New Progress in Anisotropic Grid Adaptation for Inviscid and Viscous Flows Simulations
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. New Progress in Anisotropic Grid Adaptation for Inviscid and Viscous Flows Simulations M.J. Castro Dı´az F. Hecht and B. Mohammadi. N˚ 2671 Octobre 1995. PROGRAMME 6. ISSN 0249-6399. apport de recherche.
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(76) lq]}k_i
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(80).
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(85) m_kp~ m:lkdqlqilo:~ qXl q o=qli~z
(86) lq]}k_i ~k_idqilk:qR|d~Cxö~¤ inZlq:l q>~dlqil]kZx{nZ~k_¸qiliIk_i» ùí ,õpnh;éli ñ âåjõ!âLká õ jåë nyíür mé l s å$mâ XKn z íÒã ë ð$ ç à êìákëîj,ñ¹åë pá oLêöé %ò á¶qå í å á q ñlélëìê í åLñòOâL{ìçá¶õ íÒëöë é ä,ç å÷Iõ ê å ë çXí8|Gká é j õ¶ëîåjíjõ¶í åå à ñ êöá éIë âñ åç:ä}ká v~j o'é á {ìá¶r\åá¶õ s é j$ XKåät õ¶Oã âLuUç í8å| ã â vá éë ç:ká jå íüé ðåõ¶å í ù ê á íjë nwräå xyç,å í2é $ %?X&$ æ . .
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(89) xC~ qnpinp ilk,¼d~oÃnp~k_=~}2np dxC~q=´ÜkRi gpl q }qq ¡ np] l q
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(91) lq]}k_*i + znpO¤q/dqL] q¤G¾ + 0^2 4 6 P $ P 9 2@: $ 4 l 6 ! ±'G|]}Ïl q>qililk_iµkyÁRqi'n%oq}Oqdm_
(92) q Q znp¸¤qUzLk_o | lqn_} . . . Q LQ OQ ( LQ `Q 4 _kG³q>³npd2lkzLk_]}lil|]zLn:oq} ³ qilq>I q
(93) x·q mRlOkp´n,xCx(I q qdm_q}w~·l q
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(95) kp´*+ np] ³~lfxq m_l]} ¡ 6 $ np]¡ 6 ! ilq}]qzL~·ÁRqLx·¾_»±d~}>qLxCxC~·]}q ~})l qÃ| d~X¤]n,xCxw~ Id~{}'m_qk_oqliI¾_» .
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(97) l qUzn_}q}kp´!| ilqn,xC~}~zXoqli~z}³d~zI¸}k_oq~·o=q}³q o:~m_dm_q» ±d~}~} kRwiIqn,xCx·¾fnXilq}li~zL~k_fn_}|]}|]nyxöx¾³wq
(98) ]n,Á_qUn:m_kdk =~{dqn=kp´j³]npI q}qU_|]nZG~~q}
(99) } k_|dxO¤q_» wÎ>}I qUÀ_q¾Okp~G
(100) ~fl qUo=qli~zUdqL]d~~k_f~}
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(102) ]d~lqUqLxqoqd }kpxÁ_qi» ± qilqL´¶k_ilq_]n%³wqnZÀ:´ÜkRilo:|dx{nZ~k_ ü¤d¾
(103) ilqq ´ÜkRilo:|dx{n 2]n_}lk:¤qU|]}q Ik%zLkRo | lqUI q q}l}~{nZ]}» !. . "!$# %'&)( *,+.-0/21+3(415-768-9( *: _| kG}q: kp³Ul]np }qÁRqin,xjÁynpi~np¤dxq}}g $ < g ! < 4"4$4 < g4;UnpilqUmZ~·ÁRq ]l q: ilkR¤dx·qo ¤qzLk_oq} < 0
(104) > 0 C= 0 )>0
(105) +-
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(108). ]m» ¡ »_| kG}q:l]np'~·d~~n,xCx¾7k_dx¾O³kÁynZi~np¤dxq} /g $ np]@g ! npilqU iIkyÁR~dq» q>]]fnpnZ ilk,¼G~oÃnp~·kR kZ´(l qUkR ~oÃn,x~GlqiL}qzL~k_¸qLxCxC~]}q¤d¾ÃI q
