Reduced-order modeling: new approaches for computational physics

TL;DR: In this paper, the authors review the development of new reduced-order modeling techniques and discuss their applicability to various problems in computational physics, including aerodynamic and aeroelastic behaviors of two-dimensional and three-dimensional geometries.

About: This article is published in Progress in Aerospace Sciences.The article was published on 2004-02-01 and is currently open access. It has received 732 citations till now. The article focuses on the topics: Computational resource & Aeroelasticity.

Summary

Both l i ~~l i t a t i o ~~s

• The use of point vortices to sirril11;ttr tlie nonliiiea~ tlvriarnics of vorticity gcncrating systcrris is a simple exa~riple of lolr-ortlcr ~riodeling.
• Lonr fidelity implies that a complltational model is rriiss~~ig key physical behaviors that render the model highly inaccurate in certain regimes, wilereas high fitfelity implies a broader range of model applica-1)ility.
• These terms alone are of ambiguous meaning, as they are dependent on the class of problems to which modcls arc applied.
• Thus, use of liigh-fidclitv motlcls usually leads to accurate soltitioris.
• Point simulations lisi~ig liigli-fitlelity equation sets (e.g., Navies-Stokes cquations) typically cannot be obtained fast enough to permit design.

Volterra Theory of Nonlinear Systems

• This transition has been described by Silva,' which the authors si~mmarize here.
• Early mathematical models of 1111steady aerodynamic response capitalized on tlie efficiency and power of superposition of scaled and shifted fi~ndamental responses.
• Attempts to atidress the problcm of high comput<ltiorial cost inclridc thc dei~eloprnent of transonic indi-cia1 responses.
• There ;ire several ways of iclrntifying \'olterra kernels in tllc!.
• The appeara,nce of discrete-time rnet,hotIs has great implications for aeroelast,icit,y and aeroservoelast,icity 1)y providing a means for efficiently ~r~otleli~ig nonlinear acrotlynamics.

The Proper Orthogonal Decomposition

• V'ith spectral methods, field variables are approximated iising expansions involving c.hosen sets of basis functions.
• HIoin and r\loser4 ilsed data from a niiincrically si~iiulatcd channel flow to compiltc rharactcli\tic.
• Through this allpioaclt, for example, flt~icl-dynamical svstrrn5 ale first simulated with CFD tcchniqnes, samplrs arc taken.
• As dcscril)etl al~ovc, tlie POD is being apl)lic+d in Illany diffprent scie~itific aiid e~igi-iieeri~ig discipli~ics, i~iclutlirig aeroelasticity \.
• Tinie-depentlent vectors of spat,ially tliserc.tizctl ficlld v:iriables, referecl t,o as full-syste~ii vectors!.

Volterra Theory

• \Ye begill by revicn,i~lg key features of tlle V.oltcrra.
• "7,'2,58 the authors follow the prese~~tatiotis of Silva" 59 and Ravell, Levy and I<arpeIL3 to capture issues relatrtl to aeroclastic analysis.
• Furtherrnorc., this section will c.o~icr>~itrate oli tinic-cloniaiii VVolt,csra for-rtitllatio~is, co~lsistellt with tlie il~ipliecl al)l)lic,ztiorl to tinie-dornai~l.
• The first response term represents tlie convolution of the first-ortlcr ker~iel with the systc~ii i~iljiits for ti111c.s between 0 a~tcl t, ivherc.
• Funrlamental difference between a continilous-tinit unit impulse response and a discretetime unit irnp~llse response. '. j9.

Lineul-(F7~cquency Dorrlain) P O D Fo~,rrl,ulatiorl

• Hall et al. ol ,tain an cqriation with strllct~ne equivalrnt to that of (15) llsing an explicit, cell-ccntered, finite-vohlme God~lnov rrirtliotl, ltilt tlrrivc an expression for a Lax-lY~ri(11off scherne that contains additional terms which are second orclrr in u.jO.
• The rornplcx mode matrix 9 is computed 11y f i ~s t solving the cornplex form of (8), and then forming tlie product SV.
• To predict the timedependent response about the equilibrium state, q is approxirnat~d bj-9q ( q is the array of redurcd-order variables) ant1 substit~rted into the small-disturbanrr equation ( 15 15) arc rcquiretl for different forcing conditions, the niatrix A may be CZA-tlecompos~rt or may be analyzecl for eigcnrriotles that will tlomiriate in the predicted response.
• 1)1it hecornc impractical when A' becomes sufficiently large, sincc the complrtational effort grows at a much faster rate than the number of equations.
• The POD approach is well-s~litccl to ~lillltldisciplinary analysis invol~~ing repeated interactions hetween equation sets.

