Design of Minimal Dispersion Fluidic Channels in a CAD-Free Framework
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(4) tlccc. ISSN 0249-6399. ISRN INRIA/RR--3982--FR+ENG. 1 . apport de recherche. 7.
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(65) pq_=be{ ivzp|£k=rC{pqh srlhvÒtF x kC{ p "Rvpnpubên"r x sb wa#bpuqho{nm* k©n"r[puhNpqhNbv Pmonusb|hov @ ¯[°/b pupub hv4pu_=bn"tFt= }[hNanpuh sr ¯ hjphov x b²©r=b x n"ppq_=b ° sk=r x nuyi¯ pq_=b{k=u¨snpuk=qb |pu_Fb±° sk=r x nuyhovvan"mjm P_Fbuber c{kFu¨sn"pqk=ubn"pên"mjm ¦ ^4_=b w n x hjbr[p " hNpq_ubvt/bµ{pp ¥hov4shN¨ibrc°[y . x. mx. x. x. [x. x. mx. tx. *x. x. rx. x. x. 2 5 N u ; u ; @ [. ; . 2N5 & @ [ L & ; ;A. 4^ _=b vzbµ{ wr x pubqa °Cbp4bb r tCn"qbr[pu_=bvhov hv pu_=b vzp n"pqb¸mNhjr=bnuhNµn"puh wr{ wr[puuhj°=k=pqh wr_=h{_hv r=bsmNb{ pub x hjrchNr©{ wactFmNbpqbvzbrCvhNpqhN¨[hNpuhjbv¦Ub{n"r vzb bepu_Cnppu_=bvhNwr pq_=bew n x hjbr[phovn"mjnyFv { suubµ{pn"r x pq_=bn"t=t= }hjan"pqh wr#hov4°Cbpupqb4 wmonusb @ ¦ ÃêvX4bvqn"h x ¯w{ ivzp£k=rC{ puh sr ud Xhv2vk=hNp n"°=mjb w0pu_=b¥n"tFt=mNh{n"pqh wr ChNrC{ wact=mjbpubRvzb rCvzhjpuhN¨[hNpqhNbv ¦ Ube{nrchNr©{ubnsvzb x hNqb{p sb wa#bpuqho{nm={ srpquhj°=k=puh wrCv|°ypq_=bnw{p|pu_©n"p4pq_=b¨ibm [{hjpy hov tCnnmNmNb m p ªpu_Fb± n"mjmov¦R^4_=b{ ivzp4£kFrC{puh wr4b{ srCvzh x b4 s x buhj¨sn"puh wrhov4pq_=bqb wq b S S < u . ! ab & acb A _=bqb hv4pu_=b m [{n"mk=rFhNppnr=wb rp¨ib{ p wp pu_=b n"mNm¦ ^ ¸b ¨wnmNkCnpubcpq_=b#nw{{kFnw{ y "pu_Fbvzbªw n x hjbrp v¯04b{ sactCnubpu_Fblqbvzk=mjpv w°=p n"hNrFb x hNpq_ pu_Fhovcnt=t= }[hNanpuh wr pu_=b wn x hNb rp vlhNpq_ pu_ vzbU{ sachNr=£ sa ²CrFhNpub x h !\bqbrC{bµv¦ ^4_Fhov hNr©{ wactFmNbpqb¸vbrCvhNpuhj¨hjpy b¨sn"mjkCn"pqh wrAt ivqvzhN°FhNmNhjpyGv_ vcpu_=bhjact wup n"rC{ b " ub x b²Cr=hjr={ vzp £k=rC{ puh srCv|nwv|° sk=r x nuyªhNr[pubsnmov _=brªt ivuvhN°=mjb ®nsv4v_ rcn° ¨ib Òn"r x p °=uhjr=nwv¥{m vzbênwv t ivqvzhN°FmNb|pu_=b¥{ vzpX£k=r©{puh wrn"r x { srpq sm x b²Cr=hjpuh wrem [{n"pqh wr©v¦ ^4_FhovXhovXhjr tCn"qpuho{ k=monXhNa#t sup n"r[p w wÄ { sr²CwkFn"pqh srCv n"r x hNpRn"mov etCbua#hNpvXp { srCvzh x bRn_ wmjb{hj{kFhNptCnpupub urªn"r x r wp sr=mNy n±tFhNb{b 0v_Cn"tCbi¦¥Á®rKns{p¯ wtFpuhNa#hNnpuh sr°Cb{ wa#bv2t ivqvzhN°FmNb w4pq_=bê{ wr²©wk= n"puh wrCv w¥_=ho{_ vzhjalkFmon"pqh sr±hv2n ! w x n"°=mjbensv2pu_=b{ ivzp "vzhjalk=mn"pqh wrnr xx bvhNwr°Cb{ wacbvXb k=hj¨sn"mNb rp ¦2Á®r x bb x ¯ vzb rCvzhjpuhN¨[hNpynrCn"mjy=vhovehovr (b ik=hN¨snmNbr[pp pu_FbmNhNrFbn"qhNnpuh sr pu_Fb wmNm hNr=nt=t= "}[hNa¬n"pub vzhjalkFmon"pqh srlm wt r !