Hybrid simulation is used to investigate the experimental dynamic response of a com- ponent or sub-assembly of a prototype structure using a hybrid model. The latter comprises both physically-tested and numerically-simulated substructures interacting with each other in a real-time feedback loop. In this study, we extend our previous work on metamodel-based sensitivity analysis of deterministic hybrid models to the practically more relevant case of stochastic hybrid models. The aim is to cover a more realistic situation where the physical substructure response is not deterministic. A generalized lambda surrogate model recently developed by some of the authors is proposed to surrogate the hybrid model response, and Sobol’ sensitivity indices are computed for substructure quantity of interest response quantiles. The eﬀectiveness of the proposed hybrid simulation global sensitivity analysis framework is demonstrated using an experiment. Keywords: Hybrid simulation, stochastic hybrid model, global sensitivity analysis, Sobol’ indices, stochastic surrogate modeling, generalized lambda model. 1 Introduction 1.1 Background and motivation Hybrid simulation (HS) is used to investigate the experimental dynamic response of a structural component or sub-assembly subjected to a realistic loading scenario, which includesthedynamicinteractionwithacredibleyetvirtualstructuralsystem. Coupled physical and numerical substructures (PS and NS) form the so-called hybrid model. Speciﬁcally,thePSistestedusingservo-controlledactuatorsprovidedwithforcetrans- ducers, while the NS is instantiated in a structural analysis software. A time-stepping analysis algorithm computes the hybrid model response on-the-ﬂy. This ensures dis- placement compatibility and force balance between NS and PS throughout the entire 1 DRAFT experiment. When the PS response is rate-independent, it is good practice to perform HS with an extended time scale from 50200times slower than real-time in order to improve the accuracy of position/force control. HS is used to investigate the inner workings of a structural component beyond the linear regime without testing an entire structuralassembly. Asaresult,thecostofexperimentationissubstantiallyreduced. A reportbySchellenbergandco-workersprovidesacomprehensivereviewofHSmethods and algorithms [1]. In earthquake engineering, HS is the only viable solution for testing large struc- tures such as bridges (e.g., [2, 3]). Similarly, HS has been recently proposed to test mooring lines of oﬀshore structures for which hydrodynamic tests with sizeable scale are prohibitive (e.g., [4, 5]). HS is gaining popularity for component-level testing in ﬁreengineering. Thereasonisthatinternalforceredistributionoccurringatthesystem level heavily inﬂuences the failure modes at the component level (e.g., [6, 7]). More recently, HS has been combined with centrifuge testing for investigating soil-structure interaction problems[8]. In all the cases reviewed above, NS and related excitation are conceived as de- terministic. However, in the majority of cases encountered in structural engineering, loadingisstochastic,whiledampingandboundaryconditionsarehighlyuncertain. An exhaustive exploration of all possible load cases is clearly not an option given the ex- perimentalcostassociatedwithasinglehybridmodelevaluation. Accordingly,Abbiati andco-workers[9]proposedsurrogatemodelingtocomputethevariance-basedglobal sensitivity analysis (GSA) of the response quantity of interest (QoI) of a given hybrid model with respect to a set of input parameters that characterize both substructures and loading excitations. In detail, polynomial chaos expansion (PCE) was used to construct a surrogate model (a.k.a. response surface) of the hybrid model response. Sobol’ sensitivity indices of the QoIs were obtained as a by-product of polynomial coeﬃcients as explained in [10]. The goal of such an approach was to reveal what inﬂuences what within the HS: This entails uncovering the