**3. Application**

The methodology presented in Section 2 is applied now to a relevant example in mechanics of deformable solids, namely, Taylor's impact test [21,22]. In what follows, we will study the calibration of the three material models of Section 2.4 based on the outputs obtained from this well-known test that consists of a high-velocity impact of a metallic anvil onto a rigid wall. As illustrated in Figure 2, the impact creates irrecoverable deformations in the anvil that, due to the symmetry of the problem, can be macroscopically quantified by measuring the changes in the diameter and length of the impactor.

**Figure 2.** Schematic of Taylor's impact test.

Figure 3 illustrates the procedure advocated for our numerical analysis: starting from a prior distribution for the material parameters, a meta-model of Taylor's impact test is constructed based on anisotropic RBF. The meta-model, once completed, is cheap to run and can be used to perform sensitivity analyses and to update, via Bayesian calibration, the probability distribution of the original parameters. If deemed necessary, the latter probability distribution can be reintroduced in the Bayesian calibration, this time as prior, as illustrated in Figure 3, until the parameter distribution converges to an (almost) stationary function. In theory, one could use the posterior probabilities to start the whole process, helping to build a better meta-model that will be later employed in the GSA and calibration. This route, however, might be too expensive in real life applications.

**Figure 3.** Iterative process for a two-stage approach of screening and calibration of model parameters.

To build the meta-model, five impact velocities are selected over a typical range of Taylor's bar experiments: namely, 200, 230, 260, 290, and 320 m/s. Then, the different tests for each impact velocity are simulated considering a Cr-Mo steel as the anvil's material. Each impact velocity point consists of 612 simulations for the Johnson–Cook and Zerilli–Armstrong models, and 1800 for the Arrherius-type, since the latter involves a larger number of material parameters and requires more data in order to get reliable levels of accuracy when constructing the meta-models. The parameters fed to the simulations have been sampled from uniform distributions centered at nominal values taken from the literature [19,33] with ±10% ranges, varying them according to a Low Discrepancy Design method (LDD), or Sobol sequence [34]. The latter is obtained with a deterministic algorithm that subdivides each dimension of the sample space into 2 *<sup>N</sup>* points, while ensuring good uniformity properties.

The QoIs selected for the meta-model are ∆*R* and ∆*L*; that is, the changes in radius and length of the anvil after impact. Using the methods described in Section 2.2, an RBF-based meta-model is obtained for each material and impact velocity. The meta-models now serve as the basis for the Global Sensitivity Analysis that will identify the most significant parameters in each model, ruling out from the Bayesian calibration those whose influence on the QoIs is relatively small. Finally, for each of the material models, a full Bayesian analysis will be done based on the concepts of Section 2.3, providing a fitted Gaussian process per model and QoI. This last step demands standard but cumbersome operations and has been performed using a freely available R package [35].

To complete the Bayesian calibration, we need meta-model predictions for arbitrary velocities of the impactor. Since the available meta-models are only defined for five selected velocities, we will interpolate linearly their predictions for the QoI at any intermediate velocity (see Figures 4–6). This strategy will speed-up the generation of data for the Bayesian analysis. To validate it, we will first confirm that the error made by this interpolation is negligible. For that, we will compare solutions obtained with FE simulations at arbitrary velocities of the anvil against interpolated meta-model predictions.

**Figure 4.** Linear interpolation of meta-model predictions of ∆*R* for the Johnson–Cook constitutive relation. Each piecewise linear interpolation connects predictions with the same model parameters.

**Figure 5.** Linear interpolation of meta-model predictions of ∆*R* for the Zerilli–Armstrong constitutive relation.

**Figure 6.** Linear interpolation of meta-model predictions of ∆*R* for the Arrhenius-type constitutive relation.

Once accepted, this strategy for combining meta-models will result in an extremely cheap source of simulated data that will be used to study the material models. For each of the latter, the fitting data will consist of *<sup>n</sup>*<sup>1</sup> = 20 sets of observed points <sup>∆</sup><sup>1</sup> = {*x*1, . . . , *<sup>x</sup>n*<sup>1</sup> }, plus *n*<sup>2</sup> = 500 sets of computational outputs, derived from the meta-models interpolation, ∆<sup>2</sup> = (*x* ′ 1 ,*t*1), . . . ,(*x* ′ *n*2 ,*tn*<sup>2</sup> ), where *x<sup>i</sup>* and *x* ′ *i* are the experimental impact velocities and interpolated impact velocities acting as the variable input, respectively, while *t<sup>i</sup>* are the parameter inputs to the meta-model. To assess the results of the meta-models interpolation, the FE cases against which are to be compared will be generated employing the same parameter inputs *t<sup>i</sup>* and impact velocities *x* ′ *i* .

In this work, we have chosen to calibrate CrMo steel because the parameters for the JC, ZA, and ARR models could be found in the literature for this material. However, no experimental measurements are available for Taylor tests with anvils of this material. Hence, we follow an alternative avenue to obtain data, one that is often employed in statistical analyses [36,37]. The idea is to generate data from finite element simulations (20 in our procedure) using a fixed material model with nominal parameters, exploring all impact velocities and adding white Gaussian noise to all the measured QoIs, consistent with Equation (18).

To complete the problem definition, it remains to choose prior probability distributions for the complete set of material parameters *θ*, the variance of the global observation error *λ* 2 , and the hyperparameters *χ<sup>δ</sup>* of the discrepancy function. Tables 1 and 2 describe the probability distributions chosen for each parameter in the three models and the references employed for their choice.


**Table 1.** Prior probability distributions.


**Table 2.** Mean values for the parameter distributions according to literature [19,33].

## **4. Results**

We now present the results of the GSA analyses, the meta-models interpolation and calibration procedure for the three material models described in Section 2.4 based on the results obtained from the experiments of Taylor's anvil impact. These are obtained from the RBF meta-model whose construction is detailed in Sections 2.2 and 3.
