Chapter 6 Finite Element Method (FEM)

Transcription

1 MEE7 Computr Modling Tchniqus in Enginring 6. Introduction to FEM Histor nd Dvlopmnt Bsic Finit Elmnt Concpts Elctromgntic Anlsis Progrm Tchniqus nd oftwr Pckg 6.. Histor nd Dvlopmnt Chptr 6 Finit Elmnt Mthod (FEM) Turnr, Clough, Mrtin, nd Topp introducd th finit lmnt concpts 956. Almost simultnousl, Argris nd Kls dvlopd similr concpts in sris of publictions on nrg thorms. Cournt, Hrnnikoff nd McHncr r lso rl prcursors of th finit lmnt mthods. trting in 967, mn books hv bn writtn on th finit lmnt mthod. Th thr ditions of th book uthord b Prof. Zinkiwicz rcivd worlwid diffusion. Gllghr, Rock t l., s wll s th books writtn b Absi nd Imbrt. Th first FEM book (in English) for lctricl nginr writtn b P. P. ilvstr is publishd in 98.. Who nd Whn An computr cods in th rl 96s for solving structurl nlsis problms in mnnr similr to wht is now clld th finit lmnt pproch istd onl within th ircrft industr, which ws on of th rlist motivtd to hv improvd nlsis mthods. Turnr, Clough, Mrtin, nd Topp introducd th finit lmnt concpts 956. Almost simultnousl, Argris nd Kls dvlopd similr concpts in sris of publictions on nrg thorms. Cournt, Hrnnikoff nd McHncr r lso rl prcursors of th finit lmnt mthods. trting in967, mn books hv bn writtn on th finit lmnt mthod. Th thr ditions of th book uthord b Prof. Zinkiwicz rcivd worldwid diffusion. Gllghr, Rock t l., s wll s th books writtn b Absi nd Imbrt. Th first FEM book (in English) for lctricl nginr writtn b P. P. ilvstr is publishd in 98.. Wht nd Wh Th principls usd to stblish vlid qutions dscribing th bhvior of th nginring problm t hnd includ quilibrium, Nwton's id rgrding forc cting on mss, Chptr 6 Pg

2 MEE7 Computr Modling Tchniqus in Enginring potntil nrg, strin nrg, consrvtion of totl nrg, virtul work, thrmodnmics, consrvtion of mss, Mwll's qutions, nd mn mor. Th problm smd to lws b tht onc ll th hrd work of formulting th problm ws complt, solving th rsulting mthmticl qutions (somtims linr nd oftn nonlinr prtil diffrntil qutions), ws lmost impossibl. Ritz hd tndd his id to us diffrnt gomtric rgions, stblish sprt pproimting functions in ch rgion, nd thn hook thm togthr. This is prcisl wht finit lmnts r bout. Th id hd to wit until modrn digitl computrs took w th fr of lrg numbrs of lgbric qutions. oon mtrics nd mti mthods of orgnizing lrg numbrs of lgbric qutions whr brought into th finit lmnt pproch; rcll tht th word "mtri" ws in th titl of th importnt Air Forc Confrnc in 965. Now mthmticins rntrd th scn to rmind th nginrs of ll th mthods of solving mtri qutions from linr lgbr. Rsttd, th finit lmnt mthod is on whrin th difficult of mthmticll solving lrg compl gomtric problms is trnsformd from diffrntil qution pproch to n lgbric problm, whrin th building blocks or finit lmnts hv ll th compl qutions solvd for thir simpl shp (s tringl, rod, bm, tc.) Th rprsnttion of th rltionship of th importnt vribls for th littl, but not infinitsiml, lmnt is dtrmind through Rligh or Ritz pproch just for ch lmnt. Onc this is don, mtri of siz qul to th numbr of unknowns for th lmnt cn b producd which rprsnts th lmnt. Infct, th mthod cn b usd to solv lmost n problm tht cn b formultd s fild problm. Th dvlopmnt of dditionl softwr products nd us within industr hs tkn th lst 5 or rs. Th r still undr th rl phs of us is probbl th lctromchnicl nd lctromgntic r, probbl owing to dld rcognition of nd nd trnsfr of knowldg from structurl to lctricl nginrs. tod's incrsd world comptition, th prssur to dsign for lctromgntic comptibilit, nd nw lctronic nd communiction dvics. 6.. Bsic Finit Elmnt Concpts. Fild Anlsis Enginring dsign is idd b nginring nlsis, th clcultion of prformnc of tril dsign. To prdict th prformnc it is oftn ncssr to clcult fild, which is dfind s quntit tht vris with position within th dvic nlzd. Chptr 6 Pg

