Hybrid simulation is a tool for discovering the inner workings of a tested substructure beyond the linear regime. Hybrid simulation is conducted to reproduce the response of a prototype in scaled or real time using a hybrid model that combines physical and numerical substructures interacting with each other in a feedback loop. As a result, the tested substructure interacts with a realistic assembly subjected to a credible loading scenario. The obtained low-quantity-high-value experimental data is used to conceive and calibrate computational models for nonlinear structural analysis in the current practice. Instead, this paper extends the scope of hybrid simulation to constructing a safe/failure state classiﬁer for the tested substructure by adaptively designing a sequence of parametrized hybrid simulations. Such a classiﬁer is intended to compute the state of any physical-substructure-like component within system-level numerical simulations. The proposed procedure is experimentally validated for a three-degrees-of-freedom hybrid model subjected to Euler buckling. Keywords: Hybrid simulation; meta-modeling; Kriging; active learning; classiﬁer; buckling. 1 Introduction Hybrid Simulation (HS) is a dynamic simulation paradigm that merges physical experiments and computational models. Inmechanicalandelectricalengineering, suchanapproachisknownashardware-in-the-loopsimulation. The report by Schellenberg and co-authors [1] provides a comprehensive overview of state-of-art HS methods and algorithms. In detail, HS is conducted using a hybrid model, which combines physical and numerical substructures (NS and PS, respectively) that interact with each other in a real-time feedback loop, to simulate the time history response of a prototype structure subjected to a realistic excitation. The PS is tested in the laboratory using servo-controlled actuators equipped with force and displacement transducers. The NS replaces those substructures that can be reliably simulated (e.g., masses or components whose response remains in the linear regime) or would exceed the testing capacities of the experimental facility. A time integration algorithm coordinates the HS ensuring interface force balance and displacement compatibility between NS and PS. As a result, HS reproduces the time history response of the prototype structure subjected to a realistic excitation. When the PS response is rate-independent, it is good practice to perform HS with an extended time scale from 50-200 times slower than real-time in order to improve control accuracy. Time scaling enables full-scale experimentation without the need for large oil-ﬂow demand. In earthquake engineering, HS is used for testing large structures such as bridges (e.g., [2, 3]) and buildings (e.g., [4, 5]). Similarly, testing of ﬂoating structures with a sizeable scaling factor in hydrodynamic laboratories is challenging due to the extent of mooring line footprints, which can range from two to four times the water depth. For this reason, HS has been recently proposed to truncate mooring lines whose missing parts are simulated as NS (e.g., [6, 7]). Hybrid ﬁre testing Corresponding Author: Dr. Giuseppe Abbiati, abbiati@eng.au.dk , ORCID: 0000-0002-5048-8505 . 1 DRAFToriginated in response to limitations of standard ﬁre tests to account for a realistic internal force redistribution caused by thermal expansion or partial collapse of tested assemblies. (e.g., [8, 9]). Regardlessofthespeciﬁcdomainofapplication, HSisatoolfordiscoveringtheinnerworkingsofastructural system in the nonlinear regime with few signiﬁcant experiments. The information carried by low-quantity-high- value experimental data produced via HS is used to conceive and calibrate computational models for structural analysis [10]. In this context, the set of loading scenarios explored via HS corresponds to the domain of validity of developed computational models. The usability of those computational models outside such domain is tightly related to the generalization error concept, which indicates the prediction errors for unobserved combinations of loading histories and boundary conditions [11, 12]. The damage pattern characterizing civil engineering structures typically involves debonding of heterogeneous materials (e.g., concrete and steel), crack opening (e.g., steel beam-column connections, concrete members or masonry elements), buckling and yielding (e.g., steel rebars or members). To reproduce such damage features, highly nonlinear expensive-to-evaluate computational models are necessary. The computational cost of such models is seldom aﬀordable beyond the scale of a single structural component (e.g., beam-to-column joint). Also, usually, little physical justiﬁcation supports some of the parameters, which are tuned to match available experimental observations. Instead of supporting the development of a computational model of the PS, this paper proposes to expand the scope of HS to training a classiﬁer to determine the failedornon-failed state of the PS. The parameter space deﬁning loading and boundary conditions