# Investigating Bayesian Parameter Identification Using Non-Standard Laboratory Specimens
## 5 Results
### Identification of Elastic Parameters  
The identification process begins with the training of a proxy model to ensure a fast yet precise approximation of virtual test results of the numerical model. To train the gPCE proxy model, a dataset of 800 parameter combinations was generated from the prior distribution using QMC sampling, with an additional set of 200 for accuracy testing. The computation of the results for this parameter sample by cluster computer took around 5 min to complete. Training the optimal gPCE model involves solving Equation (20) and selecting the polynomial degree that minimizes the error metrics between the proxy model and the generated test sample results from the numerical model. In this work, the mean Maximum Absolute Error (MAE) averaged over the timestep domain is used. A fourth-degree polynomial with 15 coefficients per timestep was found to provide the best fit.  
With the trained proxy model, the next step is to define the error distributions. First, we consider the measurement error distribution. For the force-displacement data, the standard deviation \(\sigma_{i}\) is set to 0.167% of the measured value. This is based on the maximum error specified by the UTM machine (class 0.5), assuming the maximum error represents 3\(\sigma\) under a normal distribution. For the lateral deformation measurements, the standard deviation is determined in the same way using the error specified by the LDTV manufacturer (class 0.5), and is also set to 0.167% of the measured value. Additionally, a constant component of 0.0001 mm is included to account for other sources of noise, such as those originating from the measuring circuitry, thermal fluctuations, and similar effects, which manifest as spikes due to the small magnitude of measured values (see Figure 7b).Since the identification of elastic parameters focuses on the material's linear elastic behavior, modeling errors are not considered at this stage. The constitutive law used in the numerical model is sufficiently accurate for this purpose. The standard deviation of the proxy model error is defined based on the differences between the predictions of the trained gPCE model  
\begin{table}
\begin{tabular}{l c c c} \hline \hline
**Parameter** & **Specimen** & **Measurement** & **Equipment** \\ \hline \(E\) & Rectangular & Force\(-\)displacement & UTM \\ \(\
u\) & Rectangular & Lateral deformation & LVDT \\ \(\sigma_{0},H,\sigma_{n},G_{fn}\) & Single\(-\)notched & Force\(-\)displacement & UTM \\ \hline \hline \end{tabular}
\end{table}
Table 2: Identification of elastic, elastoplastic, and fracture parameters.  
and the results obtained from 200 independent testing samples. These error distributions, along with the collected laboratory measurements, are presented in Figure 7. Here, only the measurement errors are visible, as the proxy model errors are very small due to the linear nature of the problem.  
The MCMC analysis is conducted using 100 Markov chains and 500 simulation steps, totaling 50,000 computed results for the combinations of the elastic parameters. The entire analysis was completed in less than a minute. The results of the identification are shown in Figure 8. The results shows a successful identification of both elastic parameters, as their posterior distributions have minimal variation, which ensures a high degree of certainty. The maximum a-posterior (MAP) value--the point estimate of the parameter, maximizing the posterior distribution--for [PERSON]'s modulus \(E\) is 66,542 N/mm\({}^{2}\), while the Poisson's ratio \(\
u\) is 0.319. These values are considered certain and are used as constants in the next phase of the identification.  
Figure 8: Identification of [PERSON]’s modulus \(E\) and Poisson ratio \(\
u\) parameters.  
Figure 7: Identification of elastic parameters\(-\)95% CI error distributions (rectangular specimen): (**a**) Measurement of force\(-\)displacement measurement (**b**) Lateral deformation measurement.