inner workings of the PS, the unknown part of the hybrid model in both epistemic and aleatory sense. In that study, the PS was treated as deterministic, that is, aleatory uncertainty was neglected by assuming that nominally identical specimens have identical responses (plus some negligible measurementnoise). This assumption, however, is still far from a realistic scenario in structural testing. Nominally identical specimens are, in practice, never actually identical. Also, some sources of loading exerted through the PS are inherently stochastic (e.g., ﬁre or hydro- dynamic loading). As a result, uncertainty, both aleatory or epistemic, always aﬀects the PSstructure behavior. 1.2 Scope This paper extends the GSA framework for HS proposed in [9] to the case of PSs with non-deterministic behavior. Similarly to the original framework, the idea is to surrogate the hybrid model response as a function of the input parameters that can be controlled by the experimenter and originate from substructures and loading (physical and numerical). However, latent variables that do not appear in the input parameter vector make the hybrid model response stochastic. To account for the latter, the 2 DRAFT generalized lambda model recently developed by Zhu and Sudret [11, 12] is used here todirectlysurrogatetheprobabilitydensityfunction(PDF)oftheresponseQoI.Thisis achievedbymeansofthefamilyofgeneralizedlambdadistributions,whicharesuitable to approximate a wide class of distributions commonly found in engineering contexts [13]. Theparametersofthegeneralizedlambdadistributionsarecastasfunctionsofthe input parameters of the hybrid model and approximated via PCE [14, 15]. Normally, severalrepetitionsofeverysinglesampleoftheinputsparameterswouldberequiredto replicatetheresponseofastochastichybridmodel. However,acquiringsuchrepetitions wouldbeimpossibleinaHS.Instead,byusingthegeneralizedlambdamodelpresented in[12]wecansolvethisproblem,asthegeneralizedlambdamodelcanbecomputedin a non-intrusive manner and does not require repeated HSs for a single sample of input parameters. Forthesereasons,thegeneralizedlambdamodeliswell-suitedtosurrogate theresponseofa stochastichybridmodel. Variance-basedGSAisuniquelydeﬁnedfor deterministic simulators [16]. In the context of stochastic simulators, three alternative variants of Sobol’ sensitivity indices are discussed in [17], namely classical, quantile- basedandtrajectory-based Sobol’indices. Inthiswork,quantile-basedSobol’indices are usedfor theGSA ofthe HSresponse QoI. The eﬀectiveness of the proposed framework is demonstrated for a 3-DoF hybrid model subjected to mechanical and thermal loading. Speciﬁcally, thermal loading is experimentallyexertedonthePSsothattemperatureﬂuctuations(outofcontrolofthe experimenter)entail astochasticresponse of thehybridmodel. This paper is organized as follows. Section 2 introduces the generalized lambda model and describes the GSA framework for stochastic hybrid models. Section 3 presents the 3-DoF hybrid model used to test the framework. Section 4 discusses the resultsoftheGSAofthestochastichybridmodelresponse. Finally,Section5presents the overallconclusions ofthis study. 2 Global sensitivity analysis framework Let a random variable vector x=fG1G "g2D ^R", whereD^denotes the range of deﬁnition of x, represent the input parameters to a HS. Due to the random natureofthehybridmodelresponse,foragivensetofparameters x,thecorresponding QoI.¹xºis a random variable rather than a deterministic value. This is because some latent variables zcannot be identiﬁed or measured in the process, which makes it impossible to include all the relevant variables in x. Therefore, a stochastic HS can be expressed asa mapping: MB:D^R x7!M B¹x`º(1) The latent variables zare grouped into a random vector `. Note that with a ﬁxed x and`varying according to some probability distribution, the HS output .¹xºremains random. The input parameters of vector xare treated as random, modeled by known prob- ability distributions and grouped into a random vector ^=f-1- "g.