## 3 Comparison and Verification  
The finite element strength reduction method could achieve the ultimate failure state of the deeply buried tunnel construction failure model when the karst cave is located at the bottom of the tunnel by establishing the elastic-plastic finite element model and continuously reducing the internal friction angle and the cohesion of the rock and soil mass, so as to obtain the failure state of the deeply buried karst tunnel model and calculate the corresponding safety factors. Firstly, the two-dimensional numerical simulation model of deeply buried tunnel construction when the karst cave is located at the bottom of the tunnel should be established. The tunnel is 100 m long, the buried depth of the tunnel is 50 m, the diameter of the tunnel is 6.7 m, the diameter of the karst cave is 3 m, and the clear distance between the karst cave and the tunnel is 3 m.  
Secondly, the upper boundary of the numerical simulation model of the deeply buried karst tunnel is a free surface, the displacement constraint perpendicular to the boundary direction should be imposed on the side boundary, and the bottom boundary should be subjected to displacement constraints in two directions. The numerical simulation model only considers the self-weight effect of the rock and soil masses. Thirdly, the numerical simulation model of the deeply buried karst tunnel could adopt the physical property indices of the moderately weathered dolomite, as shown in Table 1. The internal friction angle and the cohesion of the rock and soil masses could be reduced once, and the finite element calculation model could be automatically iteratively calculated once until the construction of the deeply buried karst tunnel is simulated and destroyed, so as to obtain the corresponding safety factor. The non-intrusive stochastic finite element method for deep-buried karst tunnels could combine the finite element numerical simulation batch-processing calculations with the stochastic analysis theory, and it should establish the interface framework between the instability probability analysis in the construction of deep-buried karst tunnels and the universal finite element software after taking into account the effects of the random distribution of the parameters of the rock and soil bodies, so as to calculate the safety coefficients of the construction of the deep-buried karst tunnels. Firstly, the finite element strength reduction method could be applied to calculate the safety factors for the construction of the deeply buried karst tunnels. Secondly, the geotechnical body parameters could be simulated by means of the random field, thus obtaining n samples of the input variables and generating n finite element run codes by means of the editing program. Thirdly, these n code files of the finite element models could be operated, so as to calculate the ultimate stress value under the damage state of n deep-buried karst tunnel construction and compare with the ultimate stress value when the safety factor is 1. Eventually, the instability probability in the construction of the deep-buried karst tunnel could be statistically determined. Through comparing and analysing with the safety coefficients calculated by two methods, such as the non-intrusive stochastic finite element method and the finite element strength reduction method, therefore, the reasonableness of the calculation method of the instability probability of the deeply buried tunnel construction could be verified when the karst cave is located at the bottom of the tunnel. And the safety factors calculated by the four methods, namely, the non-intrusive stochastic finite element method, the finite element strength reduction method, the PH\(-\)1 failure mode, and the PH\(-\)2 failure mode, should be compared. In addition, in the non-intrusive stochastic finite element method, the random sample of 1500 was drawn from the range of internal friction angle 42\({}^{\circ}\)-52\({}^{\circ}\).