REVIEW Lecture 14: Elliptic PDEs, Continued. Parabolic PDEs and Stability

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1 9 Nmrcal Fld Mchancs prng 015 Lctr 15 REIEW Lctr 14: Ellptc PDEs, Contnd Eampls, Hghr ordr fnt dffrncs Irrglar bondars: Drchlt and on Nmann BCs Intrnal bondars Parabolc PDEs and tablty Eplct schms (1D-spac on Nmann Implct schms (1D-spac: smpl and Crank-Ncholson, on Nmann Eampls Etnsons to D and 3D Eplct and Implct schms Altrnatng-Drcton Implct (ADI schms 9 Nmrcal Fld Mchancs PFJL Lctr 15, 1

2 TODAY (Lctr 15: FINITE OLUME METHOD Intgral forms of th consraton laws Introdcton to F Mthods Appromatons ndd and basc lmnts of a F schm F grds: Cll cntrd (Nods or C-facs s Cll rt; trctrd s Unstrctrd Appromaton of srfac ntgrals (ladng to symbolc formlas Appromaton of olm ntgrals (ladng to symbolc formlas mmary: tps to stp-p F schm Eampls: on-dmnsonal ampls Gnrc qatons Lnar Concton (ommrfld qn: conct fls nd ordr n spac, 4 th ordr n spac, lnks to CD Unstady Dffson qaton: dffs fls 9 Two approachs for nd ordr n spac, lnks to CD Nmrcal Fld Mchancs PFJL Lctr 15,

3 Rfrncs and Radng Assgnmnts Chaptr 94 on Th control-olm approach for llptc qatons of Chapra and Canal, Nmrcal Mthods for Engnrs, 014/010/006 Chaptr 4 on Fnt olm Mthods of J H Frzgr and M Prc, Comptatonal Mthods for Fld Dynamcs prngr, NY, 3 rd on, 00 Chaptr 5 on Fnt olm Mthods of H Loma, T H Pllam, DW Zngg, Fndamntals of Comptatonal Fld Dynamcs (cntfc Comptaton prngr, 003 Chaptr 56 on Fnt-olm Mthods of T Cbc, J P hao, F Kafyk and E Larnda, Comptatonal Fld Dynamcs for Engnrs prngr, Nmrcal Fld Mchancs PFJL Lctr 15, 3

4 Intgral Consraton Law for a scalar (from Lctr 8-N d d d d ( n da C q n C n da s d fd fd CM C C C C Adct fls Othr transports (dffson, tc (Ad& dff fls = "conct" fls m of sorcs and snks trms (ractons, tc C, fd ρ,φ s Φ q Applyng th Gass Thorm, for any arbtrary C gs: ( q s t For a common dffs fl modl (Fck s law, Forr s law: q k Consrat form of th PDE ((k k s t 9 Nmrcal Fld Mchancs PFJL Lctr 15, 4

5 trong-consrat form of th Nar-toks Eqatons ( 9 Nmrcal Fld Mchancs PFJL Lctr 15, 5 C, fd s Φ q q ρ,φ p g t g g p p p p p p p p Applyng th Gass Thorm gs: Eqatons ar sad to b n strong consrat form f all trms ha th form of th drgnc of a ctor or a tnsor For th th Cartsan componnt, n th gnral Nwtonan fld cas: ( C C C C C F C d d n da p gd p g d F gd d d ( n da p ( ( n da p ( d d d d ( n da p ( ( n da p ( ( n da p ( ( n da p ( ( n da p ( ( n da p ( ( n da p ( ( n da p ( ( n da p ( ( n da p ( ( n da p ( ( n da p ( C C C F gd C C C C C C C C C C p p p p C C C C C C C C C C C C C C C C C C C C C C C C C C p C C C C p C C C C p C C C C p C C p p p p p p p p F gd p p p p p p p p F g d g d Wth Nwtonan fld + ncomprssbl + constant μ: For any arbtrary C gs: ( 0 p g t g p p p p 0 t Momntm: Mass: 3 t g g Wth Nwtonan fld only: (from Lctr 8-N Cons of Momntm: Cachy Mom Eqn

