2.8 Variational Approach in Finite Element Formulation [Bathe P ] Principle of Minimum Total Potential Energy

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1 .8 Varatona Approach n Fnt Emnt Formuaton [Bath P.-6] In unrstanng th phnomna occurrng n natur, w ar qut us to ffrnta quatons to scrb th phnomna mathmatca bas on basc phsca prncps, nam consrvaton aws n man mchanca ngnrng probms. Insta of ths ffrnta formuaton, phsca phnomna can b scrb n trms of mnmzaton of tota nrg (or functona assocat wth th probm, whch s ca varatona formuaton. Fnt mnt formuaton can b rv b ths varatona formuaton as ong as thr sts a varatona prncp corrsponng to th probm of ntrst..8. Prncp of Mnmum ota Potnta Enrg hr s a vr mportant phsca prncp to scrb a formaton procss of an astc bo, nam Prncp of Mnmum ota Potnta Enrg, whch can b summarz as bow: Π U +V : ota Potnta Enrg U V : Stran Enrg : Potnta Enrg u to trna oas (kpt constant Π s mnmum wth rspct to th stat varabs or functon varabs at th qubrum stat hs prncp can b as app to formaton of astc bos b ntfng th stran nrg an potnta nrg u to trna forcs whch ar assum to b f urng th formaton. h tota potnta nrg can b a functon of stat varabs or a functon of functons, whch s ca functona. For nstanc, Π Π u, u, L, u wth u bng th stat varabs Π ( n Π(, f (,, f ( ( n Π u,, L,n f L How to mnmz wth rspct to functons f (? subjct of cacuus of varaton (It ma b not that thr ar othr varatona prncps than th Prncp of Mnmum ota Potnta Enrg. In ths ntrouctor cours of FEM, our scusson w b mt to ths prncp on. 5 b. H. Kwon 38

2 Eamp : Sprng Probm [Bath P.86-87, E.3.6] h formaton of a sprng s takn as th smpst amp to monstrat that th varatona formuaton s th sam quaton as th consrvaton aw,.., forc qubrum quaton. h probm of a nar sprng sstm s pct n th foowng fgur. k P Equbrum concpt : k P Enrg concpt: Stran nrg : U k Potnta nrg : V P ota potnta nrg : Π k P Mnmz Π wth rspct to th varab : Π k P Eamp : ght Strng Probm hs s th frst amp of a functona. A tght strng wth a tnson app at th n was s unr a strbut oa w( as pct bow: w( ( Fn ( for a gvn w(. 5 b. H. Kwon 39

3 Equbrum concpt : + w( Enrg concpt: Stran nrg : U Potnta nrg : V w Π ( ota potnta nrg : ( w Fn ( to mnmz Π ( (, whch s a functon of a functon,.., Functona hr ar man ntrstng amps of functona. o nhanc motvaton, on mor amp w b ntrouc, nam Brachstochron Probm b Brnou 696: A B On wants to fn ( for th mnmum fang tm for f A an B ponts. t + s ( v : a functona to b mnmz. g B B ( 5 b. H. Kwon 4

4 .8. Cacuus of Varaton Cacuus of varaton as wth such probms to mnmz a functona. At ths pont, t w b pan n a concs mannr so that a bgnnr can start wth a varatona formuaton for FEM wth no ffcut. As th frst amp of cacuus of varaton, consr a functona of th foowng form: I ( ( F(, φ, φ, φ : functona φ φ wth φ, φ. Fn φ ( whch mnmzs I( φ (. φ( : act souton φ ~ φ φ + φ φ : sma varaton Consr an appromat souton φ ~ whch has a sma varaton φ ovr an act souton φ (. hat s, ~ φ ( φ( + φ( h appromat souton s substtut to th functona prsson ng I ( φ( ~ ~ ~ ~ F(, φ, φ, φ I( φ + I In orr for φ( to b th souton, I for an φ. 5 b. H. Kwon 4