^is 3 DRAFT characterized by its joint distribution with the PDF denoted by 5^. Furthermore, we assume that -8’s are mutually independent, and thus the joint PDF is the product of marginalPDFs, i.e. 5^¹xº=Î" 8=15-8¹G8ºwith5-8beingthe marginalPDF ofthe 8-th variable. For a given set of input parameters x, the QoI.¹xºis a random variable charac- terized by an unknown conditional probability distribution. Therefore, representing the stochastic behavior of a HS consists in estimating the response distribution for any parameters x2D ^. However,onesimulationfor xdoesnotprovidethewholeproba- bilitydistributionbutratherasinglerealizationof .¹xº. Hence,itisusuallynecessary to repeatedly conduct experiments for the same x(called replications) to have enough insight into the resulting hybrid model response probability distribution. This quickly becomes intractable when the number of x’s to be investigated increases. To alleviate the burden, surrogate models can be constructed to emulate the stochastic behavior of a HS. Once a surrogate model is constructed, we can perform further analysis of the hybrid model response at a low cost, namelythe GSA. The simplest surrogatemodel ofa stochastic HS involves additive Gaussian noise: .s¹xº=¹xº¸/ /N¹ 0f2º (2) To build such a surrogate, one needs to estimate the mean function and the noise variancef2. In this case, PCE [14, 18] and Gaussian processes [19] with a regression setup can be directly applied. However, Eq. (2) can be rather restrictive. To cover a wider group of problems, we choose to use the recently developed statistical model called the generalized lambdamodel [11,12]. 2.1 Generalized lambda models A generalized lambda model (GLAM) assumes that the probability distribution of hy- bridmodelresponseQoI .¹xºcanbeapproximatedbyageneralizedlambdadistribution (GLD). The latter is a highly ﬂexible four-parameter distribution family, which is able to approximate many common distributions such as normal, lognormal, uniform and extreme valuedistributions[20]. A GLDis deﬁnedby its quantilefunction &¹D;,º=_1¸1 _2D_31 _3¹1Dº_41 _4  (3) whereD2»01¼and,=f_;:;=1 4garethefourdistributionparameters. More precisely,_1is the location parameter, _2is the scaling parameter, and _3and_4are the shape parameters. To have valid quantile functions, _2should be positive. From Eq. (3), we can derive the associated PDF: 5.¹H;,º=_2 D_31¸¹1Dº_411»01¼¹DºwithD=&1¹H;,º (4) where 1»01¼istheindicatorfunction. Fromtheaboveequation,itisclearthatevaluating the PDFfor aparticular Hrequiressolvingnumericallythe equation D=&1¹H;,º. 4 DRAFT Under this setup, varying xwill lead to .¹xºfollowing a GLD with diﬀerent distribution parameters ,. In other words, _;’s are functions of x, which allows us to express the QoI as .¹xºGLD¹_1¹xº_2¹xº_3¹xº_4¹xºº (5) Recallthattheinputparameters xaremodelledasindependentrandomvariables ^with jointPDF5^¹xº=Î" 8=15-8¹G8º. Underappropriateassumptions[21],eachcomponent of,¹xºadmits a polynomial chaos (PC)representation _;¹xº_PC B¹x;cº=Õ "2A ;2;"k"¹xº ;=134 _2¹xº_PC 2¹x;cº=exp Õ "2A 222"k"¹xº! (6) wherefk":"2N"gis a basis of multivariate polynomials that are mutually orthog- onal with respect to the probability measure of ^,"is a multi-index that identiﬁes the polynomial degree in each of the input variables, A;N"is a truncated set deﬁning a ﬁnite set of basis functions for _;¹xºandc=f21"2 ;"gdenotes the associated coeﬃcients (see [11] for details). Note that the polynomial chaos expansion for _2¹xº is built on the logarithmic transform so as to ensure that _PC 2¹xºis always positive. Combining Eq. (5) with Eq. (6), we deﬁne the generalized lambdasurrogate model .GLAM¹xºGLD _PC 1¹x;cº_PC 2¹x;cº_PC 3¹x;cº_PC 4¹x;cº (7) TobuildaGLAM,weneedtodeterminethetheassociatedcoeﬃcients c. Toavoid the need for replications, which may result in a large number of experiments, we use themethoddevelopedbyZhuandSudret[12]. Weﬁrstgenerateasetofrealizationsof the