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

79 Chuang5 : AP DT)NGL!P TRINH TINH TOAN HINH THDe VAo MOT SOBAI TOAN CO HOC 5.1Bili tmin 1: Tinh toaDhlnhthucchobili toaDbienkhuye'chtankhi thai theomohlnhbachi~u. Bflitminbien truy~nva khue'chtankhithai (xem[35],[361): TImrpE C2(0) thoa : C(p &p &p ocp (Olcp /::~<:J: 01rp- +u-+v-+w-+a-rn-I[ -+---;- -v-= f~ ~ ~ ~ 't" r ~J ~ ~JIA ax: 0' oz ax:- (1'- 0::;- trang Q x (O,T) * £)i~uki<$nbien' * Di~uki<$nd§u . . Dangbie'nphan . Voicaekhonggianham: v={V/E D( 0) : V/If =o} H =~1.2(01 va v = ~HJ(Q) uiy V/Ev, nhan phudngtrlnhtrenvdi u. richphantrenQ , r6i apdl:111geongthti'c GreentaduQc: r~dO+a(rp,fjI)=ffV/dOn Q trangd6 : f(&p orp c<oJ fa(rp,If/) = u--:::-+v- +\I" ;.. V/dO +(J rplf/dO + Q ex 0' (- Q f,/ o(poV/ co' forp olf/ f+J-l [--:::-:::- +--=-:-dO +\" ~ -:::- dO +vfJ rpV/dr 0 Q .ex ax: ('. n oz oz f" Tadide'nphatbi~ubie'nphaneuabairoan: Chofer) EL1(O,T,V ), rim rp(t)EL1(O.T.:O)n L"(O.T,H) vdi rpoEH, thoa : rp=0 tren f x (O,T) &p, tren fox(O,T)=fJrpoz orp tren fh x (O,T)-=0Oz rp(G) = rpo trong Q 80 (d:;t),1f/);.IX,.+a(tp(t),If/)= (f(t),If/)vXVI,"dlf/E V <p(G)= tpO Roi r<,lemi~nxaedinhva xa'pxi trenm6iph~ntll'([36])tadi thudu<;1eh<$phuong trlnhd<,lis6 xaedinheaegia tri cpt<,licaedi€m nut: . J1ct>+Kct>=F e=1 <"=1 lie va F =I '(,IFJ.. . F e=1 M M tron!!d6 . _VI ="'\'T.'m1.: l\.="'\'T'k.T~ ~, " ~ e " m, = fN'NdQ, e 0, va k =k +k +k +k +ke e, e, <", e, e, ke2= a fN' N dQe 0, k" =lN' N[(U,,)(~)+(v"t;)+(w,,{~)]dQ, k., ; PQ{(~n~)+(ZHZ)]dQ, k" = v1[(~n ~)]dQ, ke,=v/3fN' N drOer", va F: =fN' N dQe 0, caematr~nTeehilacaematr~ndinhvi (l~prap),khongdn phaixaedinh.Cae richphantrend~udu<;1ctinhtrencaeph~ntll'th1,icQt:, Khi tinhroantrenmay,ta chuy€nd6ichocaeph~ntll'quyehie'uOr.(tuonglingvoiph~ntll'th1,ic).Ne'ucae ph~ntll'th1,icla ph~ntll'kh6i Sailm~t,ph~ntll'quychie'ue6d<,lng: ChQneosa xa'p Xl : ! -l'-l'l~ -l..l.l: '.II , 1 ,. . '. ,., I ~---:----- -1.-1.-1 ' I 8-n-i-_u_", I ,I.'.-1.-1 p=(1 ~ 11 ~ ~1l 11c; ~C; ~11c;) (C6 th€ thayd6i d~dangtrongchuangtrinh) Ma tr~nJacobi cuaphepbie'nd6i nay: '~1.1.-1 " £)~t : Q= J-I ex cy e=- - e; c; 13; ex cy c=J=I - CTJ CTJ cry ox dy c=- - ct; at; ot; 81 (o~)ox B\, = Il/ O~V)oy (~) (~) B,=I(~:) (~~) va v =((UII) (VII) (WJ ) Khi d6 ta c6 : me= fN'.\ [dct(J)1dO,. n, k, = fN' X VQB:Idet(J)ldO,. n, k" =a J N' Vi det(J)1dO,. Q, k",+k",= fBe'Q'DQB" Idet(J)1dO,. 0, k'r = vfJ IN'.V J, d~d'7 fll' troll (1 do .b [ Jl 0 0 ] D= 0 Jl 0 0 0 v ( 0'0= 0=Ov) 2 O=m: &:C:- ) 2 [,nry rym:) 2 J, = 11 01;dry- 02;oOry T 02;dry - o~cry + ot: dry- 02;dry Xu dl:lilgthu vi~n(Package)FEMI ta c6 th~tie'nhanhroanb(>caetinhroanly thuye'td trend~di de'nke'tquacu6iclingla caematr?n me, ke ' Ce , Fe dd<;lng chuy~nd6i qua ngon ngu so truy~nthong(Fortran) ( 1t?P tin tren ilia) : >eq:=Diff(phi,t )+11*Diff(phi,x )+v*Diff(ph i,y)+w*Diff(phi,z)+sigma*phi- mu*(Diff(p hi,x$2)+Diff(phi,y$2))-nu *Diff(p hi,z$2)=f; "a"~, ( '3 ) la\ '0\ 13q:= l~ It 1+u -.; 1>.+V l;:",1» +W i. -:::-IP)+ G 1> - ~Lut) .jX v.,Y \ az -v ( ~q,I=j az2 .J ( " J( '-" lJ a<::. a<::. ax 2 1>+ 0-,2 1>/ >ini:= philgamma[1]]=0,Di.ff(phi[gamma[2J],::.)=beta*phi,Diff(plzi[gamma[3]],z)=0); ini :=:; =0, ~ 1> = ~1>,~ 1> =0 "f az"ll az "II '1 (2 (3 82 >read'FEM1.m' : >with(FEMI) : >ep:=bienphan(eq,ini); j ' j ' ;=1 'a a " a " 8P= r~t1Jdo.+ (u(ax(P)+ v (~ 1Jj + w (IE (P)) lV do. + (5J 1J lV do. +~f(a: ~)(a:'")+( ~~)dO +-i f(~~)(~\If )a'O + v ~fh aT 2~ J1' ltfdo. > P:.=[1,xi,eta,zeta,xi*eta,eta*zeta,xi*zeta,xi*eta*zetaJ; P:= [1,';,lJ,~,';lJ,lJ~,~';,';lJ~] >matran(ep, P , Lagrange.,8) ; m[e],k[e]luutrendia-->MMvaKK Ch£ngh~n,ffiQtdo~ncuat~ptinKK ( chuamatr~nKe) cod~ngd ngonngu Fortran): tl =dy**2 t3=dx**2 t4=nuy*tl*t3 t6=dz**2 t7=muy*t3*t6 t9=muy*t1*t6 tl0 =tl *t6 t12= tl0*u8*dx t13= t3*tl KK(1,l) = -(t35+t64)*t66*t70/864 KK(1,2) = (t72+t73)*t66*t70/864 KK(1,3) = (t77+t78)*t66*t70/864 KK( 1,4)= -(t82+t83)*t66*t70/864 KK(1,5) = -(t87+t88)*t66*t70/864 KK( 1,6)= (t92+t93)*t66*t70/864 KK(1,7) = (t97+t98)*t66*t70/864 KK( 1,8)= -(tl02+tl03)*t66*t70/864 KKr2,J) = -(tl07+tl08)*t66*t70/864 83 KK(2,2)=(tl12+tl13)*t66*t70/864 KK(2,3)=(tl17+tl18)*t66*t70/864 KK(2,4)=-(t122+t123)*t66*t70/864 KK(2,5)=-(t127+t128)*t66*t70/86 MQtvi d1,lC1,lth€: tinhto<lnchomi~nkh6nggiannhusau:(moi buckla 1m) M6 hinhtfnhbaitoankhuyechtan khf1000nut 49 10 E 0- il 0-x E H) ~ lmx9=9m I -/ il- I I 1 4011 -" 403:;z:::: 405 407 /./ // 410 4Q x .. 10 500 Ngu6nt<;1inut455.Ngu6nca'pkhi thaivaom6itruong(trongffi~tph~ng401-500) Haitruongh<;p: 1./Kh6ngcogi6 2.1C6giGffiQtchieutheophuongx \"~nt6c:O.2n1Js;O..+mJs:O.6m/s '/ -I , :1, / ./ / / ./ / / / / / / / / I 10 t y / / / v / 1/ v )(IV v ( V V '1n I)() 7n lnn ",- SV THAY 001 GIA TR!HAM <pTHEa THOI GIAN - va V N TOC GIG Boc tu do 10 291 - Nut so: 453 Bvac thol ion W=O w=0.2 w=0.4 w=0.6 24 5.55207 12.5541 19.6337 25.30451 1 5:57E-D1 5.57E-D1 5.57E-D1 5.57E-D1 25 5.609416 12.9021 20.2132125.88121 2 1.06372 1.13216 1.19896 1.26577 26 5.6615 13.2337 20.7564 26.3 3 1.525419 1.72139 1.9155142.11240 27 5.708796 13-.5491 21.2639 26.8337 4 1.945823 2.31994 2.69537 3.08037 28 5.751738 13.8489121.7365727.2153 5 2.32859 2.92382 3.5279 4.15258 29 5.79072 14.133422.1751 27.53821 6 2.677058 3.52946 4.403241 5.311481 30 5.826102 14.403122.5807 27.80604 7 2.994271 4.13371 5.311 6.53951 8 3.283004 4.73377 6.24412 7.81942'1 ........... 