` !, q Y !` _=h{_acbnrCvDpq_Cn"p4b sr=mNy nw{ { wk=r[p4 wpu_Fb±a x hT²\{ n"puh wrhNrpq_=b wb sacbpuqho{nmXtCnup x b²CrFb x ¨ib4pq_=b hNr=rFb{_©n"r=r=b mXnmNm¦ x. x. x. ux. mx. x. x. mx. x. x. x. x. x. mx x. x. x. x. x. +. mx. ux. kx. x. x. x. x. mx. mx. x. x. x. z. x. mx. ux. cx. mx. $x. x. x. x. x. tx. x. x. mx. mx. tx. x. x. x. x. x. x x. x. x. mx. x. mx. x. x. x. x. x. mx. x. x. x. x. x. tx x. x. . x. x. x. . x. x. x. x. . x. mx. ex. x. x. x. x. x. x. x. mx. x. x. x. ux. xcx. +. x. +. +. x. +. x. +. "x. +. x. x. x. . . g(
(66) ^
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(70) EKJ. ¾= sa _Cnpvunh x wrUhNrC{ wact=mjbpubvbrCvzhjpuhj¨hjpuhNbµv¯=hNp hve{mNbµn"epq_Cn"p hNpêhovea wqbvk=hNp n"°=mjbp _Cn¨ib n"r nw{{kFn"pqbcvpnpubcb¨snmNkCnpuh sr nr x nrn"t=tF }hjan"pqbw n x hjbrp pq_Cn"r p puuyp { sact=k=pqbªn"r nw{ {k=npub w n x hjbr[p°Cnwvb x sr¸n sr=#vzpnpub w°=p n"hNrFb x wrnª{ n"vb acbvz_¦ x. x. x. x. x. mx. x. x. x. ux. x. x. x. ux. x.
(71) ¿ srCvzh x b|ne°=hjmNhjr=bn"0{ ivzp £k=r©{puh wr hjr¨ smN¨[hNr=êvzp n"pqb nr x wb sacbpquho{ n"m ! { wr[puqhN°=k=pqh srªn"r x x b²©r=b x ¨ib4pq_=bvun"a#beubsh sr¸pu_©n"rpu_=b{ srpq sm ¤¦ u ! u A ! u { in" vzbO ;A ! u { invb ! A u A ! & u ²Cr=b & [ ^4_=bmNb£pR_Cn"r x vzh x bhvÒpq_=b x h !\bqbrC{b¥°/b pz¥bbrªb}Fnw{p|nr x hjrC{ wa#t=mNbpqb|wn x hNb rp¥{ sact=k=pqb x wr n ²Cr=b acbv_¦ ^4_Fhovbu w0hov "£pub r±tFubvbrpRn"r x hov x k=b|p epu_=b ns{p2pq_Cn"p0pu_Fb{ vzp "Chjpub n"puhj¨ib achjr=hNach npuh wr n"r x sn x hNbr[p b¨sn"mNk©n"puh wrCvemjhNachjp4pu_FblkCvbep ¸{ in" vzb acbµvz_=bvpu_Cnr«_Cnp wkFm x _Cn¨ib °Cbbr kCvb x wên t=k=ub vzhjalk=mn"pqh wr¦ §r=bªt ivqvzhN°FhNmNhjpy¸p n¨ wh x pq_=hov x h { k=mNpyhovep kCvb x h !\bub rpemjb¨ibm "qb²Cr=ba#brp4 s pq_=b vzp n"pqbn"r x pq_=blsn x hNbr[p¦ ^4_=hv±hov pu_=b#h x bnc°Cb_=hjr x aªk=mNpqhT¡mNb¨ib mRvz_Cnt/b st=puhjachNnpuh wr_=bqb pu_Fbsn x hNbr[p hov sr=mNy { wactFk=pub x wrcpq_=b{ n"vbmjb¨ibm "¤nÀaªk=mjpuhT¡wqh x { wrCvpuqkC{puh wrªnr x _=bqb pu_Fb vpnpub{ wa#bvÒ£ wa pq_=be²Cr=bmjb¨ibm 4bkF} 5Äbu¨[hNb k=} u
(72) u A ! ²Cr=bêmNb¨ibm ! u { n" vzb mNb¨ib m A !, ²Cr=b [ u ²©r=b mNb¨ibm ! ²CrFb ;A ^4_=b²©vzp¥pubqa "0pq_=bemNb £pD_©n"r x vzh x bhovDpq_=behjrpqbut wmn"pqh wr "0pq_=besn x hNbr[p{ sact=k=pqb x wrpq_=b { n"vb suh x ¨ib¥pu_=be²Cr=b mjb¨ibm®¦ Sx. x. x. x. cx. mx. x. . . x. . +. . . . mx. (+. x. . . +. x. . . +. +. x. x. . +. x. ux. x. x. ux. x. x. ux. x. x. x. x. x. [x. mx. x. x. cx. x. x. x. ex. x. x. x. x. x. x. x. ux. {:{. . .
