An application of cyber-physical testing to the empirical estimation of diﬀerence-frequency quadratic transfer functions is presented. As an alternative to today’s procedure based on hydrodynamic tests with broad- bandedorrealistic(e.g., JONSWAP)wavespectra, testsinbichromaticwavesareconsidered. Thelaboratory setup is the one developed by Sauder & Tahchiev (2020) that enables magnifying the sensitivity of the ﬂoater responsetothelow-frequencywaveloadingbyadjustingthestiﬀnessanddampingparametersofavirtualsoft mooring system. Bayesian experimental design is proposed to optimize the selection of the control variables (frequencies in the bichromatic wave and properties of the virtual mooring system) for a batch of cyber- physical tests. The experimental design algorithm is based on the recent work of Huan & Marzouk (2013). Inavirtualyetrealisticcasestudyusinganuncertainparametricquadratictransferfunction, wedemonstrate how the uncertainty of its describing parameters and other calibration parameters (low-frequency added mass and hydrodynamic damping) can be reduced. Results indicate that the proposed procedure has the potential for reducing experimental cost for calibration of hydrodynamic models. Keywords: Hydrodynamic testing; cyber-physical testing; Bayesian experimental design; model calibra- tion; quadratic transfer function. 1 INTRODUCTION 1.1 Background and motivation The design of moored ﬂoating structures requires validated computational models to predict wave-frequency (WF) and low-frequency (LF) hydrodynamic loads [1]. In general, ﬁrst-order potential ﬂow theory provides suﬃciently accurate estimates of WF loads when viscous eﬀects are negligible. LF loads, which are crucial for determiningextremeoﬀsetsandtensioninmooringlines,aregenerallydescribedbyQuadraticTransferFunction (QTF) that represent the relationship between the amplitudes of two monochromatic waves of frequencies ff1;f2g, and the LF load exerted to the ﬂoater at a frequency jf1f2j. See the previous reference for details. An example of QTF is shown in Figure 1 for a ship-shaped ﬂoating structure in head seas. QTFs obtained with numerical tools accurately predict LF loads in mild sea states and in the absence of current. However, since the existing numerical methods are based on potential theory, perturbation methods, and treat the presence of current only through approximations, they generally fail at predicting LF loads accurately in moderate-to-extreme sea states where current eﬀects, viscous eﬀects, and beyond-second-order potential eﬀects become signiﬁcant. Therefore, empirical methods play a central role in improving such QTF estimates [2]. As of today, state-of- the-art model-tests practice is to install the oﬀshore structure under study in an ocean basin, using a so-called soft mooring system . The stiﬀness of the mooring system in surge, sway and yaw is ﬁxed and designed to be low, leading to natural periods well above the WF range. This in order not to disturb the WF motions of the ∗Dr. Giuseppe Abbiati, abbiati@cae.au.dk , ORCID: 0000-0002-5048-8505 . †Dr. Thomas Sauder, thomas.sauder@sintef.no , ORCID: 0000-0001-7445-7239 . 1 DRAFT Figure 1: Example of QTF amplitude: the red box indicates the domain of wave frequency pairs ff1;f2gleading to a LF excitation with frequency f=f1f2tuned with the ﬂoater eigenfrequency fn. ﬂoater, which play an important role for the LF wave loads. The system is then subjected to incoming waves representing either a realistic sea-state (wave energy concentrated around a peak period) or a broad-banded wave spectrum. Based on input wave amplitude and ﬂoater motion, a linear deconvolution and cross-bi-spectral analysis are performed to compute a non-parametric estimate of the QTF for the net LF wave load exerted to the ﬂoater [3]. Besides the statistical uncertainty inherent to this type of estimation, two main limitations can be mentioned: i) when LF wave loads and ﬂoater transfer function are detuned (i.e. the oscillator does not respond signiﬁcantly to the wave load at the studied frequency), no information on LF load is present on the ﬂoater response and, therefore, deconvolution is not applicable; ii) the accurate identiﬁcation of the QTF requires that signiﬁcant wave energy is allocated for any frequency pair ff1;f2g, which is diﬃcult to achieve in practice for suﬃciently broad range of frequency pairs. In response to the ﬁrst limitation, Sauder & Tahchiev [4] developed a