Tài liệu Lập trình tính toán hình thức trong phương pháp phần tử hữu hạn giải một số bài toán cơ học môi trường liên tục

DAI HQCQuac GIA THANHPHa HO cHI MINII TRUONG DAI HQC KHOA HQC TV NHIEN :;::, ~? NGUYEN DINH HIEN L!P TRINH TINH TOAN HINH THUC TRONG PHUONG PHAP PHAN T\J HOU HA.N GIAI M(rr s6 nA.I TOAN CO HQC MOl TRUONG LIEN T{)C Chuycn nganh: CO HQC V~T RAN BIEN D~NG Ma 86: 1.02.21 T6M TA'l' LU~N AN TlEN SY ToAN LY Thanh ph6 1-16CHi MINH - 2003 LPr (~l), Ie; NCr ~ H 10-03 tong tr}nh duQc hoan thal1h t~i Khoa Toan Til1 , TrU<1Iigf)~i hQc khoa hQc tlf nhien, D~i hQc q'u6c gia th1inh ph6 H6 chi Minh. Nguoi huang dfin khoa hQc : 1./ pas. TS. NGG THANH PHONG TrW!118D~ti h()(,'Khoa h()(,'TV l1hiel1Tp.HCM 2./ TS. NGUYEN DONG Vi~n Co h()(,'Ong dfl/18. Vi~n KH&CN Vi~t Nmn Phan bi~n I: PGS.TSKH. NGUYEN VAN GIA Vi~l1 Co h(J(,'0118 dfwg. Vifl1 KH&CN Viiit Nan? Phan bi~n 2: PGS.TSKH. CHtJ VI~T C00NG P!uln vi~n G3ng I1gM TMng tin c BQP Phan bi~n 3: POSTS. NGG KIEtJ NHI D~lih()(,'BelchK!lOaTp.HCM Lu~n an se duQc baa V9trltac H0i dOng chtm lu~n an dp nhil nuac hQpt~i: vao hOi giO ngay llulng Ham co th@tlm hi@u Juan an tai: A" ,{! THV VI~N KHOA HOC TONG H("'J IILI .1 "'r'~liIC,I: ! ',I r. . I".r., ~ I.:.,:... ,'. i i ""'.~ Md DAU ~y i b~ r-~'--" .~ 1. ~,i'.K' It' ~ ~;; i~'.;~ ( - ..... Trong phu'dng pilar ph~n ta hii'uh~n ta phai Ihlfc h~~ncae huck sau - v~ Lhlfehi~n d u~nggiclilicIt: . L J ~ du'a hili luein v6 d~ng yc'u (d~ng hic'n pilau) . lhie'tl~p cae M lhue roi r~e hoa mi~n xac dinh eua bai loan, . tich phan tren cac mi~n con (phh La), . thie'll~p cae ma lr~n de>cung, ma tr~n khcSi1u'<;1ng cho ph~n la Toi day la .mOico lh6 l~p lrlnh cho may linh lhlfe hi~n. Thlfe ra, may tfnh ehll1\m ding vie;e lifp ghcp cae ma Ir~n ph~n \l't v;) gi;li h(- pillidng Lrlnh d~1i s6 llly6'n tfnh KX=F hay he; phll\1ng L!'lnh vi ph;ln M X+ ex+ K X = F. Cong vi~e Hnh loan chufi'n hi cho I~p lrlnh (1('jih(>i di'Lnhi~u Lhtfigian va eong sue. Thl}'nhung, khi sO'd~lngehudng Iduh . Hnh loan nay, ta l~i bi gidi h~n rdt nhi~u VI:u~ng phfin La u,~ng ham xflp xi, cae di611kie;n lien l\,le, klul vi dc'n dip k nao d(i . . . Jii dlt~1cm;}c djnh san lro~g chu'dng lrlnh, Vi<;e ung u\,lng £)~i s0 m;ty Hnh (Compuler Algehra) hay Hnh Loan hlnh lhue (Symbolic Compulalion) sc giup chung la giiH quyc'1 cae khlS khan lren. Liinh vlfe E)~i s0 m,iy llnh khac vdi nhii'ng ehu'dng ldnh hi~n hii'll dti~1e hlnh lhanh lren ncSn Lang Hnh loan s0. Nhii'ng M lh6ng d~i sO'may Hnh co lh6 thao lac lren nhii'ng d6i lu<;1ngloan hQe hlnh thuc hen e~nh