9 3.5457885.3272247.1925179.134641 I 10 3.784931 5.91193 8.149124 1O.4693 11 4.002538 6.48606 9.10697 11.80897 141 6.164285118.312 25.3834726.2734 1 12 4.200527 7.04807 10.0597 13,1398 142 6.164283 18.312 25.3835 26.27347: 13 4,380647 7.596641 11,0018414.4497 143 6.164281 18.312 25.3835 1 26.27352, 144 i 14 4.544493 8.130674 11.9282115.72781 6.16428 18.3123 25.3835 26.273 15 4.693519 8.649287 12.8344 16.9644 145 6.164279 18.3123325.3836326.2735. I 16 4.829048 9.15177 13.7168 18.15154 146 6.164277 18.3122 25.383 26.273621 17 4.95229 9.637604 14.5721 19.28224 147 6.164276 18.3122 25.3836 26.2736 18 5.064343 10.1063 15.397720.35107 148 6.164275 18.312L 25.383726.2736 19 5.166212 10.5578 16.1912 21.35377 149 6.164274 18.3121 25.3837 26.2736 20 5.258809 10.9918716.9511422.2872 150 6.164273 18.3121 25.3837426.2736 21 5.342968 11.4084 17.676 23.1496 22 5.419447 11.8075 18.365123.939 23 5.488938 12.189419.0177424,6578 100 Me;hinhtfnhbai toan khuyech tan khf3600nut 3030 57 1mx29=29m 3035 1800 TRlJONG HQP 1:MAT 1201-1800CHUA NGUON THAI vAo I\t'IOITRl.IC1NGT~INOT1480 BAITOANEX..t:1CGIAivd! cAc TRLeJNGH.?P: 1.KhOngc6 gi6 2 .C6 gl6phUongX.v<)nt6c :O.Amjs 1800 NGU6N - 1485 - 1487 T~NG HQP2:~ T 1201-1800CHUA4 NGLON THAI vAo MOlTRlfJNG T~IcAe NOr 1425-1427-1485-1487 BAtToAN Dl..1OCGlJ\I ~I TRI.JO>JG!-pp KHCNG cO Gl6 E to " ~ E 107 Nhanxetvake'tluanchob~iiloan1: Vdi cacchudngtrinhtinhloanhinhthuctrenmaytinh(l~ptrinhSymbolic) c6the'chophepgiai cacbaj loanr!t phuct~p,cacdi6uki~nyell cftucao(Cl , C 2 ...)vavdidQchinhxaccaomad d6khongthe'chQncacphftntitddngianduQc(vi d~v~tli~ucomposite,cdhQcphahuy...). Tit cacthu'vi~nchuyend1;lng(Packages)c6the'ti€n tI!dQnghoal~ptrInh (t~omangu6nchocacchu'dngtrinhgiaisO'). K€t quabaiLoanthayd6ithencaevi tringu6n,v~nt6cgi6.va otnhtinh nh~nth!y hoanloanhQply. C6 the'mdrQngvi dl;lthanhffiQttht!cnghi<%mtrenmay tlnh. 108 5.2Hilitoan2: Phudngphapphftntii'hUllh~nghHbai toaDbien ? ., vai v~tli~lldaDnhat-danghuang-dangnhi~t 5.2.1./TiDhcha"tcdhoct6D2QUatcuabaitmiDdaDnhot: Quilllijt llng xii ailnnh6'tclla vijtreind~tatrell caccdsdsall:[8l[25] . Tiend§ :giatritlicthaicliatensorlingsuatphI,!thu()cv2wroanbQlichsubien thienclia tensorbienJ~lng. . Tien d§ bQnhdtatd~n:slfphu[huQcCULlgiLltri tlic thaicuabien sC;thlinhat VaGgiLltri CULl biensO'thlihai lmngthaigiantniacdayGl((jCxacdinhnhamOt hamtrQngluQng,hamnayd~il11baaslfph~lthuQcgiiimlien tucvaocaeslfki~n trangquakhli k~tuthaidi~mLiangxettrdv€ truac.Tien denay th~hi~nrhea df c:;Id(C;ik,(f)) dt khi t, >t2>0bi~uthlicsau: d(C;lki(r)) '=1, 1=1. . Nguyen 19dQcl~plac dl;lng. . &(t) lahamlientl;lctheothoigian. Phuongtdnhcobandn thietd~gi~iibairoanlingsuat-biend~ngtlfatInhtuyen tinhcho\.~th~batd~nghuangl3.baphuongtdnhcanbang: a(5 if + F, = 0 ax J (5.2.1) trangd6 Fr la cacthanhph~ncliJ 'ector llfc kh<5i,sau quanh~chuy~nvi biend,:lllg I ~ l(Oll;,OU,(t)= -; --:- Gij ,; ax ax;- " (5.2.2) clingvaisauquanh~lingsuatbiend:;mg oc ~ (j if = fC ijkl (t - r, Xi) CE:kl d r 0 ar (5.2.3a) trangd6 giathietcachamling su{,biend~ngtri~tlieu khi bienthai gian t<D. TrangtruanghQpv~tdn d6ngchar.d~nghuangC""I(t) chiph~thuQct. Cijkl(t) : la modulh6iph~c(relaxationfunction). 109 00 (J = fe (t ) O£k!ij ijk! - r ~ d r 0 OT (5.2.3b) Vie'tlC;li(5.2.3) rhea trong truongh<;1pd~nghuang: "' f ( )0£ (T) f oGier)(Jij(t)=5ijAt-r ~ dr+2JJL(t-r) : dT 0 OT 0 or (5.2.4) hayvie't lC;li(5.2'-+)rhea modun E(t)va "(I) "' f" E(t - r) ( ) dt- r) )}0";;(1)= ( v( -)) £,,(r+ "){ )c5ii£kk(r ir0 1+ I-, 1-..;.lt-r (5.2.5) trangd6: va .E(t) ,u(t)=2(1+v(t)) E(t).v(t) ;.(t) = (1+v(t)XI- 2v(t)) (5.2.6) (5.2.7) Di;lngkhaccuaquailh~Ung su:1t-bie'ndq.ng: YO E:ij = fSijk! (t - r) Cc: k! dr 0 cr (5.2.8) Clingnhuhamh6iphl,lcvahamchaych~mS ijk! (t) d~ctrungchocactinhchatco hQCcua v~tli~u.Vai truongh<;1pd~nghuangtac6 dC;lngt6ngquatcuatensorhC;lng 4d~nghuang: (jk/(t)=~[c;(t)-C (t)~~5k/+~[~(t)K5;k5j1+c;CjJj ..;. (5.2.9) B~t(5.2.9) VaG(5.2.3) ta thudu<;1c: :c , oe Sij = fC1 (t - ")~ dr 0 o. (5.2.10) 00 ~ (]' kk = fC 2 (t - :,)c~,dr 0 c:' (5.2.11) trangd6: 1 Sf;=OJ, -::, I5kk0; .J i tensor l~ch ling suat) (5.2.12) 110 1 eij =G ij - ""3G kk 5'1 (tensorlc%chbie'nd~ng). (5.2.13) ! vatac6 h~thlicchaych;%mgiii'abiend~ngva lingsuattu'dngling: X f ( ) as eij = J1 t - r --Ldr 0 or (5.2.14) YO fJ ( ) oakk Gkk = 2 t - r ---;::-dr 0 or (5.2.15) Trangd6: ~(t),~(t): hamh6i pht,lcvahamchaych;%mtu'dnglingvai tr~ngtheljbiend~ng tru'Qt. C2(t),J2(t):hamh6i pht,lcva,hamchaych;%mtu'dngling vai tr~ngthaibiend~ngthti tich. A.pphepbie'nd6i Laplacevao(5.2.10),(5.2.11),(5.2.14),(5.2.15): S =sC] eI) I) e =sJ ] S I) 1J (5.2.17) 0k =S~&kk Gkk= sJ2 Gkk Tu'dngtVnhu'ly thuyetdanhbi.caehamh6i pht,lcd£nghu'angkhi truQtthucintily vagiannd th~rich: J1(t)=~c](t) ; k(t)=XC2(r1 (5.2.18) So sanhyai (5.2.17)tathayrang:Khi thaymodundanhbi bangphepbiend6ide hamhbi pht,lctu'dngling trongde h~thliccualy thuye'tdanh6i tase thuduQcly thuyetdannhat. Ta c6cac bie'nd6i Laplacecua: ModunYoung 111 ~(s)= 3~'C2(s) = 9170l.kW ~+2C2(s) j.l(s)+3kW (5.2.19) h~soPoisson: ;;w- ~- ~ - 3fG)- 2;0) - s(~+ 2~) - ~u (.I')+ 3k(s)] (5.2.20) vaHamh6ipht,lc: m=~[c2(.I')-~] =k (.I')- y;P (.I') (5.2.21) Vdi v~trandannhdtthaythe(5.2.2)(5.2.4)vao(5.2.1)tathuduQc: ."' f - ( ) au, i (r) oo f ~ , ( ) ( ) ell k ki (r ) pt-rl dr+ At-;j.lt-r . dT+F=O 0 aT 0 aT (5.2.22) rheonguyen19dQcl~placdt,lngtac6th€ matacacchuy~nvi,biend~ng,vaung sU<ltnhusail: Ui(Xi,t)= uiO(xJ.u(t) E ij (.