# A Response-Adaptive Method for Design of Validation Experiments in Computational Mechanics
## 4 Case Study  
In real engineering applications, it is almost impossible to know the mother distribution of the experimental data since this would require an infinite number of experimental samples. To overcome the challenge, simulation samples generated by numerical models are often used. For instance, the use of numerical models is considered as a natural way in the study of phenomena arising in applied sciences [20; 21] such as the diffusion and transport of pollutants in the atmosphere [22], the dissipation of nonlinear waves [23; 24], and the growth models in ecology and oncology. In this case study, a numerical model is employed to quantify type II error and to show the effectiveness of the proposed RAED method. The numerical model (\(\psi\)) is devised to have a generic exponential form with one random variable (\(\beta\)), one operating variable (\(\zeta\)), and two model parameters (\(p\) and \(q\)). \[\psi=(p\times\beta)\times\zeta^{q} \tag{9}\]  
The exponential form well represents various engineering applications, such as friction of a rubber material [25], fatigue of a solder joint [26], and transient thermal conduction [27]. For example, the friction model (\(\mu\)) of a rubber material [25] is a function of contact pressure (\(\varphi\)) with two model parameters: (1) friction coefficient \(\mu_{0}\) at a reference contact pressure \(\varphi_{\mathit{ref}}\) and (2) pressure exponential parameter \(v_{0}\). Two model parameters can be figured out with experimental data obtained at different operating conditions of contact pressure. \[\mu=\mu_{0}\times\left(\frac{\varphi}{\varphi_{\mathit{ref}}}\right)^{-v_{0}} \tag{10}\]  
To demonstrate the proposed RAED method, we assumed that the predicted model and the mother function of the experimental results have different parameter values. \begin{table}
\begin{tabular}{c c} \hline
**Cases** & **Type II Error** \\ \hline Condition I & 0.35 \\ Condition II & 0.00 \\ \hline \end{tabular}
\end{table} 
Table 2: Calculated type II errors. Predicted model:  
\[\psi=(p\times\beta)\times\zeta^{q};p=1.0,\,q=1.0 \tag{11}\]  
Mother function of experimental results:  
\[y=(p\times\beta)\times\zeta^{q};p=0.896,\,q=1.1 \tag{12}\]  
where \(\beta\) is the random variable that follows a normal distribution with the mean of 1.0 and the standard deviation of 0.5; \(\zeta\) is the operating variable that is assumed to be between three and seven (i.e., \(3\leq\zeta\leq 7\)). The confidence intervals of the mother function of the experiments (\(y\)) and the predicted model (\(\psi\)) are in the validation domain, as depicted in Figure 7. The predicted model represents the experimental results when \(\zeta\) is set to three. In contrast, the difference between \(y\) and \(\psi\) stays larger as \(\zeta\) is increased to seven. With an assumption of having eighteen samples for model validation, three different experimental designs were considered to show the benefits of RAED, including:  
Experimental Design A: Consider three levels of different operating conditions (\(\zeta=3,\,5\) and \(7\)) and obtain six data points at each level. Experimental Design B: Obtain eighteen data points at operating conditions evenly distributed between \(3\leq\zeta\leq 7\). Experimental Design C: Consider three levels of different operating conditions (\(\zeta=3,\,5\) and \(7\)) and obtain eighteen data points using RAED. The PDFs of the area metric (\(U\)) were constructed according to the three designs by following the steps outlined in Section 3.1. It was assumed that the virtual sampling size (\(k\)) was 10,000 and the number of sampled data (i.e., the number of test specimens) was eighteen. The PDFs of individual area metrics are shown in Figure 8. Experimental data were sampled based on the three designs A, B and C. It was confirmed that the \(f_{u,18}\) of Design C shifted further to the right than the distributions of Designs A and B. Table 3 summarizes the statistical information and type II error of the PDFs.