Linear stability analysis in fluid-structure interaction with transpiration. Part II: numerical analysis and applications
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Linear stability analysis in fluid-structure interaction with transpiration. Part II: numerical analysis and applications Miguel-Ángel Fernández — Patrick Le Tallec. N° 4571 Septembre 2002. ISSN 0249-6399. ISRN INRIA/RR--4571--FR+ENG. THÈME 4. apport de recherche.
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(19) Yjwj_Æ Γ ~vKu_y ~ ©cxzÇvf!j:pqy}tqjvÇDxzr}y u uy Γ ©cxz(vf}j)xr}vtjvMxzr!y u_zÆ}~wjj Í[zr}wj&z± s. 3. 3. 3. s. f. in. out. in. out. in. out. γ. Ωs. Γin. Γout. Ωf Ω. Ù pqzr}wj& gegf!jnxzi:|!r}vKuvpqxy[ut xzikupy Ω Æ jÍ!y!j ¥vf!j~G~vwji pyÏp»v~{jÑTr!pqtpq ¬ wpqr}i,xy}Í[zr!wu_vpxzy ¦ j u~w~r!ikjvwf!jª!r!p vx§Mjy}j°gvxy!pIuy<®Gpq~xzr!~Æ(f!xzi:xzzjy!jxr!~:uy pqy!xzi ¬ |!wj~w~wp!tqj_±ËÅöv~$Djf[uP®GpqxzWpq~ j~pqMj ¥kpv~®zjtqxGp»vÈ u uy |}wj~~wr!j p ±9(v$jÑTr!p ¬ tqpq}wpqr}i Ævwf!j~jÍ[jt ~s~wu_vp~©µbvwf!jk©cxzttqx¼°pqy}py!xik|!j~~wpq}tqj:¯uP®Gpqj ¬ oTv x _j~njÑGr!u ¬ vpqxy!~Ë°wp»vwvjykpy&jr!tjwpquyxy!~wj®_u_vp»®zj©cxzwisr!tIu_vwpqxzyspqyvf!jr!&y Gy}xî°yxzy!p»©cr!uvpqxy 0. Ω = Ωf ∪ Ω. s. . °pvf. 1 div u0 ⊗ u0 − σ(u0 , p0 ) = 0, ρ div u0 = 0, u0 = uΓin , σ(u0 , p0 )n = 0, u0 = 0, ε(u0 ) =. ßß=ê$#&% ('Pÿ. 0. 1 ∇u0 + (∇u0 )T , 2. py p y zx y xy xy. Ωf , Ωf , Γin , Γout , γ,. σ(u0 , p0 ) = −p0 I +2µε(u0 ),. +,.
(20) . γ $[¶Ä ´ G¶Ä 6¶¹-´ ´!G¶wº "À(¸½B¹-³¿Á ²¶ ¸ÄµÄq¶KÁ . . . . .
(21) . . ~vuy ~n©cxz)vf!j%®xztqr}ikjª!r!p jy!~pvÈzÆ µ ©cxzvf!j Tpqy}jvpk®Gpq~xz~pvÈ2x©gvf!j ª[r!p Æ ©cxz0vf!jkr!y!pv)y!xwikut9®zjvxzxzy Γ |MxzpyTvwpqy!py!~wp j Ω ÆËu_y u ©cx vwf!j Í} dj ®zjtqxGp»vÈxy Γ ±9oGpqy!jÆ¥u_v"jÑTr!ptqpq}wpqr}i vf!j$~vwwr!vr!j(pq~"u_vËj~vÆvf}j(pqy¥vj©Buj xzy pvpxzy'+ , j d|!j~w~j~vf!jxyTvwpqyTr!p»vÈÇx©Gvf!j9®jtqxGp»vÈÍ[jt u_vvwf!j˪[r!p d¬ ~vwr}vr}wj pqy¥vj©Buj±9ÅÈyvf!j ~wjÑTr!jt°$j)°ptqt·~r!|!|Mxz~wj vf!u_v u uy p uj ~wi:xdxvwf©cr!y!vwpqxzy}~± egf}j0jtIu_~vpY~xztqp r!y j(tquwzj p~w|!tqujikjyTvw~(pq~ j~pqMj ¥p»v~$®jtxdpvÈuy pv~~vwwj~w~Ëvjy!~wx±"Y~Ëpqy »P ¿Æpqyvwf!pq~6|[u|Mj°ÚjÚ°ptqtT~wr}|!|Dx~wjÚvwf[u_v"vwf!j pq~w|}tIuji:jy¥v x©vf!j~vwr}vr}wjÆuxzr!y u Ty!x¼°y§xzy}Í!zr!uvpqxy Ω Æpq~Çzp»®zjy§¥Îutpqy!juxzi ¬ !pqy[uvpqxyÏx©$ukÍ[y}pvjsyGr}inDjÇx©®Gpq!wu_vpxzyÏi:x j~ Æ ≤n pqyÏ~r!Kfbuk°ÚuPvf[u_vYvf!jikx_vpqxyx_©Ëvf!j~vwwr!vr!jnϕPuy : DΩj)°−→ wp»vwvRjyÎ,u~ 1I ≤+i Φs pqy ÆM°p»vf s ∈ R uy Φ = [ϕ |ϕ | . . . |ϕ ] pq~Çu 3 × n iku_vp» ~vKuy pqy!:©cxz{vwf!j Ω wj r!j i:x u_t"[u~pq~±:ÅÈy¨vf}pq~Ç°Úu¼Ævwf!j&~vwwr!vr!wutDjf[uP®dpxz0pq~ wp®jy§¥¨zp®jy iu~~ uy ~vwp Dy!j~~xz|Mjwu_vxw~Æ M uy K j~w|Mjvp»®zjt»z±:egfTr!~Æ'vf!j&jÑGr!u_vpxzy j ¬ ~wpq!py! vf!j{~vwjPu :jÑGr}pqtqp!wpr!i x_©vf!j{~vwwr!vr!jÇ~wr}GjvÚvx)ª[r!p (j Djv~$pq~Wzp®jy T Z +] , Ks = − Φ σ(u , p )n da, ρ > 0 n. s. Γin. in. 5. 0. 0. s 0. s 0. i. ns. s 0. 1. T. 0. s. Ωs0. s. ns. 2. 3. 0. 0. °f!jj s ~vKuy ~W©cxvwf!j{zjy!jutpqÌj xGxz pqy[u_vwj~x©Dvf!j~vwwr!vr!wut pq~|!tIu_ji:jy¥v± ÅövjÑTr!pqtpq!wu_vj~(vf!j0ª[r}p tqx¥u u_vgvf!j pyTvwj©Buj γ ± %Ó 2¢!¡ ´ ½ [¶ÚÂ}¸¹-½B³¿ÁdÄ»¸¹ Áw¸ ¼¶
(22) [¶¹K¶½ [¶ÁK¾´ $G¹¸½]³¿¾´Ï¸½Ë½ [¶¶G³µÄ ³¹-³ ³ ´·¾
(23) 9´ ÂM¹$¾ Äq¶ + , ¸!´ +] , ¸¹ ¶ G´·ÁK¾ îÂMÄq¶ ¼¶K¶:´ ¶¹-³¿Áw¸ÄÚ¶Â[¶¹³ ¶´[½ Ω ¶Äq¾
(24) !)¾
(25) Ú¶ "¶¹ Y³µ´ G¶´·¶¹w¸Ä Y ½ [¶¶ d³µÄ ³ ¹-³ ÁK¾´ $d¹¸½B³¿¾´ Ω G¶ÈÂ[¶´! ¾´ ½# [¶ ½] ¹ [Á½ G¹¸Ä ³ ÈÂMÄ»¸GÁK¶ ¶´[½ s ¸!´ #½ [¶´nÂM¹$¾ Äq¶ 5+ , ¸!´ +¿ , ¸¹¶ ½]¹¾´ Ä kÁK¾ ¼ÂMÄq¶ ¼¶K% ¶ $'&() ÅÈykvf!p~Ú|[u|MjW°ÚjY©cxdr!~Úxzykvf!jYyGr}ikjwpqut~wxztr}vpxzy:x©vf!j{©cxztqtx¼°pqy! ÑTr[u wu_vp jpqjy¥®_utqr!j:|!xz!tqji nÍ!y vf!jk|!r!t~u_vwpqxzy λ ∈ C Æ6vf!jk|Djvr}w[u_vwpqxzy®zjtxdpvÈ u : Æ·|!j~~wr!j uy Î pq~|!tIu_ji:jy¥v|[uu_ikjvwpÌPu_vwpqxzy s ∈ C Æ Ω −→ C °pvf (u, p, s) 6= 0 Æ!~r!Kpf :vwΩf[u_v −→ C γ. 0. . . . . 0. f. 3. f. ns. ÝcÞßÝcà.
