## 1 Introduction  
Vent holes, pin holes, bolt holes, and other holes are widely distributed in aero-engines, airplanes, automobiles, ships, and other mechanical products. The shapes of these holes may be different because of the diversity of service requirements. Circular holes are the most common, and complex shapes such as elliptical and polygonal holes are also widely used. Due to sudden changes in geometry and load, the areas near the holes are prone to stress concentration [1], which may become a potential inducement to reduce the product's overall life. Especially for the high-temperature components of an aero-engine, the alternating loads at high temperatures make it easy to produce fatigue cracks in the high-stress areas around the holes [2, 3]. These cracks may extend to nearby load-bearing structures and eventually lead to structural failure. Therefore, it is essential to obtain an accurate stress distribution near the holes. The stress distribution around the holes has been extensively studied by analytical methods [4, 5], finite element methods (FEM) [6], and experimental methods [7, 8, 9]. [PERSON] al. [10] introduced a general shear-stress solution to the anti-plane problem of an arbitrarily shaped hole that was reinforced with a functionally graded layer in a homogenous plate. [PERSON] et al. [11] investigated the high-temperature tensile behavior for film-hole plates and found that the strain distribution obtained in situ by the digital image correlation technique exhibits an X-like concentration around the hole. The FEM-based simulation was also conducted to explain the in-situ experimental results. Using a self-developed theoretical model, [PERSON] et al. [12] investigated the circular hole-edge stress concentration of the long glass fiber reinforced polypropylene composite. The corresponding physical tests were also conducted under tensile loading monitored by a digital image correlation system. The results showed that the theoretical predictions agree well with the experimental results. The analytical methods mainly utilize the theory of complex variable functions and the conformal mapping technique to obtain the stress distribution of a rectangular or infinite plate with holes [10]. It may be insufficient to deal with the holes with complex shapes or particular positions in practical engineering structures with advanced materials [13; 14]. The experimental methods could measure the stress results closest to the actual situation. However, the operation procedure is complex and inconvenient for newcomers, and the test cost is relatively high and unbearable for most design tasks. With the rapid development of computing technology, the accuracy and efficiency of FEM have been continuously improved and recognized by academia and industry [15]. The FEM-based optimization has become popular in the shape optimization of holes to reduce the stress concentration with the help of numerical optimization techniques [16; 17]. [PERSON] et al. [18] carried out a FEM-based optimization for the shape and pattern of holes and slits of a brake disk, which minimizes the concentrated stress and simultaneously improves the cooling performance. [PERSON] et al. [19] proposed an equilibrium multi-objective optimization model with self-regulated weighting factors for the optimum design of a non-circular clearance hole on the front flange of a turbine disk. The stress decrease around the hole and the minimal hole's profile variation were considered in the equilibrium design. [PERSON] et al. [20] optimized a uniaxial symmetry non-circular bolt clearance hole on the turbine disk based on FEM by selecting the maximum first principal stress as the objective function. The results showed that the stress concentration is relieved, and the maximum first principal stress and the maximum von Mises stress on the critical area are reduced by 30.39% and 25.34%, respectively. The author of [21] proposed a four-arc hole design method for vent holes and adopted the traditional Multi-Island Genetic Algorithm to optimize them.