\: i , t )=E ~ (xJ. u(t) (j ij (xi , t )=(J" ~ (xi ). F (t) (5.2.23) f)~c6th€ tachbienduoid~ngt6ngquatnhutrenhc$soPoissondannhotphaila hangsotht;tc[45].f)i~unayc6th€ th<1yratll'phuongtrlnhdin b~lllgclingnhudi~u kic$ntuongthich. f)~tIIi (xi ,t) tll'(5.2.23)vao(5.2.22)vad~t~=0 tathayrangphuongtrlnhdin bangchi thoamankhi: A(t) -"- ,LL(t) =K)J(r ) trangd6:K=hangso A.pdt,lngcachc$thlic (5.2.20)va (5.2.21)vaodangthuctrenta timduQc: v (5)=~ 5 nghlala ( ) J v '\ V t =L- ! - =v =canst<s (5.2.24) 5.2.2.1V~motphuongphapghHbai toaDdfmMot tm-entinhd~ngnhiet: 5.2.2.1.1Ngllyenlj tztdngIlng:[25] Theonguyen19tuongling,nghiemcuabairoanbiendallnhotruyentinhc6 th€ thu duQctll'nghi~mcuabAiroanbi~ndanh6i.trangd6cachangsodanh6iduQcthay 112 b~ngcactm1ntlthampht,lthuQcthaigian(modunchungling Stittho~chamchay ch~m). Dl,l'atrennguyen19tuongling, taapphepbie'nd6i Laplacetheobie'nthaigian thl,l'ct~0 vaocacphuongtdnhcanb~ng,h~thlicCauchybi~uthiquailh~chuy~n vi- bie'nd~ng,dinhlu~tHooket6ngquatvacacdi€u kit$nbien. ClCphuongtrinhthuduQcchliahamJnh cilaphepbiend6iLaplacehoanloan tu'dngtt,tv€ m~tlOanhqcvai caephuongtdnhcua19rhuye'tGanhM ruye'ntinh.TJ n6ieacphuongtdnhnaymorabairoanGanh6i ke'thQp. D~thuduQcnghi~mcuabai lOanbiendan-nhattaphaiapphepbiend6i Laplace nguQcvaonghi~mcuabai loandanh6i ke'thQP. 5.2.2.2/Phat biill biLiloan biendiLnnhottllytn tinh: Phuongtdnheobanc~nthietd~giai bai loan ling SU<lt-bie'nd~ngtt,tatInhtuyen tinhehov~tth~btt d£nghuanglabaphuongtrlnhcanb~ng: C(J"if F. =0-+ I cx) xem (5.2.1) rrangd6 F; la caethanhph~ncuavectorIt,tekh6i,Sailquailh~ehuy~nvi biend~ng - ] ( aui aui )E. -- -+- ). 2 ax. ax.I I xem (5.2.2) clingvai sau quailh~lingStittbiend~ng cY:! J ( ) a[; k! iJ ij = C ijk! t - 7:,x! _8 d7: 0 7: xem (5.2.3a) Di€u ki~nbiendannhat: /I. =U. ,X E fI' " \5.2.25) G ;1/=T;.x; EfT \5.2.26) rrangd6 l1j: eosinehihuangcuavectorphapdonvi huangrangoaicuam~tbienf. Theo[9]h~thlie(5.2.3a)e6th~viet:- - cr =Cijk/&k/ (5.2.27) 113 ijk/ =P .Cijk/ (5.2.28) ..anhcila bie'nthaigianthvct quaphepbie'nd6i Laplace. ijk: la anhcila ~jk quaphepbiend6i Laplace. .2.2.3./Bizi toandizllhiJi ktt h{lp: uabiend6i LaplacetathuduQcbai roanuanh6i ke'thQptit(5.2.1).(5.2.2): cau+FI=O ax) (5.2.29) ~=~ [ au . ; + aui I lj } a a I- X i XI ) (5.2 .30) Vabie'nd6i eae di~uki<$nbien: - - UI=Vi tren C (5.2.31) O".dn=Lx ErriJ J I I (5.2.32) Trangd6gia thie'trangcac m~tbienkh6ngph1,lthuQcthaigian( ehuy~nvi be). Phuongtrinh(5.2.29).(5.2.30),(5.2.31),(5.2.32)clingvoi quailh<$ung sua't-bien d<;1ng(5.2.27),(5.2.28)la Svtu'angduanghlnhthuGvoibairoandanh6itrenmi~n hinh Qcgi6ngnhau.chilltaGd1,lngchuy~nvi ui =ui(Xj'p) .lve m~tr, =r,(x;,p) val1,1'ckh6i F;=F;(x .p), tagQid6la bairoandanh6ikefhQp. Khibienkh6nggian\-abie'nthaigiane6th~bi~udi€n banghamphanly bien: U; =u;o(XjP?V), F; =F;°(..JF;I/(t), ... (5.2.33) [hIsvphanb6kh6nggiancilachuy~nvi vaIvctaGd1,lngtrongbairoandanh6iket hQpgi6ngnhu'bai roanaan-nhotg6c.Theo (5.2.24)ai~ukiend~tachbie'nnhutren la: v(t) =const 114 5.2.3.Giai bai toaDta'mchii nhiit. vat lieu daDnhot b~ngphu'dDgphap giai tich. (TAMCH~UTAl PHAN BO VUONG GOC VOl MA.T PHANG GIU A TAM.) Vi€t l(,1icaephuongtrlnh(5.2.29),(5.2.30),(5.2.31)chobai loanta-mmong hlnhchITnh~tb~d~yh chili lionvoi h,tcphanb6 q(x,y): °j-L., --,"" //0 //- . b//I T /',/ / /-1 / /."." I ./ /' / I"' / ./ / , I I /( ./ T -'- 1\/16hTnngiGi bai toan tam kfchthL16ca x b u' I / / xa -~------- So do bieu dien each tfnhu'rhea Zva C( NhaDxet (XemhlDh) : Chuy~ndichu',v' t(,1idi~mtrenlOpsongsongva each m~tgiITamQtkhoangz , t(,1ithaidi~mt -do51,1'u6ncongta-mt(,10ra- co d(,1ng: ? u'=-z sin(a(r))::::-zrg(a(t))=-z o-waxel (5.2.34) ? v'=-2sin(a(t))::::- ::tg(a(t))=-z o-w Oyot , OW Laplace(u ) =- zp-::;-,ox - ow Laplace(v')=-zPa;: 115 Tlinh~nxettrenke'th<;ip(5.230)chota: a,- ",,- ,--w - o-w - a-w E:tr=-zp ;- , fi,"V=-::p :;- , fin =-zp- . o."c" oy" oxay (5.2.35) Tll (5.2.28)chota: - "-E I (o:~' 0";'\(Jxx=-p :;-::~ +v~ I 1- v- \ 0- ~v- ) (5.2.36) - "- I ( o"~' 01~ ] (J =-p E-z -+v- XI' I-v" m': ax" ~=-p"E - I ( 02~\,\ xv -7 I. I _-I +V axm'! Vie'tl:;1i(5.2.29)chobai roanta'm(lingsua'tph~ngsuyrQng)[IJ taco: a~ or", o~~+~+~=O Ox av oz (5.2.37) ar", aCT\1 a~ ~ + '-'--+~ =0 Ox 01' oz arx. oil~ oa--- 0~+--'-+--=- = Ox Oy oz f)ua(5.2.37)VflO(5.2.36)chota: a I '"' ~"- ",1-, I 0'x= 2-E 0 cwo W I 2-E-= p -z- -+- =P -7-L1 W az I-v" ax ex2 ay") I-V2 -ax I (5.2.38) 8",= ,- I 0 82; 02~' ,- 1 0 --'- =p' E :;-Z_. --;- +---;- = P' E --;- z - ~I).V oZ 1- v- or or ay') I-v- Oy Tichphan(5.2.38)rheaz: - ,- I Z1 a - ,.-=p"E ---~,W+Ip (X,1') '- I-y- 2 ax . .. (5.2.39) - ,- I Z1 a - r,==P" E , -~~I w+ If/(x,y) . 1- y- 2 oy 117 1 - 1 h3 a - Q,=-p- E I-v212 axLl1W => -8 -QI =-D-:::-~,wox - 1 h3 a A -;1 --0, 0 =-p- E~ 12~'-2 1- V uy => -8 - Q2=-D 0' ~IW T d' D 1 E 1 h3 ( . I' d A' kh' A' rang 0: =- p- 1- V2 12 gQI a Q cling tn,1 I lion) (5.2.47) Phu'dngtrinh(5.2.46)c6 d<;ing: ( ~4- 4- --4-\ - a wow c w - - D --:! +2 ~ 2 1 +~ J =DI1/11 H' =0 ax ox CJy- c~v (5.2.48) Phu'dngtrinhtrenIa phu'dngt:rinhSophie-Germaintrangbairoandanh6ichophep xacdinhdQvangcuabairoandanh6ike'th<;1p. TImnghi~mphu'dngtrinh(5.2.46)voi caedi~'uki~nbien (ta'mc6d<;inghlnhchi1'nh~t0:$x:$ a va 0.::::y:$ b): }-- 0 ' a-W 0 ' . 0 ,w= va - = VOlx= vax=a ax2 (5.2.49) - 0 ' a2; 0 ,. 0 , b w= va 0'2 = VOly= vay= sadl;1ngphu'dngphapNavie- (dungchu6ilu<;1nggiackep): f)i~uki~nbien(5.2.49)serheane'utabi~uthide)vangquachu6iFourier: x '" - ~ '" ' milX . mrv W =L L- amll SIn - SIn ---::.- 111=111=' a b (5.2.50) Takhaitri~nhamrai trQngq(x,y)thanhchu6iFourier: 00 '" q(x,y)=~~ q sin m1lX . I1tryLL 11111 SIn- m=ilI=i a b trangdo theo[1]gia tri gm'n' Ii: 4 a b " ff ( ) ' m ilX . 11m' .dqm'II'= - q x,y SIn-SIn ~cl..:r: y ab 00 a b (5,2.51) vanghi~mcuabairoanIa: L 118 W- - ~~ q11l1l 0 mm 0 n7r)l- ~L..J sm-sm- 11/:1//:1 ..- ( m~ n~ J 2 a b JT:D -+- a2 b~ (5.2.52) Trrtullghdp q(x,V) =qf)=cons!: 4QoUf h f 0 mm 0 nJrY dxd 16qoqlllll== - Sln-Sln- y==.,-ab 0 0 a b If-mn (5.2.53) - 16Qo 'f- f' 1 . mTCX. nm' 11'=---=- L L SIn-SIn---'-ir"D -,. -' ( ~ ~ J ~ a b /1/-1.0.0..//-1.0.5 m, n mnl- T - \. a~ b" (5.2.54) Ta chi e~ntinh ~khiffi, n lanhungs61e.Vdi taingoaiphanb6d~u,ffi~t giuakhiu6nphaid6i xling,caes6h~ngffi,n chaotudnglingydi de)yangkhong d6ixungnenchungphaib~ngkhong. Thea nguyen19tudnglingchungtase thuduQcloi giai cuabai tmindan nhOttuloi giai cuabai tmindanh6ik€t hQpb~ngphepbiend6i LaplacenguQc. w=Laplace-'6~) 11'=Laplace -I r 16qo : Jr6 D I I " I '" I'" I 0 111TCX 0 n:ry ( ' , J "Sin-Sin - b/1/:1.3.5...