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Modeling of topological derivatives for contact problems

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(1)Modeling of topological derivatives for contact problems Jan Sokolowski, Antoni Zochowski. To cite this version: Jan Sokolowski, Antoni Zochowski. Modeling of topological derivatives for contact problems. [Research Report] RR-5599, INRIA. 2006, pp.36. �inria-00070408�. HAL Id: inria-00070408 https://hal.inria.fr/inria-00070408 Submitted on 19 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Modeling of topological derivatives for contact problems J. Sokolowski — A. Zochowski. N° 5599 16 Juin 2005. N 0249-6399. ISRN INRIA/RR--5599--FR+ENG. Thème NUM. apport de recherche.

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(22). N@. 9,. ρ. I. 7M. @. B(ρ). 21. 21. P3. ρ. @. 2.. )@. ρ. ρ = 0+. ρ ∈ (0, ρmin ] Ωρ = Ω \ B(ρ) ρmin. 01. ͐é =Í.

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(26). Γs. .. Ω ⊂ IR2. u(Ω). u = u(Ω) ∈ K :. @.

(27). Ωρ. B(ρ). ρ} ⊂ Ω O u = u(Ωρ ). . B(ρ) = {x : |x − O| <. ,. ρ. .. u(Ωρ ). Γ , ρ. Ω. . ∇u · ∇(v − u) ≥ 0 ∀v ∈ K. K(Ω) = {v ∈ H 1 (Ω)|v = g on Γ0 , v ≥ 0 on Γs } .. @. A3. Z. @. @. O. B(ρ). u(Ωρ ) ∈ H 1 (Ωρ ). Ωρ ρ ≥ 0 ρ. 1 E(Ωρ ) = 2. Z. ρ → 0+. k∇u(Ωρ )k2. kqs8KM`®YuXYaHJLkqXrVWKMj8Y[fhYa`ct‚L £8jlkp\lXr`nfp\ y  Jj=• pypyJ  l•· DD FE. RU. Ωρ. S L#E. ρ2 π |∇u(Ω; O)|2 + O(ρ3−δ ) 2. `n\—fq_Y[HJXafpLKMLEL\JL*_[ pVˆ]J\lt“Y[`‘fq•\lkhGm(Hlt*L kp\—l_u”4XZYL„fq_aLE_[jJsJ_[L*LE_Xat L*fq\d_aYa_[L;LEs¡t Ya`‘\—`nfp\Y[HJYaLLE_aL;K bd]J`‘’#kpm‘LE\qYPfp_[K fpBkm‘`n\JL„`‘\dYaLE p_ukhm=f#’qL*_Pk t*`‘_ut mnL `®Y[H Y[HJLt LE\dYa_[L khY E(Ωρ ) = E(Ω) +. T.. 0<δ <1. ΓR = {x : |x − O| = R}. ρ2 b(u(Ω), u(Ω)) = −ρ2 π|∇u(Ω; O)|2. @. 1 b(u, u) = − 2πR6. O. "Z. ux1 ds. 2. +. Z. ux2 ds. 2 #. 4jGkhHl_uL*kh_[KML L fpY[_[L*Lp_     L,t*kp\ sJL l\JLY[HJL\JL LE\JL*_[ pV ]J\4t“Ya`nfp\4khmls8L l\JL;s„fq\ kh\4s sJL*jxL*\ls8`n\J fq\Y[HJL,XrKkpm‘ GXa`nfpHl\lLX%]Jfh\lFt LEYam‘n`n`fpj8\lYakp`ct‚m fpjxL*_uk#Yafq_[XEXr  ]J”4HlXZ`nYutukhH \dYa`‘\l`ckht*mnm‘‘]4V¡s8s8LE`X%xYaLEHl_[L‚XPXa_[fhfp¢ït*K kpm‘YanLEHJs:L„XaL*L*\Jj4LEkh_a_u qk#VYaL;s8s:L lkp\JXaVWL;KMs¡j8fpYa_‚fpYaXr`cLEt*m®X åkqs¨gZfp6 `n\d·YPL 6£W 8YakhLE\4\8s ¢ @. .. W@. ρ. E0 (ρ ; w) =. E0 (ρ ; w). Πɓɓ“ ].

(28) @. 1 2. ΓR. Z. 21. .. Ω. ΓR. Ω. k∇wk2 dx + ρ2 b(w, w). .. =D SE D LE.