5 I φ + φ φ + + φ ( φ +, ( φ ( s not to b ncu. Intgrat b parts s I φ + φ + On mor ntgra b parts gvs φ φ ( φ I + + φ + φ φ (.6 I for arbtrar φ mps that +. < < : Eur-Lagrang quaton an th foowng st of bounar contons an or φ at an or φ at an Natura B.C. Essnta B.C. 5 b. H. Kwon 4

6 Proprts of oprator h aws of varatons of sums, proucts, an so on ar compt anaogous to th corrsponng aws of ffrntaton. ( F + G F + G, ( FG ( F G + F( G, ( F n nf n F ntrchangabt btwn an : F ~ F F ntrchangabt btwn ~ ( F F an : F F F ~ φ φ + ( φ ~ φ φ φ + φ ( φ Eamp : ght strng probm Π Eur-Lagrang quaton: ( (.7 ( w B.C.: w w : confrm or at an 5 b. H. Kwon 43

7 Eamp : Bnng bam probm w( ( E: Young s mouus I : Momnt of Inrta h tota potnta nrg can b prss as Π ( (.8 ( EI w Eur-Lagrang quaton: w + Bounar contons: EI EI or at an an EI or at an Natura bounar conton Essnta bounar conton Rmn: M V w( V+V M+M M V, V EI M, w 3 EI 3 V, 4 EI 4 w 5 b. H. Kwon 44

8 Summar of Eur-Lagrang Equatons for Varous Forms of Functonas A Functons nvovng hghr orr rvatvs: I F(, φ, φ, φ, L, φ ( n n ntgraton b parts s th Eur-Lagrang quaton: n ( ( (.9 B Functons nvovng on npnnt varab but svra pnnt varabs an thr frst rvatvs: I F(, φ, φ, L, φn, φ, φ, L, φ n hs nvovs n ffrnt varabs φ n an s n Eur-Lagrang quatons:, L, n (. C Functons nvovng mor than on npnnt varabs an on pnnt varab an ts frst rvatvs: Ω I F(,, φ, φ, φ Usng Gauss horm (Grn s horm n pan ˆ ˆ (. whr ˆ F F F F, tc. * Gauss horm (Grn s horm n pan r r VΩ V ns ˆ Ω A Ω A φn s, A A φn s 5 b. H. Kwon 45

9 Eamp 3 : Hamton s Prncp an Lagrang s quaton n namcs Dnamc moton of a rg bo sstm can b scrb n svra was. Nwton s n aw of moton s on of thm. Othr mthos ar bas on nrg concpt. Nwton s quaton of moton s rprsnt b r F m & r Lagrang s quaton can b wrttn as t L L wth L V bng Lagrang functon q & q (. (or kntc potnta whr q : gnraz coornat : kntc nrg V : potnta nrg On can as rv Lagrang s quaton va cacuus of varaton from Hamton s prncp that s rprsnt b th foowng (cas B n th summar. t t ( V t (.3 or mnmz I ( V t. t t Whch mps that actua path foow b a namc procss s such as to mak th ntgra of (-V a mnmum. Hamton s prncp for formab bo can b stat as t t ( t Π t Lt (.4 t wth L Π, Π : tota potnta nrg ( Π U + V 5 b. H. Kwon 46

10 .8.3 Bounar Contons n Varatona Prncp It ma b not that a varatona prncp gvs rs to not on Eur-Lagrang quatons but aso bounar contons assocat wth th probm. In fact, th functona tsf ncu th ffct of bounar contons accorng. W w pa attnton to th mtho of ntroucng bounar conton ffct to th functona. Consr th tght strng probm as an amp for ths purpos. w( ( h functona for ths probm was Π (. (.5 ( w At ths tm, w w rv th Eur-Lagrang quaton an th assocat bounar contons b usng th oprator to th varaton of th functona wthout rfrrng to th formua rv bfor. ~ + Π( ~ Π( + Π Π w w ( w Eur-Lagrang quaton: B.C.: w + + w, < < or (.. s prscrb at an 5 b. H. Kwon 47

