The determination of the design variables strongly depends on the purpose of model updating. Since the aim of this contribution is damage identification, specifically damage localization and quantification, the parameterization should be able to identify the geometric position of the damage and its intensity.

As damage mainly manifests itself as a change in stiffness, the general approach for most FE model-updating procedures with the aim of damage identification is to alter the stiffness properties of the model at hand . This approach is also applied in this work. As no prior knowledge of the defect location is assumed, the updating of the stiffness properties of all Nel elements is performed, independent of the element type. This is implemented by adapting the initial Young's modulus E0 of each FE with a corresponding scaling factor θk 3Eθ,k=E0θkwithk∈[1,Nel]. The stiffness scaling factors θk are calculated on the basis of the design variables. In this work, the design variables parameterize a so-called damage distribution function, previously introduced and applied by the authors . By formulating the mapping of the considered structural properties to the FEs using a distribution function, a smooth and realistic spatial distribution is ensured. As a result, the model-updating process is forced to focus on global structural dynamics instead of over-fitting local deviations. Moreover, the model-updating procedure is independent of the FE mesh resolution and of prior assumptions about the defect location while only needing a few design variables. However, applying the damage distribution in its current form restricts damage identification to only one damage position. In this work, a spatial Gaussian damage distribution function is considered based on the assumption that the damage roughly follows this type of distribution. In addition, this representation can cover a wide range of damage scenarios, from highly localized defects affecting only a few FEs to one FE to a smoothly distributed stiffness degradation over an extended region. Furthermore, the smooth spatial distribution of the damaged area leads to a comparatively smooth objective function, which, in turn, promotes faster and more stable convergence of the utilized optimization algorithm.

The determination of the design variables additionally depends on the specific model used in the updating procedure, as its element type and geometry inherently influence their configuration. In this work, two different numerical models of the large-scale rotor blade are considered, which necessitate at least two different design variable configurations. In the following subsections, firstly, the parameterization of a one-dimensional damage distribution function for the model updating using a rather simple beam model of the rotor blade is presented. This definition, based on three design variables, has already been introduced and successfully employed in several preceding publications , where deterministic model updating was performed on simulated and experimental application examples. In addition, in this work, the one-dimensional parameterization is extended by a fourth design variable, which independently adjusts the stiffness properties in the edgewise and flapwise directions of the rotor blade, allowing for a distinct consideration of both directions. Secondly, the parameterization of a two-dimensional damage distribution function for the model updating using a more detailed shell model of the rotor blade is introduced. This configuration is based on five design variables and allows for damage identification in the longitudinal direction and circumferential direction.

As uncertainty is inevitable in real-world model-updating applications, the formulation of the model-updating problem is uncertain as well. To tackle the issue of input uncertainty propagation within the model-updating procedure, the authors recently introduced the sample-based deterministic model-updating (SDMU) approach . The key idea behind the SDMU approach is to exclude the input uncertainty from the design-variable-dependent part of the objective function formulation, i.e., from the actual model-updating procedure. Instead, the input uncertainty is incorporated indirectly by generating multiple discrete input samples. Following the sample provision, a fully deterministic model-updating procedure using a numerical optimization algorithm is executed based on each input sample. Thus, multiple deterministic model-updating procedures are performed based on the input samples.

Due to the separation of the input uncertainty incorporation and the model-updating procedure, both the type of sample provision and the choice of the optimization algorithm are arbitrary. This interchangeability allows the approach to be fully adaptable to the specific problem at hand. Consequently, the SDMU approach can be realized to deliver fully reproducible results, which is also applied within this work. Therefore, the global deterministic optimization algorithm global pattern search (GPS) is chosen. As the two modal parameter eigenfrequencies f and eigenmodes φ are considered to be the two objectives for the model-updating procedure of the large-scale rotor blade, multi-objective optimization is required. Hence, the multi-objective extension of the deterministic optimization algorithm is utilized, namely the multi-objective global pattern search (MO-GPS) . Accordingly, the present work employs a fully deterministic SDMU realization for the damage identification on the laboratory rotor blade.

