Luận án tiến sỹ 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 GIATHANHPHa HOcHI 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 dfinkhoa hQc:
1./ pas. TS. NGG THANH PHONG
TrW!118D~tih()(,'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~l1Co h(J(,'0118dfwg. 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 duQcbaa V9trltac H0i dOngchtm lu~n an dp

nhil nuac hQpt~i:
vao hOi giO ngay llulng Ham
co th@tlm hi@uJuan an tai:
A" ,{!
THV VI~N KHOA HOC TONG H<!P TP.HCM
. -\
>("'J IILI .
1
"'r'~liIC,I:
!',I r. . I".r., ~ I.:.,: ,'.i
i ""'.~ ~ y 1. ~,i'.K' It' ~
Md DAU i .~ b ~ ~;; i~'.;~ (
r-~' " -
Trong phu'dngpilar ph~n ta hii'uh~n ta phai Ihlfc h~~ncae huck sau - v~
Lhlfehi~nd 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'<;1ngcho 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'tv;) 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+ KX =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~lngehudngIduh
Hnh loan nay, ta l~i bi gidi h~n rdt nhi~u VI:u~ng phfin La.u,~ngham
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 ncSnLang 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.ptrillh tillh loall IIlllh tlui'c, Lam cd si'Jtie" 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/(1/1va 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(1nt~linghi~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,ntrong 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 la111S,\ng It>
y luting
Hnhloan d~i so tlOngHnhloan 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 to<lncd h<;>cnoi 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 vffnde 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 lO<lnnao do, nguoi ta pbl;l
thuQcvao Lhuvi<$nki€u ph§n tu da dinh sKucua ehudng trinh, khong
ehu dQngduQe v6 d~ng ph~n tu cIlia, d~ng ham xffpxi, va dc yell du
rieng clla Lungbai loan (tinh lien t1,lC,kha vi den cffpk nao do ), cac
tich pMn dftu Vi v~y, ta thu(lng phiii thic't I~p l<,licae tinh loan v<3mo
hlnh, xay dt!ng It~ide h9 thue phftn lii'va I~p lrlnh It.li,till v~y, h9 llnh
loan mdi vftnIwnchc'd cdcd~l1lgdii X(IYdlfng,va di nhiennguoita
khong th€ dua vao thu vi<$ntinh to<\ntfftd de d<,lngxffpxi, tfftca cac
d1:J,ngph§n tIT, VIs6 luQngdo la vo h<,ln. .
3
Pho, \7"",,$"'Qn
tU'd~
, '
lap",p ,""g;i!; .<I'h.
"'."ngtlnh [\SIT
~t~~1.:.~. PTVP
Chudng nay dIng nhtln m1.lnh vi~c quay tnf l~li ede khdi nif-111hall allu
eita elf h(JCkhi Hnh loan hlnh thuc, chAng h1.lnvi~c sii dl,lng nguyen 19

eong khcldT trang hiti lOcin h(: nhi0u v~t 111;1.J.Willenhurg (1;11;1111:
~ [ ,,; [ m,;~ - i;) , ",;', [ J, ,;,- Ai:)] - 0
Voi cae bai loan Cd h()c stYdl,lng phu<lng phap phftn Ill' hull h1.ln,chung
toi t<;\mphilo chia thilnh hai dOi tu\lng :
II. Cac hili WclllxuJ't phcll tiYplllf<lngIrluh vi philn:
L(u)+ j;: =0 tren mien V
Voi :('
(
II
)
=
/
' , Iren bien 8
"
va ('
(
11
)
=-::
f
' , Iren bien 8
',. , ,. " , " I
dc;tOgbie'nphiln: W(u)= flfl[L(u)+}; ]dV=O trongdu VIj/ thuOcm(H
,.
khong gian ham xac dinh nilo d6, vii cae dieu ki(:n biellnhu Iren
4
2/.Cac bi\i loan dl,i'a lrcn m(H nguycn 19 bie'n pMn cua cd hQc,
chAngh~n lrong cac bi\i loan cd V~lrAn: gO (u) = 0 '
vdi: D(Il)=! JCi/k/l'ij(II)l'k/(II)dV- JIll; dV - JF;II;dS
2" r ,l"

MQl b~li loan bien lrong cd 11gesau khi dua v~ d~ng bie'n phan ho~e da: d
df;lng cae nguyen Iy bic'n phiin lrong Cll hQe (Lagrange, Casliano,
Reisne-Henliger, ) (l~m ehua xel dC"nslf l<SnIf;liduy nha't nghi~m lrong
cae khong gian ham), va vdi cae xffp xi phftn ltl huu ~f;lnta dU<;1edi de'n
mo hlnh eua phudng phap phftn ttl huu Iwn di~n la b~ng M phudng trlnh
vi pMn :
Truong h<;1pbai loan bien tuye'n Hnh : /If :\'+ ex+ K X =F
Truong h<;1pbili loan bien phi tuyC'n : /If
i + (' ,¥+ K.Y + G(X. X) =F
.
trong d6 G(X,X) lit mQtham phi tuyC"n.Trong e<lhai lruong h<;1p:
'1=~T' T'
K=~T'kT' (
:=~T' ( 'T va F=f-T'FT
J'L ,I/!" Lo L " L "."
1 1 ,.1 ".1
Trong d6, Te ehi lit ma tr~n dinh vi. Hip rap ma tr~n phftll ttl vao ma tr~n
tdng th~. Cae ma tr~n me ' k,. , Ce ' Fe ph", thuQe vao d~ng bie'n phan,
vao cd sa cua da thli'c xKp xi, s6 nut va tinh chKl ph~n ta (Lagrange,
Hermite C I , C 2 , ) .
Trollg lufj.Il all da 1fj./1trilllt trollg Maple de'tilllt toall Itilllt tltue elto eae
plll/dllg trilllt vi /1llt111va cae d{lIlg bie'll pltl1l1, tl(dllg ILligfa cae lira trfj.1I
m keF d trell, tllllllg li'llg Vln cae kiill pllall tit tilY j 1, 2, Itoiic
e' e' e' e .
3 cltid", dcplg Lagrallge, Hermite C J, C 2, sf;'mit IllYj,
Vdi luu y r~ng: la't cii.cae linh lmin dn thiC'l: dua v~ d<~ngyC'u,xa'p xi
biC"nphan, eae tich phan, cae ma tr~n hiC'nd6i , d6u e6 lh~ liC"nhilnh
tren n:uiyd d~ng giai Heh. Chi dn ehQn IlIa trude da lhue cd sd eua xa'p
xi , va VIcae tinh loan - du la d,~nggi,li rich- du,1elie'nhilnhireDmay
Den e6 th~ ehQn IlIa cae da lhue h~e eao. Ngoili ra cac lieu chugn baa

