Keywords: Surrogate modelling, data fusion, incomplete factorial DoE, tensor approximation

Transcription

1 Buiding Data Fusion Surrogate Modes for Spacecraft Aerodynamic Probems wit Incompete Factoria Design of Experiments Miai Beyaev 1,, 3, a, Evgeny Burnaev 1,, 3, b, Erme apusev 1,, c, Stepane Aestra 4, d, Marc Dormieux 5, e, Antoine Cavaies 5, f, Davy Caiot 5, g and Eugenio Ferreira 5, 1 DATADVANCE, c, Porovsy bvd. 3, Moscow, 10908, Russia Institute for Information Transmission Probems, Bosoy aretny per. 19, Moscow, 17994, Russia 3 PreMoLab, MIPT, Institutsy per. 9, Dogoprudny, , Russia 4 Airbus Group, St. Martin du Touc, 316 route de Bayonne, Tououse Cedex 9, France 5 Airbus Defence and Space, Route de Verneui, Les Mureaux Cedex, France a miai.beyaev@datadvance.net, b evgeny.burnaev@datadvance.net, c erme.apusev@datadvance.net, d stepane.aestra@eads.net, e marc.dormieux@astrium.eads.net, f antoine.cavaies@astrium.eads.net, g davy.caiot@astrium.eads.net, eugenio.ferreira@astrium.eads.net eywords: Surrogate modeing, data fusion, incompete factoria DoE, tensor approximation Abstract. Tis wor concerns a construction of surrogate modes for a specific aerodynamic data base. Tis data base is generay avaiabe from wind tunne testing or from CFD aerodynamic simuations and contains aerodynamic coefficients for different figt conditions and configurations (suc as Mac number, ange-of-attac, veice configuration ange) encountered over different space veices mission. Te main pecuiarity of aerodynamic data base is a specific design of experiment wic is a union of grids of ow and ig fideity data wit consideraby different sizes. Universa agoritms can t approximate accuratey suc significanty non-uniform data. In tis wor a fast and accurate agoritm was deveoped wic taes into account different fideity of te data and specia design of experiments. Introduction Airbus Defence and Space, in te frame of te deveopment of various spacecraft veices, is producing aerodynamic modes. Different inds of data are used to buid tese modes. It can be experimenta resuts derived from WTT campaigns, wit a ig eve of confidence. Anoter type of data is numerica data resuting from CFD simuations, wit a ower eve of confidence, wic depends aso on te used numerica metod (Euer / RANS, fow regime, etc.) CFD computations and even more WT campaigns are very costy and it is sometimes not conceivabe to perform a figt configurations (surfaces defections, e.g.) on te woe figt range. An inter-extrapoation process is ten necessary to buid a compete aerodynamic mode. Wit a ot of input parameters (Mac, Ange of Attac, Ange of Sidesip, surfaces defections, etc.), tis process can be quite time consuming and may ead to inconsistent resuts wit a cassica approac. Furtermore avaiabe data can constitute anisotropic grids wic ead to severey arden te inter-extrapoation process. Te objective is terefore to buid a consistent mode taing into account a avaiabe data, wit a fast and rationae metod. Airbus Defence and Space is interested by te generation of surrogate modes from given spacecraft aerodynamic database. Te AErodynamic DataBase (AEDB) provides te aerodynamic coefficients of te veice for te different figt conditions and veice configurations (notaby aerodynamic contro surfaces defections) encountered over te woe mission domain. Tis type of database is typicay incuded witin goba veice beaviour modes.

