## 2 Bidirectional FSI Numerical Model
### Experiments and Validation
#### 2.4.2 Validation  
The numerical model was experimentally validated using an indoor simulated drilling platform. Figure 9 demonstrates the variation trend of driving torque with feed pressure. It can be observed that the experimental and simulated values of driving torque exhibit similar patterns, with the driving torque increasing as feed pressure rises. The maximum error of approximately 12.98% occurs at a feed pressure of 15 N, which can be attributed to the fact that the majority of the driving force is predominantly consumed in overcoming frictional resistance induced by manufacturing tolerances within the experimental setup.  
Figure 10 illustrates the variation trend of annular outlet velocity with the flow rate of drilling fluid in the wellbore. It can be seen that the experimental values and the simulated values have a high similarity, with a maximum error of approximately 6.87%, indicating that the model has a high accuracy.  
Figure 8: Indoor simulated drilling experimental platform: (**a**) overall view of the experimental platform; (**b**) schematic diagram of the sensor connection; (**c**) data acquisition system; (**d**) software.  
Figure 9: Variation trend of driving torque with feed pressure.  
Figure 11 illustrates the centroid trajectory at the midpoint of the drill string. The experimental and simulated values exhibit good agreement in motion trajectories and share the same trend, with the drill string primarily moving in the lower-right region of the wellbore. At higher rotational speeds, the simulated errors increase significantly. This indicates that when the rotational speed increases, the movement amplitude of the drill string increases while the randomness of the movement gradually increases.  
To further validate the high efficiency of the recursive algorithm adopted in this paper, a comparison was made with a typical explicit dynamics analysis method (LS-DYNA). The hardware specifications used in this study were an Intel Core i7-14700 KF CPU, an NVIDIA GeForce RTX 4090 graphics card, and 64 GB of RAM. The benchmark case used for this comparison was a drill string model with a total length of 20 m, simulated for a physical time of 3 s. The relevant comparison results are shown in Table 2. Comparative analysis demonstrates that the recursive algorithm employed in this study exhibits a significant computational efficiency advantage in solving drill-string dynamics with nonlinear contact. It substantially reduces computational time; for instance, under a mesh size of 1.8 mm, the recursive algorithm reduces the computational time by approximately 59.3% compared to conventional explicit methods.  
Figure 11: Trajectory of the drill string center at different rotational speeds: (**a**) 3.67 L/min, 256 rpm; (**b**) 3.67 L/min, 435 rpm.  
Figure 10: Variation trend of annular outlet velocity with flow rate of drilling fluid in the wellbore.  
In summary, systematic validation has thoroughly confirmed the effectiveness, accuracy, and significant computational advantages of the bidirectional FSI numerical model and its integrated recursive algorithm developed in this study.