Explicación de la propuesta

Where Dk is the set of current known simulator outputs η for a given set of param- eter combinations θ, and H(p (θ)) = −R p (θ) log p (θ)dθ, the negative differential entropy, which for BHM is defined as in Eq. (6.24).

H(p (θi ∈ θnI)) = − Z

p (θi ∈ θnI) log p (θi ∈ θnI)

− (1 − p (θi ∈ θnI)) log(1 − p (θi ∈ θnI))dθ (6.24)

Equation (6.23) in practice proves demanding to evaluate as the entropy does not have an analytical solution and the distribution p θk_s| Dk∪ {θ, η}
must be calculated for numerous combinations of θ and η.

In contrast to ES, Predictive Entropy Search (PES) targets the mutual information between θk_s and η given Dkleading to a sequential design criteria defined in Eq. (6.25) [164]. P ES(θk_s) = H(p (η | Dk, θ)) − E_p(θk s| Dk) p η | Dk, θ, θ k s

(6.25)

This formulation leads to calculating posterior distributions (and their entropies) which for a GP have analytical forms or can be approximated more easily, simplifying the sequential design process. These approaches are likely to improve the efficiency and effectiveness of sequential BHM and are left as areas of further research.

6.2.3

Model Discrepancy

BHM accounts for model discrepancy by defining a prior variance Vm, stating an assumption of uniform additive discrepancy across the space. As stated in Section 6.2.2, BHM is a subcategory of ABC and therefore has the property of performing exact Monte Carlo inference for a uniform additive model discrepancy. In order to illustrate this result a numerical example is outlined.

6.2. METHODOLOGY 153 200 300 400 500 600 700 800 900 1000 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

Figure 6.6: BHM model discrepancy numerical example where the red (·) are the observational data with ±√Vo+ Vm bounds and the red (−) realisations from Eq. (6.27).

A simulator is constructed from a mass, tensioned wire system from Section 2.1.3 redefined in Eq. (6.26), where M is mass, T is tension l = 1m is the length and wn is the natural frequency.

η(x, θ) = wn(T, M ) = 1 π r T M l (6.26)

In the example model discrepancy is considered additive and sinusoidal, i.e. δ(x) = 0.5 sin(2π × 0.01x + φ) where φ ∼ U (0, 2π) is a random phase. The observational process is defined in Eq. (6.27) with a ‘true’ mass ˆθ = 5.43kg and observational uncertainty e ∼ N (0, 0.012_{). A comparison of the simulator and experimental data} is displayed in Fig. 6.6.

z(x) = η(x, 5.43) + δ(x) + e (6.27)

The uncertainties used in BHM are Vo = 0.012 from the noise and Vm = 0.5 due to the maximum and minimum of the discrepancy δ(x). A multivariate implausibility metric is implemented with a threshold calculated from the 99% quantile from a 10

(a) (b)

Figure 6.7: BHM model discrepancy numerical example. Panel (a) shows the approximate posterior over the parameters from importance sampling when κ = 2; the black (−) is the modal estimate and the red (−) the ‘true’ parameter value. Panel (b) presents Monte Carlo realisations of the simulator output given the parameter posterior where the blue (−) is the modal estimate, the red (·) are the observational data with ±√Vo+ Vm bounds and the red (−) realisations from Eq. (6.27).

degree of freedom χ2-distribution. The emulator is constructed from a linear mean function m(x, θ) = [x, θ]Tβ and Mat´ern covariance (where p = 2) and a nugget term ν = 1 × 10−8. A non-sequential approach is used where the parameter domain is uniformly sampled with 50, 000 samples. The parameter domain bounds were [2 20]kg and the experimental data was obtained at 10 equally space points from 200-1000N when φ = 0.

BHM reaches the stopping criteria after one wave and the approximate posterior from importance sampling is presented in Fig. 6.7a along with Monte Carlo realisations of the simulator output in Fig. 6.7b. It can be seen that the ‘true’ parameter value ˆθ = 5.43 is within the central probability mass with a modal estimate being θmode= 5.53 showing good agreement.

Another scenario of interest is when the model discrepancy is not a sum. This may occur in most practical engineering scenarios, where the missing physics are coupled with the known physics. In this scenario BHM should not be expected to perform exact inference, but will result in an inflated parameter posterior where there should be a portion of probability mass where the ‘true’ parameter occurs. In order to demonstrate this scenario a numerical example using the mass, tensioned wire system is demonstrated — as shown in Fig. 6.8. Here the observational process is an offset mass, tensioned wire system defined as in Eq. (6.28); where a = 0.2 and b = 1 − 0.2.