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(30) 2[  XF [. u(Ωρ ). E0 (ρ ; w). H 1 (Ω). K = K(Ω)

(31) u(Ωρ ). ?@. @ uρ. @. 67# @ . uρ = u(Ω) + ρ2 q + o(ρ2 ) .. @. . . 01.  (  ?;  67   

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(58) KHJ–L—GHJjxLL—fpmn_aVWHlL;HJk¨XrLE’q]ls8L—m®Y_[`cYatf fpmnt*’qffpL*,\W_[’p`®XL*V ”d£ V YaXrHlkhL*khY:\—Yˆ`‘kh\Y[”lHJXrLYa_u`‘Xakp_[L `nt tuY YHl_[m‘L*LEYˆXa]JXaXrm‘khjlY,Yakq`cfpXZt \LqV • YaHJjJGL_[fpfsJjx` khL*3jJ_aL*YZjl_[V L*m‘V\dYa{;Y[`ckhHJ”l:L—`‘mn p_[`®`nYZLE’pV XaL*]J\fhm®YM”xK L*fqL*mn_Mf Ya_[Ya`nHJt• L¦jlTW_afp`‘`n_u pgZXZLE\lY;t“fp  Y[_[`‘%`‘fq\J\ L ` `n\dYa_[f8s8]lt L‚Y[HJL\JfhYuk#Y[`‘fq\=•‚`‘’qL*\ %Ls8L l\JL Xr]4tuH Y[Hlk#Y {E {p{ t`c*Xfp\lk XrYa_u`nkh_a`n`c\d`ctuXHJY[Xt*mnL kp`nY\—m‘nXrLEjlsˆ]Jkq\4Yat HJt“LqYaL W`nfpkpYu\\lkh\JsXr qjlY[L*HJkq\dt L Y,L;XrX*t fp lfq¢ Ya\JtEHJLPkhLmnY[‘L;fY[s:kh\lt* pkpL*jl\dkpYPt*`®t*YZfpk#V\JY t*L;kpXP\ˆkh_[”xLLs8`n\dL*• YaYaJLE_[f8fq_aKM_%s8]lY[`n\JHJt L;LELs:s¡t `nfq`n\\ \W’pXaƒJL ]l£ˆ{tuHXa•PL kGYuX XaHJjlL kp`®t*XrY[jlLpH¡•kq]Jt ,L\Jf `nmkhL*YaL*’q_uL*kh_;m  jxfqL,Xakh`‘Ya_[`nL’pL pfqKM`‘\JLE ‚kqXrY[]Jf_[Ls8L l\lfhLPlkh\lmn8`®Y[\JL,LEt*L*LE\JX[LEXa_akp q_aV VfpƒJ”8{ gZ3L;mnt“`‘’WY[X`n\J `®fpY[HJ\MfpY[]8HJY&L_[t*L fpLE`n\l_a_[t `n`c\Js8 ‚L*\4Yaft L,YaHlXaL LPY t*kpjlkpt*`®YZVfp3XaL YuX*• Jfq_Bkh\WV L `‘\dY[_af8s8]lt*L‚YaHJLt*fp\JL. H 1 (Ω). u ∈ K ⊂ V = H (Ω). @. ρ=0. 01. @. u0 ∈ K. @. . K.. @. CK (u0 ) = {v ∈ V ; ∃t > 0. . u0 + tv ∈ K},. . TK (u0 ) = CK (u0 ) .. TK (u0 ). K⊂V. @. .. @. µ. .. u0 ∈ K. 

(59). @. D E. 1 I H @ (Ω). @. 5D E. 9@.

(60).

(61). Ξ = {x ∈ Γs |u0 (x) = 0}. Z. [Ymn`Y‚HJ’dL`n`n\JX‚ MfpXamnHJfpf f \`‘\J\—   `‘\ t fq\lƒls8{ `‘YaY[`nfpHl\:k#Y‚`cX`‘\€XakhYaYaHJ`cXL lPL;s`n_[`nfqtuHJ_mnkpL m‘Y‚Xrjlkqt Lq  khHJ\l`ctus H¦khkpmnxjJjxjJfqmn`‘XaL;`‘XYa`nYa’pfˆLPYaKMHJL;LMkpTWXa]Jfp_a”xL;fpX mnL*’—fhXa jllkp\lt*L`®Y[LLE\JL*_[ pV   {¨ Ls8LE\JfhY[L kp\ls¡s8L l\lL YaHJL jxfqXa`®Y[`‘’qLK L;kpXa]J_[L fh l\J`‘YaLL*\lL*_[ pV 3. @ @. D E. O(µ) = µ⊥ = {v ∈ HΓ10 (Ω)|hµ, vi = µ[v] =. .. Ξ. ,. u0 = u(Ω). @. u0 ∈ K. µ. hµ, vi =. Πɓɓ“ ]. µ. TK (u0 ) ∩ O(µ) = CK (u0 ) ∩ O(µ) .. .. Z. Ω. v dµ = 0}. ∇u0 · ∇vdx .. 4.. .. HΓ10 (Ω) .

(62) ;{  GHW]lXE JkhjljJm‘VW`n\J YaHJL_[LEXa]Jm‘Y[Xfp ƒJ{ L Hlk¨’pL RD. 