nhii'ng phep tinh'tren d~i lu<;1ngsO' hQe, Th~t v~y, vi Ilguyell tdc, bitt ky toall tii toall h{IC 1l{1OClI ca,l trllC a(1i st{ crill!: ClI tld t/r(t'e I';fll (111{/etrell /rf d<,; sfI may tillh. M(lc dich cda 11Iq.1lall Ilay la Ilghiell cli'll cae gidi thuq.1 eda D(1i ,'iflmay lillh, ktt IU/p voi gidi thuij.t s{;'truyill tho"g Ilhdm xtiy dlfllg mQI hf c/llidllg trillh t{llh toall hlllh tlllt'c gidi cac bai loall cd h{lc. Cling dn n6i them Iii I~p lrlnh llnh lmin hlnh lhue khong e6 nghia Iii phil nMn Hnh luetn s6. Hflll hc't cae lru'ong h~jp, tinh luein s01a gicli pluip duy nhi\t di de'n Wi giiii eu6i cling. Vi~c tlm du'<;1enghi~m giai lich chinh xae cUa me>tbai loan ed la ri\l hic'm. Kef quit cda luij.II all La dii xtiy dlfllg qua trillh lillh toall cac ma Irij.II pha'lI lit cila pIUMII!:pllap pJlli.Il I,t Ju1uh(lll d't'(n d(1llg cae toall t,~ d(1i so: qua do, dii ell Ihi Iq.p trillh tillh loall IIlllh tlui'c, Lam cd si'J tie" lai 2 ml)t moi tn/dllg tif dl)lIg /rOlllljp trill/r (allto-codillg). Nc'u xiiy dt!ng giao tiC'pt6t, cae cbudng trlnb Hnh lmln hlnb thue sc kc'l hl.lpvdi ehu(lng tdnh tinh loan s6 truyen th6ng (Fortran) l~p thanb h~ chudng trinh tinh loan mQtldp cac bai loan. CHt!ONG 1 MOT SO KY HItU, DJNH NGHIA VA CAC KHAI NItM CO BAN Chudng nay trlnh bay de ky hi~u t()(lnh{JC.Hldl,ln/{tron/{ luc;invan, cac dinh n/{hfa, cac c()n/{thue bie'n ddi tEch1'1/(1/1 va cac djnh Iy c(1biin cua gidi tEchham, d~c bi<$tkhai ni<$mv~ d~l1lgye'u (d~ng bic'n phan) cua bai loan bien va cac xffp xi lam cd sa clIo pht/(Ing phdp pl/(1ntii hilu h~m. Cae djnh ly v6 .qtl/(}i tl,l, t(1n t~li nghi~m va cac ddnh /{id sai sf} cling duQc nh~c l~i. Nh~m ung d1,lngphuong pIlar ph§n tu huu h1:J,n trong cac bai loan bien cua cd h<;>c,Ben cac khai ni~m cd hiln Clla If thuyet dcln '/(1;, dllll '/(1; nl/(Jt cling duQc d<3c~ p. Ngoai fa, cae eau true de;; sf) I~m cd sa clIo cac t[nh todn hinh thue (symbolic computation) cling dul.1Cnhf{e I~Ii nhltm la 111S,\ng It>y luting Hnh loan d~i so tlOng Hnh loan hlnh LInk. CHUaNG 2 TONG QUAN VE CAC TiNH TOAN HINH THUC Chudng nay lrlnh bay t6ng quaIl, vai lro va dc khai ni(;m cd sa eua Hnh loan hlnh thuc trong vi~c giai hili toc noi chung va trong phuong pIlar ph§n tu hall h~n noi ricng. Sd d6 clIo vi9c giai mOtbili loan cd h<;>c co th€ di~n ta nhu sd d6 sau (xem trang 3): Cac vffn de nhu v~y phai duQc tic'n hanh trudc khi I~p trlnh lren may Hnh,do v~y khi xu dl;lngmOtchudng trlnh tinh lOcvao each chQn cd s(1cua xflp xi: u(x,y,z)=(N1(x,y,z)N2(x,y,z)...Nm(x,y,Z)XIl, 112 ...lImY =N(x,y,z)(llm) Plll/dllg pluip xap xi bdllg plutll tit Illl11lu;l1lla phudng pIlar xa'p xi nut tren cae mi~n eon, nhung dn dap ung eae yeu du sau : Xa'p xi nullren m6i mi~n con yc co stf lham gia cua cae biC'nnul tudng ung voi cae di€m nullren yc va, e6 lh€, ca lren bien eua yc . Ham xa'p xi uC(x) lren m6i mi6n con yc du(je xay d1!ng la lien t\,le tren yc va thoa man cae dieu ki~n lien t\,legiii'a cae mien eon khae nhau. Nhu' v~y vi~e xap xi bd"g plitt" Iii IlllllluJ.l' sc d(it ra hai Va-IIdt! : . . . Dinh dS}.nghinh hQc cua cac pIlau to' cIlia . XAy dtfng cac ham nQi sur N; (x) tu'dng dng tren m6i phdn to'. Chu v: Cae dii11l 11(J;,fUy klu)ng I1I/(ttthilt chlla diim mit hll1h 11{)e,ma can C()thi cd cae diim bell trol1g 17'1£111 tt~. Tuy nhien, xAy dtfng cac xa'p xi tren cae pIlau to' thtfe, thi ma tr~n nQi sur N se phI) thuQc vao tQa dQ cae di~m nut cua phftn to', nghia la phl! tIllIQCvao d{lllg hlllh h{)c, va cIlllllg kluic llhall vai miJi plUtll tit. Ne'u xa'p xi lren du(je lh1!ehi~n lren cae phall tii Iham chie'" sao eho ma tr~n ham N fa dl)c lijp vai d{lllg hlllh hvc c,;a plUtll Iii Ih~lc,thi cae ham nay co lh€ xu d\,lngeho mQiphfintu e6 ph5n lu tham ehie'ugi6ng nhau (Cae ph~n lu lh1!ese e6 clIng phfinlu tham chie'ukhi chung gi6ng nhau l1ut 11(J;suy). Sau d6 la xay d1!ng phep bic'n d6i tu(ing dU(ing (hinh hQc) giii'a ph~n tu lh1!c va ph~n tu v~: [o(ti hinl1 d(l11g, s(j' nut IIll1h h()c )1([ ,W;' 7 tham ehie'u. GQi phep bie'n d6i hlnh hQe giua ph~n tu tht1c va phfin tu tham ehie'u la: r:f ~ x(f) = N(f)(xm) trong do: ~=(~,17,(), x=(x,y,z) Yi~c tinh loan ham xa'p Xl N du'<;1etie'n hanh blnh thu'ong nhu' cae xa'p Xl ~ phfin tu hUll h<;1nquell thuQe vdi cae tQa dQ nut ehi6u H\ ua niC't, qua cae nu'de : C/u!n da thlic ("(1,Wlcua xap xl : . . TEnh ma trrJn nut P(~)=(PI eua phfln tu tham P2 ... 1'" =(PI(~»),i,j= Pm) I,...,m . Tinh N(~) theoc(jngthlic : N(?)=/'(?)I',,-I Phep bie'n d6i hlnh hQe tren eho phep ta ehuy~n cae Lichphan eua mOt ham f tren ph~n tii' th\fc thanh tich phan don ghln bon, tren mQLph~n tii' tham ehie'u. 3.3 XAI» xi TRftN PIIAN 'I'D TIIAM Cillfiu Nh~m lam don gicln cae tinh loan d~c tru'ng cua mQt ph~n tu co d<;1ng phue t<;1p,ta ~u'a vao khai ni~m v8 ph~n' Lii'tham chie'u quell LhuOc: phh tu tham ehie'u yr la mQt ph~n tu co d<;1ngra't don gicln, d~t Lrong . mQth~ tham ehie'u , c6 Lh6hic'n d6i LhanhmM phfin tu OWeyc qua m0t phep bie'n d6i hlnh hQc r" . Phep bie'nd6i r e bie'nd6i mQtdi~m co tQadQ ~ eua phh tu tham ehie'u thanh mQtdi~m eua ph~n tu th\fe co LQadQ x: r" : ~ ~ x' = x" a) Phep bie'n d6i "e nhu' v~y, phl,lthuQe vao d~ng va vi tri CUBphfin ttl thlle, nghia la vao tQadQeua cae di~m nut hlnh hQe. Nhu'v~y , vdi m6i thl' Phh tii'thu'c . . r trong do XI' xi ' Xk Lht1c yo . . . ):- - - ): --'" -" = -.. ( " -r X X ",XpX"Xk"" ) , la tQa dQ eua cae nut hlnh hQc cua ph~n tii' . D~ dap ung cae yell du tren, mQt phep bie'n d6i hlnh hQc "e t6ng quat phclithoa man ba di8u sau day: i. La mQtsong 3nh tit ph~n tu tham ehie'u vao ph~n tii'th\fe. ii. Cac dinIt!a de ham p;(;f) la mOLva'n d~ ed ban eua phu'dng pMp phftn HI hifu h<.tn: u(:[) = (PI (:[)P2 (:[)...Pml (:[)XOI 02" .a"d Y = P(:[)(a,,) T~p h~p de ham lrang I'(;f) L<.t°Den cd .'111 cLia xap xi . SII sInu;l1lg ciia 110phdi blillg .'IIIbif/" mit hay .'IIIbq.c t{t dO"d trIa phih, tit . Tom tift cac blluc xiiy d{tllg ham (11latrq.1lhillll) » Ch{JIlda tl"ie C(f,ffl I' (;f ) » Tinh17latrQnnut » Ngh;ch ddo ma trQII nut p" » Tinh N(:[) N(~): Pn=(Pj(';») ,i,j=I,...,nJ theocvngthue N('t)=P('t)Pn-1 \J 3.4 PHEP BIEN DOl CAC TOAN 'l'(J DA.O HAM vA TicH PHAN Trang cae bai loan Cd hQe , V~t Iy ,dn phai tlm cae ham ehua bi€t un va cae d~o ham eua no ai" alex, tren mOt mi~n xae djnh nao do. a ' cy ,... N€u dung cae xa'p xi tren eae ph~n tii'tham chi€u tIll Tat cd cac bie"lltIllfC lien qllall de'n II va cac d(lo Illim ctla II dlfi VtJi x, y, z, se dll{ICddi thll1lh d(lo ham theo .;, 1] va (, tIllIng qua ma trQ" Jacobi (J) clla phep bie" diJi. B!€n d6i eae d;o. ?a~ ~~e nha't. Sii' dl,lllg eae eong thlie d~o ham eua ham h<;lp, ta eo . ~a~)- J(a J Trong d!,\ng d!,\i86, J la tkh eua hai m8 tr~l1: giUa ma tr~n cac O!,lOham cua eae ham bie'n 06i hlnh hQc theo cae biGn ~ , 11, va <;, va ma tr~n cac tQa dO cua ne nul hlnh hQc cua ph~n Iii'. J= ~ ( o~ ~ ~ 017 o~ ) 7(XYZ)= ;: - ((XI1XYI1XZJ) [ Nc ] Voi o~o ham eflp cao: Cae ph6p biGn 00i <.1<.10 ham cflp ( i ) co Ihd linh theo d~o ham d"p (i-i) lheo phep qui h6i. Cu6i cung, ta chuy~n lich philo cua mOt ham f tren ph~n tii' lhl!c ve thanh mQt lfch philn lrcn ph~n tii' lham chie'u vr bAng cong Ihlie quell thuoc : ff(x)/ 1" (x)/'(x),..dxdydz = f f(x(~) )/(x(~) )(("(~) ),.ldcl(.J)ld~d!!£It; 1" voi J la ma lr~n Jacobi cua phcp bie'no6i Ngoai ra chudng nay con oua ra mOLsCSofnh nghla cua chuii'ncua sai sCS, nh~m sii'dt;lllgcae Hnh eha't eua ham xa'p xi dS ti€n hanh Hnh loan hlnh lhlie eho cae sai s6 nay lren may linh. CHUaNG 4 L~P TRINH TINH TOAN HINH THDC CHO BAI TOAN CO HQC. CluJ'cJngnay Irlnh bay de ed sa eua cae tinh loan hlnh lhlic (symbolic) tren may Hnh, dl!a tren n~n tang eua cae tinh loan hlnh thlie tren eac da thue. Cae khai ni<$mdin ban eua cae ca'u Irue o!,lis6 oil ou~1e06 c~r nh~m d~n dAt de'n cae linh loan phuc I~p h<.1n eho cae ma lr~n. cae loan tii' d~o ham, Heh philo, bi€n d6i Laplace, ... 