## 1 Introduction  
Aerostatic spindles are widely employed in various kinds of precision equipment for their distinct advantages of ultra-high precision, negligible friction, and wide rotating speed range [1]. However, the drawbacks of low capacity, low stiffness, and poor high-speed stability restrict their possible application significantly. Many factors have a great impact on the overall performances of aerostatic spindles, such as the geometrical parameters of the aerostatic bearing, the supply pressure, the rotating speed, and the restrictor type, etc. Hence, it is difficult to optimize the design parameters of aerostatic spindles comprehensively. Numerous research studies have been conducted to facilitate the design of aerostatic bearings by investigating the impact of the air film geometrical dimensions on their static performance [2, 3]. [PERSON] et al. [4] numerically studied the performances of the orifice-type restricted aerostatic thrust bearings with varying geometrical parameters based on computation fluid dynamics, and the effect tendency of geometrical parameters to the bearing performances was acquired. [PERSON] [5] and [PERSON] et al. [6] investigated the stiffness, load-carrying capacity (LCC) of aerostatic thrust bearings and journal bearings with pressure-equalizing groove (PEG), and found that the PEG can enhance the static performance of aerostatic bearing dramatically. Many studies also have been carried out on the optimal design approach of air bearings [7, 8, 9]. [PERSON] et al. [10] proposed an optimal design approach based on a genetic algorithm, and a hybrid selection scheme was adopted when selecting the mating groups, which is more efficient. [PERSON] et al. [11, 12] optimized the orifice diameter, orifice position, and orifice number of inherently compensated rectangular aerostatic pad bearings based on a multi-objective genetic algorithms optimization approach. To optimize the design parameters and supply pressure of a rectangular aerostatic thrust bearing, [PERSON] et al. [13] proposed a Pareto optimization method to avoid the variation of the optimal results with different initial populations; the trial points are therefore distributed in the design space uniformly. However, the structural deformation caused by pressurized air film was neglected in the aforementioned researches. Since the performance of the aerostatic bearings is significantly impacted by the air film thickness, this phenomenon may induce great error when calculating the performance of the aerostatic, and the initial design requirements may not be satisfied by the actual performance of the aerostatic bearing. This phenomenon is especially dramatic in the design of aerostatic bearings with high stiffness and high LCC, such as the spindle of fly cutting machine tools, rotary tables, etc., and has drawn increasing attention in recent years [14; 15]. [PERSON] et al. [16; 17] numerically and experimentally investigated the performance of an air spindle considering the fluid-structure interaction (FSI) effect. They found that the LCC curve declines slightly due to the FSI effect, and the axial stiffness declines with the decrease of the thrust thickness plate. In a study by [PERSON] et al. [18], the performance of an aerostatic spindle was investigated by a two-way FSI simulation model. It showed that the air film thickness changes about 7.65 \(\upmu\)m due to the structure deformation and the stiffness of the spindle declines about 34%. According to the above literature review, it can be seen that the design of aerostatic bearing includes the design of the air bearing parameters and the design of the crucial dimensions of the solid parts. Even though many optimal design approaches of air bearings have been proposed, the FSI effect is seldom considered, and the crucial structural dimensions are usually decided empirically. Therefore, the aerostatic spindle cannot be optimized comprehensively from its LCC, volume flow rate (VFR), stiffness, natural frequency, etc.