//:1.3.5 111- n , - a mn -+- a~ b" (5.2.55) voi(5.2.47)chu 9r~ng: ( 1 J 1 ( 12(1- v2) ] ( 1 J 12(1- v2) Laplace-I = ==Laplace- - , ==Laplace-I ---= ---:- D p2 EhJ p~E h- D~t: WI ==Lanlace -J! ~ )r , p2 E (5.2.56) f ! ,, -i12CI-\o:)16Qo f' ~ 1 . !l1TCX,.n:ry11'0_ I ' 13 6 L L. ( ' , \ ~ Sin sin b1 If -'- -,- - - am-l.o.J...//-L.J ,In }/ )m11\--'--. . a"- . b~\ \ (5.2.57) 119 taco: W=WIWO (5.2.58) Vidu: Gia SItvq.fli~udilll Ilhflt !nOfa Ilhztsan: v=canst.. ( 1 J 1 --'71 E =Eo 1- 2e 2 (5.2.59) Giiiibai loan(theo(5.2.58»: E=Laplace(E)=E{ ~- 2(P~~)] '7' WI=Laplace -I ( ~ J - 2e-2 coshTJ! ,- - " p-E -Eo Nghi~mcuabailoandannhot(theov~tli~u(5.2.59)): w= ( 2e-f cosh~t J ~r12(1~V')16Qo - Eo l 11' Jr6 x L x L I . 111m:. 111rV] ( 1112 n= ) 2 sm~smb I mn ,--+~ )a- b- m=U.5...//=U.5,.. (_'E- \ f ' W=I2e 2 coshTJl I 12(1- v-) 16q0 ~ 2) E h3 .".6 / l 0 '" ~ ~ 1 . mm . "1ZY 1 '-' L. ,sm-sm- i I " I " ( ' ' J - a b ' In= .0.)...11=.0.0 111- 11- , mill - + - j ~a2 b2 ) W=WTJWE (5.2.60) trongdo: ( _'7t t JW1/= 2e .::cosh ~ (5.2.61) 12(l-~':)I6qo ~ ~ 1 . mm . mry W =1 ~ ~ Sin-Sin- E E /1: ,76 -' -' . ( 2 2 J 2 a b 0 1n-1.~.5...II-I.~.) m 11 mil - +- al b2 (5.2.62) 120 2 1.8 \\ \, \. '\ \\ \\ \ \. '\ "",",,-,. 1.6 1.4 1.2 """"", -""'..,., .~ -'- '- '--' ~- 1 i:J 2 4 6 8 10 ( - "t t JD6 chicuahambien&;mgnh6tw" = 2e 2 cosh ~ ' v6i: 17=0.5 5.2.4.Giai bili tmindiln-nhot tllye'ntinh binS!phu'dngphapphftntti'hull hall: Qua vi d~trentathefynghit$mrimduQcla nghit$mcila b~lito£lntefmph~ng. Trangthl;tcte',s6cacbairoancdhc,cod~ngnghit$mgiiiirich,Ia reftit. Cachgiii trenkh6ngnhungkh6ngmarQngchonhi~ubairoanmaconkh6ngth~l~ptrinh hinhthuc. (j daychungt6i xin giai thit$uffiQtphu'dngphapgiii khat6ngquatcho nhi~ubai roandannhat,d~nghu'ang,d~ngnhi~tb~ngcachl~ptrinhph~ntd'huu h'.lnke'thQpnguyenly tu'dngung. Tatimchuy~n\'i theod~ngham: u(t) =Uo . u"(t) (5.2.63) Bie'nd6i Laplace: u(p) =Uo . u,,(p) f;:-;.~. dV =J~t . f v . dV + f;'f rdf r I' r (5.2.64) (5.2.65) Trangdo: (j"t =[~11, (j :.::, (]' 33, (j 12, (j 23' (j 13] (5.2.66) 121 ~t= [~11' {;22 ,{;33 ' {;12,{;23 , {;13] (5.2.67) - (J=C'{; xem (5.2.27) C=p.C xem (5.2.28) C=Laplace (C) c= E(t}C(1+v)(1-21')1 (xemchtidng1-19 thuye'tdanh6i) (5.2.68) Quanh~bie'nd£!.ngva chuySnvi (xemchtidng1): - [;=Du (D tmlntU'vi phan trongcong thuc Cauchy) (5.2.69) Apd~ngphtidngphap ph~ntU'hUll h£!.n,ta xet xa'pXl tren m()tphan tU'e ba'tky, tli' (5.2.64)ta suy ra chuySn vi Ue(t) tren ph~ntU'1a: ldt)= UOe.ul] (t) bie'nd6iLaplace====> Ue(p) =uoe.ul](p) (5.2.70) !u'u9ding: . UOe: 1achuy~nvi trenph~ntU',vala thanhph~nchIph~thuQcbie'nkhong gian.khongph~thuQcbie'nthaigian.Daychinhla thanhph~nchuy~nvi danh6icuaph~ntU'. . UlJ(t): lahamchuy~nvi theathaigian,vala thanhph~nkh6ngph~thuQc bie'nkhonggian.Daychinhla thanhph~nchuy~nvi nhdtcuav~tran. GQi: . Ue(t)1achuy~nvi t£!.inutcuaph~ntU'e t£!.ithaidi~mt.Anh cuano quabie'n d6iLaplacela U,(p). . UOela gia tri chuyenvi danh6it£!.inutcuaph~ntU'e t£!.ithaidi~mt. . Se(t) : la hamxapXlchuySnvi quagia tq chuySnvi nut- trenph~ntU' e. Tll(5.2.70),(5.2.69),(5.2.27)suyra : ue(p)=Se. Ue(p) = Se . UOe. Ul7(p) (5.2.71) 122 fie= Due = DS eUOeU'l (5.2.72) (Je=C . fie=CDSeUOeU'l (5.2.73) Vie'tl<;iinguyen19congao chobai lOandaTIh6i ke'thQp,vdi nela sophftntli'cua bailoan [4]: n, L F;U~eS;D1C{DSeUoeu'ldV= e=\ v~ ne - - n, - - =L fll"U~eS:~.dV+L fU"U~eS:/fdr e=1I'e e=l fe (5.2.74) D~t: R =DS" " Va: la vectorchuy€n vi loanh~- la matr~nh~ng. ~:Matr~ndinhvicuaphftntli'e lIenloanh~. tasuyra : UOe= I;Uo TU(5.2.74)ta co: n IfUTJU~I:'R~C'R):UouTJdV=IF:U~T/S~/vdV +:tfuTJU~Te'S;frdr (5.2.75) ,=1V, e=!V,. e=1r, Nh?nxet rang: u,?' U~co th€ duara ngoai da'utich phan va da'ut6ng VI chung khongph\!thuQcto<;idQkhonggian.Ta khli'cacthanhphftntuongling a2 ve': nc ~ - n, - n, - L. fTe{R~C'ReTeUoll"dV= :L fTelS;lvdV + I fT/S;frdf e=!Ve e=1Ve e=1r, (5.2.76) Bie'nd6i Laplace nguQc(5.2.76), chti9r~ngchico C{,u'?'Iv, Ir lachilabie'np: (kyhi~uLaplace -I (I) 1abie'nd6i1ap1acenguQchamcuahamf) tT:[lR;(LaPlacdc'.uJ'jR,dr- T,Uo= 123 11. II. =I fT/S;J;,dV+ ! fT/S;frdr e=1,'" e=1f" (5.2.77) Ky hit%u: Ke= fR~(Laplace-l(Ct.;;))RedVlamatr~ndQclingphgnnY. v, (5.2.78) 11" K="T1.K .1'L c e e (5.2.79) e=l 11. 11. F=! fTetS~f,;dV+ I fT/S~ffdr e=ll', e=1f,. (5.2.80) tadi denbi~uthlicsail: K .U 0 =F (5.2.81) Giaiht%phu'dngtrlnhdi.;lituyentrentatimdu'QcVa la cacchuy~nvi daTIh6i ti.;licac nut.Do do:nghit%mdaTInhdtla : u(t) =Ua ,ulJ(t) (5.2.82) £)~timham IITJ (t) tahillyden giathietValanghit%mdaTIh6i,tZfdosuyralil ma trQnKphil;lil matrijll hlillgdolva;t.Daychinhlamatr~ndQclingdanh6i.£)i~u naycho ta ht%thuc sail : (Laplace-l(Ct.uJ) phailamatr~nhangs6d6ivdit,nghlala: (5.2.83) - ) A (e' .u, = p trongdoA : lamatr~nhangs6d6iydip ,pia anhcua t qua bie'nd6i Laplace. 124 Xetmothiii toaDenth~: z 9 xO.4m=3.6m 61 / / 51" .'<:,;. // 31' '\ "" \ \ '\ ~+Q~ ;~~\ \~"-~, y (~.' A ~ ~, , K MO HINH GIAI BAI TOAN TAM 70 x ~ \ \\". >(~" M~,TKHA,O \ \ "" SAT\,~. \ \ \\ 10 Tamday: O.O3m VatlieuEo=2xlOE8KN/m2 TOitrQngq=-lOKn/m2 Trztdnghdv 1: ( I J 1 --'If E(t)= Eo 1- ie 2 (giong vi dl,ltren) (5.2.84)) Til (5.2.68)vadi~uki~n(5.2.24)tasuyra : C E(t) rt- (l+v)(l-2v)~ vdiv=canst (5.2.85) rhea(5.2.28)thl Ct =[p . Laplace (E(t ))] [(l+V)(\-2V)C: ] (5.2.86) X6th~thuc: Laplace-I ((if.u,J vdinh~nxet:matr~nKe(trong(5.2.78))phaiIii matr~ndoeun2daDh6iphftn tll',tatha'yr~ng: Laplace-I ((if.Laplace(U7(p)))= ( - . - ))~ 1 C' =[Laplace-lp.