Application of a Dynamic Step Size Iterative Method for Parameter Inversion in the Unified Hardening Models  
[PERSON]  
1 China Academy of Railway Sciences Corporation Limited, Beijing 100081, China; [EMAIL_ADDRESS]  
2  
[PERSON]  
3 State Key Laboratory of Highspeed Railway Track System, China Academy of Railway Sciences Corporation Limited, Beijing 100081, China  
34 Infrastructure Inspection Research Institute, China Academy of Railway Sciences Corporation Limited, Beijing 100081,  
railways, and other engineering projects [1; 2; 3; 4]. The Modified Cam-Clay (MCC) model [5; 6], a classical elastoplastic constitutive framework, has been widely successful in capturing the mechanical behavior of normally consolidated clay. However, the MCC model shows limitations in capturing the behavior of overconsolidated clays, such as abrupt transitions in stress-strain curves between elastic and plastic responses and an overestimation of peak strength [7]. To better characterize the strain softening and dilatancy behavior of overconsolidated saturated clay, [PERSON] et al. [8; 9; 10] introduced the Unified Hardening (UH) model based on the MCC model. By modifying only the hardening parameter of the MCC model, the UH model can uniformly describe strain hardening and softening, as well as contraction and dilatancy, in clays with different densities (both normally and overconsolidated clays), demonstrating broad applicability. In engineering practice, not only clays but also granular soils such as sands, rockfill materials, and railway ballast are encountered. However, the UH model, which is designed for clays, cannot adequately describe the stress-strain behavior of granular soils. To address this, [PERSON] et al. [11] further developed the Unified Hardening model for Clays and Sands (CSUH) based on the UH model. The CSUH model is capable of describing both clays and granular soils and can degenerate into the UH and MCC models, representing a theoretical enhancement and innovative development of the MCC model [12]. Before applying the CSUH model to laboratory experiments and engineering simulations, it is essential to accurately calibrate its parameters. When using inverse methods for parameter calibration, it is necessary to compute the error between the theoretical predictions of the constitutive model and the experimental observations, which requires numerical integration to obtain the theoretical response curves. [PERSON] et al. [13] imported experimental data into Isght and used its optimization algorithms to automatically adjust the model parameters in the finite element software ABAQUS 2025, thereby minimizing the error between the simulated and experimental stress-strain curves for parameter calibration. In recent years, adaptive step-size control in the field of finite element computational methods has received considerable attention in the field of computational mechanics. For instance, [PERSON] et al. [14; 15] proposed an adaptive step-size scheme based on local truncation error estimation, where each substep is computed using both a formula of order \(p\) and another of order \(p+1\). The difference between the results is used to estimate the local error, which then guides the automatic adjustment of the step size. Such methods are known for their robustness and general applicability, especially in finite element integration where the shape of the stress-strain curve is not known a priori. However, these approaches typically require multiple computations and error analyses at each step, resulting in increased computational effort. Other adaptive schemes, such as embedded Runge-Kutta or arc-length methods, are also widely used in related fields. An alternative approach for computing theoretical constitutive model curves during parameter inversion is explicit iteration [12; 16]. Common iterative methods include the forward Euler method and the fourth-order Runge-Kutta method. The fourth-order Runge-Kutta method provides high computational accuracy and stability, but it is computationally expensive.