The force arid moment fiinrtiori is written as F

• For a given airfoil configuration, tlie flutter speed car1 be brac.kctc~d by systerri;ltically varying retlucetl velocity as a pararnetcr tuitil the eigenvalue with largest rral part c,hariges sig11.
• Tioll, allows fluttvr speeds to be predictctd at a con~putatiorial ratc compura1)le to that of solving the rio~llinc~ar ecluatio~~s for the static I)asc> solutiori.
• ~l-sion~,~\vIiile niuch aclclitio~ial work in this tlirectio~~ is requiretl for the direct approach.

2." \Vitl~ this ap-

• Roach, tlie Jacobiari is nul~~erically corti1)uted about a specified state, wo! arid then frozen i11 the iterative proccdt~re j ( w O ) (wVf1 -wV ) = -R, where the superscript index denotes iteration.
• As drsc,ribecl furtlier below, results 1i;ive r e c c ~~t l y bc.c.11 preseiiteti7"or integration with the second-ordcraccurate Crank-Nicolson schenle: '" where the supclscript IIOW drriotes tinlrl level.
• The direct approach involves solving y = O through Newton 's method: where t,he correction il"+'cr is t,ypically relaxed with the paramet,er whopf: a"+' = cru +.

Nonllnpnr POD Forrnnlnt~on -Gnlerkzn P r o ~p c t ~o n

• T h ~s approach is the ~tiost comnion tecliniq~lc usetl t o obtain nonlinear RORIs tli~ougli the proper orthogonal decomposition, ~nclutling applications involving eq~lations of fluid motion."".
• 2 , which is evaluatetl nr~merically, since the modes are available in discrete form.
• There ale advantages and disadvantages associated with eacli approach.
• Hoa.ever, tlie low romp~~tatiorial cost of solving the ROhI is accompanied by a high cost in the construction of the ROhI.
• T h ~s method is free from tlie drawbacks associated with the Galerkin projection method.

S'SV~R

• The rccluircriic~it for slriall, global tir~ic steps arid accurate iritc~gratiori over rluriierous cycles increases contl)utational cost over stcatfy-state analysis, for wliicll large, local time steps can be used.
• And which can be solvetl with l)set~do-til~ic~-ii~tt~gratiorl usir~g standard accrlt>ratioli tecli~iiclut~s (local tirncl stcq)pitig and ritulti-grit1 accelrratiorl).
• Tliis technique does yield an efficient and low-order represerit,atiori of the tenij)oral variatio~is of cor~lplex systerns experiencirig cyclic behavior in tittle.
• A for111 of tliis HB nicthotl that allon's tlit' period T to 1)e explicitly treated as all u~lknown (i.r.., for an autorlo~ltous systellt) is currcrltly bcirig tlr~.c.lopc~f by Hall and liis colleiiguc~s.
• First five components of the second-order kernel for plunge, also known as Right.

Volterra Series Analysis

• First-and second-order kernels for the Kavier-Stokes solution (with Spalart-A1111iaras turbulence model) of an RAE airfoil in plunge at a transonic hlach nun]her using the CFL3D cot1e8%re presented in Fig. 1 .
• On the left are two sets of first-order kernels due t o two different sets of excitation amplitudes.
• CTsi~lg this nlrtllotl, there is no need for the ropc~atod~ allel costly, exc'cutio~l of tlte CFD code for tliffexrerlt i~lputs.
• By c.o~rlpariso~l, tlic \'olterrab;isc~l ROhI response show^^ recluirrtl a1)out a nlillute.
• But ratlier tllau transfor~~iirig the t,inle-tlo~tiain GXFs irito thc frequency d o t t i a i ~~, discrete-tilnc., statesp;lce systenls can be created using tlie \Toltcrra pulsc responses tfircctly.