3 MEE7 Computr Modling Tchniqus in Enginring Thr r mn kinds of filds,nd ch fild hs diffrnt in flunc on th dvic prformnc. Tbl. Vrious Aspcts of Prformnc Fild Potntil Ht flu Mchnicl strss Elctric fild Mgntic fild Fluid vlocit Tmprtur Displcmnt Voltg Mgntic vctor potntil Fluid potntil Vrious Problms in Elctromgntic Filds: ttic Filds Elctrosttic nd mgntosttic fild clcultions for both linr nd non-linr problms Qusi-ttic Filds Tim-dpndnt filds, including th trnsint nd std stt bhvior of lctromgntic dvics, dd currnts nd skin ffct Wv Propgtion Wv propgtion problms including microwvs nd ntnns, cttring nd rdition Optimiztion Optimiztion using dtrministic nd stochstic mthods, invrs problms, AI pplictions, nurl ntworks Mtril Modling Modlling of mtril proprtis covring suprconducting, composit, nd microwv bsorbing mtrils nd th numricl trtmnt of nisotrop, smi-conductor, hstrsis, prmnnt mgnts Chptr 6 Pg

4 MEE7 Computr Modling Tchniqus in Enginring Coupld Problms: Moving boundr problms, s wll s lctromgntic filds coupld to mchnicl, lctronic, thrml nd/or flow sstms. Finit Elmnt Modlling Clcultion of ll th bov filds nd potntils cn b prformd using finit lmnt nlsis. Th nlsis bgins b mking finit lmnts modl of th dvic. Th modl is n ssmblg of finit lmnts, which r pics of vrious sizs nd shps. Th finit lmnt modl contins th following informtion bout th dvic to b nlzd: gomtr, subdividd into finit lmnts mtrils cittions constrints Mtrils proprtis, cittions, nd constrints cn oftn b prssd quickl nd sil, but gomtr is usull difficult to dscrib. Figur shows tpicl nginring problm tht hppns to b sttic thrml or ht trnsfr problm. Rubbr shth Coppr wir Fig. Cross-sction of currnt-crring coppr wir with rubbr covr shth. Th wir nd its currnt tnd into nd out of th pg. To prform finit lmnt thrml nlsis, th finit lmnt modl shown in Fig. ws constructd b dividing th dvic into finit lmnts. Chptr 6 Pg 4

5 MEE7 Computr Modling Tchniqus in Enginring Fig. Two-dimnsionl finit lmnt modl of Fig... Enrg Functionl Minimiztion nd Glrkin's Mthod All th dsird unknown prmtrs in th finit lmnt modl is b minimizing n nrg functionl. An nrg functionl consists of ll th nrgis ssocitd with th prticulr finit lmnt modl. Th lw of consrvtion of nrg is tht th totl nrg of dvic or sstm must b zro. Thus, th finit lmnt nrg functionl must qul zro. Th finit lmnt mthod obtins th corrct solution for n finit lmnt modl b minimizing th nrg functionl. Thus, th solution obtind stisfis th lw of consrvtion of nrg. Th minimum of th functionl is found b stting th drivtiv of th functionl with rspct to th unknown grid point potntil to zro. It is known from clculus tht th minimum of n function hs slop or drivtiv qul to zro. Th bsic qution for finit lmnt nlsis is df dp (.) whr F is th functionl (nrg) nd p is th unknown grid point potntil to b clcultd. Th bov simpl qution is th bsis for finit lmnt nlsis. Th functionl F nd unknown p vr with th tp of problm. In vritionl clculus th functionl is shown to ob rltionship clld Eulr's qution. ubstitution in th pproprit Eulr's qution ilds th diffrntil qution of th phsicl sstm. Thus, th finit lmnt solution obs th pproprit diffrntil qution. Chptr 6 Pg 5

6 MEE7 Computr Modling Tchniqus in Enginring Th Eulr's qution cn not b lws found in som diffrntil qutions. In this cs, w hv to us Glrkin's mthod to find th discrtizd qution. G * P N dd (.) i whr N i is intrpoltion function. () on-dimnsionl lmnts i Cubic(4) Linr() Qudrtic() (b) Two-dimnsionl lmnts Tringulr lmnts Linr() Cubic(9) Qudrtic(6) Qudriltrl lmnts Linr(4) Cubic() Qudrtic(8) (c) Thr-dimnsionl lmnts Ttrhdronl lmnts Linr(4) Qudrtic() Cubic(6) Chptr 6 Pg 6