input random vector ^= x¹1º x¹#º , called the experimental design (ED). For each point of the ED, we conduct a HS and collect the corresponding QoI into _= y¹1º y¹#º . Note that each HS may correspond to a diﬀerent realization of the latent variable `, which does not need to be explicitly known in the analysis. In a second step, we estimate cby maximizing the conditional likelihood, i.e. minimizing the negative log-likelihood, that is ˆc=arg min cL¹cº=arg min c#Õ 8=1log 5GLD H¹8º;,PC x¹8º;c (8) where5GLDis the PDF of the GLD deﬁned in Eq. (4). To solve the optimization problem, it is necessary to determine the support of c, which is equivalent to ﬁnding the truncation setA;for each_PC ;. To this end, we plug the hybrid-LAR algorithm [15]intothemodiﬁedfeasiblegeneralizedleast-squaresframework[12]. Thelatterﬁts the mean and the variance function in an alternative way. The basis functions selected for these two functions are then used to represent _PC 1¹xºand_PC 2¹xº, respectively. As _3and_4mainly control the PDF shape of a GLD, which is expected not to change 5 DRAFT muchwhen xischanged,wecanpickpolynomialswithlowdegree,namely0or1,for _PC 3¹xºand_PC 4¹xº. After specifying the basis functions, we solve Eq. (4) to build the associated GLAM. 2.2 Sobol’ sensitivity indices Variance-based sensitivity analysis has been extensively studied and successfully de- veloped in the context of deterministic models [9, 16]. For a random vector ^with independent components, any deterministic mapping .=MB¹^ºwith Var».¼¸1 can bedecomposed as[22] MB¹^º=M0¸"Õ 8=1M8¹-8º¸Õ 8 9M8 9¹-8-9º¸¸M 1?¹G1G "º =M0¸Õ uf1"g u<;Mu¹xuº(9) whereM0is a constant, andMu¹xuºis a function of the subset of input variables xu speciﬁed by u<;uf1?g. This decomposition is unique, and the elementary functions aredeﬁned byconditional expectations Mu¹xuºdef=Õ vu¹1ºjujjvjE».j^v=xv¼ (10) wherejujgives the cardinality of u. Following this deﬁnition, M0=E».¼is the expected value of .. Moreover, the various terms Muare orthogonal with each other w.r.t. theinnerproductinducedbytheinputPDF.Thuswecandecomposethevariance of.as: Var».¼=E ¹.M 0º2 =Õ uf1"g u<;Var»Mu¹^uº¼ (11) Additionally, the deﬁnition in Eq. (10) allows for calculating the variance of Mu¹^uºby +udef=Var»Mu¹^uº¼=Õ vu¹1ºjujjvjVar»E».j^v¼¼ (12) The Sobol’index (uis deﬁnedas theratio of +uto thetotal variance Var».¼ (udef=+u Var».¼ (13) Forjuj=1, we obtain the ﬁrst-order Sobol’ indices f(8:8=1?g, which represent the main eﬀect of each input variable. Higher-order indices quantify the interactive eﬀect within a given group of input variables. The total Sobol’ index ()8 account forall theeﬀect relatedto -8 6 DRAFT ()8def=Õ u38(u (14) Duetotherandomnatureofstochasticsimulators,decompositionsimilartoEq.(9) is generally impossible. Therefore, it is necessary to represent a stochastic model by a deterministic function to obtain the associated Sobol’ indices. Based on the choice of thedeterministicrepresentation,wecanhavediﬀerentextensionsoftheSobol’indices [17]. The most straightforward way is to include the latent variables within the input variables [23]. This leads to the underlying (yet unknown in practice) deterministic modelMBdeﬁned in Eq. (1). Decomposing this function, we have .=MB¹^`º=M0¸Õ uf1?g u<;Mu¹^uº¸M `¹`º¸M ^`¹^`º(15) Asthedeﬁnitionof MuisthesameasEq.(10),theequation(12)for +ustillholds. Thisimpliesthat (ucanbedeterminedbythestatisticaldependencebetween .and^u. Thedeﬁnitionofthetotalindex ()8requiresincludingtheinteractiveeﬀectbetween -8 and the latent variables `. However, interactions with `cannot be determined by the responsedistributionbutratherdependontheprecisedatagenerationprocess(i.e. how `is present in thefunction MB). Since the latent variables `are generally impossible to characterize and control in a real experiment, the total Sobol’ indices cannot be assessed. Alternatively to Eq. (15), certain summary quantities of the response random vari- able.¹xºcanbeemployedasadeterministicrepresentationof .