(109) ´ÜkpxCxkp³~ mà iIk zLqd| ilq D ~ö~{´}XlzL k_qUdn,³~ kqqLÇxCxC~~·]d}lqk¸}UnpzL | kRillÁR qqE} i $ l< q:! k_] n,q:Á_³qX~ lk_7 }nZoÃGn,¾=xCx~qG}lUqiLnp}iIqqzLn:~~k_}>fLnp]À_kpq~d nR ü}µld~·d~}µlq]inp}q zL]q~k_]7}
(110) ³np] qfl k_q qX}|dqL~xCxönp~]¤d}xqq oqli~z
(111) ~}µl qUn_}l}k zI~nplq kR q D k_±l q>qil~³dl~q}q_i}&qxzLq ~k_6FQ onp]qlH i~G z:QF
(112) +ÍSmJ <$IR~}
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(114) ¨. C
(115)
(116) 5* 1 > 0 0 . µkp³UR~C´ T pÁ npi~np¤dxq}
(117) npilq>mp~Á_q l q
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(119) } D + J 0b~·dlqi}qzL~k_ + $ < + ! 4 D ´Ü+ k_J i 0bS 0 ~·dF lq< i4$}4$q4 zL<UT ~k_ + J < + Q 4. First ellipse Second ellipse Approximated intersection ellipse. 0.15. 0.1. 0.05. 0. -0.05. -0.1. -0.15. Ä ~m_| ilq ¡ dÎ iIk¼d~oÃnplqÇk_ ~oÃn,x qLxCxC~]}qkp´j³wkR¶qLxöxC~]}q)~·dlqi}qzL~k_ » -0.15. -0.1. -0.05. 0. 0.05. 0.1. 0.15. . "!. . . 1 .+ 46 6* /$: 1 *: *,+ ( .
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(121) _|d~lq)x·kRmp~zn,xnp] oqnZ]}l]npqLxqoqd} dk= k_ zI]np m_q
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(123) l q:oqli~zXk_¸I qU³n,xCx}
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(125) ³n,xCxcl q
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(128) zLk_o | lq LnpÀ_~ m l q>oÃn,¼d~o:| okp´(l q>³k: ilk_qzL~k_ xq m_l]}µk_ / jkp´I .q + | d~
(129) qL~m_q ¶Á_qzLlk_iL}~oq}µl qUzLk_ililq}]k_]R~ m qL~·mRq ¶Áyn,x| q}» ¹!~}! qzLq}l}lnpil¾>Ik À_qq:l q~d´Ük_iIoÃnp~k_zLk_o:~ m
(130) ´¶ilk_oI qwkRi~mp~]n,x o=qli~zk_) , lk
(131) ¤]qnp¤dxqlk
(132) ilqzLk_m_d~ q }~·I|]np~k_]}}|]zInR}} k zlÀd¶¤]k_| ] nZil¾ xn¾Rqi2~dlqin_zL~·kR]}» ΢ilqLxn,¼ÏnZ~k_ ilk zLqd| ilq>znpà kp³¤q np dxC~q lkX ilk_]npmGnZlqld~}2kRill k_m_kR]n,xC~·¾X ilk_]qiI¾X´¶| ill qi~· l q o=q} » üOId~{}µ]npqi ld~}µ]n_}¤]qqfdk_ q>´¶k_ik_dx¾O³k/ k n,x xn,¾_qi}\»]Ä ~m_| | ilq ¡_¡ } ky³
(133) }l qUR~}np]zLqXkp´ I q>]kp~d}¤]qLxk_ mp~ mOlkI q}q
(134) xn¾Rqi} lk=l qU³n,xCx ´¶k_i n}| ]qi}kRd~{zXÁR~}lzLk_|]}'º]ky³¢kyÁRqi'nZn,~i´¶kp~Cxü»c± kmp~Á_q nZ~dqn:k_ ³]np³q
(135) m_q³~l k_| ld~}¤]kR| ] npil¾OzLk_]R~~k_¸k_ l q
(136) oqli~zp I q zLk_iIilq}]kR]R~· m=ilq}|dx³~l kRdx·¾= k_ok_m_q qkR|]}wµq| oÃnp O¤k_| ] npil¾ zLk_]R~~k_ ´ÜkRi2l q>oqIi~z n,xk_ m:l q
(137) ³n,xCxc]n_}n,x}k:¤qq¸} kp³~ I qU}lnpoq
(138) ]m_| iIq_» $ %?X&$ æ !.