Regarding the sample provision used in this work, the available measurement data were segmented into a number of Nsets datasets. For each dataset, the eigenfrequency and eigenmode mean values were identified using BayOMA. Subsequently, these Nsets datasets for each considered reference state and analysis state were cross-combined, resulting in a total number of Nsets2 model-updating scenarios for each combination of reference and analysis state. After applying this specific sample provision, a resulting number of Nsets2 modal parameter samples fM,i,s and φM,i,s with i∈[1,Nmodes] and s∈[1,Nsets2] represent the input samples for each subsequent deterministic model-updating procedure. The subscript M denotes measured data, and i denotes the mode index, whereby Nmodes is the number of modes investigated, and s is the sample number. As a result, the two objective functions are each vectors with Nsets2 entries, the first comprising the mean squared eigenfrequency error εf,s and the second comprising the mean squared eigenmode error εφ,s: 13εs(x)=εf,s(x)=1Nmodes∑i=1NmodesfSA,i(x)-fSR,ifSR,i-fMA,i,s-fMR,i,sfMR,i,s2εφ,s(x)=1Nmodes∑i=1Nmodes1Nsensors∑u=1NsensorsφSA,u,i(x)-φSR,u,i-φMA,u,i,s-φMR,u,i,s22withs∈[1,Nsets2],i∈[1,Nmodes],u∈[1,Nsensors]. The subscript M indicates a measured quantity, whereas simulated modal parameters are denoted by the subscript S. The subscripts R and A indicate the reference state and analysis state, respectively. The presented definition of the two objective functions was introduced by for damage localization on the LUMO benchmark structure using a stochastic multi-objective model-updating approach. Both objective functions are based on a relative formulation, which takes into account a possible constant discrepancy between the simulated and identified responses for both the analysis and the reference states of the examined structure. The goal is to obtain a numerically efficient, well-formulated optimization problem, which can handle irreducible modeling errors.

One way the Nsets2 two-objective functions ε(x) can be handled is by performing independent model-updating procedures on each of the two-objective functions. However, for high sample numbers and more complex numerical models, this approach will eventually become very expensive in terms of computing time. This is why the use of a meta-model is proposed, whereby the process of setting up and integrating a considered meta-model is incorporated into the SDMU approach and is referred to as metaSDMU . The objective is to derive a meta-model using a single representative deterministic model-updating procedure in order to generate a sampling pattern within the design variable space, which is dense in the area where the solution(s) are expected. The design variable samples generated during this optimization procedure and the corresponding objective function values calculated provide the training data for the two meta-models – one meta-model for each input, i.e., each modal parameter. Subsequently, these meta-models replace the actual numerical model such that 14fmeta,S,i(x)≈fS,i(x)andφmeta,S,i(x)≈φS,i(x)withi∈[1,Nmodes] holds true for each (simulated) eigenfrequency fS,i and eigenmode φS,i.

This way, the potentially computationally expensive model evaluations required in every iteration step only have to be performed during one model-updating procedure. For all other samples, the model updating with consideration of input uncertainty (cf. Eq. ) is performed using the meta-models, which are computationally much more efficient. More detailed information regarding the (meta)SDMU approach is given in .