¹xº. Thisisparticularly helpful when the selected summary quantity itself is of interest. Typical quantities are mean<¹xº=E».¹xº¼, varianceE¹xº=Var».¹xº¼[23], andU-quantiles@U¹xº[24]. As these functions are well-deﬁned as deterministic functions of x(since the eﬀect of the latent variables `has been marginalized), the associated Sobol’ indices follow directly from Eq.(9). A generalized lambda surrogate model emulates the response distribution of a stochastic model, which fully captures the statistical dependence between the input variables ^andtheQoI.. Therefore,suchasurrogateallowsforevaluatingbothtypes of Sobol’ indices mentioned above. More precisely, we can apply either Monte Carlo simulations or polynomial chaos expansions to the easy-to-evaluate emulator (see [17] for more details). 3 Experimental illustration of the proposed GSA frame- work 3.1 Stochastic hybrid model The proposed GSA framework is illustrated using a stochastic 3-DoFs hybrid model subjected to both thermal and mechanical loading, illustrated in Figure 1. As can be 7 DRAFT appreciated from Figure 1a, the hybrid model consists of a simply-supported beam provided with rotational elastic restraints. In this regard, D1andD2indicate the two rotational DoFs, while D3is the axial DoF. Figure 1b describes the substructuring of the hybrid model into PS and NS. The NS comprises the two elastic rotational restraints, the axial spring and the linear dashpot whereas the PS coincides with the beam element. Speciﬁcally, the axial spring is characterized by a constant stiﬀness of 3=8100103N/m. Twolumpedrotationalmassesaredeﬁnedby 1= 2=10kgm while the lumped translational mass is deﬁned by "3=5000kg. The linear dash-pot is characterized by 3=1129103Ns/m. The PS consists of an aluminium plate of020002m rectangular cross-section and length !=047m. Accordingly, the cross-section of the plate is characterized by an area =4104m2and a moment ofinertia =66671012m4. TheYoung’smodulus,densityandthermalexpansion coeﬃcient of aluminum are =695GPa,d=2700kg/m3, andU=23106°C-1, respectively. SinceHSareconductedin pseudodynamic modeusingatestingtimescale equalto50,thePSmass(rotationalandtranslational)doesnotcontributetothehybrid model inertia. For the sake of clarity, all equations and plots in the following refer to simulation time, which is virtual and 50 times slower than the wall-clock time. K3M3F(t) θ(t)C3E,I,A,L,ρ,α,K2J2 ,K1J1 (a) ,K2J2u2 u3 u3u2 u1 u1E,I,A,L,ρ,α NS NSPS+ +K3M3 C3,K1J1 (b) Figure 1. Prototype structure (a) and its hybridmodel (b). Mechanical loading is supplied as a bending moment history ¹Cºapplied to the rightrotationalDoF D2whilethermalloading,appliedbyaheatinglamp,isdeﬁnedby a ramp & hold temperature history \¹Cº. The expressions of both read: ¹Cº=( <0Gsin¹c¹CC0º )º,C0CC0¸)2 0, elsewhere(16) 8 DRAFT Parameter Description Lower Bound Upper Bound Units 1 Rotational stiﬀness left spring 11058 110580Nm/rad 2Rotational stiﬀness right spring 11058 110580Nm/rad <0G Bending moment pulse peak 4000 5000 Nm ) Bending moment pulse period 050 200 sec \<0G Temperature plateau 10000 13000 °C Table 1.Input parameters of the hybrid model. \¹Cº=( ¤\C,\¹Cº\<0G \<0G, elsewhere(17) where<0Gand)are peak value and period of the half-side bending moment pulse applied to DoF D2with a time shift C0=01s;¤\and\<0Gare temperature rate and plateau characterizing the temperature history imposed to the PS. For the sake of this example, Figure 2a depicts the bending moment history computed for <0G=45Nm, )=1s. Similarly,Figure2bdepictsthetemperaturehistorycomputedfor ¤\=215°C/s and\<0G=120 °C. 0 1 2 3 4 5 6 70515253545 (a) 0 1 2 3 4 5 6 720406080100120 (b) Figure 2. Bending moment history for <0G=45Nm,)=1s (a); Temperature time history for¤\=215°C/sand\<0G=120 °C(b). ConsistentwiththemotivationsunderlyingthedevelopmentoftheGSAframework, the stiﬀness of the two elastic rotational springs, which play the role of boundary conditionstothePS,aswellastheloadingparameters,areselectedasinputparameters