of the hybrid model is the support of the classiﬁer, which can be reasonably assumed as the domain of validity for numerical predictions. To greedily construct a safe/failure domain classiﬁer at the lowest experimental cost possible, an initial random sampling of the parameter space is adaptively enriched using kriging meta-models [13] and active learning [14]. The proposed method, namely active-learning-kriging hybrid-simulation (AK-HS), relies on an adaptation of the active-learning kriging Monte Carlo simulation algorithm from the ﬁeld of structural reliability [15]. The idea of combining the active-learning kriging Monte Carlo simulation algorithm and HS was originally proposed in [16] and applied in [17] to compute the probability of failure of a hybrid model with uncertain parameters. However, the computed failure probability has no utility outside the context of the examined prototype structure. Besides providing a comprehensive experimental validation of the AK-HS method, the main contribution of this paper lies in the direct link established between structural testing, performed using HS, and the computation of a safe/failure state classiﬁer for the PS. Such a classiﬁer is intended to determine the state of any component of the same type of the tested PS within a system-level, rather than a component-level, numerical simulation. In doing that, one must verify that, for every PS-like component, boundary conditions and loading must stay within the range explored via HS. The eﬀectiveness of the AK-HS method is illustrated for a 3-degrees-of-freedom (3-DoFs) prototype struc- ture, which consists on an elastically restrained beam subjected to constant axial loading. Failure is associated with Euler buckling, which may occur or not depending on the rotational stiﬀness of the restraints. Experi- mental results are validated against the analytical model proposed by Newmark [18]. The paper is organized as follows. Section 2 describes the AK-HS method. Section 3 describes the experi- mental validation of the method, the results of which are discussed in Section 4. Finally, conclusions and future outlook are given in Section 5. 2 Introducing active learning into hybrid simulation Let us denote with X=fX1;:::;XMganM-dimensional vector of input variables to parametrize the NS, the PS and the loading excitation (or a subset of these) of the hybrid model of the prototype structure, and with Ya generic response Quantity of Interest (QoI) of the hybrid model. A mapping Mbetween input variables and a single QoI reads, X2D XRM7!Y=M(X)R (1) It can be argued that the inherent variability of the hybrid model response obtained with nominally identical PSs, as well as measurement noise, requires a stochastic mapping between input variables and QoIs. However, the scope of this paper is limited to the case of PS with almost deterministic behavior. Moreover, relevant studies testify that measurement noise of standard structural testing equipment has a negligible eﬀect on the prototype structure response evaluated via HS with modern hardware [19, 20]. Therefore, Mis assumed to be deterministic. 2 DRAFTThe failure domain Df2D Xis deﬁned as the region of the input parameter space where the hybrid model response QoI is greater or equal than a given threshold value yadm, x2Df:M(x)yadm (2) ComputingDfmeans solving a classiﬁcation problem where every sample xis categorized either as failed or non-failed. However, since it entails testing of the PS, M(x)can be assimilated to an expensive-to-evaluate black-box model. In order to make such a classiﬁcation problem tractable, kriging meta-modeling is introduced to surrogate the hybrid model response, Y^Y=^M(X) (3) The kriging meta-model adaptively drives the choice of the next sample xto evaluate via HS so that the convergence of the limit state surface, which separates failure and non-failure domains, requires a minimal number of experiments. The choice of samples to be evaluated can be considered as a dilemma between the exploration and the exploitation of the input parameter space. Active learning [21] addresses this problem by providing algorithms for selecting the most informative sample against the solution of a speciﬁc supervised learning problem, which, in this case, corresponds to the identiﬁcation of the boundaries of the failure domain. After recalling the basics of kriging surrogate modeling, the AK-HS procedure is outlined. 2.1 Basics of kriging meta-modeling Kriging is a surrogate modeling technique that considers the computational model to be a realization of a Gaussian process, ^M(x) = Tf(x) +2Z(x;!) (4) where f(x) = [f1(x);:::;fp(x)]are regression functions, is a vector of coeﬃcients, which compose the mean value of a Gaussian process. 