# Optimal Design of Droplet Ejection for PZT Printhead Based on Surrogate Model
## 1 Introduction  
Microdroplet ejection technology has the ability to deposit materials at the micron level [1], which makes the technology widely used in industrial production, such as electronic printing, bioengineering, display devices and sand mold production [2; 3; 4; 5]. In the microdroplet ejection technology, the printhead is the key structure, and the piezoelectric (PZT) on-demand printhead is the most commonly used [6; 7]. Depending on the printing requirements, the PZT printhead can spray droplet deposits to produce the product. However, in droplet ejection technology, the generation of satellite droplets can easily cause surface defects such as model pores and sand inclusion. Some scholars have shown that through the optimization design of printing parameters, such as droplet characteristics or droplet speed, the generation of satellite droplets can be effectively reduced, and the molding accuracy of the model can be fundamentally controlled [8; 9; 10]. The essence of the optimized design to reduce satellite droplets is to improve the stability of the droplet ejection. Therefore, many scholars take stable ejection (that is, no obvious satellite dropletsare generated) as the research objective to optimize and study the ejection process and to set the parameters of the droplets [11; 12; 13]. However, the PZT printhead droplet ejection process is a complex fluid-structure interaction process, which is affected by many parameters, such as the frequency of the driving waveform, the physical properties of the ink and the size of the nozzle. It is very difficult to use or achieve traditional modeling methods to optimize the above parameters and obtain the optimal solution. Therefore, seeking a method that can replace the traditional model and accurately express the relationship between the influencing parameters (driving waveform, ink characteristics, etc.) and the ejection droplet is a prerequisite for designing optimized printing parameters, which is also a difficult problem to be solved. In recent years, the optimization design of printing parameters was mainly based on the simulation of the CFD numerical model, and the simulation results were used to optimize the design of individual parameters. For example, [PERSON] et al. [14] established a one-dimensional (1D) digital model between the deformation characteristics of the PZT membrane and the ejected droplets according to the working principle of the PZT printhead, which helped optimize the design of the PZT printhead and greatly improved the simulation speed. However, the study focused only on the kinetic effects of the PZT membrane, ignoring the effects of fluid dynamics on the jetting droplets. In order to improve the injection quality of droplets, [PERSON] et al. [15] established a two-dimensional (2D) CFD model of the PZT printhead to study the effects of operating parameters such as ejection aperture, driving waveform and fluid characteristics on the injection results and verified that high-quality droplets could be obtained by setting reasonable operating parameters. In order to improve the dynamic accuracy of the single droplet injection process of PZT, [PERSON] et al. [16] established a three-dimensional (3D) simulation model to study and analyze the relationship between the driving pulse, ink characteristics, nozzle diameter and spray droplet speed, volume and shape through the simulation results, and explained the influencing mechanism of process parameters on droplet injection. In studies of droplet injection, many scholars advocate that the characteristics of droplet injection are mainly determined by the driving pulse, so they are committed to optimizing the driving pulse to improve its stability. For example, [PERSON] et al. [11] designed three different driving waveforms and verified the effect of driving waveforms on droplet morphology through the volume of fluid (VOF) simulation model. To determine the optimal duration of the waveform and improve the droplet ejection accuracy, [PERSON] et al.