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(140).
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(142) znpO¤]q
(143) ´¶k_| ]=~ F np] ¨ ¹» £]zLqbl qfoqli~{z~{}Çmp~Á_q I q7oq}Am_q qinZ~k_ ilk zLqd| ilq ~{}np dxC~q» ±'d~{}n,xm_k_i~l o |]}q}=]Á_q R~ qilqdx·k zn,x(IkGkpx}´¶k_ild~}2Ln_}À ) ?! = ) ?! =
(144) .[ ! =' ++-$
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(149) n,xm_k_i~l o }kpxÁ_q}I q
(150) ´ÜkpxCxkp³~ mà iIk_¤dxqo . . . . . . . . >. ,*
(151)
(152) 0 +,0 0 ( .[ ' $
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(157) ³~l¸I qL~·iUnR}l}k zI~{nZlqOnZ m_qG
(158) Á_qzLlkRi}d ¼ qOkp~G
(159) xk zn,xC~ nZ~k_ qzp» ΢lk_]kZx·kRmp~zn,x(Á_qi~Ccznp~·kR¸}lqOlk_mRql qi³~l¸nXli~np mpxq>k_i~qdnp~k_ ~}o n_dq
(160) ~Ok_idqilk:m_|]npinZGlqq
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(162) ,1Æ L {®]C :-,_® $¬ :{Ë 9%$ ¿
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(164) ~Ok_idqilkX~o ilkpÁ_q
(165) l q
(166) ~d~~n,xjoq}¸R|]n,xC~·¾_»]µ| oqi~zn,x!qL¼ qi~q]zLq}>} kp³ I]npl q
(167) ]]n,x ilq}|dx ~}¤qllqiµ~C´ld~}'~d~·~{nyxk_ ~·o%~ nZ~k_O~}µoÃn_dq_» &OC :Ì ®]Ì ® D ¬® $¬ :{Ë 9%$ ± q Ik_]kpxk_mZ~{znyx]oq} qLx·qo=qG}ilqLxnplq=³~l. $ npilqµk_idqilqX~On>]npiI~zL|dx{nZiw npn. . ) > + / C. 0 + # 0. 2 1
(168) }lil|]zLl| ilq¸znyxöxq »j± ilqq¸dk_| ¤dxqxC~{}}ÃnZilq zLkR]}¹~dqilq üÁ_qil~zLq}qdm_q}np] Ii~np mpxq} »Ï_| ]kG}q
(169) l]npl q]i} S d ¡µqdm_q}npilqliIqnplqpl q l qqdm_q Q ~}npÀRq »G± q iIk zLqd| ilq
(170) l]np ³q> ilkR]kG}q
(171) ~}µl q
(172) ´¶kpxCx·kp³~ m k_ q ¡R» q[ N`Q!¤]qUI q OQ xq m_lzLkRo | lqO³'~·IOl q>oqli~z)lq]}k_*i +Í» 7q>]n,Á_q>I ilqq>]kG}I}¹~¤d~öxC~~q} D ¶±´ NOq Q1 > q³Ab43 o65 q} Ãb43 qL6x5 qo qd¡ 4ì¨ } j ülÁ_ qqil~{zL `qQ(}\_o:q|]d}m_q}¤]qUnp]zL=| 'li~·~np ³wmpxkXq} q 2dnpm_iIqq }n_|] }¹~d qm Inp 2qUln_ q