laboratory apparatus that enables magnifying the sensitivity of the ﬂoater response to the LF loads generated by the incoming wave by adjusting the stiﬀness and damping parameters of a virtualsoft mooring system. This type of Cyber-Physical (CP) testing method has been introduced earlier in hydrodynamic testing, under the denomination Real-time Hybrid Model testing∗[5]. A CP model comprises a physical subsystem tested in the laboratory, and a numerical subsystem simulated numerically. The two subsystems interact with each other in a closed-loop as parts of a unique prototype. In hydrodynamic testing, the physical subsystem (the ﬂoater) is installed in an ocean basin, and boundary conditions computed from the numerical subsystem (the virtual mooring) are prescribed using a real-time control system. Here, CP testing is performed in force-control, that is, a set of actuation lines impose the virtual mooring forces to the ﬂoater, and the corresponding LF velocity and position feedback close the simulation loop. More details can be found in Section 4.1 and in [4]. In response to the second limitation, one could perform a suﬃciently large number of CP tests based on bichromatic waves to cover the entire domain of the QTF ff1;f2gwith a suﬃciently dense grid. However, it requires about 3040min to complete a single experiment before starting a new one; therefore, a brutal mapping of the QTF is not feasible due to time constraints. 1.2 Scope Instead of scanning the whole grid of frequency pairs systematically to compute a non-parametric estimate of the QTF, this paper proposes to calibrate a parametrized QTF model based on a minimal set of CP tests. Bayesian Experimental Design (Bayesian-ED) is proposed to optimize the selection of the control variables for an entire batch of CP tests so that the expected reduction of uncertainty of the QTF calibration parameters is maximized. Noteworthy such control variables include both the frequencies of bi-chromatic waves and stiﬀness and damping of the active mooring system. As a result, the ﬂoater transfer function can be tuned to the incoming LF wave load (limitation i)) and the entire domain of the QTF can be explored (limitation ii)). ∗ReaTHM®testing is a registered trademark of SINTEF Ocean. 2 DRAFTModel calibration is a cornerstone of engineering science provided with a mathematically sound framework rooted in Bayesian inference [6, 7]. In principle, a forward model, some experimental observation, and a measure oftheirlikelihoodareusedtoinfersomequantityofinterestnotdirectlyobservable(i.e., calibrationparameters). A set of control variables determines the experimental conditions from which observations are taken. Bayesian- ED provides a decision-theoretic methodology to optimize the selection of such control variables [8, 9]. Since the outcome of an experiment is uncertain, Bayesian-ED is performed by maximizing the expected value of a utility function. The latter is usually expressed in terms of a measure of the information gain on calibration parameters following the experiment. In this work, the Bayesian-ED algorithm presented in a recent work of Huan & Marzouk [10] is adopted to design a batch of CP tests. The selected utility function is deﬁned as the Kullback-Leibler divergence [11] from prior to posterior probability density functions (PDF) of calibration parameters. The eﬀectiveness of the proposed approach is demonstrated for a virtual hydrodynamic application where a QTF model is calibrated. In this example, the QTF model consists of a mixture of bivariate Gaussian functions [11]. The paper is organized as follows. Section 2 provides the basics of Bayesian model calibration. The aim is to introduce the fundamental terminology and to provide a self-contained mathematical notation. Section 3 describes the Bayesian-ED procedure introduced by Huan & Marzouk [10] and utilized in this work. Section 4 presents the application example. CP testing is used to calibrate the parameters of the QTF for a typical moored system. Main conclusions are summarized in Section 5. 2 BAYESIAN MODEL CALIBRATION This section recalls the basics of Bayesian calibration of computer models in its simplest form, that is, for the case of a deterministic computational model and statistically independent observations. The aim is to provide a self-contained notation for the proposed Bayesian-ED procedure, which is described in the next section. For a more generic treatment of the Bayesian calibration problem and a comprehensive review of the state of the art, the reader is addressed to the work of Nagel & Sudret [7]. The relationship between an observation yi2D yRnyand the corresponding forward model prediction M(;di)reads, yi=M(;di) +i (1) where the index i2f1;:::;ngindicates a speciﬁc experiment of a batch. 