(73) x. x. . . . . +. x. +. . x. x. x. mx 1x. +. x. x. x. ¸!# ±-F ¸#0K! $4 K0& 0
(74) ¿ wr©vzh x b pu_Fb4 wmNm hNr=epqhNacb x b t/b r x br[p2b kCnpuh wr w|pq_=bvz_CntCbtCnnacbpub uhNµn"puh wr ¦ b ubi¯ pu_Fb±pqhNacbhovD²/{puhjpuh skCvn"r x hovvhNachjmonDp pu_=b x bµvu{br[ptCnn"a#bpub ¦ [ & ^
(75) N 5 ;" A ¨ibhv£puk=_Fr©b{pun h x wacr ho"vqvzhNpu°F_=mNbb b v}=tCnsnw{{p b sn"|r hNx rC{ pus_=acb t=vzmja bpub sspu_=n hjr=x hNbr[wptC¯sbhNp np p n"s³i b v2^ hjr p nw ¦{ { ^4w_=k=hr[vp2v y=sv pupub_=a.btFub t=uqbb{ vzb puhrsp r v a vzpeachjr=hNa#hTµn"puh wr n"mj suhNpq_=av ¦ÁP H* ¯4bªub{ ¨ibpu_Fbªvzpub btCbvzp x bvq{br[p n"tFt= nw{_¦cÁP * ¯ pq_=hov hovepu_=b#_=bn¨y°©n"mNm¥acbpq_ x ÖÃpup wkC{ _ ¿ sachNrFbpupqh q
(76) ê^4_=b¬n"hNa hNrUpu_Fhov n"tFt= nw{_hv±p nw{ {bvuv x h !\bqbr[p±a#hNr=hjan epq_=bct= w°=mjba n"r x r wp wrFmNypu_Fbcr=bnubvp m [{ n"m achjr=hNak=a ¦2¿ sruk=inpub w n x hjbr[pn"r x ikCnsvzhT¡ bp wrca#bpu_ x v4{nr¸nmov °Cb{nwvp4hNr#pu_=hvÒ wqa Þ w_Cnacaca#n x h (~ hN wr=r=bµn"k wsw
(77) ¦ ^ n x ¨sn"rC{ b±hjrlpqhNacb ¯4bkCvb n{br[pu n"m x "h !\bubr©{bevq{_Fbacb x br wpqbvD°[y pu_=b v_Cn"tCb x b suanpuh wrnpevpubt ] < [ L 5 < & " A u. ^4_Fbªvz_Cnt/bt©n"nacbpqbuhjn"pqh wr °Cbhjr= x b²Cr=b x ¯Xpu_=b x y[rCn"ach{n"mn"mj wqhNpq_=a ¥b kCvb hovênwv wmjm v i £¼ ç·[ £¼ Sx. x. x. x. mx. +. x x. x. x. ux. xcx. x. 1x. x. Sx. x. w. ux. mx. . {}{. ux. rx x. ux. x. mx. x. x. ~. . +. x. x. +. mx. x. x. x. ux. Ux. Sx. x. x. x. ux. x. +. x. :x. x. x. x. +. x. x. x. ux. +. +. . :x. . mx. . . . . . . .