6 FINITE OLUME METHOD: Introdcton Fnt Dffrnc Mthods ar basd on a dscrtzaton of th dffrntal forms of th consraton qatons Fnt olm Mthods ar basd on a dscrtzaton of th ntgral forms of th consraton qatons: d fd Basc das/stps to st-p a F schm: Grd gnraton (Cs: d ( n da C q n C n da s d C C C C Adct fls (Ad& dff fls = "conct" fls Othr transports (dffson, tc Dd th smlaton doman nto a st of dscrt control olms (Cs For mantnanc of consraton, sally mportant that Cs don t orlap Dscrtz th ntgral/consraton qaton on Cs: atsfy th ntgral form of th consraton law to som dgr of appromaton for ach of th many contgos control olms ol th rsltant dscrt ntgral/fl qatons 9 Nmrcal Fld Mchancs PFJL Lctr 15, 6 fd m of sorcs and snks trms (ractons, tc

7 F METHOD: Introdcton F approach has two man adantags: Ensrs that th dscrtzaton s consrat, locally and globally Mass, Momntm and oftn Enrgy ar consrd n a dscrt sns In gnral, f dscrt qatons ar smmd or all Cs, th global consraton qaton ar rtrd (srfac ntgrals cancl ot Ths local/global consratons can b obtand from Fnt Dffrncs (FDs (strong consrat form, bt thy ar natral/drct for a F formlaton Dos not rqr a coordnat transformaton to b appld to rrglar mshs Can b appld drctly to nstrctrd mshs (arbtrary polyhdra n 3D or polygons n D In or ampls, w wll work wth d s d ( n da q n n da s d n da q n n da ( t ( t ( t ( t whr (t s any dscrt control olm W wll assm for now that thy don t ary n tm: (t= 9 Nmrcal Fld Mchancs PFJL Lctr 15, 7

8 F METHOD ral Appromatons Ndd To ntgrat dscrt C qaton: d A tm-marchng mthod nds to b sd to ntgrat d to th nt tm stp(s d Total fl stmat F s rqrd at th bondary of ach C F n n da ( n da q n da g F = adcton + dffson fls Total sorc trm (sm of sorcs mst b ntgratd or ach C s d Hnc cons qn bcoms: s d d ( n da q n n da s d d d d F n da Ths nds lad to basc lmnts of a F schm, bt w also nd to rlat and 9 Nmrcal Fld Mchancs PFJL Lctr 15, 8

9 F METHOD ral Appromatons Ndd, Cont d Tm-marchng mthod for C qaton: Th arag of or a C cll, d, satsfs for fd n tm 1 d F n da Hnc, aftr dscrt tm-ntgraton, w wold ha pdatd th cll-aragd qantts For th total fl stmat F at C bondary: Rconstrcton of from Fls ar fnctons of => to alat thm, w nd to rprsnt wthn th cll Ths can b don by a pc-ws appromaton whch, whn aragd or th C, gs back Bt, ach cll has a dffrnt pc-ws appromaton => fls at bondars can b dscontnos Two ampl of rmds: Tak th arag of ths fls (ths s a non-dsspat schm, analogos to cntral dffrncs 9 Fl-dffrnc splttng Nmrcal Fld Mchancs PFJL Lctr 15, 9 d F n da d d 1 (snc d ( d

10 F METHOD Basc Elmnts of F chm 1 Gn for ach C, constrct an appromaton to (, y, z n ach C and alat fls F (,y,z Fnd at th bondary sng ths appromaton, alat fls F Ths gnrally lads to two dstnct als of th fl for ach sd of th bondary Apply som stratgy to rsol th fl dscontnty at th C bondary to prodc a sngl F or th whol bondary F n da 3 Intgrat th fls F to obtan n da : rfac Intgrals 4 Compt by ntgraton or ach C: olm Intgrals 5 Adanc th solton n tm to obtan th nw als of d F n da 9 Nmrcal Fld Mchancs PFJL Lctr 15, 10 Tm-Marchng