£lp)'u)p 'J(1-vXl-2v) I (5.2.87) 125 thi: [Laplace-l(p.E(P). up(P»)J=Eo Laplace-l{Et.Laplace{UT/(P»))=Eo (I+VXI-2V) c; (5.2.88)DeU (5.2.89) BaychinhIa matr~nCot,la matr~ndanh6iv~tli~u.Tit dotha'ydudedingK TITbi€u thuc(5.2.88)tafundu'<Jchambitn d(;lngnhot: up (1)=1+e-l1t (5.2.90) Vanghi~mcuabai loanla: u(t)=Uo(1+e-l1/) Trongd6UoIakttquacua(5.2.81), d6lacaechuy€nvi dotinhcha'tdaDh6iv~t li~uvalachuy€nvi t(;licaenut. E(I)=E.(l- ~e-",)00TH!E(t)THEOTHill GIAN,vUicaeh? sd TJ E(t)=y.Eo 1.1 2 4 6 8 10 12 14 16 Thai giant 11= 1.0 11=0.7 11=0.5 11=0.2 0.9 0.8 0.1 0.6 0.5 0 126 V6 thjbiend~ngnhdtWTJ(t) , theothoigianva h~8611 E(t)=E.(l- ~e-~'") w =1+e-111 11 V~tli~uco tinhchiftOOdt: Ham Wll (1)It!: W,/t) 2 3 .. 5 6 7 8 9 10 1 1 Thoi giant 11=0.2 11=0.5 11=0.7 11=1.0 KET QuA BAI TOAN(tru'itnghf/p1): TINH TAM, v~T LItU DANNHOT BANGPHUONGPHAP PTHH 00 tillchuyenviZtheotholg/ont NUT ODOE-t{)O 31 33 34 35 36 37 38 40 ~ -5.00E-D6> ~'" 1i -1.00E-D4 -1.60E-D4 ------------------ t=10 t=8 t= t=4 t=2 t=0 MAT CAT GIUATAM QUAcAc NUT31-40- 2.15 1.95 1.75 1.55 1.35 1 .1 5 0.95 0 O.ooE+OO ~ -5.~> c:. i -l.t1OE-O4u -1.5OE-Gt o.aEtOO ~ c.>- ~ -l.~ 127 H~s6nMt eta=0.2( Xems6li~uph&nph~l~c) DoltIichuyenviZIheoIhoigtanI NUT 31 4034 35 36 37 t= 10 t= 8 t= 6 t= 4 t= 2 t= 0 M~T CAT GIUA TAM QUA'CAC NUT 31-40- H~s6OOoteta=0.5( Xems6li~uph&nph~l~c) Dolfi c:I1u\8IwZIMIoIx8 gemt NUl" 31 4)34 35 36 37 t=10 t= 8 t= 6 t= 4 t= 2 t= 0 M~TCAT GIUATAM QUA'CACNUT31-40- H~s6OOoteta=1.0(Xems6li~uph&nph~l~c) 128 ( ) ( 1 -1-111\ TnilI1lghdp2: ViiI Ji?uc6: E" =Eo 1+2:e 2 ) DO TIQ E(t) THEO THOI GIAN, YcHcaeh~S611 E(t) =Y x Eo 1.6 1.5 1.4 1.3 1.2 1.1 0.9 0 2 3 4 5 6 7 8 9 10 11 11=0.2 11=0.5 11=0.7 =1.0 D6tbjbitn dfngnhdtW"(I). theothaigianvaht;sf)11 1 (171) HamW17(t)Ui:wl1(t)=l-3"e3 WJt) 1.05 0.95 0.9 0.85 0.8 0.75 0.7 0.85 0 2 3 4 5 7 11=1.0 11=0.7 11=0.5 =0.2 --, I 12 13 14 15 16 . Thoigiant 9 10 11 Thoi giant 129 K ."" , K A A" , KETQUA BAI TOAN (trli<tnghejp2): TINHTAM, V~TL~UDANNHOT BANG PHUONG PnAP PTHH Do tIll elu,yoll vlZ thoo thol gmll I NUT O.OOE.OO, .,.00E-OS3: 36 3731 34 35 30 N -2.00E-OS -s ~ -3 OOE-O5 } .. -4.00E-Q5 -S_OOE-OS -b_OOE-OS -,--.~-~----- t=O t= 2 t= 4 t= 6 t= 8 t= 10 MAT CAT GIUA TAM QUA CAC NUT 31-40- He.s6nhoteta=0.2(Xerns6lie,uph~nph~l~c) DothichuyenviZtheothoigiant O.OOE+OO -J.Q()8)5 N -2.00E-05 ~ -3.00E-05. ?; -4.00E-05r. u -5.00E-05 -5.OOE-05 -7_OOE-OS~-_._------_._-------- ---- --------- t=O t= 2 t= 4 t= 6 t= 8 t= 10 MAT CAT GIUA TAM QUA CAC NUT 31-40- He.s6nhoteta=0.5(Xerns6lie.uph~nph~l~c) 130 Do IhI c:hIyen 111Z Iheo thai ~ t NUT 0.00800 -1.OIE-OS N -2.0IE-ffi "> -3.0IE-ffii ~ -4.OIE-OS.&:; u -6.OIE-OS ~.OIE-ffi -7.0IE-ffi 1=0 t= 2 t= 4 t= 6 t= 8 t= 10 M~T CAT GIUA TAM QUA CAC NUT 31-40- H~s6nhdteta=1.0( Xems6li~uph~nph~l~c) 5.2.5.Nhanxetvak6t loin CUBvi do2: Phu'dngphapph~ntithii'uh~nchobaitoaDdaDnhd~diinghu'dng,diingnhi~t giuptagiii du'<Jebaitoantu'dngd6it6ngquat.Co theapd~ngchocaebailoan mQtchi€u,haichi€u,bachi€u.Bi€u ki~nlit: h~s6Poissoncuav~tli~ugii thittla hAngs6deapd~ngphanly bitn. QualiJi giii tren,vi~exaedjnhbienthde u,.(t)di du'<Jcl~ptrlnhtinhtoaD hinhthue,ph~thuQcvitohamE~)t6ngquatcuav~tli~udaDnhdt. B6i vdi caenghi~mdaBh6i Do, t1mb~ngphu'dngphapph~ntithii'uh~n,ta d1icogiii thu~tl~ptrlnh[4],[6],[31]. Trongvi d~e~the()tren,tathiy hamehuylnvi u,.(t),trongliJi giii giii tich vaph~n1eualiJi giii b~ngphu'dngphapph~ntithii'uh~n,la nhu'nhau. 131 ~ Tinh tminbaitmincohQcngftunhienb~ngphuong phapphftntll'hUllh~nngftunhien. 5.3.Bai tmin 3: 5.3.1/.Phuong phapph~ntit hUllh~nngftunhien [21]: TrangCo HQc,cacbai roanv€ ngaunhienthuanglien quaildenmQth~phuong tdnhvi phantuyentinhvoi cach~sO'ngaunhien.Cac h~so'nayd~ctrungchocac tinheha'tcuah~, coth~hi~ulacaebienngaunhien,hayehinhxachonlacacqua trinhngaunhienvoi mQtCalltrucxacsuatnaodo.V€ m~ttminhQc, bairoanc6 d~ng: Au=f (5.3.1.1) trangd6Ala mQtroantuviphantuyentinhngaunhien, ula phanungngaulimen, f lakichdQngngaunhien MQtI;' thuyetroanhQcdftduQcxay dlfngchotruanghQp(5.3.1.1)c6d?ng Ita(cach~sO'ngaunhiencuaroantuA duQcgia thiet1acaequatdnh<5ntrang [29]),Wigiaila mQtquatdnhMarkovmaphanb6xacsuathoaphuongtdnh Fakker-Planck- mQtphuongtdnhd?ohamriengnoichungkhangconghi~mgiai tich. Tuongtlfphuongphapph~ntuhUllh?n (pp.pthh)ti€n dinh,trangd6 cacham so'duQcbi~udi~nbdimQt?Prair?ccacthamsO'g<5mgiatfihamvad?ohamcua not?icacdi~mnut,trangtruanghQpngaunhien,mQtquatrinhngaunhiencling du'Qcbi~udi~nbdi mQtt?PGemduQccacbienngaunhiennhamrai r?c hoaqua trinh.V€ m~tungd1,mg,matacacquatdnhngaunhientheoeachnay co th~thu du'QcIDQtbi~uthucph~ntuhUllh?ncuabai roanV?tI;'. 5.3.1.1Biiu di~llquatrzllhllgiiunhienvakhaitriill Karhzmell-Loeve MQtrongnhungketquaquailtrQngnhatcuaI;' thuyetquatrinhngaunhienla vi~e bi~udi~nph6cuacacquatrlnhngaunhien: OJ(x,e)= fg(x)d,LL(e) (5.3.1.2) 132 COJOJ (XI ,X~) = fg(x, )g(x~)(d/11(8)d.u~(8») (5.3.1.3) Fongd6 co(x,8)la mQtquatrlnhng~unhienva CCOCO(XI,X::)la hamhi~pphuong i (covariance)cuan6,g(x)la mQthamti~ndinh.va d~(8) la mQtdQdong~u hientn!cgiaoxacdinhtrenmQtcr-tniongcacbienc6ng~unhien. M!)tbiiu diinphilkhacla kIwitriin Karhlllien-Loevesaltday,ifdom!)tqua 'inkngdunhienm(x,B) cothi khaitriin theocacs6'h(PlgcllaIll!)tt{jphtfpdtlll 'l'(lCcacbitnngdllnhien tr~'cgiao.dov{jyco thi xelllllhlt Lal1l!)triJi rt:J-clla qua 'nhngdullhien. Qi OJ(x) la ky vQngcuaco(x,8)va, khid6 : OJ(x,B) =OJ(x) +f ;,,(B).[i: f,,(x) . ,,=0 (5.3.1.4) Itrongd6 Anvafn(x)lagiatqriengvavectoriengcuanhanC(x,,x::)nghlala nghi~mcuaphuongtrlnhrichphan: fC(XpX2)/,,(x)dYI =).,111,,(X2) D (5.3.1.5) Dlami~nxacdinhcuax, va {~n(8)}lacacbienng~unhienrhoa: = 0 va = Onm (5.3.1.6) TtldngtV'nhuco sdroanhQccuapp.pthhti~ndinh.Devijverva Kittler ([22])da:chI fatinhcha'tsai s6cV'cti~uva tinhcha'tduynha'teuakhaitri€n Karhunen- Loeve rheah~tQadQsuyrQnglacaevectoriengcuaOx, , X2) trong (5.3.1.5). 