsss.. Idt t tt odto o t oow ts d t t ts o t dtd odto. d w dw d w d d d d d d tt Ass L. o. H L ttp://st.s..
28 pt. o C d Eot E... Ot d o t p o t Wo s wpot I do od s q d Q-o sst d s q w t s std s t doto t t t wo do t Q os t o t od s q to t t t wo o doto do t t Q-stsss. Qd d w Mtt wpot - Cotos o I A sod od o sst t t oow sttt sod od. H υ d t od o A sod t s s. A d - Ep : B po υ o qto o po : psso o t wo : d w d w d w w EI q d EI d d d d υ: A pop to sp d w EI q d wqd I w s t dspt dd t t od ppd od t t psso o t pp o t wo os s oows. µ M M EI Q d wqd w d w w M µ s t ot dd t t od ppd t d s d dt to ppd t. tt Ass L. o. H L ttp://st.s..
29 pt. o C d Eot E. tt Ass L. o. H L ttp://st.s.. Mtt wpot - st o υ A A υ: A pop to sp - Ep : ss po Eq Eqtos t ots Y X H o t Wo Epsso Y X H s - os θ θ Y X θ θ Y X s os Y X Y X θ s θ os θ θ
30 pt. o C d Eot E. os θ µ os θ µ EA s θ s θ µ EA X Y X Y µ µ I µ o sst ossts o t od ppd t -t ot t dto t µ X µ Yµ X osθ osθ EA.5. t Et stzto o stzto... od Π d d d d d d d d d d d d d d d d d d o dss Itpoto o spt d Et L L L tt Ass L. o. H L ttp://st.s..
31 pt. o C d Eot E. tt Ass L. o. H L ttp://st.s.. L L L L L L L L L stzd o o to ttt 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 Π B B L d d d d d d d d d d d d d d Coptt Codto Cott qt L L C C L Go st Eqto Go tss Eqto. C C C C C C C C Π dss o
32 pt. o C d Eot E. tt Ass L. o. H L ttp://st.s.. -tz p Itptto 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 -
33 pt. o C d Eot E. tt Ass L. o. H L ttp://st.s.. t Et od. Go qtos t do od odtos o t od.. w o o t G.E. d B.C. t to pp o qt.. stz t do d od wt t ts.. Ass t dspt d sp tos d od s wt t. 5. Ct t t stss t d ss t od to t optt. d B B 6. Ct t qt od o d ss t od to t optt. d 7. App t dspt od odtos d so t stss qto. 8. Ct st stss d to o. Ep wt Ets d o ods - p to Mt L L L L L ] [ L d d d d B - Et tss Mt ] [ ] [ d
34 pt. o C d Eot E. tt Ass L. o. H L ttp://st.s.. - Coptt Mt C L C L C L - Go tss Mt C C C C C C C C
35 pt. o C d Eot E. tt Ass L. o. H L ttp://st.s.. - o d C C C C - st Eqto - Cs : d d sp- t o tss
36 pt. o C d Eot E. tt Ass L. o. H L ttp://st.s.. - Cs : t stzto t Eqto d d w s d -od t d opto. sp- t o tss
37 pt. o C d Eot E. tt Ass L. o. H L ttp://st.s.. t Opto Ct - ppos s ppotd d -od po d d t Eqtos o to ods t Eqtos o Bod ods wt spt BCs I s tt dspt BC s spd t od ods t t d qtos d to st p o o to ods. BC ppd to t t d qto o t od dt to t od ods. - Ep Cs t Eqtos o Bod ods wt to BCs I s tt tto BC s spd t od ods sp ttt o od odto s s ost od s todd.
38 pt. o C d Eot E. tt Ass L. o. H L ttp://st.s.. - t d qto t t od : - Appoto o t tto BC t t d opto. d d - sttto o t tto BC to qtos o t od od. t oto o t ost od s t ssd wtot oss o t. qto o t od od os - Ep Cs Howo. s t ott qts o pos t -od OE osdd owo popos st tpoto sp tos o t d d t t stss t o t t.. M qto o t st to ottd od t t d. opos M qtos o od odtos o pos s pop ost ods. stz t wt 5 ods d two od ods. o ot so t sst qtos.