(26) ²"³µ´·¶w¸¹.½¿¸³µÄ ³µ½%¸´M¸Ä ³ ³µ´d³ 4 ½B¹ Á½d¹¶³µ´[½È¶¹w¸GÁ½B³¿¾´. p y pqy xy xzy xy. 1 ∇u0 u + ∇uu0 − 2ν div ε(u) + ∇p = λu, ρ div u = 0, u = 0, σ(u, p)n = 0, u = −λΦs − ∇u0 Φs, Z 0 2 λ M s + K + B s = − ΦT σ(u, p)n da,. Ωf , Ωf , Γin , Γout , γ,. + ,. °f!jj M u_y K jy!x_vjÆTj~w|Mjvp»®zjt»zÆzvwf!j(vKuy}zjy¥vpqutGiu~~9uy ~vp Dy!j~w~9iku_vpqj~ x©}vwf!jÚ~vwr}vr}wj(u_y B pq~6u n ×n wjPu_tGjxzi:jvwwpqiku_vp» DÆzp®jys¥ vf}j$©cxzttqx¼°pqy! j d|!wj~w~pqxzy Z +cl , B = ∇σ(u , p )ϕ n + σ(u , p ) I div ϕ − (∇ϕ ) n · ϕ da, γ. 0. 0 ij. 0. s. 0. s. 0. j. 0. j. j. T. i. ©cxz 1 ≤ i, j ≤ n ± oG|Djvwut9|!xz!tqji + , u_wpq~j~Ç©cxzi vf}j:tpqy!ju0~vu!ptqpvÈÏuy[u_tG~wp~0x©$vwf!j&ª[r!p G¬ ~vr!vwr!wjjÑTr!pqtpq!pqr!i=~vKu_vwj (u , p , s ) Æ·~u_vwpq~©µGpy!%jÑTr[uvpqxy!~ + , uy +] , ± egf!p~ jpqjy!|!xz!tji °Úu~ jp®j pqy"uv{Å ¼dÆModjvwpqxzy }± +B~jjutq~x »¼dÆMÚf[u|}vwjl$ , T:xzin!py!pqy! vf}j #py!jPupqÌu_vpxzy:pqy!pq|!tj{u_|!|!wxzuKf +B~jj z , °pvf:vf!jYwjjy¥v tqpqy}jPupqÌPuvpqxy ikjvf!x j®zjtqxz|Mj py Pl!Æî ¿Æ(|!uvwpqr}tIut ~r!pvwj ©cxk|!xz!tji:~ pqy¥®zxzt»®Gpqy!&i:xî®Gpy!&Dxzr}y upqj~±ÚÅÈyvf!p~°ÚuPzÆvf!j)uMx¼®zj~vju jÑGr}pqtqp!wpr!i ~vKuvj °pqtqtMMj)xzy!~p jwj tpqy!juwt»u_~Gik|dvxvwpqPu_tqt:~vKu!tjÆ!p»©jpqjy!|!xz!tji + ,Ú xGj~gy}xv f[uP®zj%jpqzjyT®_utr!j~ °pvwf2y!j¥u_vwp®zjkwjut|[uv±
(27) Yx¼°Új®zjÆ9vf!p~~vju ~vKuvj&°pqtqtMj vjikj r!y!~vKu!tjgp©[vwf!jjgj dpq~v~ÆGu_vËtqjPu_~vPÆxzy!jgjpzjy¥®_utr!jÚ°pvwfy!jzu_vp»®zj(jPut!|!uv± %Ó 2¢!¡ [¶{½B¹¸´ ÈÂM³µ¹w¸½B³¿¾´ ¾ G!´ ¥¸¹ &ÁK¾!´ ³µ½]³¿¾´ + , ¸!´ n#½ [¶Ç¸ ¾" ¶ B2¸ 2G¶ ½]³ ´·¶ :¸½]¹-³ B ÁK¾ ¶ -¹¾ Ê#½ [¶ G¶K¾ ¶½B¹³¿Á{³µ´[½È¶¹¸GÁ½]³¿¾´ ¶½
(28) Ú¶K¶´#½ [¶ d³ &¸!´ #½ [¶ ½] ¹ [Á½ G¹¶ ¹ ¶ P¶¹n½ ¾ $
(29)
(30)
(31) ( {¸! ´ ½¿¸ ¥¶³µ´[½È¾¸GÁKÁK¾ G´[½(#½ [¶YÂ[¾ ³Äq¶ ¾½B³¿¾´ ¾ 0#½ [¶)³µ´[½È¶¹ ¸GÁK¶ γ. s. 0. 0. 0.
(32) . ßß=ê$#&% ('Pÿ. 5. . 0. .
(33) γ $[¶Ä ´ G¶Ä 6¶¹-´ ´!G¶wº "À(¸½B¹-³¿Á ²¶ ¸ÄµÄq¶KÁ . . . . .
