Adaptive Parameterization using Free-Form Deformation for Aerodynamic Shape Optimization
Texte intégral
(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Adaptive Parameterization using Free-Form Deformation for Aerodynamic Shape Optimization Régis Duvigneau. N° 5949 Juillet 2006. ISSN 0249-6399. apport de recherche. ISRN INRIA/RR--5949--FR+ENG. Thème NUM.
(3)
(4)
(5) !#"$% & (')"*+ ,- &./% ./0 121 354167 98: ( $; <>=@?BA$CDEF?G=@HIKJLD MN)OPRQSUTUVXWZY\[K]_^`OaPRQa]&bKcdPRefhgjikc)Qa] lmfhnpoqQ^rtsau vkw xzy|{d{}nkf_^z~\QfhQaN)QfNdQbL+|p kWZkcdgQ^||kW k{)y||Qa] ∗. *a/d|| l+ypfypPRQ^`Qfhgay/^hgnkb^hQaNdb)gi>cdQ]$nkPPRnkbd[c)]`Qa~gbypQfhn\~\[Kb)ypPRgj]_N)y|{}Q nk{\^`gPRgy/^hgnkbypfhQQ]`]`Qb>^hgy|[ kQbdQafhy|y|b)~¡Ptcd¢^hg¢£¤{dcdfh{Lnk]`Qy|{d{dfhnky|N)Qa]a¥7¦]1y§n|b)]`Qai>cdQab)Qk¨%^`NdQa[y|bdbdnp^©}QªQ&]_cdg^`Q~¡^hn7y§{)y|f_^hgcdy|ft]`N)yp{LQ nk{\^`gPRgy/^`gn|b{dfhn|©dQP¥«MNdQ{)f`Q]_Qabk^]q^hc)~\[¬{dfhn|{Lnk]`Qa]tybdQaª®PRQ^`N)nK~¡^`N)yp^Ryk~dyp{d^h]ypb¡gbdg^`gjyp&ypb)~¡{}Qaf`N)y|{)]tb)yp¯G°|Q {)y|fhy|PRQ^`Qaf`gayp^`gn|b n|±(ypby|Qfhn\~\[Kb)ypPRgj]`N)yp{LQ©K[^`N)Q²)f`QaQ£³²dn|fhPµ´UQ±¶n|fhP1y/^`gn|b·¸²%²%´º¹U^`QNdbdgji>cdQ|¨@^hn$^`N)Q{}ypf`^`gjcdjypf nk{\^`gPRgy/^`gn|b{dfhn|©dQP®^`n1]_nk°kQ|¨\y|ankfh~\gbd^`n1yt»)fh]_^&yp{d{dfhn½¼\gP1yp^`gn|b*n|±F^hNdQ]_nkc\^hgnkb¾¥+MNdgj]{)ypfypPRQ^hQfhgy/^hgnkb*y|~dy|{\£ ^hgnkb7PRQ^`Ndn\~7g]gb)c)~dQa~§gb¬y^`NdfhQQ£¤~dgPRQb}]_gn|b)y|+ypQfhn\~\[Kb)ypPRgjt]`N)y|{}QRn|{d^`gPgayp^`gn|b7{dfhnKQa~\c)f`Qt±¶n|f¿mc)Qaf`gjypb§À)n/ªz] y|b)~g]&~\QaPRn|b)]_^`fy/^hQa~*±¶n|f^`N)Q~\Qa]`gkbnk{\^`gPRgy/^hgnkbn|± ^hNdQª&gb)n|±yR©dc}]_gbdQa]h]&ypgffy/±Á^a¥  Ã>Ä@Å`ÆtÇ /È| l+ypfypPRQ^hQfhgy/^hgnkb¾¨|²dfhQQ£¤²)n|fhPÉ´Q±¶nkf`P1yp^`gn|b¾¨\¦&~dy|{\^`gn|b¾¨\YKN)y|{}Qºn|{d^`gPgayp^`gn|b¾¨dÊ(n|PR{dcd^hy/^hgnkb)yp@²c)g~ ´[Kb)ypPRgj]a¨)¦Qfhn\~\[Kb)ypPRgj]. ∗. ËÌ_ÍkÎ
(6) ÏÐ}ѸÒÓ¸Ô_Õ³Ö¸×ÁØ\Ô_ÙÚ Unité de recherche INRIA Sophia Antipolis 2004, route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex (France) Téléphone : +33 4 92 38 77 77 — Télécopie : +33 4 92 38 77 65.
(7) %F% *
(8) 61 - &! t. § _* ./+ 121t
(9) ]`n|b>^t{} ypf Q ]`]` QQb}]%Q$^`QkNdebdbdeagji>fhy|cdQaQ]m]~dQQz^{)Pypfcdyp^`PRgn|e©d^`fhoqgQa]h
(10) y/^h^hg°kgnkQab]a¥c\^hl+gypg]`fReQ]mnkb)N)]_y|e©di>gcd^`cdQQab>^a¨%QPRQaQQab>]^ bdQab1Qn|{L{\Q^hc)g°|PRQgjb>]`^yp^`{)gn|y|b1] ~\^`Q&fh±¶Q*nkf`©)PRgQaQzby|y|e~dfhnKy|~d{\[>^`eab}Qayp] PRgi>cdcdb Q {dfhn|©)OaPQ$~ nk{\^`gPRg]hy/^hgnkb¡~\Q1±¶n|fhPRQ1Qb¬{)y|f_^hgcdgQafa¥ Ê(Q^_^hQ$e^hc)~\Q${)f`nk{}n>]_Q1cdb)Q$bdn|c)°|QQ1Pe^`Ndn\~\Qikc)g(y|~)yp{\^hQ1cdbdQ {)y|fhy|PRe^`fhgj]`yp^`gn|bgbdg^`gjypQQ^U{}Qac\^_£ ^hf`Qb}yp¯G°|Q~ cdbdQ±¶nkf`PRQy|efhnK~d[>b}ypPRgi>cdQ{)y|fzjyRPe^`Ndn\~\Qt~\Qa]U©}nkg¢^hQa]zQabd|n|©}ypb>^`Q]¨ y|c${dfhn|©dOPRQ~ nk{\^`gPRg]hy/^hgnkb${)ypf`^`gjc)gQfi>cdQU nkb*]`n|cdN)y|g¢^hQUfhea]`n|c)~df`Qk¨k]`cdf(jy©}y|]`QU~ cdbdQº{df`QaPRgOaf`Qºyp{d{dfhn½¼\gP1yp^`gn|b1~\Q jyt]`n|c\^hgnkb¾¥ Ê(Q^_^hQPe^`Ndn\~\Q~ y|~dy|{\^hyp^`gn|b~\Qjyt{)ypfypPRe^hf`gj]hy/^`gn|b1Q]q^gb)cdQ~dypb}](c)bdQ{)f`n\e~\cdfhQ~ n|{d^`gPgj]hy/^`gn|b~\Q ±¶nkf`PRQty|efhn\~\[Kb)ypPRgjikc)Q^`fhg~dgPRQb}]_gn|bdb)QQt{}nkcdfº~dQa]eankcdQPRQb>^h]¿mcdQfhgQb)]Q^PRg]`QQabnKQcd°KfhQ{Ln|cdfº n|{d^`gPgj]hy/^`gn|b ~\Qjyt±¶n|fhPRQ~dQ ypgQ~ cdboqQ^º~ "y !@y|gfhQ|¥ #SUcdÇ PRaeÅfh%gi>$ cd RQk¨) ¦el%fhn\y|fh~\y|[KPRb)ypQPR^`fhgjgj]`ikypc)^`Q gn|b¾ ¨ &"n '^hQa]mQabd|n|©}ypb>^`Q]¨>¦&~dyp{\^y/^hgnkb¾)¨ ({\^hgPRgayp^`gn|b*~\QU±¶n|fhPRQ|¨kVeay|bdgjikc)QU~\Qa]²c)g~\Q].
(11)
(12) !"$#% &'"($ !)+*-,.. 0/1
(13) 2 & !". 3. t &t A 576 ÃKÃkÅ 6 Ç Å8Ã:9¤Ç \; Ç=<?> \/ à à ;@|\&; Ç< F BDC \/ à à ;!@k\; Ç=< dÈ > ; Ç=<E> Ç Ã ÈF à d¥ G ²)fhy|PRQªn|fIH ¥¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥º¥¥¥ d¥ l f`gb)g{dQa]¥¥¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥º¥¥¥ d¥ 3 Qayk]q^`£¤]hi>c)ypfhQa]yp{d{)f`n½¼\gPRyp^`gn|b ¥º¥¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥º¥¥¥ J d¥ Vyp{d{dgbdR{)y|fhy|PQ^`Qafh] ¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥º¥¥¥ K d¥ ¦&~)yp{\^hgnkb*±¶cdb}
(14) ^`gn|b}yp5¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥º¥¥¥ d¥ ¦&~)yp{\^hgnkby|kn|fhg¢^hNdP®¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥º¥¥¥LGa dM¥ J SUcdPRQfhgayp gPR{dQaPRQb>^hyp^`gn|b ¥¥¥º¥¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥º¥¥¥LGa N ÆÇ)Å OÈ ; :à < ; =Ç < $P; $ $ a/\ ; =Ç < 55 Q >1> $!;B|\& ; Ç < Ç N 8 Ã Ç È RÄ < S;¸& T >(à UÇ > ; S;!@k\ ; =Ç < 5A }¥ G ¦Qaf`n\~\[Kb)y|PRgº]_N}yp{LQn|{\^hgPRgay/^hgnkb9¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥º¥¥L ¥ G }¥ ²n/ª ]`n|°|Qaf ¥¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥º¥¥L ¥ G }¥ d¥ G VnK~dQgbd ¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥º¥¥L ¥ G }¥ d¥ Y\{)y/^hgy|F~\gj]`f`Q^`gay/^hgnkb«¥¥º¥¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥º¥¥L ¥ Ga }¥ d¥ 3 MgPRQgb>^hQ|fy/^hgnkb5¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥º¥¥L ¥ Ga }¥ 3 VQa]`Ncd{@~dyp^`Q ¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥º¥¥L ¥ Ga }¥ ({d^`gPgayp^`gn|byp|nkf`g^`NdP ¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥º¥¥L ¥ Ga A à $¶a B=V d¥ G M Qa]_^_£³yk]_Q~\Q]`f`g{\^`gn|b0¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥º¥¥¥µp d¥ xzQa]`cd¢^]±¶n|fzyk~dyp{\^hgnkbyp^Unkb>°kQfh|Qab)Qª&g¢^hNfhQ£¤gbdg¢^hgy|gayp^`gn|b ¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥º¥¥¥µp d¥ 3 xzQa]`cd¢^]±¶n|fzyk~dyp{\^hgnkbyp^Unkb>°kQfh|Qab)Qª&g¢^hNn|b>^`gbKc)y/^hgnkb ¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥º¥¥¥µ| d¥ 3 ¥ G Ê(n>ypf]_Qº{)y|fhy|PQ^`Qaf`gayp^`gn|b ¥¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥º¥¥¥µ| d¥ 3 ¥ VQa~\gcdP {}ypfypPRQ^hQfhgy/^`gn|bµ¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥º¥¥¥µ 3 d¥ 3 ¥ 3 ²gbdQ{)y|fhy|PQ^`Qaf`gayp^`gn|b ¥º¥¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥º¥¥¥µ/ d¥ xzQa]`cd¢^]±¶n|fzyR{dfhn|kf`Q]`]`g°|QUyk~dyp{\^hgnkb¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥º¥¥¥µp d¥ }¥ G Ê(n>ypf]_Qº{)y|fhy|PQ^`Qaf`gayp^`gn|b ¥¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥º¥¥¥µp d¥ }¥ VQa~\gcdP {}ypfypPRQ^hQfhgy/^`gn|bµ¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥º¥¥¥µp d¥ }¥ 3 ²gbdQ{)y|fhy|PQ^`Qaf`gayp^`gn|b ¥º¥¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥º¥¥¥µ K d¥ Y\{}Qg»}f`Q]_c)¢^]±¶n|fzyPRQa~\gcdP {}ypfypPRQ^hQfhgy/^`gn|b ¥¥º¥¥¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥º¥¥¥ 3 G d¥ d¥ G Ê(nkPR{)ypfhg]`n|bnp±^`NdQP1y|{d{dgbd±¶cdb)
(15) ^hgnkb)] ¥¥¥º¥¥¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥º¥¥¥ 3 G d¥ d¥ Ê(nkPR{)ypfhg]`n|bnp±^`NdQ]`N)y|{}Q] ¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥º¥¥¥ 3 d¥ d¥ 3 Ê(nkPR{)ypfhg]`n|bnp±^`NdQÀ}n/ªz]¥¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥¥º¥¥¥¥¥¥¥¥¥¥¥º¥¥¥ 3 4. W1WYXZU[\]\.