39 pt. o C d Eot E. Cpt Mtdso os Estt os tt Ass L. o. H L ttp://st.s..
40 pt. o C d Eot E... o to t E ν Go Eqtos d Bod Codtos Eq Eqto : Costtt Lw : : t-spt tosp : spt Bod odto : o to Bod Codto : o t C s to o t Bod : o t E to pp. - 6 o o o Estt oso Π t d d d ot ott E Π d d d tt Ass L. o. H L ttp://st.s..
41 pt. o C d Eot E. tt Ass L. o. H L ttp://st.s.... Eo Mzto Eo Estto : Π d d Lst q Eo L d d d d d d d d d d d Π Π E to ot ott 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 t Π Π
42 pt. o C d Eot E. tt Ass L. o. H L ttp://st.s.. Mzto os L Π Π Π M M M w..t. υ Π M : -tz Mtod o p o M ott E d υ s tt z Π w } o { < d d d d d υ - υ υ : t soto. - υ υ : A ppot soto. Π : to p o p o t Wo Π d d d d d d d d d d d d d d d t t t t t Asot M opt o t ot ott E d d d d d d d d d d d d d d d d d d d t t t t t Π
43 pt. o C d Eot E. tt Ass L. o. H L ttp://st.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 t t t t Π Π Π Π Π Π Π Γ s post dt > / d d d o. o??. qt s ods o o Π Π E.. p o t Wo I t oow qt s d o α t pp o t wo ods. υ Π α Π dss o d d d d d d d d d d d d d d d t t t t υ α α α α α α α α α α α Π α
44 pt. o C d Eot E. tt Ass L. o. H L ttp://st.s.. I t pp o t wo ods t t pp o pott ods s t od qto o t tot pott ss dt. ppot so o t pp o t wo s d d d t dss o Eq to t E - Et o υ d d t - Appot o d d t υ. υ d d t.. Eq to t Wtd sd Mtod - stzto : d d d d t t o poss o.. qss o soto I two sotos sts t pp o t wo t d d d d d d t t dss o dss o B stt two qtos dss o d
45 pt. o C d Eot E. Cpt 5 stzto tt Ass L. o. H L ttp://st.s..
46 pt. o C d Eot E tz p stzto 5.. t Et stzto 5.. t Et o L stt s tt Ass L. o. H L ttp://st.s..
47 pt. o C d Eot E. tt Ass L. o. H L ttp://st.s tz p stzto Appoto p p p p p p p p p o M ott E d d d t p p p p q q p p Π M o Π 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 t t t t d o Π p o t Wo Τ 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 t t t Π o d o p p q pq
48 pt. o C d Eot E. tt Ass L. o. H L ttp://st.s.. Mt o t Wo Epsso d d d d d γ γ γ d d d t spt t t γ γ γ
49 pt. o C d Eot E. tt Ass L. o. H L ttp://st.s.. t t - Mt o B γ γ γ tss-st dspt to B γ γ γ ν E st Eqto B B d d d d d d d d d t t t B B o dss o d d d t
50 pt. o C d Eot E. tt Ass L. o. H L ttp://st.s.. Ep spt d B t t soto t dspt d o t stt s ssd s 6 ν ν ν ν 6 E B tss Mt ν ν ν ν ν ν ν ν ν ν ν ν ν ν E dzdd E dzdd E
51 pt. o C d Eot E. tt Ass L. o. H L ttp://st.s.. Lod dd st Eqto ν ν ν ν ν E EI EI E 5 ν ν ν spt d tss 6 5 EI EI ν ν I I M I M I 6 5 τ ν
52 pt. o C d Eot E. Ho Wo 5. dspt d o t pt s pssd s oows. w w z z z z w w t st opot s t dspt d. Ass d t pt s d p stss odto d stss opots. pssos o t tot pott d t t wo s t pt s st to ts od o t pp s. t: po t tto t dto o t tss z t t o qto d poss od odtos o t o ds.. Az t stt sow t oow d t p stss odto -tz tod q tt Ass L. o. H L ttp://st.s..