dam tinh CI , c2
ciia xa'p xi cling e6 th~ du<;1eki~m chung d~ dang.
Cae ma tr~n ph§n ltl d~u du<;1exac djnh ireD may d d~ng bi~u thue lrude
khi ehuy~n d6i (tlf dQng) qua eae ngon ngu s6 truy~n th6ng. M",e dich
eu6i eung eua bai loan vfin la cae kC"tqua s6, vi~e I~p tdnh tinh loan
hlnh thue nh~m xay dlfng la't cii cae bi6u lhue giai lieh trung gian, vdi
5
ham xa'p xi va d<;lngphfin tli kha tiiy y ( 20, 32 nut ), tie'n de'n tt,i
di)ng boa xfiy d1,fngcae ehudng trlnh can. Cae ky thu~t Hip nip ma tr~n,
giai cae M phu'dng tr1nh d<;lis6 v~n lien hanh t~cae ngon ngu s6 truy~n
th6ng.
CHUONG 3
MQT HIE U DIEN CUA PHUONG PHAP I)HAN TD HUU H~N
CHO TiNH ToAN HINH THUC
Chuong nay xfiy thIng m(Hbi~u oi6n kluic eua phuong pluip ph~n tll hITu
h<;lnphii h<;1peho Hnh loan hlnh thue tren may Hnh.
3.1 PHUONG PHAP PHAN TO HUU H~N TONG QUAT
Xct m<)ibi) ba (K,P,L) trong do cae m6i lhanh phjn K, P vaL co cae th~
hi~n va quaDh~ nhu sau:
KIa mQtt~p con compact thuQc R", khae r6ng co bien lien t\,leLipschitz,
P la thong gian vector huu h<;ln(m) ehi~u , g6m cae ham xae dinh IreD
K, co gia tri tht,ie.T~p h<;1pL = {oi},i= I, ,m,eua cae di(~m G;E KIa P-
duy nhflt giai ne'u: '\fa; E R,i = I, ,m , 3!p E P sao eho
p(a;)~a;,'\fi = 1, ,111.Ph~n tll hITulu~n li'I IIl~Ji00 oa(K.P.~) sao eho
P-duy nha't giiii. Ham P; E PIa ham cd sd eua philo ta huu h<;ln,xae
dinh bhg : Pi (OJ)= OJ;' 1:$ i,.i:$ III
3.2 L!P TRiNH TINH TOANHINH THDc
M\,Iclieu la I~p tr)nh Hnh loan h1nh thlie clIOc:ic pllli<fng Ir1nh vi phfin va
cae o~ ng oie'n pIta n. IUt1ngung lit cae ma tr~n 111,.,k". C", F" (el11ow)'e /III)
ta cae 111atr(in nay cua cae pluin tu lulu "<In)