5.3.1.2.Giiii slfphztdngmnh richphiin (5.3.1.5) Vi~capdl:111gkhai tri€n Karhunen-Loevetrongpp.pthhph\,!thuQcVaGpt. richphan (5.3.1.5)(la mQtphuongtrlnhFredholmthuftnnharlo(;li2.Devijver. Kittler ([22]), vaGhanemR.G., SpanosP.D. ([40])da:chIra nghi~mgiai richtrongffiQts6 d<:1ng d~cbi~tcuahamm~tdQph6cuaquatrlnhng~unhien.d daychungtoixU'd\,!ng pp.Galerkind€ timnghi~mb~ngs6.CacvectoriengduQcrimquad.c hamxa'pXl hiCx) 133 fk (x) = N L di(k>hi(x) i=1 (5.3.1.7) oisais6tu'ongling dovi<$cxffP xiia : ~d;"'[IC(X"x,Jh;(X,Jdx, - ""h,(X,J]& N = (5.3.1.8) ~sais6benhfft,tu'dngdu'ongvaivi<$csais6naytn!cgiaovaikhonggianxffpXl: f:N, hi (x» =0 , i =1,..., N (5.3.1.9) ongd6 (.,.) chItichvohu'ang: fd/*> [ f [ fc(xl,X2)h;(X2)dX2 ] h/XI)dXI -A" fh;(X)h/X)dX ] =0 ,=1 D D D (5.3.1.10) f)~t: Cij= f fC(XI'X2)h;(X2)dx2hj(x,)dxl DD (5.3.1.11) Bij = fhi(x)hj(x)dx D Dij =dij va Aij =<5ij Ai khid6«5.3.10)trdthanh: CD=ABD (5.3.1.12) trong.doC=(Cij), B=(Bij),D=(Di)lacaematr~ntu«5.3.11).Tanh~ndu'Qc (5.3.12)labaitoangiatririengd(;1.is6blnhthu'ong. 5.3.1.3.Caedathuehiin lol}-Il(Chaos): GQi {~ih=l,oola t~pcaebiSnngfiunhienGausstnfcgiaova gQi rp la t~p h<jpcacdathuctrong{~i}i=l,00b~ckhongquap, va trt!cgiaovai r p-l. Khi d6rp du<jcgQila t~ph<;Jpcacda thach6nlo(lnb~cp. Kakutani(xem[40])dii chIrar~ng ffiQihamngfiunhien( blnhphu'ongkhatich) d~ucoth€ biSudi€n quacacdathuc h6nlo(;1.n: '" '" i, Ji (e) =a0r0 + ~ a; r1(;; (e» + ~ ~ aii r 2(;; (e),; i (e»L.., IlL.., L.., 12 , , ;,=1 i,=1;,=1 '" i, i, + L L L ai,i,i) r3(~il (O)'~i, (O)'~i) (0» i, =1. i, =I i)=1 134 ~ i, iz i) +L L L L ailiZi)i~ r (~i,(O)'~iZ(O)'~i)(O)'~i~(0)) i)=1 iZ=1 i)=1 i~=1 + (5.3.1.13) Dotinhtn!cgiaovaGausscuacacbienng~unhien~i , cacdathlicrpcoth€ tinh du'Qc[40],(cactinhtoancoth~tienhanhtu'angtvquatrlnhtrvcgiaoboaGramm- SmithtrongB,!i s6tuyentinh) : r II L (-IY L Il ~ik<Il~iI)nchan r=1I ;r(i"...,i.) k=1 l=r+1 r (j::'.,-,~. )=' rchullP ':>', 'n I r II L (-Iy-I L Il ~ik<Il~iI)nle r=1I ;r(i"...,i.) k=1 l=r+1 rchull (5.3.1.14) chungt6i dil l~ptrlnhsymbolictrenmaytinhkhi tinhtoancacdathlicnay( xem giaithu~tchuang4) : r .=1 O' r 1(~i):=~i r ( l'- J<. ) '=J<. J<. -0 2 ';>, , ':0. ' ':0, ':0. ,. z1 z2 z1 z2 z1z2 r ( J<. J<. " ) '=J<.J<. "_J<.o -" 0 -I< 0 3 ':0. ,':0, ,~, ' ':0. ':o.~. ':0. .. ~, .. 1". .. z1 z2 z3 z1 z2 z3 z1 z2z3 z2 z1z3 z3 z1z2 .. , . . . . . , 3 r3:=~1 -3~1 2 ~1 ~2- ~2 2 ~1 ~3- ~3 ~12~3- ~4 2 ~1~2 - ~1 .. . . . . . . . Tadanhs61,!ithlitvtrong((5.3.13)d~thu~nti~nchocachviet: 135 00 /-i(B) = L a;¥Ij[{q(B)}] ;=\ (5.3.1.15) 5.3.1.4Xii d{lngcac da th,lC h{jn lo{lntrongphltdngphap philn tit hllil h{ln ngdll nhien Co nhi~ucachxaydvngm6hlnhphc1ntii'hUllh<;lngftunhien( xii'd\mgpp. perturbation,khaitriSnNeuman,...[12]va[40]), songd daychungt6ixii'd\mgcac dathuchanlo<;lnVInhi~ubiSuthucvah~s6cothStinhtoanqual~ptrlnhsymbolic. Bai toan(5.3.1.1)co thSvi€t du'oid<;lng: [ L(x) +II(x,B)] u(x,B)=j(x,B) (5.3.1.16) trongdoL la toantii'vi phanti~ndinh,ilIa toantii'vi phancocach~s6la cacqua trlnhngftunhientrungblnhkh6ng.Kh6nglamgiamtinht6ngquattacothSxem Il(x,B)=a(x,fJ)R(x) trongdoR lamQtoantii'ti~ndinh.KhaitriSnKarhunen- Loevechoa(x,fJ)tadu'QC: AI [L(x)+ L: ~lIall(x)R(x)] u(x,B) = j(x,e) 11=1 (5.3.1.17) KhaitriSnKarhunen-Loevechou(x,fJ)taco : /, u(x,B) = L: ej X j(B)bj(x) j=1 (5.3.1.18) Sud\mgdathuchanlo<;lnchobi€n ngftunhienXj(8)(dllngdath,lCh{jnlo{lntt..irdi rlJchoam/)tqllatrinh ngdllnhien) : p lj(e) =. L: Xj()1fI j[{~II(e)}] ;=1 (5.3.1.19) Thay (5.3.19) VaG (5.3.18) ta du'QC: u(x,e) = p L: 1fI;[{~r}]d;(x) ;=0 (5.3.1.20) f trongdo: I I M P [L(x)+ ~';"a,,(x)R(x)] 'f, 'I'Ag.}]dj(x) = f(x,O) di (x) =}; xP} ej bj(X). Bay gio (5.3.17)trd thanh : (5.3.1.21) 136 Loi giai u(x,8)sexacdinhduQcn€u bi€t dj (x).B~ngpp.phfinti'thuuh£:lnti~ndinh, xtp Xl di(x) theocaehamcdsci gk(x) , k =1,..., N, N la s6b~ctl!docuah~ : N di(x) =I dki gk(X) k=I (5.3.1.22) Thay(5.3.1.22)vao(5.3.1.21)taduQc: p N [ M ]I ~~dkj [\f/j[{qJ]] L(X)gk(x) +t;qi(8)\f/){qr}]ai(X)R(x)gk(X) = l(x,8) (5.3.1.23) Nhanhaiv€ cua(5.3.1.23)voi gh(x) r6ilty tichphantrenmi~nxacdinhtaduQc: P N [ M ]~t;dtg[VlJ {~r}]]I L(X)gk(x)gh(x)dx+~~i(e)VlJ {~r}]lai(x)R(x)gk(x)gh(x)dx = ff(x,8)gh(X)dx , h =1,...,N D (5.3.1.24) D~t : Lkh =fL(X)gk (X)gh(x)dx D Rikh = fai(x)R(x)gk(X)gf1(x)dx /) va 1;, = ff(x)gh(X)dx I) (5.3.1.25) Khi d6(5.3.1.24)trCithanh : P N [ !vi ]~B VIj[{~r}]Lkh +-8 ~i(e)VI j[{~r}]Rikh dkh h = 1, ... , N = Ih (5.3.1.26) Nhanhaiv€ (5.3.1.24)voi \Pm[{~r}], lty kyvQnghaiv€ valu'uy r~ng: = 0m tanh~nduQC: N p]V M I L kh dk/ll +I I dkj I Rikl C ij/ll k=l j=O k=1 i=1 = <Ih , VI /II [{~r }]) h =1,.. .,N m=l,...,M (5.3.1.27). trongd6 : Cijm= , hay: [G+R]d=h (5.3.1.28) trongd6 : 137 Gmj = J mj L , M Rn!i=LCijm Rj i=1 va hm= Phu'dngtrinh (5.3.1.28)1ftmQth~phu'dngtrinh d<;1is6bInh thl.long,tim dl.l<;1ccae dki, duavao(5.3.1.24)va(5.3.1.22)taxacdinhduQcth€ hi~ncuau(x). 5.3.2.Ghii bili tminkhuye'chtankhithaibachi~utheoroohinhngfiunhien. Baitoanbientricuast!khuye'chtankhong ian3chi~u:(xem[35],[36]): ,TimrpEC2(Q) thoa: I orp orp orp orp ( 02rp 02rp J 02rp Of +u & +v 0; +w & +arp-fl &2 + 0;2 -D &2 =1 trong .ox(O,T) f)i~uki~nbien: rp=0 tren Tx (O,T) " orp a;=fJrp on I;)x (O,T) (5.3.2.1) Giatri d~u orp=0 & <;0(0)=<;00 on Ih x (O,T) In .0. - Q famQtkho'itrl;lco bienfa 00: b~m(ittrl;lT,ddyT{)(z=O),dlnhlh(z=h). cp(x,y,z,t)famgtdQkhfthaigayra batngu6nf(x,y,z,t)va ~=(u,v,w)fa vgntoegio, wii u,v,wfahamcua x,y,z,t. 51!t6nt~ivaduynhfftcuaWigiaicuabaitoan(5.3.2.1)dii(duQcchungminh trong[35],[36]) ~ Xemdingv~nt6cgio v =(u,v,w)vahamngu6n j(x,y,z,t) 1aquatrlnhng~unhien, vadodom~tde>rp(x,y,z,t)ciingtrllangngdunhien. D~ngbie'nphancuabaitoan(5.3.2.1) nhusau:(xem[36]) fZdo.+a«;O,If/)= fllf/do.Q n trangdo: f( orp orp orp J fa(rp,lf/)= U& +v .rit+W& If/do.