11 Equvanc + w, at < < an ( Mnmz Π( w or Π w for an Now, consr a cas whr on has to ntrouc a natura bounar conton at on n of th strng as pct n th foowng fgur. w( ( (prscrb ownwar oa If s prscrb at, th functona shou b mof as foows: Π( ( w (.6 It shou b not that th atona trm rprsnts th potnta nrg u to th trna oa. In ths cas, th varaton of functona bcoms sght ffrnt from th prvous cas to th sam Eur-Lagrang quaton wth 5 b. H. Kwon 48

12 ffrnt st of bounar conton as rv bow. Π whch mps that w + w + for an + w, at an < < at It ma b not that th varatona formuaton ncus th natura bounar contons n th functona form: Π w for an (.7 In othr wors, as ong as th tota potnta nrg s corrct foun ncung th ffct of natura bounar contons, on os not hav to worr about th ta procur of ntroucng th natura bounar conton n th FEM formuaton an mor. hs pont s th mrt of FEM ovr othr numrca anass tchnqus n ang wth th natura bounar contons whch nvov hghr orr rvatvs than ssnta bounar contons. hs aspct of mrts w b scuss atr agan. h tght strng probm scrb abov s anaogous to othr phsca probms rprsnt b on-mnsona scon orr ffrnta quaton, for nstanc, onmnsona sta hat conucton n partcuar. S th nt summar for ths anaog. 5 b. H. Kwon 49

13 Anaog btwn ght Strng probm an Hat Conucton ght Strng Hat Conucton θ + w, < < k + q, < < Π ( ( w Π( θ( k qθ θ whn at an whn θ θ at an Π ( ( w ( θ( θ Π k qθ k θ θ whn at at whn θ θ at θ θ k k at Π( ( w + ( θ ( θ θ Π k qθ + k θ whn at at whn θ θ at θ θ k k at Ra: [Bath E. 3.8, E. 3.9, E. 3.] for natura bounar contons. Quston: Can on fn a functona for a varatona prncp for an ffrnta quatons? h answr s No. It s not awas foun. (On ma tr to rvrs th rvaton of Eur-Lagrang quaton to obtan th corrsponng functona, vn though not awas succssfu. 5 b. H. Kwon 5

14 .8.4 Fnt Emnt Mtho vrsus Ragh-Rtz Mtho Fn ( φ to mnmz ( φ( Π. On wants to obtan an appromat souton to mnmz a functona Π ( φ(. On of th hstorca famous appromat mthos for ths kn of probm s Ragh-Rtz Mtho, an th othr morn mtho s th Fnt Emnt Mtho. Hr w w scuss both mthos wth th comparson n mn. Ragh-Rtz Mtho: ~ n φ( ψ ( : appromat souton satsfng th ssnta B.C. c hn, whr ψ ( : tra functons (fn ovr th who oman Π ~ φ Π( c, c, L, c to b mnmz w.r.t. c. ( n hrfor Π( ~ φ,, L, n c : n quatons for n unknown c s. hs mtho s vr smp an as to unrstan. Howvr, t s not as to fn a fam of tra functons for th ntr oman satsfng th ssnta bounar contons whn gomtr s compcat. h souton to ths troubsom pont can b foun n th Fnt Emnt Mtho. Fnt Emnt Mtho ~ φ ( N ( φ φ N ( : noa vaus : shap functons 5 b. H. Kwon 5

15 hn, Π ~ φ Π( φ, φ, L, φ to b mnmz w.r.t. φ. ( n Π,, L, n : n quatons for n unknown φ s. In ths cas, th shap functons can b foun mor as than th tra functons wthout havng to worr about satsfng th ssnta bounar contons, whch maks FEM much mor usfu than Ragh-Rtz Mtho. In ths rgar, th Fnt Emnt Mtho s a mornz appromaton mtho sutab for computr nvronmnt. Eamp: ght strng probm va two mthos w( ( Spca cas: w( w : constant Π Ragh-Rtz Mtho ( to b mnmz. ( w ~ π π A sn + A sn (Not that tra functons satsf ssnta B.C. Π Π( A, A ~ wa + A π 4 w ~ π + A π Π A Π A π -w + A π A π 4 w A, A 3 π 5 b. H. Kwon 5