Vie't l<;li nguyen 19 eong kha oj theo d<,\ng ma tr~n
f£'TdV = fulf/ldV + fu"f'dS
" " ,\"
Trang d6: & la bie'n o<;lngkh:l oj ung vOiehuy~n vi kId 01U
X:I'pxi v~t Ih0 h~ng de phan 1ftr<'1in,le, hlfll 11<,ln.lien kel qua c;\c di0m
nut. Chuy0n vi trong h~ din phlt<1ngcua lIli'jiphfin ta dude gi:"iIhi61 IiI
m(Jtham eiia ehlly6n vi N nut ciia loan h/;. nghIa la. v(1iph;in lit Illia c6:
(11"'1(.1'.,1'.::)= N("'I(X.,v.::)(I
trong d6: N(III) la ma tr~n ni)i suy (hay ham xa'p xl),
ehuy~n vi loan M
u la vector
(j
H~ e6 N di€m nutthll6ng qual U co 6N lhflnh phdn :
U'=(U1V1"~U1'P1'I'1 UNVNWNUN'PN'I'N)
Ta e6th€ vie't: ell/l)(x,y,Z) = DII(I/I)(x,y,=)= DN11/I)(x,y,z)U
Trang d6 D la ma lr~n eae loan lu d~o ham. Yie'll~i nguyen Iy eong khii
di vdi ky hi~u: R =U'[ IJN(I/I"f'I/l'/IdV'I/I) + IlNII/II'f(I/I)'dSII/I']
Ta co: [:LJU/ N,m,'/J'c1m'DN1mludVlm,]= R
T~ day ta tha'yr~ng xay d1!ngcae xa'p xi eua u tren cae ph~n tu, nghia la
xay d1!ngma lr~nnOisuyN , tily lhu(>cvao each chQncd 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!ngla 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 .
. XAydtfng 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£111tt~.
Tuy nhien, xAy dtfng cac xa'p xi tren cae pIlau to' thtfe, thi ma tr~n
nQisur N se phI) thuQc vao tQa dQcae 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,;aplUtll 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 clIngphfinlu tham chie'ukhi chung gi6ng nhau
v~: [o(ti hinl1 d(l11g, s(j' nut IIll1h h()c )1([,W;' 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
7
tham ehie'u. GQiphep bie'n d6i hlnh hQe giua ph~n tu tht1cva 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 XlN du'<;1etie'n hanh blnh thu'ong nhu' cae xa'p Xl
phfin tu hUll h<;1nquell thuQe vdi cae tQa dQ nut ~ eua phfln tu tham
ehi6u H\ua niC't, qua cae nu'de :
. C/u!n da thlic ("(1,Wlcua xap xl :
.
TEnhma trrJnnut
P(~)=(PI P2 Pm)
1'"=(PI(~»),i,j= 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,lthuQevao d~ng va vi tri CUBphfin ttl
thlle, nghia la vao tQadQeua cae di~mnut hlnhhQe. Nhu'v~y, vdi m6i
P
hh tii'thu'c thl' . . ): '" -" =
(
):- - -
)
. . r . " -r X X ",XpX"Xk""
trong do XI' xi ' Xk , la tQa dQ eua cae nut hlnh hQc cua ph~n tii'
Lht1c yo .
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 di<!mmH hlnh IH)Cciia phfin (l?(rham chic'lI hMng l~ng
vdi cae di<!mnut hlnh hc)cciia ph~1ILllLht1c.
8
Hi. Bao toRn vi tr{ cac bi~n: bien cua phdn lU'lham chic'u xac
djnh bdi eae diegmnul hlnh h9C,se lu'dng dng vdi bien ciia phh Lt!'lht!e
xac djnh bdi cac diegmnul hlnh h9Clu'dngdng qua r (,.
Begddn gian ky hi~u, Labo qua chi s61ren e d~e lru'ngcho ph~n Lt!'.
Phep bie'n d6i hlnh hge giifa ph~n It!'lham ehic'u va ph§n It!'lht!c ma La
st!'dl,lng11\mQt phep bie'n d6i tuye'n tinh, ph thuQc cac tQa dQ hlnh

hQc(x )cua cac di~m nut cua philn hi tht/c Vc.
Nhll I'QY ta C() thi xiiy d1!lIg C£lC11£111111; cita plu!p billl ddi hillh h{JCtheo
euIIg 17l{Jte£lch tlllle nhll x£iy d1!ng c£le 11£1111nOi suy Ni (~), I'a lllu yrling
ham N (C; ) lil d(jc lq.p vui d(l1lg hillh h{Jc cLiaphall tLi:tlll.jC V. Do vQY
eae ham nay C()thi xii'dl:lngclIO /1/{Jiplul11Iii'C(}pluln Iii' thWll ehilu diJe
tn(llg qua: Hinh d(lIIg , ,ffJ'mit hillh h{Jc, S(J'nut n{Ji .wy :
Xa'p xi lren ph~n It!' lham ehie'u :
uex(~) = u(~) = U; = (NI(~)N2(~) NII, (?;)XUI U2" . u"J1'
-
{
o khi i oF j
Nj(l;;)=
I
kl
" " .
11 1= J
Xa'p xi lren ph~n It!'lham ehie'u ph~lid~lmbao Hnhlien ll,lelren phh It!',
va Hnh lien ll,legiifa de phdn It!'
Ta s~ biegudi~n II(~) lrcn ph~n It!'lh~\lnehic'u du'di d~ng mOll6 h~p .
luye'n Hnh eua eae ham s6 dOe I~p dii ch9n lru'de PI(;f) ,P2(;f), ,
lhOng lhu'Clngnha'll1\ eae d(/n tlulC dOc I~p. Vi~e eh<;>nIt!a de ham p;(;f)
la mOLva'n d~ ed ban eua phu'dng pMp phftn HIhifu 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 .'111cLia 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) N(~):
» Ch{JIlda tl"ie C(f,ffl I' (;f )
» Tinh17latrQnnut Pn=(Pj(';») ,i,j=I, ,nJ
» Ngh;ch ddo ma trQII nut p"
» Tinh N(:[) theocvngthue N('t)=P('t)Pn-1

Lrangd6 :
\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 mOtmi~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 VtJix,
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=
(
~ ~ ~
)
7(XYZ)=
[
;:
]
((XI1XYI1XZJ)
o~ 017 o~ -
Nc
Voi o~o ham eflp cao: Cae ph6p biGn 00i <.1<.10ham 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)/ (x)/'(x), dxdydz = f f(x(~) )/(x(~) )(("(~) ),.ldcl(.J)ld~d!!£It;
1" 1"
voi J la ma lr~nJacobicua phcp bie'no6i
Ngoai ra chudng nay con oua ra mOLsCSofnh nghla cua chuii'ncua sai sCS,
nh~m sii'dt;lllgcae Hnheha't eua ham xa'p xi dS ti€n hanh Hnhloan 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'uIrue o!,lis6 oil ou~1e06 c~r
nh~m d~n dAtde'n cae linh loan phuc I~p h<.1neho 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 biLitoe", 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(l1lggidi 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 TRINH TfNH ToAN HINH THUC CHO PP.PTHH
. ,
Bil i tm! n cd hQc
Lll =f
Nguyen 1:9
com! khii di
ChQn philo tl't- xac t1inh philo tl'fthalli 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(~») ,i,j= 1, ,11"
Nghjch dao ma tr~n nul p"
Tinh N(~) the a cong thuc N('t) = P('t) p,,'1
.
Tinh loan xftyd~rngilia tr~n ham: N(~)va
N(~)
Thie't I~p ma tr~n Jacobi ciia phep bie'nd6i,
I~p ma tr~n ]"1,tinh dct(J).
Nguyen 1:9
bie'n phan,
v6 dang ve'u
IWi r~c mien
xac t1jnh-
philo tl't hull
h:,\11
Xap xi tren
philo tl't thalli
chiC'lI
Tinh ma tr~n
philo tl't
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 = Jf(x(~)~et(J)d~d17d(
v"" v'
Tinh Heh phfin ilia tr~n
K(m) = Jf(x(~»)Id~t(J)ld~d1]d~
v'