+a rplf/do.+n vy n 138 f(orpOljl orpJ forpOljl f+j.1 -- +;;I, do.+D - :-1. do.+D/3 rpljldfo0 Ox Ox l/'y 0 & UL. f" Roir'.lcmiSn xac dinh va xa'pXl tren tungph~ntli' : rp(x,y,z,t,B) = Ne(x,y,z) <De(t,B) <De(t, B) = T: <D(t,B) chungtathuduQch~phuongtrlnhmanghi~mla gia tri cua rpt'.linutcuacacph~n tli': . M<D(t,B)+K<D(t,B)=F(t,B) (5.3.2.2) voi: /Ie M =L T;,ImeT;, e=1 /Ie K = L T;,IkeT;, e=] /Ie va F = "" T,I F,L < < e=1 (5.3.2.3) me = fNI N dQe 0, and k, = k, + k, + k" + k" + k,< " <2 ., .. <r k" = J N' [U(X,y,z,! '/1{ '.:)+v(x,y,Z,!Ii{:) +w(x,y,z,!,Ii{ '.:) ]dQ, ke2= (J fNI N dQe 0, k" =p)[(~n~)+(~)t~)]dQ, k" ~ vl[('.:')'( '.:') ]dQ, , k., ~ DfJ,[N' N dro, ' F, ~jN'j(x,y,z,/)dQ. ~ - Khi v =(u,v,w)va j(x,y,z,t) la quatrlnhngaunhien,chungtachIchuyde'nma tr~nke,vaF. GQiV lacuongdQcuav~nt6cgiG:V=II~" va aI, a2,a3lagoc ~ , giuav va cactn,lctQadQx, y, z , IanluQttaco: u=vcas(a] ) v =V cas( a2) w =Vcas(a3) (Cacgoc a] , a2,a3co thSxemla quatrlnhng~unhien,nophanb6gi6ngnhau rheacachuangchinh) V la truongng~unhien. KhaitriSnKarhunen-Loevechoquatrlnhnayvachohamngu6nj(x,y,z,t) : 139 ,iii V(x,y,z,t,B)=V(x,y,z,t)+Ic;k (B)~Ak(t)fk(x,y,z,t) k=1 M f(x,y,z,t,B) =f(x,y,z,t) +I c;k(B)~j3k(t)gk (X,y,z,t) hI trongdofk lavectori{~ngtu'onglingvoitririengAk'tu'ongtl!chogkvaak,khido chunglanghit%mcuaphu'ongtrinhrichphan: f C(X"Xl )In (Xl )dXt = An In (Xl) 12 vaiXj=(Xj'Yi'Zj) Vachungtaco th~stYdl,mgthutl,lCGalerkinchoWi giai 56. Vi V?ymatr?n ke,va Feco d~ng: k" ~ 1N'V (x,y,z,t) [ cos(a,{~)+cos(a,{:) +COS(a3{'.:') ]dQ, +I;k(B) ffi:fk(X,y,z,t)NI [cOS(al{eN) +cOS(a2)( av ) +COS(a3)( av )] dOe k=1 no ex 0' & M Fe= fN1](x,y,z,t)dOe +Ic;k (B) fN1 fi{gk(x,y,z,t)dOe n, k=1 no Phuongtrlnh(5.3.2.2)tro thanh: . M M M~(t,B)+K,~(t,e)+L;kKY) ~(t,e)=~ +L;kF2(k) k=1 k=1 (5.3.2.4) . lie vai: M nhu'trong(5.3.2.3)va K, =Ir.1fer.. . pI ke= ke,+ ke2+keJ+ke. +kef ' trong do : Eo, ~1N'V (X,y,z,t)[ cos(a,J( ::)+cos(a,>(:)+cos(a,)(~) ]dQo lie K(k) ="Tlj( T2 L eee e=1 k. =IF.f,(x,y,z,I)N'[cos(a,{-:) +cos(a,{:) +cos(a,{~) ]dQ, 140 /Ie F;=L r:1F'"I voi Fe = fNI ](x,y,z,t)dQe e=1 fl. /Ie va F;(k)=L r:1 F',,2 vOi Fe2 = fNI -J7i;gk(x,y,z,t) dQe e=1 fl. Giai phu'dngtrinh (5.3.2.4),chungtaco nghi<%mcua <D(t) Trang tru'onghQpham ngu6n Ia ti€n dinh , phu'dngtrlnh (5.3.2.2)co d,;l.llg(h<%s6 () duQclo(;lira chobi~uthucddngian): . M M<D(t)+K, <D(t)+L qkK?) <D(t)=F(t) k=l (5.3.2.5) Bie'nd6i Fouriercuaphu'dngtrinh(5.3.2.5): [iWM+K, +t;,K~" ] Cw)= FCw) hayla mQtd(;lngkhac: [H(W) +~~Kj')](W)~F(w) (5.3.2.6) voi: H(OJ)=iOJM+K, lamatr~nti€n dinh. Phudngtrinh(5.3.2.6)co th~vie'tla: [I +~~R!"](UJ) ~ P(UJ) trongdo: Rik)=H(OJ)-' KY> va P(m)=H(m)-'F(OJ) Dodoph6cua <D(t)co d(;lng: [ M ] -1 [ M ] -H S(OJ)= I +~qkRY> Spp(OJ)I +~qkRik> (Chi s6H th~hi<%nbie'nd6i Hermite) ho~cla sudl;lngkhaitri~nNeumanncholoantungu'Qc, taco: 00 00 S(CU)=LL(-I)i+)[qkRjk)f Spp(cu)[qkRjk)Y i=\ )=1 L 141 5.3.3.AP dQngtInhtminchobili tminv~HrAnbie'nd~ngvOiv~tli~ucocae thams6ngftunhH~n: B~ngeachxU'dvngkhaitri~nKarhunen-Loeved~hlnhthanhmahlnhchophu'dng phapphffntU'hUllhC].nngfiunhien,chophepl~ptrlnhtinhroanhlnhthlic(1~ptrlnh kyhi<%u,symbolicprogramming)chohffuh€t cacbi~uthlictru'ockhichuy~nd6icho cacHnhroans6truy~nth6ng. Taxetcacbairoank€t ca'ucocacthams6ngftunhien,ch~nghC].n0 dayta-xetcac v~thi codQcL£ngfamQtdqifu(1ngngtiunhien.Moduldanh6icuav~tli<%ud 'Qcgia thi€t,ch~nghC].n,lamQtquatrlnhngfiunhienGauss-3chi~uvoikyvQngE vaham hi<%pphu'dngsai(covariancefunction)C(Xj,X2).Tu'dngtl,l'nhu'phffntru'oc,tac6th~ giathi€t hamkichdQngla ti~ndinhhoi.icngfiunhien,tuynhiend~ddngiancho eachvi€t sauday,tahayxemkichdQngla ti~ndinh.Ta apdvngphu'dngphapphffn tithullhC].ngfiunhienchocacbai roanyoi dC].nghlnhhQcba'tky. GiasU'mQtta'mQ, du'QCchiathanhN phffntU'hUllhC].n.Th€ nangbi€n dC].ngdanh6i V cuam6iphffntU'De codC].ng: ve=~ fa-T(X)£'(X)dQe fl, (5.3.3.1) trongd6 a(X) va £'(X) la cacvectorlingsua'tva bi€n d,:lllg, X ky hi<%uthaycho (x,y,z). i./Voibairoandanh6ituy€ntinhtacocacquailh<%giualingsua'tvabi€n dC].ng: a = De(£,- G,,)+ a" aT = (ax ay az Txy Ty=Tzx) va £'T =(£' x £'y £'= rxy ryz rzx) De=Ee(X,B)Re (5.3.3.2) 142 1 C) C1 1 C) 1-1/ I 1 Re = re (1+,lie )(1- 2,lie) C2 C2 C2 ,lie c=- ) 1 - ,lie 1- 2,lie C2 = 2 (1- ,lie) :ongdo Re IamQtmatr~nti€n dinh, f.1eIah~s6Poissoncuarheintii'e, va ;;ex,B)la moduldanh6icuarheintii'e. i./Voi baitminlingStittph~ngtaco: a r = (a x a y T xy ) va & T =(&x & y rxy) C2 Re=C1IC2 (I-C2) 2 1 C1=1- ,lie2 C2 = ,lie ii./ Voi baitoanbi€n dangph~ngtacocaehethlictu'ongtu': 1- /I C = re I (1+,liJ (1- 2,liJ ,lie C2 = 1- ,lie IXii'dl:mgphuongphaprheintii'huuhl;ln(ti€n dinh) voi hamnQisuyif , vector. chuy€nvi u(X, B) co th€ bi€u di€n quacaechuy€n vi nut If( B) : u(X,B)=if(X) if( B) (5.3.3.3) quailh~giuabi€n dl;lngva chuy€n vi : &(X, B) =S u(X,B) trangdo S lamQt(matr~n)toantii'viphan, . (5.3.3.3)trothanh : &(X,B) = Be U(B) (5.3.3.4) trongdo Be=Sif . Thay(5.3.3.3),(5.3.3.4)vao (5.3.3.1)tadu'<Jc: 143 V e = ~ VeT fE (X , B)BeT R eBe do. eVe Qe vadodo : 1 N v =-2: VeT fE(X,B)BeTReBedo.eVe 2 e=1 Qe (5.3.3.5) Quanh~giii'avectorchuy6nvi diaphuongif vavectorchuy6nvi loancl,lcU thongquaphepbie'nd6i : if =LeU vavi the'tavie'tl(;li (5.3.3_.5)dudid(;lng: 1 TV=-U KU 2 trongdo N K =L Le1'Ke Le e=1 Ke = ~ fE(X ,B)Be T ReBe do.e Qe (5.3.3.6) Bu'ocke'tie'ptaphai giai bai loangiatri riengtuonglingvoi nhanhi~pphuongsai di;lng(5.3.1.7).Vdi ffiQts6nhand~cbi~t,dffco ffiQts61Oigiai giai tich. Trang tru'onghQpt6ngquat (5.3.1.7)cod(;lng: N AnIn (X, y) = I fc(x, y ;u, v) In (u, v) dO e e=1ne (5.3.3.7) Xudl,lngnQisuy : In (X, y) = Heine (5.3).7) trdthanh : N Anln(x,y)=I fC(x,y;u,v)HedOe Ine e=1ne (5.3.3.8) (5.3.3.8) cod(;lngla ffiQtbai loangia tri riengd(;lis6 t6ngquatnhu (5.3.1.14). Khaitri6nKarhunen-Loeved6ivdiffioduldanh6i,vathayvaobi6uthucthe'nang bie'nd(;lngdaTIh6i (5.3.3.5)taduQC: 144 1 m V =- uTI C;k(e)K k u 2 k=1 (5.3.3.9) HiygiOtatinhC6ngcualIfcldchdQng: n W=L fUeT(X,e)j(X)dOe e=10, n =uTI reTfHe Tj (X) dOe=U TF e=1 0, (5.3.3.10) )0sanh (5.3.3.9)va (5.3.3.10), tadi de'nphudngtrinh: m ( I !;k(e)K k ) U = F . k =I (5.3.3.11) giltirataduQccacth€ hi~ncua U tudngtlfnhuml;1ctruac. 