For the visualization of the results, cumulative distribution functions (CDFs) are utilized. In this context, a CDF is calculated based on all Pareto-efficient solutions calculated during an SDMU procedure. Following , this means that for the number of Nsets2 model-updating scenarios used as input samples in this work, a set of Pareto-efficient solutions Y^s exists for each of these individual model-updating scenarios. Consequently, each Pareto-efficient solution set contains an individual number of |Y^s| (optimal) design variables xs,l with s∈[1,Nsets2] and l∈[1,|Y^s|]. The empirical probability p(xs,l) of a Pareto-efficient solution xs,l for a singular input set can be determined as 15p(xs,l)=Nsets2|Y^s|-1forallxs,l∈Y^s⇒∑l=1|Y^s|p(xs,l)=Nsets2-1withs∈[1,Nsets2],l∈[1,|Y^s|]. Due to the deterministic properties of the MO-GPS optimization algorithm utilized in this work, a given design variable xs,l is often among the Pareto-efficient solutions for several model-updating scenarios. Hence, the total probability of an optimal design variable x* is given as the sum 16p(x*)=∑s=1Nsets2∑l=1|Y^s|0forxs,l≠x*(Nsets2|Y^s|)-1forxs,l=x*withs∈[1,Nsets2],l∈[1,|Y^s|]. By evaluating the probability of every (optimal) design variable x* being Pareto efficient for at least one model-updating scenario, the empirical probability distribution function and, based on this, also the CDF can be approximated.

In general, a CDF provides a comprehensive and complete representation of how probabilities are distributed across the range of the considered (design) variable without discretization bias, i.e., the choice of bin sizes and locations. A notable feature of CDFs is the rate at which they increase. A steep section in a CDF indicates a rapid accumulation of probability over a small range of values. This indicates that a significant portion of the data points are concentrated around that region. Consequently, the probability density function (PDF), which is the derivative of the CDF, will be high in this area, pointing to a high density of occurrences.

In this work, Bayesian operational modal analysis (BayOMA) is utilized for the identification of the modal parameters from the measurement data of the dynamic tests under broadband white noise excitation. The in-house implementation of the BayOMA was developed in MATLAB programming syntax and strictly follows the methodology described in detail in and . To ensure correctness and reproducibility of the implementation, the developed code has been thoroughly tested and compared with the established covariance-driven stochastic subspace identification (SSI-COV) .

During the measurements, the sampling rate was set to 100 Hz. Figure  illustrates the frequency spectrum including the first Nmodes=5 eigenfrequencies of an example acceleration time period of the large-scale rotor blade recorded in state I. The frequency ranges farea,i per eigenfrequency i∈[1,Nmodes] utilized for the BayOMA are highlighted.

To visualize the stability and consistency of the recorded measurement data over time, the measurement data were segmented into a number of datasets Nsets. The evaluation time period was set to Tdata=400 s, and a moving window with an overlap of Toverlap=385 s was applied. For approximately 20min=1200s total measurement time, this results in Nsets=53 datasets. For each dataset, BayOMA was applied, resulting in 53 outputs which comprise the mean value and standard deviation of each eigenfrequency and the mean value and covariance matrix of each eigenmode. Figure  shows box plots of all eigenfrequency mean values f‾M,i,j identified from all 53 measurement datasets in the three different rotor blade states with i∈[1,Nmodes] and j∈[1,Nsets].

First of all, this analysis reveals that the interquartile ranges of the eigenfrequency mean values f‾M,i remain stable for each observed rotor blade state. This indicates consistent measurements with no significant fluctuations or outliers due to the stationary laboratory conditions.

Looking more closely at the variation in the mean values of the five different eigenfrequencies across all states, it is clear that the eigenfrequencies f‾M,1 and f‾M,3, corresponding to flapwise bending mode shapes, are not significantly influenced by the damage that occurred during the fatigue test. In contrast, eigenfrequencies f‾M,2, f‾M,4, and f‾M,5 exhibit a gradual reduction in magnitude, whereby the second and fourth eigenfrequencies correspond to edgewise bending mode shapes. The relative deviations of the first five eigenfrequencies between states I and III range from 0.02% to 0.6%.

The measured acceleration time series together with the BayOMA results of the three rotor blade states are published as open-access resources alongside this work within the public data repository of Leibniz University Hanover .