(16) . " . t F1 3m ({\^hgPtcdP5]_N}yp{LQ*~\Qa]`g|b¬gbypQaf`n\~\[Kb)y|Pgj]N}y|]©LQQabypb¡yk
(17) ^hg°kQ$f`Q]_QypfN^hn|{dgj1±¶nkft]`Q°kQfyp+[|Qypf]¥ MNdQ*f`QQb>^gP£ {dfhn/°|QaPRQb>^h] np±FbKcdPRQfhgjyp}PRQ^`N)nK~)](gbÊ(nkPR{dc\^hyp^`gn|b)y|L²cdgj~*´[>b}ypPRga]·Gʲ%´º¹(ypb}~$gb*n|{d^`gPgayp^`gn|b¾¨\yk]mªQ@yk]m^hNdQ ~\Qa°|Qank{}QaPQab>^ np±¾Ndg|N1{LQf`±¶n|fhP1ypb)QUnkPR{dc\^`gbdayp{)y|©dgg^`gQa]a¨k[KgQj~1y]`gkbdg¢»Lypb>^ fhQa~dc)
(18) ^hgnkb1np± ^`N)QU~\Qa]`g|bR^`gPRQ[KQk¥ MNdQaf`Q±¶n|fhQ|¨ ^`NdQ*^`NdfhQQ£¤~dgPRQb}]_gn|b)y| ypQaf`n\~\[Kb)y|PRg1]`N)yp{LQ$nk{\^`gPRgy/^`gn|b¬n|±z{)ypf`^h]np±Uypb¡y|gffy/±Á^gj]^hn\~dy½[7yn|PRPRn|b Q¼KQafhg]`Qgb y|Qfhn|b)y|c\^`gj]a¥ ¦ fhg^`gjypgj]`]`cdQgb ]_N}yp{LQnk{\^`gPRgy/^hgnkbgj]^hNdQ Ndn|gjQnp±^`NdQ ]`N)yp{LQ{}ypfypPRQ^hQfhgy/^`gn|bF¥«MN)Qnk©\oqQa^`g°|Qnp±U^hNdQ {)y|fhy|PRQ^`Qaf`gayp^`gn|bgj] ^`n~\Qa]hfhg©}Q(^`NdQ&]`N)yp{LQ|¨/nkf^`NdQ&]`N)yp{LQ(PRn\~\g»}yp^`gn|b¾¨/©K[yº]_Q^+np±){)y|fhy|PRQ^`Qafh] ª&NdgjNty|f`Qn|b}]_gj~\QfhQa~ yk](~\Q]_g|b*°½y|f`gjyp©)Q] ~\cdfhgbd^`N)Qnk{\^`gPRgy/^`gn|b*{df`n\Qa~\cdfhQ|¥l+ypfypPRQ^hQfhgy/^hgnkbt^hQaNdb)gi>cdQ] gb]`N)yp{LQUnk{\^`gPRgy/^hgnkb1N)y½°|Q ^hnR±¶cd¢»}F]`Q°kQfyp@{dfy|^`gjyp f`g^`Qaf`gj y.
(19) ^`NdQ {)ypfypPRQ^hQfhgy/^hgnkb]`Ndn|cdj~º©}Q(yp©dQ%^`n^y H|Q+gb>^`nzyknkcdb>^FnkP{)Q¼U|Qan|PRQ^hf`gQa]a¨{Lnk]h]_g©d[Ugb)c)~\gbdUn|b)]_^`fypgb>^h] ypb)~]`gbdkcdjypfhg¢^hgQ]
(20) ^`NdQbKcdP©LQfnp±z{)y|fhy|PRQ^`Qafh]]_Ndnkcdj~¬©}Qy|]t]_P1ypy|]{Lnk]h]`g©dQ|¨m]`gb)Q1^hNdQ]q^hg !@bdQa]h]tnp±&^`N)Q]`N)yp{LQ*nk{\^`gPRgy/^`gn|b bKcdPRQfhgayp@±¶n|fhPc)yp^`gn|b*gb)f`Qy|]`Qa]yp©dfhcd{\^h[*ª&g¢^hN^hNdQbKcdP©LQf&n|±{}ypfypPRQ^hQf]
(21) ^`NdQ{}ypfypPRQ^hQfhgy/^`gn|b]_Ndnkcdj~y|n/ª ^hnRn|b>^`fhn|@^hNdQ]`PnKn|^`NdbdQ]`]n|± ^hNdQf`Q]_c)¢^hgbd1]`N)yp{LQa]
(22) ^`NdQ*{)ypfypPRQ^hQfhgy/^hgnkb7]`Ndn|c)~7©LQ*nkb)]_gj]_^`Qb>^ª&g^`N¬kf`gj~7cd{@~dy/^hQ*yp{d{)f`n>y|NdQ]ypb)~7ª&g^`N~\nkP1ypgb7{}ypf`^`g^`gn|bdbdgbd PRQ^`N)nK~)]¥ ¦]_cdfh°|Qa[Un|±\]`N)y|{}Qm{}ypfypPRQ^hQfhgy/^`gn|b^`QNdbdgji>cdQa]a¨½ª&NdgNy|f`Q(ypb)y|[KQ~y|n|f~\gbd^`nz^hNdQm{dfhQ°Kgn|c)]Ffhg¢^hQfhgy)¨agj]¾{dfhn|{Lnk]`Qa~ ©K[7YdypP1ypfhQN G\P¨ G GG¥ ³b¬y|n|f~dypb}Qª&g¢^hN¬Ndg]n|b)c)]`gn|b)]a¨¾^hNdQ$²dfhQQ£¤²)n|fhP-´UQ±¶n|fhP1y/^`gn|b·¸²%²%´¹^`QNdbdgji>cdQ G½ gj] yk~\n|{\^hQa~*gb^`NdQ{dfhQa]`Qb>^]q^hc)~\[|¨d]`gb)Qºg¢^&{dfhn/°Kg~dQa]ypb*Qy|]`[1ypb)~*{}n/ªQf`±¶cdL±¶fhy|PRQªn|If H±¶nkf(^hNdQ~\Q±¶nkf`P1yp^`gn|b$n|±nkPR{dQ¼ ]`N)yp{LQa]a¨dy|]^hNdnk]`QQb}n|c)bk^hQfhQa~*gb ypQaf`n\~\[Kb)y|PRga]¥ N)yp^`Qa°|Qf^hNdQ{)ypfypPRQ^`Qfhgay/^hgnkb$PRQ^hNdn\~N)nk]`Qb©>[$^`NdQc)]`Qf&gj]¨d^`NdQP1ypgbg]h]_c)Qºgb]_N)y|{}Q{)y|fhy|PRQ^`Qaf`gayp^`gn|b*±¶n|f nk{\^`gPRgy/^`gn|b{dc)f`{Lnk]`Qg]&^`NdQf`g^`gjy|¾~\Q{LQb}~\Qb)[n|± ^hNdQnk{\^`gPcdP0]_N}yp{LQ±¶n|cdb)~nkb^`NdQ{)y|fhy|PRQ^`Qaf`gayp^`gn|b¾¥ ³b}~\QQ~ ¨ ^hNdQ*NdnkgQtn|±(^hNdQ1{)ypfypPRQ^`Qfhgay/^hgnkb§~\Q^`Qaf`PRgbdQa]^hNdQ$]`Q^np±]_N}yp{LQa]^hN)y/^aypb7©}Q$fhQaykNdQa~§~\cdfhgb)^hNdQ$nk{\^`gPRgy/^`gn|b {dfhn\Q~\cdfhQ|¥YKgb)Q^`N)Qn|{d^`gPcdPµ]`N)yp{LQ±¶nkcdb)~ ©LQn|bd>]&^hn$^`N)g]º]_Q^a¨@g^º~\Qa{}Qab)~d]n|©K°Kgnkc)]_[*nkb ^`N)Q{)y|fhy|PQ^`Qaf`gayp^`gn|b¾¥ ¦]y7nkb)]`Qai>cdQb}Q|¨%^hNdQNdn|gjQnp±z^hNdQ{)ypfypPRQ^hQfhgy/^hgnkb¬gb\À}cdQb)Qa]t^`N)QfhQa]`cd^h]tnp±z^hNdQn|{\^hgPRgay/^hgnkb¾¨+gb^hQfhPR]tnp± ykc)fhyk[n|± ^`NdQ]_nkc\^hgnkb·¶gG¥ Qk¥%^`NdQº~\gj]q^ypb)Qz©}Q^qª(QaQbR^hNdQUnk{\^`gPcdP]`N)yp{LQ&±¶nkcdb)~$y|b)~^`NdQº]_nkcd^`gn|b}¹+y|b)~$n|bK°|Qaf`kQb)Q fy/^hQR·¸f`Qayp^`Q~$^hnR^`NdQ]_^`g !Lb)Qa]h]np±^`NdQ{dfhn|©)QaP1¹¥ MNdQ{dfhQa]`Qb>^]_^`c)~d[{)f`nk{}n>]_Q] yPRQ^`Ndn\~1^hnty|c\^`nkPRyp^`gjy|[yk~dyp{d^ypb*gbdg^`gjyp@ypb}~${LQfhN)yp{)] b)yp¯G°|QU{)ypfypPRQ^hQfhgy/^hgnkb n|±Fypb$ypQaf`n\~\[Kb)y|PRg&]`N)yp{LQz©K[^`NdQº²%²´ ^`QNdbdgji>cdQ|¨k^`n^`NdQ{)ypf`^`gjcdjypf n|{\^hgPRgay/^hgnkb1{dfhn|©dQP ^`n]`n|°|Qk¨kyknkfh~dgbd^hnty »)f]_^yp{d{)f`n½¼\gPRyp^`gn|bnp±^`NdQ]_nkcd^`gn|b¾¥MNdQ{dfhn|{Lnk]`Qa~PRQ^hNdn\~g]Ugb)]`{dgf`Q~±¶f`nkP {dfhgn|fzªn|If H\]©)y|]`Qa~n|b &eagQfzc)f`°kQa] y|b)~*y|{d{dgQ~$^hnytPRn|fhQ]`gPR{dQ|QnkPRQ^`fhgjyp)^qª(n|£¤~dgPRQb}]_gn|b)y|}{)f`nk©dQa P MJG¥ ¤^ªyk](±¶n|cdb)~ @^`N)yp^^hNdQf`Qa|cdjypfhgay/^hgnkb n|±m^`NdQRnkb>^`fhn|F{Ln|[K|nkb np±m^hNdQ &eagQafºcdfh°|Q[KgQj~d]y*]`gkbdg¢»Lypb>^UgPR{dfhn/°|QPRQabk^gb^`QfhP1]Un|±(y|acdfy|[|¥UMN)Qt{dfhQa]`Qb>^ ªn|If H$ypgP1]zy/^&Q¼K^hQb)~\gbd^`N)Qa]`Q~\Q°kQn|{LQPRQb>^h]^hn 3 ´ y|QfhnK~d[>b}ypPRgº]`N)yp{LQn|{\^hgPRgay/^hgnkb¾¥ MNdQ{)ypfypPRQ^`Qfhgay/^hgnkb1y|~dy|{\^`gn|b1PRQ^hNdn\~1gj](~\Q]`f`g©LQa~1gb1^`N)Qz»)f]q^]_Q
(23) ^hgnkb¾¥+MNdQb¾¨k^`N)QU^qªnp£³~\gPQab)]`gnkb)ypd{}ypfypPRQ£ ^hQfhgy/^`gn|bnp±%ypby|gf`±¶n|g@g]c)]`Qa~*^`ngc)]_^`fy/^`Qº^hNdQ{)f`nk{}n>]_Q~*yp{){df`n>y|N¾¥%MNdQ]_Q~\Q°kQn|{LQPRQb>^]y|f`QU^`NdQabn|b)]`gj~\QfhQa~*gb ^hNdQ±¶fypPRQaª(nkf Hnp± y|QfhnK~d[>b}ypPRg]_N)y|{}Qn|{d^`gPgayp^`gn|b¾¥¦&±Á^hQfº^`N)Qt©)yk]_QagbdQtnp±m^hNdQt{)f`n\Q~\cdfhQgj]fhQPRgb)~\Q~¡·¶¿mc)Qaf`gjypb À)n/ª ]`n|°|Qf¨K|fhg~cd{@~dy/^hQ|¨dnk{\^`gPRgaQf
(24) ¹
(25) ¨\^hNdQy|~)yp{\^hgnkbPRQ^hNdn\~g]&y|{d{dgQ~^`nR^`N)QypQfhn\~\[Kb)ypPRgj]`N)y|{}Qnk{\^`gPRgy/^`gn|bnp± ^hNdQª&gb)1np±y©dc}]_gbdQa]h]&ypgffy/±Á^a¥. 1W "!.