2is the corresponding variance. Z(x;!)is a zero-mean, unit-variance, stationary Gaussian process, which is characterized by an autocorrelation function R(jxx0j;)and its hyper-parameters . The kriging model is trained with a set of realizations X=fX(i);i= 1;:::;Ngand the corresponding responses of the computational model Y=fM X(i) ;i= 1;:::;Ng, which together form the so-called Experimental Design (ED) fX;Yg. Kriging parameters are obtained by generalized least-squares solution: () = FTR1F1FTR1Y (5) 2 y() =1 N(YF )TR1(YF ) (6) where Rij=R jX(i)X(j)j; is the correlation matrix and Fil=fl X(i) . In practice the correlation hyper-parameters are unknown and their values shall be inferred by e.g. maximum likelihood estimation. Having determined the kriging parameters, the prediction value of the computational model at a point x2D X is a Gaussian variable with the following mean value and variance: ^Y(x) = Tf(x) +r(x)TR1(YF ) (7) ^Y(x) =2 y 1r(x)TR1r(x) +u(x)T FTR1F1u(x) (8) where ri(x) =R jxX(j)j; andu(x) =FTR1r(x)f(x). 3 DRAFT2.2 Adaptive design of hybrid simulations The purpose of the kriging meta-model is to accurately locate the limit state surface, which separates failed and non-failed states of the PS, regardless the probability measure assigned to the input parameter space. Therefore, all input parameters in Xare assumed to be statistically independent and uniformly distributed. Corresponding ranges must comply with displacement and force capacity of the testing facility. The kriging meta-model obtained with the procedure stated in Section 2.1, is capable of exactly reproducing the points in the ED set. Indeed, the kriging meta-model is an exact interpolator, meaning that ^Y X(i) = M X(i) ;i= 1;:::;N, exactly. This, however, does not mean that such a meta-model accurately reproduces the boundary of the failure domain. Adaptively enriching the ED set in a guided way can improve the accuracy of the trained kriging meta-model in predicting the failure domain of the hybrid model. The proposed learning rule is an adaptation of the adaptive kriging algorithm proposed by Echard et al. [15] for failure probability estimation problems and later extended to quantile estimation problems in Schöbi et al. [22]. The main steps of the AK-HS procedure are listed here: 1. Generate an initial ED by sampling Xwith a space ﬁlling technique (e.g., Sobol’ sequence or LHS) and evaluate the corresponding hybrid model response QoIs Y(i)=^M X(i) . 2. Train a kriging meta-model ^Mbased on the initial ED fX;Ygas described in (5) and (5). 3. Generate a large set of samples of the input parameter space S=fx1;:::;xj;:::;xMgand predict the response values of ^M, i.e.,^Y(x)and^Y(x), as described in (7) and (8). 4. Enrich the ED by the taking the sample x2Sthat maximizes the probability of misclassiﬁcation, x=arg max xj2S j^Y(xj)yadmj ^Y(xj) (9) where  ()is the cumulative Gaussian distribution function. As can be argued from (9), best candidate samples lie either where the variance of the meta-model is high (exploration) or close to the boundary of the failure domain (exploitation). In both cases, the probability of misclassiﬁcation tends to its upper bound, which is equal to 0:5. 5. Evaluate the corresponding QoI y=M(x)via HS and add the pair fx;ygto the ED. The procedure loops between Step #2 and Step #5 until a maximum allowed number of experiments Nmax is reached. ThestoppingcriterionoriginallyproposedinthepaperofEchardetal. [15], wasdeﬁnedbasedonathreshold value for the ratio between conﬁdence interval and average of the failure probability estimate associated with the failure domainDf. However, as highlighted in the discussion of the results of the validation example (Section 4), the number of experiments necessary to meet such a convergence criterion are not reasonably aﬀordable. The safe/failure domain classiﬁer is obtained as by-product of the last computed kriging meta-model, 8 < :p(Z=Fjx) = 1^Y(x)yadm ^Y(x) p(Z=Sjx) = ^Y(x)yadm ^Y(x) (10) whereFandSindicatefailureandsafestates, respectively. 3 Experimental validation of the method 3.1 Prototype structure In order to validate the proposed procedure, a benchmark experiment was conceived. It consists of a simply- supported beam restrained with linear elastic rotational springs at both ends and subjected to a axial load applied with a linear ramp. This type of structure is common in practice, as most beam-columns are neither truly pinned-pinned nor ﬁxed-ﬁxed. The prototype structure is depicted in Figure 1. 4 DRAFT Figure 1: Prototype structure: a) undeformed conﬁguration; b) buckling deformation for the limit case K1;K2<<EI; c) buckling deformation for the limit case K1;K2>>EI. whereK1andK2deﬁne the stiﬀness of left and right rotational restraints. Polar moment of inertia J1= J2= 90000 kgmm2and massM= 20kg provide inertia to the three DoFs of the system, which correspond to rotations of both beam ends and translation of the right end. In addition, the horizontal translation of the right support activates a linear dashpot characterized by viscous coeﬃcient C= 3000Ns/mm. The simply-supported beam consists of an aluminum plate with a nominal 200500mm footprint and 2mm thickness. However, the simply-supported length of the beam Lwas actually shortened to 460mm, due to the end clamps of the testing rig. Accordingly, the cross-section is characterized by area A= 400mm2and a moment of inertia I= 