Lire la seconde partie
90
pages
Français
Documents
Obtenez un accès à la bibliothèque pour le consulter en ligne En savoir plus
Découvre YouScribe et accède à tout notre catalogue !
Découvre YouScribe et accède à tout notre catalogue !
90
pages
Français
Documents
Obtenez un accès à la bibliothèque pour le consulter en ligne En savoir plus
Niveau: Supérieur, Doctorat, Bac+8
Lire la seconde partie de la thèse
Voir
Lire la seconde partie de la thèse
- modele
- methode de regularisation ponderee
- poids convenables dans la definition de l'erreur dans la fonctionnelle cout
- poids
- pod
- differentes techniques de calibration
- basee sur la sensibilite globale
Lire la seconde partie de la thèse
Chapter
4
Integration
Introduction
and
calibration
of
ROM
Cechapitreconcerneessentiellementlesdiff´erentestechniquesdecalibrationutiliserpourstabiliser lemode lesd'ordrer´eduit.Ilestsouventdifcilederepre´senteravecsufsamentdepre´cision,mˆeme sur un temps court la dynamique du syste me initial, ce qui interdit l'utilisation des mod e les d'ordre reduitspourdese´tudesdesensibilite´,d'optimisationetdecontroˆleoptimal.L'id´eeprincipaledela ´ calibrationestd'identierlescoefcientsdumode lePODGalerkin(ROM)desortequesadynamique propresconcideavecladynamiquetemporelledelaPODissuedessimulationsnum´eriquesdirectes ¨ etquiestconnuea l'avance.Lesraisonsdelamauvaisepre´cisiondumode leROMpeuteˆtreattribu´ee a latroncaturedesbasesPODou les´echellesdissipativesinclusesdanslesmodesPODd'ordres ´eleve´snesontpasprisesencompte.C'estunproble meanaloguea celuirencontr´eenSimulationde Grandes Echelles (LES) ou la dissipation des petites e´chelles est manquante. Meˆme si tous les modes de la projection de Galerkin sont pris en compte, on peut aboutir a un mauvais attracteur a cause desinstabilit´esstructuralescommecellesobserve´esparRempfer(2000),Noacket al.(2003). Les techniquesdecalibrationpeuventˆetreclasse´esendeuxcate´gories,lapremie reconsistea traiterle proble medecalibrationcommeunproble medefermeturedelaturbulence.Lasecondeconside reun proble med'identicationdecoefcients,cequirevienta uneproble med'optimisationoudecontroˆle optimal.Coupletet al.(2005israpaapnsdantseresetneriuqsruereln´eraledesdiff´e)odnnueenuvgee´ ROMetproposeunetechniquedecalibrationbas´eesurlaminimisationd'unefonctionnelleline´aire del'erreur.R´ecemmentuneam´eliorationdespropositionsdeCoupletet al.(2005´tpea)e´ne´t´rseee dansCordieret al.(2009uqinneearitnouspp´lmenetairedecettetech)orpsuoN.unnssopoioelm´ea introduisantdiff´erentstypesdematricedeponde´rationdanslad´enitiondel'erreur.Danslapremie re me´thodelespoidssontde´termin´esa partird'uneanalysedesensibilite´del'e´nergie.Danslaseconde approchelespoidssontd´eniesa partirducontenue´nerg´etiquedelarepre´sentationenROM.
89
4. Integration and calibration of ROM
D´enitiondeserreurs
L'actionnementdanslesyste mesROMestd´etermin´eparuneproce´dured'optimisationL2se retrouve principalement restreint de a un mode spatial. Par conse´quent il est sufsant de pre´senter les princi-paleside´essurlacalibrationassoci´eeLesdynamiquestemporellesauxcassansactionnement.Les erreurspouruneidenticationpolynˆomialedumode led'ordrer´eduit(ROM)peuventeˆtreessentielle-ment classees en: ´
1.moduntiecefcoeslu o,tate´'dnoitibraacalLcetnesvareurrepr´eti´espoostndine dleReMO pre´cisionl'´etattemporelduROMenaccordaveclescoefcienttemporelsdelaPODdusyste me original.