7 Hhdron lmnts MEE7 Computr Modling Tchniqus in Enginring Linr(8) Cubic() Qudrtic() Prismticl lmnts Linr(6) Cubic(4) Qudrtic(5) 4. Two Dimnsionl Finit Elmnts Two-dimnsionl finit lmnts connct thr or mor grid points ling in twodimnsionl pln s shown in Fig.. W will brifl driv th qutions for tringulr finit lmnt modling phsicl sstm tht obs Poisson's diffrntil qution. Poisson's qution in two dimnsions is d d k dt d d + d k dt P (.) d whr nd r two dimnsions nd k is mtril proprt. Eqution (.) govrns sttic tmpr-turns T, in which cs P is powr input pr unit volum. It lso govrns sttic lctric or mgntic filds, in which cs T is potntil ( or A) nd P is chrg dnsit (Q) or currnt dnsit (J), rspctivl. Th nrg functionl for ll ths phsicl problms is F k dl T ds PT d ( ) (.4) s s Chptr 6 Pg 7

8 MEE7 Computr Modling Tchniqus in Enginring whr is th surfc r in th two dimnsions. Th first trm of Eq. (.4) is th nrg stord in th css of lctric or mgntic filds nd is rltd to powr dissiptd in th cs of thrml filds. Th scond trm is th input nrg in th css of lctric or mgntic filds nd is rltd to powr input in th cs of thrml filds. Th first trm involvs th grdint G d l T T (6..5) T G u T (, ) u + (6..6) whr u nd u r unit vctors. Th simplst tp of two-dimnsionl finit lmnt ssums linr, or firs-ordr, vrition of th unknown potntil T ovr th lmnt. Fig. Tringulr finit lmnt in th pln. Within this first-ordr lmnt T is rltd to th thr unknown T vlus t th thr tringulr grid points ccording to T T ( + b + c ) (.7) k k k k k l, m, n Evluting Eq. (.7) t th thr vrtics givs th solution for th, b, c cofficints: l m n bl bm bn c c c l m n l m n l m n (.8) ubstituting Eq.(.7) in Eq. (.5) givs Chptr 6 Pg 8

9 MEE7 Computr Modling Tchniqus in Enginring G(, ) [ b T u + c T u ] (.9) k k k k k k k l, m, n Thus, th tmprtur grdint is constnt within prticulr tringulr finit lmnt. Th grid point potntils Tk cn b found b minimizing th functionl (.4), whr dsdd. ubstituting Eq. (.9) nd (.4) in Eq. (.) nd considring on tringulr finit lmnt ilds kg PT ds j l m n T,,, (.) s j Crring out th intgrtion ovr th tringl cn b shown to ild th -b- mtri qution [][T][P] (.) whr th "stiffnss" mtri is ( bb l l+ cc l l) ( bb l m+ cc l m) ( bb l n+ cc ) l n [ ] k ( bb m l+ cc m l) ( bb m m+ cc m m) ( bb m n+ cc m n) (.) ( bb + cc) ( bb + cc ) ( bb + cc ) whr is th r of th tringl, n l n l n m n m n n n n l l m m (.) n nd th right-hnd sid is th "lod vctor": n P [ P] P (.4) P Eqution (.) solvs for th potntil T in rgion contining th on tringl with l, m, n in Figur. For prcticl problms with N nods (grid point), th bov procss is rptd for ch finit lmnt, obtining stiffnss mtri or lmnt cofficint mtri [] with N rows nd N columns. [P] nd [T] r thn column vctors contining N rows. 5. Aismmtric Finit Elmnts Chptr 6 Pg 9

10 MEE7 Computr Modling Tchniqus in Enginring All dvics dsignd b nginrs r in rlit thr-dimnsionl. A spcil cs of thr-dimnsionl dvic is on tht hs il smmtr. Th bsic qution cn b writtn s k r r r ( T )+ r T k P (.5) z Fig. 4 Aismmtric finit lmnt. 6. Finit Elmnts in Thr Dimnsion Thr dimnsionl Poisson's qution: ( k T ) + ( k T ) + ( z k T ) P (.6) z A thr-dimnsionl finit lmnt hs t lst four grid points, nd th do not ll li in on pln. Th lmnt forms solid shp contining volum of mtril. Th Two-dimnsionl nd ismmtric finit lmnts dscribd bov r usd whnvr possibl, bcus th r simplr to dscrib nd uss thn thr-dimnsionl finit lmnts. Fig.4 shows th thr most common thr dimnsionl solid finit lmnts. Chptr 6 Pg

11 MEE7 Computr Modling Tchniqus in Enginring Fig.4 Thr-D finit lmnts: Th drivtion of th mtri qution for th ttrhdrl finit lmnt is rthr similr to tht of th tringulr lmnt. Thus in th cs of quntitis obing Poisson's qution th ttrhdrl lmnt is similr to th two-d tringulr lmnt. Etnding Eq. (.7) givs T T k( k + bk+ ck+ dkz) (.7) k l, m, n, o Th drivtion of th mtri qutions procds in fshion similr to tht dscribd in - D. 7. Empl of -D FEM. V V Boundr condition:. 56. For th tringulr lmnt, th qutions tht must b stisfid for this linr rprsnttion to gr with th functionl vlus t th vrtics r Chptr 6 Pg