2 Tese beaviour modes are ten used as input to various system studies, suc as trajectory and performance anaysis, or anding quaities and figt contro system anaysis. From a practica standpoint, te AEDB generation process raises te foowing two fod caenges: Mutidimensiona interpoation/extrapoation: ow to cover a prescribed fu figt enveope defined in te mutidimensiona space of figt conditions/veice configurations, on te basis of scattered discretized input data? Mutipe data combination: ow to buid a consistent and omogeneous aerodynamic database on te basis of mutipe input data sets wit different eves of fideity? Te fina goa is to obtain a surrogate mode tat can automaticay interpoate mutidimensiona data, union of anisotropic grids wit non-uniform data and different eve of fideity, and terefore to buid a consistent and omogeneous aerodynamic mode tat can cover a intermediate points witin te figt enveope. Last but not east, te surrogate mode sa ensure tat te exibited outputs aways remain witin reaistic imits (i.e. remain meaningfu from aerodynamics beaviour standpoint), watever te input vector content. Surrogate Modeing One approac to soving probems of engineering design activey deveoping in recent years is te surrogate modeing [1]. In tis approac a compex pysica penomenon is described by a simpified (surrogate) mode constructed using data mining tecniques and a set of exampes representing resuts of a detaied pysica modeing and/or rea experiments. Te probem of approximation of a mutidimensiona function using a finite set of pairs point - vaue of te function at tis point is one of te main probems to be soved during construction of te surrogate mode. d Approximation Probem. Let us consider continuous function g: D R R, were D is a compact set. Let us refer to a set of points Σ and a set of function vaues at points from te set Σ as N a training set: S = { xi Σ, yi = g( xi)} i= 1 = { Σ, g( Σ)}. Set of points Σ we wi ca a design of experiments (DoE). In tis paper we consider approximation probem in te foowing statement. Given te training * set S, cass of functions F and penaty function P : F R find suc f F tat minimizes te error function R( f, Σ, g( Σ )) λ + f = argmin ( g( x) f( x )) + P ( f ) = argmin R( f, Σ, g( Σ)). (1) * f F λ f F x Σ Introduced penaty function aows to contro variabiity of te approximation mode. For exampe, it can be a norm of te second derivatives of f (see, for instance, smooting spines []) or norm of f in some Hibert space (erne ridge regression [3]). Data Fusion. Te data sets considered in tis wor contains output vaues obtained from two different sources (experimenta measurements and CFD simuations). Bot sources mode te same pysica process. However one of tem (experimenta measurements) is supposed to be more accurate tan anoter (CFD simuations). We wi refer to te more accurate one as ig fideity (HF) mode and denote it by g ( x ). We wi refer to te second source as ow fideity (LF) mode and denote it by g ( x ). So te training set S is spit into parts: ig fideity sampe S = { x, y = g ( x )} = { Σ, g ( Σ )} and ow fideity sampe S = { x, y = g ( x )} = { Σ, g ( Σ )}. i i i A data fusion tas is given ig fideity and ow fideity data sets approximation f ˆ( x ) of g ( x ). i i i S and S construct an

3 Usage of LF points aows to buid more accurate surrogate modes as tey contain information about te pysica mode in regions wic don t ave HF points. To measure accuracy of HF and LF vaues we introduce confidence eves of te sampe w (for HF points) and w (for LF points), w + w = 1, w, w > 0. Tey express our confidence about te accuracy of te source tat produced output vaue at given point and can be tougt of as a probabiity of being te true vaue of te pysica caracteristic at given point. Design of Experiments. One of te pecuiarities of te probem considered in tis wor is a specific n DoE. First of a, et us introduce some definitions. Let us refer to sets of points σ = { x i D} i = 1, D d R, = 1, as factors. A set of points Σ fu is referred to as a factoria design of experiments if it is a Cartesian product of factors: eements of a factors Σ fu are vectors of a dimension = 1 Σ = σ σ σ = x x = = 1 fu 1 {[ i,, ], 1,, 1, }. 1 i i n Te d = d and te sampe size is a product of sizes of = 1 N = n. If a te factors are one-dimensiona Σ fu is a mutidimensiona grid. A reasonaby arge subset of Σ Σ fu we wi ca an incompete factoria design of experiments. Te DoE considered in tis paper is a union of severa anisotropic grids (ere anisotropy means significanty different sizes of te grids), see Fig. 1. It is an incompete factoria DoE. Fig. 1: Design of experiments. Fig. : Exampe of degenerate GP regression mode. Suc designs are compicated for approximation metods wic don t use any nowedge about te data structure. Universa approximation agoritms, ie Gaussian Process (GP) regression [4], expicity or impicity assume tat te DoE is rater uniform (tere are no big oes in te DoE). However, union of anisotropic grids is sufficienty a non-uniform DoE. In big regions witout training data te mode smootness soud be controed in order to avoid osciations and oversoots. GP regression as some ind of smootness contro (not direct!) but it wors ony in te neigborood of te training data (see Fig. ). To sove tis probem we use a specific approximation agoritm described in te foowing section. Tensor Product of Approximations Te approximation f ˆ( x ) wi be modeed by te inear expansion in a dictionary of parametric functions. To construct te approximation we ave to coose cass of parametric functions F, coose penaty function P λ and ten sove optimization probem (1). p Te dictionary of functions is cosen in te foowing way. Let = { ψ } = 1 be a dictionary of functions defined on j j D (for a = 1, ). Te dictionary of functions defined on D1 D D