2.retnese´rperruopesdmphascle,o ulescoefcienstostndineit´seLalicaatbrdnoice´'eluotnem vecteurs de l'integration temporelle du ROM en accord avec les champs de vecteurs obtenus par la POD.
Troiserreurspeuventeˆtred´eniesparlese´quations4.2,4.4et4.6 fois l'erreur choisi,. Une ´ l'i dentication polynoˆmiale est obtenue par minimisation des fonctionnelles couˆt basees sur cette erreurcommedonn´eesparlese´quations4.3,4.5,4.7 la fonctionnelles couˆt tous les modes. Dans sontdonn´esdepoids´egaux.Laminimisationdesfonctionnelles4.5,4.7n'undioutcal´rselonoudtia syste meline´airepuisqueleserreurs(´equations4.4et4.6) sont des fonctions afnes des coefcients polynoˆmiaux.
M´ethodedecalibrationdeCouplet
La me´thode deCoupletet al.(2005) introduit la minimisation de fonctionnelles base´es sur la combi-naisonline aireconvexedestermesquirepr´esententl'erreurnormalis´eeetunemesuredelavariation descoefcientsdumode leparrapportauxvaleursobtenuesparlaPOD.Cependant,Ils'ave reque lesyste meline´aireestmalconditionne´etdivergeapre suntempscourtcommeonpeutlevoirsurla gure4.1lechoixdependantrtdeepuramae´arulelisonlaegr´ˆocnC.tunofaoitcerecosdurre´P.uo re´gularisationde´penddel'utilisateur.Pouryreme´dierCordieret al.(2009u)ilittnes´malhoetdede re´gularisationdeTikhonovpourmieuxconditionnerleproble me.
Approchepond´ereederegularisationdeTikhonov ´ ´
Danscetravailunenouvellem´ethodedecalibrationbas´eesurl'ame´liorationdelam´ethodede Cordieret al.(2009) est utilise´e en introduisant des poids convenables dans la de´nition de l'erreur dans la fonctionnelle couˆt, en donnant ainsi une importance aux modes qui le sont. Deux voies pour d´enirlespoidssontpropos´ees: 1.ruseesstructl'effetditasednodomaile´tdesslaneserntsaueleonsqint´plusuocssnn,´drenoisDa e´nergetiques et ainsi les spectres propres servent comme une mesure de l'importance des modes. ´ C'estlechoixleplusnaturelpourlad´enitiondespoidsdansl'erreur.
90
2.
Dans , l'erreur est ba
se´esurunesensibilit´eglobaled
e la foncionnelle couˆt.
Lesdeuxmatricespoidscommerepr´esent´eessurlagure4.10tnatnnoderiae´,squ´eeselontsimils poidssontbase´ssurl'utilisationd'uncrite ree´nerge´tique.
Application aux ecoulements de cavite´ ´
Lam´ethodedere´gularisationpond´ere´ebase´esurlasensibilit´eglobaleae´te´appli´ucasde quee a l´ecoulementdecavite´.Lame´thodereproduitlesdynamiquestemporellesdelaPODcommeonpeut le voir sur les gures4.11et4.12'autvecd´eeampare´oc´ateohed´mteLa.bilaitarnostrehnecueiqecsd comme on peut le constater dans4.4et sur la gure4.13eTik´eedvhonooitasirare´dnopn.ulegr´La surpasse les autres techniques de calibration en termes d'erreurs normalis e´es et d'erreurs modales. Laprincipaleforcedecetter´egularisationre´sidedanslefaitquelesparam´etresder´egularisation sontd´etermin´essansaucuneinterventiondel'utilisateur.Finalementnousv´erionsl'ade´quationdu mode le de calibration pour de longues pe´riodes d'int g´ ration temporelle comme on peut le voir sur la gure4.14.nsAimiquespotlesdynaelrpe´idlimedoe uppe auresr 4'dsedoirtallicsoeiond´ep l'´ecoulement,maisdivergerapidementquandoninte greau-dela .Ceciestdˆuaufaitquelesmodes ne´glige´s (tronque´s) ne sont pas pris en compte et le proble me de fermeture reste ouvert, meˆme en d´eterminantcorrectementlescoefcients.Nousdevonsdonccalibrersurplusdepe´riodessinous souhaitonsutiliserlemode lepourdese´tudesdecontrˆole.