Mode 4

• Reduced-order models of the flow equations are constructed from solution snapshots resulting from two independent airfoil movements about base states: pitch oscillation and plunge oscillation.
• Using the procedure detailed in the Analysis section, snapshots are computed from (15) for reduced frequencies (nondirnensionalized by freestream velocity and airfoil chord) cvenly distributed between -1 and 1.
• Ii'itl~ this approach, only two s~iapsllots are rccluired for cac:h natural niode of tlie structure, i ~i ;rtlditio~i to ;i srt, of "funtlamelital" niodes, to construct the ;tt~oelastic ROA1.

Naslli7lear POD A ~~u l y s i s -SuILbspuce Projectio7l

• POD is built at each hlacli riu~liber by co~~iputing solutiom of (15) at reducecl freclrie~icit.~ evelily tlistributed I~etween 0 and 0.5 (conjugate solutio~is are assoc:iatcd ~vith negative freque~icies) for tlir first 5 riatl~ral ~r~oclcs of the ~vilig structure.
• For a given hlach number, there is a critical bump arnplitudc beyond which the shock attached to the leading edge of the bump detached, forming a bow shock.
• In the steady-state analysis, modes are retained that are ntuch smaller in magnitude (eigenvalues of STS as small as 10-lo) tlii~n that of the dynamic analysis.
• A total of 200 s~~al)sl~ot,s, eacl~ representing a col1oc:itioii of tlic coi~sclr~ed i~aria1)lrs over thc cotl~put;ttional (lo-111ain. arc, collectetl at 10-iterate intervals.
• 11l(~tlioc1 yields ~ir~~nt~ric;~lly tlivergc~~t results wlic~i tlic rlunil)rr of retailled ~lloclrs esrc~c~ds 19.

Nonlir~ear POD A ~A ( L ~Y S Z S of Lirrlit-Cycle Oscillatior~

• To assess the applicability of POD-based ROhIs to differerltial equations that c>sliil)it liltlit-cycle oscilla- Right: Time histories of computed surface pressure at bump midpoint for different integrations of 10-mode ROM.
• Preliminary results obtained by Cizrrias and Palacios demonstrate the ability of the POD to capture efficiently thc energy content in a gas/solid niixtlire.
• The governing transport equation5 are much nlore complex than the Navier-Stokes equations; 3 gas and 3 solids equations cwnprise the set.
• Illl~strated in Fig. 11 are ten snapshots of the y-component of vclocity taken at equal intervals in time that partially represent the enscmhlc of snapshots over a11 systcrn vatiables.

Concluding Remarks

• The. basic objective of the theory is tlie icler~tification of linearized arid rionlil~ear kerriel functions t h a t capture the do111- are routine.
• Another important rcslilt rcvie\i~eti was an Euler solution of t h ~ AG.ARD 115.6 Aeroelastic U'irig reccritly coml)lited by Raveh clt a1.13 l~sirig tlic EZKSS CFD code.
• Linearizt~d state-space models are being developed using the CFD-based pulse responses.
• Otlier techrlicl~ws, suc.11 as collocatio~~, sIloul(l 1)c. esl)lorc~l that may allow the P O D rilodos t o be used ill a Illore efficient rnallrier than subsp:tcc projectiol~, but wit11 perhaps greater flexibility tlla~l Galc~kiii projr>c.tior~. .
• There are several clialleriges that need t o be overcome 1)efore ROLI methods can be routinely applied to practical problems.

Figures

Fig. 14 Imaginary part of first harmonic of the blade pitching moment as a function of interblade phase angle (frortl Hall et al.77 with permission).

Fig. 3 Comparison of direct and convolved responses to sinusoidal excitation at 5 Hz (Mach 0.96).

Fig. 8 Comparison of full-system and POD-based ROM predictions of aerodynamic response to bump oscillation (b l = 0.005). Left and Middle: Computed density contours at time t = 20 (14-mode ROM). Right: Time histories of computed surface pressure at bump midpoint for different integrations of 10-mode ROM.

Fig. 7 Minimum local Mach number achieved by full-system and ROM solutions at selected freestream Mach numbers and bump amplitudes: Left: All 104 modes; Middle: 60 modes; Right: 40 modes.

Fig. 9 Conlparisorl of full-systerrl and ROM solutiorls for an 8-mode ROM trained at p0 = 0.16. Left: Eyuilibriurrl and LC0 branches. Middle: ROM system stability. Right: Sensitivity to 11 variation.

Fig. 1 Volterra kernels for CFD analysis of RAE airfoil. Left: first-order kernel and effect of ID amplitudes. Right: First five components of the second-order kernel for plunge.