53 pt. o C d Eot E. tt Ass L. o. H L ttp://st.s t Et stzto o stzto Γ Γ Γ Γ Γ Γ Τ Τ d d d d d d t t dspt d t w X.
54 pt. o C d Eot E. tt Ass L. o. H L ttp://st.s.. t st d t B γ γ γ stss d t E B γ γ γ ν ν ν ν ν ν ν ν ν ν ν ν ν ν ν ν ν ν ν ν ν
55 pt. o C d Eot E. tss Eqto B Bd Coptt Codtos d t d o dss t Et od. Go qtos t do od odtos o t od.. w o o t G.E. d B.C. t to pp o qt.. stz t do d od wt t ts.. Ass t dspt d sp tos d od s wt t. 5. Ct t t stss t d ss t od to t optt. B Bd 6. Ct t qt od o d ss t od to t optt. d d 7. App t dspt od odtos d so t stss qto. 8. Ct st stss d to o. t tt Ass L. o. H L ttp://st.s..
56 pt. o C d Eot E. 5.. t Et o L stt s o tt I - Ass od Lod to - C. o stto A - C. o Bd wdt poss Cto o Et stss t d Lod to Loop o ts Go tss Mt d Lod to Ass E..M d E.L.. o Go M Gss Eto Bd so oposto t ostposs C. t & tss C o to o sp t tt - Coto t : # o ods # o ts # o sppot # o os ppd t ods... - Got t : od Coodts Et oto p Mt opts Ids - Mt opts - Bod Codto to od odto d os ppd t ods spt od odto - Msos optos tt Ass L. o. H L ttp://st.s..
57 pt. o C d Eot E. 55 to o W o Π p o M ott E d d t d o dss t Eqto & Bod Codtos d p o t Wo d t d o dss Π o Et oto Mzto o Eo d o dss Wtd sd Mtod G Mtod d d o dss t Appot oto -tz tp stzto Π p o M ott E d d t d o dss d p o t Wo d t d o dss p o M ott E Π o dss d d d - t o dss - t Et stzto o. H L ttp://st.s.. d p o t Wo d t d o dss o dss tt Ass L.
58 pt. o C d Eot E. 56 Cpt6 wo-so Estt os z 6.. tss 6.. t 6.. Ast tt Ass L. o. H L ttp://st.s..
59 pt. o C d Eot E. 57 tt Ass L. o. H L ttp://st.s..
60 pt. o C d Eot E. tt Ass L. o. H L ttp://st.s tss tss : t : ν γ γ Modd tss-st to γ ν ν ν ν ν ν ν ν E E E E E γ ν ν ν ν Itpoto o spt t-spt to B γ p o t wo γ A A A A t td tda tda td tda tda B B
61 pt. o C d Eot E. tt Ass L. o. H L ttp://st.s t t : γ γ tss : ν tss-st to E γ ν ν ν ν ν Itpoto o spt t-spt to B γ p o t wo γ A A A A t td tda tda td tda tda B B
62 pt. o C d Eot E. tt Ass L. o. H L ttp://st.s Ast t : γ γ γ θ θ θθ z z zz z z tss : θ θ z tss-st to z zz z zz E γ ν ν ν ν ν ν ν ν ν θθ θθ Itpoto o spt t-spt to z zz z z z z z z z z B γ θθ p o t wo γ π π π π π π θθ θθ θ θ A zz zz A z z A A da da d da da d t B B
63 pt. o C d Eot E. 6 Cpt 7 os ps o Ets 7.. Costt t C Et 7.. Isopt oto 7.. B Isopt Et 7.. H Od t Et - L tt Ass L. o. H L ttp://st.s..
64 pt. o C d Eot E H Od t Et - dpt 7.6. Isopt Et tt Ass L. o. H L ttp://st.s..
