Intelligent Calibration of the Cycle Liquefaction Constitutive Model Parameter Using a Genetic Algorithm-Based Optimization Framework

Abstract

Earthquake-induced soil liquefaction poses significant geotechnical hazards, including sand boiling, loss of foundation bearing capacity, lateral spreading, pipeline flotation, uneven settlement, and slope instability. While cyclic liquefaction constitutive models can effectively simulate and predict site liquefaction behavior, their reliability hinges on the accurate calibration of constitutive parameters. Traditional calibration methods often fail to capture the comprehensive material response, are labor-intensive, time-consuming, and susceptible to subjective judgment. To overcome these limitations, this study develops an intelligent calibration framework for a cyclic liquefaction constitutive model by integrating a numerical solver for unit tests with the genetic algorithm (GA)-based optimization framework. The proposed method is rigorously evaluated in terms of calibration accuracy, convergence, repeatability, uncertainty, and computational efficiency. Validation via a series of laboratory unit tests on materials from an extremely high earth-rock dam project confirms the method’s effectiveness. Results demonstrate that the intelligent calibration approach achieves a high accuracy of 91.84%, offering a reliable, efficient, and robust solution for parameter determination.

1. Introduction

Soil liquefaction during earthquakes remains one of the most critical and challenging problems in geotechnical engineering. When subjected to dynamic loading, saturated sandy soils can experience a rapid increase in pore water pressure and a consequent loss of shear strength and stiffness, leading to catastrophic failures such as sand boiling, foundation failure, lateral spreading, and slope instability [1,2]. The accurate prediction of these phenomena is paramount for mitigating seismic risk.

Numerical simulation has emerged as a powerful tool for achieving refined modeling of both the initiation of liquefaction and the subsequent response process, offering high computational efficiency and excellent repeatability. Among various approaches, nonlinear analysis based on dynamic consolidation equations represents the mainstream and a key developmental direction in current research. This analytical framework, rooted in the dynamic consolidation equation [3] and the corresponding theory and the dynamic consolidation theory [4,5,6], possesses the distinct advantage of explicitly accounting for the nonlinearity, large deformation, and elastic-plastic behavior of the soil skeleton. Crucially, it fully couples the generation, dissipation, and diffusion of pore water pressure with soil deformation.

Over the years, a series of specialized liquefaction analysis procedures built upon this framework have been developed, including but not limited to DIANA [7], SWANDYNE IV [8], DIANAFLOW [9], SUMDES [10], SUMDES2D [11], OpenSees [12], PLAXIS [13,14], ABAQUS [15,16], FLAC 2D [17], FLAC 3D [18,19].

However, the simulation capability and predictive accuracy of any analysis procedure are not inherent but are contingent upon several factors: the completeness of site investigation data, the level of modeling refinement, the validity of the adopted liquefaction constitutive model, and—most critically for this study—the rationality of the selected constitutive model parameters. A significant advancement in constitutive modeling was achieved with the development of a cyclic elastic-plastic liquefaction constitutive model [20], which provides a unified description of the mechanical behavior of saturated sandy soils both before and after liquefaction. The numerical implementation of this model (hereafter referred to as the “Cyclic Model” or “CycLiq”) on the OpenSees platform [21] has made it readily applicable, and its validity has been substantiated through extensive engineering practice. The CycLiq model integrates cyclic plasticity and critical state soil mechanics, characterizing sand dilatancy under monotonic and cyclic loading through its separation of volumetric strain into reversible and irreversible components, which makes it particularly suitable for simulating saturated sand during earthquake-induced liquefaction.

The model has been rigorously validated against a broad suite of laboratory element tests under various loading and drainage conditions, as well as through physical and centrifuge model tests [22,23,24,25]. Beyond these foundational validations, CycLiq has been successfully adopted and extended in diverse complex engineering and research contexts. For instance, Maheshwari and Firoj [26] integrated it into seismic response analyses of combined piled raft foundations, where it effectively captured liquefaction-induced settlements and soil-structure interaction. In offshore geotechnics, Li et al. [27] applied the model to assess the dynamic response of liquefiable seabed foundations reinforced by lattice-type cement-mixed soil, confirming its robustness under cyclic loading in marine environments. These independent applications attest to the model’s versatility and reliability across a range of boundary value problems.