2D Rnare deterministic albeit unknown calibration parameters and, therefore, associated with epistemic uncertainty. Accordingly, calibration parameters are thought as of a realization a priorprobability density function (PDF) (), which reﬂects prior belief on possible values. di2D dRndcomprises all those variables that can be controlled by the experiments and therefore indicated as control variables . Finally, the discrepancy term i2D Rny incorporates model inadequacies and measurement errors. Commonly, the residual term is assumed as of a realization of a normal distributions iN(0;i)with zero mean and experiment-speciﬁc symmetric positive- semideﬁnite covariance matrix i. The set of all available experimental observations Y=fy1:::yi:::yngand the corresponding samples of control variables D=fd1:::di:::dngform the so-called Experimental Design (ED)fD;Ygfor the Bayesian calibration problem. In this paper, (1) refers to a single CP test of a batch of nCP tests. The forward model M(;di)includes a computational model of the PS of parameters . The remaining known parameters of the CP model are stored indi. For the sake of simplicity, both NS, PS and external loading models and therefore M(;di)are assumed as deterministic. Given a forward model in the format speciﬁed in (1), an ED acquired from a batch of nexperiments, a probabilistic model for discrepancy and the prior PDFs of calibration parameters, the Bayes’ rule for conditional probability is used to compute the posterior PDF of calibration parameters, (jD;Y) =L(;D;Y)() (YjD)(2) where the likelihood functionL(;D;Y)measures the evidence of provided by the ED. For simplicity, in this paper, experimental observations are assumed as statistically independent. Still, for simplicity, discrepancy terms are assumed as uncorrelated between diﬀerent forward model outputs. Therefore, iis diagonal. The expression of the likelihood term of (2) reads, 3 DRAFTL(;D;Y) =nY i=11p (2)nydet(i) exp1 2h (yiM (;di))1 i(yiM (;di))>i(3) The denominator of the right-hand side of (2) is called marginal likelihood orevidence and it is obtained by applying the total probability theorem, (YjD) =Z DL(;D;Y)()d (4) The solution of (4) is typically avoided and the unnormalized posterior PDF of is sampled numerically using Markov Chain Monte Carlo (MCMC) methods. 3 BAYESIAN EXPERIMENTAL DESIGN This section describes the Bayesian-ED procedure proposed by Huan & Marzouk [10] and its application to CP testing. This combination is the essence of the novelty carried by this paper. 3.1 Theoretical derivation of the utility function The Bayesian-ED procedure proposed by Huan & Marzouk [10] follows the decision-theoretic approach as origi- nally introduced by Lindley [12]. The review paper of Chaloner & Verdinelli provides a comprehensive overview on this subject [8]. In a decision-theoretic approach, an utility function u(Y;D)quantiﬁes the information gain provided by a speciﬁc ED. The utility function reﬂects the goal of the experimental campaign. Noteworthy, for a given D, experimental observations collected in Ypropagates the uncertainties of calibration parameters  and model discrepancy . Accordingly, u(Y;D)is also uncertain. Therefore, Bayesian optimal ED is performed with respect to the expected value of the utility function, U(D) =Z DYu(Y;D)(YjD)dY (5) where(YjD)represents the probability measure associated with a speciﬁc ED fD;Yg. It is clear from (4) that the averaging of (5) adds less weight to those EDs for which experimental observations Yare unlikely either because corresponding calibration parameters are unlikely (small prior ()) or discrepancy is unlikely (small likelihoodL(;D;Y)). The solution of the Bayesian-ED problem reduces to the following maximization problem, ^D= arg max DU(D) (6) Forthespeciﬁccasewhereexperimentalobservationsareusedtocalibratetheparametersofacomputational model (see Section 2), Huan & Marzouk [10] proposed to adopt as utility function the Kullback–Leibler (KL) divergence [11], a.k.a. relative entropy, from the posterior to the prior PDF of calibration parameters, u(Y;D) =DKL((jY;D)jj()) =Z D(jY;D) ln(jY;D) () d(7) The intuition behind (7) is that a large DKLdivergence from posterior to prior implies that the data in Y decreases the entropy in and hence this data is more informative. The expression for the expected utility is obtained by substituting (7) in (5), U(D) =Z DYZ D(jY;D) ln(jY;D) () (YjD)ddY(8) 4 DRAFTStill in [10], it is demonstrated that (8) is equivalent to the mutual information between andY. Mutual information is a measure of the mutual dependence between two random variables [11]. 