(173) dq¶]Á_=qilkplqL´(¼Ol qLil~k izLzLqk_dil| ilqilq_} » ]k_]R~ mO¿>¿ j»]µky³X? `Q(]n_}µzl]np mRqOnp] ³q npmdn,~¸zLk_o | lq
(174) ~}xq m_I % NOQ¹» ¶´ NOQ > b43 65 l q ³wq>iIq]qnp
(175) l qU}lnpo=q> ilk zLq}I}| G~öx ~·L} n_}I}k zI~npIq=x·q mRl N`Q b43 65 » D ¶zl ´ qN zIQ À 7 ~C´wb l3 Q#qL8 ~iµ x·b q3 Q#mR8 l] } nZP il4îrqU
(176) ¤dl~ m_m_qqi
(177) lQ ]np~} }| 9 5 ildq~}lf}ql]jnp Idz~{n_})}xqUqn_l R~qX mbilqlkfÁ_~k_ |]q}µ³ qildk mRzLqq}l}/}'zL~{iI}>qnpnZ ~dk_xö]~q}» 7k_qi }oÃn,xCxqi I]np b-3 Q#8ó qiIq>ld~} ilk zLqq}'~{}µb-3 npmGn,~Oilqqnplq» D ~ö´ b43 Q#8 NOQ b43 65 `Q!~}À_q » Ï»¶´j³wq
(178) ]n,Á_q>dk_ qUn)x·k zn,x oq}Oo=k R~Ccznp~k_ n>xk zn,x k_ ~o:~ np~k_f}lq o:|]}¤]qUzLk_]}¹~dqilq»ÏüOld~} zn_}q_qdm_q}³np d~ m¸n,xm_k_i~l o1~}>np dxC~qflk l q: q³ kRi ok R~C]qfqdm_q}» ±'d~{}
(179) kR ~o:~ np~k_7}lq ~·]zLiIqn_}q}µl qUoq}¸_|]n,xC~¾_» ±'d~{}µ ilk zLq}l}~}µilq]qnpIqOkpÁ_qin,xCx(l qXqdm_q}kp´2¿>¿ j» : : 9 ® !Ë D $¬ : E:4,_® $¬ :Ë 9;$ Ä ~]n,xCx¾_ n
(180) mpxk_¤]n,x]k_ ~·o%~ nZ~k_à iIk zLq}l}~}2zLk_]}¹~dqilq»dεxCxc k_X ¼ q_ k_:~dlqil }| ¤cdk_oÃny~·)~Glqi~·kRi(Á_qiI~zLq} ³~l
(181) xq}l} l]np ¨qLxqoqGL}(zLk_ qzLIqUIk2~jnpilq2}| iIq}l}qUnZ]Un¤]npiI¾ zLqdlqi~ m ù í åt=õ <ú äj,å *åxyç,xpåõ ä í áwí ù âLvpçäå íâLð ñ âLéõ¶ë ç õ¶åí íã àyâLçäOá¶â ë çá¶åõ í å ñ á ë âLç í mâ nõ¶å ÷ ù êöé õ í ùpõ n éIñ å í
(182) é ç,äÃká j,å í å ñ âç,äÃâLçå íñ âLõ¶õ¶å í àyâLçäOá¶â ë çá¶åõ í å ñ á ë âLç í mâ n XYX ç[Z ý\]^ . . .
(183). !.
(184).
(185).
(186).
(187). . . . . . . . . .
(188).
(189).
(190).