133:33mm4. An elastic modulus E= 69500 MPa a density = 2700kg/m3and a Poisson ration = 0:32 characterized the material properties of the plate. Furthermore, the eﬀective buckling length of the beam is sensitive to the stiﬀness of each elastic rotational restraint. In the limit case of K1;K2<< EI, the eﬀective lengthL0of the beam tends to L= 460mm i.e., pinned-pinned conﬁguration. The opposite case corresponds toK1;K2>> EI, the eﬀective length L0of the beam tends to L=2 = 230 mm i.e., ﬁxed-ﬁxed conﬁguration. The following approximate equation, developed by Newmark, is used to interpolate the buckling load between these two extreme conﬁgurations [18], Pcr=EI L2(0:4 +1) (0:4 +2) (0:2 +1) (0:2 +2) (11) where1=EI K1Land2=EI K2Laccount for the relative bending stiﬀness of the beam compared to the rotational restraints. The lower bound of the buckling load Pcr;min = 432:2N corresponds to the pin-pin conﬁguration while the upper bound Pcr;max = 1729 N to the ﬁxed-ﬁxed conﬁguration. Figure 2 depicts the dynamic substructuring scheme adopted for HS. Figure 2: Dynamic substructuring scheme adopted in HS. With regard to Figure 2, the PS coincides with the 3-DoFs beam, which is tested in the laboratory, while the NS comprises the remaining part of the prototype structure. The rotational and translational DoFs are denoted byu1,u2andu3, respectively. The out-of-plane displacement of the beam at its center point, denoted by u4, 5 DRAFTis activated by buckling and measured with a laser displacement sensor. It is important to stress that this out-of-plane displacement is not included in the equation of motion of the hybrid model. 3.2 Hybrid simulation setup The 3-DoFs HS test rig used to conduct the HSs is a stiﬀ loading frame equipped with four electro-mechanical actuators interfaced to an INDEL real-time system [23]. The 3-DoFs HS test rig is designed to test plate specimens with an approximate footprint of 200500mm and thickness varying between 1and3mm. Figure 3 illustrates the architecture of the HS setup, including a close-up view of the plate specimen accommodation. The GINLink bus connects the actuator servo-driver INDEL SAC4 and all data acquisition modules INDEL 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 PS GINLink bus 𝑢𝑢1,𝑢𝑢2and 𝑢𝑢3 𝑟𝑟1,𝑟𝑟2and 𝑟𝑟3Host PC Figure 3: Architecture of the 3-DoFs HS test rig. COP-ADA to the real-time computer INDEL SAM4, which runs the HS software. The latter is developed in the MATLAB/SIMULINK environment and downloaded to the real-time computer INDEL SAM4 via Ethernet from the Host-PC. At each simulation time step, the HS software imposes displacements u1,u2andu3to the plate specimen, the PS, reads the corresponding restoring forces r1,r2andr3measured using force transducers, and solves the coupled equation of motion of the hybrid model. In addition, a laser sensor measures the out-of- plane deﬂection at the mid-span of the plate specimen. A detailed description of the time integration scheme adopted to solve the equation of motion is reported in [24]. Figure 4 shows two axonometric views of the 3-DoFs HS test rig, including the main hardware components. In this ﬁgure, the gray-colored parts are ﬁxed to the reaction frame, which is not visible, while the moving parts of the 3-DoF HS test rig are colored in yellow. The two rack-pinion systems (10) convert motion along the vertical actuator axes y1andy2(1) to rotations u1and u2, respectively, which are imposed to the short edges of the plate specimen (6) through aluminum clamps (3). Horizontal actuators along axes x1andx2(2) control the position of the moving frame (4), which is mounted on two proﬁled rail guides using ball bearings (5). As a result, the position of the moving frame (4) equals the axial elongation of the plate specimen u3(6). The repeatability of the HSs was assessed by comparing the results of three experiments characterized by the same values of rotational stiﬀness. The force vs. out-of-plane displacement curves for these experiments are shown in Figure 5a. It is clear that the HSs were highly repeatable. In addition, these results conﬁrmed the assumption that the output of the hybrid model is almost deterministic with negligible variability in the PSs. Figure 5b also shows results from three HSs using diﬀerent rotational stiﬀness combinations. In general, the experimental data matched well with the theoretical critical buckling loads predicted by (11). 6 DRAFT 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; (10) rack-pinion systems. The moving parts are colored in yellow while the grey parts are ﬁxed to the reaction frame, which is omitted in this ﬁgure for clarity. Figure 5: Validation of experimental assumptions: (a) test repeatability and (b) agreement with Newmark’s equation predictions (dash lines). 