(78) s ¦4{ sact=k=pqbpq_=b w n x hjbrp s ¯ hT w n vzp wt¦ ¦ x b²CrFb±pq_=b r=b n x a#hovuvhN°=mjbev_Cn"tCb x b wqan"pqh srckCvzhjr= u [¦4va wpq_pq_=b x b suanpuh srCv4kCvhNr= ¯ dF¦ x b wua.pu_=b acbv_¦ 5 ¦¥{ wa#t=k=pubêpu_=b r=b vzp n"pq b u
(79) 5 ¦ ¦|{ sact=k=pqbepu_=b r=b { vzp ¦ [¦ ] ] [ L nr x wp ¸s¦ ½ ¼ À¼ 1 i P¼ /£¼¼ A x. x. 3x. + . . ~. z. ux. . . xcx. . x. x. mx. . mx. |. x. x. . x. x. . }x ux. . . . . . . . 2 $4 0- . ®Á r n x=x hjpuh wrp ªpu_=b{_Cn" nw{pqbuhvzpqho{v|t=ubvbr[pub x n° "¨ib¯=4b kCvzbnªÄbmn"k=rCnyca#bvz_n x n"tFpn"pqh sr pubµ{_=rFhik=b°[ycm [{n"macb puuh{{ wr[pu smXh x bmNyªkCvb x hjrc¨sn"uh wkCvvzhjalkFmon"pqh srCv hNr[¨ wmN¨[hNrF pu_Fb v wmjk¡ puh wr " ~|Ä ÖvÖ¾=qby 5Âb wqwb q
(80) ¯ b{_p + w_Cnacan x h wsw
(81) ¦^4_=bhjactCnw{ p " pu_Fhov4{ wk=t=mjhNr=_©nwv4°Cbbrv_ r wr n pu_=bÀn x ¨ib{ puh sr "Rn t©nwvuvhN¨ib vq{n"mn"°[ylpq_=b²Cb m x o²Cwk=qbv. ¦|Á®p4hov¥{mNbµn"|pu_Cnpp _Cn¨ibepu_=bÀvun"a#b ikCn"mjhNpyhNpq_ wkFp4acbv_cn x n"tFpn"pqh srhjact=mjhNbv2pq_=bekCvb " nlqbwk=mn"²Cr=bacbvz_b¨ib uy[_=bub¯_=h{_hov wkFp "|qbnw{_ ssbr=b n"mRn"t=tFmNho{ n"puh wrCv ¦^4_=b qb¡ acbv_=hNrFhv2nmov hjact wup n"r[p0n"r x n"°©v wmjk=pub mNy r=b{bµvuvunuy nwv2pq_=bmonuwb x b wqan"pqh wr©v¤hjrpq x kC{ b x w|pq_=bvz_CntCbean"³ibµv¤pq_=bacbv_p x hovzp wqpub x p ±°Cb-b !\b{pqhN¨ibD s ²Cr=hNpqbbmNb acbr[pvzhjalkFmon"pqh srCv¦ Ub v_ pq_=bev³ibvDtF x kC{ b x °y wn"r x s x bsubbv|puk=qrCv4hNrªt=h{puk=qbv £d"¡ ¦DUbepu_Fbr n"tFt=mNhjb x sk= st=puhjachNµn"puh wrUnt=t= nw{_p ¸pq_=bvbc{ sr²Csk=npuh srCv¦ vzy[acacbpquy nwvqvzk=actFpuh sr _CnsvK°Cbbran x bi¦U^4_=bc²C vzpª{mnwvuv st=puhjachN b x v_Cn"tCbv wpq_=b wUn"r x w x b wub bcpuk=qrCv ®¾0hN=¦ µ¡ R{ suubµvzt wr x ik=hNpqb4p _Cnp2 wkFr x °[ynrhNr[puk=hjpuhj¨ib x bvzhjwr Þ wmj_ bpn"m¦ sww
(82) ¦ ^4_=hv±hov hjact wqpn"r[pnwv hjp tCbua#hNpv v wacb#{ wr[² x brC{ b wrpu_Fbcwm w°Cnm x bvzhjwrnt=t= inw{_¦U^4_=b vzbµ{ wr x {monsvuvbv " wt=pqhNachjb x v_Cn"tCbvl wªpu_Fb w nr x s x bsubb puk=qrC v ®¾0hN=¦ ¡ _Cn¨ib °Cbbr s°=pnhNr=b x nwv³hjr= wªpu_=bU{ ivqvY¡Yvzb{pqh sr r wpcp ub x kC{b p alk©{_nr x n"mv w#mNbvqv ub wk=mn"uhjpy± w¥pu_=bêvz_©n"tCbv¦ 4k=p ¯ pq_=hov mjbn