65 pt. o C d Eot E. tt Ass L. o. H L ttp://st.s Costt t C Et spt Itpoto 6 5 α α α α α α α α α α α α α α α α α α α α α w t Copots B γ tss Mt B B B B A tda t
66 pt. o C d Eot E. 6 Howo 6 - Ct B wt t od ss - Ipt o ow t t po o - stt pos s C t. W o d o po osd pdt so tt o s dd ot tps o ts to o po o t owos. sss ow to st two od odtos t osos s oo qto d p o d s od s so t po o ot od odtos wt C ts. o ot od ss s o od odto o o o wt C ts d o t o tst wt t st 5 dt s ots o t d s od s. s t od odto o o o. sss o ts t-t t osso t stss otto t. st st pots d ts o dspt d stss to st o o dsssos. Copso o o sts wt ot sotos s s t sotos odso sotos s sto odd o o dssso. Ass E. ν.. tt Ass L. o. H L ttp://st.s..
67 pt. o C d Eot E. tt Ass L. o. H L ttp://st.s Isopt oto Itpoto o Got η η η η η η η η X
68 pt. o C d Eot E. tt Ass L. o. H L ttp://st.s.. 66 Itpoto o spt t Et η ts o t spt p tos η η η η η η η η η η η o J J X J η η η η η η η > : ppt t : Isopt t < : pt t η η η d d J t tda A B B B B
69 pt. o C d Eot E B Isopt Et η η η.5 η.5 η.5..5 p tos t pt oodt sst. η η α α αη αη η η α5 α6 α7η α8η η η η η η α η η η α α α α α α α α α α η α η α η η η α α α η α η η α α α α α α α α α α α α α α α α α α α η η η η tt Ass L. o. H L ttp://st.s..
70 pt. o C d Eot E H Od t Et - L p to o -t Od o -t od O so η η η η η η η η η η η η η η η η η η η η η η IJ I J tt Ass L. o. H L ttp://st.s..
71 pt. o C d Eot E. 69 Q9 Et ot o ods t : ot o ts -t od poos : ot o t pst ts : s oos η Copt oo η η η η η η η η η η η st η tt Ass L. o. H L ttp://st.s..
72 pt. o C d Eot E H Od t Et - dpt Q8 Et 5 η 8 η ˆ η ˆ 5 ˆ 5 8 tt Ass L. o. H L ttp://st.s..
73 pt. o C d Eot E Isopt Et ot o ods o sds o t t o -t od..: ot o ts -t od poos : ot o t ods : - s oos η η η η η η η η η A Coodt st α ostt A A α ostt A α ostt A A A α α α α α α A A A p tos - C Et α α α - L Et α α α α 5 α α α α 6 α α α α tt Ass L. o. H L ttp://st.s..
74 pt. o C d Eot E. tt Ass L. o. H L ttp://st.s.. 7 Itpoto o Got α α α α α ~ α α α α α ~ ] ~ [ ~ ~ ~ ~ ~ ~ X X Itpoto o spt t Et [ ] [ ~ ] α α α α α ts o t spt p tos ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ α α α α α α α α α α α α α α α α α α α α J ~ ~ o α J X J α α α α α α α α α α α α α α α ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ Howo 7 sp tos q sdpt t d t t t pt oodt.
75 pt. o C d Eot E. 7 Cpt8 Itto 8. Gss Qdt 8.. dd Itto 8.. pos Zo E od 8.. t Itto tt Ass L. o. H L ttp://st.s..
76 pt. o C d Eot E. 7 tt Ass L. o. H L ttp://st.s..
77 pt. o C d Eot E. tt Ass L. o. H L ttp://st.s Gss Qdt O so d W I t to s poo t s poss to ostt t qdt tt ds t t tto. - s ostt: W W d - s st od: O pot s ood o. - s sod od: W W W W W W d W W W W W W W W W W W W W 5775 / s td od: wo pot s o. - s ot 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
78 pt. o C d Eot E. 76 W W W W W 5 W / W W W W W W W W Bs o t st odto w d to dd o ows o -pots G.Q.. - W tt --t poos t wt -pots G.Q. o -t od poos w odtos o G.Q. -w s w dt -pot G.Q.. - tss Eqto B Bd B BAd B B A J d W B B A J 5 9 d Ad A J d W A J wo-dso Cs t Ets - Qdt η ddη W η dη W W η WW η - tss qto A B BtdA B η B η t J ddη W W B η B η t J A tda td η t J ddη η p t d W W W η p η t t J wo-dso Cs Ets A B BtdA α B α α B α α t J dα dα W B α α B α α t J tt Ass L. o. H L ttp://st.s..
