*3.1. The Forward Problem*

We consider a 3-D Aluminum beam with mean elastic properties of *E* " 70 GPa, *ν* " 0.3 and *ρ* " 26.25 kN/m3. For the damping representation, we consider Rayleigh damping as **C** " *ηm***M** ` *ηk***K** with the constants *η<sup>m</sup>* " 0 and *η<sup>k</sup>* " 0.001. The stiffness **K** in the Rayleigh damping is based on the mean properties. We utilize FEniCS [25] for the forward finite element simulations. Figure 2 shows a 2D projection of the beam geometry, where we parameterize the inner cylinder by length *li* and radius *ri*, and the outer cylinder by length *lo* and radius *ro*. For the reference case, the inner and outer dimensions are *li* " 0.45 m,*ri* " 0.025 m and *ro* " 0.05 m and, *lo* " 1.0 m, respectively. The beam is subjected to an impact force defined as: *<sup>F</sup>*p*t*, **<sup>x</sup>**q"r0, 0, *<sup>F</sup>*0*t*{*tcδ*p*<sup>t</sup>* ´ *tc*qs*T*, where *F*<sup>0</sup> " ´5.0 GN and the ramp time is *tc* " 0.5 ms. The beam is fixed at both ends and subjected to zero initial displacement and velocity.

**Figure 2.** Schematic showing a 2D projection of a typical beam. For the reference case the inner and outer dimensions are (*li* " 0.45 m,*ri* " 0.025 m) and (*ro* " 0.05 m and, *lo* " 1.0 m), respectively.

We consider the vertical deflection at the mid-span to be the quantity of interest (QoI) in identifying the underlying beam geometry. Figure 3 shows the mid-span displacement and velocity for a the reference case.

**Figure 3.** The displacement and velocity at the mid-span of the reference case for the mean material properties *E* " 70 GPa, *<sup>ν</sup>* " 0.3 and *<sup>ρ</sup>* " 26.25 kN/m3.

#### *3.2. The Surrogate Model*

In order to infer the beam geometry from measurement of the QoI, many runs of the forward model, the 3D finite element dynamical code, are required. A surrogate model can overcome this issue by utilizing a limited number of a prespecified runs. The design of computer experiments concept can be used to optimally select the required runs [26–28]. For multi-fidelity simulations, where a high-cost high-accuracy and a low-cost low-accuracy simulators are available, a balance between the computational cost and accuracy can be achieved in designing the numerical simulations experiments [29].

The surrogate model is constructed based on samples that can represent the variability in the beam geometry due to different values of the inner dimensions. We define the variability of the inner dimensions by assigning a uniform random distribution with a specified bounds as *li* " *U*p0.25, 0.75qm and *ri* " *U*p0.01, 0.05qm. Using Latin hypercube sampling technique, we generate 50 independent samples for the inner dimensions. Using these samples, we generate the geometry of the beam followed by constructing the corresponding finite element mesh, and executing the forward model to calculate the mid-span deflection (QoI). Samples of the training geometries are shown in Figure 4. Clearly, the samples span a wide range of the probable geometries of the beam. The corresponding scatter of the

mid-span vertical displacement of the 50 samples is shown in Figure 5. The variability of the inner dimensions not only affect the geometry, but also the location and magnitude of the bouncing deflection at around time *t* " 0.002 s and *t* " 0.005 s.

**Figure 4.** Four samples showing the variability in the beam geometry due to different values of the inner dimensions (*li*,*ri*).

We randomly split the 50 samples into two groups as follows—40 samples for training and 10 for testing. For numerical implementation and to mimic the real world, we add a Gaussian random noise of strength (e.g., 10´<sup>3</sup> ˆ maxp*u*q) to the deflection measurements. Figure 6 shows samples of observed and predicted responses with confidence bounds for different values of the inner dimensions. The errorbars (based on two standard deviation) are indistinguishable within the scale of the graph. The maximum and minimum values of the mean squared error between the prediction and the observed response are 2.10 ˆ <sup>10</sup>´<sup>7</sup> and 5.35 ˆ 10´9, respectively. Given the fact that the testing samples are not seen by the model during the training phase, the GP model can predict the unseen data within the given accuracy.

To summarize the quality of the prediction, in Figure 7, we show the *L*2-norm of the observed and predicted QoI. The observed/predicted validation plot indicates that the coefficient of determination between the prediction and observation is *r*<sup>2</sup> " 0.98, and the corresponding mean squared error is 2.53 ˆ <sup>10</sup>´6. These statistical metrics indicate that the GP model can estimate the unseen geometry from a noisy measurement of the QoI within a given accuracy.

Once the GP model is validated, it can be deployed as a low-cost surrogate for the 3D finite element analysis code. The prediction of GP model takes only a fraction of the time that is needed by the finite element code to estimate the QoI with a fair accuracy.

**Figure 6.** Observed and predicted quantity of interest (QoI) for different testing samples. The test samples are not part of the training set. The errorbars are indistinguishable within the scale of the graph.

**Figure 7.** The observed/predicted validation plot showing the norm of the observed (test data) and the corresponding model predictions.

#### *3.3. The Backward Problem*

In the backward problem, we try to estimate the inner dimensions (*li*,*ri*) of the beam from noisy measurements of the QoI. We assume that a noisy measurement for the QoI is available as shown in Figure 8. The synthetic data is generated using inner dimension *li* " 0.313 m and *ri* " 0.055 m plus (*σ<sup>n</sup>* " 0.1 ˆ maxp*u*q) Gaussian noise to mimic a real experiment setting.