91
4. Integration and calibration of ROM
4.1
Introduction
As demonstrated in the previous chapter it is often diff cult to represent with suff cient accuracy even the short time dynamics of the original system which bars the utility of the Reduced Order Model for applications in sensitivity studies, optimisation and optimal control. Methods in the literature that pertain to improving the accuracy of the Reduced Order Model is termed as calibration The main idea of calibration is to identify the coeff cients of the POD Galerkin model so as to match the tem-poral dynamics of the POD which are known in advance. This strategy is usually called as a system identif cation or black-box model in control literature when the dynamical system is determined with respect to an identif able dynamics of the process. The reasons for the inaccurate behaviour of the ROM can be attributed to the truncation of the POD bases. An analogous problem occurs in the Large Eddy Simulation (LES) of f ows where there is lack of dissipation, due to truncation of the smaller scales. Even including all the modes in Galerkin projection may still lead to the wrong at-tractor due to structural instability as has been demonstrated inRempfer(2000),Noacket al.(2003) . Other problems may arise due to the contribution of pressure at the boundaries of the domain, which is usually neglectedNoacket al.(2005). The stability properties of the compressible POD-Galerkin approximation has been studied byIolloet al.(2000) and shows that just the stability of the numerical scheme is not suff cient for the stability of the ROM and a suitable numerical stabilisation is required. Consecutively a Sobolev inner product has been def ned for the norm to improve the accuracy of the ROM. The calibration techniques can be broadly classif ed into two categories,
1. To treat the problem of calibration similar to the closure problem of turbulence
2. As a process of system identif cation for the coeff cients, leading to an optimisation problem.
Regarding the calibration techniques for the ROM, based on treating them as a closure problem, the earliest attempt is due toAubryet al.(1988). In this work inter-modal transfer of energy between the truncated POD modes and the resolved POD modes by means of an artif cial viscosity are modeled. Podvin(2001) proposes a connection between the closure problems encountered in the large eddy simulation of turbulence and the truncation terms of the ROM where one has to model the missing terms. An approach to model the inter-modal transfers by means of artif cial viscosity assuming the conservation of the average kinetic energy in the ROM can be found inCazemieret al.(1998). A spectral vanishing viscosity method has been proposed byKaramanos & Karniadakis(2000), which was initially developed for LES to improve the long term integration of the ROM. The time dependent modal eddy viscosity can be found as a solution of an optimisation as given in Bergmann & Cordier(2005), for the stabilisation of ROM for wake f ows. Regarding the problem of calibration, based on an optimisation procedure, most of the calibration techniques tries to identify the system coeff cients, so as to minimise the error between the POD tem-poral dynamics and that predicted by the ROM. The methods mainly relay on the def nition of the error leading to an optimisation problem, which can be solved iteratively as found inGalettiet al.(2004). Coupletet al.(2005) gives a general frame work of the various error which arises in the ROM and proposes a calibration technique based on minimising a linear functional of error. Stabilisation of the ROM by means of a polynomial identif cation of the ROM independent of the physical system can
92
4.2. Def nition of errors
be found inPerretet al.(2006 ). Amethod called as ”Intrinsic stabilisation” has been proposed by Kalb & Deanne(2007error with respect to the reference) which takes into account the instantaneous dynamics, and is obtained by replacing the original ROM, with another ROM with the polynomial coeff cients obtained from the temporal POD dynamics. Recently an improvement of the above ideas ofCoupletet al.(2005) has been presented inCordieret al.(2009). In this work we evaluate different methods of calibration based on the solution of the optimisation problem. We begin by introducing various def nitions of errors, between the calibrated dynamics and the original temporal dynamics. The optimisation problem of minimising the error leads to a solution of a linear system, in case the errors are aff ne functions of the predicted dynamics. The linear system is ill conditioned and needs to be regularised.Cordieret al.(2009) have used the method of Tikhonov regularization to solve the ill-conditioned problem. We present a further improvement of this technique by introducing the various type of weight matrix used in the def nition of error. The f rst method is by performing a sensitivity analysis of the ROM with respect to a given cost functional, to determine the weights of the relevant dynamics of calibration. The second method is by using the energy content of the representation in forming the weight matrix to be used in calibration.