(26) .
(27) !"$#% &'"($ !)+*-,.. 0/1
(28) 2 & !". $" % & (')"*+ '), &./+ Z% % ( MNdQR²dfhQQ£¤²)n|fhP£¤´Q±¶nkf`P1y/^hgnkb¬·¸²%²%´º¹U^`QNdbdgji>cdQtnkf`g|gb)yp^`Qa]&±¶f`nkP ^hNdQ$Ê(n|PR{dc\^hQfºfhy|{dNdgj]U»)Qj~ G¤¥¤^y|n/ªz]&^hNdQ ~\Q±¶n|fhP1y/^`gn|b n|±ypbnk©\oqQa^*gb y|´-nkf 3 ´ ]`{)y|Q|¨&fhQky|fh~dQ]`]Rn|±^`NdQ§f`Qa{df`Q]_Qab>^hy/^hgnkbnp±^`N)g]n|©\oqQ
(29) ^¥ ³b)]_^`Qy|~np± P1ypb)g{dc)yp^`gbd*^hNdQ1]_c)f_±¸ykQn|±m^`N)Qtnk©\oqQa^~\gfhQa
(30) ^h[k¨ ©K[ c)]_gbdjy|]h]_gjy|&(£qYK{dgbdQ]Unkf &(eagQfº{)ypfypPRQ^`Qfhgay/^hgnkb np±m^hNdQ ]`cdf`±¸y|Qk¨\²²%´ ~dQ»)bdQ]y~dQ±¶n|fhP1y/^hgnkb$»)Qa~*n/°|Qf(^`N)Q]`{)ykQºQP©LQa~)~\Qa~gbytyp^_^`gjQª&NdgjNg]©dcdg^Uypfhn|cdb)~1^`NdQn|©doqQa
(31) ^¥ &~\Q[1]`^hf`fhgy|{\b)^h]qgnk±¶nkb¾f`¥ PRgbdt^`NdQ]`{)y|QnKn|f~\gb}y/^`Q] gb)]`g~dQ^`NdQjy/^_^hgQ|¨)²%²%´ ~\Q±¶n|fhP1] ^hNdQn|©doqQa
(32) ^¨\f`Qaky|fh~\Qa]h]mn|±g^h]kQnkPQ^`fhgayp V nkf`Qt{dfhQagj]`Q[|¨¾nkb)]`g~\Qafy$^hNdf`QaQ£³~\gPQab)]`gnkb)ypNdQ¼dy|Qa~\fypyp^_^hgQtQaP©LQa~d~\gbd^`N)Qnk©\oqQa^^hn©LQ1~\Q±¶nkf`PRQ~ ¥t²gp£ cdfhQ· G¹U]_Ndn/ªz]ypb Q¼\y|PR{dQnp± ]`c)Ny1jy/^`^`gjQ©)cdg^ypfhn|cdb)~ y$f`Qypg]_^`gjª&gbd)¥U¦ nKaypnKn|f~\gb)y/^hQ]_[\]_^`QaP (ξ, η, ζ) g] ~\Q»)bdQ~$gb$^`N)Qjy/^`^`gjQ|¨>ª&g^`N ¥´cdfhgbd^`NdQ~\Q±¶n|fhPRyp^`gn|b¾¨k^`N)Qº~\gj]`{dykQaPQab>^ ∆q np±¾Qy|N {Ln|gb>^ q gb)]`g~\Qº^hNdQyp^_^hgQg](ξ,NdQaf`η,Qζ)~\Q»)∈bd[0,Q~1]©K[×y[0,^hNd1]gfh×~K£¤n|[0,f~\1]Qaf &eagQf^`Qab)]_nkf&{dfhnK~dc)
(33) ^. XXX · G¹ B (s )B (t )B (u )∆P . ∆q = ¨ B y|b)~ B ypfhQº^`NdQ &Qfhb)]_^`Qagb{}nk[KbdnkPRgy|]n|± n|f~\Qaf n ¨ n ypb)~ n ·¸]`QQº±¶n|f&gb)]_^hypb}Q KÁ¹. B ·G|¹ B (t) = C t (1 − t) . ³b· G¹¨ (s , t , u ) ypfhQ&^hNdQU²%²%´ n>nkfh~dgb)yp^`Q]+np± ^hNdQz{Ln|gbk^ q ª&N)gN1fhQa]`cd^m±¶f`nkPÉyPRy|{d{dgbdn|± ^`NdQzjy/^`^`gjQºnKn|f~\gb)y/^hQa] ¥mTU]`c)yp[R^hNdgj](P1y|{d{dgbdtgj]]_gPR{d[$~\Q»)b)Qa~*yk]m^hNdQg~dQb>^`g^q[|¥ &c\^a¨dyk] ª&gL©LQQ¼\{djypgbdQa~$yp^_^`Qafa¨>^hNdQfhQPRg|N>^ (ξ , η , ζ ) ©LQ ]_nkPQy|~d°½y|b>^hypkQa]np±Uc)]`gbd7y|bn|^`NdQaftP1yp{){dgb))¥²gb)yp[k¨ ∆P ypfhQ*ªQg|N>^`gbd7nKQ $gQb>^]¨mnkfn|b>^`fhn|{Ln|gbk^] ~\gj]`{dykQaPQab>^h]a¨ ª&NdgjN¬ypfhQ1c)]`Qa~§^`nPRnkbdg¢^hn|f^hNdQ$~\Q±¶n|fhPRyp^`gn|b¬ypb}~¬ypfhQRn|b)]`gj~\QfhQa~«y|]~dQa]`gkb¬°½y|f`gjyp©)Q]~\cdfhgbd^hNdQ ]`N)yp{LQn|{d^`gPgayp^`gn|b{dfhnKQa~\c)f`Qk¥ ni. nj. nk. ni i. q. nj j. nk k. q. q. ijk. i=0 j=0 k=0. nj j. ni i. nk k. i. n p. q. q. q. q. p p n. j. k. n−p. q. q. ijk. ² g|cdfhQ$G ¿+¼dypPR{dQºn|±%²%²%´jy/^`^`gjQ$·¶fhQa~}¹ypfhn|cdb)~yª&gbd}¥ MNdQ$²%²%´®~\Q±¶n|fhP1y/^`gn|b§^`QNdbdgji>cdQ$~\Qa]hfhg©}Q~§y|©}n/°kQgj]ªQ ]_c)g¢^hQa~§^`n nkPR{dQ¼7]`N)yp{LQRn|{\^hgPRgayp^`gn|b¾¨F^`N)y|b.H\]º^`n ^hNdQ±¶n|n/ª&gbdR{dfhn|{LQf`^`gQa].
(34) ^`NdQgb)g¢^hgy|(]`N)yp{LQ*y|b¬©}QQ¼dy|^`[§fhQ{)f`Q]_Qabk^hQa~·¶b)n~\Q±¶nkf`P1yp^`gn|b¬n\cdfh]ª&NdQby|(ª(QagkN>^`gbdnKQ $gQabk^]y|f`Q Qaf`nK¹
(35) ^`NdQ~dQ±¶n|fhP1y/^hgnkb*gj]&{}Qaf_±¶nkf`PRQ~*ª&N)yp^`Qa°|Qf^hNdQnkP{)Q¼Kg^q[$n|± ^hNdQ]_N}yp{LQR·¶^`Ndgj]&gj]zyt±¶f`QaQ£G±¶n|fhP:^`QNdbdgjikc)Q¹
(36) |Qan|PRQ^hf`gj(]`gbd|cdjypfhg^`gQa]ypb©LQ(^y HkQbgb>^`ny|ankcdb>^·¶^`NdQgbdg¢^hgy|d]_N}yp{LQgb)c)~dgbdg^h]+]`gb)|cdjypfhg¢^hgQ]Fgj]+~\Q±¶n|fhPQ~)¹
(37) ^`NdQ]`PRnKnp^`N)bdQa]h]np± ^`NdQ~\Q±¶n|fhP1y/^`gn|bgj]&n|b>^hf`nkQ~·Á^hNdQ~\Q±¶n|fhP1y/^`gn|bg]zf`cdQa~©K[ &Qfhb)]_^`Qgb{}nk[KbdnkPgjypj]¹
(38) ^`NdQºbKcdP©LQfn|± ~\Qa]`g|b*°/y|f`gjyp©dQa]~\Qa{}Qab)~d]n|b*^`NdQºc)]`Qf ]NdnkgQt·Á^hNdQ~\Q±¶nkf`P1yp^`gn|b$gj]gb)~dQ{LQb)~\Qab>^n|±¾^`N)Qº]`N)yp{LQ g¢^]_Qa¢±¹
(39) g¢^zb)gQ[*~\Qay|]ª&g^`NPtcd¢^hg¢£¤Q°|Qa f`Qa{dfhQa]`Qb>^hyp^`gn|b§·¶^`N)y|.b H\]^`n^hNdQ &egQfz~dQ|fhQQºQQ°/yp^`gn|b{df`nk{}Qaf_^q[d¹¥ W1WYXZU[\]\.