Although CycLiq is built upon a well-established theoretical foundation and has a history of successful application, its practical predictive performance depends critically on the precise calibration of its parameters. Traditionally, the calibration of constitutive model parameters has relied on a combination of analytical solutions and manual trial-and-error approaches. While capable of matching specific stress–strain states from laboratory tests, this approach often fails to capture the material’s holistic mechanical behavior across diverse loading conditions [28,29]. This limitation becomes particularly critical as modern constitutive models evolve toward greater sophistication, incorporating an increasing number of parameters that exhibit complex interdependencies. In such high-dimensional parameter spaces, traditional calibration methods struggle to identify unique and physically consistent solutions [30], underscoring the pressing need for more systematic and efficient calibration frameworks.

Advancements in computer technology and mathematical theories have catalyzed the development of intelligent, data-informed calibration methods. Pioneering work applied Genetic Algorithms (GAs) to calibrate the HISS model [28], while subsequent studies introduced parameter sensitivity analysis [29] and refined fitness functions [31] to guide the process. This research stream has become increasingly systematic, exemplified by the integration of numerical solvers with laboratory databases for calibrating various elastoplastic models [32,33] and the refinement of cost functions within hybrid frameworks [34]. Concurrently, machine learning applications in geotechnics have diverged along two prominent paths: one focuses on direct geotechnical hazard assessment, such as predicting soil liquefaction potential through comparative ML evaluations [35,36] and deep learning techniques [37], including the forecasting of its consequences like building settlements [38]. The other stream aims to enhance fundamental soil modeling, leveraging ML for tasks like determining constitutive model parameters from data [39] and simulating complex soil behavior under challenging conditions [40]. Despite their predictive power, a critical gap remains in seamlessly integrating physical principles with robust optimization to reliably calibrate advanced constitutive models, especially under complex cyclic loading where parameter interdependence is paramount.

To address this calibration challenge, various modern computational optimization paradigms have been explored, each with distinct advantages and limitations: (1) Bayesian inversion provides uncertainty quantification and incorporates prior knowledge but is computationally demanding for high-dimensional models and sensitive to prior selection; (2) Swarm intelligence algorithms like Particle Swarm Optimization (PSO) and Differential Evolution (DE) offer efficient global search with simple implementation, yet PSO can converge prematurely in complex landscapes, and DE’s performance is sensitive to control parameters; (3) Hybrid heuristic-gradient methods aim to balance global and local search but face challenges when gradients are unavailable or costly to compute, a common scenario when coupled with complex numerical solvers; (4) Purely data-driven approaches establish direct input-output mappings efficiently after training but often lack physical interpretability and require extensive training data covering the entire parameter space—a significant hurdle for sophisticated models.

These methods have shown success in calibrating relatively simple constitutive models with fewer parameters and smoother responses. However, they exhibit pronounced limitations when applied to advanced cyclic elasto-plastic models like CycLiq, which was developed to unify the mechanical behavior of saturated sand before, during, and after liquefaction. The CycLiq model contains over a dozen parameters with strong physical interrelations and exhibits pronounced path-dependency and state-dependent responses under cyclic loading. A notable previous attempt by Zhou et al. [41] to address this employed Radial Basis Function neural network to establish a direct mapping between laboratory test characteristics and CycLiq parameters. However, their purely data-driven approach demonstrated a critical limitation: it failed to accurately predict key model parameters, particularly the state parameters (n^p and n^d) and dilatancy parameters (d_re,1 and d_ir). This shortcoming reveals the insufficiency of black-box mapping strategies when dealing with the complex physical relationships embedded in advanced cyclic constitutive models, precisely highlighting the research gap that the present study aims to address through a more physically embedded optimization framework.