(63) 2[  XF [. E @. hktET8khY`‘\l\€t*L]4XrYaLMHJLkh\t fq`‘\¡kh\W”l’pY[XrL HJYa£L_ukpXaXaL t jlYYPkp_at*L;L Xr]lm®Y p`n’pLE\€`n`cXX(”W qjxV€`‘fp’qmnL*GVd\¡HlHJLEL*”Wfqs8V:_a_[LE`nY[tpKHJ #LYa:HlKML jJL _[Y[f#_akp`c’ptsJLEkps¡jJK_[`nfh\khgZ_uLEst ƒqYas8`n fp` x\¡ =LEG_afpLE\dHJ\dY[LEYaf„fp`ckh_[Y[L*mWHJK fhLxt*lK fp•n{\JL*4YaL _[ 3`ntTWL;jJt“_[Y[fh`‘fqgZL;\ t“• Y[l`‘fq•· \ L fq`‘_BYaH Y[HJLXrLE\lXa`®Y[kh`‘’W\4`‘s YZVMkp\lkhmnV8Xr`cX%fhXafpmn]8Ya•`nfpG\4X&HJLY[f„K’#kpkhjJ_[`cjJk#`nYa\J`nfp \4khmx`‘`‘\ \JL;bd{E]l} khPmn`®`cY[X`‘L;t*X*fp•@\JG`ct*HJkpLEm‘nfpV_[L*sJK'` 3L*_[`cL*X\dYa]l`cXakhLE”lsm‘Lq fq_`‘Ya} H Y[HJLs8` xLE_aLE\dYa`ckhm q`‘’qL*\”dV:YaHJL]J\l`nbd]JLXafpmn]8Ya`nfp\ Yaf Y[HJL’#kh_[`nkhYa`nfp\lkpmx`n\JL;bq]4khmn`®YZV SK (u0 ) = TK (u0 ) ∩ O(µ) = {v ∈ HΓ10 (Ω)|v ≥ 0 on Ξ(u) ,. u0 ∈ K. @. I. K ⊂ H ( Ω) H 1 (Ω). 2t1 = ρ. 2. at (·, ·) = a(ρ; ·, ·) Π0 h. a(u0 , v) = 0}. 21. Π. DD E. , S S SK (u0 )YS E. . 01. . &@. kpkq\lXaXas ]JKMj8`‘YaY[H `‘fq\lXfh&GHJLEfp_[L*K Mkp_aL fp’p_LE_akh` mnlL;skp\ls”WV • { J]J_aYaHJ`‘YaLEH _aKM{Efqz _a*Lq 8  fq%_LYaHlHJk¨L’q’¨L kp_a`ck#Y[`‘fq\lkh• m3`n,\JLELEbd\l]lt L—kpm‘` khYZmnV }  8lYafpHJ_LYaL*HJ£8LMjlt khfq\l\dXa’q`nfpL*\ \J`nL*\4%t Lfqfpm‘n&f ,YaHlX*L„•@_[GLEHJkq`cs8XL*t _ fq%KML„jJ_am‘L*L;Yat*L;khXmnYaHJHJLL*_[jJL_[Y[fWHJfhL Z• kp”lXrYa_ukpt“Y‚_aL;Xr]lm®Y HJ`ntuH`nX‚k pL*\lL*_ukhm‘¢ `nŸEkhYa `nfpL \Y fhYaHJL`nKM”4jJLm‘`ckt `‘t*YafpL\W’p]JL*\l£ˆt“Y[kh`‘\lfqs¡\t YaHJmnfqLEXafpLE_[s¡L*K Xa]J”lfqXa_L Y’¨kpfh_a@`ck#k Y[`‘,fq`n\lm ”4khm3LE_r`n\JYLEXabdjl]lkqkpt m‘L ` Ya`nLEXE l• kp\lsm‘L*Y s8LE\JfhY[L YaHJL sJ]lkhmn`®YZLVˆXrjlHlkhkp`nm‘_a=`n\Jt* Mfp\l”xXaL `cs8Y L*_%L*L*Ya\HJL‚fqm‘khnf\ls `‘\l   åkpHlK L*`n_[m LV:fhF’#s8kpLE_a\J`ck#fhY[Y[`‘LEfqX\lYakhHlmxL`‘\ls8LE]lbdkh]lm–khfpmn`‘ Ya`nLEX• s8LEj4LE\ls8`n\J Mfp\ k„jlkp_[kpK„¢ L*YaLE_     {;ƒ ”x`nL\ kM{EXrz fqm‘]JYa`nHlfp`n\tuHYaf `nXYa{EHJƒ L • tEkp XaL*LPY,]lfqX_ \J} fp*Ya•Lp JYaHlkhYfp_ kp\ls Q€L fp_[fpL*”8f#Yu’pkhLE`n_E\  dmnL Y  ™W ‹ ™  Π0 h ∈ SK (u0 ) :. @. .

(64). hA0 w, vi = b(w, v). U .