5.3.4.Vi d\lghlimQthili toaDHIm X6tmQtUlmmongc6cacdi~uki~pbienphilhQP.Giltsli'moduldanh6icuata'mla mQtquatrinhng~unhienGausshaichi~uyaigiatritrungblnhE vahamhi~p hudngsai C(x" X2) dffbie't,Ilfc ngoaila ti~ndinh(ne'uIlfc ngoaila ng~unhien clingkh6nggaykh6khangl). Chiata'mthanhN ph§n tli'tli giac,m6iph§ntli'c6 8 b~ctlfdo.NangluQngbie'n d'.lngve cuam6iph§n tli'Aec6 d(;lng: ve =.!.-fcrT(x)E(x)dAe, 2 AC (5.3.4.1) Vectorcr(x)va E(X)bi€u di~nlingsua'tvabie'nd(;lng,Iahamrheaxtrongm6iph§n tu: crT=rcr cr 't ]t x, ", "1"2 (5.3.4.2) ET = [E",E"2Y""'2] (5-;3.4.3) trongd6 cr"iIa ling sua'theohuangdQCXiva 8",la bie'nd(;lngdQCrheaclinghuang. Giltsli'lingxli'cuav~tli~uIa tuye'ntinh,lingsua'tduQcbi€u di~nquabie'nd(;lngnhu Gu: 145 cr=DeE (5.3.4.4) :rongd6 De la matr~ndaDh6i, E>6ivai bai loanling Stittph~ng,Dedu<Jchobdi : trongd6 pematr~nti~ndinh, ~ela ht%s6Poissonva Ee(x,8)la moduldaDh6i. Vectorchuy€nvi u(x,8)mieutelchuy€nvi theochi~udQcvangangt~idi€m btt ky trongm6iphftntU'du<Jcxtp Xlquachuy€nvi nutcuaphftntU'duaid~ng: u(x,8)=He(I"I2)Ue(8), (5.3..4.6) trongd6 He(I" I2)la matr~nnQisoycuaphftntU',Ue(8) la vectorchuy€n vi nut (ngaunhien), IIva I21atQadQdiaphuongga:ntIeDphftntU'.D6i vaiphftntU'chIT nh~t: (XI - Xlc) II = , Ixl (X) - x)J[- - - 2-~ x, (5.3.4.7) trongd6 Ix;la oQdai c~nhcuaphftntU'chITnh~tdQctheohuangthli i Ih , vaXic latQa dQdQCtn;lcmahuangcuag6cthtpnhttd bell trai cuaphftntU'.Ma tr~nnQisoy He(II' [2) c6 d(~lllg: [ N} 0 N 2 0 N3 0 N 4 0 ]He(I"I2) = 0 NI 0 N2 0 N3 0 N4 (5.3.4.8) I trongd6 N} =I} (1- I2) ,N2 =I2 (1- II), N 3=III2,N 4=(1- II )(1- I2) (5.3.4.9) Thay(5.3.4.5)vao(5.3.4.1)tadu<Jc: Ve=.!.fEe(x,ekT(x,e)pec:(x,e)dAe 2 A< (5.3.4.10) Quanht%giuabie'nd~ngvachuy€nvi,theochi~udQcvatheochi~ungangc6d~ng: 1 e 0 De=Ee(x,8)I 1 0 1=Ee(x,8)Pe, (5.3.4.5)1 2 e-e (1-e)0 0 2 146 (5.3.4.11) Khi d6 (5.3.4.11)duQcvi6t l~id~ng: s(x,e)=BeUe (5.3.4.12) Thay(5.3.4.12)VaGl~i(5.3.4.10)tac6 : 1 I V =~UeT.f fE(rprz,e)BeT(rprz)peBe(r"rdr IdrldrzUe0 0 (5.3.4.14) trongd6 IJeIta dinhthlicJacobiancuabi6nd6imQtphfintii'ba'tky (e)VaGphfintii' thamchi6u([18]),tichphanduQcth1!chi<$ntrenphfintii'thamchi6u, tu d6nang lu'Qngbi6nd~ngt6ngcQngV c6d~ng: 1 N ] ( V=- IUeT ffE(rl,rz,e)BeT(r(,rz)peBe(r],rz~JeldrldrzU 2 e=' 00 (5.3.4.15) Sii'd\mgmatr~ndint vi ce voi vectorchuy€n vi toanh<$: u"=,eeu (5.3.4.16) Khid6nangluQngt6ngcQngcuah<$duQcvi6tduoid~ng: v=.!..uTKU 2 (5.3.4.17) trongd6 . N ; K =IeeT Keee : e=1 L (5.3.4.18) a 0- ax, s(x,e)=I 0 lu(x,e) axz a a- - axz ax( voi - (-1 +rz) 0 (1+rz) 0 rz 0 _2 0 I Ix, I Ix,x, x, Be=1 0 (-l+r]) 0 r( 0 r, 0 (1- r() I (5.3.4.13) Ix I Ix Ix2 x, 2 2 (- 1+r() (-1+rz) r( (1+;z) r( rz 1-rl rz- - - I Ix, I I I Ix I IX2 X2 x, X2 I X2 x, 147 C6 th€ dungnhi~uky thu~tHipnip d€ tinh(5.3.4.18)makhongc§n sttd\lngma tr~ndinhvi c". Tu'ongt1,!'trongcactinhtoantru'ocday,bu'ocke'tie'pla taphaigiai bai toangia t~i riengtu'onglingvoi nhanhi~pphu'ongsai : A"/,,(Xl'Yl) = fC(x] 'Yl ;X2'Y2 )/" (X2'Y2)dx,dYI A (5.3.4.19) Tren m6iph§nttt A", phu'ongtrinh(5.3.4.19)trdthanh : N A"/,,(X,,y,)= I fC(XPYI;X2'Y2)/',(X2'Y2)dA" "=1A' (5.3.4.20) xa'p Xl qua gia tri t(;linut: /" (x,y) =H" (r],r2)/,," (5.3.4'.21) trongd6 H"(rl,r2)lamatr~nnQisuytheocacthanhph§ncuatQadQdiaphu'ongr, varz, /,,"lavectorcacgiatqt(;linut(ham§n)lingvoiph§nttt(e).d dayc6th€ chQnd(;lngnQisuysongtuye'ntinhtrenph§nttt tli giacb6ndi€m nut.Khi d6,ma r tr~nH"h,r2) c6 d(;lng: . I H"(rprz)=~[(l-rIXl-rz) (1+r,Xl-r2) (l+r,Xl+rz) (l-r,Xl+rz)] (5.3.4.22) Thay(5.3.4.21)vabie'nd6i tutQadQtoanC\lCsangtQadQdiaphu'ong,(5.3.4.20)trd thanh N A"/,,(x,y,)=I fc(x, ,Y, ;Xz'Y2)H"(r\,r2~J"ldA"/'," "=1A' (5.3.4.23) trongd6IJ"lla Jacobiancuabie'nd6itQadQ.Sais6tu'onglingtn,tcgiaovoicac hamnQisuytu'ongt1,1'm\lc(1.2)ph§n 1., tu(5.3.4.23)tathudu'Qc: CD=ABD (5.3.4.24) trongd6 cQtthlij cuaD la vectoriengthlij tinhdu'Qct(;licacdi€m nUtva A.. =8..11.. I) I) I (5.3.4.25) Matr~nC va D thudu'Qcb~ngcachHipnipcacmatr~nCefva Bef : 148 Cef = f fC(XpYI;X2'Y2)HeT(r"r2)Hf(r"r2)dAedAf, AcAf (5.3.4.26) Bef = fC(XpYI;X2'Y2)HeT(r"r2)Hf(r"r2)dAe, Ac (5.3.4.27) (j day (xpYI)va (rpr2)bi€u diSnh~tQadQtO~lllC1;1Cvah~tQadQdiaphu'dngcua ffiQtdi€m trongAe. Vi~cHipnip matr~nthvcra conhi€u cach,tu'dngtv nhu'trong PP.Phfintii'hUllh~nti€n dinh,tie"nhanhnhu'(4.26)va(4.27)lamQtphu'dngpharo Khaitri€n Karhunen-Loevechomo~uldaTIh6ir6ithe"vaophu'dngtdnh(5.3.4.15), bie"nd6i phu'dngtdnh(5.3.4.17)chota: V =!UTt~k (8)K(k)U 2 k=1 (5.3.4.28) CongcuaIvcngoaila: N V' =:LSueT(x,8)f(x)dA e, e=A' M =UT:L CeTfHeT(x)f(x)dA e=UTf k=1 Ac (5.3.4.29) (jday N f =:LCeTfHe(x)f(x)dAe. e=1 Ac (5.3.4.30) Cvcti€u hoat6ngthe"nang (V - V') chota : [K(O)+~~k(8)K(k)JU =f (5.3.4.31) Tu'dngtl!m1;1c1.,U co th€ cocackhaitri€n : p U =:LCi\VJ{~I}]' k=1 cac Cj du'Qctinhquah~phu'dngtrinhd~is6 : [~K"l ~ viA{.;,}]f>, j =l...P. 149 . 0 ...0. 5 -O~1 ..().15 .0.2 -0.26 ..o~3 0 0 Bi6nchpageiiati'mtheokYvQng95% 5.3.5.K6t I~n coobaitoaD3: Sitd'1ngkhaitrin Karhunen-Loevevadatlntchanlo~nde'xAydt1ngmQtmohinh chophu'dngphapphin tithii'llh~nnglunhiensedttthu~nti~nchocaebaitoaDcd hQCnglunhientrongkhimQtvaiphudngpbapmophong86choquatrinhngftu nhienIari'tkh6. f)~cbi~t,chungtoinhinm~nhrAng,hiDhEtcacoongth1.1'ctrongphu'dngphapnay c6the'tinhtoaDhinhthactrenmaytinhtrttdckhichuye'nsangcacngonngfi'tinh L toaDs6truy~nth6ng.