3.2 Numerical computation of the expected utility In general, there are no analytical solutions for computing the expected utility function, so the integral of (5) is solved numerically using Monte Carlo. According to the procedure described in [10], with few manipulations (which are omitted for brevity), the expected utility of (5) is rearranged as follows, U(D) =Z DYZ D[ln(Yj;D)ln(YjD)] (Yj;D)()ddY(9) Then, (9) is evaluated using two nested Monte Carlo loops. Speciﬁcally, the outer Monte Carlo loop evaluates the expected utility function, U(D)1 nMC;outnMC;outX j=1ln(Yjjj;D)ln(YjjD) (10) whereYj(Yj;D)andj(). The inner Monte Carlo loop computes the marginal likelihood of the EDfD;Yjgin (10), (YjjD)1 nMC;innMC;inX k=1(Yjjk;D) (11) where k(). Sampling and evaluating the likelihood terms of (10) and (11) entail the evaluation of the forward model. Therefore, they are the most computationally expensive tasks in the calculation of the expected utility function. Accordingly, for a given value of D, the same fresh batch of jis used as k (nMC;out =nMC;in =nMC) and corresponding—deterministic—forward model evaluations M(j;di);di2D are stored. Observations ( Yj) and corresponding likelihoods ( (Yjjj;D)in (10) and (Yjjk;D)in (11)) are obtained by sampling and evaluating the joint PDF of the the discrepancy term. As a result, the cost for computing (8) is reduced from nMC;outnMC;intonMCevaluations of the forward model. As highlighted in [10], this approach leads to a biased estimator of the expected utility. However, the bias is small. 4 APPLICATION EXAMPLE In order to exemplify the presented Bayesian-ED procedure for the calibration of QTFs, the following virtual cyber-physical testing campaign is conceived. 4.1 Description of the case study The objective is to identify empirically a QTF deﬁned on a relevant range of mean frequencies f0= (f1+f2)=2 anddiﬀerence-frequencies f=f1f2(notethat fisthefrequencyoftheloading). Forsimplicity, mimicking thepatternsthatcanbeseenonarealisticQTFssuchasinFigure1,weassumethat Qisrepresentedanalytically by a real multivariate Gaussian mixture model with three lobes (see Figure 3), namely lobe a,bandc: Q(F;f;f0) =Qa+Qb+Qc (12) where, Qa=Aaexp(f1fa)2 22a(f2fa)2 22a (13) Qb=Abexp" (f1fb1)2 22 b1(f2fb2)2 22 b2# + Abexp" (f1fb2)2 22 b2(f2fb1)2 22 b1# (14) 5 DRAFT H(a) (b) Figure 2: CP model of the moored system: a) PS with actuation lines; b) block-diagram of the simulation loop. Qc=Acexp(f1fc1)2 22c1(f2fc2)2 22c2 + Acexp(f1fc2)2 22c2(f2fc1)2 22c1 (15) Assuming that Aa,Ab, andAcare known, the parameter vector that fully characterizes the QTF reads F=ffa;a;fb1;b1;fb2;b2;fc1;c1;fc2;c2g (16) and we assume that preliminary (uncertain) estimates of Fare available, for example from numerical methods. As presented in the work of Sauder and Tahchiev [4], and shown in Figure 2a the structure under consid- eration is subjected to head waves in an ocean basin and kept in position using an active positioning system. The structure in its mooring is then assimilated to a single-degree-of-freedom second-order dynamical system, and its LF motions are described by (m+a)x+dh_x= m(x;_x) + w(t) (17) wherexis the LF surge displacement of the ﬂoater and _xits LF surge velocity. The mass term includes both ﬂoater mass mand the asymptotic value of the hydrodynamic added mass afor low frequencies; dhis the LF damping parameter, which includes, among others, wave drift damping. m(x;_x)represents the restoring force of the (virtual) mooring system, which is linear w.r.t. both velocity and position. Finally, w(t)is the LF load produced by a bichromatic wave with two wave components of frequencies f1andf2(withf1f2). The numerical subsystem of the CP model coincides with m(x;_x)while all other terms of (17) belongs to the physical subsystem. CP testing is performed in force-control, that is, a set of actuation lines impose mooring forces mto the ﬂoater, and the corresponding LF velocity _xand