(40) . " . ¦ bdQaª bdn|^hyp^`gn|bg]&gb>^`fhn\~\c)Q~$^hnR±¸y|gg¢^y/^hQ±¶c)f_^hNdQfU~dQ°|Qank{}QaPRQb>^h]a¥ Ê(n|b)]`g~dQf^`NdQ°kQa^`n|f]. n. B T =[Bini (sq )Bj j (tq )Bknk (uq )]i=1,...,ni ; j=1,...,nj ; k=1,...,nk ∈ <(ni +1)×(nj +1)×(nk +1) n n =[B0ni (sq )B0 j (tq )B0nk (uq ), B0ni (sq )B0 j (tq )B1nk , . . . , Bnnii (sq )Bnnjj (tq )Bnnkk (uq )],. ∆P T =[∆Pijk ]i=1,...,ni ; j=1,...,nj ; k=1,...,nk ∈ <(ni +1)×(nj +1)×(nk +1) =[∆P000 , ∆P001 , ∆P002 , . . . , ∆Pni nj nk ].. MNdQab¾¨>^`NdQ^`N)gf~K£¤n|f~\Qf ^`Qb}]_nkf {dfhn\~\c)
(41) ^· G½¹ aypb1©LQQ¼K{)f`Q]`]`Qa~1yk](yt]_gPR{dQgb)bdQf{dfhnK~dc)
(42) ^n|b MNdQ]_Qb)np^hyp^`gn|b)]y|f`Qc)]`Qa~gb^hNdQbdQ¼K^U]`Qa^`gn|b^`n1gP∆q{)f`=n/°kBQU^hNd∆P. QQa|g©dgg^q[$np± ^`NdQ±¶nkf`Ptcdjy|]a¥ % % * %13( & % T. . ·3 ¹ ·¸ >¹. <(ni +1)×(nj +1)×(nk +1). ·G|¹.
(43) . MNdQ²%²%´ {}ypfypPRQ^hQfhgy/^`gn|bR{dfhQa]`Qb>^hQa~$y|©}n/°kQUypn/ªz](ypb$Qy|]`[RPn\~\g»}ay/^`gn|b*np± ^`N)Qº]`N)yp{LQUn|± ypb*n|©\oqQ
(44) ^(©K[RPRn\~\g¢±¶[Kgbd ^hNdQRª(QagkNk^hgb)n>Q $gQb>^h] ∆P ~\cdfhgb)^hNdQRn|{\^hgPRgayp^`gn|b{)f`n\Q~\cdfhQ|¥Un/ª(Qa°|Qafa¨L^`N)g]ºfhQ{dfhQa]`Qb>^hyp^`gn|bgj]bdn|^_cdbdg£ °kQf]`y|
(45) ¨)gG¥ Qk¥U^hNdQf`Qy|N)y|©dQ]`N)y|{}Q]~\Qa{}Qab)~ nkb]_nkPRQtNdnkgQa]zn|±+^`NdQc)]`Qf¥²)n|fUgb)]_^hy|b)Qk¨Lg±+^`N)QtP1yp{){dgb)*{dfhn\~\c)gbd ^hNdQ²%²%´ nKn|f~\gb)y/^hQa] (s, t, u) g](PRn\~\g¢»}Qa~ ¨Kn|f g¢± &(£qYK{dgbdQ] ©}y|]`g]%±¶c)b)
(46) ^hgnkb)](y|f`QUQPR{dn/[|Qa~Rgb)]_^`Qy|~Rnp± &Qaf`b)]_^`Qagb1{Ln|[>£ bdnkPRgy|]a¨ n|fg±(©)yk]_gj]±¶cdb)
(47) ^hgnkb)]ºnp±NdgkNdQf~\Qa|fhQQypfhQtc)]`Qa~¾¨@^`NdQab^hNdQRf`Qy|N)y|©dQ]_N}yp{LQa]ºª&g+bdn|^º©LQ^`NdQ1]hypPRQtnkbdQa]a¥ ¦&^`c)y|[|¨@^hNdQN)n|gjQn|±m^`NdQ{)y|fhy|PQ^`Qaf`gayp^`gn|b ~\Q»}bdQa]^`N)QtP1y/^hNdQP1yp^`gjyp%]`cd©)]`{)y|Q·¶n|±m»)bdg^`QR~\gPRQb)]`gnkb}¹&gbª&N)gN nk{\^`gPRgy/^`gn|b*nKacdf]a¥ ¤±Fy{LnKn|fNdn|gjQzgj]mP1yk~\QU©K[t^`NdQc)]_Qafa¨k^`N)QU©LQa]_^(fhQay|N}yp©dQz]_N}yp{LQzª&gL©}QU±¸ypfm±¶fhn|PÉ^hNdQ]`n|c\^`gn|b ·¸np±¾gb\»)bdg^`Qº~\gPRQab)]_gn|bL¹mypb}~R^`NdQºn>]q^m±¶c)b)
(48) ^hgnkb$ª&gL©}Qn|PRQzPRnkf`QUn|PR{dQ¼ ¨|[KgQj~\gbdty]_^`g !@Qf nk{\^`gPRgy/^hgnkb${dfhn|©dQP¥ MNdQypgPnp±F^`NdQ{dfhQa]`Qb>^ª(nkf Hg]^hn~dQ°|Qank{*ytPQ^`Ndn\~\nknk|[^`nRy|~)yp{\^&ypbgbdg^`gjyp y|b)~${LQfhN)yp{}](b}yp¯G°|QU{}ypfypPRQ^hQfhg¢£ y/^`gn|b^hn^hNdQ&{)y|f_^hgcdy|f+{dfhn|©dQP ]_^`c}~\gQ~ ¨|fhQa~dc)gbd]_nº^hNdQ&gbdÀ)cdQb}Q&np±L^`N)Q&gbdg^`gjyp}Ndn|gjQnp±L^`N)Q&c)]_Qafmypb)~
(49) ")
(50) " ,. 0 "( ! np±^`NdQ1n/°kQfyp {dfhn\Q~\cdfhQ|¥ ³b§^hNdQR±¶fypPRQªn|If Hnp±&ypQfhn\~\[Kb)ypPRgjt]`N)y|{}QRn|{d^`gPgayp^`gn|b¾¨¾^qª(nn|cd^hnkPRQa] y|f`QQ¼\{LQa^`Qa~.
(51) f`Qy|Ny7]`N)yp{LQnp±U©LQ^`^`QfR»)^`bdQ]`] ·¸gB¥ Q|¥ ~\Qaf`Qy|]`Q$^`N)Q~dg]_^hy|b)Q©}Q^qª(QaQb^`NdQn|{d^`gPcdP ]`N)y|{}Q±¶n|c)b)~y|b)~¡^hNdQ ]_nkcd^`gn|b}¹
(52) gb}fhQay|]`Q^`NdQn|bK°|Qaf`kQb)QUfy/^`Q1·¸gB¥ Q|¥+gPR{df`n/°kQU^hNdQnkb)~\g^`gn|bdgbdRnp±^`N)Qn|{\^hgPRgay/^hgnkb{)f`nk©dQaP$¹
(53) ¥ ijk. ". #$%'&)(*%,+.-, 0/. ¦®~dgfhQa^typ{d{)f`n>y|N^hnPRn|bdg^`nkfy|~)yp{\^hgnkb«y|b)~7PRnK~dg¢±¶[§^`NdQ²²%´ {)ypfypPRQ^`Qfhgay/^hgnkb{dfhQa]`Qb>^hQa~«yp©Ln/°|QRgj]^`n c}]_Q1^hNdQ P1yp{){dgb){dfhnK~dc)gbd ^hNdQ²%²%´ nKn|f~\gb}y/^`Q] (s, t, u) ±¶f`nkP-^hNdQ*jy/^`^`gjQ*n>nkfh~dgb)yp^`Q] (ξ, η, ζ) ¥ ³b}~\QQ~ ¨
(54) ,- 10 (
(55) " )"=I324%
(56) ,U 52&) !" # ,. .
(57) "U6 ,.
(58) ,U 362 '
(59) "87 ") 9' ,. " !:2 ; gb*^`NdQ²%²%´ ±¶nkf`Ptcdjy/^`gn|b · G¹
(60) ¥ &[Pn\~\g±¶[Kgb)z^`N)g]P1yp{d{)gbd}¨½n|bdQypbtn|b>^hf`nk/^`NdQnKay/^hgnkbnp±d^hNdQ(~\Q±¶n|fhP1y/^`gn|b^hN)y/^%gj]Fnk©\^hy|gbdQ~©K[{}Qaf_^hcdf`©}y/^`gbd Qy|N$ªQg|N>^`gbdnKQ $gQb>^a¥+MNdQaf`Q±¶n|fhQ|¨k^`NdQºPRn\~\g¢»Ly/^hgnkb$np± ^hNdQºPRy|{d{dgbdt]`QQaP1]m^hn©LQºy|nKn\~1PRQaypb}]+^`nRnkb>^`fhn|d^hNdQ {)y|fhy|PRQ^`Qaf`gayp^`gn|b¾¥ MNdQ$yk~dyp{d^`gn|b7{df`n\Qa~\cdfhQ1n\cdfh]yp^]_nkPRQR]_^`Qa{)]n|±(^hNdQ*]`N)yp{LQ1n|{\^hgPRgayp^`gn|b«{dfhnKQa~\c)f`Qk¥ ¤^g]bdnp^Qypfy/^^hNdQ {dfhQa]`Qb>^^`gPRQª&NdQab¡^hNdQyk~dyp{d^`gn|b]`Ndnkcd~¡©}Q y|f`fhgQa~«n|cd^nk{\^`gP1yp[k¥¡MNdQ]`N)y|{}QN)y|]t^`n§©}Q n>]_QQbdnkcd|N¬^hn^hNdQ nk{\^`gPcdP ]`N)yp{LQgb nkfh~dQf^`n${)f`n/°Kgj~\Q]_cdg^hy|©dQyRgb\±¶nkf`P1yp^`gn|b^`n*yk~dyp{\^&^hNdQ{)y|fhy|PQ^`Qaf`gayp^`gn|b¾¥ Un/ª(Qa°|Qf¨\n|b)Qª(nkcd~ g HkQy|~dy|{\^`gn|b ^hn¬nKacdf*©}Q±¶n|fhQ^`NdQ ±¶c)ºn|bK°kQfh|Qb}Qnp±º^`NdQnk{\^`gPRgy/^hgnkb gj]$ykNdgQa°|Q~ ¨(gb n|f~\QafR^`n¬gP{)f`n/°kQ^hNdQ n|bK°|Qaf`kQb)Qºfy/^hQ|¥&MNdQaf`Q±¶n|fhQ|¨d±¶c)f_^hNdQfQ¼\{}Qaf`gPRQb>^h]Uª&gF©LQfhQai>cdgfhQa~^`n*~dQ^`Qaf`PRgbdQy$]`cdg^hy|©dQtfhg¢^hQfhgnkb^`n${LQf`±¶n|fhP yk~dyp{\^hgnkb¾¥ MNdQy|~dy|{\^`gn|b{)f`n\Q~\cdfhQmnkb)]_gj]_^h]¾gbPRgbdgPRggbdy|by|~)yp{\^hgnkbnk]_^¾±¶cdb}
(61) ^`gn|bᦻ&NdgjNPRQayk]_cdfhQa] ^`NdQcdbdQ !LQ
(62) ^`g°|QabdQa]h] n|±¾^`NdQcdf`fhQb>^{)ypfypPRQ^hQfhgy/^hgnkb}¹%©K[1PRnK~dg¢±¶[Kgbd]`n|PRQºy|~dy|{\^`gn|b${}ypfypPRQ^hQf]¨>ª&NdgjN~\Q»)bdQ^`NdQºP1yp{d{dgbd}<¥ Un/ª(Qa°|Qafa¨ ^hNdQyk~dyp{\^hgnkb1{dfhnKQa]h] ]`Ndn|c)~RQ^(^`NdQºcdfhf`Qab>^(]_N}yp{LQUcdb)y|¢^hQfhQa~ ¨>y/^Qy|]_^ gbyQay|]_^_£³]hikc}ypfhQa]+]`Qb)]`Q|¨>gb*n|f~\Qf+^hn©LQbdQ»d^ ±¶fhn|Pµ^hNdQ1n|{d^`gPgayp^`gn|b7{)y/^hN7y|fhQayk~\[ {LQf`±¶n|fhPQ~ ¥RMNdQb¾¨F^`NdQRPRgbdgPRgayp^`gn|b«np±(^`NdQ1y|~)yp{\^hgnkb7nk]_^º±¶c)b)
(63) ^hgnkb7]`Ndn|c)~ ©LQ]`cd©\oqQ
(64) ^º^`n^`NdQRnkb)]_^`fypgbk^U^`N)yp^^hNdQtfhQa]`cd^`gbd]`N)y|{}Q»d^h]^`N)Qcdf`fhQb>^]_N)y|{}Qtgb7y$Qayk]q^`£¤]hi>c)ypfhQ]`Qb)]`Q|¥ N)y/^hQ°kQf ^hNdQb)yp^`cdfhQn|±^hNdQtyk~dyp{d^`gn|b n>]q^z±¶c)b)
(65) ^hgnkbypb)~{}ypfypPRQ^hQf]¨\^hNdQtyk~dyp{d^`g°|Q]`N)yp{LQn|{d^`gPgayp^`gn|b {dfhn\Q~\cdfhQg]z»)b)yp[ nkf`>ypbdgQ~*y|]±¶nkn/ªz]. 1W "!.
