## 1 Introduction  
With the rapid growth of the city, land resources on the ground are becoming increasingly scarce, and at the same time, the demand for urban transportation is becoming greater. The rational use of underground space and the construction of subway lines became the top priority to alleviate traffic problems [1; 2]. Due to the rapid expansion of the urban rail transit network in recent years, more and more new projects have to interact with existing subway tunnels, which greatly increases the construction difficulty of the project [3; 4; 5; 6; 7]. It is common for new subway tunnels to cross under existing subway tunnels in urban construction [8; 9; 10; 11; 12]. The newly-built tunnel located beneath the existing tunnel will change the balanced stress field, cause the settlement of the strata, and change the internal force of the existing tunnel structure, which will interfere with normal operation ([PERSON] et al. [13]). In this regard, a number of scholars carried out studies. Some scholars studied the mechanical behavior of existing tunnel and ground deformation response through model tests and monitoring data ([PERSON] et al., [PERSON] et al. [14; 15]). More documents discuss the impact of underground excavations on existing tunnels and protective measures. [PERSON] et al. [16], [PERSON] et al. [17], and [PERSON] et al. [18] found that the existing tunnel is dominated by vertical deformation with insignificant horizontal deformation by means of the monitoring data and numerical simulation. There are other scholars who use theoretical analysis methods for their research. [PERSON] and [PERSON] [19] and [PERSON] et al. [20] proposed their own calculation models to calculate the deformation of existing tunnels caused by underpass tunnels by different methods and principles. The impact on existing tunnels is particularly significant when a new tunnel is underpassed for a short distance. It is challenging to ensure the serviceability of the existing tunnel and the safety of construction in the undercrossing area [21]. [PERSON] et al. [22], [PERSON] et al. [23], and [PERSON] et al. [24] mentioned the problem of new tunnels with a distance of only 2\(\sim\)3 m crossing under the existing tunnels, whose construction requirements are stricter. When a new tunnel crosses under an existing subway tunnel, there is often a need to leave a certain distance, which is helpful for safety after strengthening the strata between the tunnels ([PERSON] et al. [22]). However, there are situations where new construction projects, such as subway stations, cross under an existing tunnel at zero distance due to site restrictions or traffic demand. Close penetration faces more serious safety issues and is often closely related to the rock environment within the site [25], which poses a greater challenge to the safety of close penetration. At present, related studies are still relatively scarce. In the existing documents, [PERSON] et al. [26] introduced a protection method of using a jack to lift the existing tunnel above when closely crossing, and [PERSON] et al. [27] proposed a modified underpinning method, which is used as large-area support at the bottom to control the settlement of the existing tunnel in a subway station project in Seoul. In terms of theoretical research, [PERSON] et al. [28] proposed a superposition method based on the Winkler model to simulate the mechanical response of the new undercrossing tunnel and the existing tunnel when they are in close contact. However, this theoretical approach, which is only concerned with the initial and final engineering state of the excavation, neglects the effects of the excavation process and still has significant limitations. With the rapid development of computer technology, the numerical simulation method is becoming an important research method according to [PERSON] et al. [29]. Numerical simulations have the advantages of directness, efficiency, and low cost compared to theoretical studies and model tests.