10 Vi~c I~p trlnh Hnh loan hlnh thuc cho phlidng phap ph~n tii' hUll h~n, chu ye'u t~p lrung vao vii;c dlia biLi toe", biell vi d(l1lg bitll p"lill (d~ng ye'u) ho~c sii' dl,lng cac nguyen Iy bie'n phfin (Lagrange chdng h~n) va till" toe", d(l1lg gidi tic" cae 11latrQ-llp"OIl tll (ma lr~n cung, ma tr~n can ho~c ma tr~n khoi Ili<;1ng)sau khi xa'p xi ph~n tu hUll h~n . Sd d5 qua trlnh nay the hi~n nhli sd d6 phia sau : SO DO LAP TfNH ToAN HINH THUC CHO PP.PTHH . TRINH , Bil i tm! n cd hQc Lll =f Nguyen 1:9 com! khii di Nguyen 1:9 bie'n phan, v6 dang ve'u ChQn philo tl't - xac t1inh philo tl'f thalli chiC'lI Xac djnh ham lien tI,IcdC'n dip mil'y'! ChQn da thuc xa p xi p(:f) Tinh ma tnJn nut: Pn =(Pi(~») Xap xi tren philo tl't thalli chiC'lI ,i,j= 1,...,11" Tinh ma tr~n philo tl't Nghjch dao ma tr~n nul p" Tinh . N(~) the a cong thuc IWi r~c mien xac t1jnhphilo tl't hull h:,\11 N('t) = P('t) p,,'1 Tinh loan xfty d~rngilia tr~n ham: N(~)va N(~) Thie't I~p ma tr~n Jacobi ciia phep bie'n d6i, I~p ma tr~n ]"1,tinh dct(J). II Chuyen cae Hehphilo eua f tren phftn ta th1,fe thiinh Hehphilo tren mQtphftn ta tham ehie'u ddn giiin hdn : K(m) = Jf(x)dxdydz = v'Jf(x(~)~et(J)d~d17d( v"" Tinh Heh phfin ilia tr~n K(m) = Jf(x(~»)Id~t(J)ld~d1]d~ v' In J<.c: - B~u lien la hili loan Unh ma tr~n phh t\1cua bili loan khuyC'chtan khi thiU 3 chi~u. Trong phh phI) Il)c co trlnh bay vil httang dfin cach thlic kC'th<;1pcac ma tr~n ph~n tli' nily vito mOt chtfdng trlnh t1nh loan b~ng ngon ngu Fortran. -TiC'ptheo lil gidi thil$u phttdng phap ph~n tli' huu h<;\ngiai mOt bai loan diln Ghat ddng nhil$t,dhg httang. -Bili loan 3 lil ling dl)ng da thlic h6n lo<;\n(chaos) vilo phttdng phap ph~n t\1huu h<;\nng§u nhien giai bili loan khuyC'ch tan khi thai theo mo hIGh ugh nhien va bili loan v~t lil$uco dOcling philo b6 ugh nhien. Bai toaD 1: Tinh toaD hlnh thuc cho bai toaD bj(~nkhuy~ch tan khf thai theo mO hlnh ba chi~u. Bili loan bien truy~n vil khuC'chtan khi thai: TImIpEC2(Q) thoa: Orp+uorp +vOrp +wOrp +arn-J1 'a a 0' a 't' * Bi~u kil$nbien: n cd sa Xa'Pxi : P (1 11 = ; ~ ;11 11~ ;~ ;11~) G<;>iJ lil ma tr~n Jacobi cua phep bie'n d6i nily D~t : Q=r' 13 HI" =((:) (~) (~)r; va v = ((u,,) H{ =((~~)(~) (:)f (w,,)) (VII) Khi d6 ta c6 .: 111,. = jN'Nldct(J)ldO, n, 'k" = fN'NVQlJ{ldct(.J)lclH, ", k'2 =()" IN'Nldetl(J)dO, 