Lire la seconde partie

Nagarajan, Kaushik Kumar

Description

Niveau: Supérieur, Doctorat, Bac+8
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

Sujets

• Simulation
• Modèle

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 25
• Langue: Français
• Poids de l'ouvrage: 1 Mo

Extrait

Lire la seconde partie de la thèse

Chapter

4

Integration

Introduction

and

calibration

of

ROM

Cechapitreconcerneessentiellementlesdiff ´ erentestechniquesdecalibrationutiliserpourstabiliser lemode  lesd'ordrer ´ eduit.Ilestsouventdifcilederepre ´ senteravecsufsamentdepre ´ cision,m ˆ eme sur un temps court la dynamique du systeme initial, ce qui interdit l'utilisation des mod eles d'ordre reduitspourdese ´ tudesdesensibilite ´ ,d'optimisationetdecontro ˆ leoptimal.L'id ´ eeprincipaledela ´ calibrationestd'identierlescoefcientsdumode  lePODGalerkin(ROM)desortequesadynamique propresconcideavecladynamiquetemporelledelaPODissuedessimulationsnum ´ eriquesdirectes ¨ etquiestconnuea  l'avance.Lesraisonsdelamauvaisepre ´ cisiondumode  leROMpeute ˆ treattribu ´ ee a  latroncaturedesbasesPODou  les ´ echellesdissipativesinclusesdanslesmodesPODd'ordres ´ eleve ´ snesontpasprisesencompte.C'estunproble  meanaloguea  celuirencontr ´ eenSimulationde Grandes Echelles (LES) oula dissipation des petites e´chelles est manquante. Meˆme si tous les modes de la projection de Galerkin sont pris en compte, on peut aboutir aun mauvais attracteur acause desinstabilit ´ esstructuralescommecellesobserve ´ esparRempfer(2000),Noacket al.(2003). Les techniquesdecalibrationpeuvent ˆ etreclasse ´ esendeuxcate ´ gories,lapremie  reconsistea  traiterle proble  medecalibrationcommeunproble  medefermeturedelaturbulence.Lasecondeconside  reun proble  med'identicationdecoefcients,cequirevienta  uneproble  med'optimisationoudecontro ˆ le 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'optimisationL 2 se retrouve principalement restreint de aun mode spatial. Par conse´quent il est sufsant de pre´senter les princi- paleside ´ essurlacalibrationassoci ´ eeLesdynamiquestemporellesauxcassansactionnement.Les erreurspouruneidenticationpolyn ˆ omialedumode  led'ordrer ´ eduit(ROM)peuvente ˆ treessentielle- 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.

Troiserreurspeuvente ˆ tred ´ 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 ´ˆ ocnC.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 modele 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  .Ceciestd ˆ uaufaitquelesmodes ne´glige´s (tronque´s) ne sont pas pris en compte et le probleme de fermeture reste ouvert, meˆme en d ´ eterminantcorrectementlescoefcients.Nousdevonsdonccalibrersurplusdepe ´ riodessinous souhaitonsutiliserlemode  lepourdese ´ tudesdecontr ˆ ole.

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

a R =f(ya R ) +g(za R  γ) ˙

(4.1)

As previously shown, the temporal dynamics obtained by adding the actuation mode, determined by anL 2 optimisation 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 modesa P , and the problem of calibration is to identify the coeff cientsy Onesuch that this representation is possible. then naturally seeks to minimise the error e 1 (y t) =a P (t)−a R (t)(4.2) with the constraints

P C  a˙ R (t) =f(ya R (t)) a R (0) =a P (0)

e 1 ∈R N is time dependent and we seek to minimise I 1 (y) =  ke 1 (y t)k 2 Λ  T 0 (4.3) wherehi T 0 is a time averaging operator over[0 T 0 ], forN t equally spaced elements on[0 T 0 ]we have N X

hf(t)i T 0 = N 1 t i= t f(t i ) 1

with

t i = (i−1)Δt

and

Δt= T 0 N t −1

93

4. Integration and calibration of ROM

kk Λ is a norm onR N . For any positive def nite matrixΛ∈R N×N the norm of any vectore∈R N is given by kek 2 Λ =e T Λe The matrixΛacts as a weight function giving importance to the specif c POD modes, whenΛ =I N 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 ofI 1 under the constraintsP C leads to a non-linear constrained optimisation problem of minimising

I 1 (y) = N 1 N t k X = t 1 N i= X 1 (a P i (t k )−a i R (t k )) 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 theI 1 formulation 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 ofe 1 . By integrating the POD ROM the errore 1 can be written as t Z 0 (τ))dτ e 1 (y t) =a P (t)−a P (0)−f(ya R The nonlinear constraint is suppressed by replacinga R witha P  have a new def nition of the error. We e 2 def ned as e 2 (y t) =a P (t)−a P (0)− Z 0 t f(ya P (τ))dτ(4.4) Minimisation of the error def ned byI 2 (y) =hke 2 (y t)k Λ 2 i T 0 has 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 I 2 (y) = N 1 k N t X = t 1 N i= X 1  a i P (t k )−a P i (0)− Z 0 t f i (ya P (τ))dτ  2 (4.5)

4.2.3 Flow calibration method

A third criterion of error is obtained by taking the temporal derivative of thee 1 criterion.

