Source: ICHE Conference Archive - https://mdi-de.baw.de/icheArchiv

Kawahara, M.

Nojima, K.

English

Hydraulic Engineering Repository

Conference Paper, Published VersionNojima, K.; Kawahara, M.Drag Force Minimizing Shape Identification of BodyLocated in Compressible Fluid FlowZur Verfügung gestellt in Kooperation mit/Provided in Cooperation with:Kuratorium für Forschung im Küsteningenieurwesen (KFKI)Verfügbar unter/Available at: https://hdl.handle.net/20.500.11970/109835Vorgeschlagene Zitierweise/Suggested citation:Nojima, K.; Kawahara, M. (2010): Drag Force Minimizing Shape Identification of BodyLocated in Compressible Fluid Flow. In: Sundar, V.; Srinivasan, K.; Murali, K.; Sudheer, K.P.(Hg.): ICHE 2010. Proceedings of the 9th International Conference on Hydro-Science &Engineering, August 2-5, 2010, Chennai, India. Chennai: Indian Institute of TechnologyMadras.Standardnutzungsbedingungen/Terms of Use:Die Dokumente in HENRY stehen unter der Creative Commons Lizenz CC BY 4.0, sofern keine abweichendenNutzungsbedingungen getroffen wurden. Damit ist sowohl die kommerzielle Nutzung als auch das Teilen, dieWeiterbearbeitung und Speicherung erlaubt. Das Verwenden und das Bearbeiten stehen unter der Bedingung derNamensnennung. Im Einzelfall kann eine restriktivere Lizenz gelten; dann gelten abweichend von den obigenNutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte.Documents in HENRY are made available under the Creative Commons License CC BY 4.0, if no other license isapplicable. Under CC BY 4.0 commercial use and sharing, remixing, transforming, and building upon the materialof the work is permitted. In some cases a different, more restrictive license may apply; if applicable the terms ofthe restrictive license will be binding. DRAG FORCE MI IMIZI G SHAPE IDE TIFICATIO  OF BODY LOCATED I  COMPRESSIBLE FLUID FLOW  :  This paper presents a numerical method of shape identification of a body located in compressible viscous flow. The purpose of this research is to identify the optimal shape that minimizes the fluid forces subjected to the body. The formulation of the shape identification is based on the optimal control theory. The finite element method is used for calculation of fluid flow. In this paper, optimal control is treated as fluid force minimization. The first thing that should be carried out in the optimal control theory is to define a performance function which expresses the optimal shape. The performance function must be minimized satisfying the state equation. In the research, the Lagrange multipliers are introduced for the constraint conditions. To avoid the break down of calculation caused by destruction of elements, the finite element mesh is reconstructed in the identification process. The grid generation scheme based on Delaunay triangulation is applied to the reconstruction of finite element mesh. In this paper, an optimized shape of the body is obtained by computation.  : finite element method, shape identification, optimal control theory I TRODUCTIO  1044SHAPE IDE TIFICATIO  Compressible Viscous Flow ixU̇ + AjU,j − (KjiU,i),j = 0 in Ω,jA jiKAj =∂Faj∂U,KjiU,i = Fdj ,U =ρρu1ρu2ρu3ρe≡ρm1m2m3ρe, Faj =ρujujρu1 + δ1jpujρu2 + δ2jpujρu3 + δ3jpuj(ρe + p), Fdj =1Re0τ1jτ2jτ3jujτij − qj,ρ iu e p jqijτ ijδδij ={1 (i = j)0 (i 6= j).tee = ε +uiui2,ε1045p = (γ − 1)ρε,γij i j j i k k iju u uτ λ δ∗, , ,= + + ,λ∗ = − ,vcε θ= ,j jqPrγε ,= − ,vc Mγ γ−∞= − ,vc PrM∞ ReCCθθθ θ∞∞+= ,+C = ,θ Kθ∞ jAjiKAj =0 δj1 δj2 δj3 012δj1γ̄u2 − uju1 δj1u1 − δj1γ̄u1 + uj δj2u1 − δj1γ̄u2 δj3u1 − δj1γ̄u3 δj1γ̄12δj2γ̄u2 − uju2 δj1u2 − δj2γ̄u1 δj2u2 − δj2γ̄u2 + uj δj3u2 − δj2γ̄u3 δj2γ̄12δj3γ̄u2 − uju3 δj1u3 − δj3γ̄u1 δj2u3 − δj3γ̄u2 δj3u3 − δj3γ̄u3 + uj δj3γ̄(γ̄u2 − γe)uj δj1ε̄ − γ̄uju1 δj2ε̄ − γ̄uju2 δj3ε̄ − γ̄uju3 γ̄uj,γ̄ = γ − 1, u2 = uiui, ε̄ = γe − γ̄uiui2.1046Kji =0−{µ(m1ρ2δji +miρ2δ1j) + λ∗ mjρ2δ1i}−{µ(m2ρ2δji +miρ2δ2j) + λ∗ mjρ2δ2i}−{µ(m3ρ2δji +miρ2δ3j) + λ∗ mjρ2δ3i}−{µmkmkρ3δji +mjmiρ3(µ + λ∗) + λ∗mimjρ3+ γµPr( eρ− mkmkρ3δji)}0 0µρ(δji + δ1jδ1i) +λ∗ρδ1jδ1iµρδ1jδ2i +λ∗ρδ2jδ1iµρδ2jδ1i +λ∗ρδ1jδ2iµρ(δji + δ2jδ2i) +λ∗ρδ2jδ2iµρδ3jδ1i +λ∗ρδ1jδ3iµρδ3jδ2i +λ∗ρδ2jδ3iµm1ρ2δji + µmjρ2δ1i + λ∗ mkρ2δ1jδki −γµPrm1ρ2δji µm2ρ2δji + µmjρ2δ2i + λ∗mkρ2δ2jδki −γµPrm2ρ2δji0 0µρδ1jδ3i +λ∗ρδ3jδ1i 0µρδ2jδ3i +λ∗ρδ3jδ2i 0µρ(δji + δ3jδ3i) +λ∗ρδ3jδ3i 0µm3ρ2δji + µmjρ2δ3i + λ∗ mkρ2δ3jδki −γµPrm3ρ2δjiγµPr1ρδji.BΓU = Û(xi, 0), in Ω,OΓ BΓ(u2, τ2j , qn) = (0, 0, 0) on ΓS1 × I,(u3, τ3j , qn) = (0, 0, 0) on ΓS2 × I,(p0, τ1j , qn) = (p̂0, 0, 0) on ΓO × I,(ui, qn) = (0, 0) on ΓB × I,I pIΓut = ût, on ΓI × I,R∞ = un −2cγ − 1, on ΓI × I,c2 = γ(γ − 1)(e −12uiui), on ΓI × I,S = ln(pρ−γ), on ΓI × I,nu tu c R∞ S1047ΓS2ΓS2ΓS1ΓS1ΓIΓBΓOFig.  1   Computational domain.  Performance Function JJ =12∫I(Fi − F̄i)Qij(Fj − F̄j)dt,Fi = −∫ΓBtidΓ,ti = (τij − pδij)nj ,it iF iFijQ IJ ∗J∗ =12∫I(Fi − F̄i)Qij(Fj − F̄j)dt+∫I∫ΩΛT · {U̇ + AjU,j − (KjiU,i),j}dΩdt,1048Λ =ξη1η2η3ζ.ξ iη ζAdjoint Equations J ∗ JJ ∗Jδ ∗ = .J ∗δJ∗ =∫I∫ΩδU · {−Λ̇ − (ĀjΛ),j + Āj,jΛ− (K̄1jiΛ,i),j + K̄1ji,jΛ,i − K̄2ji,jΛ,i+∫I∫Γξ̄δρdΓdt +∫I∫ΓS1η̄1δm1dΓdt +∫I∫ΓS1η̄3δm3dΓdt+∫I∫ΓS2η̄1δm1dΓdt +∫I∫ΓS2η̄2δm2dΓdt +∫I∫ΓO+ΓBη̄iδmidΓdt+∫I∫ΓS1+ΓS2+ΓOζ̄δ(ρe)dΓdt +∫I∫ΓBζ,jδqnjdΓdt−∫I∫ΓIηiδτijnjdΓdt −∫I∫ΓS1η2δτ2jnjdΓdt −∫I∫ΓS2η3δτ3jnjdΓdt−∫I∫ΓIζδqjnjdΓdt −∫I∫ΓIζuiδτijnjdΓdt+∫I∫ΓB{−ηi − (Fj − F̄j)Qij}δtidΓdt+∫ΩΛ(xi, tf ) · δU(xi, tf )TdΩ = 0,Āj =0 12δj1γ̄u2 − uju112δj2γ̄u2 − uju212δj3γ̄u2 − uju3 (γ̄u2 − γe)ujδj1 δj1u1 − δj1γ̄u1 + uj δj1u2 − δj2γ̄u1 δj1u3 − δj3γ̄u1 δj1ε̄ − γ̄uju1δj2 δj2u1 − δj1γ̄u2 δj2u2 − δj2γ̄u2 + uj δj2u3 − δj3γ̄u2 δj2ε̄ − γ̄uju2δj3 δj3u1 − δj1γ̄u3 δj3u2 − δj2γ̄u3 δj3u3 − δj3γ̄u3 + uj δj3ε̄ − γ̄uju30 δj1γ̄ δj2γ̄ δj3γ̄ γ̄uj,1049K̄1ji =0 −{µ(m1ρ2δji +miρ2δj1) + λ∗mjρ2δ1i} −{µ(m2ρ2δji +miρ2δj2) + λ∗ mjρ2δ2i}0 µρ(δji + δj1δ1i) +λ∗ρδj1δ1iµρδj2δ1i +λ∗ρδj1δ2i0 µρδj1δ2i +λ∗ρδj2δ1iµρ(δji + δj2δ2i) +λ∗ρδj2δ2i0 µρδj1δ3i +λ∗ρδj3δ1iµρδj2δ3i +λ∗ρδj3δ2i0 0 0−{µ(m3ρ2δji +miρ2δj3) + λ∗mjρ2δ3i}µρδj3δ1i +λ∗ρδj1δ3iµρδj3δ2i +λ∗ρδj2δ3iµρ(δji + δj3δ3i) +λ∗ρδj3δ3i0−{µmkmkρ3δji +mjmiρ3(µ + λ∗) + λ∗mimjρ3+ γµPr( eρ− mkmkρ3)δji}µm1ρ2δji + µmjρ2δ1i + λ∗ mkρ2δj1δki −γµPrm1ρ2δjiµm2ρ2δji + µmjρ2δ2i + λ∗ mkρ2δj2δki −γµPrm2ρ2δjiµm3ρ2δji + µmjρ2δ3i + λ∗ mkρ2δj3δki −γµPrm3ρ2δjiγµPr1ρδji,K̄211,1 =0 0 0 0 (µ − λ∗)mlρ2{(m1ρ),l − (mkρ),kδl1}0 0 0 0 −(µ − λ∗)1ρ{(m1ρ),1 − (mkρ),k}0 0 0 0 −(µ − λ∗)1ρ(m1ρ),20 0 0 0 −(µ − λ∗)1ρ(m1ρ),30 0 0 0 0,K̄222,2 =0 0 0 0 (µ − λ∗)mlρ2{(m2ρ),l − (mkρ),kδl2}0 0 0 0 −(µ − λ∗)1ρ(m2ρ),10 0 0 0 −(µ − λ∗)1ρ{(m2ρ),2 − (mkρ),k}0 0 0 0 −(µ − λ∗)1ρ(m2ρ),30 0 0 0 0,K̄233,3 =0 0 0 0 (µ − λ∗)mlρ2{(m3ρ),l − (mkρ),kδl3}0 0 0 0 −(µ − λ∗)1ρ(m3ρ),10 0 0 0 −(µ − λ∗)1ρ(m3ρ),20 0 0 0 −(µ − λ∗)1ρ{(m3ρ),3 − (mkρ),k}0 0 0 0 0,1050K̄212,1 = K̄213,1 = K̄221,2 = K̄223,2 = K̄231,3 = K̄232,3 = 0,ξ̄ = {−uiuj +12γ̄ukukδij)ηi + (γ̄ukuk − γe)ujζ}nj− {µ(ηi,j + ηj,i) + λ∗ηk,kδij + µ(miρζ,j +mjρζ,i) + λ∗mkρζ,kδij}miρ2nj ,η̄i = {ξδij + (ukδij − γ̄uiδkj)ηk + ηiuj + (ε̄δij − γ̄uiuj)ζ}nj+ {µ(ηi,j + ηj,i) + λ∗ηk,kδij + µ(miρζ,j +mjρζ,i) + λ∗mkρζ,kδij}1ρnj ,ζ̄ = (γ̄δijηi + γζuj)nj .−Λ̇ − (ĀjΛ),j + Āj,jΛ − (K̄1jiΛ,i),j + K̄1ji,jΛ,i − K̄2ji,jΛ,i = 0, in Ω × I,(ηi, ζ) = (0, 0) on ΓI × I,(ξ̄, η̄i, ζ̄) = (0, 0, 0), on ΓO × I,(ξ̄, η̄1, η2, η̄3, ζ̄) = (0, 0, 0, 0, 0), on ΓS1 × I,(ξ̄, η̄1, η̄2, η3, ζ̄) = (0, 0, 0, 0, 0), on ΓS2 × I,(ηi, ζn) = (−(Fj − F̄j)Qij , 0), on ΓB × I,Λ(xi, tf ) = 0, in Ω,nζ jζ , jnJ ∗ ixδδmi = δ(ρui) = uiδρ + ρδui = uiδρ + ρui,lδxl.J ∗grad(J∗)l = {ρξδij + (ρe + p)ζδij + µ(ηi,j + ηj,i) + λ∗ηk,kδij}ui,lnj .Minimization Method K1051Bml l l l l li i ij j j e kIeK J x x W x x d dt a { A x A }∗ + + ∗ +Γ== − − − Γ + − ,∑∫ ∫l lij ijW W δ= ,l lWa∗Kδ =ml l l l lij j ij j i e keiW x W x G a A xx+ ∗ +=∂= + + .∂ ∑lixUMERICAL EXAMPLE and CO CLUSIO  iF iFQ Q Q . .. Θ = . t = . ε = .W = . E −=Fig.  2   Finite element mesh of a car. 1052Fig.  3  Pressure distribution at front nose. REFERE CES J. Fluid Mech 59J. Fluid Mech 64J. Fluid Mech 72Technical ReportVol.1Int. J. Comp. Fluid Dyn 17 19, Int. J. umer. Meth. Fluid., Comp. Meth. Appl. Mech. Eng.,Aerodynamic shape optimization using the adjoint method1053

Drag Force Minimizing Shape Identification of Body Located in Compressible Fluid Flow

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Drag Force Minimizing Shape Identification of Body Located in Compressible Fluid Flow

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