(66)
(67) !"$#% &'"($ !)+*-,.. 0/1
(68) 2 & !" J. G|¥ ³bdg^`gjypgy/^`gn|b. Ê(NdnKnk]`Qypb gbdg^`gjyp]_N}yp{LQ nkf`fhQa]`{Ln|b)~\gbd«^hn¬^`NdQ7gbdg¢^hgy|º~dQa]`gkb °½y|f`gjyp©)Q] ∆P = 0 ±¶n|f*^`NdQ§gb)g¢^hgy| P1yp{d{dgbd M ·¶gj~\Qabk^hg¢^q[*±¶n|f&Sgb)]_^hypb}Q¹¥ \¥ &Q|gbdbdgbdRnp± ^`NdQnKn|{n/°|Qafyk~dyp{d^`gn|b]q^hQ{)] ]_Q^ l ← 0 3 ¥zYKN)y|{}Qnk{\^`gPRgy/^`gn|b l%Qf`±¶n|fhP k nk{\^`gPRgy/^`gn|b]_^`Q{}][>gQj~\gbdR^`N)Q]_N)y|{}Q S ~\Q»)b)Qa~©K[1^`NdQ~dQa]`gkb°/ypfhgjyp©dQa] ∆P ¥ )¥l+ypfypPRQ^hQfhgy/^`gn|by|~dy|{\^`gn|b Tz{@~dy/^hQ^`NdQRP1yp{d{)gbd^hn ©K[PRgbdgPggbd^`N)QRyk~dyp{d^`gn|b§nk]_^±¶c)b)
(69) ^hgnkb¾¨F]`cd©\oqQa^^`n^`NdQ1nkb)]_^`fypgbk^º^`N}y/^ ^`NdQ]_N}yp{LQ S f`Q]_cd^`gbdM1±¶fhn|P0^hNdQP1yp{){dgb)N)y|bd|Qy|b)~ ~\Q»)bdQa~©K[*^hNdQ~dQa]`gkb °½y|f`gjyp©)Q] ∆P gj]z^hNdQ Qy|]_^_£³]`i>c)y|f`Q](y|{d{dfhn½¼\gP1y/^hgnkb*n|± S ¥ \¥¿mb)~np±^`N)QnKn|{ ]_Q^ l + 1 ¤±%]_^`lnk← {d{dgbd1fhg¢^hQfhgnkb*fhQaykNdQa~*^`N)y|b]_^`nk{¾¥ ¿mj]_Q ºn|^`n$\¥ &Q±¶nkf`Q^hNdQUyk~dyp{\^hgnkb$n>]q^+±¶cdb}
(70) ^`gn|b$y|b)~R{)y|fhy|PQ^`Qafh]+y|f`Qz~\Q»)bdQ~ ¨>^`NdQzQayk]q^`£¤]hi>c)ypfhQa]%y|{d{dfhn½¼KgP1y/^hgnkbtgb1^`N)Q&±¶fhy|PRQ£ ªn|fIH1np±^`NdQ²%²%´ ~\Q±¶n|fhP1y/^`gn|bgj]&~\Qa]hfhg©LQa~gb^`NdQbdQ¼K^&]`Qa^`gn|b¾¥ (0) M0. (0) M0. 0. (kopt ) Ml. opt. (kopt ) Ml. l+1. (0) Ml+1. (0) Ml+1. (kopt ) Ml. .
(71) /3/ < 0/ +)+ %'
(72) %, & . ¨¾^hNdQ$]_N}yp{LQ$]_Ndnkcdj~fhQP1ypgb«y|]cdb}yp^`QfhQa~ NdQab«^`NdQ$cdfhf`Qabk^P1yp{){dgb) gj]cd{@~dy/^hQa~7^`n ykg°kQb7PRy|{d{dgbd ky ]{}n>]`]`g©)Qº^`n1©LQb)Q»d^&±¶fhn|P^hNdMQn|{d^`gPgayp^`gn|b{)y/^hNy|fhQayk~\[1{LQf`±¶n|fhPMQ~ ¥ YKcd{){}n>]_Qº^`N}y/^&^`N)Qcdfhf`Qab>^]`N)yp{LQ S gj] ~\Q»)bdQ~*©K[R^`N)Q~dQa]`gkb*°/ypfhgy|©dQa] ∆P ¥mMNdQab¾¨K^`N)Qº]`N)yp{LQ S f`Q]_cd^`gbdt±¶f`nkP:^`NdQP1yp{){dgb)N)y|bd|Q]`Ndn|cdj~$©LQ ^hNdQzQay|]_^_£³]hikc}ypfhQa]%yp{d{)f`n½¼\gPRyp^`gn|btnp± S ¥MNdgj]my|{d{dfhn½¼KgP1y/^hgnkbtgj]+n|©d^hypgbdQ~t©K[n|b}]_gj~\QfhgbdU^hNdQU²%²%´ ~\Q±¶n|fhP1y/^`gn|b ~\Q»)bdQ~¡©K[«^`NdQ~\Q]_g|b¡°/y|f`gjyp©dQa] ∆P ª&NdgjNPRgbdgPRgQ]^hNdQkyp{©}Q^qª(QaQb¡^hNdQcdfhf`Qab>^]`N)yp{LQ S ±¶nkf^hNdQ P1yp{){dgb) M ypb}~*^hNdQbdQaª ]`N)yp{LQ S ±¶n|f&^hNdQP1yp{d{dgbd M. V gb)gPRgQ Z Z 1 [∆q − ∆q ] dS, ·¸>¹ 2 ª&NdQaf`Q ∆q y|b)~ ∆q ypfhQ^`NdQ~\gj]_{djy|QPRQb>^]Un|± y1{}nkgb>^ q ±¶f`nkP ^hNdQgbdg^`gjyp%]_c)f_±¸ykQ S ¨L±¶nkfz^`N)Qt²%²%´É~dQ±¶n|f`£ P1y/^hgnkb)]{dfhn\~\c)gbdf`Q]_{LQa^`g°|Qa[^hNdQ1cdfhf`Qab>^]`N)yp{LQ·¶P1yp{d{)gbd M ¹y|b)~ ^hNdQ1bdQª yp{d{dfhn½¼\gP1yp^`Qa~§]_N}yp{LQ·¸PRy|{d{dgbd ¹¥ zn/ªQ°kQf¨)^hNdg]gb>^`Qa|fyp%ypb)bdnp^©}QRQa°½y|c}y/^`Q~gb^hNdQ1²²%´É±¶fypPRQªn|If H]_gb)QRbdnc)f`°KggbdQay|fº~dQa]hfhg{\^hgnkbnp± M g]y½°/ypgy|©dQ|¥%MNdQfhQ±¶nkf`Qk¨|^hNdQ~\g]hfhQ^hQz±¶nkf`Pn|±¾^`NdQQayk]q^`£¤]hi>c)ypfhQa]my|{d{dfhn½¼\gP1y/^hgnkb1g]{dfhQ±¶Qaf`fhQa~¾¨>©K[RPRgb)gPRggbd. S X1 · Jp¹ [∆ q −∆ q ] δS , J = 2 ª&NdQaf`Q ypb}~ fhQ{dfhQa]`Qb>^^`NdQ~\gj]_{djy|QPRQb>^]np±^`N)Q$PRQa]`N¬bdn\~\Q ±¶f`nkP^hNdQ$gbdg¢^hgy|(]`cdf`±¸y|Q ±¶nkf^hNdQ∆²%²%q´ ~\Q±¶nkf`P1y/∆^hgnkb)q]{)f`n\~\c)gb)f`Q]_{LQa^`g°|Qa[7^`N)Q*bdQaª®]_N)y|{}Qypb)~¬^`NdQcdfhf`qQab>^t]_N}yp{LQ|¥ δS g]y ªQg|N>S^`gbd n>Q $gQb>^zQ¼\{dfhQa]h]_gbdt^`NdQ~dg]hfhQ^hQºgb>^`Qa|fy/^`gn|bnp± ^`NdQ{dfhQ°Kgn|c)]gb>^`Qa|fypG¥ T]_gbd ·Bk¹
(73) ¨\^hNdQ±¶n|n/ª&gbdQ¼\{dfhQa]h]_gn|bg]U~\Qfhg°kQa~. X1 · Kk¹ ] δS . J = −∆ q [B ∆P 2 MNdQ]`n|c\^hgnkbn|±F^hNdg]z{df`nk©dQP aypb©}QQy|]`g[$nk©\^hy|gbdQ~©>[Q¼\{dfhQa]h]_gbdt^`N)yp^&^`NdQkfhyk~\gQb>^&np±^`NdQ±¶cdb}
(74) ^`gn|b}yp¾°/ypb)g]`NdQa]a¨ ª&NdgjN[KgQa~d]^hNdQ±¶n|n/ª&gbdRgb)Qaypf&]`[\]q^hQP n|± bdnkf`P1y| Qai>c)y/^hgnkb)]. # " ) ( X X ·Bk¹ δS . B ∆ q = δS ∆P B B . l. l+1. (kopt ) Ml. (kopt ) Ml (kopt ) Ml (0) Ml+1 (0) Ml+1. l. (0) Ml+1. (kopt ) Ml. l+1. (0) 0. SM. (0) Ml. (kopt ) 2 Ml. (0) Ml+1. (kopt ) Ml+1. (0) M0. l. l+1 (0) M0. N. n (0) Ml+1. LS. n (kopt ) 2 Ml. n. n=1. n (0) Ml+1. n (kopt ) Ml. n. n. N. T n Ml+1. LS. n (kopt ) 2 Ml. (0) Ml+1. n. n=1. N. N. n Ml+1. W1WYXZU[\]\. n=1. T n Ml+1. n. (0) Ml+1. n Ml+1. n=1. n (kopt ) Ml. n. (0) 0.