(34) . . ; ( "#( < Y©µvjWvwf!jiku_vwf!jiku_vpPut!uy[ut»d~pq~Wwj|Dxzvj pqy"uv(Å ¼}Æ}odjvpxzyl¿ÆT°Úju ! j~~Æ pqy%vf}pq~g~wjvwpqxzyÆdvf!j0Í[y!p»vjÇjtqjikjyTv pq~wwjvpqÌu_vpxzyx©'vf}j)jpqzjy¥®u_tqr!jÇ|}wxz!tji + , ± ¦ jÍ[w~v"j°pvj9vwf!j öª[r!p |!uv x© + , py)®_upIu_vwpqxzy!ut_©cxzi uy vf!jy°$j$|}wxz|Mxz~wj$u ~vKu_!pqtpqÌj Í[y!pvwjjtqjikjy¥v(u|}|!wxP dpqiku_vpxzy± Ù pqy!utqt»zÆ_°$j|}wx¼®Gp jvf!j{xwwj~w|Mxzy py! iu_vwwp ©cxzinr!tqu_vpxzyÆ¥°f!pqKftqju ~gvx:u~w|!uw~jzjy!jwutqpÌj jpqzjy¥®u_tqr!j0|!xz!tji± . . .
(35)
(36)
(37) . ÈÅ y vf!j ~wjÑTr!jtÇ°ÚjÏxzy!~p j©cr!y!vpxzy!~ jÍ[y!j pqy Mxzr!y j ~wr}!~wjv~x© R Æ{uy vKu$Tpqy}(®u_tqr!j~py0vf!jxik|!tj Í[jt C ±egfGr}~Æzu_tqt L uy odxzMxztqj®Ç~w|!uj~"u|}|Djuwpy! pqyÎvf}pq~0~wjvwpqxzy§ujnvKu$_jy§u~ xzi:|!tj ®zjvxz ~|[uj~ x©$©cr}y!vwpqxzy!~Ç°p»vfbxzik|}tqj ®u_tqr!j~Æ¥~wjj ¼ ¿± #jv Ω Dju_ykxz|Mjy&Mxzr!y j ~wr!!~jv$x© R Æz°p»vf&tqxGPutt #pq|!~Kf!pvwÌ xzy¥vpyTr!xzr!~Mxzr!y u_ ± ¦ j¨u_~w~wr}ikjbvf!u_v pq~u y!xy ¬ jik|}vÈ xzy!y}jvwj x|Djy ~wr!}~wjvΓ x=© ΩΓÆ(°∪p»vf«Γ tqxGPutt #p|!~wKf!p»vÌxzy¥vwpqyTΩr!xzr!~:Mxzr!y u_ γ Æ uy ~wr!Kfvf[uv Ω ⊂ Ω Æ[~wjj Í[zr}wj&z± ÅÈy¥vwwx r!py!vf!j&Kf[uy}zjkx_©$®_upIu!tj~ z = −λs Æ'jpqjy!|!xz!tji + , vK$u _j~ vf!j ©cxztqtxî°py!Ïvwu p»vpxzy[utW©cxzwi +c~wjjÎut~wx§"uvÅ »P}Ægodjvwpqxzy«l!±Ãd±( , Í[y λ ∈ C Æ Æ uy s, z ∈ C Æ!°p»vf (u, p, s, z) 6= 0 Æ[~wr}Kf vf!u_v u : Ω −→ C p : Ω −→ C py Ω , ρ ∇u u + ∇uu − 2µ div ε(u) + ∇p = λρu, py Ω , div u = 0, xzy Γ , u = 0, xzy Γ , +] , σ(u, p)n = 0, xy 3. p. 3. in. s. out. s. f. 3. ns. f. 0. f. 0. f. in. out. u = Φz − ∇u0 Φs,. K+B. 0. . s+. Z. γ, −z = λs,. ΦT σ(u, p)n da = λ M z.. ¦ j xzy}~wp jgvf!j0©cxzttqx¼°pqy}nxzi:|!tj oGxzDxtqj®~w|[u_j~( xzy Γ H (Ω ) = v ∈ H (Ω )|v = 0, xzy Γ H (Ω ) = v ∈ H (Ω )|v = 0, γ. 1 Γin. f. 1. f. 1 Γin ∪γ. f. 1. f. in in. ,. ∪γ ,. ÝcÞßÝcà.
(38) ²"³µ´·¶w¸¹.½¿¸³µÄ ³µ½%¸´M¸Ä ³ ³µ´d³ 4 ½B¹ Á½d¹¶³µ´[½È¶¹w¸GÁ½B³¿¾´ uy u:tpqy!ju(xzy¥vpqyTr!xr!~tqp©µvÚxz|Mjuvxz. +B ,. 1. R : H 2 (γ)3 −→ HΓ1in (Ωf )3 .. $<inr!t»vpq|}tGpqy!bjÑTr[u_vpxzy +] , ¥ Æ(pqy¥vjzuvpqy}¥<|[uv~uy vKu$Tpqy}kpyTvwx%uxzr}yTvYvf!j)Mxzr!y u_vxz∈y Hpvwpqxzy!~Æ[(Ω°Új)zjv{vf[uvjpqzjyT®_utr!j)|!wxz}tqji +¿ , uy§Djn°pvwvwjyÆr}y j0®_upIu_vwpqxzy[u_t'©cxwiÆpqybvf}jn©cxztqtxî°py!°ÚuP ÇÍ!y λ∈C uy (u, p, s, z) 6= 0 py H (Ω ) × L (Ω ) × C × C ~wr}Kf vf!u_v 1 Γin ∪γ. . 1. 1. f 3. 2. f 3. ns. f. ns. u − R(Φz − ∇u0 Φs) ∈ HΓ1in ∪γ (Ωf )3 , a(u, v) + b(p, v) = λd(u, v),. ∀v ∈ HΓ1in ∪γ (Ωf )3 ,. +] ,. b(q, u) = 0, ∀q ∈ L2 (Ωf ), −z = λs, K+B. °pvfy!x_vKu_vwpqxzy. 0. . s+. Z. ΦT σ(u, p)n da = λ M z, γ. a(u, v) = ρ(∇u0 u + ∇uu0 , v¯)0,Ωf + 2µ(ε(u), ε(¯ v))0,Ωf , d(u, v) = ρ(u, v¯)0,Ωf , b(p, v) = −(p, div v¯)0,Ωf .. ¯{xvj0vwf[u_vgvf}j)~r!©Bu_j pqy¥vjzu_tpqy ¿+ , Æ 5. inr!~vÇDjnvu$jy§pyÏvf}j~wjy}~)x_© jÍ[y}j u_~ ©cxz. Z. γ. Z. γ 1. H − 2 (Γ ∪ γ)n. uy °pvf. χγ (ϕ) =. ßß=ê$#&% ('Pÿ. s. Æp]± j_± vwf!j i¬ vf¨xzik|Mxzy!jy¥v0x©. σ(u, p)n · ϕi da = hσ(u, p)n, χγ (ϕi )iH − 12 (Γ∪γ),H 12 (Γ∪γ) ,. i = 1, . . . , ns. + ,. ΦT σ(u, p)n da,. . 0 ϕ. xzy xzy. Γ . γ. + ,. p~.
(39) ¼. γ $[¶Ä ´ G¶Ä 6¶¹-´ ´!G¶wº "À(¸½B¹-³¿Á ²¶ ¸ÄµÄq¶KÁ . . . . .
