## 1 Introduction
[25] reported an experiment of a drop impact on pillar-arrayed surfaces fabricated by polydimethylsiloxane (PDMS). They found that low solid fractions can expand the range of impact velocity for complete bouncing. Other researchers pay attention to the effects of ridges on droplet impact [26; 27; 28]. The ridge is, in a sense, many pillars that are tightly packed together. The ridge causes the non-axisymmetric spreading and retraction of the droplet, which can reduce the contact time significantly [29; 30]. For simplicity, the majority of investigations in the literature have centered on the droplet impact dynamics on the stationary surfaces. In practical applications, however, the surfaces are moving when the droplet impact happens, such as aircraft wings, car windscreens, and turbine blades [31; 32; 33]. In the study by [PERSON] and [PERSON] [34], the dynamics of drop impact onto moving hydrophilic and hydrophobic surfaces was compared systematically. They reported that the moving surfaces mainly affect the lamellar extension and retraction stages due to spreading and stretching asymmetrically. Furthermore, [PERSON] et al. [35] observed that the stretching of lamella in the moving direction of solid surface is more obvious for low-surface-tension liquids, such as silicone oil and ethanol. [PERSON] [36] studied the impact of microdroplets on a moving solid surface by the lattice Boltzmann method (LBM). The droplet obtains the momentum imparted by the moving surface, resulting in a breakup of symmetry in the processes of the spreading and recoiling stages. This phenomenon is more pronounced when the droplet impacts the moving surface obliquely. Theoretically speaking, the asymmetric elongation of the drop impacting on a horizontally moving superhydrophobic surface can be attributed to the increased viscous force induced by the entrainment of the air layer [37]. Besides horizontal motion without reciprocating, the solid surface may also vibrate vertically or horizontally when the droplet impacts [38,39]. Under this mode of motion, many other factors should be discussed further, such as vibration frequencies, amplitudes, and phase angles [40, 41, 42, 43, 44]. Based on the study by [PERSON] et al. [45], the phase angle of surface vibration plays a vital role in influencing the droplet impact dynamics, and the rebound velocity is significantly correlated with the vibration frequency and phase angle. To sum up, previous research often idealizes the conditions of the droplet impact process to simplify the discussion. The combined influences of multiple factors should be proposed and analyzed further. As mentioned above, the regular structured surfaces with an array of micro- or nanoscale pillars are kinds of commonly used superhydrophobic materials. In a real work scenario, the pillar-arrayed surfaces may move, resulting in the destruction of the air layer between the droplet and the pillars. In this study, the effects of the stationary and horizontally moving superhydrophobic micro-pillar surfaces on the droplet impact dynamic behavioral characteristics are investigated numerically, focusing on the droplet morphology, spreading diameter, contact time, and energy conversion. This study aims to deepen the understanding of the interactions between the droplets and solid surfaces, which is beneficial for the optimization of multifunctional surface designs for actual applications.