for the surrogate modelling phase. In line with the procedure described in Section 2, theinputparametersaredescribedbyindependentuniformdistributions,whosebounds are summarized inTable 1. It is important to remark that, in order to reduce the experimental eﬀort required to validate the proposed framework, the hybrid model and loading excitation were 9 DRAFT designed such that the PS always remained in the linear response regime. As a result, HS were conducted using a single aluminum plate. The response QoI selected for the GSA corresponds to the maximum absolute out-of-plane deﬂection of the tested aluminum plate, which is denoted as D!<0G. 3.2 Hybrid simulation setup The 3-DoFs HS test rig used to conduct HS is a stiﬀ loading frame equipped with four electro-mechanicalactuatorsandaninfrared(IR)lampmoduleinterfacedtoanINDEL real-time computer [25]. The 3-DoFs HS test rig is designed to test plate specimens with an approximate footprint of 200500mm and thickness varying between 1and 3mm. Figure 3 illustrates the architecture of the HS setup, including a close-up view of the plate specimen accommodation. Two axonometric views of the 3-DoF test rig, consisting of the main hardware componentsare shownin Figure 4. The moving parts of the test rig are colored in yellow, the plate specimen in brown and the ﬁxed parts in gray. The latter are ﬁxed to a reaction frame. In order to impose theD1andD2rotations, two rack-pinion systems (10) are installed along the vertical actuatorH1andH2(1). The rack-pinion systems aim at transforming the commanded displacements from the actuators to rotational DoFs, applied to the short edges of the plate specimen (6) through aluminium clamps (3). The two horizontal actuators G1 andG2control the position of the moving frame mounted on proﬁled rail guides (4) and corresponding to the axial DoF D3of the plate specimen (6). A linear variable diﬀerential transformer (LVDT) measures the out-of-plane deﬂection at the mid-span oftheplatespecimen(labeled D!inFigure3). ATypeKthermocoupleinstalledatthe center of the plate provides the feedback signal for the control of the IR lamp, which imposes the temperature history \¹Cº. The GINLink bus connects actuator servo-drivers INDEL SAC4, IR lamp and all DAQmodulestothereal-timecomputerINDELSAM4,whichexecutestheHSsoftware. The latter is developed in MATLAB/SIMULINK, compiled, and downloaded to the INDELSAM4fromtheHost-PC.Ateachsimulationtimestep,theHSsoftwareimposes displacements D1,D2andD3to the plate specimen, the PS, reads the corresponding restoring forces A1,A2andA3measured using force transducers, and solves the coupled equationofmotionofthehybridmodel. Also,theHSsoftwaregeneratesthetemperature commandfortheIRlamp. Acomprehensivedescriptionofthetimeintegrationscheme used for HS is reportedin [3]. ForcesweremanuallysettozerobeforestartingHS.Anelectricfancoolsdownthe PS at the end of each test. Room temperature was quite stable and equal to 30°C for the entire testing campaign. 4 Results and discussion TheresponseofthehybridmodeldescribedinSection3.1wasevaluatedusingtheHS setup described in Section 3.2 on 200 samples of the input parameter vector generated using latin hypercube sampling [26]. The resulting ED ^_was used for computing surrogate models. In this regard, Figure 5 reports out-of-plane and axial displacement 10 DRAFT u1 u2u3uL θ(t)INDEL SAM4 Real-time computer Execute the HS software INDEL SAC4 Actuator servo-drivers Impose displacements to the PS INDEL COP/ADA Data acquisition modules Read restoring forces from the PSINDEL COP IR lamp driver Impose thermal load to the PSHost PC GINLink bus,u1r1 ,u2r2,u3r3 PS, and u1u2 u3 , and r1r2 r3θ(t)Figure 3. Architecture of the 3-DoFs HS test rig. Figure 4. Axonometric views of the 3-DoFs HS test rig with main components: (1) vertical actuators, (2) horizontal actuators, (3) installation clamps, (4) moving frame, (5) proﬁled rail guides, (6) plate specimen, (7) hinges, (8) vertical actuator load cells, (9) horizontal actuator load cells and (10) rack-pinion systems. The moving parts are colored in yellow, the plate specimen is brown, while those parts ﬁxed to the reaction frame are gray. The latter is omitted in this ﬁgure. 