In J<.<m!ra file CJ clan
N6i ke't vOi cae ehu'dng tr'lnh con kh:lc: nh~p s6
liell. IllIh ilia IrQII KillIheo s{) liC;lIlIh~l'. 1:11'nip
ma lr~n loanel;1e-,glMI!~ KU=F
In ma tr~n U (U Iii loi giiii sO'eua ke'tqua din
tlm)
Chu'dng nily cling d6 c~p d6n ghli thu~t hlnh thuc cho phu'dng plulp phitn
Iii'hull h<.1nugh nhicn d~ giai cae bili to:ln bien co ehua dc tham 86
hay qua trlnh ngau nhicn, trong do viOc xflp xi c~tc da thuc X~ICc.1inh
tu'dng ung nhu' trang ph§n Iii' hull h<.1nxae c.1inhc.1u'<;1ethay bang cae da
thue chaos thong qua khai tri6n Karhuncn-Locvcn, cae c.1alhue nily
trang tru'ong h<;1pt6ng quat co th~ lien hanh tu'dng It! nhu' giiii thu~t
Gram-Smith, da thuc chaos clip n co d<.1ng:
(
0 ,
(
"
)
2)-1)' I n~'1 fI~"
rpk ""'~I)= ,." n(I", J,P'I
.
/.,11
I , ~(-Ir-I"
fg / n ~I
)
I'(Ji11Ie
L L . \,-,.,1 I
"-I< .(1,., '.) I. I
trang do n-(.) Iii phep hoan vi (nghia Iii t6ng thlfe hi~n ln~n cae phep
hmin vD, cae d<.1nge\l th~ se du'<;1ctrlnh bay trong ehu'dng sau,

wti11cluJn
CHUdNG 5
Chttdng nay ling dl;lOgUnh loan hIGh thlic vilo mOt s6 bili loan phlic t<;\p
trong cd h<;>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 t1nhloan b~ng
ngon ngu Fortran.
-TiC'ptheo lil gidi thil$uphttdng phap ph~n tli'huu h<;\ngiai mOt bai loan
diln Ghatddng nhil$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'chtan khi thai theo mo hIGh
ugh nhien va bili loan v~t lil$uco dOcling philo b6 ugh nhien.
Bai toaD 1: Tinh toaD hlnh thuc cho bai toaD bj(~nkhuy~ch tan khf
thai theo mOhlnh ba chi~u.
Bili loan bien truy~n vil khuC'chtan khi thai:
TImIpEC2(Q) thoa:
Orp+uorp +vOrp +wOrp +arn-J1
(
o2rp +02rp
)
-VO2<p =1
'a a 0' a 't' a2 0'2 a:2
* Bi~ukil$nbien: <p= 0 tren f x (O,T)
o<p= pcp tren fo x (O,T) ; olp = 0 ireD fh x (O,T)
Oz Oz
* Bi~u kil$n d~u : lp(O) = lpo trong Q
Btta v~ d<;\ngbiC'n philn, foi r<;\cmi~n xac dinh vil xa'p xl ireD m6i ph~n
tli' ta di thu dtt<;1chl$phttdng trlnh d<;\is6 xac dinh cac gia tri <pt<;\icac
di€mnut: M<iH-K<P=F

Khi Unh loan tren may, ta chuy€n d6i cho cac ph~n t\1thalli chiC'u Qr
(tttdng ling vai ph~n t\1 th\fc). NC'u cac ph~n t\1th\fc lil phh t\1kh6i sau
m~t, ph~n t\1thalli chiC'u co d<;\ng:
Ch<;>ncd sa Xa'Pxi : P =(1 ; 11
~ ;11 11~ ;~ ;11~)
G<;>iJ lil ma tr~n Jacobi cua phep
bie'n d6i nily
D~t :
12
trong Q x (O,T)
Q=r'
13
HI" =((:) (~) (~)r; H{ =((~~)(~) (:)f
va v = ((u,,) (VII) (w,,))
Khi d6 ta c6 .:
111,.= jN'Nldct(J)ldO, 'k" = fN'NVQlJ{ldct(.J)lclH,
n, ",
k'2 =()" IN'Nldetl(J)dO, 'k., +k., = JB/Q'DQB~ldet(J)ldO,
0, 0,
d
'
(
I' 0
Iron 0:
k,.,= vfJIN'N.J,dl;d77' g f) = () II
I", 0 0
I
[
)
'

(
)
'
l
)
'
.J - Oy~- ~ Oy + 13z 13x- 13x!! + 13x~- Oy13x
0 131; m, 131; m7 131; m7 131; ml 131; 13'1 131; 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,ohlnh 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
fj
"
~
,I
40
~]
. "."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~uke'tquadii tlm ouQC,trong t6m t~t chI giOithil;u vai k0t qUIt
dieD hlnh.
"
1
)
-§
I
9
30 !'D 70
100
lmx9=9m