11 Dffrnt Typs of F Grds Usal approach (sd hr: Dfn Cs by a stabl grd Assgn comptatonal nod to C cntr Adantags: nodal als wll rprsnt th man or th C at hgh(r accracy (scond ordr snc nod s cntrod of C Othr approach: Dfn nodal locatons frst Constrct Cs arond thm (so that C facs l mdway btwn nods Adantag: CD appromatons of drats (fls at bondars ar mor accrat (facs ar mdway btwn two nods Nod Cntrd C-Facs Cntrd 9 Nmrcal Fld Mchancs PFJL Lctr 15, 11

12 Dffrnt Typs of F Grds, Cont d Othr spcalzd arants Cll cntrd s Cll rt trctrd: All msh ponts l on ntrscton two/thr lns s Unstrctrd: Mshs formd of tranglar or qadrlatral clls n D, or ttrahdra or pyramds n 3D Clls ar dntfd by thr nmbrs (can not b ndntfd by coordnat lns, g, Rmarks Dscrtzaton prncpls th sam for all grd arants prngr All rghts rsrd Ths contnt s cldd from or Crat Commons lcns For mor nformaton, s orc: Cbc, T, J hao, t al Comptatonal Fld Dynamcs for Engnrs: From Panl to Nar-toks Mthods wth Comptr Programs prngr, 005 => For now, w work wth (a: Cll cntrd (, s th cntr of th cll, smlar to FD In 3D, a cll has a fnt olm (for trdd msh, gn dstanc to plan s sd bhas as D What changs ar th rlatons btwn aros locatons on th grd and accracs 9 Nmrcal Fld Mchancs PFJL Lctr 15, 1

13 Appromaton of rfac Intgrals Typcal (cll cntrd D and 3D Cartsan C (s conntons on fgs Total/Nt fl throgh C bondary s sm of ntgrals or for (D or s (3D facs: F n da f da k k for now, w wll consdr a sngl typcal C srfac, th on labld To compt srfac ntgral, s ndd rywhr on srfac, bt only known at nodal (C cntr als => two sccss appromatons ndd: Intgral stmatd basd on als at on or mor locatons on th cll fac Ths cll facs als appromatd n trms of nodal als 9 Nmrcal Fld Mchancs PFJL Lctr 15, 13 z k y y +1 y y -1 y WW NW W W W nw w sw N n n P n s s NE E y E n W s Notaton sd for a Cartsan D and 3D grd Imag by MIT OpnCorsWar b n t P B T n EE y n N E z

14 Appromaton of rfac Intgrals, Cont d 1D srfacs (D C Goal: stmat mplst appromaton: mdpont rl ( nd ordr F F s appromatd as a prodct of th ntgrand at cll-fac cntr (tslf appromaton of man al or srfac and th cll-fac ara f da y y +1 y y -1 WW NW W W nw w sw N n n P s s n NE E y E Notaton sd for a Cartsan D and 3D grd Imag by MIT OpnCorsWar EE F f da f f O y f ( f ( y f ( y f '( y f ''( y R y y! nc f s not aalabl, t has to b obtand by ntrpolaton Has to b comptd wth nd ordr accracy to prsr accracy of mdpont rl 9 Nmrcal Fld Mchancs PFJL Lctr 15, 14

15 Appromaton of rfac Intgrals, Cont d Goal: stmat F f da y +1 NW N NE Anothr nd ordr appromaton: Trapzod rl F s appromatd as: ( fn fs F f da O y ( y y -1 nw n n WW W w P sw s s E EE y y W E Notaton sd for a Cartsan D and 3D grd Imag by MIT OpnCorsWar n 9 In ths cas, t s th fls at th cornrs f n and f s that nd to b obtand by ntrpolaton Ha to b comptd wth nd ordr accracy to prsr accracy Hghr-ordr appromaton of srfac ntgrals rqr mor than ponts / locatons on th cll-fac mpson s rl (4 th ordr appromaton: als ndd at 3 locatons To kp accracy of ntgral: g s cbc polynomals to stmat ths als from P s narby Nmrcal Fld Mchancs ( fn 4 f fs F f da O y 6 4 ( PFJL Lctr 15, 15