x v p a sub°Cnr xcx hvztCbvh srnny £ sa5n"mjmov ¦ ^4_=b t=qb¨h wkCv wt=pqhNachjn"pqh sr4bqbtCb wua#b x kFr x b¥pu_=b { wrCvpunhNr[pD s4pu_Fb wk=pqb4 n"mjmXp uba¬n"hNr k=rC{_Cnr=wb x ¦^ n¨ wh x p caªkC{_cubvpuqho{pqh wrhjr¸pu_Fb{_Cn"r=rFbm0{ vuv®¡vb{puh wr¯nKpq_=hN x {mnwvuv " vz_©n"tCbª{n"r °Cb w°Fpn"hjr=b x nwvz³[hNrFl w° wpu_pq_=b hjr=r=bnr x wkFpubnmNmovp ªa ¨ib ¾2hNF¦e¡s ¦ 4b ¨ib¯wpq_=hov4pqk=urhvDany°Cb r wp st=puhjan"mCr=bhjpu_=b nwv4¥be sk=m x mjhN³ibn"mv ªp ªn¨ wh x w¥p {m ivzb w x bsubb#puk=qrCv p hjrpqbz£bqbi¯_=ho{_ wk=m x hNactFmNyn¸mn"qwbªvk=zns{bl wpq_=bc_ smNb {_Cn"r=rFbm®¦ x. x. x. x. x. x. . x. x. x. x. x. z. {}{:{. ex. mx. x. ~. x. ux. x. x. x. x. x. mx. x. ux. mx. Sx. mx. x. ux. xcx. x. x. x. |. x. x. [x. x. x. x. x. x. x. x. x. x. {. x. mx. mx. x. ~. ux. |. x. xcx. x. x. x. x. cx. mx. |. |. x. ux. x. x. x. 4{. x. x. x. {. x. x. x. x. Ew. x. x. x. x. mx. x. x. #x. mx. x. x x. x. x. |. x. x. x. x. ux. "{. x. x. [x. x. xcx. x. x. mx. x. x. x. . cx. x. x. x. x. mx x. x. x. mx. x. x x. x. cx. x. x. x. [x. x. (% $¥¸!# 06¬ Ub_Cn¨ibvz_ "r_ "p { waª°=hNr=bRhNrC{ sact=mjbpub|vbrCvhNpuhj¨hjpy n"rCnmNyFvzhv n"r x tCvbk x ¡®k=r©vzpubµn x y st¡ puhjachNµn"puh wr#n"t=tF ins{_ p x bµvzhNsrub x k©{b xx hovtCbvh wrbmjb{pq wtF_ wqbpuh{ ach{ [{_Cnr=r=bm x b¨[ho{ bv¦. rx. x. x. rx. ux. x. x. x. x. mx. x. x. x. x.
(83) ¾2hNsk=ub ¥ à x nt=puhj¨ib vzhjalkFmon"pqh srlp c{nt=pukFubnw{{ k=npubmjylpq_=bvz³ib ¦ z. mx. ux. ¾2hjwk=qb4d RÁ®r=hNpqhon"m[v_Cn"tCb w pq_=b s x bwqbbvpuk=qr 2b !Xbµ{p \pu_=b¥puk=qr wrpu_Fbn x ¨ib{ pub x vztCb{ hNbv ¦ x. {. x. x. ¾2hjwk=qb U ÁYrFhNpuhn"m vz_Cnt/b¸ slpq_=b w x bwqbbvpuk=qr ¸b-!\b{p pu_Fb¸pukFur sr pu_=b n x ¨ib{pqb x vztCb{ hNbv ¦ x. |. x. x.
(84) ¾2hjwk=qb 4¾0hNvp{monsvuv " wt=pqhNachjb x vz_CntCbv wpu_Fb s x bwqbb puk=qr¦4^4_Fb anwr=hjpuk x b "|pq_=b vz³ib _Cnsv °Cbb rqb x kC{ b x °[y wr=b s x b ¦ . x x. x. x. {. >x. x. ¾2hjwk=qb |¾2hN vzp¥{monsvuv " wtFpuhNa#hNb x v_Cn"tCbvD s4pq_=b w x b wub bpqk=ur¦|^4_Fbean"sr=hNpqk x b "0pq_=b vz³ib _Cnsv °Cbb rqb x kC{ b x °[yca wqbepu_©n"r sr=b w x b¦ . Sx x. x. x. x. x. |. yx.
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