.~

.~
.~

.~
.~
.~

.~
.~

.~
.~
.~







.~
.~
.~

-~
-~
.~

.~



-~

.~
-~
.~
.0-
.M-

.~



.0-


14
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
)-
el
~I
'Oi
Ei
,
;,
1m,29 .29m
Bi~u d6 ding tri ke't qua IreDmi,it1201-1800 - ngu6n du'akhi vao m6i tru'ongt~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<1nI:
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 dOngboa
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 Hnhnh~n Ihily hoiln loan h<;lply. Co Ih~ md rOngvi dl,llhanh mOl
th\fe nghi~m IreDmay 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'ongling-Bai tmin daD h6i ke't hop:
Thea nguyen ly luong ling, nghii;m cua hai loan hi0n dan nhdl
tuye'n Hnhco 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'<;1cbai loan dan h6i k61h<jp:
aO" if + Fi == 0 ; e,; == ~
(
au, + auJ
]
ox;
- 2 ax; ax;
- -
ai; =Cijk}EkJ ' trong do : CijAi==p' C'ikl
p: anh cua bie'n thai gian lh1fct qua ph6p bi6n d6i Laplace.
Cijk: la iinh ciia ~;kJ qua phep bie'n d6i Laplace.
Va bie'n d6i cac di~u ki~n bien:
- -
U
I

==
v
,
' ireD r; ; (J'.n -==T
,
X. E f
7
'
", "
Khi v
(t) == const la co lh€ tach bie'n nhu'sau:
(J, ~ (/,"(I",)/;'(t), 1';:- 1,;u(rJ/';"(t),
5.2.21Ap dung giai b~liloan Ulmchil'nhal. vat lieu dan nhdl bang
phuong phap giiii tich.(Ta'm chju tai pMn b6 d~u vuong goc m~t ph~ng
giil'ata'm)
Vie't l~i cac phlfdng Ir.nh cd han cua hai loan dan h6i clIo hai
((I,lnlfll11l11iinghlnh chii'IIh~1h~ d~y h chjnn6n Vt1iI~fcphlln h6 q(x.y):
' 1
J
- /
.«(::::::~
_
V-
-
-

-
-
"-:-
,1-"'-"-'

x.


/v
Nhao xet (Xem hloh) : Chuy€n dicit u' ,v' tl.lidi€m ireD Idp song song
va each m~t giil'a mOLkhm\lIg z , ll,i thili didm I - do s1futIli colig lfi'm
tl.lOra - co dl.lng :
82w
u'==-zsin(a(t)):::::: -ztg(a(t)) ==-z-
8x8t
16
a2w
v'= -zsin(a(l)) ~ -zlg(a(l)) =-z-
ayal
I OW Ow
Laplace(u) =-zp-, Laplace(v') =-zp-
OX oy
Ti:rnMn xet teen ke't h<;lpquaIl Mbie'n u<.1ng-ehuy~nvi cho La:
- 2- 2-
- 02 W - 0 W - 0 W
Ii =-z
p
-, Ii =-z
p
-, Ii =-z
p
-
rx OX2)J' oy2'.'. oxoy
Ti:rquaIl Mung sua't-bie'n di;\ng cho La:
(

- -
) (
2- 2-
)
- 2- 1 02W 02W. - 2- Iowa w
a =-
p
E-z -+v-' a ,=-p E-z -+v-
rx 1- V2 (k2 0'2 JJ 1- v2 0'2 (k2
- 2- 1
(
02;
)
rxy= -P E 1+ v Z oxOy
Giai bili loan diln h6i ke't h<;lpteen yoi mo hlnh bili loan ta'm (ung sua't
phAng suy rOng), y~n UI,l11gWi giai da co, ta tlm du'<;Icphu'ong trlnh
Sophie-Germain:
-
(
04; 04; 04;
)
- - ,. . - 2- I hJ
D
-+2-+- =Df: f: w=O, YOlo D=-p E
(k4 OX20'20'4 II l-v212
VOi pIll/dill! vllifv Navier- Trlidlll! IUfDo(x, v) =00. =COllst :
4q '
f
' h
f

' m1lX . 111ry 16q
qmn = 2 sll1-sll1-dxdy= L
ab 0 0 a b ,,2"111
- 16qo ~ ~ 1 . 11111X . I17ry
11'=-= ~ ~ 5In-5In-
1f6 D - -
(
2 2
)
2 a b
m-I,J,3 "-I,U, 111 11
/11/1 -+-
a2 b2
Ta chi dn tinh w khi m, n lil nhung s6le. Vol tai ngoili philn b6 o~u,
m~t giua khi u6n phai d6i xung, cae s6 h<.1ngm, n chii'lltuong ung yoi dO
yong khong d6i xung Hen chung phili b~ng khong.
Thea nguyen 19Luongung chung ta se thu du<;lcll1igiiii cua bili loan oiln
nhot ti:rWigiiii cua bili loan oiln h6i ke't h<;lpb~ng phep bie'n deSi
Laplace ngu'<;Ic.
w = Laplace-I(;)
17
1 . //I l1X . mcv
](
//I2~
)
2SII1-;-'Sll1h
//III +
a2 b2
[
16 '" '"

II'= l_aplace-I ~ " "
tr'O D -L L;
m.U.J ,,1,.1,
chti yding:
(
I
)
1
(
12(I-v2)
)
I
(
1
}
12(I-V2)
Laplace-I
= =Laplace- 2- J =Laplace- ;-= ~
D P Eh P F- h
Cu6i ding la co:
Vdi:
-I
(
I
J
WI =Laplace =
p2E
_
[
12(I_V2) 16q" ~ ~

