## 1 Introduction  
With the continuous progress of urbanization, the development and utilization of urban underground space resources have also been intensifying day by day. Among them, subway construction can relieve urban traffic congestion effectively, enabling more scientific and rational urban planning and reducing pollution, which helps to achieve green travel. The interaction between subway tunnels and surface buildings was the main technical problem encountered in the construction of early urban subway projects. A series of studies have been carried out around this topic by many scholars [1; 2]. As the urban population grows rapidly, the capacity of existing subways is no longer able to meet people's travel demands. As a result, the development and utilization of underground space resources have further intensified. Since the continued subway construction in the crowded urban space were only challenged in depth, the construction of subways at the present stage gives rise to another technical problem, namely the interaction between new and existing tunnels. In recent years, the interaction of overlapping tunnels in urban underground spaces has gradually become a hot research topic. [PERSON] et al. [3] analyzed the deformation mechanism of excavated caverns in the existing tunnels through numerical simulations. Their study took into consideration the effects of different tunnel depth/diameter ratios and circular cavern diameter/depth ratios, and verified the numerical model by utilizing centrifugal test results. In the past few years, they have explored the influences of factors like excavation geometry, excavation depth, ratio of overburden to tunnel diameter (depth/diameter ratio), and sand layer density on the tunnel invert excavation response based on the results of centrifuge tests via a numerical approach, and put forward a rather simple but practical evaluation method [4]. [PERSON] et al. [5] investigated the soil stress variation and settlement evolution resulting from the orthogonal passing of a planned two-line earth pressure balance (EPB) shield tunnel underneath an existing tunnel and concluded that the shield exterior grouting in the construction process was effective in controlling the settlement. They also performed a parameter sensitivity analysis with Spearman's correlation coefficients, further finding that the elastic modulus of the filling material was a sensitive parameter affecting the settlement of existing tunnels and the ground surface. In a study by [PERSON] et al. [6], the influence caused by the oblique crossing of a newly built double-line tunnel above an existing tunnel was studied by simplifying the new tunnel into a continuous Euler-Bernoulli beam with a certain equivalent flexural stiffness. They also studied the effects of gap distance, propulsion distance, and multi-tunnel construction on the existing tunnels. Deformation characteristics attributed to the oblique passing of a newly built twin tunnel underneath an existing tunnel were studied by [PERSON] et al. [7] by focusing on discussing the variation trends of a ground settlement trough and the soil pressure acting on an existing tunnel during the shield tunneling. Then, they further derived the lateral deformation, internal force, and torsion characteristics of existing tunnels caused by a new tunnel excavation. Finally, they explored the influence of the skew angle variation of spatially overlapping tunnels on the deformation of existing tunnels and found that the skew angle was significantly influential on the maximum settlement of the existing tunnel floor, but was only a little influential on the maximum settlement of the existing tunnel roof. Employing PLAXS software numerical analysis, [PERSON] and [PERSON] [8] studied the surface settlements and bending moments of circular and rectangular twin tunnel linings, where the key research parameters included the relative position, critical distance, volume loss, overburden, and size of the twin tunnel. [PERSON] et al.