4 is formed as tensor product of functions from : = ψ ψ ψ = = 1 tensor { j j j, j 1, p, 1, }. 1 Te cass of functions F wi be a inear expansion in a dictionary tensor. It means tat te 1 mode f( x) F can be written as f( x) = α j1, j,, j ψ ( ) ( ) ( ) j x ψ 1 j x ψ j x. j1,, j For te penaty function we wi use variabiity of a function aong some factor (or group of factors). For exampe, to penaize te variabiity aong te first factor te foowing penaty function is used 1 f ψ j1 1 1 α j1, j,, j 1 i ψ 1 j i ψ j i x Σ i1,, i j1,, j x x x x Pλ ( f) = = ( ) ( ) ( ). ( ) ( ) Note tat te cosen penaty function is quadratic over decomposition coefficients A = { α, j = 1, p, = 1, }. Tis penaty function can be generaized to contro variabiity over j, j, j 1 severa factors, for te detais see [5]. Here we describe a simpified version of te ast step since it is important for te data fusion probem. Due to te coice of penaty function te origina optimization probem (1) can be reduced T T to minimization of te function R (A) = (Y Ψ A) W(Y Ψ A) + A ΩA, were Y is an extended vector of training vaues, Ψ is an extended matrix of regressors, Ω is a square penaty matrix (see 1, if xi Σ detais in [5]), W is a diagona weigting matrix (sizes are N* N ): Wii, =. 0, if xi Σfu \ Σ. Optimization probem m in R (A) can be soved in a very efficient way using specia structure of te training set, tensor cacuus and conjugate gradients [5]. Data Fusion Based on Tensor Product of Approximations In tis section we wi describe 3 different approaces to data fusion probem. A of tem are based on tensor product of approximations on incompete factoria DoE (ita). We suppose tat bot HF points ( Σ ) and LF points ( Σ ) are incompete factoria DoE. In tis section we wi use te foowing notation: Σ=Σ Σ and Σ bot =Σ Σ. Merged Soution. In tis approac te mode is constructed using ita wit modified training set and specific weigting matrix W. New training vaues we define by y = w y + w y, i i i were w = 1 w, w and W are as foows if x i Σ\ Σ ten w = 1 and Wii, = w; if x i Σ\ Σ ten w = 0 and Wii, = w; ese w = w and Wii, = w + w. Tis approac intends to fit LF mode in regions were ony LF vaues are given, HF mode in regions wit ony HF vaues and weigted sum of LF and HF vaues in oter regions. Fused Soution. Tis approac uses modified training set. To construct approximation of g ( x ) a training set { Σ, Yˆ } wi be used, were Σ is a set of estimated HF vaues at points Σ. To estimate unnown vaues ŷ Yˆ we wi use te foowing mode

5 yˆ = g ( x) + g ( x), x Σ. () diff were gdiff ( x ) is a bias of te LF function g ( x ) wit respect to te HF function g ( x ). If gdiff ( x ) is nown ten Eq. () gives exact vaues g ( x ) for a x Σ and yˆ = g ( x ). However, gdiff ( x ) is unnown. Tat s wy we wi use te approximation of gdiff ( x ). Let us denote Ydiff = { g ( x) g ( x): x Σbot}. Now construct bias mode g diff approximating te data set { Σ bot, Ydiff }. Ten we can cacuate Y ˆ using Eq. () for x Σ\ Σ and setting yˆ = g ( x ) for x Σ. Loca Fideity Soution. In previousy described approaces we used te goba fideity, i.e. te confidence eve doesn t depend on te ocation of te point and its neigborood. However, if region contains ots of HF experimenta points ten in tis region CFD points soud be treated as LF, wereas in region witout experimenta points CFD points become HF data (see Fig. 3). Terefore we come to te idea of oca fideity. Fig. 3: Loca fideity of data Te difference between oca and goba approaces can be rougy expained using simpified data (see Fig 4.). Te goba fideity mode try to catc te difference between HF and LF and use it to estimate HF vaues in [0.8, 1]. Te oca fideity mode doesn t trust HF data in [0.8, 1] (because tere is no points nearby) and prefers to use LF data instead. Fig. 4: One-dimensiona sice of constructed mode Fig. 5: Loca fideity vs. Goba fideity Te concept of oca fideity can be impemented as foows. For eac point te output vaue y as x Σ we estimate w ( x) w ( x) y ( x) = g( x) + g( x), w ( x) + w ( x) w ( x) + w ( x) (3)