16 Appromaton of rfac Intgrals, Cont d D srfac (for 3D problms Goal: stmat F f da for 3D C T mplst appromaton: stll th mdpont rl ( nd ordr F s appromatd as: F f da f O y z (, Hghr-ordr appromaton (rqr als lswhr g at rtcs possbl bt mor complcatd to mplmnt for 3D C Intgraton asy f araton of f or D srfac s assmd to ha spcfc asy shap to ntgrat g assm D polynomal ntrpolaton or srfac, thn complt (symbolc ntgraton z k y W n W s Notaton sd for a Cartsan D and 3D grd Imag by MIT OpnCorsWar b n t P B n y n N E z 9 Nmrcal Fld Mchancs PFJL Lctr 15, 16

17 Appromaton of OLUME Intgrals Goal: stmat mplst appromaton: prodct of C s olm wth th man al of th ntgrand (appromatd by th al at th cntr of th nod P 1 s d d y +1 y y -1 y WW NW N nw n n W w P NE n E y sw s s W E T EE P appromatd as: n t N s d s s P P P Eact f s p s constant or lnar wthn C nd ordr accrat othrws Hghr ordr appromaton rqr mor locatons than st th cntr z k y W n W s Notaton sd for a Cartsan D and 3D grd Imag by MIT OpnCorsWar b P B n y n E z 9 Nmrcal Fld Mchancs PFJL Lctr 15, 17

18 Appromaton of OLUME Intgrals Goal: stmat Hghr ordr appromatons: Rqrs als at othr locatons than P Obtand thr by ntrpolatng nghbor nodal als or by sng shap fnctons/ polynomals Consdr D cas (olm ntgral s a srfac ntgral sng shap fnctons B-qadratc shap fncton lads to a 4 th ordr appromaton (9 coffcnts 9 coffcnts obtand by fttng s(,y to 9 nod locatons (cntr, cornrs, mddls For Cartsan grd, ths gs: 9 1 s d d s(, y a a a y a a y a y a y a y a y a3 a4 a8 P s d y 0 a y y Only 4 coffcnts a (lnar dpndncs cancl, bt th a stll dpnd on th 9 nodal als y +1 y -1 Nmrcal Fld Mchancs y y WW NW W W nw w sw N n n P s n NE E y E Notaton sd for a Cartsan D and 3D grd Imag by MIT OpnCorsWar s EE PFJL Lctr 15, 18

19 Appromaton of OLUME Intgrals, Cont d D and 3D D cas ampl, Cont d For a nform Cartsan grd, on obtans th D ntgral as a fncton of th 9 nodal als: y P s d 16sP 4ss 4sn 4sw 4s ss ssw sn snw 36 nc only al at nod P s aalabl, on mst ntrpolat to obtan als at th nodal locatons on th srfac Has to b at last 4 th ordr accrat ntrpolaton to rtan ordr of ntgral appromaton 3D cas: Tchnqs ar smlar to D cas: abo 4 th ordr appro drctly tndd For Hghr Ordr Intgral appromaton formlas ar mor compl Intrpolaton of nod als ar mor compl 9 Nmrcal Fld Mchancs PFJL Lctr 15, 19

20 Appro of rfac/olm Intgrals: Classc symbolc formlas rfac Intgrals 9 D problms (1D srfac ntgrals Mdpont rl ( nd ordr: Trapzod rl ( nd ordr: mpson s rl (4 th ordr: 3D problms (D srfac ntgrals Mdpont rl ( nd ordr: Hghr ordr mor complcatd to mplmnt n 3D olm Intgrals: D/3D problms, Mdpont rl ( nd ordr: D, b-qadratc (4 th ordr, Cartsan: F f da F f da f f O y f ( ( fn fs F f da O( y ( fn 4 f fs 4 F f da O( y 6 F f da f O y z 1 s d, d (, s d s s P P P y P 16sP 4ss 4sn 4sw 4s ss ssw sn snw 36 Nmrcal Fld Mchancs PFJL Lctr 15, 0