(40) . . egf!pq~ jÍ[y!p»vpqxyiku$_j~~wjy!~wj)DjPur}~wjÆ[©cxzi ¿+ , Æ 2. + ,. div σ(u, p) ∈ L2 (Ωf ),. uy vwf!jy ÆG~wjj{©cxzWpqy!~vKuy!j $ ]±ËÅÈyu p~wjvwjY©cu_ikj°Úxz xGj~"∈y}HxvËzj(Γy!jw∪utqγ)t»)f!xzt ±6egf!jwj©cxzwj_Æz°Új(f!u¼®j(vxÇ~|Djp»©µpqzxwxzr!~t vf!p~"xy pσ(u, vwpqxzy p)n vf!j pq~wjvjxzr}yTvwj|[uv%x© + , ± (u~wpPuttÆWvwf!jÏp jPu§xzy!~pq~v~py«vwwju_vpy!§vwf!j ~wr!©Buj§pqy¥vjzu_t u~Ïu ®_uwpqu_vpxzy[utYwj~wp r[ut¿Æ~wjj ]±ÅÈy jj Æ ¥Éinr!t»vpq|}tGpqy! +¿ , ¥ Æpqy¥vwjzwu_vpy! ¥@|[uv~uy vK$u Tpy! pqy¥vx«uxzr}yTvvf}jbDxzr}y u R(ϕ ) xzy pvpxzy!~Æd°$jzjv Z + ¼ , σ(u, p)n · ϕ da = a(u, R(ϕ )) + b(p, R(ϕ )) − λd(u, R(ϕ )). ÚtqjPu_wtÆDvwf!j&wpqfTvf[uy ~p jx© +¼ , pq~ÇMjvvj u u|}vwj vxu%Í[y}pvjsjtqjikjy¥v pq~ ¬ wjvpÌPu_vwpqxzyvf[u_yÏvf!jtqj©µvxzy}j±e6$u Tpqy}pqy¥vwx%uxzr!y¥'v + ¼ , ÆD°$jsuyÎwj°wp»vj +] , py vf!j9©cxzttqx¼°pqy!$°(uP Í[y λ ∈ C uy (u, p, s, z) 6= 0 pqy H (Ω ) ×L (Ω )×C ×C ~wr!Kfvf[u_v − 21. . 1. i. i. i. i. i. γ. 1. f 3. 2. f. ns. ns. u − R(Φz − ∇u0 Φs) ∈ HΓ1in ∪γ (Ωf )3 , ∀v ∈ HΓ1in ∪γ (Ωf )3 ,. a(u, v) + b(p, v) = λd(u, v),. b(q, u) = 0, ∀q ∈ L2 (Ωf ), −z = λs, 0 a b K + B s + F (u) + F (p) = λ M z + F d (u) ,. °pvf. s. zp®jyÆ}©cxzi + ¼ , Æ!¥vf!jÇ©cxzttqx¼°pqy}nj d|!j~w~pqxzy}~. F a (u), F b (p), F d (u) ∈ Cn a b F (u) i = a(u, R(ϕi )), F (p) i = b(p, R(ϕi )),. ©Bu. . i = 1, . . . , ns . + ,. ±. d F (u) i = d(u, R(ϕi )),.
(41)
(42) . ÈÅ yvwf!j ~wjÑTr!jt¿Æ[°$j)°ptqtu~~wr!i:j0vf[uv Ω ⊂ R p~u:|Mxzt»dxzy[ut xziu_pqy%°pvwf°f!pqKf °Új)u~~wxGpIuvj)uwjr!tIu$©Buikpt:x©vpIuy!r!tIu_vwpqxzy}~ {T } +B~jj à , Æ}~wr!Kfvf[u_v f. 3. h h>0. f. Ω =. [. K,. ∀h > 0,. K∈Th. ÝcÞßÝcà.
(43) ²"³µ´·¶w¸¹.½¿¸³µÄ ³µ½%¸´M¸Ä ³ ³µ´d³ 4 ½B¹ Á½d¹¶³µ´[½È¶¹w¸GÁ½B³¿¾´. z. °f!jj h pq~ jÍ[y!j ¥ h = max h Æ!°pvwf h vwf!j pIu_ikjvjÚx© K ± ÅÈy xz jvx u|!|!x¼ dpiu_vwjbvf!j§xzy¥vwpqyTr!xzr!~~w|!uj~ H (Ω ) uy pqy¥vx r}jÇvf}j Í!y!pvwj pqi:jy!~pqxzy!ut·~|[uj V jÍ[y!j ¥ k. K∈Th. k. 1. f. L2 (Ωf ). ÆÇ°$j. h. n. f. Vh = vh ∈ C 0 (Ω )| vh|K ∈ P1 (K),. ° f!jj P (K) ~vuy ~g©cxzÚvwf!j ~w|[u_j x©6|Dxzt»Gy!xzi:pIut~$xzy vxz±9egfTr!~Æ[°$j jÍ[y!j vf}j ©cxtqtqx¼°py! p~wwjvwj~|[uj~ 1. Xh = Vh ∩ HΓ1in (Ωf ), . Q h = Vh ,. ÈÅ y<u pq~wjvj©cwui:j°$xz Æ9vf!jxz|Dju_vwxz xz|Dju_vwxz. R. o ∀K ∈ Th , K. x© jzjj tqj~w~gxzgjÑTr[ut +¼ ,. Vh,0 = Vh ∩ HΓ1in ∪γ (Ωf ).. py ]+ , pq~nj|!tquj ¥ u p~wwjvwjtqp»©µv. Rh : Tr(Vh )3|γ −→ Xh3 .. ÅÈyvf!j~ui:jY°ÚuPzÆ}°$jÇpqy¥vwwx r!jÇu xzy γ P. P1. #'uzu_y!zj. ¬ |!pqjj°pq~wjYpqy¥vjw|MxztIuvpqxykxz|Mjwu_vx. h. Ph : C 0 (γ)3 −→ Tr(Vh )3|γ ,. jÍ[y}j u~{vf!jwj~vpqvwpqxzyxzy γ x_©Ëvf!jtqu~w~pqPu_t P #'uzwuy!zj ¬ |!pjj°pq~jpy¥vj|Dx ¬ tIu_vwpqxzy%x|Dju_vwxz Π pqy Ω ± ¦ p»vf<vf!p~&y!xvKuvpqxyÆW°$jÏuy u|!|!x¼ dpiu_vwj|!wxz}tqji +z , Æ(¥<wj|!tIu_pqy}¨vwf!j xzy¥vpyTr!xzr!~{~w|[u_j~ H (Ω ) u_y L (Ω ) ¥vf!j p~wjvwj)~|[uj~ V u_y Q Æ+B~jj à , ±
(44) {jy!jÆ9°$j%xz}vupqy2vwf!j©cxzttqx¼°pqy}Ïu|}|!wxP dpqiku_vjk|!wx!tqji nÍ[y uy λ ∈ C p y ~ ! r K f w v [ f _ u v (u, p, s, z) 6= 0 V × Q × C × C 1. f. h. 1. 3 h. f 3. h. 2. ns. f. h. h. ns. 3 u − Rh Ph (Φz − ∇u0 Φs) ∈ Vh,0 ,. a(u, v) + b(p, v) = λd(u, v),. 3 ∀v ∈ Vh,0 ,. b(q, u) = 0, ∀q ∈ Qh , −z = λs,. °pvf. K + B0 s + Fha (u) + Fhb (p) = λ M z + Fhd (u) , s. zp®jyT. Fha (u), Fhb (p), Fhd (u) ∈ Cn a Fh (u) = a(u, Rh Ph (ϕi )), b i Fh (p) = b(p, Rh Ph (ϕi )), d i Fh (u) i = d(u, Rh Ph (ϕi )),. ßß=ê$#&% ('Pÿ. + ,.
