*4.3. Bayesian Calibration*

Finally, we proceed to perform a Bayesian calibration of the material models employed in the sensitivity analysis, keeping fixed at their nominal value those parameters that have been found to be non-influential in the sensitivity analyses. The calibration results considering both QoIs, ∆*R* and ∆*L*, are shown in Figures 13–18.

Specifically, Figure 13 shows the prior probability distribution functions provided for the three most relevant parameters, *A*, *B*, and *C* of the JC model, and their posterior probability functions. Similarly, Figure 15 illustrates the same probability functions, now for the most relevant parameters of the ZA model: namely, *C*0, *C*3, *C*5, and *n*.

**Figure 13.** Prior vs. posterior probability distribution functions of parameters *A* ∼ *θ*<sup>1</sup> , *B* ∼ *θ*2, *C* ∼ *θ*3, in the JC model considering ∆*R* as the QoI.

**Figure 14.** Prior vs. posterior probability distribution functions of parameters *A* ∼ *θ*<sup>1</sup> , *B* ∼ *θ*2, *C* ∼ *θ*3, in the JC model considering ∆*L* as the QoI.

Based on the results of the calibration, we can make general comments on the calibrated models. In the case of the JC constitutive law, the calibration process has notably sharpened the probability density function of the three most significant parameters (see Figures 13 and 14), eliminating a great part of the uncertainty linked to the variance in the prior probability distributions. Comparing the calibrated values of the parameters in the JC model obtained for the two QoIs analyzed (see Table 6), we note that both are similar. This suggests that the JC model is a good constitutive model for capturing the physics behind Taylor's test. In turn, the posterior probability distributions for the ZA model barely reduce the variance of the priors (cf. Figures 15 and 16). As a consequence, the calibration does not reduces significantly the uncertainty in the parameters. In addition, some of the calibrated parameters for the two QoIs under study have large disparities in their means. This is a consequence of the fact that, in our experiments, simulations carried out with the ZA model predict softer results, irrespective of the impact velocity and QoI observed. This fact, far from being a negative result, proves the potential of the method and illustrates that when the experimental and the simulation data are not in full agreement, the outcome of the calibration alerts of more uncertainties in the model and/or the data, or even the inability of the constitutive model to capture the physics of the problem.

**Figure 15.** Prior vs. posterior parameter probability distributions for *C*<sup>0</sup> ∼ *θ*<sup>1</sup> , *C*<sup>3</sup> ∼ *θ*2, *C*<sup>5</sup> ∼ *θ*3, *n* ∼ *θ*<sup>4</sup> , of the ZA model considering ∆*R* as the QoI.

**Figure 16.** Prior vs. posterior parameter probability distributions for *C*<sup>0</sup> ∼ *θ*<sup>1</sup> , *C*<sup>3</sup> ∼ *θ*2, *C*<sup>5</sup> ∼ *θ*3, *n* ∼ *θ*<sup>4</sup> , of the ZA model considering ∆*L* as the QoI.

Regarding the calibration of the Arrhenius model, it can be observed from Figures 17 and 18 that the variance reduction in the posterior probability distribution of its parameters is not as strong as for the Johnson–Cook constitutive law, although it is still significant when compared to the Zerilli–Armstrong case. Something similar happens when analyzing the (mean) calibrated parameters when considering the two QoIs. Even if there is a good agreement among them, the calibrated parameter *α*<sup>3</sup> obtained for the two QoI is fairly different. A potential explanation can be found, again, in the complexity of the constitutive equation and the effects of ignoring a large number of

the model parameters. While this conscious choice saves much computational cost, it causes the loss of information in the model behavior, that, either way, could be countered to a large extent with additional simulated data at a low computational cost.

**Figure 17.** Prior vs. posterior parameter probability distributions for parameters *A*<sup>2</sup> ∼ *θ*<sup>1</sup> , *A*<sup>3</sup> ∼ *<sup>θ</sup>*2, *<sup>α</sup>*<sup>3</sup> ∼ *<sup>θ</sup>*3, *<sup>n</sup>* ∼ *<sup>θ</sup>*<sup>4</sup> of the Arrhenius-type model considering <sup>∆</sup>*<sup>R</sup>* as the QoI.

**Figure 18.** Prior vs. posterior parameter probability distributions for parameters *A*<sup>2</sup> ∼ *θ*<sup>1</sup> , *A*<sup>3</sup> ∼ *<sup>θ</sup>*2, *<sup>α</sup>*<sup>3</sup> ∼ *<sup>θ</sup>*3, *<sup>n</sup>* ∼ *<sup>θ</sup>*<sup>4</sup> of the Arrhenius-type model considering <sup>∆</sup>*<sup>L</sup>* as the QoI.


**Table 6.** A posteriori mean values obtained for each parameter, considering both QoIs, ∆*R* and ∆*L*, for the calibration of the models.