6 weigts w, w are cacuated according to te idea of oca fideity: x x w ( x) = exp, x Σ σ were = or =. If tere are a ot of HF points in te neigborood of point x ten te vaue of w ( x) wi be arge, if tere are few points in te neigborood ten w ( x) wi be sma. Te same ods for w ( x ). Terefore, in regions wit arge amount of HF points and ow amount of LF points te vaue y ( x) wi be cose to g ( x ) and vice versa. If for some x Σ g ( x ) or g ( x )) is not nown an approximation of tis vaue constructed using HF or LF data can be used instead. Comparison of te Proposed Soutions. Now et us compare tree proposed soutions on rea data. Training sampe as te foowing caracteristics: Input dimension: 3 Output dimension: 3 Hig fideity sampe size: 1846 Low fideity sampe size: 180 Fig. 5 iustrates one-dimensiona sice of approximations. As it can be seen Merged soution as sma singuarities at points were bot ig and ow fideity vaues are given. Fig. 6: Probem wit fused soution. Usua view (eft) and enarged view (rigt) of approximation of difference between HF and LF modes. Approximation taes arge vaues in region witout training points. (3) Merged soution (3) Fused soution

7 (3) Loca fideity soution Fig. 7: D-sices of approximation. Te Fused soution is based on te difference between HF and LF mode. Te difference mode is buit using points for wic bot CFD and experimenta vaues are nown. Suc points don t cover te woe domain, so we ave to extrapoate in oter regions. Tis procedure is rater inaccurate and introduces arge uncertainties. Particuary, approximation can tae vaues of te same magnitude as te HF and LF vaues. Suc beavior is not reasonabe from pysica point of view. Fig. 6 iustrates tis probem. Fig. 7 depicts two-dimensiona sices of obtained surrogate modes. In tis figure bue points denote HF points wic were removed from te training sampe in order to see te beavior of approximation in regions wic ave HF points but don t ave LF points. Loca fideity surrogate mode provides smoot approximation witout oversoots and osciations. One can see tat merged and fused soutions can cange teir beavior significanty in suc regions wic eads to oversoots. Tus, oca fideity soution doesn t ave bot mentioned disadvantages of merged and fused soutions and provides rater smoot and reasonabe approximation. However, in some cases oca fideity soution can be ess pysica tan fused soution. Loca fideity mode interpoates te HF data in regions were ony HF data is avaiabe wie tere is difference between LF and HF modes and te bot vaues g ( x ) and g ( x ) soud be used and taen wit corresponding confidence eves. So te fused soution can be more suited for suc situations from pysica point of view. Concusions In tis wor we considered te data fusion probem were te design of experiments is a union of severa anisotropic grids of HF and LF points. Severa approaces (merged soution, fused soution, oca fideity soution) based on ita approximation tecnique as been deveoped to sove tis probem. Te ita tecnique taes into account structure of te data set and tus buids accurate approximations in a very efficient (in sense of computationa compexity) way. Te approac based on idea of oca fideity can provide good approximations but its beavior is not aways pysica. Te fused approac is te most promising soution for now as it is abe to foow pysics of te process. Te merged approac is mainy used for bencmaring purposes. Te wor on te bot promising approaces (oca fideity soution and fused soution) is not finised. Furter deveopment soud be put on removing discussed disadvantages.

8 References [1] Forrester A., Sobester A., eane A. Engineering Design via Surrogate Modeing. A Practica Guide. Wiey [] de Boor C. A Practica Guide to Spines, nd edition. Springer-Verag. Berin 001. [3] Scöopf B., Herbric R., Smoa A. A generaized representer teorem Computationa Learning Teory. (001) V P [4] Rasmussen, Car E. and Wiiams, Cristoper. Gaussian Processes for Macine Learning. MIT Press, 006. [5] Beyaev M. Approximation probem for factorized data. Artificia inteigence and decision teory. (013) V P