**Figure 8.** Noisy measurement of the QoI.

For the Bayesian calculation, we use non-informative prior for both the parameters *θ* " r*li*,*ri*s to assess the robustness of the inversion process. An adaptive MCMC method (DRAM) [16,17] is utilized to estimate the posterior density. In Figure 9, we show the estimated posterior density of the parameters *θ* " r*li*,*ri*s . We also show the prior density and the true value of the parameters. Note that the true parameters are not part of either the training nor the testing data sets. This highlights the robustness of the framework. The mean of the estimated values are *li* " 0.310 ˘ 0.048 m and *ri* " 0.054 ˘ 0.004 m (the confidence bounds are based on two standard deviation).

**Figure 9.** The estimated posterior density function of the inner dimensions *θ* " r*li*,*ri*s. The sold line is the posterior PDF, the dotted line is the prior PDF and the bullet dot represents the true value *li* " 0.313 m and *ri* " 0.055 m.

Next, the uncertainty in the parameter estimation represented by the posterior density in Figure 9 is propagated forward through the surrogate model to estimate a confidence bounds on the prediction of the QoI. In Figure 10, we show the model prediction and the 95% confidence interval as well as the true measured response. The *L*2-norm of the discrepancy between the mean model prediction and the measured data is 0.005 m. This conforms that the response due to the estimated parameters uncertainty agrees reasonably well with the true response. Note that, in the estimation of the localized region of interest, the material properties are assumed deterministic. For uncertainty propagation, the Maximum A Posteriori (MAP) estimation is used for the inner dimensions, while assuming random material properties in the region of interest. The relativity small errorbars (within

the scale of the graph) indicates that the single point estimation MAP can be used to set the inner dimension sufficiently.

**Figure 10.** The prediction of the surrogate model and its confidence interval due to uncertainty propagation of the variability in the estimated inner dimensions.

#### *3.4. Localized Uncertainty Propagation*

The QoI is confined within the core cylinder defined by inner dimensions *θ* " r*li*,*ri*s. Once these dimensions are available, the effect of the random variability in the material properties of the inner subdomain can be estimated using PC expansion. Without loss of generality, here we assume that for the inner cylinder, the Young's modulus and material density are random quantities, while Poisson's ratio is deterministic as

$$E(\mathbf{x}, \xi\_1) = \begin{cases} E\mathbf{e} (\mathbf{1} + \sigma\_E \xi\_1^{\mathbf{x}}), & \text{for } \mathbf{x} \in \Omega\_2 \\ E\_{0\prime} & \text{otherwise} \end{cases} \tag{31}$$

and

$$\rho(\mathbf{x}, \mathbf{\tilde{y}}\_2) = \begin{cases} \rho\_0 (1 + \sigma\_\theta \, ^\circ \mathbf{\tilde{y}}\_2), & \text{for } \mathbf{x} \in \Omega\_2 \\ \rho\_0, & \text{otherwise} \end{cases} \tag{32}$$

where the artificial boundary for Ω<sup>2</sup> are defined by MAP estimation of the inner dimensions *θ* " r*li*,*ri*s, *E*<sup>0</sup> " 70 GPa, *ρ*<sup>0</sup> " 26.25 kN/m3, *σ<sup>E</sup>* " 0.25 and *σρ* " 0.15 and *ξ*1, *ξ*<sup>2</sup> are standard normal random variables. Note that, not only the solution over Ω<sup>2</sup> is stochastic, but also over all the whole domain since the spatial finite element and stochastic basis functions are continuous across the domains interfaces. We use second order PC expansion to propagate the localized uncertainty due to the random Young's modulus and material density as shown in Figure 11.

To verify the PCE order, Figure 12 shows the error between the predictions of both the displacement and velocity using second and third order expansion. The error measure is defined as errorp'q " p'q3*rd* ´ p'q2*nd*. The relatively small values of the error confirm that the second order expansion is sufficient for uncertainty propagation for this problem.

The uncertainty bounds follow the trend of the response, with a higher value near the shock location. Although not explored here, high spatio-temporal resolution solver can be directed toward the region of interest, while a less resolution alternative can be assigned to the regions away from the QoI. As demonstrated in References [7–9], PASTA-DDM-UQ approach leads to a customized solver for localized uncertainty propagation with less computational cost.

**Figure 11.** The Polynomial Chaos (PC) prediction of the displacement and velocity at the mid-span. The uncertainty bounds represent two standard deviation.

**Figure 12.** The error between prediction of the 2nd and 3rd PC order for the displacement and velocity at the mid-span. The error measure is defined as errorp'q " p'q3*rd* ´ p'q2*nd*.

#### **4. Conclusions**

We present a data-based partitioning scheme for localized uncertainty quantification in elastodynamic system. The localized region of interest is identified using Bayesian inference framework. Measurement of the system response at one location in conjunction with a physics-based computational model is used to infer the localized features of the region of interested. A data-based surrogate model for the physics-based simulator is constructed using Gaussian process regression in order to reduce the computational cost of the Bayesian inversion. Material uncertainty in the region of interest is propagated through the system using polynomial chaos. We exercise our framework on a three-dimensional beam with localized feature and subjected to an impact load. The presented framework can facilitate quantifying the effect of the confined uncertainty in a localized region of interest within the global computational domain. Proper assessment of uncertainty at various level can accelerate the adaptation process of a new component introduced to an existing system.

**Author Contributions:** Conceptualization, W.S.; methodology, W.S.; software, W.S.; validation, W.S.; writing—original draft preparation, W.S., S.G., P.P. and Y.Z.; writing—review and editing, W.S., S.G., P.P.; supervision, L.W. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.