(45) î. γ $[¶Ä ´ G¶Ä 6¶¹-´ ´!G¶wº "À(¸½B¹-³¿Á ²¶ ¸ÄµÄq¶KÁ . . . . .
(46) . . ©cxz i = 1, . . . , n ± Y~6°$jÚf[uP®jgutqjPu |Dxzpy¥vj xzr}v6pqys"uv"Å ¼dÆ_|[uwuzu_|!fsl!±Ãd± ]Æzuy u~"p»v'°pqtqt Dj&xy}Í[wi:j Djtqx¼° +cpqy§~wjvpxzy§l!± l , Æ'vf!j:~wxztr}vpxzy¨x©$|!wxz}tqji + , Æ6py¥®zxzt»®zj~u i:p» dj Í[y!p»vjjtji:jy¥v6u|!|!xP }piuvpqxyÇx©!tpqy!juwpÌj ¯YuP®dpj ¬ oGvwx j~6|!xz!tji:~ P /P °pvfpqzf¥vf[uy ~p j)uy Ìjwx:xz jgjPuvpqxyvwji Æ pqy Ω , ρ ∇u u + ∇uu − 2µ div ε(u) + ∇p + rρu = ρf, pqy Ω , div u = 0, xzy Γ , +l , u = 0, xzy Γ , σ(u, p)n = 0, xy γ, u=u , °pvf r ∈ R Æ f u_y u zp®jy u_vKud± egf}j |[u_pqnx©~w|[u_j~ ÆWKf!xz~wjy vwx p~wwjvwpqÌj%vf!j®zjtqxGpvÈ uy |!wj~w~r!wj Í[jt ~Ƽ©Bupt_vxg~wu_vp~©µYvf!j # P /Pbxik|[uvpq}pqtqp»vÈxzy pvwpqxzy + #'u GÌf!jy! ~ uP¥u ¬ (u}r! ~ u ¬ ÚwjÌÌpx pqyd© ¬ ~wr}| Çxzy p»vpxzy , Æd~wjj $ ]±Åövpq~(°$jtt Gy}xî°yvf[uvPÆ!p»©vwf!pq~gxzy p»vpxzy xdj~ky}xvf}xzt Æ$vf!jyTr!ikjwpPut~wKf}ji:jÎ|!wx r!j~kxz~wpqtqtqu_vpy!b|!j~w~r!wj~ +B~jj©cxz pqy!~vKuy!j î , ±Ëegf!j È~vu!pqtpqÌj Í[y!p»vjjtji:jy¥vWikjvf!x ~ gxî®jxzi:jYvwf!pq~|!xz!tqji ± egf!jszxzutpq~gvxkjy!f!uy!j~vu!pqtpvÈ +Bpqy¥vwwx r!py! p Dr!~wpxzy , °p»vf!xzrdvYr}|!~wjvwvpy!kvwf!j xzy!~pq~vjy} +B~jj , ± ¦ j(xzr!t r!~wjguYxzin}pqy[u_vwpqxzy)x©~|[uj~Ë~uvpq~©µGpqy!vwf!j # xzy p»vpxzyÆ!r}v"vwf!j yGr}ikjwpqutMu|!|!xP }piuvpqxykx©xzy¥®zjvwpqxzy ¬ö p Dr!~pqxzy +BxzÚ¯uP®Gpqj ¬ oGvw x j~ , jÑTr[u_vpxzy!~Æ °pvf«tqx¼° ¬ xz j|!pqjj°pq~wj|MxztGy!xzi:pIu_tq~Æ9ikuP<ut~wx¨|!wx r!j ~j®jwjt«xz~pqttIu_vwpqy! ~wxztr}vpxzy!~±&ÊyTr!inMj)x©(~vKu}pqtqpÌPu_vwpqxzyvjKf!y!pqÑTr!j~sf[uP®jMjjy j®jtxz|Dj ©cxzÇvwf!j vwju_vi:jy¥vgx_©6vwf!pq~(|}wxz!tjiÆ}~jj0©cxz(py!~vuy!j dÆGÆ!lT ¿±WodpikptIu p :r!t»vpqj~(iku¼ u|!|MjPu°f!jy jPutpqy! °pvwf xikpy[u_vpy!0wjuvwpqxzy:vjwi:~±ËY¥upqy·ÆG~xzi:j{vwjKf!y!pÑTr!j~ x©Ë~vKu!ptqpÌPu_vwpqxzy%f!u¼®jsMjjy j®jtxz|Dj ©cxzgvwf!pq~{|!r!|Dxz~jÆM!r}v©cxz{|[uvpqr!tIu_(Pu~j~ °f!jj)vwf!j0wjuvwpqxzywj r!j~Yvwx:u&~Putqugxy!~vuy¥v l }ƼGÆM_ldÆ ¿± egf}js~vu!ptqpqÌj ~wKf}ji:j~Çu_wj zjy}ju_tqt%xz}vKu_pqy!j Æ!©cwxzivwf!j)tIu_~w~wpPut )utj Gpy ikjvf!x +cy!xv ¬ ~vu!pqtpqÌj - , ÆM¥Ïu ! py!u } pvwpqxzy[u_tvjik~Ypqy¥®zxt®Gpqy!:vf!js|!wx r!vÇx© vf!j{wj~wp r[utx©vwf!jjÑGr!u_vpxzy:°p»vfuy!j°@vj~vÚ©cr!y}vpxzy&°f!pqKf j|Mjy ~$xzy%u)tqxGPut ~vKu_!pqtpqÌPuvpqxy |[uu_ikjvjÆ ± Yyd©cxzvwr!y[u_vwjtÆÚvxxzr!:|!wj~wjyTv Gy}xî°tj jÆ vf!jwjpq~Ëy}xu0~vKu!ptqpÌj Í!y!τpvwjgj>tqjik0 jy¥vikjvf!x ©cxzËvf}j pq~wjvpÌPu_vwpqxzyx©·uÇzjy!jwut |!wxz}tqji x©vÈd|Mj + Pl , Æg~wjj lzl ¿± egf!j p~wjvwpqÌPuvpqxy ~Kf!ji:jÎvf[uvk°$jb|!wx|Dxz~j f!jjpqy©cxz +l , pq~ pqjvt%xz}vKu_pqy!j ©cxzivf[uvY|}wxz|Mxz~wj py lT '©cxzgvwf!j Ç~jjy·Û ~ s. 1. 1. 0. f. 0. f. in. out. γ. γ. 1. . 1. . . . . . . . . K. ÝcÞßÝcà.