(65). K ⊂V. ,. a(Π0 h, v − Π0 h) ≥ hh, v − Π0 hi ∀v ∈ SK (u0 ) ,. @. 2. w, v ∈ V WL @. @. V. 0. . @. ft = 0. =@. V. @. ?@. QI. C@. TI. V. V V. 0. h·, ·i. t ∈ [0, δ) δ > 0. yt ∈ K :. yt = Pt (ft ) y = Π0 (−A0 y0 ). at (yt , ϕ − yt ) ≥ hft , ϕ − yt i. @. . @. 0. . ∀ϕ ∈ K .. ft = 0. . yt = Pt (0). .  6  67(N  .  Y'  "&  (    T8. .  F Y"& "< "66 H67"  () :  . . . at (·, ·) : V × V → IR   '    "&   8B

(66)  "&    2 [ t ∈ [0, δ) At ∈ L(V ; V 0 ) 6 H.    C6F

(67)  N ?R      >$  0 at (φ, ϕ) = hAt φ, ϕi ∀φ, ϕ ∈ V  A ∈ L(V ; V 0 ).  . At = A0 + tA0 + o(t). t > 0, t. ". . K ⊂V. S. (NX"&6

(68) R    "V Y60 :   ft = f0 + tf 0 + o(t).  ? . {. L(V ; V 0 ) .. ". {;y. . V 0,. ft , f0 , f 0 ∈ V 0. ! "> " 5 2"&;   ?  67< "A8N ?! < " X "& 6 <: Πf = P0 (f ) ∈ K :. a0 (Πf, ϕ − Πf ) ≥ hf, ϕ − Πf i. {;}. ∀ϕ ∈ K. ͐é =Í. .

(69) {q{. 

(70)  

(71)    ! "#$ %& $'  ()  J 2 "  ; "< %8

(72) '  <:   67  2 ∀h ∈ V 0 :. Π(f0 + εh) = Πf0 + εΠ0 h + o(ε). ". {. U . V.    (N7"&6

(73)    ?   ?(T$& "V 0 9 "< "66 R" J? <   : 0 ε > 0, ε. ?(N $"&6 )"& P67"  (    < Π#: V → 0VR" ( 

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(75) . {;ƒ. t. t=0.  . t > 0, t. yt = y0 + ty 0 + o(t). . ™ Œ8‹ . ". {. FL . V,. y 0 = Π0 (f 0 − A0 y0 ) .. . }qy. {Ez.  X6 9$' X ?4 ? J[ $ C   < "  = "  

(76) :J67" < "  "&! ?     A )'  " J ( ;

(77) " ':5 67 2"9>    "  . . a(ρ; v, v) =  .  ? "& :5$ " :. . Z. Ω. ev (O) = |∇v(O)|2.  

(78)    . k∇vk2 dx − πρ2 ev (O).  68/R ?  0

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(80) . .  . u = u(Ω) = (u1 , u2 ). Ω. σ = (σ)ij i, j = 1, 2. . − σ=f Cσ − (u) = 0. u=0. uν ≥ 0,. I. σν ≤ 0,. σν uν = 0. στ = 0. . Ω, Ω,. . . Γ0 ,. Γc ..  2 2 σν = σij νj νi , στ = σν − σν = στi i=1 , σν = {σij νj }i=1 , 1 ij ( ) = (ui,j + uj,i ), i, j = 1, 2, ( ) = (ij )2i,j=1 , 2 {Cσ}ij = cijk` σk` , cijk` = cjik` = ck`ij , cijk` ∈ L∞ (Ω).. Πɓɓ“ ]. [@. Ω. @.

(81). Γc. @. .

(82) ¨{  GHlL PfWf pL  XY[L*\lXafp_ XakhYa`cX 4LEXYaHlLL*mn‘`nj8Ya`ct `‘YZV t*fp\lsJ`®Y[`‘fq\  kp\ls %HJL LEHl\ k¨Ya’qHJL‚L]4YaXrfqL;j4s:fqm‘Y[fqHJ pLVXrfp]l K K`ckhX@Yatu`nHlfp\¡kp\Jt pfpL;\Ws–’q  L*%\dLYa`nfpHl\k¨’qf#LB’pY[LEHJ_%L_[L*fpjxmnLE‘f khYaLE`ns\J `n\lt s8fp`c\dt YuLEkpXEt“• Y(jJ_[fp”JmnL*K `‘\„YaHJLs8fqKkh`n\ `®Y[H ŠEY[ŒlHJhLŠ XaKM‹ kp m‘‡–Hl™ fp mnL • `n\ls kh\ls    JXr]4tuHˆY[Hlk#Y ž `‘`n\\ ph}y fqfq\\  fp\ hz ”l`‘mnLB`‘\lkqLEXakhXa_]JKMfqL_aK fp_FkpXaX[`‘XrKMf8t jJ`cmnk#`ct Y[LE`‘YZs VPY[Hl`®Y[k#H Y=Y[Y[HJHJLBLtE”xkpfpXa]JL\lfpsl8kh`nXa_[fhV:Y[_a’#fqkhjJmn]J`ntL LEjJmnkq_aXZfqY[”J`nt*mnL*`®YZK V`nXht  fpï\4¢ Xrh`cs8ƒ L*_[`cLEXs= p  `nYa’pHWLE]l\ X–”WYaHJV LBXaVWKMK L*Ya_[`nt ƒp HlL*G_[LHJLMY[HJjJL_[fp\J”lfpm‘Y[LEk#K Y[`‘fq\ˆqy fqï¢ _p`nXaz fhY[_a`nfqXjJkp`cjJtPjJLE_[mnf¨kq£8XZY[`‘K`nt*`®k#YZY[V:LEs—`cX ”WJV—£8LEY[sHJL`n\€jJTW_[fpLE”Jt YamnL*`nfpK \¡ƒJ•`‘YaH KMf8s8` lLEs”J`nm‘`n\JLEkp_ fp_[K `n\ Y[HJL † fpmn f ‹ `‘\J  Œ8k¨Š Vp•  

