## 4 Dynamic Iteration Method
### Proposed Dynamic Iterative Method  
The slope of the stress-strain curve of soil is not uniform. As shown in Figure 3, the slope of the curve is steep in the front part and gentle in the latter part. By segmentally adjusting the model's iterative increment step size, the total number of iterations can be reduced, thereby accelerating the speed of parameter inversion. When the curve slope is large, a small axial strain increment, \(\Delta\varepsilon_{1}\), can cause a significant stress increment, whereas when the curve slope is small, a larger axial strain increment, \(\Delta\varepsilon_{1}\), induces only a minor stress increment. If the same iterative increment step size, \(\Delta\varepsilon_{1}\), is used regardless of the curve slope, \(\Delta\varepsilon_{1}\) must be set sufficiently small to ensure accuracy when the slope is large, which results in a high number of iterations and slow computation. To enhance the numerical computation efficiency of the constitutive model, a dynamic iterative method that adjusts the iterative increment step size in segments is proposed to reduce the total number of iterations. A smaller iterative step size is adopted when the stress-strain curve slope is large, while a larger step size is used when the slope is small. This approach ensures accuracy while reducing the number of iterations, thereby accelerating the computation speed. Generally, the axial strain distributed in an arithmetic sequence accounts for about 30% of the total axial strain, while the uniformly distributed axial strain accounts for about 70%. The segmentation position dynamically changes with the total axial strain. As shown in Figure 3, the total axial strain, \(\varepsilon_{\text{1 max}}\), is divided into two parts. The first part adopts an arithmetic sequence distribution, with its axial strain, \(S_{\text{1n}}\), accounting for a proportion, \(p_{\text{t}}\), of \(\varepsilon_{\text{1 max}}\), where the strain increment, \(\Delta\varepsilon_{1}\), follows an arithmetic sequence distribution. The second part adopts a uniform distribution, with axial strain \(S_{\text{2n}}\) accounting for a proportion (\(1-p_{\text{t}}\)) of \(\varepsilon_{\text{1 max}}\), where \(\Delta\varepsilon_{1}\) is uniformly distributed. Given a total number of iterations, \(N_{\text{s}}\), the number of iterations in the uniform distribution part is \(N_{\text{s2}}=p_{\text{t}}N_{\text{s}}\), and the corresponding axial strain increment, \(\Delta\varepsilon_{1}\), is as follows:  
\[\Delta\varepsilon_{1}=\frac{S2_{n}}{N_{\text{s2}}}=\frac{(1-p_{\text{t}}) \varepsilon_{\text{1 max}}}{p_{\text{t}}N_{\text{s}}} \tag{42}\]  
Figure 3: Segmented adjustment of the iterative increment step size \(\Delta\varepsilon_{1}\) in the constitutive model. Assuming that the initial iterative step size, \(a_{1}\), in the arithmetic sequence part is \(1/10\) of the uniform distribution step size \(\Delta\varepsilon_{1}\), we have:  
\[a_{1}=\frac{\Delta\varepsilon_{1}}{10} \tag{43}\]  
Given the total number of iterations \(N_{s}\), the number of iterations in the arithmetic sequence part is \(N_{\mathrm{s1}}\) = (\(1-p_{\mathrm{t}}\))\(N_{\mathrm{s}}\). The common difference \(d\) of the arithmetic sequence and the iterative increment step size, \(a_{n}\), can be expressed as follows:  
\[d=\frac{2(S1_{n}-a_{1}N_{\mathrm{s1}})}{N_{\mathrm{s1}}(N_{\mathrm{s1}}-1)} \tag{44}\]  
\[a_{n}=a_{1}+(n-1)d \tag{45}\]  
By selecting an appropriate proportion, \(p_{\mathrm{t}}\), and a reasonable total iteration count, \(N_{\mathrm{s}}\), a highly accurate stress-strain curve can be obtained with a reduced number of iterations.