94

d d t e 1 (y t) =˙a P (t)−f(ya R (t))

4.2. Def nition of errors

the error is given by replacinga R (t)witha P (t)in order to suppress the nonlinear constraint, to obtain the def nition of error given by

e 3 (y t) =a˙ P (t)−f(ya P (t))

(4.6)

The corresponding minimisation can be def ned for the error def ned byI 3 (y) =hke 3 (y t)k 2 Λ i T 0 . 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

I 3 (y 1 ) N t N  X (τ))dτ  2 = N t X =1 i=1 a˙ i P (t k )−f i (ya P k

4.2.4 Afne function of error Fori= 2and3, we havee i an aff ne function with respect toy∈R N y ,i.e.we def ne

e i ( t) :R N y −→R N y−→E i (t)y+e i (0 t)

where fori= 2 E 2 (t)y=− Z 0 t f(ya P (τ))dτ

and fori= 3

E 3 (t)y=−f(ya P (t))

and

and

with

E i (t)∈R N×N y

e 2 (0 t) =a P (t)−a P (0)

e 3 (0 t) =a˙ P (t)

Assuming a symmetricΛ, we have fori= 23 I (i) (y) =  ke i (y t)k 2 Λ  T 0 =y T hE (i) (t) T ΛE (i) (t)i T 0 y+ 2he (i) (0 t) T ΛE (i) (t)i T 0 y +he (i) (0 t) T Λe (i) (0 t)i T 0 =y T A (i) y−2b (i) T y+c (i)

where

IfΛis symmetric then so i solving the linear system:

A (i) b (i) c (i)

sA (i) b

= = =

hE (i) (t) T ΛE (i) (t)i T 0 ∈R N y ×N y −hE (i) (t) T Λe (i) (0 t)i T 0 ∈R N y he (i) (0 t) T Λe (i) (0 t)i T 0 ∈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

J α (i) (y) = (1−α)ε (i) (y) +αD(y)fori= 23

(4.8)

α∈[01]is a regularising parameter.ε (i) (y)measures the normalised error between the actual temporal dataa P (t)and that predicted by the modela R (t).Dis a measure of the difference between the coeff cients of the modelycoeff cients obtained from the Galerkin projectionand y GP .ε (i) and Dare def ned as

and

ε (i) (y) = h hke k ( e i) (i ( ) y ( G y P t  ) t k ) Λ 2 k 2 i Λ T i 0 T 0

I (i) (y) = I (i) (y GP )

D(y) = ky− ky G y P G k P Π 2 k Π 2

wherekk Π semi-norm on the polynomial vector space and for anyis a y∈R N y is def ned as

kyk Π 2 =y T Πy

(4.9)

(4.10)

(4.11)

whereΠ∈R N y ×N y is a non-negative symmetric matrix. ForΠ =I N y it means that all the coeff cients are given equal importance in the calibration. A partial calibration for different values ofI N is y possible as reported byCoupletet al.(2005). The functional in (4.8) can be written as

where

J i α (y) =χ A α I (i) (y) +χ α Π ky−y GP k 2 Π { } { } f 1 (y)f 2 (y)

α I (i 1 ) ( − G α P nd χ A = y) a

χ α Π = ky G α P k 2 Π

(4.12)

As demonstrated in section§4.2.4whenΛis symmetric we havef 1 (y) =y T A (i) y−2b (i) T y+c (i) . Similarly for a symmetricΠit can be shown thatf 2 (y) =y T Πy−2y GP T Πy+y GP T Πy GP . For the quadratic functions,f 1 andf 2 one obtains the gradient as ∇f 1 (y) = 2  A (i) y−b (i)  and∇f 2 (y) = 2Π  y−y GP  (4.13)

Minimisation of the functionalJ α (i) is obtained aty (αi) where∇J α (i) (y (αi) ) = 0and is equivalent to solving the system of linear equations fory (αi) given by

96

A (i α ) y (i α ) =b ( α i)

(4.14)