11 DRAFT historiesobtainedviaHSforasinglesampleofinputparameters,where D!<0G,D3<0G, D!0andD30scalar quantities are highlighted. Speciﬁcally, D!<0Gcorresponds to the QoI (see Eq.(18)) and D3<0Gto the absolute maximum axial displacement while D!0 andD30indicate the initial position of the out-of-plane displacement and axial axes respectively, relative to the value measured during the ﬁrst HS. Figure 6 describes the evolution of D!0andD30scalar quantities over the entire experimental campaign. 4.1 Drift observed in measurement data From Figure 6 it is clear that both D!0andD30quantities have a constant drift, which results in a total accumulated out-of-plane and axial displacements of 30mm and 03mm, respectively. Such a drift can be reasonably ascribed to cumulative slippage of plate ﬁxtures produced by heating/cooling cycles. This drift occurs regardless of the type of analysis that the HS setup was used for, and hence it should be removed from the acquired raw data before any further post-processing. Accordingly, prior to the calculation of surrogate models, the eﬀect of drift on D!<0Gwas eliminated via linear detrending with respect to D!0. The detrended QoI is referred to as ˆD!<0G and compared to original values in Figure 7. Notably, D!0is independent from the parameters of the hybrid model since the initial position of the PS was set by zeroing actuator forces. Consistent with the notation introduced in Section 2, surrogate modeling was per- formed considering the following input parameter vector and response QoI, ^=f 1 2<0G)\ <0Gg _=fˆD!<0Gg(18) 0 1 2 3 4 5 6 7-10123456 (a) 0 1 2 3 4 5 6 7-0.100.10.20.3 (b) Figure 5. Time history response of the hybrid model with 1=224454Nm/rad, 2=118235Nm/rad,<0G=48Nm,)=0618sec and\<0G=11673°C: (a) out-of-plane displacement; (b) axial displacement. 12 DRAFT 0 50 100 150 200-101234(a) 0 50 100 150 200-0.100.10.20.30.4 (b) Figure 6. Drift in the hybrid modelresponse: (a) D!; (b)D3 50 100 150 20033.544.555.566.5 (a) 0 50 100 150 20033.544.555.566.5 (b) Figure 7. EﬀectofdetrendingontheQoI:(a)originalvalues( D!<0G);(b)valuesafter detrending ( ˆD!<0G). Dashed lines indicate a linear trend of data. 4.2 Global sensitivity analysis framework results A GLAM of the hybrid model dynamic response QoI was computed as explained in Section 2. In this study, we set the candidate degrees up to 5 for _PC 1and 3 for_PC 2. In order to validate the GLAM, 10 repeated HS were performed for two validation ED points, namely 58 and 157, associated with diﬀerent regions of the input parameter space and characterized by appreciably diﬀerent QoI values. For each validation ED point, Figure 8 compares the GLAM prediction to the empirical distribution of the 10 related repetitions. It is observed that the GLAM correctly captures the empirical distribution of the QoI for both points. It is interesting to note that the computed GLAM model converged to zero-order polynomials for _PC 2, and that the two PDFs of Figure 8 are similar in shape, thus suggesting an homoscedastic stochasticity of the hybrid model, which was further veriﬁed using a PCE surrogate model. Speciﬁcally, a residual analysis was performed on the diﬀerence between the measured QoI and its PCE. 13 DRAFT As highlighted by Torre and co-workers [27], in presence of noisy data, PCE is a powerful denoiser,whichnaturallyprovidesasurrogateoftheaveragemodelresponse E/».jx¼. In this regard, the Tukey-Anscombe plot [28] of Figure 9a compares the PCE output to the corresponding residual for each sample of the ED. The Q-Q plot of Figure9bcomparestheempiricalquantilesoftheresidualsnormalizedtounitstandard deviation to the theoretical values of a standard normal distribution N¹01º. The zero-average uniform scattering of residuals highlighted by the Tukey-Anscombe plot and the fairly good agreement between empirical and theoretical quantiles highlighted by the Q-Q plot conﬁrms that the hybrid model response was aﬀected by a Gaussian homoscedasticadditive noise. 