# A Response-Adaptive Method for Design of Validation Experiments in Computational Mechanics
## 3 Response-Adaptive Experiment Design
### Procedure of Experimental Design  
The relationship between the observed response (\(y\)) and the predicted response (\(\psi\)) of a computational model can be defined.  
\[y(\zeta)+e=\psi(\zeta,\beta)+\delta,\zeta\in\Omega^{s},\beta\in\Omega^{l} \tag{6}\]where \(\zeta\) is the vector of operation variables \(\{\zeta_{1}\), \(\zeta_{2}\),..., \(\zeta_{s}\}\) (e.g., operational conditions and environmental temperature); \(\beta\) is the vector of model variables \(\{\beta_{1}\), \(\beta_{2}\),..., \(\beta_{l}\}\); \(s\) and \(l\) are the number of the operation and model variables, respectively; and \(e\) and \(\delta\) are the measurement and modeling errors, respectively. In HTVC, \(e\) and \(\delta\) are considered to be encompassed in \(y\) and \(\psi\), unless the true functions of \(e\) and \(\delta\) are quantified explicitly [18]. The effect of bad and missing data on classification problems was discussed in [13].  
For the model validation, additional experimental data in a validation domain are required. Experimental conditions are typically designed with model variables and operation variables in a validation domain. Thus, the vector of design variables (\(\theta\)) can be defined.  
\[\theta=\{\beta_{1},\beta_{2},\ldots,\beta_{v},\zeta_{1},\zeta_{2},\ldots,\zeta _{w}\},\theta\in\Omega^{(v+w)},v\leq l,w\leq s \tag{7}\]  
The procedure for RAED consists of six steps as shown in Figure 3.  
Step 1: Determine design variables and levels for factorial design. The level of individual design variables can be determined based on characteristics of the predicted model (e.g., linearity, nonlinearity, statistical variance), characteristics of design variables (e.g., continuous variable, discrete variable, boundary of the variables), the number of available test specimens, and expert opinions. Then, define experimental conditions based on the selected factorial design (e.g., full factorial and fractional factorial). The total number of experimental conditions (\(N\)) is calculated based on the number of design variables  
Step 4: Calculate the area metrics (\(U\)) of each set in Step 3 using all u-values except \(u_{i}\). For example, \(U_{1}\) is calculated using {\(u_{2}\), \(u_{3}\),..., \(u_{N}\)} and \(U_{N}\) is calculated using {\(u_{1}\), \(u_{2}\),..., \(u_{N-1}\)}. The sets of the area metric, \(u\)-value, experimental datum and test condition are {\(U_{1}\), \(u_{1}\), \(y_{1}\)}, {\(U_{2}\), \(u_{2}\), \(y_{2}\)},..., {\(U_{N}\), \(u_{N}\), \(y_{N}\)}, \(e_{N}\)}.  
Step 5: Make groups of \(U\) values that have an identical experimental condition and calculate the mean of the \(U\) values for each group. The sets of means of \(U\) values and test conditions are {\(\overline{U}_{1}\)}, {\(\overline{U}_{2}\)},..., {\(\overline{U}_{N}\)}, {\(e_{N}\)}. Find the set having the smallest \(U\) value mean and define the (\(N+j\))th experimental condition, \(e_{s(N+j)}\), as the same experimental condition of the selected set. The subscript \(s\) indicates the number of the experimental condition, which is between 1 and \(N\). If untested specimens remain, go to Step 6. Otherwise, terminate.  
Step 6: Obtain (\(N+j\))th experimental datum at the experimental condition, \(e_{s(N+j)}\). The sets of an experimental datum for each given experimental condition are {\(y_{1}\) | \(e_{1}\)}, {\(y_{2}\) | \(e_{2}\)},..., {\(y_{i}\) | \(e_{N}\)}, {\(y_{N+1}\) | \(e_{s(N+j)}\)}. Set \(j=j+1\). Go to Step 3.  
To present the procedures step by step, an example is shown in Figure 4. In this example, a two-level factorial design with two design variables is shown. However, any variation of the design can be made.