12 MEE7 Computr Modling Tchniqus in Enginring { } α α α α α α ( ) + + (.8) In mtri nottion, α α α (.9) thn α α α (.) { } (.) Th prssion for (,) is quivlnt to { } b b b c c c d d d ( ) (.) whr b i j k k j c i j k } (.) d i k j nd b b b c c c d d d ( ) (.4) Chptr 6 Pg

13 MEE7 Computr Modling Tchniqus in Enginring ( ) (.5) ubstitut intrpoltion functions ( ) N i i i (.6) whr Ni r intrpoltion or shp functions. Ni ( bi + ci+ di ) (.7) ( ) ubstitut Eq.(.6) to th nrg functionl corrsponding to Lplc's qution,, ε W ε d + dd (.8) nd thn Eq.(.8) cn b writtn s N N N W i j + ( ) i j N i j i j dd (.9) whr ij N N N + ( ) N i j i j dd (.) Ni/ nd Ni/ r givn s N i N i c ( ) (.) i d ( ) (.) i ( ) dd ( ) (.) Th nrg functionl W corrsponding to i, W/ i, w driv n ij j (i,,...,n) (.4) j Th mtri form is shown s follows: Chptr 6 Pg

14 MEE7 Computr Modling Tchniqus in Enginring []{}{} (.5) whr (.6) [ ] nn L L M O M { } M n (.7) whr [] is clld lmnt cofficint mtrics. For th singl lmnt, th qution cn b writtn s W W W ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (.8) whr cc dd ij i j i j ( ) ( ) + 4 (.9) ubstituting ll of ij nd in Eq. (.5), nd thn sstm mtri qution cn b solvd b computr progrm. For th four-lmnt mpl, N6, so th mtri qutin s writtn s Φ Φ Φ Φ Φ Φ ε Chptr 6 Pg 4

15 MEE7 Computr Modling Tchniqus in Enginring Trnsposing ll known voltg vlus to th right-hnd sid of qution, thn th unknown nodl voltgs r obtind s th solution of this qution. Th rsults r Φ Φ 4 If ε ( ) ij ( d ) ij / V cic j + did ε ( ) 4 ε cic j + ε d ( ) 4 j i d j uprmtri nd bnd mtri Th mtri cn b stord in rctngulr rr of dimnsion n(b+) s shown blow: Chptr 6 Pg 5

16 MEE7 Computr Modling Tchniqus in Enginring ˆ 4 ˆ ˆ ˆ Progrm Tchniqus Th procdur of numricl tchniqus:. Pr-procssor Msh gnrtion nd mtril dscription b visul tchniqus Dimnsions nd smmtricl modls Linrit nd nonlinrit. Numricl nlsis FEM progrm flowchrt. Fig. FEM progrm flowchrt. Mtri solvr. FINITE ELEMENT DATA EVALUATION OF ELEMENT MATRICE AEMBLING OF THE TOTAL YTEM OF EQUATION INTRODUCTION OF THE EENTIAL BOUNDARY CONDITION OLUTION OF THE YTEM OF EQUATION FOR ALL EVALUATION OF FLUXE PER ELEMENT REULT OUTPUT Chptr 6 Pg 6

17 MEE7 Computr Modling Tchniqus in Enginring Elctromgntic Anlsis. Lplc's Eqution. Poisson's Eqution A spcil cs of thr-dimnsionl dvic is on tht hs il smmtr. Th bsic qution in clindricl coordint sstm cn b writtn s r r r A A µ r µ θ ( ) + ( z ) J r z z stm mtri qution cn b prssd s []{A}{K} For singl lmnt, K hs th form of θ K i J θ whr J θ is th currnt dnsit. Th cofficint mtri for singl lmnt is Chptr 6 Pg 7

18 W W W ( ) ( ) ( ) / u / u / u MEE7 Computr Modling Tchniqus in Enginring ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) A A A θ θ θ K K K whr ij cn b prssd s ( ) ij ( ) r cc + dd ε π ( ) 4 i j i j whr c z z i j k d r r i k j r ( ( r ) ( + r ) ( + r )). Hlmnts' Eqution Progrm Tchniqus nd oftwr Pckgs uprmtri nd bnd mtri. Msh gnrtion nd confirmtion. Dvloping our own progrm. Post-procss Chptr 6 Pg 8