(75) " K. ² gb)yp[k¨F±¶nkfy|g°|Qab§P1y|{d{dgbd M y|b)~«y cdfhfhQb>^]_N)y|{}Q S ¨F^`NdQ²²%´ ~\Q±¶nkf`P1y/^hgnkb«ª&NdgN7»d^] S gb¡y Qayk]q^`£¤]hi>c)ypfhQa]]_Qab)]`Q±¶nkf&^`N)QºP1y|{d{dgbd M gj]{dfhn/°>gj~\Q~*©K[*]`n|°Kgbdt^`N)Q{df`Qa°Kgnkc)]gbdQypf&]_[\]_^`QaP®±¶n|f ∆P ¥ (kopt ) Ml. l+1. (kopt ) Ml (0) Ml+1. l+1. . . +)+)%& + <
(76) /. MNdQRP1yp{){dgb){dfhn\~\c)gbd^`N)QR²%²%´®n>nkfh~dgb)yp^`Q] ±¶fhn|Pµ^hNdQRyp^_^hgQn>nkfh~dgb)yp^`Q] gj]c)]`Qa~ ^hnPnkbdg^`n|f ^hNdQyk~dyp{\^hgnkb1{dfhnKQa]h]¥%MNdQaf`Q±¶n|fhQ|¨p^hNdgj] P1yp{){dgb)N)(s,y|]mt,^`nu)©LQU{)f`Qgj]_Qa[R~\Qa]hfhg©}Q~1ypb)~R^hNdQ(ξ,yk~dypη,{d^`ζ)gn|b1°/ypfhgy|©dQa]m~\Q»)bdQ~ ¥ M n$]_gPR{dg¢±¶[*^`NdQ{dfhn|©)QaP¨K^hNdQ±¶n|n/ª&gbdRy|]h]`cdPR{\^`gn|b)]&y|f`QºP1y|~\Q. · Gk¹ s = φ(ξ) t = ψ(η) u = θ(ζ). MNdgj]Ndn|gjQUn|fhf`Q]_{Ln|b}~d]%^`ntypb1cdb)n|cd{)Q~1f`Qayp^`gn|b)]`Ndg{1©}Q^qª(QaQbR^hNdQjy/^`^`gjQUnKn|f~\gb}y/^`Q]mypb}~R^`NdQº²%²´ n>nkfh~dgb)yp^`Q]¥ MNdQ]_Qyk]`]`cdPR{\^hgnkb)]&ypfhQf`Qy|]`n|bdb)y|©dQºyk]nkbd1y|]^`N)QPRy|gb ~\gfhQa
(77) ^hgnkb)]np±^`N)Qyp^_^`gjQy|f`QªQF]`cdg¢^hQa~^`n^hNdQ]`N)yp{LQº^`n nk{\^`gPRgaQ|¥ MNdQab¾¨}±¶cdb)
(78) ^hgnkb)] ¨ y|b)~ ypfhQ]_c){d{}n>]_Q~^`n$©LQ{Ln|[Kbdn|PRgjypj]znp±m~\Qa|fhQQ] n ¨ n ypb)~ n ¨Ly|b)~ ypfhQQ¼\{dfhQa]h]_Q~gb ^hNdQ &Qfhb)]q^hQgb{Ln|[Kbdn|φPRψgjypj]©)y|]`θgj]. 0 i. n0. 0. φ(ξ) =. ni X. n0i. Bi (ξ)φi. ψ(η) =. n0 Bj j (η)ψj. j j=0,...,n0j. i i=0,...,n0i. · G G¹ ypfhQºn|b)]`gj~\QfhQa~y|]&yk~dyp{\^hgnkb°/y|f`gjyp©dQa]a¥ ypfhQ]`Q^U^hn*gb)Qaypfº°/ypcdQa]a¨Lgb§n|f~\Qf. θ(ζ) =. j=0. i i=0,...,n0i. 0 k. 0. j X. ² gb)yp[k¨dªQg|N>^`gbdRnKQ$gQb>^] (φ ) ¨ (ψ ) py b}~ (θ ) ³bdg^`gjyp[|¨@ª(QagkN>^`gbd*nKQ 1gQab>^h] ¨ (ψ ) py b)~ (θ ) ^hn$~\Q»)b)Q^`NdQgj~\Qb>^hg¢^q[$P1y|{d{dgbd (φ ) i=0. 0 j. n0. Bk k (ζ)θk .. k=0. k k=0,...,n0k. j j=0,...,n0j. nk X. k k=0,...,n0k. i i = 0, n0i , n0i j ψj = 0 j = 0, n0j , nj φi =. k n0k. θk =. · G½|¹. k = 0, n0k .. MNdQab¾¨^hNdQy|~)yp{\^hgnkb {)f`n\Q~\cdfhQ¡nkb)]_gj]_^h] gbPRnK~dg¢±¶[Kgbd ^`N)Qy|~dy|{\^`gn|b °/ypfhgjyp©dQa] (φ ) ¨ (ψ ) ypb)~ ` ^ 1 n R P g d b g R P g Q h ^ d N Q | y d ~ | y \ { ` ^ g | n b > n q ] ^ ¶ ± d c } b
(79) ` ^ g | n F b ¥ (θ ) ³bn|f~\Qf^hn~dQ»)bdQ&yºn|b)Q£G^`np£¤n|b)Q P1yp{d{)gbd}¨/±¶cdb)^`gn|b)] ¨ ypb)~ ]`Ndn|c)~t©}QNdnkPRQn|PRnkf`{dN)g]`P1]F±¶fhn|P ^`n ¥ YKc 1gQab>^+n|b)~dg¢^hgnkb)]^`ny|NdgQ°kQ ^hNdgj]+fhg¢^hQfhgnkby|bt©LQ&~\Qφfhg°kψQa~±¶f`nkP θ ^`NdQ^`Qaf`PRgb)y/^hgnkb{dfhn|{LQf`^`gQa]%y|b)~t~dQ[0,fhg°/y/1]^hgnkb[0,fhcd1]Qa] n|± &Qfhb)]_^`Qagb{}nk[KbdnkPRgy|] K ¤¥ YKc $gQb>^Unkb)~\g^`gn|b)]&aypb©}QQ¼\{df`Q]`]`Qa~y|]. i i=0,...,n0i. j j=0,...,n0j. k k=0,...,n0k. φ0 = ψ0 = θ0 = 0,. ·G3 ¹. φn0i = ψn0j = θn0k = 1, φi < φi+1. i = 0, n0i − 1,. ψj < ψj+1. j = 0, n0j − 1,. θk < θk+1. k = 0, n0k − 1.. MNdQ]_Q n|b)~\g^`gn|b}]1ypfhQ^yHkQb gb>^`n¬yknkcdb>^Ryk]RgbdQypf$nkb)]_^`fypgbk^]R~\cdfhgbd7^hNdQPgbdgPRgy/^`gn|bn|±U^`N)Qy|~dy|{\^`gn|bn>]q^ ±¶cdb}
(80) ^`gn|bF¥ MNdQ&~\Qa|fhQQ] ¨ ypb)~ f`cdQ^`NdQzn|PR{dQ¼\g^q[np±}^`NdQ&fhQa]`cd^`gbdPRy|{d{dgbd)¥ ¤^ ]_QaQP1]%c)]`QQa]h]^`n~\Q»)bdQzyºP1yp{d{dgbd ª&Ndn>]_Qn|PR{dQ¼\g^qn[*gj]&nNdg|NdQaf&n^`N)y|b^`N)yp^Unp±^`N)Q~\Q±¶nkf`P1yp^`gn|b¾¥MNdQb¾¨)^`NdQt~\Qkf`QaQa]np±%^hNdQP1yp{d{dgbd n ¨ n ypb}~ n y|f`Q ]`gPR{d[]`Q^zQi>c)yp@^hnt^hNdnk]`Qnp±^`NdQ~dQ±¶n|fhP1y/^hgnkb n ¨ n y|b)~ n ¥ (bdQ&aypbb)np^`gjQ^hN)y/^%^`N)Q&PRy|{d{dgbd~\Q»)b)Qa~t©>[· Gak¹%ypb)~· G Ga¹n|PR{Lnk]`Qa~tª&g¢^hN^hNdQ&²%²´ ~\Q±¶nkf`P1y/^hgnkb· G¹fhQa]`cd¢^]gb 2 )
(81) " ,. +
(82) "U
(83) !32O !# !" I3 1 # "() !" "
(84) "U32 '"$ !32 '0
(85) ,U !2 !# !" 3 1P# "() !" #,: ,. 2 ª&g¢^hNf`Q]_{LQa^&^`n^`NdQjy/^`^`gjQn>nkfh~dgb)yp^`Q] (ξ, η, ζ) ¥ 0 i. 0 j. 0 k. 0 i. i. . j. 0 j. 0 k. k. 1W "!.
(86) .
(87) !"$#% &'"($ !)+*-,.. 0/1
(88) 2 & !" . <+ %' & )& ( %, &
(89) -. . MNdQtyp{d{dfhnkykNc}]_Q~*^hn*~\Q»)bdQ^`N)Qy|~dy|{\^`gn|b±¶cdb}
(90) ^`gn|b}ypFgj]&gb)]`{dgfhQa~±¶fhn|P yRPRQ^`N)nK~ ~dQ°|Qank{}Q~±¶n|f&^qªnp£³~\gPQab)]`gnkb)yp {dfhn|©)QaPR]a¨ ©)yk]_Q~nkb§y{)ypfypPRQ^hQfhgy/^hgnkb©K[ &egQfcdf`°kQa] d¨¾\¨¾ ¤¥ ³b§^`N}y/^]_gPR{dQfyk]_Qk¨ g¢^ªy|]º]_N)n/ª&b^`N}y/^ypb yk~dyp{\^hgnkb¬{dfhnKQa]h]f`Qa[Kgbd n|b«^hNdQ*fhQkcdjypfhgy/^`gn|b§np±^`N)Qnkb>^`fhn|+{Ln|[>kn|b«gj]{)y|f_^hgcdy|f`[Q $gQb>^a¥ ³b)~\QaQa~ ¨%g^ªy|] nk©)]_Qaf`°kQa~^`N}y/^z^hNdQ]`N)yp{LQn|{d^`gPgayp^`gn|b {dfhn\Q]`][KgQj~d]Uy1N)gkNd[gfhfhQ|c)y|fzn|b>^hf`nk¾{}nk[Kkn|b¾¥MNdgj]z{dNdQabdn|PRQabdn|bªy|] y|]`n ~\QPRn|b}]q^hfhyp^`Qa~§©K[§y]`{}Q
(91) ^`fypmy|b)yp[\]_gj]np±^`N)Q$yp|Q©)fhy|g]_[\]_^`QaPgb ¤¥ (b7^`NdQ1n|^`NdQafN)ypb}~ ¨Fg^gj]ª(QaP HKbdn/ª&b ^hN)y/^Ug±%^`NdQtbKcdP©LQfzn|±mn|b>^hf`nk {}nkgb>^h]&^`Qb}~d]&^hn$gbd»)bdg^q[|¨d^hNdQn|b>^`fhn|¾{Ln|[K|nkb*^hQb)~d]z^`n1^hNdQcdf`°kQ|¨dª&N)gNgj]zfhQ|c)y|fa¥ MNdQaf`Q±¶n|fhQ|¨@ypb gf`fhQkcdjypfUn|b>^`fhn|¾{Ln|[K|nkbypb©}Qn|b}]_gj~\QfhQa~ yk]Uy|b gb)~dgay/^`nkfUn|± y$n/ª P1yp^`cdfhg¢^q[{)y|fhy|PQ^`Qaf`gayp^`gn|b¾¥ MNdQab¾¨Fy|b§Q 1gQab>^yk~dyp{d^`gn|b§y|kn|fhg¢^hNdP ªyk]º©dcdg¢^¨ ^`N)yp^f`Qa|cdjypfhgaQa]U^`NdQ1n|b>^`fhn|{}nk[Kkn|b©K[ PRgbdgPRggbd^`NdQ^hnp^hy| °/ypfhgjy/^`gn|bnp± ^`NdQn|b>^`fhn|@{Ln|gb>^h]zn>nkfh~dgb)yp^`Q] {y }. Z X · Ga >¹ |y (t)|dt, |y − y | ≈ T V ({y }) = ª&NdQaf`Q y(t) gj]zyf`Qa|cdjypf±¶cdb}