(191) r. C
(192)
(193) 5* 1 > 0 0 . }lqO~}UnZ dxö~q !n,xCx k_ ¼ qfÁ_qil~{zLq}>nZilqU |
(194) ~7l q:zLqGIqi>kp´2m_in,ÁR~·¾Okp´2I qL~·iX qL~m_G¤k_i}
(195) ³~lfilq}]qzL Ik/I qUoqli~{z"+Í» ±'G|]}!n=]]n,xwo=q} n_ np Iq´ÜiIk_o ( ³~l½iIq}]qzLXlkOl qoqIi~z:lq]}k_i}+ ~{}Xk_¤ Ln,~ q7np]~ iIq}]qzL}Xl q=m_qk_oqli~{znyx'$ np]fIk_]kpxk_mZ~{znyxµzLkR]}lin,~d}»Îµ´ÜIqiUqn_zI nR np np~k_b}lq (nZf~d~·~{nyx }kpx| ~k_ ~})k_¤ n,~ q kyÁRqi>l qà q³uoq}7¤d¾~GlqiI]kpxnp~k_kZ´wl qO}kpx| ~k_bk_7l q= ilqÁ_~k_|]}Xoq} »(±d~}>kp~GX~} zLiI|]zI~{nyxwn_}
(196) ³qU³~CxCx'}qq:q}qzI~n,xCx·¾ ´¶k_i | ]}lqn_d¾zLkRo | np~k_]}\»Î }
(197) ³qznp7}qq_l q:ilqoq}d~· m¸nyx·mRk_i~l o nyx·³n,¾ }2³k_iIÀ }xk znyxöx¾_» ±d~}
(198) }linZlqm_¾Ãn,xCxkp³ }|]}Ikilqoq}¸}| i´ nRzLq}³~l¸kRdx·¾¸n/x·k zn,x(]nZinpoqli~ nZ~k_ » 3; w 2$
(199) 6Ã /$ ; 6 ¹ld~}2}qzL~k_³qdq}lzLi~¤]q>l iIqq'zLkRd]m_| inp~k_]}kp´&zLkRo ilq}l}¹~¤dxqµ~·dÁR~{}IzI~{OnZ]:Á_~}lzLkR|]}!º]kp³ }»_ÄÏk_i2n,xCx]l q}q znR}q}dl q>~d~~n,xjoq} q}µ]n,Á_q ¤qqfm_q qinplq |]}~· mO2 vw ü» .
(200). . !$#. . . * - 1+8/ - : * ( /$+ 2* (. . . " . . ±' q¸]iL}¸zn_}q½zLkR]}¹~}}OkZ´%np²~GIqil]n,x/}| qi}k_d~z º]ky³<nZ¸nRzl F ~ n}lzLinZo>q ~dxq»
(201) ¿
(202) q}d~lq7l q zLkRd]m_| inp~k_f¤qL~· m7}¾dooqIi~zp l q:³ kpxqOdk_oÃn,~7]nR}
(203) ¤]qqzLk_o | lqfIk¸}qq:~C´l qÃ}kpx| ~k_ ilqoÃn,~]} }¾Goo=qli~z>k_iµ k_»]±d~}>zn_}q>~} dqÁRk_lqOIk npOqÁpn,x|]np~k_7kZ´k_| i
(204) o:|dx~¶Áynpi~np¤dxqÃ}linplqmR¾´¶k_il qXoqli~z dqL]d~~k_ »± q:}qzLk_] ok R~Ccznp~k_~} k_|]}qÇ qilq/nR}~~} np ~dÁ_~}lzI~Ǻ]kp³U» Ä ~Á_qOnp| lkRoÃnp~z:n_ np nZ~k_]}>]n,Á_q¤qq nZ dxö~q» qÃ} kp³8l q oq} q}Xnp] l qÃzLk_iliIq}]k_]R~ m~}k_ fn_zI7zLk_dlk_| i}µ´¶ilk_o l qX~d~~n,xlkOl q:xn_}Un_ np np~·kR }lqf~f]m_| ilq}X F ¨¸np] ¦ » 7qznp½}qq= ky³ nRzzL| inplq