6.2. METHODOLOGY 155 200 300 400 500 600 700 800 900 1000 1 2 3 4 5 6 7

Figure 6.8: BHM model discrepancy numerical example with coupled discrepancy, where the red (·) are the observational data with ±pVo+ Vm(x) bounds and the red (−) realisations from Eq. (6.27).

z(x) = 1 2π

s

T (a + b)

M (ab) + e (6.28)

In this example the same settings are used as the previous example, however the model discrepancy uncertainty is defined as Vm(x) = [1, 1.11, . . . , 2], describing a linear increase in the model discrepancy. The offset produced by the model discrepancy in Eq. (6.28) will affect the ability of the BHM process to approximate the posterior parameter distribution. This is because the calibration process is still limited to the incorrect functional form defined by the simulator. In addition the offset will cause a bias in the posterior parameter distribution as is shown in Fig. 6.9a. Although the parameter posterior distribution contains probability mass at the ‘true’ parameter value there is a significant discrepancy between the modal estimate θmode= 3.54 and the ‘true’ parameter value ˆθ = 5.43. Furthermore it can be seen in Fig. 6.9b that the modal parameter solution produces an output that closes matches the observational data, with a NMSE of 0.17. This indicates that the method will try to calibrate the simulator given the modelling assumptions of a model discrepancy that is additive. In contrast, the result in Fig. 6.9a shows that the ‘true’ parameter is within the

(a) (b)

Figure 6.9: BHM model discrepancy numerical example with coupled discrepancy. Panel (a) show the approximate posterior over the parameters from importance sampling when κ = 2; the black (−) is the modal estimate and the red (−) the ‘true’ parameter value. Panel (b) presents Monte Carlo realisations of the simulator output given the parameter posterior where the blue (−) is the modal estimate, the red (·) are the observational data with ±pVo+ Vm(x) bounds and the red (−) realisations from Eq. (6.27).

probability mass, and given that in most real applications the model discrepancy is completely unknown, BHM can be a practical tool given the modeller limited knowledge.

6.3

Representative Five Storey Building Case Study

Calibration of five bending modes of a representative five storey building structure was performed using BHM in order to demonstrate the approaches applicability for forward model-driven SHM. Modal testing was performed on a representative five storey building structure made from aluminium 6082 under different pseudo-damage extents as shown in Fig. 5.10. These pseudo-damage extents were added masses m = {0, 0.1, . . . , 0.5}kg fixed to the first floor of the structure demonstrated in Fig. 5.10b. The structure was excited with a 409.6Hz bandwidth Gaussian noise via an electrodynamic shaker, with sample rate and sample time chosen to allow a frequency resolution of 0.05Hz. Accelerometers were placed at each of the five floors in order to obtain the first five bending modes. 40 averages were acquired for each measurement and ten repeats were performed for each damage extent in order to obtain an understanding of the underlying modal frequency distributions.

6.3. REPRESENTATIVE FIVE STOREY BUILDING CASE STUDY 157

Parameter Lower Bound Upper Bound

Elastic Modulus E 63.9GPa 78.1GPa

Poisson’s Ratio ν 0.297 0.363

Density ρ 2493kg/m3 3047kg/m3

Table 6.1: The prior parameter bounds for BHM on the five storey representative building structure.

The observational data z(xz) used within the calibration process were the mean natural frequencies when xz = {0, 0.3, 0.5}kg. The unseen validation set were the full repeat measurements of z(xz) as well as those from the {0.1, 0.2, 0.4}kg pseudo- damage extents, with the inputs collectively denoted as x∗. This highlights that with a small subset of damage data predictions can be made using BHM for forward model-driven SHM.

The simulator η(x, θ) was a modal FEA model where the five bending natural frequencies were extracted as a set of outputs y. Evaluations of the simulator were acquired for the six damage extents x = {0, 0.1, . . . , 0.5}kg and a range of parameter θ values within a set of prior bounds; set as ±10% of typical material properties for aluminium 6082 as shown in Table 6.1. Simulator runs for parameter combinations determined by a fifty point, three dimensional GMLHC, were implemented as training data for five independent GP emulators with a separate ten point three dimensional GMLHC used to generate validation data.