# Optimal Design of Droplet Ejection for PZT Printhead Based on Surrogate Model
## 4 Optimization of the PZT Printehead Based on the Surrogate Model
### Sensitivity Analysis and Parameter Optimization  
The formation of the injection droplets in the working process of the PZT printhead involves the interaction of multiple parameters and physical fields. Therefore, in the early parameter optimization design, the main parameters affecting the droplet ejection results should be found for optimization. In this study, the Sobol method in the Global Sensitivity Analysis (GSA) algorithm is mainly used to complete the selection of the main parameters. The Sobol calculation method is mainly based on the Monte Carlo method of variance [27], and the combination of individual and other parameters can be expressed through different simulations. The process of estimating the sensitivity of the ejection parameters using the Sobol algorithm is as follows:  
1. Generate two groups of sample values, which can be expressed as; \[A= \begin{bmatrix}x_{11}\cdots x_{1i}\cdots x_{1d}\\ x_{21}\cdots x_{2i}\cdots x_{2d}\\ \vdots\\ x_{n1}\cdots x_{ni}\cdots x_{nd}\end{bmatrix} \qquad B= \begin{bmatrix}x^{ \prime}_{11}\cdots x^{\prime}_{1i}\cdots x^{\prime}_{1d}\\ x^{\prime}_{21}\cdots x^{\prime}_{2i}\cdots x^{\prime}_{2d}\\ \vdots\\ x^{\prime}_{n1}\cdots x^{\prime}_{ni}\cdots x^{\prime}_{nd}\end{bmatrix} \] 2. Switch the ith line parameters of matrix A and B to get the new matrix A and B; \[A_{i}= \begin{bmatrix}x_{11}\cdots x^{\prime}_{1i}\cdots x_{1k}\\ x_{21}\cdots x^{\prime}_{2i}\cdots x_{2k}\\ \vdots\\ x_{n1}\cdots x^{\prime}_{ni}\cdots x_{nk}\end{bmatrix} \qquad B_{i}= \begin{bmatrix}x ^{\prime}_{11}\cdots x_{1i}\cdots x^{\prime}_{1k}\\ x^{\prime}_{21}\cdots x_{2i}\cdots x^{\prime}_{2k}\\ \vdots\\ x^{\prime}_{n1}\cdots x_{ni}\cdots x^{\prime}_{nk}\end{bmatrix} \] 3. Substitute the values of each row in the new matrices A and B into the regression model as the input to obtain the corresponding response results; 4. Utilize the Formulas (13)-(16) to calculate the response of the vibration prediction; \[f^{2}_{0}=\frac{1}{n}\sum_{r=1}^{n}f(x_{r1},x_{r2},\cdots,x_{rd})f(x^{\prime} _{r1},x^{\prime}_{r2},\cdots,x^{\prime}_{rd})\] (11) \[\bar{V}(y)=\frac{1}{n}\sum_{r=1}^{n}f^{2}(x_{r1},x_{r2},\cdots,x_{rd})-f^{2}_{0}\] (12) \[\bar{U}_{i}=\frac{1}{n}\sum_{r=1}^{n}f(x_{r1},\cdots,x_{rd})f(x^{\prime}_{r1 },\cdots x^{\prime}_{r(i-1)},x_{ri},x^{\prime}_{r(i+1)},\cdots,x^{\prime}_{rd})\] (13) \[\bar{U}_{-i}=\frac{1}{n}\sum_{r=1}^{n}f(x_{r1},\cdots,x_{rd})f(x_{r1},\cdots x _{r(i-1)},x^{\prime}_{ri},x_{r(i+1)},\cdots,x_{rd})\] (14) 5. Calculate individual effect values \(x_{i}(\hat{S}_{xi})\) and overall influence effect values \(x_{i}(\hat{S}^{T}_{xi})\) of parameters, the process of which is as follows; \[\hat{S}_{xi}=\frac{\bar{U}_{i}-\hat{f}^{2}_{0}}{\bar{V}(y)}\] (15) \[\hat{S}^{T}_{xi}=1-\frac{\bar{U}_{-i}-\hat{f}^{2}_{0}}{\bar{V}(y)}\] (16) where the sum \(\hat{S}^{T}_{xi}\) of all input parameters is \(1\). The higher the \(\hat{S}^{T}_{xi}\), the higher effect of the input parameter. In this study, ink density \(\rho\), viscosity \(\eta\), surface tension \(\sigma\) and driving pulse period T values were selected for sensitivity analysis, and the results are shown in Figure 12. According to the calculation results, the individual and global sensitivity index of the ink density value is the highest, which means that the density value has the greatest influenceon the liquid column length and satellite droplet. The second is the ink surface tension and viscosity values, which have the same effect on droplet stability. The last is the driving pulse period value, which has the smallest influence. The four above parameters are the main parameters that have been confirmed to affect the accuracy of droplet injection, but why the density value affects the proportion the most needs further research and judgment. The stable ejection of droplets along the density and viscosity directions is plotted to verify this, as shown in Figure 13.  
Figure 14 shows the trend diagram of the influence of design parameters \(\eta\), \(\rho\), \(\sigma\) and T on the number of satellite droplets and droplet height Hd.