To bridge this critical gap, this study proposes a novel physics-embedded intelligent calibration framework. Its core innovation is the tight coupling of the CycLiq numerical solver directly within the optimization loop of a Genetic Algorithm (GA). GA was selected as the optimization engine due to its particular suitability for this complex task: its population-based evolutionary mechanism provides robust global exploration in high-dimensional, non-convex parameter spaces; as a gradient-free method, it integrates naturally with numerical solvers, enabling the essential “physics-embedded” paradigm; it balances exploration and exploitation through configurable genetic operators, mitigating the risk of local minima; and its search process maintains interpretability, facilitating checks for physical consistency. The proposed framework thus prioritizes not only data-fitting accuracy but, more critically, the mechanistic consistency of the calibrated parameter set. The core of this paper is organized as follows: Section 2 details the numerical implementation of three key unit tests using the CycLiq model. Section 3 elaborates on the developed GA-based calibration methodology, including the population strategy, the Fréchet distance-based cost function for evaluating curve similarity, and the overall workflow. Section 4 presents a comprehensive analysis of the method’s accuracy, convergence, repeatability, and computational efficiency. Finally, Section 5 validates the method’s effectiveness through a series of laboratory tests on materials from an extremely high earth-rock dam project, demonstrating its superior performance and practical utility in engineering applications.

2. Numerical Unit Tests Conducted by Cycle Liquefaction Constitutive Model

To reliably calibrate the 14 parameters of the cyclic elastic-plastic liquefaction constitutive model [21,42,43], which comprehensively describes the mechanical behavior of saturated sandy soils before and after liquefaction, a suite of numerical unit tests was designed and implemented. The CycLiq model is founded on a theoretical framework that integrates critical state soil mechanics with kinematic hardening plasticity. Its formulation relies on several core assumptions: (1) the existence of a unique critical state toward which the soil converges under sustained shearing; (2) the use of a kinematic hardening rule within a bounding surface framework to capture cyclic hysteresis, memory effects, and path-dependency; (3) the decomposition of total volumetric strain into reversible and irreversible components to unify the description of small-strain stiffness, cyclic accumulation, and post-liquefaction deformation. These assumptions enable the model to comprehensively describe the mechanical behavior of saturated sandy soils both before and after liquefaction, but they also inherently define its primary applicability to shear-dominated loading paths, as represented by conventional triaxial and torsional shear tests. Consequently, the designed test suite encompasses three fundamental types strategically chosen to decouple and capture the distinct soil responses governed by different parameter subsets within the CycLiq model: the conventional triaxial compression drainage test, the undrained cyclic torsional shear test, and the drained cyclic torsional shear test. A numerical solver for these unit tests was developed in C++ based on the CycLiq model implementation, forming the physical core for the subsequent evaluation of parameter sets during intelligent optimization.

2.1. Conventional Triaxial Compression Drainage Test

The conventional triaxial compression drainage test is a standard laboratory procedure for determining the stress–strain relationship and shear strength of soil under saturated, drained conditions. The specimen is consolidated under a specified confining pressure before an axial deviatoric load is applied at a controlled rate, allowing for full dissipation of excess pore water pressure. This test primarily characterizes the monotonic stress–strain-strength behavior and critical state of the soil. For the CycLiq model, it serves as the primary source for calibrating the following parameters:

Elastic parameters: G₀ (small-strain shear modulus) and κ (elastic bulk modulus parameter). Plastic hardening parameters: h (plastic modulus). Critical state parameters: M (critical state stress ratio), λ_c (compression index), e₀ (critical void ratio at reference stress), and ξ (critical state line shape parameter). State-dependent stiffness parameters: n^p and n^d (exponents controlling the dependence of plastic modulus on the state parameter).

2.2. Undrained Cyclic Torsional Shear Test

The cyclic torsional shear test is an effective method for investigating the dynamic characteristics of sandy soil under simulated in situ stress conditions. In this test, a hollow cylindrical specimen is subjected to independently applied internal and external pressures, axial load, and torque, generating a complex stress state involving normal and shear stresses. Under undrained conditions, the specimen develops shear strain without volume change, leading to a progressive buildup of excess pore water pressure under cyclic loading. This test is crucial for capturing the soil’s response leading to liquefaction, characterized by cyclic mobility and accumulated shear strain. It is particularly sensitive to the following CycLiq parameters governing irreversible dilatancy and post-liquefaction deformation:

Irreversible dilatancy parameters: d_ir (irreversible dilatancy coefficient), α (irreversible dilatancy exponent), and γ_d,r (reference shear strain for irreversible dilatancy).