4 Experimental results and discussion The AK-HS procedure described in Section 2 was applied to train a classiﬁer for the buckling domain of the prototype structure described in Section 3 within the range of the input parameters deﬁned in Table 1. In detail, ﬁrst, a reference axial load Pref= 2000N was selected to ensure a full development of buckling for any pair offK1;K2g2D Xin the given range. This reference axial load was applied to the hybrid model as a linear increasing ramp of 10sduration in simulation time. Since hybrid simulations were performed with a testing time scale= 50, each experiment lasted 500sof wall-clock time. The critical buckling load was taken as the plateau value of the resulting measured force-displacement axial response curve. Such critical load estimated with HS corresponds to the QoI Y, as speciﬁed in (3), and accounts for the stiﬀness of the numerical rotational 7 DRAFTTable 1: Input random parameters. Variable Distribution Lower bound Upper bound Unit K1Uniform 10000 1000000 [Nmm rad] K2Uniform 20000 1200000 [Nmm rad] restraints. It should be noted that all of the experiments conducted were destructive, and thus diﬀerent (but nominally identical) aluminum plates were used for each hybrid simulation. The AK-HS procedure described in Section 2 was executed considering a target buckling load of yadm= 1200N. An initial 10-sample ED was generated using a Sobol’ sequence and the corresponding buckling loads were estimated using HS. Then, the ED was adaptively enriched with 30 additional samples using the AK-HS procedure. The kriging meta-models were estimated using the UQLab software framework developed by the Chair of Risk, Safety and Uncertainty Quantiﬁcation at ETH Zurich [25]. Figure 6 depicts the meta-models obtained using 10,20,30and40samples. (a) (b) (c) (d) Figure 6: Results of the AK-HS procedure for (a) 10-sample ED (b) 20-sample ED (c) 30-sample ED and (d) 40-sample ED. Diamonds represent non-failed samples, squares represent failed samples, the dash line indicates the theoretical failure surface predicted by the Newmark equation and the dash-dot line indicates the failure surface predicted by the trained classiﬁer. As shown in Figure 6, the classiﬁer trained using the initial ED did not accurately reproduce the buckling domain of the hybrid model. However, after 20 experiments, the failure surface estimated by the classiﬁer 8 DRAFTalready converged toward the analytical solution of provided by Newmark (i.e., using (11)). This result is consistent with the plots of Figure 7, which depicts the convergence of the normalized failure domain area f and related 95% conﬁdence interval. The latter corresponds to a failure probability if one assumes a uniform probability measure over DX. Furthermore, by the end of the experimental campaign, the ED points mostly accumulated along a well-deﬁned failure surface. Figure 7: Convergence of the normalized failure domain area fand related 95% conﬁdence interval. Finally, Figure 8 depicts the safe/failure state classiﬁer for two orthogonal sections of the support of the kriging meta-model. It is predicted from the Newmark model of (11) that for a given load of yadm= 1200N, andK2= 600000 Nmm/rad the system buckles for K1<98995Nmm/rad; and similarly when K1= 500000 Nmm/rad the system buckles when K2<107040Nmm/rad. As can be appreciated from the plots, the trained classiﬁer fairly agrees with such analytical predictions and it is characterized by a sharp transition between failed and non-failed states. (a) (b) Figure 8: Safe/failure state classiﬁer evaluated for two orthogonal sections of the support of the kriging meta- model: a)K2= 600000 Nmm/rad; b) K1= 500000 Nmm/rad. 5 Conclusions This paper extends the scope of hybrid simulation to training a classiﬁer that determines the failedornon-failed state of the physical substructure. Kriging meta-models and active learning are used to greedily construct a safe/failure domain classiﬁer at the lowest experimental cost possible. The resulting procedure, namely active- learning-kriging hybrid-simulation, relies on an adaptation of the active-learning kriging algorithm from the 9 DRAFTﬁeld of structural reliability. The eﬀectiveness of the procedure is illustrated for a 3-degrees-of-freedom hybrid model consisting of an elastically restrained beam subjected to constant axial loading. Failure is associated with Euler buckling, which may occur or not depending on the rotational stiﬀness of the restraints for a given axial load. Experimental results demonstrated the eﬀectiveness of the proposed procedure, which succeeded in training an accurate safe/failure state classiﬁer with about 20 hybrid simulations. Out future work will address the extension of the proposed procedure to the case of PS with stochastic behavior. 6 Data availability statement Some or all data, models, or code that support the ﬁndings of this study are available from the corresponding author upon reasonable request. Acknowledgments This work was supported by the Swiss Secretariat of Education Research and Innovation (SERI) - Swiss Space Oﬃce(SSO)[THERMICSProject(Thermo-MechanicalVirtualizationofHybridFlax/CarbonFiberComposite for Spacecraft Structures), MdP2016, grant number REF-1131-61001]. Also, the ﬁrst author acknowledges Dr. Roland Schöbi for his comments and suggestions at the earliest onset of this research.