As both production and testing were carried out by Fraunhofer IWES, an outstanding feature of this rotor blade fatigue test was the complete documentation of the manufacturing process in addition to material and geometric data. Based on this detailed information, two different finite-element (FE) models were developed and employed for the model-updating procedures presented in this work. The simulations were performed using the FE analysis software Abaqus. Both numerical models are illustrated in Fig. , and summarized information regarding the element types, number of FEs, and the computing time for the modal analysis is given in Table . Here, the computing time comprises the setup of the model itself (i.e., the input file) and the modal analysis.

Firstly, a beam model was created based on the available cross-sectional characteristics of the rotor blade. Two-node linear beam elements, available in Abaqus as B31 elements, are selected, whereby a total of 251 beam elements are utilized. The B31 elements are based on the Timoshenko beam theory, allowing the representation of bending, axial, and torsional deformations along the beam axis. The varying sectional properties were assigned to the beam elements using general cross-sectional parameters. The load shears were simplified as point masses according to the information listed in Table  and assigned to the structure using concentrated mass elements in Abaqus. As the model-updating procedure applied in this work is based on the first five eigenfrequencies and corresponding eigenmodes, which are predominantly bending dominated, the rotational inertias of the load shears are not included. However, it should be noted that a more detailed representation of the load shears including the moments of inertia becomes increasingly relevant when torsional and higher-order modes are taken into account, which are not explicitly targeted in this work.

Secondly, a detailed shell model was set up based on the geometric data and composite layup available from the manufacturing process documentation. For this numerical model, S3R (three-node triangular) and S4R (four-node quadrilateral) shell elements with reduced integration were utilized. These elements efficiently capture bending, membrane, and transverse shear behavior, making them suitable for modeling thin to moderately thick shell structures. The load shears were modeled in detail using C3D4 tetrahedral solid elements and were included in all subsequent calculations, as they were attached during the whole experiment. In total, the shell model comprises approximately 310 000 FEs, of which around 81 700 are shell elements.

A comparison of the first five eigenfrequencies obtained from the beam model (fS,Beam,i) and the shell model (fS,Shell,i) is provided in Table . For an additional comparison between the numerical results and the modal parameters identified from the measurement data, the median eigenfrequencies f̃M,I,i computed from all mean eigenfrequencies in state I (cf. Fig. ) are listed in Table . For a comparison of the corresponding eigenmodes, the modal assurance criterion (MAC) defined by , 17MAC(φ1,φ2)=φ1Tφ22φ1Tφ1φ2Tφ2, is evaluated between the respective eigenmodes and included in Table . The MAC quantifies the similarity between two (eigenmode) vectors, returning a value of 1 for linearly dependent vectors and 0 for linearly independent ones. In addition, a visual comparison of the eigenmodes of the beam and shell models and the eigenmodes identified from the measurement data is provided in Fig. . Again, the “measured” eigenmodes are represented using the median value computed from all mean eigenmodes identified in state I.

To begin with, the damage identification results obtained using the beam model of the rotor blade are presented for the three- and four-dimensional design variable configurations, as introduced in Sect.  and , respectively. Before the application of the SDMU approach, preliminary studies are conducted to determine suitable settings for the input data and optimization algorithm hyperparameters. These studies are carried out using the three-dimensional design variable parameterization defined for the beam model with upper and lower bounds according to Table .

Here, the damage identification results of the (meta)SDMU approach applied to the shell model of the laboratory rotor blade are presented. In this case, the two-dimensional damage distribution function parameterized by five design variables is utilized with upper and lower bounds according to Table .

In the initial deterministic model-updating run for each combination, the same settings as listed in Table  are employed. Based on the samples and corresponding eigenfrequencies and eigenmodes, the respective meta-models are created using the same settings as before.