position xfeedback close the simulation loop. Figure 2b reports a block diagram of the CP setup. Assuming bichromatic waves, the complex amplitude Fof the (second-order, LF) hydrodynamic force w at frequency fis given by the following expression F= 2Q(F;f;f0)a1a 2 (18) wherea1anda2are the complex amplitudes of the two incoming waves. Accordingly, the complex-valued LF ﬂoater displacement Xobtained at steady state can be approximated by the following scalar frequency-domain equation X=H(f)F (19) whereH(f)is the transfer function of the moored system obtained from (17). The analytical expression of H(f)is H(f) =1 km1 1 f fn2 + 2i h+dm 2p (m+a)km f fn(20) whereiis the imaginary unit, and kmanddmare the stiﬀness and damping coeﬃcients of the (numerical) mooring system, respectively. The ﬂoater eigenfrequency reads fn=1 2r km m+a(21) Note that the total damping ratio =h+mincludes both the hydrodynamic damping hof the ﬂoater and the mooring-induced damping dm= 2mp (m+a)km. Notably, kmanddmare related to the numerical subsystem while m,aandh, are related to the physical subsystem. The ﬂoater mass mcan be easily measured whereas the hydrodynamic added mass aand damping hneed to be calibrated based on experiments. 6 DRAFT4.2 Setting of the benchmark study Since the CP testing campaign is virtual, the "experimental" observations are also produced numerically using the same equation utilized as the forward model. Accordingly, and in line with the format of (1), the model calibration problem is stated as follows, y:=j^Xj=j^H(f)^Fj (22) M(;d) :=jXj=jH(f)Fj (23) N(0;) (24) where= 0:3m (at full-scale) is the standard deviation of the discrepancy term. As can be appreciated from (22), the amplitude of the ﬂoater displacement is selected as experimental observation. The absolute value of the frequency-domain model of (19) is utilized both as forward model in (23) and to generate experimental observations in (22). Calibration parameters and control variables in (23) read, =fF;a; hg,d=ff0;f;km;dmg (25) The corresponding PDF are reported in Tables 1 and 2, respectively. The coeﬃcients of variation (CoV) are chosen arbitrarily in this example, but should reﬂect expert judgment. Table 1: Calibration parameters . Parameter UnitDistribution CoV fa HzGaussian 0:090 0:05 fb1HzGaussian 0:110 0:02 fb2HzGaussian 0:120 0:02 fc1HzGaussian 0:140 0:02 fc2HzGaussian 0:150 0:02 aHzGaussian 0:008 0:02 b1HzGaussian 0:008 0:02 b2HzGaussian 0:004 0:02 c1HzGaussian 0:004 0:02 c2HzGaussian 0:004 0:02 a tGaussian 11770 0:10 h –Gaussian 0:030 0:10 Table 2: Control variables d. Control variable Unit min max f0 Hz 0:050 0:200 f mHz 4 13 km kN/m 100 422 dm kNs/m 143 1284 Noteworthy,thehatsymbolsin(22)areintroducedtoemphasizethatexperimentalobservationsareobtained considering the truevalues of calibration parameters, which correspond (here, for the sake of simplicity) to the mean values of Table 1. All the remaining parameters of the transfer function and the LF load in (19) are constant and reported in Table 3. 7 DRAFTTable 3: Constant parameters. Parameter Unit Value m t 178400 a1 m 2 a2 m 2 AakN/m2150 AbkN/m2100 AckN/m280 In order to benchmark the accuracy of the proposed Bayesian-ED procedure, two EDs are produced and utilized to solve the calibration problem stated by (22), (23) and (24). A ﬁrst ED is constructed using a Sobol’ sequence [13], which is a space-ﬁlling design technique, which maximizes the coverage of the control variable space, and it is referred to as SS ED. The second ED is constructed following the procedure proposed by Huan & Marzouk [10] summarized in Section 2 and it is referred to as HM ED. Speciﬁcally, a single evaluation of the expected utility function required nMC= 1104Monte Carlo samples. 4.3 Results and discussion The solution of the optimization problem of (6) subjected to the constraint of Table 2 took about 2hours of elaboration using 4 CPU cores on the PRIME computer cluster of Aarhus University. Both EDs collect observations from 9 experiments, that is, fD;Yg=ffdi;yig;i2 f1;:::; 9gg. Corresponding samples are reported in Table 4 and overlaid to the trueQTF in Figure 3. Note in this ﬁgure both EDs are near the bisecting line of the f1f2plane. The reason is that the range of fis small compared to the range of f0(see Table 2). Table 4: EDs computed with Huan & Marzouk procedure (HM) and with Sobol’ sequence (SS). EDf0Hz fmHzkmkN/mdmkNs/mfn[mHz] SS0.13 8.50 261.00 713.50 5.90 0.09 10.75 180.50 