79 pt. o C d Eot E dd Itto Q8 t η η η η 5 5 η η 6 6 η η η η 7 7 η η η η 6η 7η 6 7 5η 6 7η 5η 6 7η η η s stss t - o opt poos: η η η - o pst ts: η η η η η η η dd Itto d t tto od o to t t t o pst ts t stss t Et s stss od s tpotd o Gss pot Gss pot t dd tto tt Ass L. o. H L ttp://st.s..
80 pt. o C d Eot E pos Zo E od Idpdt spt Mods o B Et d Bod oto zo od pos Zo E Mod Ho ss od tt Ass L. o. H L ttp://st.s..
81 pt. o C d Eot E. 79 Zo E od o Q8 Et Zo E Mods o Q9 Et Zo E Mods tt Ass L. o. H L ttp://st.s..
82 pt. o C d Eot E. tt Ass L. o. H L ttp://st.s t Itto E E E ν ν ν ν ν ν ν ν ν ν ν ν ν ν ν ν ν ν ν ν ν ν ν ν ν ν ν ν ν ν ν ν ν ν ν ν ν tto dd tto d d d d B B B B B B B B Howo 8 o t t po owo 6 o t d s od s. s L ts d Q8 ts. o Q8 t t t d t dd tto s. Cop o sts d t dspt d d t stss d wt tos C ts d t soto.
83 pt. o C d Eot E. 8 Cpt 9 Co Ct t Isopt Et tt Ass L. o. H L ttp://st.s..
84 pt. o C d Eot E. 8 tt Cott odto s odto s stsd s t ot d t dspt d dd q o t tt od wt t oodts d dspt d o t t od spt. Coptss Codto φ φ φ φ φ η H η φ η H η η H η H η η A H η w A φ φ φ φ A A φ φ φ φ H.O.. A A tt Ass L. o. H L ttp://st.s..
85 pt. o C d Eot E. 8 Cpt Msos ops. tss Eto oot d Lo Itto.. Iopt Et - Q6 d QM 6.. tt Codsto & stt.. t o tt tt Ass L. o. H L ttp://st.s..
86 pt. o C d Eot E Costts st os.6. Costts Cotos os tt Ass L. o. H L ttp://st.s..
87 pt. o C d Eot E tss Eto oot d Lo Itto tss to - tss opots sod td t t G s t ot t ods. - stss d s ot q dtd o t-t ods. tss soot t ods - Cotos stss d otd tpot stsss t t G s to ods d t. - sp to o t Q9 sp to tzd o tpoto o stss to ods dpd o t tto ss. - Md-sd ods ttd s t dpdt ods o dpdt ods o t stss d Lo tto B Bd B Bd B Bd B d d Γt dγ B ~ d B ~ d B ~ d B Bd w ~ dots t tpotd d d stss d. tt Ass L. o. H L ttp://st.s..
88 pt. o C d Eot E Iopt Et - Q6 od sp o B Et o p d Cot dod sp p d Bos o B Et o Bd - spt d η η η η η - t & tss d η γ E νe E ν ν ν -t E - Itto Π γ dd E E dd ν ν E ν ν -t E - t Itto Π s E ν E ν E ν dd tt Ass L. o. H L ttp://st.s..
89 pt. o C d Eot E. 87 Bos o Bd - t & tss d γ E E - spt d E E γ E E C C C C C E C C E C C C At E C C E -t E Et oto Π E C C C dd E dd γ E to o st E Π Π s Π Π ν ν ν. o. t o pst s os dsstos s t spt to o t s. Q6 Iopt Et - p to B η B η d s t odss ds o do. tt Ass L. o. H L ttp://st.s..
