Construction et analyse mathématique d'un modèle approché pour la propagation d'ondes acoustiques dans un tuyau mince parcouru par un fluide en écoulement.

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(1)Construction et analyse mathématique d’un modèle approché pour la propagation d’ondes acoustiques dans un tuyau mince parcouru par un fluide en écoulement. Anne-Sophie Bonnet-Ben Dhia, Marc Duruflé, Patrick Joly. To cite this version: Anne-Sophie Bonnet-Ben Dhia, Marc Duruflé, Patrick Joly. Construction et analyse mathématique d’un modèle approché pour la propagation d’ondes acoustiques dans un tuyau mince parcouru par un fluide en écoulement.. [Rapport de recherche] RR-6363, INRIA. 2007. �inria-00189320v2�. HAL Id: inria-00189320 https://hal.inria.fr/inria-00189320v2 Submitted on 21 Nov 2007. 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. Construction et analyse mathématique d’un modèle approché pour la propagation d’ondes acoustiques dans un tuyau mince parcouru par un fluide en écoulement. Anne-Sophie Bonnet-Ben Dhia — Marc Duruflé — Patrick Joly. N° ???? Novembre 2007. ISSN 0249-6399. apport de recherche. ISRN INRIA/RR--????--FR+ENG. Thème NUM.

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(67) ( 42+# .),

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(70) .

(71)

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(104)

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(111) L. M %./ $ % .

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(116) } './ $ %G# '#+$' C

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(129) È#-{~] mo]} = 2. u=. Ì ~l[{@®}~]¢d}®QÏo-}~o¼6o-}6¯|QQ|}~o-} Eu. h. ¹ 1

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(132) !:#"./+ + ./, # %+ A Im

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(134)

(135)

(136) !" # %$& $'# $( )*

(137) '+-, ./+0. 66. È%Q{*[ mo-}]¯

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(142) (LA./+0A + '. *.!./R $&

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(144) ,+$ ,# %$. λ+. Z. +. λ− < M −. 2 > 2. 1. dy. −1. λ− − M (y).

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(157) = mes(N ) > 0 u oÂ mtm~Qmo2 mo λ o[{6}moÂdEo-mAm(u, 0)~o2|o =Aλ o-}Ï(u, QmoÀ0)oÂ{~, {*¼Lo[{*YommoÀ{{*|2}®o]Q} L½ o[{*Qo¹ºmo|®n%o]{*|¿mo[»@|o[{@°±Y}~Y{,·n%dyQo-mmozm®o/·{~mmI~}#mE{ N ´ wzmY{*Q{9 mo o}% mo M o[{6} C n%Q-o]EmÃÈŃw|®} (u, v) -Qo-}~mo]8Qo]}~o- mmo|o Ì ¯mQmes(N }d6Qm)={ 0  λ ∈ σp (A) ∩ Im M. . ⇐⇒. mes(Nλ ) > 0.. λ. λ. t. t. λ. 1. λ. µ o-À|]mm¶}t mo/I. t.  (λ − M )2 u = Eu . ¯ y∈ /N λ. Á8o-²n%m morm]-o]{{~®o-n%o-}8 mo ´ mes(N ) = 0. O}~o/|o]E~}~-¢Qmo´

(158) È#{~m¶}o. v = (λ − M )u u=. Eu = 0 Eu = 0. λ. Eu (λ − M )2. -8{~®mQ¯ M -}Q} C En%o]|Ã¾¯ u morI~E®}Q{ o]}~Emo (λ − M ) u = 0 -o mo-}®mo u = 0 mm²{~ mo 1. 2. . Ï Å ©m¨ Å C

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(163) 8 A m]{~®¢Qmo L½ Qb6},mo Ì ´ µ ,QQ®},λ ∈mo¥σ¹º®(A){* %}t|oo-nE mo-, mλo ∈E/ σo]{*}t(A)E|}QQbλ6∈σ}@m(A{ )L ([−1, »~» 1] . r. p. . p. ∗. ∗. 2. . A∗ = . M. E.  . Ì m}~o-n%o-}t|®}]¯{~¾mQ{,Q}~||{~{L½ QY]E}~o]m, Q~[E{~oi½ ]km¢o/|o u o-} v ¯m{~®} mo[{6}#Em}~QQb6},o}#mm®}®o. . (S2. S=. I. M. 0 I I. 0.  . ¯mQ]®²{~o mo= = I). A∗ = S A S.. â±Þ!Üâãá.

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(168) 6Q¯ 6[QF±»´ 5o9~[{*m®}E}t{*®dE}tn%m moQ¯I-n%|}~o}o-zN|2}km-Q~l]n%o/´ o}r|N®o]nOn%oY/´ 66%¯I moQ¯o-Âmo-kmQ{#mo L½ EÃzo-o]i¯|o{*Io]}~o/|o A ½ o]{*},-{*}~®}~ Qomo/b®o]m{@m~Qmo]{@{*n%mo]{]´ zÊ ¨ Å 0. ! . % ' σ (A) ⊂ Im M. ÍÎ¾{* %},|otn%QQ}~o]8 mo{* λ ∈/ Im M o-} ½ o[{6}{8mmodEo-mmmot|o A ¯|E{ A − λI o]{*}#Qo-{*mo´ wzQ¶} (f, g) ∈ L ([−1, 1]) ¯|{,kmo]k{@·}~mQo- (u, v) ∈ L ([−1, 1]) }o-²{# mo ∗. . c. . . t. 2. 2. 2. (A − λI) (u, v)t = (f, g)t ,. E½ o]{*}t·%|~o o/ m n%l-mo/·Oi½ ] E}~2o- Á8n%n%o. 2. t.   (M − λ) u + v = f . u. Eu + (M − λ) v = g. Eu − (M − λ)2 u = h,. ¯|o]{*},®²®}~omo|®z²{*o],E λ∈ / Im M. h = g − (M − λ)f.. (M − λ)2. ¯mo m mmmo=. h Eu −u= . (M − λ)2 (M − λ)2. Ì m®² mY{,i½ I-d}o-m E ·%o-}*}~o/]¢Q®®}~Q¯|®¾zo-} Eu. . E. . 1 −1 2 (M − λ). . =E. . h 2 (M − λ). µ #Qn%nOo λ ½ o[{6}#Q{@mmo/dEo-,m~Qmo|o A Q22¹º®o-n%n%o 11 » . 1 6= 1. 2 (M − λ) −1 h 1 E −1 Eu = E (M − λ)2 (M − λ)2 −1 h h 1 1 E −1 − . u= E (M − λ)2 (M − λ)2 (M − λ)2 (M − λ)2 E. µ o-À|]mm¶}t mo. m{t Qo. p#Q{,bQ{@|Y{@nOQ}~ mo²%~[{*Q®dQ}omoL½ I-d}~o]m A }E®}#|m-o/ bo[. (A − λI)−1 (f, g)t = (u, v)t.  −1 E[g/(M − λ)2 ] − E[f /(M − λ)] g − (M − λ)f 1   − ,   u = E (M − λ)2 − 1 (M − λ)2 (M − λ)2 −1   E[g/(M − λ)2 ] − E[f /(M − λ)] 1 g   v= − E − 1 . (M − λ) (M − λ)2 (M − λ). zÊ ¨ . Å. M (yλ ) = λ. !. ./ %$. Ü ÜVUWmæGX*äYZ+[çå\]. yλ ∈ ] − 1, 1 [. . . M. + !

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