IV" - , (, L, L,
h :r ",'1.'.' " ",.
W= WIW()
I . IIITTX. n:rv
,5\11 5\11 '-
(
1111 /11
)
II h
111/1 '1-
a2 b'
Giii sii'v~lli~u dltn nhdl mo tii nhu'sau:
v
=consl; E = Eo(l-~e+'} lhco loi gidi teen ta c6:
(
'I' /
)
trong d6: W = 2e 2" cosh !L-
'I 2
_
[
12(I-V2) 16q" ~ ~ 1 . 1Il11X . nny
IV,,- J 6 L, L 2 25111 5111-
E"h 71: m~I.Un~I.U
(
Ill /12
)
a b
1Il/1 02 + b2
W = W" W"" '

Nhan xet : WE:la [(Iigidi hai toall dall hili clJdie'll
W'1:zadOllggop cua tillh llhut vijt Ii?u vao [(Iigidi cuo, cIlIlg
5.2.3/ Giiii blti loan dltn-nhdt tuye'n Hnhbling pp. ph~n Iii'hii'llban:
Qua vi d\,lteen ta lha'y nghi~m tlm du'<;Icllt nghi~m cua blti loan ta'm
ph~ng. Trong th\fc Ie', so'cac blti loan cd hQc,co d~ng nghi~m giiii lich,
Iltra't it. Cach giiii teen khong nhii'ngkhong md rOngcho nhieu blti loan
mlt con khong th~ l~p tr'inh hlnh lhuc. d day chung toi xin giOi thi~u
IX
m9t phuong phap ghli kha t6ng quat cho nhi~u bai loan dan nhat, d~ng
huang, d~ng nhi~t b~ng cach I~p trlnh ph~n tU'hii'llh~n ke't h<;Jpnguyen
ly tuong ling,
Vie't I~i nguyen Iy cong ao cho bai loan dan h6i ke't h<;Jp,vai nela s6
ph~n tti'cua bid loan :
"
I F,U~eS;DICIDSeUoeu'ldV=
e=1v,
=f JU,/U:)eS;fvdV + t J~u ~eS;frdi
e=1 v" e=1 r,
D~t :
Re = DSe
Vo : la vector chuy~n vt loan h~ - la ma tr~n h~ng.
1;: Ma tr~n dtnh vt cua ph~n tU'e tren loan h~,
ta SHYra :
Tit do ta co:
UOe = T"Uo
" II, II, -' -
!F,V~T:Ri:'R.r.vou,/v = L fu,p~Te'S~J;,dV + L fu,P~Te'S:frdr
e=lv. e=I", e=lr,
NMn xetr~ng: uTJ' V~ co th<idua ra ngoai dffu tich phan va dffu t6ng vl
chung khong phI,!thu9c to~ 09 kh6ng gian, Ta khU'cac thanh ph~n tuong

ling 0 2 ve':
" ,,",
I Jr.'R;C'RJ:Uou,/dV = I fTe'S;j;,dV+ I JTe'S;frdi
e=1v, ('=1", e=1 r,
B\e'n 06i Laplace ngu<;Jcbi~u thlic nay, chu yr~ng chi co (;', u,/,fv,fr
11\chlia bie'n p:
(ky hi~u Laplace -I (f) la bie'n d6i laplace ngu<;Jcham cua hamf)
tT:[}R;(Lapla" -,(C''u" ))R.dV] T,,U"= ~ t' S;/,dV + ~ ,f T: S;fcdr
Ky hi~u :
Ke = JR;(Laplace-I (C' . U"))RedV 11\ma tr~n 09 cling ph~ntU'.
v.
19
K=~1'f.K .1" F=~
J
T')l
j
'dV+
~
I
r'S'
t
',/I'
L ,e " L ,,',V L "~,Ie
<=1 e=1 "0 e=1 ro
ta dj d611hi011
thuc sau: K .U0 =F
Ghli M phU'dng tdnh d~i tuye'n tren ta Om dU'Qc Uo Hi cac chuyen vi clan
h6i t~i cae nut. Do d6: nghil$mGan nhdt la :u (I) = U 0 . U" (t)
D~ Hm ham u1J(t) ta 100y de'n gia thie't Uola nghi~m dan hai, tit do
.my ra 10:11latrQIl K fJIld; 10:11latrQIl /zlillg dfil ya; t. f){iYchinh la ma tr~n

dQcung dan h6i. Dieu nay cho ta M thuc sau :
(Laplace-I(C' .uJ) pluli Iii ma tr~n hang 56 d6i v<3it, nghia Iii:
(C'.;-)=.:i trong d6 A : la ma Iran h~ng s6 d6i v<3ip, pia anh cua t
~ P .
qua bie'n d6i Laplace.
Xct mQt hili tO1\n ql
'"
the:
E(l)= Eo(1 + ~ e-t"')
,
t
,
.,
'
x" "
,
m """"
-, , - 1
""-,. "'::'<:Z: :\ ,""".
MO HINH <;IAI eAI TOAN T)I;M
"""",'v ,
~"'~""'-_'."~~KN","
'""." "'K""",
C la ma tr~n v~t li~u,
v =COlist. ta c(): C = (,ifjffli;;,G; Tlrtllly,la 11mdlrt.1elIill1lhiC:"
( "')
d~ng nhdt: W'I(t) = 1- ~e -J Va nghi~m cua bai loan Hi: u(t)= Un w"
Trang d6 Un la ke't qua cua h~ phU'dngtdllh K .U 0 = F , d6 la cae
chuyen vi do linh chat daB h6i v~t li~u vii Iii chuy6n vi t~i cac nut.
DO THI E(l) THEa THaI ClAN, wJi ede hf S()''7

E(1) =y x Eo
I: :

! :
TIHli giall t
B6 thi bie'n d~ng nhat W,/(t) .
1
-: ;; __d -
.:: '-

I
20
theo thai gian vii h~ s61']
'L=do Z
i
" I
,
.
Thui gian t
Du'ai day Iii bi~u d6 bi~u di€n st! thay d6i chuy~n vi cua Him theo thai
gian. Biii loan du'<;1ckhilo sat vai cae gi,i tri cua 1']=0.2, 0.5, 0.7, 1.0 ,
trang l6m l~t chi th~ hi~n tru'ong h<;1p1']=0.2
0 I4T QuA BAI ToAN: TfNH BIEN DA.NGTAM, VAT LI~U DAN NHOT
BANG PHUONG PHAp PTHH
D. Ihl<"", ,,, lheo Iholglo. ,
g." ,
NUl
",." '
. .""."
.