(92) ^`gn|bgb>^`Qfh{Ln|jy/^`gbdt^`NdQn|b>^`fhn| {Ln|gb>^h] {y } ¥ MNdgj]yp{){df`n>y|Ngj]Q¼>^hQb)~dQa~^`n ²%²´ ©K[gb>^hf`n\~\c)gb)yPRQayk]_cdfhQnp±^`NdQR°/ypfhgyp^`gn|b)]ºn|± ^`N)Q$~\Q±¶nkf`P1y/^hgnkbgb«Qay|N ~\gfhQa
(93) ^hgnkb. Z Z Z . · Gk¹. ∇δP (ξ, η, ζ) dξ dη dζ, ª&NdQaf`Q gj]ºy$fhQ|c)y|f&±¶cdb)^`gn|b±¶fhn|P ^hn gb>^`Qaf`{Ln|jy/^hgbd$^`NdQRnkbk^hf`nkF{Ln|gb>^h]º~\g]`{djy|QPRQb>^h]ypb)~ ∇ gj]&g^h]&kδPfhyk~\(ξ,gQab>η,^(^hζ)Qb)]`n|f¥+MNdQº±¶n|fhPcdjy· G|¹g]^hNd[0,QP1]c)¢^hg~\gPR< Qb)]`gn|b)y| Q¼K^`Qab)]_gn|bnp±^`N)Qgb>^`Qa|fyp+· Ga k¹
(94) ¥ YKgb)Q gj]n|bd[ HKbdn/ª&byp^ {}nkgb>^]¨¾y~\gj]`f`Q^`Qt±¶n|fhPnp±· G|¹Ugj]ºc)]`Qa~§gb {dfy|^`gjQº^`δPn1Qa(ξ,°/ypc)η,y/ζ)^hQ^hNdQy|~)yp{\^hgnkbn>]q^&(n±¶c)b)
(95) +^hg1)nkb × (n + 1) × (n + 1). XXX · Gk¹ 1. J = ∇δP . i i=0,...,ni ni. 1. 0. i. i. i−1. 0. i=1. i i=0,...,ni. 1. 0. 1. 0. 1. 0. 3. i. 3. j. k. ni. AD. ni nj nk. nj. nk. ijk. i=1 j=1 k=1. g]yPRyp^`fhg¢¼ bdnkf`Py|b)~ y|bQ]q^hgP1y/^hQtn|± (ξ, η, ζ) n/°|Qf^`N)QtQaQaPQab>^hypfh[°|nkcdPRQnp± gb)~\gjQ] ijk ¥MNdQ c)]`Qnp±^`N)Q²dfhn|©LQbdgc)]bdnkf`P0∇δP ypb)~yt»)bdg^`Q£¤~dg !@QfhQb)Q∇δP yp{d{)f`n½¼\gPRyp^`gn|bnp±^`NdQkfhyk~\gQb>^n/°kQf^hNdQQQPRQb>^ypfh[$°|nkcdPRQnp± gb)~\gjQ] ijk [KgQa~d]. k.k. ijk. 3 X. n2i. l l l l (∆Pijk − ∆Pi−1jk )2 + (∆Pij−1k − ∆Pi−1j−1k )2 ∇δPijk = 4 l=1. l l l l + (∆Pijk−1 − ∆Pi−1jk−1 )2 + (∆Pij−1k−1 − ∆Pi−1j−1k−1 )2. n2j l l l l (∆Pijk − ∆Pij−1k )2 + (∆Pi−1jk − ∆Pi−1j−1k )2 + 4 l l l l + (∆Pijk−1 − ∆Pij−1k−1 )2 + (∆Pi−1jk−1 − ∆Pi−1j−1k−1 )2 n2k l l l l + (∆Pijk − ∆Pijk−1 )2 + (∆Pij−1k − ∆Pij−1k−1 )2 4. +. l (∆Pi−1jk. −. l ∆Pi−1jk−1 )2. +. l (∆Pi−1j−1k. −. l ∆Pi−1j−1k−1 )2. . · GJp¹. . 21. ,. ª&NdQaf`Q ∆P gj]^`N)Qn|PR{Ln|bdQab>^ l n|± ^hNdQ~\gj]_{djy|QPRQb>^&np±^`N)Qn|b>^hf`nkL{Ln|gb>^ ∆P ¥ ²gb)yp[k¨>^`NdQy|~dy|{\^`gn|b*{df`n\Qa~\cdfhQUn|b)]`g]_^h](gb*PRgb)gPRggbd^hNdQºyk~dyp{d^`gn|b*nk]_^ ±¶cdb)^`gn|b Q°/y|c)yp^`Q~1c)]`gbd· G|¹ y|b)~«· GJp¹
(96) ¨dª&g^`N^`N)Qn|b}]q^hfhy|gb>^^`N)yp^&^`NdQcdf`fhQb>^z]`N)yp{LQf`QaPRy|gb}]cdb)yp^`Qaf`Q~ ¨\gbyQayk]q^`£¤]hi>c)JypfhQ]`Qb)]`Q|¥ l ijk. ijk. AD. W1WYXZU[\]\.
(97) G. " . . <+ %' &
(98) - % ). . ²)f`nkPµ^`NdQ$~dQ°|Qank{dPRQb>^h]~\Q]`f`g©LQa~7yp©Ln/°|Q|¨@^hNdQ1y|~dy|{\^`gn|b«yp|nkf`g^`N)Pypb7bdn/ª ©LQ$~\Q»)bdQa~§gb«~dQ^hy|gG¥1²dnkfycdf`fhQb>^ ypb)~ (θ ) ¨dypb}~*ypbcdfhfhQb>^]`N)yp{LQ P1yp{){dgb)~\Q»)bdQ~*©K[R^`N)Qyk~dyp{d^`gn|b*{}ypfypPRQ^hQf] (φ ) ¨ (ψ ) ~\Q»}bdQa~R©K[t^`NdQz~\Q]_g|bR°/ypfhgjyp©dQa] ∆P ¨|^hNdQUyk~dyp{\^hgnkbR{df`n\Qa~\cdfhQg](~\Qa]hfhg©}Q~t©K[^`NdQ&±¶nkn/ª&gbdyp|n|fhg^`NdP. S G|¥¿ ]_^`gPRyp^`Q^hNdQ»d^`b)Qa]h] J np±F^hNdQc)f`fhQb>^&P1yp{d{dgbdRc)]`gb)· Gp¹ \¥ &Q|gb^`NdQnk{\^`gPRgy/^`gn|bn>nk{ 3 ¥¿m°/ypc)y/^hQ^`NdQ kfhyk~\gQb>^1np± J ª&g¢^hN fhQa]`{LQa
(99) ^1^hn«^`NdQyk~dyp{d^`gn|b{)ypfypPRQ^`Qf] (φ ) ¨ (ψ ) ypb)~. (θ ) )¥&Tz{@~dy/^hQ (φ ) ¨ y|b)~ (θ ) c)]`gbd§^`N)Q~\Qfhg°/yp^`g°|Qa]·¶kfhyk~\gQb>^_£¤©)y|]`Qa~¡n|{\^hgPRgay/^hgnkb}¹ [KgQa~\gbd1ybdQª P1yp{d{)gbd(ψ ) \¥UÊ(n|PR{dc\^hQ^`NdQb)Qª ²%²%´ nKn|f~\gb}y/^`Q](±¶n|fzyp bdn\~\Q]n|b S c)]`gbd · G 3 ¹
(100) ¨F· GGa¹y|b)~7· Gak¹ d¥UÊ(n|PR{dc\^hQ^`NdQb)Qª °kQa^`n|f] B ~\Q»}bdQa~©K[· 3 ¹(±¶n|f^hNdQ^`Qab)]_nkf`gjyp@{dfhn\~\c)^ ±¶nkf ^hNdQºbdQª P1yp{){dgb)©K[1]_nk°Kgbd^`N)Qgb)Qaypf]`[K]_^`QaPµ·¸k¹¨ JK¥¿m°/ypc)y/^hQz^hNdQQayk]q^`£¤]hi>c)ypfhQa] yp{d{dfhn½¼\gP1yp^`gn|b1np± S [KgQa~\gbd^`N)QbdQª nkbk^hf`nk@{}nkgb>^h]&~dg]`{djy|QaPRQb>^h] ∆P Kd¥¿ ]_^`gPRyp^`Q^hNdQ»d^`b)Qa]h]np±^`NdQbdQaª P1y|{d{dgbdRc)]_gbd · G½p¹ d¥¿mb)~np±^`N)Qn|{\^hgPRgay/^hgnkbnKn|{ ¤±%]_^`nk{d{dgbd1fhg¢^hQfhgnkb*fhQaykNdQa~*^`NdQab ]q^hn|{ ¿mj]_Qknp^`n 3 ¥ i i=0,...,n0i. (kopt ) Ml. j j=0,...,n0j. k k=0,...,n0k. (kopt ) Ml. AD. i i=0,...,n0i. AD. j j=0,...,n0j. k k=0,...,n0k. i i=0,...,n0i. j j=0,...,n0j. k k=0,...,n0k. (0) 0. (kopt ) Ml (0) Ml+1. . . . %'(
(101) -% +.-, 0 0&
(102) %, &. M NdQ&kfhyk~\gQb>^n|±}^hNdQzy|~)yp{\^hgnkbtn>]q^%±¶cdb}
(103) ^`gn|b J ª&g^`NRf`Q]_{LQa^+^`nº^`N)Qzy|~dy|{\^`gn|b°/ypfhgy|©dQ] (φ ) ¨ (ψ ) y|b)~ N)y|]^hn$©LQQ°/ypc)yp^`Qa~ yp^UQaykN ]q^hQ{ np±%^hNdQPRgb)gPRgay/^hgnkb np± J ¥&²dnkf&^`N)Q]`yH|Qn|±+]`gP{)gjg^q[|¨)^`Ndgj] kfhyk~\(θgQab>^) gj]&nkPR{dc\^`Q~c)]_gbd1yt»)bdg^`Q~\g !@QfhQb)Qyp{d{dfhn½¼\gP1yp^`gn|b¾¥ MNdQUcd{L~)y/^`QUnp±@^`NdQy|~dy|{\^`gn|bR{)y|fhy|PQ^`Qaf ¨ ypb)~ ^`nPRgbdgPgQ g] ©)y|]`Qa~ nkb*y]_^`QaQ{LQa]_^~\Q]`Qb>^(PRQ^hNdn\~1gb)c)~\gbdty(φgbdQº)]_QypfNR^`nt(ψ~dQ^`)Qaf`PRgbdQy]_cdg^hy|(θ©dQ)]_^`Q{*Qabdp^hN¾¥ gbdQay|f(n|Jb)]_^`fypgb>^h]my|f`Q ^y H|Qabgb>^`n$yknkcdb>^c}]_gbdR{dfhnpoqQa^`gn|b)]gb>^`n^`N)Q±¶Qay|]`g©dQ]`cd©)]`{)y|Q|¥ MNdQgb)Qaypfz]`[\]q^hQP ·¸|¹gj]z]`n|°|Qa~*c)]`gbd^hNdQ T~\QnkP{Lnk]`g^`gn|b¾¥ i i=0,...,n0i. AD. k k=0,...,n0k. j j=0,...,n0j. AD. i i=0,...,n0i. j j=0,...,n0j. k k=0,...,n0k. AD. 1W "!.