(205) l q>R~{}IzLk_G~·d|d~~q}iIq}kpx| ~k_]}npiIq´¶k_ild~}º]ky³np}lqO¦ »dü~}~GIqilq}~ mlkX k_~zLq>l]npl qilq ~} kUkG}lzI~CxCxnp~k_=~Ãl q
(206) }kpx| ~k_Ãnp]=l]npjI qº]|d~Ã}kZx·ÁRqiw}qqoÃ}!IkUd~zLqLx¾Ãn_zzLq w}|]zl oq} q}»_k_iIqkyÁRqi ³qU}qq>I]npl qU}kZx·| ~·kR}n,¾ }]qi´¶qzLx¾¸}¾dooqli~z)qÁ_qOk_¸nzIxqnpix¾O k_ Âü}¾Goo=qli~z>oq} » ±' q~d~~n,xjoq} ]n_}npilk_| ] P P P
(207) Ii~np mpxq}np]=l qµ]]n,x oq}Onpilk_| ]=1¨ P P PP »_±d~}mp~Á_q}np=~dqn:kp´ I q d| o:¤]qikp´jqLxqoqd}³q o%~·mRG qq=~C´2³q
(208) ³'nZGnp q_|d~Ápn,xqG>ilq}kpx| ~k_Ok_¸n:}lil|]zLI| ilq m_i~ }» ± q zLkRGÁ_qiIm_q]zLq>d~}lk_il¾=~} } ky³µ ~·Ç]m_| ilqUr » . . !. . * - 1+8/ 8/$- 1 .- 1 1 *. . . .
(209) . . )#. . Î f n_zI ½}| qi}k_d~z º]ky³1npÃq¾d kpx }f¡$P P PPO~} }l|]R~q²kpÁ_qiÃn7zIxn_}l}~{znyx)
(210) n_znEP PÏ¡7n,~i´¶kp~Cxü»2¹ ld~} zLkRd]m_| inp~k_ ¤]k_l²l qO q³~· mRilqR~qG} ³q ilk_]kd}qO qilqnZilq¸|]}q»± qR|]n,xC~¾kZ´U³'nyxöx>zLkGq zI~qd} ¹q}]qzI~n,xCx¾Ol q´¶i~zL~k_¸zLkdq zI~qG !~}}l|]R~q lkdk]»püO]nZil~zL|dxnpid³q³npGjIk/}qq
(211) kp³ o:|]zIl q¾Ãdqq] kROl q>oq} » ±' q%nR np np~k_xkdk_x·q mRlf~}>r:´¶k_i>ld~}>zn_}q_»c± qzLk_dÁ_qilmRq]zLq/d~}Ik_il¾O~}>} kp³~f]m_| iIq "¡ P » 7q znZÃ}qq
(212) l]np2I q qill| il¤]np~·kR d| q
(213) lk:l q n_ nZ np~k_kRÃl q
(214) ilq}¹~d|]n,x~}ok_ilq'~·o=]k_ilLnpGl]np~·OI qxn_} znR}q_»_±d~}2zLk_o=q} ´ÜilkRo ]i}jl q´ün_zLjl]nZjl qº]kp³b~}jÁR~{}IzLk_|]}2np]=}qzLk_]:l]nZj³wq]n,Á_qk_| ij q³²¤]k_| ] npiI¾ zLkR]R~·~·kRO´¶k_il q>oqIi~zUn,xk_ mI q>³'nyxöxü» ±' qÃoq} q}U³q k_¤ n,~d| i~ m7l qOn_ np nZ~k_fxkGkR npiIq } kp³ ~·b]mR| ilq np] l qOzLk_ililq}]k_]R~ m ~}kRÂünRzlR~}li~¤ | ~k_]}X~b]m_| iIq ¥ » 7qOznp }qqÃl]nZ/I