## 4 Dynamic Iteration Method  
### Common Iteration Methods  
The inversion of constitutive model parameters using optimization algorithms requires extensive computations, with the most time-consuming part being the iterative stress-strain increment calculation, similar to Equation (20). The process of determining constitutive model parameters through optimization involves providing different parameter sets to the constitutive model. The model then computes the theoretical stress-strain curve corresponding to the given parameters and initial conditions from the experiment. The computed theoretical stress-strain curve is compared with the experimental stress-strain curve, and the error is fed back to the optimization algorithm. Based on this error feedback,the optimization algorithm adjusts and attempts to provide improved model parameters. Consequently, the inversion process requires frequent evaluation of stress-strain increments to generate the stress-strain curve. As previously discussed, different constitutive models can adopt a unified iterative stress-strain increment formulation, similar to Equation (20), for stress-strain curve calculations. Several numerical methods can be applied for iteration, including the forward Euler method, the fourth-order Runge-Kutta (RK4) method, and implicit iteration methods. The RK4 method offers higher computational accuracy and better stability but faces challenges in practical applications due to its complex implementation and high computational cost. Implicit iteration methods, such as the backward Euler method and the Newton-Raphson method, allow for larger incremental steps but require solving nonlinear equations iteratively, which may lead to convergence issues. The semi-implicit return mapping method performs well in handling elastoplastic problems, but its implementation is complex, and its computational efficiency is relatively low. Due to its simplicity, intuitiveness, and ease of implementation, the forward Euler method is widely used. This method is based on a first-order Taylor expansion, where the stress state of the next step is directly computed using the tangent modulus of the current step. The core concept is to discretize the continuous stress-strain relationship into an incremental form. According to the forward Euler method, the stress-strain increment in Equation (20) and subsequent stress paths can be simplified as \(C_{n}d\epsilon_{1}\):  
\[\left[\sigma,\epsilon\right]_{n+1}=\left[\sigma,\epsilon\right]_{n}+C_{n}d \epsilon_{1} \tag{41}\]  
where \(\left[\sigma,\epsilon\right]_{n+1}\) represents the stress and strain at step \(n+1\), \(\left[\sigma,\epsilon\right]_{n}\) represents the stress and strain at step \(n\), \(C_{n}\) is the compliance matrix computed based on the stress state at step \(n\), and \(d\epsilon_{1}\) is the incremental maximum principal strain. Figure 2 illustrates the accuracy comparison between the RK4 method and the forward Euler method. While RK4 provides higher accuracy, its computational cost is significantly higher than the forward Euler method. However, it was observed that when the total axial strain is uniformly divided into 3000 increments, the integration curves obtained from both methods with 3000 uniform steps are nearly identical, as shown in Figure 2. This indicates that using 3000 uniform steps can yield an accurate stress-strain curve. Since RK4 requires solving the compliance matrix at four different substep increments, whereas the forward Euler method only requires one calculation, the computational cost of the forward Euler method is significantly lower. Therefore, when the iteration step count reaches 3000, the forward Euler method not only reduces computational effort but also provides satisfactory results, making it the preferred choice for computing the constitutive model stress-strain curve. Figure 2: Stress–strain curves for the CU stress path obtained using different integration methods with 3000 uniform step sizes.