(83)   ŠEŒJpŠ ‹  ‡  ™  t*]lfp\lL€_[_at“L;Y[s8t“`‘L*fqY[Ya\l`‘L*fqkh_[\¡KMm6 –Ya`‘YaL*\lHl_[L K¡L Y[4 =HJ_[s8L—XrL YY[KML*Y[_[L*f8KM_[s8K ` `‘4\JLEL;s8s s€L 4”J`n\J\ `nLEm‘`nX‚TW\JL;LEY[t“kpHJY[_LM`‘fqL*\fpmckp_[ƒJKXr•Yaˆ`ctkp”WXLEV¡\Jk L*fq_[Xa_a p]JK„V—K ]J`n\€mnkfpPY[HJY{;LJ{fs8*fp•Y[KL*G_[khKHJ`n\ L X* t*kpfp 3XM_[Ya_aHJL;`‘Y:t“LY[`n`‘XaX fqLE\¡t fqfqYa_M\lL*s—_[YaK HJY[L—L*`c_[X‚LEK \JbdL*]J`c_[X‚`‘ pYaVkL t*sJfpL KMl\JjJL;mns` tEk#”WY[V LEsYaHJYaLf¦fpL*_[’#Kkhmn]J]lmck#khY[LPLp`‘ (\¡khTW\lLEs t Ya`nfpLˆ\ˆs8ƒ8f¦•&\JGfhHlYLPjl’#_akhf#mn’W]J`cLEs8X%Lfh`®–YuXY[HJL*L £WXrjlVWm‘`cKMt `‘KMYL fqYa_[_a`cK¡t, (”J`nXrm‘]l`n\JtuL;H kh_&k—fqfp_a_[K K `cXkqt“Ya]4khkpmn_a‘VL q`‘’qL*\ ”dVˆYaHJLL*£8jJ_aL;XaXa`nfp\ ƒl{ `‘YaH_aL;XrjxLEt YPY[f k#Y `cX‚ p`n’pL*\ ”WGVHlL„Y[HJs8LLE_aL `n£8’#jJk#_[YaLE`n’pX[XaL `‘fq\ fh@Y[HJL„”l`‘mn`‘\lLEkh_fp_[K ƒq #Hlƒ L* _[  L khmn%“  YaHJL#} bq*]4  kh\d{;YaJ`‘{Ya*`nLE  X zdkh_[Lˆ“  LE’¨kpLˆm‘]4jJk#_[Yaf#L;’Ws `nsJLfp_„YaHJYaL HlL m‘`n\Js8L `cXrjl`‘\dmnYakqLEt pL*_uKMkhLEmcX \qY lHJL*`ctumcHs s8L lkp\JtEt L;fpX„_us8kpm‘`nB\J —YaL*Y[_[f€KXfp`n_[\ K]lƒqmnkp L ”xPL*L*mn\lf t*Lp•   Lt*kp\s8L*YaLE_aKM`n\JL‚YaHJL”J`nm `‘\JL;kh_fq_aK fp_,kpm‘  8_[fpK"YaHJLL;bd]lkhmn`®YZV. 

(84) 2[  XF [. AI. /@ , @. Ω ρ

(85) 

(86).

(87) G. .. C. @. Ω B(ρ)   

(88)  Ωρ u = u(Ωρ ) = (u1 , u2 ). @. σ = (σ)ij i, j = 1, 2 . − σ=f Cσ − (u) = 0. uν ≥ 0,. ,. σν ≤ 0,. σν uν = 0. στ = 0 . @. @. . . . . @. . . Z. Ω. . . 0.. ).. . .. @. .. . .. @.  . Ωρ. S. @. Ω. . a(ρ; v, v) = a(u, u) + ρ2 b(v, v) .. @. a(ρ; v, v). b(v, v) = −2πev (0) −. U  U S @ I ?@. U . . . @. . 2.. a(ρ; ·, ·). b(v, v). . Γc ..  2 (λ + µ)(11 + 22 )2 + µ(11 − 22 )2 + µγ12 ,. .