I::::::
"",
1;=fO
I
I . -
MAT CAT GIGA TAM QUA cAc NUT 31-40. H~ s6 nhal eta =:=0:2 0
5.2.4/ Nhan xet vii ke't luau cua biii loan 2:
Phu'dngphap ph~n Iii'hO'uh~n eho bid loan diln nhuI, d,ing hu'ang, ddng
nhi~t giup ta ghHdu'<;1ebAitmin tu'dng dt>it6ng quat. Co thlSap dl,!ng
cho cae biii loan mOtchieu, hai ehi6u, ba ehi6u. Di6u ki(:n Iii: h(: s6
Poisson cua v~t li(:u giil lhi6lla hling 56 d6 ,ip dl,!ngphfin Iy bi6u.
Qua lai giili lren, vi~c xac dinh bi6u thue up(t) dff du'<;1eI~p ldnh Huh
loan hluh thuc, phI,!thuOcvao ham E(t) t6ng quat cua v~t li~u diin
nhot.B6i vai cae nghii;1l1daB h6i Un. tim bling phu'l1ngpluip ph~n Iii'
hO'uh~n, ta dffc6 giili lhu~t I~p trlnh .
5.3. Bai toan 3 :Tinh toun hid toan cIIh(}cngfiu nhien hiing plllMng
pIlar phfin hi huu'h~n ngfiu nhien.
5.3.1/ Phu'dng pilar rh~n Iii' hO'uhall ngfiu nhicn:
2\
Trong Cd H<;>c,cac bili lmin v~ ng~u nhien lllliong lien quaIl dC'nm(Hh<';
phttdng tdnh vi phan tuye'n llnh vdi cac h9 s6 ng~u nhien
A(x,B)ll(X,B) = f(x,B) (*)
Khong H\mgi<lmlinh 16ngquat, (*) e6 Ih6 vie't d~ng :
[L(x) + a(x,e)R(x)]u(x,e) = j(x,e) ,x ED
L la mQtloan 01vi phan xae dinh, n la mQtloan Iii'vi phan co cae h~ s6
la eae qua tdnh ng~u nhien trung btnh khong, R(x) la mQt loan hI xae
djnh , dttdi I11Qts6 gii\ lhie-tta e6 th6 khai lri6n:
N
u(x,e) = Lu;(e)g; (x) trong do: g;(x) la cae da thue lien tl,le lung
;=1

khue eo dtp dil de sinh fa khong gian ham Ihii'.(de ddn gian each vie'lla
sc bo qua ky hi~lIe ) :
N
L u;[L(x)+ a(x)R(x)]gj(x) = lex)
;=1
Nhan 2 vfSeho gj(x) vii lich phan Iren Illi~n D s :
t[I~L(X)g;eX»)g/X)dx+ Ia(x)[R(X)g;(X)]g/X)dx}; =yex)g;(X)dx
Khai trien a(x,O) lheo dIng cd s() nhtt u(x,O) , khi do vC'Inti se baa g6m
mOt ma lr~o co de philo Iii'Il(dng qual1.
I)~ roi n.IC haa mOL qua Ir1nh ngftu nhien, la ap u~lng kiwi IriOn
AI
a(x) = I AII~II(/II(X)
Kafhllnen-Loeve rho a(x,O) :
II ~ I
Anvii an(x) Iii cae gi;, Iri ricng va vector ricng coa ham llt't1ngquan coa
ham a(x, e) , va ~"fa tijJ1cae bie'i, IIgiillll!riell tqlc ciao. ~" fa cae bie'll
trollg cac da tltuc cltaos (ltall lo{lll). Cac da tltuc lIay dii du{lc xay d1!llg
dum d{lllg gidi licIt qua 11lf!tcltUdllg trillit tillIt toall Itillit thuc.Nhit do,
vi~c riti r{lc 11lf!tqua trillit Ilgau Illtiell dii th(lC lti?1l dU{lc.
N
[
M
]
" K +"J:K.(."j 1I.=
(
.;j=l, ,N,
L '1 L ,"'1 ' . 1
;= I ,,~I
, K;j = HL(x).!:;(x)Fj(x)dx; K;~") = ja,,(x)[L(x)g;(X)Fj(x)dx
tfong do : I) /J