qÃo:|dx~¶Áynpi~{nZ¤dx·qÇ~GlqiL}qzL~k_ ilk zLqd| ilqO³k_ilÀ } ³qLxCx´ÜkRiUl q ~dqd~Ccznp~k_ kZ´ } k zlÀ ¤k_| ] npil¾ xn¾Rqi%nZ]b³'nZÀ_qilqmp~k_]}\(qÁRql qóqnpÀb} k zIÀ }/nZUl q Iin,~CxC~· m qdm_q>]n,Á_q>¤qq dqIqzLlq» Ä ~m_| iIq ¡R¡U} kp³ }µl qU k_iIoÃn,xR~}nZ]zLq/kZ´2l q>]i} np]}qzLk_] k dq}µx{n,¾_qiL}lk=l qU³n,xCxü» 7q/znZ}qq ]iL} I]np´¶k_i>l q)]i}
(215) xn,¾_qi>ld~}>R~}np]zLq%~{}
(216) qL¼]n_zLx¾l q}lnpoqX´¶k_i nyxöx2I q/ k dq}
(217) ³d~zI7oqnp]}
(218) l]nZ k_| i ¤k_| ] npil¾:zLk_]R~~k_k_Xl qoqli~z³k_ilÀ } ³qLxCxcnp]X}qzLk_]:l]nZ(l qilqLxn,¼]np~k_: ilk zLqd| ilq³q2]n,Á_qnp dxC~q Ik/ iIk_]npmGnpIq ld~} iIk_]qil~·q}µlk:l q> qL¼ xn,¾_qi}n,x}k=³wkRilÀ }R|d~lq>³qLxöxü»Ä k_il q}qUzLk_o | np~·kR]}³q>]nÁRq |]}X ilk_]npmGnZlqfl q ilk_qil¾kZ´'¤qL~ m½npUI q }lnZoq R~}Lnp]zLqÃIk¸l q ³n,xCxwk_dx¾7lkl q=]i}U³kOxn,¾_qi} ¤ | µl qU}lnpoq> iIk zLqd| ilqUznZ¸¤]qUnZ dxö~qk_Ol q>kRl qixn,¾_qi}»dük_idqilkÃ} ky³A³]nZ³q k_¤ Ln,~¸³~l k_| Id~{})¤]k_| ] nZil¾7zLk_]R~~k_½kR7l qoqIi~zp ³q]n,Á_qn,x}k} ky³µI qÃzLk_ililq}]k_]R~ m7R~}np]zLq}XlkOl q=³'nyxöx ´¶k_il qU}lnZoq
(219) xn¾Rqi}n,´¶lqi }³'~·Lzld~ m k k_| i
(220) zLk_ililqzL~k_ ~ » q_» k_ok_mRq qk_|]}µq| oÃnp ¸¤ » zp»d´ÜkRiwI q>oqli~z $ %?X&$ æ . . . .
(221) zLkdk_ilq}]k_]R~ milq}|dx·L}2¤qzLk_oq}Á_qil¾:kGk_i,» ±'d~{}qL¼ dxn,~]}l q
(222) ]kdk_i_|]n,xC~¾kp´ l q
(223) m_in_R~qd}jqÁpn,x|]nplqOnp I q>³'nyxöx!| G~Cx( kp³Ak_fn_ np lq oq} q}np]OI q
(224) ´ n_zLl]nZd~zLq>´¶i~zL~k_zLkdq zI~qGL}³qilq>]npiL lkk_¤ ny~· » Ä ~m_| iIq¡U} kp³ }l q
(225) ilq}l}| ilqUnp]%´Üi~zL~k_¸zLkdq zI~qd}k_¤ ny~· q kRÃl qµ~d~~n,x(oq} dk_ l q>oq} np }lq ]Á_qUnp]Çk_Ol q
(226) xn_}kR q_»]q}|dx}'´ÜkRi }lq]}¦=np]Ornpilq
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