2.3. Drained Cyclic Torsional Shear Test

Similarly to the undrained test, the drained cyclic torsional shear test subjects a hollow cylindrical specimen to the same complex stress path. However, under drained conditions, the specimen is permitted to drain freely throughout the cyclic loading process. Consequently, the specimen exhibits not only shear strain but also normal strain and volumetric changes (compaction or dilation) in response to cyclic shearing. This test captures the cyclic densification (or dilation) behavior of the soil, which is a key aspect of its cyclic response under drained conditions. It provides the primary data for calibrating the parameters controlling the reversible part of the dilatancy behavior in the CycLiq model:

Reversible dilatancy parameters: d_re,1 and d_re,2 (coefficients governing the reversible dilatancy magnitude and its evolution).

3. Intelligent Calibration Framework Based on Genetic Algorithm

To address the challenges of calibrating the high-dimensional and interdependent parameters of the CycLiq model, this study develops an intelligent calibration framework that embeds the numerical solver (the physical model) directly into an optimization loop driven by a Genetic Algorithm (GA). Unlike black-box mapping approaches, this ensures that the optimized parameters are not only data-fitting but also physically and mechanistically consistent. The core of this framework is the iterative evaluation of candidate parameter sets against the target unit test data, guided by a specially designed cost function.

3.1. Overall Workflow

The overall calibration process, illustrated in Figure 1, integrates the CycLiq numerical solver with the GA optimizer and comprises the following key steps:

Figure 1. Flowchart of the Genetic Algorithm (GA)-based intelligent calibration framework.

STEP1. An initial population of candidate parameter sets is randomly generated within predefined bounds (see Section 3.2).

STEP2. Each parameter set in the population is passed to the CycLiq numerical solver. The solver simulates the suite of unit tests (conventional triaxial, undrained cyclic torsional, and drained cyclic torsional shear tests). The simulated results are compared against the target experimental data using a cost function based on the Fréchet distance (see Section 3.3). This step constitutes the physical model embedding.

STEP3. If the convergence criterion (e.g., a cost threshold or maximum iterations) is not met, the GA generates a new population through selection, crossover, and mutation operations (see Section 3.4).

STEP4. Steps 2 and 3 are repeated until convergence is achieved, and the best parameter set is returned.

The detailed pseudocode for the main optimization procedure, population initialization, fitness evaluation, and population update is provided in Appendix A (Algorithms A1–A4).

3.2. Population Initialization Strategy

The population is represented as a matrix (Pop) where each row corresponds to an individual, i.e., a potential constitutive model parameter solution. The number of individuals is denoted by N_i. The initialization search space is defined by the vectors P*_min and P*_max, which contain the minimum and maximum plausible values for each parameter, respectively. These bounds were established based on a synthesis of prior literature, preliminary numerical trials, and physical admissibility. For instance, the range for the normalized shear modulus coefficient G₀ (100–400) encompasses typical values reported for clean quartz sands under comparable confining pressures [44,45]. Preliminary forward simulations were conducted to exclude parameter combinations that lead to non-physical model responses (e.g., negative stiffness) or numerical instability. The bounds also respect internal constraints within the CycLiq framework to ensure physically consistent parameter sets.

A hybrid initialization strategy is employed to enhance both global exploration and local exploitation. The initial population comprises three distinct sub-populations: a uniformly distributed population (P_U), a Gaussian distributed population (P_G), and an ANN-predicted population (P_A). The ANN used to generate the sub-population P_A is a standard feedforward neural network trained offline on the same comprehensive dataset generated for this study. This dataset consists of synthetic unit test results produced by the CycLiq solver across a systematic sampling of the parameter space. The ANN learns the mapping from salient features of the test curves (e.g., initial stiffness, peak strength) back to the CycLiq parameter sets that generated them.

During initialization, the trained ANN takes features extracted from the target experimental data and predicts an approximate parameter set. This set is included as part of the initial GA population, implementing a knowledge-based initialization strategy aimed at starting the search near a potentially good solution to accelerate early convergence.