./; = ff(x).!:j(x)dx
J)
Vie't l~i h9 tfen dttdi d~ng Illa Ir~n. vii Sl?(d~lng khai Iri0n Neumann d(,)i
22
voi tmin tii' ngu'<;1c, ta c6 :
[
M M N
I U= I-~;IIQ(II'+~~;II;fIlQ(II)Q(fIl)-
AI AI M
- """ ;: ;: ;:
Q
(I)
Q
(fIl'
Q
(II) +
~~~~~'~I
1=1.,=1 11=1
]g
Nhan xet: C6 hai qua trlnh song song trong bai loan. Qua trlnh xac djnh
va qua trlnh ng~u nhien. Voi qua trlnh xac dinh ta c6 th~ roi r1).cbling
cac phu'ong pIlar Galerkin. Con qua trlnh ngfiu nhien ta ap d\lI1gkhai
tri~n Karhunen-Loeve, trong do, dn thie't phai sii' dl,lI1gcac da thuc
chaos (d d1).ngghli tkh). Hai qua trlnh roi r1).c11Oanay la co sd cua ph§n
Iii'huu h1).nng~u nhien.
5.3.2/ Ap dung giai bai loan khuye'ch tan khl thai ba chi~u theo m6 hlnh
ng~u nhien.
Voi m6 hlnh tu'ong It! bai tmin ti~n djnh d ph§n tren, trong do gia thie't
~
them fling: v~n to'c gio v = (u, v, w) la qua trlnh ngfiu nhien, va do do

m~t dQ <p(x,y,z,t) cunfj tnti'lng ng/iu nhien.
Roi r1).cmi~n xac dinh va xflp xi trcn tung ph§n tll :
cp(x,y,z,t,B) = Ne(x,y,z)<P,,(t,B) voi <D,,(t,B)= ~<D(t,B)
chUng ta tho du'<;1cM phu'ong trlnh ma nghi~m la gia trj cua cp t1).inut
. .
ciia cac ph§n Iii': M<D(t,B) +K<D(t,B) = F(t,e)
Khai tri~n Karhunen-Loeve clIo d1).iIU(1ngngfiu nhien v~n to'c gio:
- - - AI . - (*)
V(x,y,,",/,B) - V(x,y,z,/) + Lc;k (B)JAk(/).fk(X,y,-,t)
kd
Ap dl,lngphu'ong phcip phfln IIIhull h~\l1ngiill lIllicH,ta co ph6 cua <bet)
co d1).ng:
[
AI
]
-1
[
AI
]
-"
S<I><I>«(») = J + t;;kR~k' SI'I,«(v) J + h;k Ril)
(Chi sO'H th~ hi9n bie'n d6i Hermite)
ho~c la sii'd\lI1gkhai tri~n Neumann clIo to<intll ngu'<;1c, ta co:
if) if)
S'f>Ij'(lV)= II(-I);+i[;kR~k)r SI",(OJ)[;kR~k)r
;=1}=I
23
5.3.3/ LSp-~t~IJJgJJnIUvii nJ,;h"Lb.iIi Iwi II_Y.UI ni 11biOn dilll~\{t1LyO IliOu nJ
cae tham s6 ng~u nhi6n- VIdu bili loan t[tlll:
Xct mQttffmmong eo cae di~u ki~n bien phu h<;1p.Gia sii'modul dan

IH1iella 1[111111\111()1qU:I 1I'111hIIgITuIIhi6n Gauss hai chi~lI v<1igii1 II"
trung blnh E va ham hi~p phudng sui C( xI' X2) dii bie't, h!e ngoai la
ti~n d~nh.Ap d\lllg phudng pIlar phffn tii' hUll h~n ng~ u nhien, ehia tffm
thilnh N phffn tii'tu giae, m6i phffn tii' eo 8 b~e It! do.
Sau khi tinh loan, V e6 th6 e6 eae khai tri6n :
r
U = 2:>i\jl;[{~,}],
k=1
eae Cj du<;letinh qua h~ phudng trlnh d~i s6 :
[~< c;\Wi[{.;J]lflj[{';/}]> l«(kJ, =< Wj[{,;,}]f >, .i = too.P.
5.3.4/ Ke't loan eua bili loan 3:
Sii' d\lJ1gkhai Ir6n Karhunen-Loeve vii da !IJlI'Chi\n lOi.1n d6 xfiy d~(ng
1Il(>IIIl(ihlnh cho plll(dng philp ph~n Iii' hl(u h:.ln ng~u nhiC.n se rfi'III\lI~n
lii)n eho cae bili loan cel hQe ngfJu nhi6n trong khi mOt viti phuong phap
mo phOng s6 eho qua trlnh ng~ u nhien 1ilrfft kIlo.
D~e bi~t, ehung toi nhffn m~nh r~ng, hffu he't cae eong thue trang
phu'(jng phap nily e6 Ih0 Linh toan hlnh IhlICIr6n l11:iyLinhIn(tk khi
ehuy6n sang cae ng{\n ngu IInh Imln s61ruy6n Ih()ng.
KET LU~N
V~ nguyen t5e bfft ky loan tii' loan hQe nao e6 <.:flutrue di~i sfj cling e6
th6 thJ,fehi~n du<;1ctr6n h<;d~li s6 may tinh.
Ung dl,lng eae giili Ihu~1 tinh loan hlnh thue, Iwiy tinh sc giup la tlll,l'e
hi9n mOLkh6i lu<;lngIOnnhung llnh loan, bie'n d6i de bi6u lhuc loan
hQe lrude khi dua vao l~p lrlnh s6.
Ke't h<;lpcae giili thu~1 eua d~i s6 may tinh vii giiii Ihu~1 sfj II'uy6n (htJng
(a sc xay dJ,fllgdu<;!e111(>1h<;clllMng 1I'111hHllh Imlll ({illgquill giiii 111('(
lOp cae bili loan cd hQe .
Cfn nhffn m~nh r~ng: l~p tdnh tinh loan hlnh thue eo kha Hang ehuy6n
d6i ke't qua sang cae ngon ngu truy~n th6ng (C, Fortran, ) do do, e6
th6 tie'n de'n moi truClng tJ,fdOng I~p Irlnh (aulo-eoding) eho cae bai loan

ed hQc.