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A partitioned fluid-structure algorithm for elastic thin valves with contact

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(1)A partitioned fluid-structure algorithm for elastic thin valves with contact Nuno Diniz dos Santos, Jean-Frédéric Gerbeau, Jean-François Bourgat. To cite this version: Nuno Diniz dos Santos, Jean-Frédéric Gerbeau, Jean-François Bourgat. A partitioned fluid-structure algorithm for elastic thin valves with contact. [Research Report] RR-6022, INRIA. 2006, pp.23. �inria00111991v2�. HAL Id: inria-00111991 https://hal.inria.fr/inria-00111991v2 Submitted on 21 Nov 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. A partitioned fluid-structure algorithm for elastic thin valves with contact Nuno Diniz dos Santos — Jean-Frédéric Gerbeau — Jean-François Bourgat. N° 6022 Novembre 2006. ISSN 0249-6399. apport de recherche. ISRN INRIA/RR--6022--FR+ENG. Thème BIO.

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(48) ¶ SN¸ u −u p n · v. ·v+ ρu · ∇u ·v+ 2η D(u ) : D(v) = − ρ δt ³\{erN]ƒn|Y_Œ‹mqsl| {|_ n|$q _Cˆ'n|q¥rNŽ¦“Mqs_^ºŠAruqsŽMnL¿kq¥n/qxl]0y\ Y]`ru{|_ª rNŽk“u_^Ž\q¥_LŽNnner"ºcru{|­Uºq°n|Y˜loŠWˆu _ X ¨XZYMyWl^¿ £±rNt¥tsr'ºqsŽv FIH3H!J9¿ºc_Œq¥ŽMn|{erm‹my\^_®neY_¯ ˆ~vu{ ˆ~Ž\vu_ª]0yt¥neqsŠt¥qs_^{†loŠ\ˆN _Cl  . ρ. un+1 − un + ρun · ∇un+1 + ∇pn+1 − δt. n+1. (2ηD(un+1 )) = 0, n+1. n+1 Σn+1. n+1. 1. 2. 0. c. Σn+1 Σn+1. Σn+1 1. Σn+1. 2. 1/2. n+1. Σn+1. n+1. 2. n+1. n+1. n. n. n+1. n+1. Ω. Ω. in. Ω. M L. n+1. n+1 Σn+1. Γin. = L2 (Ω), = (H −1/2 (Σn+1 ))2 ,. ^ ru{|{e_CloŠAruŽW‹mq¥Ž\vnern|Y_Unº/r¹^ruŽ\lone{ ˆ~qsŽMn|l®ºYqx Y¢‹m_ Ê Ž_`n|Y_loŠWˆu _ V (u ) ¨ Ÿ _`neY_LŽ  ruŽWloqx‹m_^{ n|Y_£±rNt¥tsr'ºqsŽv¦“'ˆ~{|qsˆ~neqsruŽ\ˆut)£±ru{|]"ytxˆ'n|q¥rNŽ¬ Ê Ž\‹ (u , p , λ ) ∈ X × M × L loyW Y¹neY\ˆ~nL¿ £±rN{‰ˆ~tst (v, q, µ) ∈ X × M × L ¿ n+1. n+1. n+1 Σn+1. n+1. n+1. n+1. Z. ρ Ω. Z un+1 − un ·v+ ρ un · ∇ un+1 · v δt Ω Z v + hλn+1 , T rΣn+1 (v)i − pn+1. ‹mqs“. Ω. Z. q. ‹mq¥“. +. Z. 2η D(un+1 ) : D(v) Ω. = −. Z. pin n · v,. Γin. u = 0,. Ω. hµ, T rΣn+1 (un+1 )i. = hµ, un+1 Σn+1 i,. ¶ HLN¸ ¶ H HC¸ ¶ HC‘u¸ ê±ë ìê å.

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