(89) . . $U  L . Γ0 , Γρ ,. 5@. a(u, u) =. . Ωρ , Ωρ ,. u=0 σν = 0. ,. S. cijk` ξji ξk` ≥ c0 |ξ|2 , ∀ξji = ξij , c0 > 0,. @. ρ2. ρ = 0+. .  2πµ σII δ1 − σ12 δ2 , λ + 3µ. a(ρ; v, w). ). @. . v. .. . v, w. 2a(ρ; v, w) = a(ρ; v + w, v + w) − a(ρ; w, w) .. ͐é =Í.

(90) {;ƒ. 

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(104). b(v, v). .. . ?@. N@. 2πev (0) =. ,. π(λ + µ) π 2 R6. Z. (v1 x1 + v2 x2 ) ds ΓR. 2. +.  2 12k (1 − 9k)(v1 x1 − v2 x2 ) + 2 (v1 x31 − v2 x32 ) ds + R ΓR Z   2 12k µ 3 3 (1 + 9k)(v1 x2 + v2 x1 ) − 2 (v1 x2 + v2 x1 ) ds + 2 6 , π R R ΓR. µ + 2 6 π R. Πɓɓ“ ]. Z. . ƒ.

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(114) S. 1  r n−1 cos n(φ − ψ)] dψ R R. n. n=1.  r n R. π. dy. cos n(φ − ψ)] dψ. S . ∞ 1 ∂ X  r n =− sin n(φ − ψ)] dψ [ u(R, ψ) πr −π ∂ψ n=1 R Z π ∞  n X ∂u r 1 (R, ψ)[ sin n(φ − ψ)] dψ. = πr −π ∂ψ R n=1. Z. S

(115) S . fqm‘Hlnf `‘mnL:`n\J`n\q Y[L*Y[_a q`c_[t khYa`n\Jfp —_ ”WV¦jlkp_rYuX  LHlk¨’pL]lXaLEs HlL*_[LMYaHJLˆt*fp\dYa`n\W]J`‘YZVfp f#’pL*_ • ,jJjJmnVW`‘\l —YaHJL ,. @. G. |t| < 1. . ∞ X. @. u. ∞ 1 X [(teiα )n − (te−iα )n ] 2i n=1   1 te−iα teiα = − 2i 1 − teiα 1 − te−iα t sin α = = H(t, α) 1 − 2t cos α + t2. Lkh_[_a`n’pL k#Y LXaHlkhmn–\JL*L;s¡khmcXrf„YaHJL’#kpm‘]JL fp @. ∂v 1 (r, φ) = ∂r πr. HlL*_[L @. Z. ∇v(0). v(r, φ) =. 1 2π. π. Πɓɓ“ ]. ∂u r (R, ψ)H( , φ − ψ) dψ. ∂ψ R. •  L Y,]4X_aL _a`‘YaL ”WV:KML;kh\lXfpe(fp`cXaXafp\ pLE_a\lL*m6  −π. Z. W@. π. u(R, ψ)K( −π. K(t, α) =. . tn sin nα =. n=1. ,. ΓR. S

(116) S. r , φ − ψ) dψ, R. 1 − t2 . 1 − 2t cos α + t2. d}. S . HG. S7U  S$L .

(117) ;{ } NPf. 

(118) 2[  XF [ @. qz.   1 ∂v ∂v ∂v (0) = lim (r, φ) cos φ − (r, φ) sin φ , r→0+ ∂r ∂x1 r ∂φ   ∂v 1 ∂v ∂v (0) = lim (r, φ) sin φ + (r, φ) cos φ , r→0+ ∂r ∂x2 r ∂φ. kp\ls. S . ∂K r 2 ( , φ − ψ) = cos(φ − ψ), ∂r R R 2 ∂K r ( , φ − ψ) = − sin(φ − ψ). lim r→0+ ∂r R R lim. r→0+. PL*\lt*Lp lkhÏYaLE_,s8` 3L*_[L*\dY[`nkhYa`n\J kh\ls¡Xa]J”lXrYa`‘Ya]8Y[`‘fq\¡`‘\dYa_[f qz “  L fp”8Yukh`n\ˆY[HJL l\lkpm=_[LEXa]Jm®Y I. 01. . S7U . ∂u (0) = ∂x1 ∂u (0) = ∂x2. L Y,]lX\lf _aL*Ya]J_[\Yaf @. S . • Lt fq\lXa`ns8LE_ ,. ER (u) Z Er (u) =. 1 π. Z. @\@. q`‘’qL*\ fp_. kp\ls. Er (u) r<R Z π ∂u ∂v u ds = u(r, φ) (r, φ)r dφ. ∂r ∂r Γr −π. π. Z. π. NPf L fp”lXaL*_[’pLYaHlkhY Kk¨V:”xLL £8jJ_[LEX[XaLEsˆkqX Er (u) =. A.. yh. Z Z π 1 1 ux1 ds, u(R, ψ) cos ψ dψ = πR −π πR3 ΓR Z Z π 1 1 ux2 ds. u(R, ψ) sin ψ dψ = πR −π πR3 ΓR. ,ÏYaLE_PXr]l”lXZY[`®Y[]8Ya`n\J q}   . S  ?@. −π. u(r, φ). −π. Γr = ∂B(r). ”WV. yJ{. HlL*_[L GHlL*_[L fp_[L yp %Ya_ukh\lXrfp_[KXBY[f @. . ypƒ. 1 ∂ H0 (t, α), 2 ∂α. . H0 (t, α) = log(1 − 2t cos α + t2 ).. . π. y. π. ∂u ∂ r (R, φ)u(r, φ) H0 ( , φ − ψ) dφdψ, ∂ψ ∂φ R −π −π Z π Z π 1 ∂u r ∂u =− (R, φ) (r, φ)H0 ( , φ − ψ) dφdψ. 2π −π −π ∂ψ ∂φ R. 1 Er (u) = 2π. . yq. r ∂u (R, φ)H( , φ − ψ) dφdψ. ∂ψ R. H. H(t, α) =. . Z. Z. S. ͐é =Í.

