## 4 Dynamic Iteration Method
### Proposed Dynamic Iterative Method
When \(\varepsilon_{\mathrm{1 max}}\) =20%, \(p_{\mathrm{t}}\) = 0.3, and \(N_{\mathrm{s}}\) = 50, the step sizes of different partitions obtained using Equations (42) to (45) are shown in Figure 4.  
As shown in Figure 5, the results of 50 steps arithmetic sequence step size (dynamic iteration method parameters \(p_{\mathrm{t}}\) = 0.3, \(N_{\mathrm{s}}\) = 50) are similar to those of 3000 steps uniform step size, indicating that the dynamic iteration method effectively reduces the number of iterations to \(1/60\) of the original while maintaining accuracy. Figure 4: Step Size for Different Step Partitions. Figure 5: Comparison of Uniform and Arithmetic Sequence Distributed Iterative Increment Step Size. Figures 2, 5 and 6 present the stress-strain curve of the CSUH model using Toyoura Sand parameters from Table 1 under a confining pressure of 500 kPa and a void ratio of 0.84 in CU tests. According to Equation (16), theoretically, the numerical computation iterative increment step size, \(\mathrm{d}\varepsilon_{1}\), should be infinitely small. However, practical calculations reveal that 3000 steps are generally sufficient for high accuracy (as shown in Figure 2). Since the step size is not infinitely small in actual computation, \(\mathrm{d}\varepsilon_{1}\) is denoted as \(\Delta\varepsilon_{1}\), and the same applies to other principal strain and stress increments. Figure 6: _Cont._  
Figure 6 shows the computational results of the dynamic iteration method with different values of \(p_{\text{t}}\) and \(N_{\text{s}}\). The red line represents the calculation curve with 3000 uniform steps. Fifty interpolation points are taken at equal arc lengths on the curve to represent the experimental values. Corresponding points to the interpolation points are found on the dynamic iteration method calculation curve. The error between the dynamic iteration method calculation curve and the curve with 3000 uniform steps is measured using mean relative error (_MRE_) from reference [12]. The dynamic iteration method with \(p_{\text{t}}\) = 0 and \(N_{\text{s}}\) = 50 is equivalent to using 50 uniform step iterations. As shown in Figure 6a,b, the dynamic iteration method (blue dots) differs significantly from the 3000 uniform steps calculation curve (red), with an _MRE_ of 0.2495 for \(q\) and 0.5378 for \(u\). The dynamic iteration method with \(p_{\text{t}}\) = 0.09 and \(N_{\text{s}}\) = 50 yields the worst computational results, as shown in Figures 6c,d and 7a. Figure 7 shows the variation in MRE and parameter sensitivity analysis for different \(p_{\text{t}}\) and \(N_{\text{s}}\) values. The sensitivity analysis method adopts a local sensitivity analysis method, namely Composite Scale Sensitivity (CSS) [19]. Figure 7c shows that pt has a larger sensitivity while \(N_{\text{s}}\) has a smaller sensitivity, indicating that the choice of pt is the most important in the dynamic iteration method. From Figure 7a, it can be seen that when \(p_{\text{t}}\) is greater than 0.12, the MRE does not change significantly. Therefore, for this typical stress-strain curve with a slope that is initially large and then small, \(p_{\text{t}}\) = 0.3 is appropriate. As shown in Figure 7, under the extreme condition where the total number of iteration steps \(N_{\text{s}}\) is reduced to 30, the proposed dynamic iteration algorithm still demonstrates excellent numerical stability, with the generated stress-strain curves maintaining a high degree of agreement with the full 3000-step iteration results (_MREs_ of \(q\) and \(u\) are 0.041 and 0.1331, respectively). In contrast, the traditional numerical scheme with equal step-size iteration \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Parameters** & \(M\) & \(\
u\) & \(\kappa\) & \(\lambda\) & \(N\) & \(Z\) & \(\chi\) & \(m\) \\ \hline Values & 1.25 & 0.3 & 0.04 & 0.135 & 1.973 & 0.934 & 0.4 & 1.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters used for calculating theoretical curve values of the CSUH model [11].