It is crucial to note that the ANN is used only for initialization and does not guide the subsequent optimization. The final parameter set is determined by the genetic operations (selection, crossover, mutation) and, most importantly, by the physics-based cost function evaluated using the actual numerical solver. This design ensures that even if the ANN’s prediction is imperfect or biased due to limitations in its training data, the GA can explore beyond this initial guess and converge to the true optimum. Thus, the strategy effectively combines a surrogate-assisted startup with high-fidelity refinement, maintaining robustness while improving computational efficiency. Each sub-population accounts for a specific proportion of the total, as detailed in Algorithm A2. The parameters for the Gaussian distribution are calculated based on the defined parameter bounds.

3.3. Fitness Evaluation and Cost Function

The fitness evaluation procedure EVAL.POP (Algorithm A3) is critical for guiding the optimization. It computes a cost value for each parameter set in the population, which quantifies the overall discrepancy between the numerical simulations and the experimental data. The core of this evaluation is the design of an effective and unbiased cost function.

Determining the cost function must consider two important aspects [46]: First, it should facilitate the quantification of the difference between numerical predictions and measured data; Second, the weights of the variables considered in the cost function must be dimensionless and have comparable impacts on the optimization, even though the original physical variables have different physical meanings and value ranges (e.g., void ratio e ranges [0.85–1], while shear stress q ranges [0–1.5] MPa).

To address the data weighting issue during optimization, the test points obtained from the triaxial test, undrained cyclic torsional shear test, and drained cyclic torsional shear test are scaled on the dimensionless planes $\left(\widehat{{\epsilon }_1}, \hat{q}\right)$, $\left(\widehat{{\epsilon }_1}, \widehat{{\epsilon }_v}\right)$, $\left(\widehat{p^{\prime }}, \widehat{{\tau }_d}\right)$, $\left(\hat{\gamma }, \widehat{{\tau }_d}\right)$ and $\left(\widehat{{\tau }_d}, \widehat{{\epsilon }_v}\right)$. The scaling range and method are shown in Equations (1) and (2), ultimately scaling the test data curves within the range (−1,1).

$$\widehat{{\epsilon }_1}={\displaystyle \frac{{\epsilon }_1}{{\epsilon }_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}}, \widehat{{\epsilon }_v}={\displaystyle \frac{{\epsilon }_v}{\max \left(\left|{\epsilon }_v\right|\right)}}, \hat{q}={\displaystyle \frac{q}{\max \left(q\right)}}$$

(1)

$$\hat{p}={\displaystyle \frac{p}{\max \left(p\right)}}, \widehat{{\tau }_d}={\displaystyle \frac{{\tau }_d}{\max \left(\left|{\tau }_d\right|\right)}}, \gamma ={\displaystyle \frac{\gamma }{\max \left(\left|\gamma \right|\right)}}$$

(2)

For a given set of parameters P, numerical integration generates the time evolution of the solution vector X_k on a uniform time grid. The deviation between measured data and model predictions is evaluated by the root mean square of the Fréchet Distance. This curve-based similarity measure quantifies the match between entire trajectories making it especially suitable for cyclic responses where the shape, hysteresis, and progression of curves are physically meaningful. Its key advantage is that it does not require point-to-point alignment in time or strain, providing an axis-independent and sampling-robust metric [47]. Although computationally more intensive than simpler point-wise error measures, its ability to capture path-dependent behavior justifies its use in the calibration of advanced cyclic constitutive models.

For a general (x, y) plane, given a set of M measured data points (x_k, y_k), where $k\in \left[1, M\right]$, and given a set of L numerical predicted values (x_k, y_k), where $j\in \left[1, L\right]$, the discrete Fréchet distance D_F is a vector of size $\min \left(M, L\right)=M$:

$$D_{\mathrm{F}}\left(k\right)=\underset{\forall j\le M}{\min }\left\{d\epsilon \left(k, r_{j, j+1}\right)\right\}$$

(3)

where $d\epsilon \left(k, r_{j, j+1}\right)$ is the distance between the measured data point k and the line segment $r_{j, j+1}$ connecting two consecutive numerical prediction points, as shown in Figure 2. Finally, the deviation between the test data and numerical predictions is calculated as $\delta ={\Vert D_{\mathrm{F}}\Vert }_2$, which is the L₂ norm.