(104)
(105) !"$#% &'"($ !)+*-,.. 0/1
(106) 2 & !". GG. º') & ¬ F M n gc)]q^hfhyp^`QR^hNdQ*y|~)yp{\^hgnkb§{dfhn\Q~\cdfhQR~dQa]hfhg©LQa~«yp©Ln/°|Qk¨Fy^qªnp£³~\gPRQab)]_gn|b}yp%Q¼dypPR{dQ$]`Ndn/ª&b«gb7»)kcdf`Q ·G|¹gj]nkb\£ ]`g~dQfhQa~ ¥¬YKcd{d{Lnk]`Q$^hN)y/^t^`NdQSU¦Uʦ0|U Gy|gf`±¶n|g gj]nk{\^`gPRgaQa~©>[¬y§]_N)y|{}Qn|{\^hgPRgayp^`gn|b¬{dfhn\Qa~dcdf`Qc)]`gbd§y²%²%´ {)y|fhy|PRQ^`Qaf`gayp^`gn|b · n = 6 ypb}~ n = 1¹¥%MN)g] {)ypfypPRQ^hQfhgy/^hgnkbtfhQgQ]mnkbR^`NdQ·¶gbdg¢^hgy|Ly|b)~Rb)y|¯B°kQ¹%gj~\Qb>^hg¢^q[P1yp{d{dgbd}¥ ´cdfhgbd^hNdQn|{\^hgPRgay/^hgnkb¾¨\ªQg|N>^`gbdRnKQ $gQb>^] ±¶nkf&yt°kQf`^`gjy|L~\Q±¶n|fhP1y/^`gn|bypfhQPRn/°kQa~1±¶fhn|PaQfhn^hn^`NdQagfn|{d^`gPRy| °/ypcdQ]¥%²dnkf+^`NdQz²%²%´ ~dQ±¶n|fhP1y/^hgnkbt^hQaNdb)gi>cdQ]¨pªQg|N>^hgbdnKQ $gQb>^h]+aypbR©LQUnkb)]_gj~\Qaf`Q~tyk]+n|b>^hf`nk\{}nkgb>^h] ~\g]`{djy|Q£ PRQb>^]Uypb}~bdnp^n|b>^`fhn| {Ln|gb>^h]nKn|f~\gb)y/^hQa]a¨)]_gb)Q^`NdQt]_N)y|{}Qt~\Q±¶nkf`P1y/^hgnkbg]&fhQ{)f`Q]_Qabk^hQa~ypb}~bdnp^z^hNdQ]`N)yp{LQg^h]`Q±q¥ MNdQaf`Q±¶n|fhQ|¨k^`NdQn|b>^`fhn|}{}nkgb>^h]n\y/^hgnkb$gj](b)np^&]q^hf`gj
(107) ^h[1~\Q^`QfhPRgb)Qa~ ¥ Un/ª(Qa°|Qf¨|g^&g]nkbK°|Qb)gQabk^(^`nn\ay/^`QU^`NdQn|b>^`fhn| {Ln|gb>^h]z]`n|PRQª&N)QfhQ|¨\gbnkfh~dQf^`n1{dfhn/°Kg~dQy°Kgc)y|gayp^`gn|bnp±^`NdQagfz~\gj]`{dykQaPQab>^h]a¥mMNdQfhQ±¶nkf`Qk¨\n|b»)|c)f`Q1·B|¹
(108) ¨d^hNdQnkb\£ ^hf`nk}{Ln|gb>^h]ypfhQzn\ay/^`Q~1y/^(^`NdQº{dNK[\]_gjy|){Ln|gbk^]mª&NdQaf`QU^`NdQgbdÀ)cdQb}Qºnp±Fy~\gj]`{dykQaPQab>^(gj] P1y/¼\gPcdP¨Kyp^`Ndnkcd|NR^hNdQa]`Q n\y/^hgnkb)]ypfhQR]_nkPQaª&N)y/^ypfh©dg^`fypfh[gb7^`N)g]°Kgc)y|gayp^`gn|b¾¥ ³b»}|cdfhQ ·Bd·Byk¹q¹¨¾^hNdQ1gb)g¢^hgy|m²²%´jy/^`^`gjQRgj]f`Qa{df`Q]_Qab>^`Qa~ ©K[y©djy| H*gbdQ|¨}ª&NdQfhQayk](^`NdQnk{\^`gPRgaQa~ ²%²%´yp^_^hgQ©>[yf`Q~gbdQ|¥ MN)Qa]`Qºjy/^`^`gjQ]znkf`fhQa]`{}nkb)~$fhQa]`{LQa
(109) ^hg°kQ[1^`nR^hNdQ gbdg^`gjypFS¦Uʦ|. G½y|gf`±¶n|g¾]`N)yp{LQ1·¶©djy| H*~)y|]`NdQa~cdfh°|Q½¹ypb)~*^`NdQnk{\^`gPRgaQa~ypgf`±¶n|g¾]_N)y|{}Q1·¸f`Q~cdf`°kQ¹¥ MNdQab¾¨½^`N)Qyk~dyp{d^`gn|b{dfhn\Q~\cdfhQmg]c)]`Qa~^`n»)b)~tyUbdQaª¬P1yp{d{dgbd}¨/ª&NdgNtPRgbdgPRgaQa] ^`N)Qnk]_^F±¶cdb}
(110) ^`gn|b ¨p]`cd©\oqQ
(111) ^ ^hn*^hNdQn|b)]_^`fypgb>^z^`N}y/^º^`NdQtf`Q]_cd^`gbd]_N)y|{}Qt»d^]z^hNdQtnk{\^`gPRgaQa~§]_N)y|{}Qtgb7y$Qayk]q^`£¤]hi>c)ypfhQ]_Qab)]`Q|¥MNdQtN)Jg]_^`nkf`[np±m^hNdQ PRgbdgPRgayp^`gn|b1np± J y|bR©}QU]`QQabRgbR»)kcdfhQ 3 ¥+¦U]maypbR©}Q]_QaQb¾¨>y|~dy|{\^`gn|bR[KgQa~)]my]q^hf`nkbdf`Qa|cdjypfhgay/^hgnkbn|±L^hNdQz²%²%´ jy/^`^`gjQ|¥mMN)QºfhQa]`cd^`gbdt²%²%´jy/^`^`gjQypb)~]_N}yp{LQºy|f`Q]_N)n/ª&b*gb*»)kcdfhQt·B)·¶©}¹h¹º·¶©dcdQgbdQy|b)~*cdf`°kQ¹¥%¦]Q¼\{LQa
(112) ^hQa~ ¨K^hNdQ n|b>^`fhn|+{Ln|gb>^h]jy/^`^`gjQ$c)]`gbd^`Ndgj]bdQaª P1y|{d{dgbd g]]_PRnKnp^hNdQf^hN)ypb7^hN)y/^c)]`gbd^`NdQ*g~\Qab>^`g^q[P1yp{d{)gbd}¥MNdQ$y|gf`±¶n|g ]`N)yp{LQgj]U]`gkNk^h[PRn\~\g¢»}Qa~ ¨LP1ypgbd[gb ^`NdQt°Kggbdg^q[np±%^`NdQ^`fypggbd$Q~\|Qk¥ (bdQaypb n|©}]_Qaf`°kQ^`N)yp^z^`NdQnkbk^hf`nk¾{Ln|gbk^] N)y½°kQPRn/°|Q~Ndn|fhgan|b>^hy|[*]`gb}Q^hNdQn\yp^`gn|b np±+^hNdQPRn>]q^Ugb\À)c)Qb)Qa~ {Ln|gb>^h]N)y|]U©}QaQbPRn\~\g»)Qa~ ª&g^`N^`NdQP1yp{d{dgbd N)y|bd|Qk
(113) ¥ zn/ªQ°|Qafa¨>^hNdQf`Qa|cdjypfhg^q[n|± ^hNdQ~\gj]_{)ykQPRQabk^]np±^`NdQn|b>^`fhn|@{Ln|gbk^]&gj]gb)f`Qy|]`Qa~ ¥ ²g|cdfhQ·¶ K¹ ]`Ndn/ªz]m^`NdQU±¶cdb)
(114) ^hgnkb ¨\ª&NdgN~\Q»)b)Qa] ^`NdQPRy|{d{dgbd)¥+MNdQU±¶cdb)^`gn|b)] ψ(η) ypb}~ θ(ζ) fhQP1ypgb*~\Q»)bdQ~ yk]zgj~\Qb>^`g^q[ ]_gb)Q n = n = 1 ¥¦U]ºφ(ξ) ypb ©LQnk©)]`Qfh°|Qa~¾¨}y|~)yp{\^hgnkb {df`n\~\c}Qa]y1P1yp{d{)gbd*±¶n|fª&NdgN ¥²gkcdfhQ*·G|¹ ]`Ndn/ªz]zN)n/ª ^`NdQtPRy|{d{dgbd*nKn H\]zg H|Q|¨@©K[f`Qa{dfhQa]`Qb>^`gbd$]`n|PRQ{)y|f_^hgcdjypfUgj]`n/°½y|c)Qnkbk^hn|cdf]&n|± s ypb)s~ ≥t gξb ^`N)Qay|]`Qnp± ^hNdQgb)g¢^hgy| g~\Qab>^`g^q[1PRy|{d{dgbd ·¶Q±Á^
(115) ¹ypb)~*^`NdQyk~dyp{\^hQa~*PRy|{d{dgbd ·¶fhgkN>^¹
(116) ¥+²)n|f&]_P1y| ξ °½y|c)Qa]a¨ s gj]`n/°½y|c)Qºn|b>^`nkcdfh]ypfhQ n|b>^`fy|^`Qa~¾¨\ª&NdQfhQayk] ±¶nkf&y|f`kQ ξ °/ypcdQa]a¨K^`N)Q[*Q¼\{)ypb)~¾¥ ²g|cdfhQ·¸k¹]`Ndn/ªz]^`N)Q~\Q±¶n|fhPRyp^`gn|b«©)yk]_gj]±¶cdb}
(117) ^`gn|b}]c)]`Qa~«±¶nkf^`N)Q²%²%´ gb¬^`Qaf`P1]np±^`NdQ nKn|f~\gb)y/^hQ|¨ ±¶n|f^hNdQ gbdg^`gjyp·B~dy|]`NdQ~§cdf`°kQa]¹ºypb)~«y|~)yp{\^hQa~·¶{djypgb¬cdfh°|Q]h¹P1yp{){dgb))¥*MNdQ &Qaf`b)]_^`Qagb«{Ln|[Kbdn|PRgjypj]ξ ypfhQPRn\~\g»)Qa~7©K[^hNdQ yk~dyp{\^hgnkb{)f`n\Q]`]a¥%¿ ]`{}Qgjyp[k¨\^`NdQagf&P1yp¼KgP1yRypfhQPRn/°|Qº^hn/ªy|fh~d]]`P1ypQaf ξ °/ypcdQa]a¥ MNdQ{dfhn|{Lnk]`Qa~y|{d{dfhnkykN]_Ndnkcdj~ bdn/ª ©LQ±¸ykQ~ ^`ny1±¶c)m]`N)yp{LQnk{\^`gPRgy/^hgnkbQ¼\Qafhg]`Q^hni>c)y|bk^hg¢±¶[^hNdQR©}QabdQ»d^] nk©\^hy|gbdQ~ ¥ i. j. AD. AD. j. W1WYXZU[\]\. k.
(118) G½. " . Ù
(119) X Ö Ù KÚzÙ . . MX. . . ÔXÖ Ö.
Documents relatifs
In infantile Pompe disease patients, the glycogen storage diffusely affects brainstem motor and sensory neurons, and the whole spinal cord sensory neurons, interneurons, and
De ce point de vue, il est possible d’affirmer que la mission est accomplie : les projets de renouvellement urbain (GPV et ORU) élaborés depuis 1998 dans une centaine de villes
cosinus (en bleu à gauche) et d’une droite (en noir à droite) sur les données simulées pour 4 années à une température de 45°C, avec un bruit de fond interne réduit de 100,
Florence CARRÉ ~ Réoccupations funéraires de sépultures collectives néolithiques dans la boucle du
2 Vue plus en détails dans le chapitre 1... 2 phénomène d’agrégation des protéines amyloïdes en fibres et d’autres facteurs impor- tants doivent être pris en compte comme
In this chapter, we prove an a posteriori error estimates for the generalized overlapping domain decomposition method with Dirichlet boundary conditions on the boundaries for
Revue française d’héraldique et de sigillographie – Études en ligne – 2020-10 © Société française d’héraldique et de sigillographie, Paris,
Dans ce cas, comme les coulées du Bras de Sainte-Suzanne appartiennent au massif de La Montagne daté à plus de 2 Ma (McDougall, 1971), l’ensemble des coulées pahoehoe