998.75 4.90 0.16 6.25 341.50 428.25 6.74 0.07 9.63 381.75 1141.38 7.13 0.14 5.13 220.75 570.88 5.42 0.11 7.38 301.25 285.63 6.33 0.18 11.88 140.25 856.13 4.32 0.06 12.44 321.38 499.56 6.54 0.13 7.94 160.38 1070.06 4.62 HM0.17 5.61 403.01 571.84 7.33 0.11 4.84 180.72 1056.42 4.91 0.15 6.91 291.66 290.72 6.23 0.15 8.71 115.81 354.30 3.93 0.12 5.77 254.35 541.94 5.82 0.08 4.97 188.26 660.11 5.01 0.14 4.00 119.65 667.57 3.99 0.18 8.78 115.78 918.91 3.93 0.19 12.90 412.42 878.26 7.41 It is clear from Figure 3 that the HM ED tends to cluster around the lobes of the QTF whereas the SS ED is uniformly distributed along the bisecting line. Noteworthy, it is in the proximity of the QTF lobes that observations carry more information regarding calibration parameters. Based on the HM and SS EDs, the posterior PDF of the calibration parameters were computed using UQLab, which is a MATLAB toolbox for uncertainty quantiﬁcation developed by the Chair of Risk, Safety and Uncertainty Quantiﬁcation of ETH Zurich, Switzerland [14, 15]. Firstly, the performance of HM and SS EDs are compared in terms of reduction of epistemic uncertainty in the calibration parameters, which is qualitatively 8 DRAFT 2468101214104Figure 3: True QTF (in N/m2) with both SS and HM EDs. assessed by looking at the dispersion of the posterior PDFs. In this regard, Figures 4, 5 and 6 collect the prior and both posterior PDFs of the location parameter related for Lobes a,bandcof the QTF, respectively, namely fa,fb1,fb2,fc1andfc2. Posterior PDFs are computed for both HM and SS EDs. Figure 4: Prior and posterior PDFs of fa, describing the location of lobe aof the QTF. (a) (b) Figure 5: Prior and posterior PDFs of fb1andfb2, describing the location of lobe bof the QTF. As can be observed from Figures 4, 5 and 6, the HM ED outperforms the SS ED in terms of reduction of uncertainty from prior to posterior PDFs. However, prior and posterior PDFs of lobe radii a;b1;b2;c1and c2are systematically similar for both HM and SS EDs. The same result is obtained for the ﬂoater damping ratiohand the added hydrodynamic mass a. Therefore, corresponding plots are omitted for brevity. 9 DRAFT (a) (b) Figure 6: Prior and posterior PDFs of fc1andfc2, describing the location of lobe cof the QTF. Notably, for none of the calibration parameters, the the SS ED outperforms the HM ED in term of reduction of epistemic uncertainty of calibration parameters. Noteworthy, (6) implies that the HM ED provides the most informative experimental design; all other solutions (including the SS ED) are sub-optimal w.r.t. the deﬁnition of the utility function (8). Finally, in order to provide a quantitative measure of the eﬀectiveness of the proposed procedure, for both SS and HM EDs, the Kullback-Leibler divergence from posterior to prior of the joint PDF of frequency parameters fa,fb1fb2,fc1andfc2are computed using the nearest-neighbour method described in [16]. Numerical results are compared in Figure 7. Clearly, the HM ED is more informative than the SS ED. Figure 7: Kullback-Leibler divergence from posterior to prior of the joint PDF of frequency parameters fa,fb1 fb2fc1andfc2. 5 CONCLUSIONS This paper investigates the empirical estimation of a quadratic transfer function based on a minimal set of cyber-physical tests. Bayesian experimental design is proposed to optimize the selection of the control variables for a batch of cyber-physical tests so that the expected reduction of uncertainty of the quadratic transfer function calibration parameters is maximized. Speciﬁcally, a Bayesian experimental design algorithm recently proposed by Huan & Marzouk (2013), is used. The eﬀectiveness of the proposed approach is demonstrated with a virtual cyber-physical testing campaign. Control variables for the experimental design include both the frequencies of bi-chromatic waves and stiﬀness and damping of the active mooring system. As a result, the ﬂoater transfer function can be tuned to the incom- ing low-frequency wave load, and the domain of the quadratic transfer function can be explored. A systematic comparison of the posterior distributions of the calibration parameters demonstrates that the experimental de- sign computed with the proposed algorithm outperforms a reference space-ﬁlling design sequence. The outcome of this study suggests that Bayesian experimental design has signiﬁcant potential for reducing the experimental 10 DRAFTcost for calibration of hydrodynamic models.