BCBC and Bayesian calibration (without bias correction) were performed on a representative three storey building as an experimental case study. The aim was to indicate the improved accuracy of BCBC for forward model-driven SHM over con- ventional Bayesian calibration. Modal testing of the structure, presented in Fig. 5.6, was performed for nine damage extents — crack lengths of xz∗ = {0, 2.5, . . . , 20}mm in the front right beam in Fig. 5.6 — and the first three bending natural frequencies obtained. The structure was excited with broadband white noise via an electrody- namic shaker and the acceleration response measured at each of the three floors. Five repeats were obtained for each damage scenario. The third natural frequency was the most sensitive to damage and therefore used as the damage feature in this analysis. The experimental training data were five repeats when x = {0, 5, 20}mm — chosen to indicate the methods effectiveness for identifying the functional form from a small number of observations. The validation data set included all five repeats for the nine damage extents.
The simulator was a modal FEA model where the saw cut was modelled geometrically, i.e. the geometry of the saw cut was included in that of the beam. The elastic modulus E was included in the calibration process. This meant that simulator evaluations for training the emulator were obtained at x = xz
∗ and t = {65, 66, . . . , 71}GPa due to a prior elastic modulus of E ∼ N (68, 0.1)GPa.
5.3. REPRESENTATIVE THREE STOREY BUILDING CASE STUDY 123
(a) (b)
(c)
Figure 5.7: Predictions of natural frequency using BCBC and Bayesian calibration for a three storey building structure. Panel (a) and (b) are BCBC predictions using Gauss-Hermite quadrature and adaptive Metropolis MCMC respectively. Panel (c) demonstrates Bayesian calibration using adaptive Metropolis MCMC. The shaded regions indicate ±3σ.
BCBC was performed using both the Gauss-Hermite quadrature (20 nodes and weights) and adaptive Metropolis MCMC inference methods. These results were compared to Bayesian calibration using a Gaussian likelihood with an unknown noise variance. The noise variance had a Gaussian prior, σn2 ∼ N (0.0044, 0.0001) where the mean was estimated from the variance of training observations V (z(x)). A GP emulator, fitted to the same simulator training data, was used to assess the likelihood — where the likelihood covariance was the summation of the emulator covariance and
a diagonal matrix of σ2
n, i.e. Iσ2n. Inference was performed using adaptive Metropolis MCMC for the Bayesian calibration approach.
For both the Bayesian calibration and BCBC approaches the adaptive Metropolis MCMC parameters were 50, 000 posterior samples after a 1000 sample burn in and an
Figure 5.8: Posterior distributions for a three storey building structure using BCBC via Gauss-Hermite quadrature (BCBC-GH), adaptive Metropolis MCMC (BCBC- MCMC), and Bayesian calibration (BC) methods.
update step size of 100. The initial proposal variance for BCBC was 0.1 for the elastic modulus. On the other hand, the proposal covariance for Bayesian calibration had zero cross-covariance terms with a proposal variance of 0.02 for the elastic modulus and 0.01 for the noise variance. All these approaches defined an emulator with constant mean and SE covariance functions with a nugget ν = 1 × 10−8. The BCBC methods were implemented with a model discrepancy prior defined by constant mean and Mat´ern (where p = 2) covariance functions.
The predictive distributions of the third natural frequency for all three approaches are displayed in Fig. 5.7. Here it can be seen that all three approaches have captured the trend of natural frequency with increased saw cut size, with the validation data lying within three standard deviations. The NMSE of the mean predictions for BCBC were both 8.07 compared to 12.34 for Bayesian calibration.
The inferred posterior parameter distributions are shown in Fig. 5.8. It can be seen that both the Gauss-Hermite quadrature and adaptive Metropolis MCMC methods produce similar posterior distributions. The variance of these distributions is large compared to the prior, and larger than the inferred posterior distribution from the Bayesian calibration approach. This difference between the BCBC and Bayesian
5.3. REPRESENTATIVE THREE STOREY BUILDING CASE STUDY 125 0 2 4 6 8 10 12 14 16 18 20 0 0.005 0.01 0.015 (a) 0 2 4 6 8 10 12 14 16 18 20 0 0.5 1 (b) 0 2 4 6 8 10 12 14 16 18 20 0 0.5 1 (c)
Figure 5.9: Validation metrics for natural frequency predictions from the three storey building case study. Panel (a), (b) and (c) demonstrate the area metric, total variation and Hellinger distances when compared to Gaussian representations of the observation data. Each panel demonstrates the distances for BCBC using Gauss- Hermite quadrature (BCBC-GH), adaptive Metropolis MCMC (BCBC-MCMC), and Bayesian calibration (BC) methods.
calibration posterior distributions is likely due to the omission of model discrepancy uncertainty in the Bayesian calibration formula, producing overconfident results. The regression parameter ρ was inferred as 0.08 from the BCBC approach, indicating model form errors due to a low weighting. This understanding should lead to model improvement where ρ should subsequently increase, reflecting a simulator that better captures the physics. It can be seen in Fig. 5.7c that these model form errors exist, noted by the functional difference between the 0 and 2.5mm damage extents, leading to under-estimation of the mean for other damage extents.
Method 0.0mm 2.5mm 5.0mm 7.5mm 10.0mm BCBC-GH 0 0 0 0 0 BCBC-MCMC 0 0 0 0 0 BC 0 1 1 1 1 Method 12.5mm 15.0mm 17.5mm 20.0mm BCBC-GH 0 1 1 0 BCBC-MCMC 0 1 1 0 BC 0 0 1 0
Table 5.1: KS-test results for the three storey case study where α = 0.05. The hypothesis tests were applied to the BCBC using Gauss-Hermite quadrature (BCBC- GH), adaptive Metropolis MCMC (BCBC-MCMC), and Bayesian calibration (BC) predictions.
Hypothesis testing using the KS-test (and a significance level α = 0.05), shown in Table 5.1 revealed that all output predictive distributions for BCBC, using both the Gauss-Hermite quadrature and adaptive Metropolis MCMC, produced the same hypothesis test results. This demonstrates the similarity in inference approximations. The null hypothesis was rejected for the 15.0 and 17.5mm damage extents only, stating a good predictive performance. The rejection of the null hypothesis for these predictions is likely due to an offset in mean prediction, as shown in Fig. 5.7a and Fig. 5.7b. In contrast, five damage state predictions using Bayesian calibration had significant statistical differences leading to a rejection of the null hypothesis. This indicates the issues due to model form errors, which are visually present in Fig. 5.7c. The area metric, total variation and Hellinger distances were quantified and dis- played in Fig. 5.9. The area metric shows the large distances for the Bayesian calibration predictions in the first five damage extents, compared with the BCBC approaches. The total variation and Hellinger distances indicate quite even predictive quality between all three methods, with BCBC using adaptive Metropolis MCMC slightly outperforming the other two approaches. As a result it can be determined that although improvements are evident from both BCBC methods over Bayesian calibration alone, they are not consistently better across all individual damage states.