# Characterization and Finite Element Modeling of Microperforated Titanium Grade 2
## 2 Materials and Methods
### Calibration Process  
RSM is a group of statistical methods that uses DoE, polynomial functions, and the gradient descent method. Nowadays, RSM has been and continues to be used to implement multi-objective optimizations of processes both experimentally [19, 20] and by models [21, 22]. The first step is to define the variables that will be studied (inputs and outputs). In this case, the inputs are the anisotropic ratios (\(R_{22}\), \(R_{33}\), \(R_{12}\)) and the outputs are the errors of FE models' results compared with Nakajima measurements. The range of the three inputs is 0.65-1.35. The flow diagram presented in Figure 7 summarizes the relationship between testing, FE modeling, and the calibration process.  
In this case, there are three inputs (factors) and therefore the number of combinations is \(3^{3}=27\) considering three values per factor. This kind of design is called a full three-level design (or 3 k) because it considers all possible combinations with three values (also called levels). There are seven geometries and three directions (except for G7) for each case; thus, there are 27-(6-3 + 1) = 513 models that must be launched. Note that geometry G7 has only one direction because of its round shape. There are several methods to reduce the number of combinations (such as Box-Behnken or Central Composite Designs [23]). However, such reduction is not needed in this case because the computational cost of Nakajima test models is relatively low: around 3 min using parallel computing with 12 cpus (Intel Xeon E5-2690v3, 192 GB RAM).  
For each case, nineteen models are launched, each one with a different geometry (as shown in Figure 2) and different direction. When all these simulations are completed, an  
Figure 6: Displacement and holder force applied to the FE model.  
Figure 7: Flow diagram of the testing, modeling, and calibration process.  
output for each case, geometry, and direction is calculated using the Root Mean Square Error (RMSE):  
\[Y_{kjo}=RMSE_{kjo}=\sqrt{\frac{1}{n_{j}}\cdot\sum_{i=1}^{n_{j}}\left(Y_{j\o i}^{ EXP}-Y_{kjoi}^{FEM}\right)^{2}} \tag{11}\]  
where  
\(Y_{kjo}\) is the output for geometry j and orientation \(o\) for case \(k\). It quantifies how different the FE model punch is compared to the experimental punch;  
\(n_{j}\) is the number of measurements (time points) of Nakajima geometry \(j\) in each curve;  
\(Y_{joi}^{EXP}\) is the experimental punch stroke of Nakajima geometry \(j\) and orientation \(o\) at time point \(i\);  
\(Y_{kjoi}^{FEM}\) is the FE model punch stroke of Nakajima geometry \(j\) and orientation \(o\) at time point \(i\) for case \(k\).  
Using all these inputs and outputs, it is possible to fit a low-degree polynomial. Quadratic functions are one of the most used functions to relate inputs and outputs:  
\[Y=b_{0}+\sum_{i=1}^{n}b_{i}\cdot x_{i}+\sum_{i=1}^{n}b_{ii}\cdot x_{i}^{2}+ \sum_{i=1}^{n-1}\sum_{j=i+1}^{n}b_{ij}\cdot x_{i}\cdot x_{j}+e \tag{12}\]  
where the first summation is the linear component, the second is the quadratic component, the third is the product of the variables, and \(e\) is the error. The values of the coefficients \(b_{0}\), \(b_{i}\), \(b_{ii}\), and \(b_{ij}\) must be calculated by the minimum squared method. Once the quadratic models are fitted, it is possible to find the optimum inputs (\(R_{22}\), \(R_{33}\), \(R_{12}\)) by using the steepest ascent method [24]. The objective of the optimization is to minimize the nineteen outputs at the same time. When a problem has multiple outputs, it is called the Multi-Response Surface Method (MRSM), which deals with conflict between responses. An optimal configuration for one output may diverge substantially from the optimal configuration for another output. [PERSON] [25] presented a compromise between responses by so-called desirability functions. In this case, there are nineteen outputs (one for each Nakajima geometry and orientation). Here, we use Harrington equations to optimize the anisotropy ratios to minimize the outputs defined in Equation (11).