Figure  shows the resulting CDFs obtained for each design variable of each combination. As before, these CDFs are calculated based on all optimal design variables identified in the course of all Nsets2 separate model-updating runs for each input sample. For combinations I–III and II–III, the correct edgewise and flapwise locations of the emerged crack in state III are marked in gray. In addition, the 1σ and 2σ spatial damage extents are highlighted for both directions. For all three self-comparisons, the zero-damage result is highlighted. Furthermore, the red crosses indicate a possible optimal design variable vector x2D,optimal for combination I–III.

For the design variable μL, shown in Fig. a, combinations I–III and II–III, with state III as the target state, clearly show that the damage localization along the blade length almost exactly matches the correct crack position. Most optimal values obtained using the shell model fall almost entirely in the gray-shaded area, representing the true damage location. In contrast, the results obtained using the beam model (cf. Figs. a and a) deviate from the damage position by approximately 3–4 m. This means that the shell model improves localization accuracy along the blade length. For combination II–III, the CDF also shows a short steep section at L=8 m. This indicates that the stiffness reduction has already begun at this location in state II, although it is less distinct than in state III. For combinations I–I, II–II, and III–III, no clear convergence of μL is observed, which is consistent with the expected zero-damage result for these self-comparisons.

Figure b displays the resulting CDFs for the damage position μP along the blade perimeter. For combinations I–I, I–II, II–II, and III–III, no clear convergence is visible, indicating that no specific position along the blade perimeter can be determined for these state combinations. For combinations I–III and II–III, a position between μP=-0.1 and μP=-0.3 is identified as the most probable solution. The negative sign corresponds to the pressure side of the blade, oriented upwards in the considered laboratory setup (cf. Fig. ). This outcome does not match the true crack location across the LE in state III, corresponding to μP=0, as marked in gray. However, closer inspection of Fig.  shows that the crack propagates slightly more on the pressure side and changes direction through approximately 90° at this position. This may explain why the model-updating results identify the damage predominantly on the pressure side. Still, the damage localization along the blade perimeter remains complicated.

The results for the damage extent along the length σL, shown in Fig. c, are similar to those obtained using the beam model (cf. Figs. b and b), with the median lying exactly between the 1σL and 2σL values. The damage extent along the blade perimeter σP, shown in Fig. d, indicates a rather small spatial extent in this direction, with the median being even lower than the highlighted 1σP value. Consequently, the optimal results for the covariance matrix Σ2D correspond to a crack-like shape extending in the longitudinal direction. While this reflects the characteristics of the real damage extent, the orientation does not correspond to the true extent across the LE of the rotor blade. However, it should be noted that an inclined damage extent, as is actually the case here (cf. Fig. ), cannot be captured by the currently applied covariance matrix Σ2D, since its off-diagonal terms are set to zero (cf. Sect. ).

The results for the damage intensity D2D, presented in Fig. e, follow a pattern across all combinations similar to that observed for the (meta)SDMU results using the beam model (cf. Figs. c and c). For the self-comparisons, D2D=0 is the most probable outcome. Combination I–II shows an initial stiffness reduction, which is visible due to the slight rightward shift of the CDF. For combination I–III, the CDF shifts further to the right, revealing a more distinct solution for a positive D2D. This indicates an even greater stiffness reduction. The results for combination II–III, also targeting state III, show a comparable solution for D2D ranging from 0.025 to 0.175.

The stiffness reduction corresponding to the optimal design variable vector x2D,optimal, marked in red in Fig. , is visualized in Fig. . Again, x2D,optimal is selected such that the first design variable, the damage position μL, is set to its median value μ̃L, defined as the 50% value of the corresponding CDF, while the remaining four design variables are chosen as the optimal values corresponding to this fixed first design variable. For the shell model, these corresponding values coincide with the respective median values regarding design variables σL, μP, and σP, whereas the corresponding optimal value for D2D is slightly below its median. For comparison, Fig.  shows the correct damage location associated with the crack that emerged in state III. It should be noted that the “correct” representation aligns with the intuitive perception of the crack extending across the LE. This does not correspond to the actual crack propagation, which runs inclined across the LE (cf. Fig. ). However, as mentioned before, this inclination cannot be captured by the applied five-dimensional design variable parameterization, as the off-diagonal terms of the covariance matrix Σ2D are set to zero (cf. Sect. ).