'k., +k., = JB/Q'DQB~ldet(J)ldO, 0, 0, ' k,.,= vfJIN'N.J,dl;d77' Iron d 0: g f) = I", ( ' .J - + 13z 13x- Oy~- ~ Oy I 0 -. [ 131; m, 131; m7 ) ( 131; m7 I' () 0 II 0 0 ~] ' 13x!! 131; ml ) ' + 13x~- l 131; 13'1 Oy 13x 131; m, ) Tinh cac ma tr~n tich philo nay ta thu du<;1ccae bieu thuc eua de ma tr~n M, K. va F. Xua't ke't qua d~ng file ,ngon ngu Fortran, ke't h<;1pvoi cac chudng trlnh con khac de Hnh lOan. Xet m,o hlnh 1000 nut: Ngu6n t~i nut 455, trong m~t ph~ng 491-500. Ke't qua Hnh loan la philn hC; n~ng dO khi thll i tl~i CI\c fiti t. M6 h1nh trnh bel toan khuyech tan khf 1000 nut " 1 fj " ~ ,I ) 40I -§ 9 30 !'D 70 lmx9=9m 100 ... "."n ",on """ .. ~... .n" .. """ ""H .. ".", "nn .. ..... u" "..~ ... "." mHO :. :::::: ".. .. .."" "." . "o~. - ' /~~~~' " ' ' ' ~~~~~ . .. . . . . . .. ..., .. """ .n.. . ..... .."n Bi€u d6 d~ng tr~ ke't qua tn~n m~t 401-500 - Gi6 =0 C6 nhi~u ke'tqua dii tlm ouQC,trong t6m t~t chI giOi thil;u vai k0t qUIt dieD hlnh. .~ .- 14 -~ -~ .~ ..~ --.-~ -.~ -~ .~ .0- .M--.~ .-..- .0-- ..~ ...~ Bi~u d6 dang Ir! k6t qua Iren m~t 401-500 MQtngu6n (+) I~i nut 455 V~n 15cgi6 :.:: D.2m/s.Hltdnggi6: ~ M6 hinh tinh bdl to6n khuy{1ch t6n khf 3600 nut .~ ...~ .~ .~ .- 30 .~ .~ ...~ .~ .~ ..- .-...-....~ ;, 1m,29 . )el ~I 'Oi Ei , 29m .~ Bi~u d6 ding tri ke't qua IreDmi,it 1201-1800 - ngu6n du'a khi vao m6i tru'ong t~i cae nUt: 1425-1427-1485-1487. So d6 tinh 30 x 20 x 6 = 3600 nut Nhan xct va ke't luan cho bai to<1n I: Voi cac chudng trlnh linh loan hlnh Ihuc IreD may Hnh (l~p trlnh Symbolic) co th~ cho phep ghli cac bai loan rill phuc t<,\p,cac di~u ki~n yell du cao (CI , C 2...) va vui dO chinh xac cao ma d do khong Ih~ chQn cac ph~n tii' ddn ghln duQC (vi d\l v~ I li~u composite, cd hQc pha huy...). Tli'de thu vi~n chuyen d\lng (Packages) co th€ lie'n t\f dOng boa l~p trlnh (t<,\oma ngu6n cho cac chudng trlnh giai so). Ke't qua bili loan Ihay ddi Ihco de vi lrf ngu6n, v~n 16egio. V~ dinh Hnh nh~n Ihily hoiln loan h<;lply. Co Ih~ md rOng vi dl,llhanh mOl th\fe nghi~m IreD may Hnh. 5.2 Bai toan 2: Phudng phap phftn hi hii'u h~n giai bai toan bien voi v~t Ii~u dan nhut diing huang - ddng nhi~t - 15 5.2.1/ Nguyen 19tu'ong ling-Bai tmin daD h6i ke't hop: Thea nguyen ly luong ling, nghii;m cua hai loan hi0n dan nhdl tuye'n Hnh co lh€ thu du<;1ctit nghi~m cua bai loan bien dan h6i, lrong do cac hhg s6 dan h6i du<;1clhay bling cac loan Iii'ham ph", thuOclhai gian (modun chung ling sua't ho~c ham chay cMm). Qua bie'n d6i Laplace ta lhu du'<;1c bai loan dan h6i k61 h - Xem thêm -