(47) ²"³µ´·¶w¸¹.½¿¸³µÄ ³µ½%¸´M¸Ä ³ ³µ´d³ 4 ½B¹ Á½d¹¶³µ´[½È¶¹w¸GÁ½B³¿¾´ jÑTr[u_vwpqxzy!~u_y °pvwj~9Í[y . (u, p) ∈ Vh3 × Qh. . ~r!Kf vwf[u_v. 3 u − Rh Ph (uγ ) ∈ Vh,0 ,. a(u, v) + b(p, v) + b(q, u) + rρ(u, v)0,Ω X + ρ(∇u0 u + ∇uu0 ) − 2µ div ε(u) + ∇p + rρu,. +¼ ,. K∈Th. . τK (ρ∇vu0 − 2µ div ε(v) − ∇q) 0,K X = d(f, v) + ρf, τK (ρ∇vu0 − 2µ div ε(v) − ∇q). °pvf. ∀(v, q) ∈. τK. K∈Th 3 Vh,0 ×. 0,K. ,. Qh ,. vf!j ~vu!ptqpqÌu_vpxzy|!uui:jvwjg|!x¼®dp j pqy! lT ]Æ. ~wp 0 ≤ x < 1 ~p x ≥ 1 . ξ(x) = oGj®zjut9jPu~xzy!~$r!~vp»©µÏvf}jnpqy¥vx r!vwpqxzyÎx_©Wvwf!j&uMx¼®zj&~wKf!jikj_± Çybvf!jxy!j f[uy Æ°$j&f[uP®zj&ji:|!txîj vf}j~ui:jsvpIuy}zr!tIuvpqxy T pqyÎvwf!j&xzi:|!r}vu_vpxzyÎx© u uy u +Bpvspq~y!xvikuy uvxzzÆ9!r}vn|!uvpPut , ±begfTr!~Æ$uy<ur!wu_vjxzi:|!r}vu_vpxzy x©gvf!j|Djwikuy!jy¥vnª!xî° uy x© Æ9wjÑTr!pwj~uÏwjÍ[y!j wp pqyvwf!j ®dppy!pvÈ&x© γ ±9Y~Ú|Mxzpqy¥vwj (uxzrd, vgp pq)y »î¿ÆGvf!p(∇u ~Úwj )r!j~(vf}jxik|!tpqu_vpxzy!~Wu~w~xdpIu_vwj °f!jy jutqpy!°p»vf xzi:pqy[u_vwpqy!§wjuvwpqxzy«vjwi:~± xwjx¼®jÆ~Kf!ji:% j +î , p~%u pqjvÇj Gvjy!~wpxzy¨x©$vwf!xz~jpqy¥vx r}j py Ü l uy lT "©cxzvf}j:oGv x _j~ jÑTr[uvpqxy!~ °pvf§xyT®jvpqxy±ÅÈy§vwf!pq~ °Úu¼Æ6°$jf[uP®j _j|}vvf!j:Kf!xzpj:x© τ ©cxz0vwf!pq~0vÈG|Dj:x© jÑTr[u_vwpqxzy!~Çu~0|!wx|Dxz~j pqy lT ¿± Çy¨vwf!jx jÇf!uy Ævwf!jnyTr!i:jwpPut6j d|Mjwpikjy¥v~ wj|Mxzvj pqy~jvpqxyÎs°pqtt·|Mxzpqy¥vxzrdv{vwf!j0|Mj©cxwikuy!j x©6~wKf}ji:j + î , ± %Ó 2¢!¡ ½W³ .½]¹¸³ ½ P¾¹
(48) W¸&¹ k½È¾ "¶¹-³ k ½ }¸½ +î , ³ nÁK¾´ ³½È¶´[½
(49) 9³µ#½ ½ [¶ ¼¾ Ä d½B³¿¾´ n¾ +Pl , W@zjy}ju_tqpqÌpqy! + î , ÆY°$j¨xzy!~p jpqy!~vju x© + , vf}jΩcxztqtxî°py! pq~wwjvj ~wKf!jikj Í!y λ ∈ C uy (u, p, s, z) 6= 0 pqy V × Q × C × C ~wr!Kfvf[u_v . ku0 k2 hK Rh eK = , 12ν. hK τK = ξ(Rh eK ), 2ρku0 k2. x 1. 0. h. 0. 0. 0 |γ. K. . . 3 h. ßß=ê$#&% ('Pÿ. h. ns. ns.
(50) Pl. γ $[¶Ä ´ G¶Ä 6¶¹-´ ´!G¶wº "À(¸½B¹-³¿Á ²¶ ¸ÄµÄq¶KÁ . . . . .
(51) . . 3 u − Rh Ph (Φz − ∇u0 Φs) ∈ Vh,0 ,. a(u, v) + b(p, v) + b(q, u) X ρ(∇u0 u + ∇uu0 ) + ∇p, τK (ρ∇¯ v u0 − ∇¯ q) + K∈Th. ". = λ d(u, v) +. X. ρu, τK (ρ∇¯ v u0 − ∇¯ q). K∈Th. . 0,K. #. ,. 0,K. 3 ∀(v, q) ∈ Vh,0 × Qh ,. − z = λs, K + B0 s + Fha (u) + Fhb (p) = λ M z + Fhd (u) ,. +P ,. °pvf . X ρ(∇u0 u + ∇uu0 ), τK ρ∇(Rh Ph (ϕi ))u0 Fha (u) i = a(u, Rh Ph (ϕi )) + K∈Th. X b ∇p, τK ρ∇(Rh Ph (ϕi ))u0 Fh (p) i = b(p, Rh Ph (ϕi )) + K∈Th. . X Fhd (u) i = d(u, Rh Ph (ϕi )) + ρu, τK ρ∇(Rh Ph (ϕi ))u0. ©cxz i = 1, . . . , n ± % Ó 2¢!¡ ¾¹k¶w¸GÁ. ³ [¶ ³µ´ K W~wjvwvpy!. K∈Th. s. . u ∈ Vh3. Ú¶ } ¸ "¶
(52). u|K ∈ P31 (K). 0,K. 0,K. 0,K. ,. ,. ¸´! ½# [¶´. div ε(u). ¸´3 ". . as (u, v) =. X. ρ(∇u0 u + ∇uu0 ), τK ρ∇¯ v u0. K∈Th. bs (p, v) =. X. ∇p, τK ρ∇¯ v u0. K∈Th. bts (u, q) = −. X. K∈Th. . 0,K. . 0,K. ,. ,. ρ(∇u0 u + ∇uu0 ), τK ∇¯ q. 0,K. ,. ,. ÝcÞßÝcà.