3.5 4 4.5 5 5.5 601234 (a) 5 5.5 6 6.501234 (b) Figure 8. PDF of ˆD!<0Gpredicted by the GLAM versus empirical distributions for EDpoints: (a)58,and(b)157. Crossmarkersdenotethe10relatedrepetitionsforeach point. 4.5 5 5.5 6-0.4-0.3-0.2-0.100.10.20.30.4 (a) -3 -2 -1 0 1 2 3-3-2-10123 (b) Figure 9. AnalysisofQoIresidualswithrespecttoPCE:(a)Tukey-Anscombeplot;(b) Q-Q plot. As reported in Section 2.2, only ﬁrst- and higher-order Sobol’ indices but not total Sobol’ indices can be obtained from the GLAM of the QoI. Instead, ﬁrst and total 14 DRAFT Sobol’indicescanbecomputedfortheQoIquantiles. Accordingly,Figure10provides ﬁrst and total Sobol’ indices of the 5, 50 and 95 % quantiles of ˆD!<0G. The results of the GSA indicate that the temperature plateau value \<0Gis the most sensitive input parameter for the selected QoI. Additionally, the equal Sobol’ indices values for each quantile unveils thehomoscedasticresponse ofthe stochastic surrogate. (a) (b) Figure 10. Sobol’ indices of 5, 50 and 95 % quantiles of ˆD!<0G: (a) First order and (b) Total. The development and implementation of the surrogate modelling, as well as the GSA, was performed using the UQLab software framework developed by the Chair of Risk, Safety and Uncertainty Quantiﬁcation in ETH Zurich [29]. 5 Conclusions This paper described a framework for global sensitivity analysis of stochastic hybrid models. A generalized lambda surrogate modeling technique recently developed by some of the authors is used to compute Sobol’ sensitivity indices for the quantiles of a response quantity of interest. The idea of using surrogate modeling to enable globalsensitivitystudieswithfewexpensive-to-evaluatehybridsimulationswasalready presentedinapreviousworkoftheauthors. Thenoveltyofthisworkliesintheextension ofglobalsensitivityanalysistothecaseofstochastichybridmodels, tocoverthemore realisticsituationwherethehybridmodelresponsefortwonominallyidenticalphysical substructures is not repeatable. The eﬀectiveness of the proposed framework is demonstrated in an experimental applicationconsistingofahybridmodelwithﬁveparametersandsubjectedtomechan- ical and thermal loading. A great advantage of the proposed framework is that the generalized lambda surrogate model does not require repeated evaluations of the same sample. The results of the demonstration study highlight that the stochasticity of the particularhybridmodelunderconsiderationishomoscedasticwithrespecttothehybrid model parameters. Accordingly, both ﬁrst-order and total Sobol’ sensitivity indices of 5, 50, and 95 % quantiles are almost identical. Moreover, the temperature plateau 15 DRAFT value of the thermal loading is the most sensitive parameter for the selected response quantity of interest. The outcome of the experiment demonstrates the eﬀectiveness of the proposed GSA framework in revealing theinner workingsof thehybrid model. Future research will address the issue of adaptive sampling of the parameter space ofthehybridmodeltominimizetheexperimentalcostnecessarytocomputeanaccurate surrogatemodel forglobal sensitivityanalysis. Data Availability Statement All data and code that support the ﬁndings of this study are available from the corre- sponding author upon reasonable request. Acknowledgments This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 764547. Thesoleresponsibilityofthispublicationlieswiththeauthor(s). TheEuropean Union is not responsible for any use that may be made of the information contained herein. The realization of the test rig work was funded by the Swiss Secretariat of Education Research and Innovation (SERI) - Swiss Space Oﬃce (SSO) [THERMICS MdP2016 Project (Thermo-Mechanical Virtualization of Hybrid Flax/Carbon Fiber Composite for Spacecraft Structures),grantnumber REF-1131-61001].