(119) { j4GkpHlX[LXYa]lf \lYat“HJY[`‘Lfq\mn` K `‘Y `‘YaH HlkqXBfq_ fp”8Yukh`n\J`n\Jkh\ `‘\dY[L* p_ukh”lm‘L Xr`n\J q]Jmnkp_a`‘YZVMfhYaHJL‚YZVWj4L  lXaf L Kk¨V ypy fpNPXaHl f kpm‘–fq%]l\ XaL L‚YakpHJ\dL qY%kq`‘’qY[XrfL*VW\=KMY[ k j8p`®YY[Lfh`‘Ya\d`c`nt‚Yam BfML tu£8kpHljltEkht kp\lfq\l p]JXrL`n\dfpYBYa\¡HJYaHls8LLLE’#_a`nkp`n\ ’pm‘l]JL;s:]JLLE`nfp\l\  t L‚ƒqfhy –6•&Y[HJNL•khKMXrïK\ L*mnkpVpfqm‘ Jx_[s8`‘HJ LEfp_‚mnL Yaf—`nXL*Ya’#HJkpL• m‘]lPXakhfpX[YamnXaL]8]JYaYaKM`nHlfp\`‘`n\JX„ „fhtu HlYaHlkh\JL q’#L khmn]JLL `‘\ fq\ fp\ yh} Y[HJL*\F lkpX[Xr]lK `n\J   L Hlk¨’pL y fq_ • PL*_[L y kp\lsˆYaHJLE_aL*fp_[L yhz T8]J”lXrYa`‘Ya]8Y[`‘\J MY[HJ`cX,t fq_a_[LEt Ya`nfp\ˆ`n\dYaf yJ{  p`n’pL;X 

(120)  

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(123) G. ux1 ds. ΓR. E(ρ ; uρ ) =. H@. uρ. E(ρ ; uρ ) =. Πɓɓ“ ]. ).. 1 2. Z. .  ∂v ∂v (0) cos φ + (0) sin φ . ∂x1 ∂x2.  ∂v ∂v (0) cos φ + (0) sin φ r dφ ∂x1 ∂x2 −π   Z π Z π 2 ρ ∂v ∂v =− (0) (0) u(r, φ) cos φ dφ + u(r, φ) sin φ dφ . r ∂x1 ∂x2 −π −π. δEr (u) = −. . Γρ ,. vρ = v + sρ + o(ρ2 ), ∂vρ ∂v ∂sρ = + + o(ρ2 ), ∂r ∂r ∂r. I. . ∂u ∂u (R, ψ) (R, φ)H0 (1, φ − ψ) dφdψ. ∂ψ ∂φ ∂u/∂n D E. ∆vρ = 0. r > 21 R. U. 2. +. Z. ux2 ds. ΓR. uρ. 1 2. Z. Ωρ. u=v 2 #. .. 7@. @ . .. S . k∇uρ k2 dx.. Ωρ = . ΩR. 1 1 k∇uρ k2 dx + ER (uρ ) + δER (uρ ) + o(ρ2 ). 2 2 ΩR. .