(53) ²"³µ´·¶w¸¹.½¿¸³µÄ ³µ½%¸´M¸Ä ³ ³µ´d³ 4 ½B¹ Á½d¹¶³µ´[½È¶¹w¸GÁ½B³¿¾´. cs (p, q) = −. X. K∈Th. ds (u, v) =. X. ∇p, τK ∇¯ q. ρu, τK ρ∇¯ v u0. K∈Th. es (u, q) = −. X. 0,K. . ρu, τK ∇¯ q. î. ,. 0,K. 0,K. ,. ,. |!wxz}tqji + ¼ , Pu_yÎDj °wp»vwvwjyÏpy vwf!j)©cxtqtqx¼°py!&ikxzj xzi:|[uvY©cxzi W Í[y uy (u, p, s, z) 6= 0 py V × Q × C × C ~wr!Kfvf[u_v K∈Th. 3 h. ns. h. λ∈C. ns. 3 u − Rh (Φz − ∇u0 Φs) ∈ Vh,0 ,. +¼ ,. a(u, v) + b(p, v) + b(q, u) + as (u, v) + bs (p, v) + bts (q, u) + cs (p, q) 3 = λ d(u, v) + ds (u, v) + es (q, u) , ∀(v, q) ∈ Vh,0 × Qh , − z = λs, K + B0 s + Fha (u) + Fhb (p) = λ M z + Fhd (u) .. .
(54) . ÈÅ y<vwf!pq~|!uuu|!f·Æ$|!xz!tji + î , p~wj©cxzwinr}tIu_vwj pqy vwjwi:~x©Yiuvwpj~± egf!p~ °pqtqt"utqtx¼° r!~0vwx j- }|}tqpqpvwt xzi:|!r}vwj:p»v~0~xztqr}vwpqxzy}~± #jv n = n (h) vf!j&yTr!inMj x©6®jvj x©6vf!j0vwwpIu_y!zr!tqu_vpxzy T xzyvwf!j ª[r!p xzikupy± ¦ j)pqy¥vwwx r!j0vwf!j0Í[y!pvwj jtqjikjy¥vjPut'[u~pq~ {φ } uy {ψ } x_© V u_y Q wj~w|Mjvp®jt»z±egfTr!~ÆDjPuKf jtqjikjy¥v (u, p) ∈ V × Q PuyDj0°pvwvwjyu~ f. 3 h. 3nf i i=1. h. nf i i=1. 3 h. u=. N. . 3n X. f. I. uj φ j ,. Γin. ± x!± © ± i qp ~gy!x_vxzy Γ ∪ γ}, ± x!± © ± i qp ~gxzy Γ }, = {i ∈ I|. ßß=ê$#&% ('Pÿ. p=. n X. p j ψj ,. j=1. ± ¦ j{pqy¥vx r}j{utq~x0vf!j©cxztqtx¼°pqy!0~r!!~wjv~$x©. I Ω = {i ∈ I|. in. + ,. f. j=1. uj , p j ∈ C. h. h. f. °pvf. f. ± x!± © ± i qp ~gxzy ± x!± © ± i qp ~gxzy = {i ∈ I|. I Γout = {i ∈ I| I. γ. I = {1, . . . , 3nf } ⊂ Γout }, γ},.
(55) ¼. γ $[¶Ä ´ G¶Ä 6¶¹-´ ´!G¶wº "À(¸½B¹-³¿Á ²¶ ¸ÄµÄq¶KÁ . . uy vf!jy °$j jy!x_vj u (I ), n = u (I ), n = W~wr!}~vp»vr}vwpqy! + , qp y +¼ , °$jzjv Ωf. Ωf. Γout. Γout. . . nΓin =. . u (I. Γin.
(56) . ),. nγ =. . u (I ). γ. . 2. f. 3n X. f. uj a(φj , v) +. j=1. n X. f. pj b(ψj , v) +. j=1. +. pj bs (ψj , v) +. = λ. 3n X. uj d(φj , v) +. j=1. uj as (φj , v). f. uj bts (φj , q). +. n X. pj cs (ψj , q). j=1. 3nf. X. 3n X j=1. j=1. 3nf. X. uj b(q, φj ) +. f. j=1. . f. j=1. f. n X. 3n X. . f. uj ds (φj , v) +. j=1. 3n X j=1. uj es (φj , q) ,. + ,. 3 ∀(v, q) ∈ Vh,0 × Qh .. ¦ j jy!xvj)¥ u ∈ C Æ u ∈ C Æ u ∈ C jv u ∈ C vf!j jzwjj~ x©(©cwjj xi x_© u xzwj~w|Mxzy py!!ÆËj~w|Mjvp»®zjt»zÆ"vwxÏ~f[u|Mjk©cr!y!vpxzy!~spy I Æ I Æ uy I ±ËÅÈyvf!j~wjÑTr!jt°ÚjÇ°ptqtDu~w~r!i:jvf[uv(vf!j~wf[u|MjÇ©cr!y!vpxzy!~ {φ } uj I xz jwj pyk~r!Kf%u0°(uPvwf[u_vvf!jgÍ[~v jwjj~$x_©·©cwjj xi4xzwj~w|Mxzy vwx u ÆTy!j Gv vx u Æ[y!j Gvvx u uy Í[y[uttkvx u ± W0vK$u Tpqy}Yp y + , Æ°pvf i ∈ I ∪I Æzu_y q = ψ Æ°pvwf i = 1, . . . , n Æ °Új xz}vupqyu (n + nv = +φ n ) × (4n iu_vwwp %j- }|}wj~~wpxzyÏx©vÈG|Dj + 2n ) Ωf. nΩ. f. nΓout. Γout. Γin. nΓin. γ. nγ. Ωf. Γin. γ. Γout. Γin. γ. Ωf. . Ωf. . f. AΩ 1 AΩ2 f f BΩ. AΓ1 out AΓ2 out BΓout. Γout. 3nf i i=1 Ωf. AΓ1 in AΓ2 in BΓin. i. f. Γout. Aγ1 Aγ2 Bγ. B1 B2 C. =. Γout. f. . f. . uΩ uΓout 0 0 uΓin γ 0 0 u p 0 0 z s. . f. DΩ 1 f λ DΩ2 f EΩ. f. i. s. DΓ1 out DΓ2 out EΓout. DΓ1 in DΓ2 in EΓin. Dγ1 Dγ2 Eγ. . f. . uΩ uΓout 0 0 0 uΓin γ 0 0 0 u . 0 0 0 p z s. +¿_ ,. ÝcÞßÝcà.