21 mmary: 3 basc stps to st-p a F schm Grd gnraton ( crat Cs Dscrtz ntgral/consraton qaton on Cs d Ths ntgral qn s: F n da d Whch bcoms for fd n tm: F n da whr 1 d and s d Ths mpls: Th dscrt stat arabls ar th aragd als or ach cll (C: P 's Nd rls to compt srfac/olm ntgrals as a fncton of wthn C Ealat ntgrals as a fncton of als at ponts on and nar C/C Nd to ntrpolat to obtan ths als from aragd als P 's of narby Cs Othr approach: mpos pc-ws fncton wthn C, nsrs that t satsfs ' s P constrants, thn alat ntgrals (srfac and olm W s ths n th ampls nt lct schm to rsol/addrss dscontnts ol rsltant dscrt ntgral/fl qns: (Lnar algbrac systm for P 's 9 Nmrcal Fld Mchancs PFJL Lctr 15, 1

22 On-Dmnsonal Eampls: Gnrc 1D F Grd gnraton (fd Cs -1/ +1/ Consdr qspacd grd: = Δ Control olm tnds from - Δ/ to Bondary (srfac als ar: ( 1/ 1/ +Δ/ Bondary total fls (conct+dffs ar: f L R L R Imag by MIT OpnCorsWar f( 1/ 1/ Arag cll and sorc als: 1 1 1/ ( t d (, t d 1/ Dscrtz gnrc ntgral/consraton qaton on Cs d F n Th ntgral form F n da bcoms: d 1/ f 1/ f 1/ s (, t d 1/ t s d s t d ( 1/ (, 1/ 9 Nmrcal Fld Mchancs PFJL Lctr 15,

23 On-Dmnsonal Eampls, Cont d Not: Cll-arag s Cntr al Wth ξ= and a Taylor srs panson 1 1/ ( t (, 1/ t d 1 / R d / / L R +1/ +1 + L R O 4 4 ( Imag by MIT OpnCorsWar ( t O( Ths: cll-arag al and cntr al dffr only by scond ordr trm 9 Nmrcal Fld Mchancs PFJL Lctr 15, 3

24 On-Dmnsonal Eampl I Lnar Concton (ommrfld Eqn: Wth concton only, or gnrc 1D qn d 1/ f 1/ f 1/ s (, t d 1/ bcoms: d f 1/ f 1/ 0 (, t c(, t t Compt srfac/olm ntgrals as a fncton of wthn C Hr mpos/choos frst pcws-constant appromaton to (: ( 1/ 1/ Ths gs smpl fl trms Th only ss s that thy dffr dpndng on th cll from whch th fl s comptd: L L R R f 1/ f ( 1/ c 1 f 1/ f ( 1/ c - -1 R R f f ( c f 1/ f ( 1/ c 1 L L 1/ 1/ -1/ L R +1/ +1 + L R 0 Imag by MIT OpnCorsWar 9 Nmrcal Fld Mchancs PFJL Lctr 15, 4

25 On-Dmnsonal Eampl I Lnar Concton (ommrfld Eqn, Cont d Now, w ha obtand th fls at th C bondars n trms of th C-aragd als W nd to rsol th fl dscontnty => arag als of th fls on thr sd, ladng th ( nd ordr stmats: fˆ 1/ L R f 1/ f 1/ c 1 c L R 1/ 1/ 1 bsttt nto ntgral qaton Wth prodc BCs, storng all cll-aragd als nto a ctor fˆ 1/ f f c c d d d ˆ ˆ c c c c f 1/ f 1/ f 1/ f 1/ d c c 0 d Φ c BP( 1,0,1 Φ 0 (whr B P s a crclant tr-dagonal matr, P for prodc Φ 9 Nmrcal Fld Mchancs PFJL Lctr 15, 5

26 MIT OpnCorsWar 9 Nmrcal Fld Mchancs prng 015 For nformaton abot ctng ths matrals or or Trms of Us, st:

Variational Approach in FEM Part II

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