4.2
Denition of errors
4.2.1 State calibration method with nonlinear constraints
We start with the polynomial form of the ROM explained in the previous chapter, which we restate as
aR=f(yaR) +g(zaR γ) ˙
(4.1)
As previously shown, the temporal dynamics obtained by adding the actuation mode, determined by anL2optimisation procedure is mainly restricted to the un-actuated space, it is suff cient to present the main ideas for the case ofg= 0 objective of the POD based model (. The4.1) is to accurately represent the dynamics of the POD temporal modesaP, and the problem of calibration is to identify the coeff cientsy Onesuch that this representation is possible. then naturally seeks to minimise the error e1(y t) =aP(t)−aR(t)(4.2) with the constraints
PCa˙R(t) =f(yaR(t)) aR(0) =aP(0)
e1∈RNis time dependent and we seek to minimise I1(y) =ke1(y t)k2ΛT0(4.3) wherehiT0is a time averaging operator over[0 T0], forNtequally spaced elements on[0 T0]we have N X
hf(t)iT0=N1ti=tf(ti) 1
with
ti= (i−1)Δt
and
Δt=T0 Nt−1
93
4. Integration and calibration of ROM
kkΛis a norm onRN. For any positive def nite matrixΛ∈RN×Nthe norm of any vectore∈RNis given by kek2Λ=eTΛe The matrixΛacts as a weight function giving importance to the specif c POD modes, whenΛ =IN it means that all the POD modes have the same importance in terms of the error. Later in this chapter we shall describe a method to utilise this weight matrix to def ne the relative importance of the error with respect to the POD mode. Minimisation ofI1under the constraintsPCleads to a non-linear constrained optimisation problem of minimising
I1(y) =N1NtkX=t1Ni=X1(aPi(tk)−aiR(tk))2
The optimisation problem can be solved iteratively as inBergmann & Cordier(2005) to f nd the op-timal eddy viscosity or using a single shot constrained optimisation problem with a pseudo-spectral discretisation of the variables as found inGalettiet al.(2004)
4.2.2 State calibration method
Coupletet al.(2005) have argued that the method based on theI1formulation does not have a unique solution and also there are problems of convergence when the well known gradient methods are used to f nd the minimum. As a result the nonlinear constraint is suppressed in the def nition ofe1. By integrating the POD ROM the errore1can be written as t Z0(τ))dτ e1(y t) =aP(t)−aP(0)−f(yaR The nonlinear constraint is suppressed by replacingaRwithaP have a new def nition of the error. We e2def ned as e2(y t) =aP(t)−aP(0)−Z0tf(yaP(τ))dτ(4.4) Minimisation of the error def ned byI2(y) =hke2(y t)kΛ2iT0has been used more recently by Bourgetet al.(2007for the study of transonic f ows) to determine the constant and linear coeff cients, around airfoils. All the modes have equivalent weights leading to the minimisation of the error def ned by I2(y) =N1kNtX=t1Ni=X1aiP(tk)−aPi(0)−Z0tfi(yaP(τ))dτ2(4.5)
4.2.3 Flow calibration method
A third criterion of error is obtained by taking the temporal derivative of thee1criterion.