(57) ²"³µ´·¶w¸¹.½¿¸³µÄ ³µ½%¸´M¸Ä ³ ³µ´d³ 4 ½B¹ Á½d¹¶³µ´[½È¶¹w¸GÁ½B³¿¾´. î. ÅÈyvf}j)~wui:jÇ°(uPzÆ¥~r!!~vwpvwr}vpy! + , qp y+ î , °$j zjv 4. f. K + B0 s +. 3n X. . f. uj Fha (φj ) +. j=1. n X j=1. pj Fhb (ψj ) = λ M z +. °f!pqKftqju ~(vxvwf!j0©cxztqtxî°py!niku_vp» j d|!j~w~pqxzy x©6~wpqÌj . FΩ a. f. . FΓa out. FΓa in. Fγa. f. j=1. uj Fhd (φj ) ,. ns × (4nf + 2ns ). . . uΩ uΓout uΓin 0 γ 0 K+B u p z s. Fb. . f. 3n X. f = λ FΩd FΓd out FΓd in Fγd 0 M. . f. . uΩ uΓout uΓin γ 0 u . p z s. +¿G ,. ge f}j(vu_y!~w|!pu_vwpqxzypqy¥vj©Buj(xy pvwpqxzy+î , pq~6vK$u _jy&j d|!tpqp»vt» xzyjPuKf&pyTvwj ¬ ©Buj®zjvj- DÆ x ÆGx©Dvf!jvpIuy!r!tIu_vwpqxzy·±egf}jwj©cxzwj_ÆG°$jx}vKupy:vwf!j©cxztqtxî°py!Çiku_vp» j d|!wj~w~pqxzy x©"~pqÌj (n + n ) × (4n + 2n ) 1. i. Γin. . 0 0 I 0 0 0 0 0 0 I 0 − G0. . ij. j. . (i). 1 G ij = [∇u0 ϕj ]. ßß=ê$#&% ('Pÿ. f. y¥®. (i) ),. (i). (x. (x. y!. s. . . uΩ uΓout uΓin 0 γ 1 u G p z s. °pvfvwf!j0©cxztqtxî°py!ny!x_vKu_vwpqxzy G = [ϕ ]y! 0. f. γ. y¥®. =. f. . uΩ uΓout uΓin 0 0 0 0 0 0 0 γ , λ 0 0 0 0 0 0 0 u p z s. (i) ),. i ∈ I Γin. j = 1, . . . , ns ,. i ∈ Iγ. j = 1, . . . , ns .. +¿ ,.
(58) . γ $[¶Ä ´ G¶Ä 6¶¹-´ ´!G¶wº "À(¸½B¹-³¿Á ²¶ ¸ÄµÄq¶KÁ . . . . .
(59) . . YjjÆy! (i) ∈ {1, 2, 3} p~Çvf!j&xik|Mxzy!jyTv)x©Wvf}j&®jtxdpvÈbxzwj~w|Mxzy pqy!vxvwf!j ®zjtxdpvÈ jzwjjkx_©(©cwjj xzi i ÆËuy y¥® (i) ∈ {1, . . . , n } vf!j:tIuMjt9x_©(vf!j&®zjvwj °f!jj)vwf!jÇ®zjtqxGpvÈ jzjj x©©cwjj xzi i tpqj~Æ}©cxz i = 1, . . . , 3n ± ÅÈy~f!xzvPÆdv$u Tpqy!spqy¥vxu_xr!y¥v +î , °pvwf +¿_ , Æ +¿d , uy +]z , °Új0xzdvKupykvf!u_v vf!j p~wjvwj)|}wxz!tji +î , pq~gjÑTr!p»®_utqjyTvgvwxvf!j0©cxzttqx¼°pqy!szjy!jutpqÌj jpzjy¥®_utr!j |!wxz}tqji x©6~wpqÌj n = 4n + 2n 9Í[y λ ∈ C uy 0 6= x ∈ C ~wr!Kfvf[uv
(60). f. f. 3. f. . f. AΩ 1 Ωf A2 0 0 Ωf B 0 f FΩ a |. AΓ1 out AΓ2 out 0 0 Γout B 0 FΓa out. AΓ1 in AΓ2 in I 0 BΓin 0 FΓa in. n. s. Aγ1 B1 Aγ2 B2 0 0 I 0 γ B C 0 0 Fγa Fb {z. 0 0 0 − G0 0 −I 0. f 0 uΩ Γout 0 u Γin 0 u 1 γ G u p 0 0 z s K + B0 | {z } }. x. A. . =. f. DΩ 1 Ωf D 2 0 λ 0 Ωf E 0 f FΩ d |. DΓ1 out DΓ2 out 0 0 EΓout 0 FΓd out. DΓ1 in Dγ1 DΓ2 in Dγ2 0 0 0 0 EΓin Eγ 0 0 FΓd in Fγd {z. B. 0 0 0 0 0 0 0 0 0 0 0 0 0 M. f 0 uΩ Γout 0 u Γin 0 u 0 uγ p 0 I z s 0 | {z } }. +¿ ,. x. ge f}j)iku_vwwpqj~ A u_y B uwj0jPut]Æ}~w|!uw~jÆy!xzy ¬ ~Gi:ikjvwpÇuy Æ!pqyi:xz~v(x©'vf!j u|!|!tpqPuvpqxy!~ÆËx_©Ytquwj%~wpÌj± Ù xztqtxî°py! vwf!j #py!jPupqÌu_vpxzy29pqy!pq|!tju|!|!x¥uKf j®jtqx|Dj py6uv'Å »PdÆ~wjvpxzy ¿Æîpyugtpqy!ju~vu!ptqpvÈYuy!utG~wp~vf!jzxzutp~vxgÍ[y jpqjy¥®_utqr!j~Y°p»vfby!j¥u_vp»®zjnjPut|[uv0pqyÏxz jvx jvwjv py!~vu!ptqpvwpqj~±{eWG|!pPuttzÆ utqi:xz~vÚuttDjpqzjyT®_utr!j~Úx© +]$ , f[uP®zj |Dx~wpvwp®jwjut·|!uvu_y xzy!t»un~iu_tqtyTr!inMj wx~w~Yvwf!jnpiupqy[uu dp~±egf!jj©cxzjÆ·vwx jvjv u:~vKu_!pqtpvÈKf[u_y!zjÆMvf!jpqy¥vjwj~v tqpqj~(pqyxzik|}r}vpy!nvf!j0©cj° jpzjy¥®_utr!j~g°pvwf~wikutqtj~v(wjPu_t|[uvP± ÅÈy<vf}jy!j Gv:~wjvpxzyÆ°Új°ptqt jPut(°p»vf vf}jyTr!i:jpqutgu|!|}wxP dpqiku_vpxzy2x©u ~wikutqt$yGr}inDj +Bxik|[u_wj vx n, x_©~xztqr}vwpqxzy}~x©Yvwf!jzjy}ju_tqpqÌj jpzjy}|!wxz}tqji +¿$ , ±9exn©cr!tqt»&j d|!tqxpvÚvwf!jÇ~|[uw~j0Kf[uwuvwj(x©vwf!jÇiku_vpqj~ÆdpvW°pqttDMjÇr!pqutvx r!~wjÇi:jvwf!x ~Ú°f!pKfxzy!t:pyT®xzt®jÇxz|Mjwu_vpxzy!~Úx_©vÈd|Mj0iku_vp» ¬ ®jvwxzW|!wx r!v +Bp]± j_± ÝcÞßÝcà.
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