Bayesian Inverse Problems by Juan Chiachio-Ruano;Manuel Chiachio-Ruano;Shankar Sankararaman;

Author:Juan Chiachio-Ruano;Manuel Chiachio-Ruano;Shankar Sankararaman; [Неизв.]
Language: eng
Format: epub
ISBN: 9781351869652
Publisher: CRC Press (Unlimited)
Published: 2022-06-07T21:00:00+00:00

During the monitoring stage, the MAP value from the calibration stage is used as pseudo-data for θ and the proposed algorithm is implemented based on ten sets of identified modal parameters 2r and r (r = 1, . . , 10) as the primary data for all damage cases. The stiffness ratios of the MAP estimates of the stiffness scaling parameters θ̃uv with respect to those inferred from the calibration stage (Config. 1) and their associated c.o.v. are also tabulated in Tables 5.6 and 5.7 for the three damage patterns. The actual damaged locations are made bold for comparison. It is observed that most of the components with non-bold font have their stiffness ratio exactly equal to one, showing they are unchanged by the monitoring stage data, and the corresponding c.o.v. for each of these components is zero, which means these substructures have no stiffness reduction with full confidence (conditional on the modelling) compared with that of the calibration stage. This is a benefit of the proposed sparse Bayesian formulation which reduces the uncertainty of the unchanged components. It produces sparse models by learning the hyper-parameter α, where α̃sv → 0 implies that Σ(sv)(sv) → 0 and so θ̃sv → ( ) sv. For the posterior uncertainty quantification of those components with bold font, the c.o.v. values in Table 5.7 for the partial sensor scenario are generally higher as compared with those in Table 5.6 for a full-sensor scenario, since the sensor data for the partial-sensor scenario provide less information to constrain the updated parameter values.

If we issue a damage alarm for a substructure when the corresponding stiffness ratio is smaller than one (stiffness loss is larger than zero), it is seen that no false or missed damage indications are produced for the full-sensor scenario. While for the partial-sensor scenario, two false detections are observed for the Config. 3.ps case in the substructures corresponding to θ2,+y and θ4,−y with 2.5% and 2.2% stiffness reductions, respectively. However, the value for no damage is around two standard deviations of 1% from the MAP values for 2,+y and 4,−y and so the probability of no damage is high. There are also two stiffness scaling parameters, θ5,−x for Config. 2.ps case and θ4,+y for Config. 3.ps case, where the stiffness ratio is larger than one; however, they show a very small increase in stiffness of 1% and 2%, respectively. For the identified damage extent, we observe that the identified stiffness ratios are close to their actual values (0.917 and 0.880) for both the full and partial sensor scenarios, and it is not surprising that the identified values are more accurate for the full sensor scenario. Therefore, the damage patterns are reliably detected in both qualitative and quantitative ways.

A more complete picture of substructure damage is given by the probabilities of damage, Pdamsv (f), with different fractional stiffness losses f ∈ [0,1], calculated using Equation (5.49). In Figure 5.5, the probability curves of damage are shown for the twenty θsv’s for various damage patterns from applying the proposed algorithm, for both the full- and partial-sensor scenarios.

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