94
ddte1(y t) =˙aP(t)−f(yaR(t))
4.2. Def nition of errors
the error is given by replacingaR(t)withaP(t)in order to suppress the nonlinear constraint, to obtain the def nition of error given by
e3(y t) =a˙P(t)−f(yaP(t))
(4.6)
The corresponding minimisation can be def ned for the error def ned byI3(y) =hke3(y t)k2ΛiT0. In this method we impose that the temporal POD eigen functions are the solutions of the f ow, given by f. This method as described inCoupletet al.(2005) has been applied to experimental data obtained from PIV measurements byPerretet al.(2006 we assume an identity matrix for). IfΛgiving equal weights to all the modes. We have the minimisation problem for the functional def ned by
I3(y1)NtNX(τ))dτ2 =NtX=1i=1a˙iP(tk)−fi(yaP k
4.2.4 Afne function of error Fori= 2and3, we haveeian aff ne function with respect toy∈RNy,i.e.we def ne
ei( t) :RNy−→RN y−→Ei(t)y+ei(0 t)
where fori= 2 E2(t)y=−Z0tf(yaP(τ))dτ
and fori= 3
E3(t)y=−f(yaP(t))
and
and
with
Ei(t)∈RN×Ny
e2(0 t) =aP(t)−aP(0)
e3(0 t) =a˙P(t)
Assuming a symmetricΛ, we have fori= 23 I(i)(y) =kei(y t)k2ΛT0=yThE(i)(t)TΛE(i)(t)iT0y+ 2he(i)(0 t)TΛE(i)(t)iT0y +he(i)(0 t)TΛe(i)(0 t)iT0 =yTA(i)y−2b(i)Ty+c(i)
where
IfΛis symmetric then so i solving the linear system:
A(i) b(i) c(i)
sA(i)b
= = =
hE(i)(t)TΛE(i)(t)iT0∈RNy×Ny −hE(i)(t)TΛe(i)(0 t)iT0∈RNy he(i)(0 t)TΛe(i)(0 t)iT0∈R
(4.7)
y def nition and minimising the quadratic functionI(i)reduces to
A(i)y
=b(i)
95
4. Integration and calibration of ROM
4.3
Calibration method of Couplet
i As mentioned inCoupletet al.(2005), the general idea is to determine the coeff cientsy(α), as a solution to an optimisation problem which minimises the cost functional given by
Jα(i)(y) = (1−α)ε(i)(y) +αD(y)fori= 23
(4.8)
α∈[01]is a regularising parameter.ε(i)(y)measures the normalised error between the actual temporal dataaP(t)and that predicted by the modelaR(t).Dis a measure of the difference between the coeff cients of the modelycoeff cients obtained from the Galerkin projectionand yGP.ε(i)and Dare def ned as
and
ε(i) (y) =hhkek(ei)(i()y(GyPt)tk)Λ2k2iΛTi0T0
I(i)(y) = I(i)(yGP)
D(y) =ky−kyGyPGkPΠ2kΠ2
wherekkΠsemi-norm on the polynomial vector space and for anyis a y∈RNyis def ned as
kykΠ2=yTΠy
(4.9)
(4.10)
(4.11)
whereΠ∈RNy×Nyis a non-negative symmetric matrix. ForΠ =INyit means that all the coeff cients are given equal importance in the calibration. A partial calibration for different values ofINis y possible as reported byCoupletet al.(2005). The functional in (4.8) can be written as
where
Jiα(y) =χAαI(i)(y) +χαΠky−yGPk2Π { } { } f1(y)f2(y)
α I(i1)(−GαPnd χA=y)a
χαΠ=kyGαPk2Π
(4.12)
As demonstrated in section§4.2.4whenΛis symmetric we havef1(y) =yTA(i)y−2b(i)Ty+c(i). Similarly for a symmetricΠit can be shown thatf2(y) =yTΠy−2yGPTΠy+yGPTΠyGP. For the quadratic functions,f1andf2one obtains the gradient as ∇f1(y) = 2A(i)y−b(i)and∇f2(y) = 2Πy−yGP(4.13)
Minimisation of the functionalJα(i)is obtained aty(αi)where∇Jα(i)(y(αi)) = 0and is equivalent to solving the system of linear equations fory(αi)given by
96
A(iα)y(iα)=b(αi)
(4.14)