90 pt. o C d Eot E Et ts Eqto - tt Codsto [ [ ] [ ] [ ] ][ ] [ [ ] ] [ [ ] ] [ Q6 ] Q6.. tt Codsto & stt tt Codsto: Et so Os po to ss. - o Gss to pot o w tt Ass L. o. H L ttp://st.s..
91 pt. o C d Eot E. tt Ass L. o. H L ttp://st.s.. 89 stt - stt - stt - Ass o
92 pt. o C d Eot E. tt Ass L. o. H L ttp://st.s.. 9 t stt - stt - stt - Ass
93 pt. o C d Eot E. 9.. t o tt t C L I p dspt Ot o p dspt tt Ass L. o. H L ttp://st.s..
94 pt. o C d Eot E. 9 At-t C L I p dspt Ot o p dspt tt Ass L. o. H L ttp://st.s..
95 pt. o C d Eot E. tt Ass L. o. H L ttp://st.s.. 9 t stts wt o-t od - G od - t od - Atst od A A A / / / /
96 pt. o C d Eot E. 9 C t tt Ass L. o. H L ttp://st.s..
97 pt. o C d Eot E. tt Ass L. o. H L ttp://st.s tt sst o st - Coptt Γ Γ I I - Eq Γ Γ I I Γ Γ I I I I - Eqto Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ I I
98 pt. o C d Eot E. tt Ass L. o. H L ttp://st.s Costts st os Mzto o Π M st to A st od Optt odto Π s ooos sppot odtos Π M o-ooos sppot odtos sppot ttt Π M M Π Π o Π Π Π Π Π Π G o-ooos L Costts do A o C C C C I
99 pt. o C d Eot E. tt Ass L. o. H L ttp://st.s.. 97 ] [ C C C C C C C C C C Π ] [ C C C C C C C C L Mtp M Π λ λ Π Π λ Π Π A A λ λ λ λ λ λ λ λ.6. Costts Cotos os L Mtp Π M st to A w Π s t o to dd o t zto pp o qt d A dots ddto st o ostts w dd so o o so s o t so pots. o t spt o dto o ostts spd o s s osdd ts ot. L A o d d d d d t Π Π M L L λ λ λ λ
100 pt. o C d Eot E. tt Ass L. o. H L ttp://st.s.. 98 B dstz Λ λ t d t dspt d s w q Λ Λ Λ L Λ Λ L Λ d d d d Π Ν Ν Ν Ν λ o t stto odto o odd to os Π Π q Λ q Λ Λ soto pod o ts pot s t t s s tt o dst pos. st d t ost pott qsto s wt d o tpoto to s to pod o t L tp. t to Π Π d M A A α } { C q L L L L L Π t d d d d d d d α α α Ν Ν q α α Π Π M
101 pt. o C d Eot E. 99 Cpt os wt H Cott qt Bs?? tt Ass L. o. H L ttp://st.s..
102 pt. o C d Eot E... C oto.. C oto.. oso B tt Ass L. o. H L ttp://st.s..
103 pt. o C d Eot E... C -oto Go Eqto EI d w d W o o t o qto w EI d w q d o dss w d wei d w d dw d EI d w d w EI d w d wqd d d d d w EI d w d d d wqd w M d w EI d w d wqd θ d d Cott qt sod dt o t dspt d s to pws-otos o t d t t oto. o w s to otos p to t st dts. d w EI d w d d d d w EI d w d d d q Ht p tos w w θ w w w θ w θ dw d w w θ dw d tt Ass L. o. H L ttp://st.s..
104 pt. o C d Eot E... C -oto E Cosdto w: tot dto θ: otto dw θ : s doto d dθ Π d EI d θ d d dw θ GA dw θ d d d w w θ θ wqd w ] [ θ w θ w θ dθ d d d d d θ [ ] d d d d B M dw d θ d d w θ [ ] d d d d d d B w [ ] Π M B M M EIB M d B GA B d [ ] qd Π tt Ass L. o. H L ttp://st.s..