In summary, the (meta)SDMU approach applied to the shell model of the rotor blade successfully localizes the damage along the blade length at L≈8 m. Furthermore, the damage exhibits an elongated, crack-like shape, which reflects the characteristics of the real damage extent. However, the found orientation is along the blade length rather than transverse to it, and the circumferential damage position was not accurately captured at the LE of the rotor blade but shifted towards its pressure side.

In this subsection, a direct comparison of the results obtained using the three different design variable configurations, defining the respective damage distribution functions applied to the two different numerical models, is presented. To illustrate this, Fig.  shows all the resulting CDFs for combination I–III. This combination updates the reference state I to the analysis state III of the laboratory rotor blade.

In general, the results for the three design variables μL, σL, and D (i.e., D1D and D2D) show a high degree of similarity. This overall consistency confirms the methodological robustness of the presented (meta)SDMU approach and the validity of the three different implemented parameterizations of the damage distribution function. Moreover, the agreement across the results underlines their reliability, particularly given that two numerical models of distinctly different levels of accuracy and detail were employed.

However, upon closer inspection, some differences can be discerned. Most notably, the use of the more detailed shell model increases the accuracy of the damage localization along the blade length, shown in Fig. a. Table  summarizes the localization accuracy of the different design variable parameterizations using the beam and shell models with respect to the true damage position μ^L at L=8 m along the 31 m blade. Therefore, the medians μ̃L – defined as the 50% values of the CDFs – were calculated for each design variable configuration. To provide a quantitative metric for assessing the accuracy of the presented CDFs, Table  lists the relative error eμL calculated in % per design variable configuration: 19eμL=100×μ^L-μ̃L31m.

The listed results demonstrate that the shell model reduces the damage localization error to less than 1% of the blade length. Due to its higher spatial resolution and more detailed representation of the blade geometry, it is significantly more accurate in capturing local damage. However, this accuracy comes with the need for detailed geometric and material information and the cost of higher computational demand, with a computing time of around 8 min, including the input file generation, model loading, and modal analysis. In contrast, the beam model provides a reasonably accurate damage localization within 11 %–15 % of the blade length despite its simplified representation, while requiring only around 20 s of computing time. This indicates that beam models can offer a practical compromise between computational efficiency and localization performance.

The damage extent is predicted nearly identically by all three parameterizations, illustrated in Fig. b. The true longitudinal extent of the inclined crack propagation is approximately 1 m, corresponding to a correct value of 2σL≈0.5 m and 1σL≈0.25 m, as respectively indicated by the vertical dashed and dotted lines in the figure. All in all, the three model-updating procedures accurately reflect the predominantly local nature of the damage.

Regarding the results obtained for the damage intensity shown in Fig. c, the beam model with three design variables predicts a slightly lower damage intensity, whereas the other two configurations yield similar results. Importantly, all three model-updating procedures consistently identify a stiffness reduction with high probability, indicating that the three presented damage parameterizations reliably capture the key structural effect of the damage. Again, it is noted that the value obtained for the design variable D cannot be explicitly validated for the considered rotor blade fatigue test, as the damage itself cannot be uniquely quantified. Although the crack, which finally occurred at the LE, can be visually identified, a quantification of the severity of this crack would require knowledge of geometric and mechanical characteristics such as the crack depth, crack tip opening displacement, and the specific damage mechanism. These parameters are typically not directly accessible and difficult to measure in a non-destructive manner. Moreover, the visible crack is likely not the only form of damage present in the blade, as it is very plausible that progressive material stiffness degradation occurred along the leading and trailing edges due to fatigue damage prior to the formation of the visible crack (cf. Fig. ).