## 1 Introduction  
Shot peening is a mechanical surface treatment that induces a surface layer of residual compressive stress (RCS). In the shot peening process, the surface of metal parts such as gears, springs, or turbine blades is sprayed by numerous small round metal balls and RCS is generated on the surface. RCS embedded in the surface plays a role in preventing the occurrence of cracks so that the fatigue strength of the pinched parts is improved [1; 2; 3; 4]. Engineers in charge of shot peening usually to conduct the Almen test before shot peening is carried out [5]. In this test, a thin strip (dimensions of 76.2 mm \(\times\) 18.9 mm with three commercially available thicknesses: 0.79, 1.29, and 2.39 mm) is peened with the same peening conditions as the actual parts of a given time period. The thin strip is called the \"Almen strip\" in the test. The Almen strip then bents since RCS induced by the peening leads the Almen strip to deflect from the peening direction [6]. The ejection of the balls is called impact, the balls are called shots. The maximum value of deflection of the Almen strip is called the arc height. In the Almen test, with the impact time increasing, the peening coverage increases and the arc height saturates. The value of the arc height at saturation of Almen strip is referred to as \"Almen intensity\". In shot peening process, it is difficult to test the RCS. Therefore, Almen intensity has been used as the standard for shot peening, which determines the shot peening parameters. [PERSON] et.al. [7] established a framework for predictive modeling of shot velocity. They found a relationship between Almen intensity and shot velocity. The effects of Almen intensity on microstructure and residual stress were investigated experimentally [8; 9]. [PERSON] et.al. [10] investigated the influence of the shot peening parameters on the hardness, residual stresses, and surface roughness. [PERSON] et.al. [11] proposed a comprehensive methodology to simulate a real shot peening process with minimal computation effort. [PERSON] et.al. [12] proposed a probabilistic methodology to evaluate the variability of the induced residual stress and Almen intensity of the shot peening parameters. Based beam bending principles, [PERSON] et al. [13] quantify the relationship between the several parameters affecting measured Almen intensity. [PERSON] et al. [14] proposed an analytic model to study the correlation of the Almen intensity with RCS, and compared the model-predictions with the measurements as a function of impact velocity, but a new problem was raised in assessing and interpreting the Almen intensity scale. [PERSON] [15] performed a 3D finite element (FE) analysis to relate the Almen intensity with the terms of shot velocity, but didn't present a relation between the Almen intensity and RCS. [PERSON] et al. [16] improved FEM-based approach that simulates the actual process of Almen strip peening by employing randomly located shots impacting on a part of Almen strip and the inclusion of strain-rate dependent target material properties. [PERSON] et al. [17] used the discrete element method in combination with the finite element method to obtain reasonably accurate predictions of the residual stresses and plastic strains. [PERSON] et. al. [18] examined the influences of the dimension and shape on the residual stress with 3D random impact FE model. In general, there have not been deeper studies about the influence of peening coverage on Almen intensity and RCS recently due to a lack of quantitative description of peening coverage. During the strip test or general shot peening test, numerous shots are arbitrarily bumped onto the Almen strip or specimen surface in a random sequence and random locations [19; 20; 21]. Therefore, the peening coverage is difficult to define in FE simulation of shot peening. [PERSON] et al. [22] proposed a 3-D finite element model that consists of some identical shots impacting randomly an aluminum target at normal or oblique incidence angles.