105 pt. o C d Eot E. os tos p Ito Π EI dθ dθ d d d GA dw dw θ θ d EI d d GA 6κ λ o t. EI t dw dw As λ θ θ d d d o dw θ d wq EI d to p t tot pott t. s odto stsd s t t t to sps o w d θ sd. Π EI o t t t tod t psso s pssd s dθ dθ GA d d d EI dw θ d dw θ d d o p t psso t s t t two tts: wq d EI dw t θ. st odto qs s ot s t t d sz sod ost t s s t tss o. sod odto qs t dt o dspt d to t t s s t otto d w s dt to sts. o p ot t dspt d t otto d tpotd sp tos w θ d. sod odto os d o w s qt to d d. o t otto d sod ostt wt t. s odto stsd o w t otto d s ostt o t t s o t ott qt o θ C w ows o d od otto o t d ds ss soto. I s o t to z s o pts do ot s sp tos w sts ss sotos. otod dt odd s od sp tos. I s sp tos o t s od o ot ds pod t st od t o t otto d sod ws s d t to sp o t otto ds s ws ssp o t dspt d. ow t otto d t dspt d tot dpdt ds s sow t t sto t soto sp o t otto d s to dd dpdt d ts t otod to sp ostt ds s-opt o d o sts. B ws w o s C -oto o t ss o s o pts. tt Ass L. o. H L ttp://st.s..
106 pt. o C d Eot E... oso B Go Eqto EI d θ GA dw d θ d GA d w dθ p d d W o d θ dw θ EI GA θ d d d d w dθ w GA d d p d θ υ θ & w υ w d θ dw dθ dθ dθ dw θ EI GA θ d θei θ θ EI d GA d d d d d d d d w dθ w GA d d dw p d wga θ d dw dw GA θ d d d wpd dθ dθ EI d d d dw dw dθ θ GA θ d θei d d d θ υ θ & w υ B ss ooos od odtos w dw wga θ d wpd dθ dθ EI d d d dw dw θ GA θ d d d wpd θ υ θ & w υ w Bod Codtos θ θ o EI d o w o GA dw θ d d Eto o t dspt B o t tto o t st qto d ssttto o t sod qto to t st qto d θ EI p O M God!!! d ott w odd od dt qto w dos ot t tsts d ts s dt to so. At ts pot w to osd t to-g tod sos!!! tt Ass L. o. H L ttp://st.s..
107 pt. o C d Eot E. 5 Cpt Md oto tt Ass L. o. H L ttp://st.s..
108 pt. o C d Eot E. 6 Wt s t d oto??? - to tss o st ds ttd d tpotd dpdt!!! Go Eqtos d Bod Codtos Eq Eqto : Costtt Lw : : t-spt tosp : spt Bod odto : o to Bod Codto : o t C s to o t Bod : o W sttt. ˆ d ˆ d t ˆ d ˆ υ ˆ υ ˆ d ˆ d t ˆ d ˆ υ ˆ υ ˆ d ˆ d t ˆ d ˆ d ˆ υ ˆ υ ˆ d ˆ d t ˆ d ˆ d ˆ υ ˆ υ Itpotos E t t dstzto ˆ Eˆ Q E B de de d Γt Bd dγ o dss Q ME Q Q M E ˆ d Eˆ Q M Q tt Ass L. o. H L ttp://st.s..
109 pt. o C d Eot E. 7 Ipott Qsto Wt t dss to sps o t dspt d d t st d?? C w oos t tpoto sp tos o t dspt d t st dpdt?? ott t sw s o. I s w oos t to sps t t sotos o t d oto o st w s sd so d to sp to. Bz-Bzz odto BB odto stts t qd tosp tw t dd to sps. s ss s ot o sop o ts ss. s w o s t EM sd o t d oto!!! oss os o to sps. ˆ H ˆ H. ˆ H L ˆ ˆ. L ˆ L W o do o? C o pop pto o o o? tt Ass L. o. H L ttp://st.s..
Introduction to Finite Element Method
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.. 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.
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