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(125) 2[  XF [. @. @. 2G. @. S . wR. ΩR. Z 1 E(ρ ; wR ) = k∇wR k2 dx 2 ΩR Z π Z π ∂wR ∂wR 1 (R, ψ) (R, φ)H0 (1, φ − ψ) dφdψ − 4π −π −π ∂ψ ∂φ "Z 2 Z 2 # ρ2 wR x1 ds + − wR x2 ds . 2πR6 ΓR ΓR. }pƒ . [YGHJHlLL G`nl\ HJ_ulL‚XZ]JY=_[LEYL*\ljJt _[fLLEY[XafpL*L*F_[\dKMYaY[HlkhXFLYa p`nfpXr`nK’p\ L@khL*}pmn–£8ƒ kqHJ%t“fqkhYFm‘mnLqL*f • £8,jJ,X%_amnL;]l=XaX%fhXa`nYafpf„Ya\HJYaLE_ufpKkh_=\lY[XrkhHJ_[fpLBL_[`nK bd\d]lY[kqYakpHJt“s8YL_uk#s8Y[Y[fqkq`nKX tPMkhfp`n_[\_[fpKK X ¨Xafhkpfp\lmns‚’d`nYa\JHl • L&YaYaHJHlL `‘_u”xsfpkh]JjJ\ljlsl_akhf¨_[£8V `nKM’#kpkhm‘Ya]JL;LX jl_afq –”Jm‘”JLE]JK Y fp_`‘YaHkhL*jl\JmnLEkq_at qLBV L;bdjl]lkpk#_[Y[kp`‘KMfq\L Ya`n_[\`nŸ*’#LEkhs¡_[`n”WkpV”JmnLB•s8fpïK\¦kpåkp`‘\ t YE –Xr`nY[\lfPt*LY[HJY[LHJXaL kplK _uLXrYjJY[_[L*fp_a”JKmnL*XK _aLE`n\jJ_[Y[LEHJXaL L*\dJYP£8LEL*s\lL*s8_[fq pKV khfp`n\ _ Y[sJHJ`nX[L t _[L Y[Hl`‘fpŸ;k#mnL€Ya`nfpfh\ fh   L€ K• k¨GV HlLE`nXM’pL*_[\ LEXak¨]J’qm‘Y[fpX `cs _[]lfpKXr`n\Jfp ”4XrLE_a’#k#khY[`‘\lfqs \ Y[Y[HJHlL¦k#Y_[L*fpKM]JY[L;XaXr`nHJs8`nL \J fp Y[HJ`n\ Lfq]J_[\ls8LEt Ya_`nfpYa\f pL Y qt*fp`‘’q`n\lL*\t `cs8”dL;V X `‘YaH p`n’pL*\”WV YaHJL—Xafpmn]8Ya`nfp\fhPY[HJL¡”4fq]J\lsJkp_aV’#kpm‘]JLjJ_[fp”JmnL*K `‘YaH ”J]Jm LE\JL*_[ pV } L*Gkq£Wt*Hlt*YaLL*fp_[]J\l\dkpLEY[km X,—%Yaffqfp_ `‘„_[\dK Yakh_[\lf8fpsMs8%]l]lY[t*XaHJ`‘`n`n\J\JX  jJ”4Xr_afqfqfq]J”J]J_umn\lL*t sJKLkpY[_aL*`cVMX _[K$t LEfqkq\lt*Xr`ns8fpm‘\l`‘V¦Yat*`nfpfpL*”8\d\lYaYuX&_ukhfpk#`n\J\Y[LELEs¡s fq”W• \—V—YaY[XHJk L W`‘JL,\J£8 ¡XrLELE’#s¦Lpkh dt _[K `c`n_[k#f8t*Yas8m‘`nLfp` 4\tEfhk#Ya`nkhfp\l\Ms€fpjl3kh”l_u]Jkhm KM (kpL L*Y[sJ\l_as8L*`nŸ*_[`n\J pL;sV ”WV G• HJLmckpXrY„fq_aK„]Jmnk€KMk¨V ”xL L*£8jJ_aL;XrL;s khmcXaf¦`n\kh\lfhYaHlL*_„fp_[K•  L YM]lX„sJL*\JfpYaL”dV YaHJL LE\JL*_[ pV:sJL*\lXa`®YZVk#YYaHlLj4fq`‘\dY   YaHlL ]J\lt“Y[`‘fq\ `nXHlkp_aKMfq\J`nt‚`n\  JYaHJLE\ Y[HJLL £8jJ_[LEX[Xa`‘fq\lX%fp_,s8LE_a`n’#k#Ya`n’pL;X .. @. 7-. ΩR. Y@. Ω. Ωρ. C3. .. .. ΩR. . Ω B(R). @. w. 1 E0 (ρ ; w) = 2.

(126) G ?@. wR. G. ρ. @. ΩR. )@. @. @. . C@. Ω. . Z. ρ2 k∇wk2 dx − 2πR6 Ω. G. "Z. wx1 ds. ΓR. ∂Ω. .

(127) G. 2. +. Z. @ ).. wx2 ds. ΓR. wR. G. 2 #. S. .. E0 (ρ ; w). 2.. 2G. ΓR. ρ. eu (x). x. eu (x0 ) = k∇u(x0 )k2 .. 3. u. 1 u/1 (x0 ) = πR3. B(R). Z. u · (x1 − x1,0 ) ds,. 1 u/2 (x0 ) = πR3. Z. u · (x2 − x2,0 ) ds.. kp_aL‚L £Jkqt“YE• ï\’W`‘L fpFY[HJ`nXE 8fq_aK„]Jmnk } tEkh\”4L_[L _[`®YaYaLE\ `n\YaHJLL;bq]l`‘’#khmnL*\dYfp_[K 3. W@. ΓR (x0 ). S . 1 E0 (ρ ; w) = 2. @. Z. ΓR (x0 ). }dy. . 1 k∇wk2 dx − πρ2 ew (0). 2 Ω. ͐é =Í.

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