16 ~ w π 4 π sn 3 Not: ~ w w.9 vs..5 Quston: Whn th natura bounar conton s app at th rght n, what shou b on n ths mtho? (Ra [Bath Eamp 3.] Π ( w s to b mnmz..., ( Hnt: On ma ntrouc, for nstanc, ~ π π A sn + A sn A3 + whch satsfs th ssnta B.C. at th ft n. You shou rcognz that th appromat souton can satsf th natura B.C. on appromat. Fnt Emnt Mtho Π s quvant to ( to b mnmz ( w Π w for an. Lt us ntrouc mnts to th sstm as pct bow : mnt numbr : no numbr : oca coornat Introuc th appromat souton va ntrpoaton functons (or shap functons for 5 b. H. Kwon 53

17 ach mnt. ( N ( hn N ( N N ( an th varaton of functona ovr ach mnt s summ to rsut n th varaton of th who sstm,.., Π Π Π ( + Π ( + Π (3 Lt us consr Π for an mnt. Π w (hncforth for convnnc Π N j ( K f j j j j N N (n an nca form [ K ] (n a matr form N f f wn j wn whr K f j N wn j N : mnt stffnss matr : work - quvant noa forc Not that ( s aso ntrpoat b th sam shap functon wth th varaton at th nos,. In fact, ths spcfc ( mts th arbtrarnss of n th varatona prncp so that th souton bcoms just an appromaton, not an act on. 5 b. H. Kwon 54

18 Assmb procur s a mattr of summaton of scaar quantts ng Π Π ( K j j F for an whch prov us wth th foowng goba matr quaton: [ K ]{} { F} { F } bng th work quvant noa forc 3. Consr spcfca a nar mnt for smpct. ( N ( + N ( whr th shap functons can b foun as N (ξ N (ξ wth N ( ξ ξ N ( ξ ξ ξ ca a normaz coornat. N N ξ ξ N ξ ξ ξ N N, ξ [ K ] { f } N w w N 3 In ths partcuar cas, thr mnts s th goba matr quaton as foows: h assmb procur s n fact th sam as bfor, ntfng th goba noa numbr an a th mnt stffnss matr coffcnt to th corrsponng row an coumn n th goba stffnss matr. 3 h work quvant noa forc obtan n ths wa s to b rgar as a concntrat trna oa rpacng th strbut forc ffct. For a constant strbut forc, n th cas of a nar mnt, th tota forc app to th mnt s qua v to two ajacnt nos as th work quvant noa forc. For th cas of othr tp mnts, varatona formuaton t us how to strbut th tota forc to nos assocat wth th mnt n contrast to rct approach. As an rcs, fn oursf th work quvant noa forcs at thr nos for quaratc mnt. 5 b. H. Kwon 55

19 5 b. H. Kwon 56 / / w (snguar as t s Bounar contons ( 4 ar ntrouc to th souton: w 3 9 : happn to b th sam as act souton Nots:. st an 4 th quatons ar not to b us (or obtan mor prcs snc an 4 ar not arbtrar, but zro. As a mattr of fact, howvr, ntroucton of bounar contons rpacs thos quatons. h racton forc F an F 4 can b obtan from th st an 4 th quatons, rspctv.. Avantags of varatona approach ovr th rct on: Us of scaar quantt (nrg vrsus vctors Eas n tratmnt of strbut oa 3. ratmnt of concntrat oas: h functona shou ncorporat th ffct of th concntrat forc: ( ( Π Π c c c c F w F w or or ( ( ( ( Π Π F w F w c c c c (th sam rsut (On can ntrouc a no at th concntrat oa pont. Othrws, what happns? F c c

September 27, Introduction to Ordinary Differential Equations. ME 501A Seminar in Engineering Analysis Page 1. Outline

September 27, Introduction to Ordinary Differential Equations. ME 501A Seminar in Engineering Analysis Page 1. Outline Introucton to Ornar Dffrntal Equatons Sptmbr 7, 7 Introucton to Ornar Dffrntal Equatons Larr artto Mchancal Engnrng AB Smnar n Engnrng Analss Sptmbr 7, 7 Outln Rvw numrcal solutons Bascs of ffrntal quatons