Figure 2. Schematic illustration of the Fréchet distance between measured data and numerical predictions from Jekel et al. [47].

All curves generated by numerical predictions need to undergo Fréchet distance calculation. Each curve provides a difference value between the measured data and numerical predictions. The final cost function can be expressed as:

$$C\left(\mathbf{P}\right)=\sum _{i=1}^T{w}_i{\delta }_i\left(\mathbf{P}\right)$$

(4)

where T is the number of test curves, w_i is the weight of the Fréchet distance for the i-th test curve, $i\in \left[1, T\right]$. This formulation renders the cost function highly scalable, enabling the flexible incorporation of different test types as required.

In this study, the weights are set uniformly as $w_i=1/T$, implementing an equal-weighting scheme. This choice is underpinned by a two-fold rationale aimed at ensuring an objective and balanced calibration. First, the dimensionless scaling procedure defined in Equations (1) and (2) projects all experimental and simulated data onto comparable axes, thereby rendering the individual Fréchet distances ${\delta }_i$ numerically commensurate. Second, the calibration dataset, summarized in Table 1, has been designed to comprehensively capture the key mechanical behaviours of the soil: monotonic strength and critical state via conventional triaxial tests, undrained cyclic pore-pressure generation via the undrained cyclic test, and drained cyclic densification via the drained cyclic test. Given the commensurate basis established by the normalization step, assigning equal weight to each curve constitutes the most transparent and unbiased strategy, avoiding any unwarranted prior emphasis on one physical aspect over another.

Table 1. Types and initial conditions of numerical unit tests in the comprehensive dataset.

Type of Laboratory Test	Confining Pressure (kPa)	Dynamic Shear Stress τd (kPa)	ein
Conventional triaxial compression drainage test 1	200	-	0.78
Conventional triaxial compression drainage test 2	400	-	0.78
Conventional triaxial compression drainage test 3	600	-	0.78
Undrained cyclic torsional shear test	200	20	0.78
Drained cyclic torsional shear test	200	20	0.78

The effectiveness of this equal-weighting scheme is corroborated by the high overall calibration accuracy achieved simultaneously for all target curves. Furthermore, its robustness was verified through a limited sensitivity analysis. Perturbing individual weights, such as doubling the weight for the undrained cyclic curve while proportionally reducing others, resulted in changes of less than 5% in the optimal parameter values and led to negligible differences in the overall fit quality. This confirms that the calibration outcome is not sensitive to moderate variations in the chosen weights, thereby supporting the reliability of the adopted equal-weighting approach.

3.4. Population Update via Genetic Operations

The population update procedure UPDATE.POP (Algorithm A4) generates a new population through selection, crossover, and mutation. This process aims to evolve the population towards regions of the parameter space with lower cost and higher fitness.

The purpose of the selection process is to pass the best-performing individuals to the next generation. The genetic ratio in this paper is n_E, and the final number of inherited populations is N_E = n_E·N_i, i.e., selecting the first N_E individuals from the initial population Pop according to the index vector ID.

The proportion of mutated individuals in the mutation process is denoted by n_M and is calculated using an exponential decay function of the iteration number, which can well balance the search for the optimal solution and the exploration of new populations when the population converges to the final distribution.

The crossover process first identifies the excellent individuals that can mate, and then defines how the information of the mating individuals is combined to produce new individuals. The number of offspring produced by the crossover process is N_N = N_i·(1 − n_M − n_E). This paper uses the ranking selection method to determine mating individuals [48], sampling random numbers from a triangular probability density function:

$$p\left(n\right)={\displaystyle \frac{2\left(N_f-n\right)}{{\left(N_f-1\right)}^2}}$$

(5)

where N_f = n_f N_i is the number of individuals allowed to mate, n ∈ [1, N_f] is the rank of the individual, i.e., the index in the sorted list ID. This distribution allows the fittest individual (n = 1) a higher chance of mating, and the last-ranked individual (n = N_f) a zero chance of mating. Once the best N_f individuals are identified, offspring are generated by crossover, mixing the characteristics of two parents:

$${\mathbf{P}}_{new}={\theta }_k·{\mathbf{P}}_{n1}+({\theta }_k-1)·{\mathbf{P}}_{n2}$$

(6)

where n₁ and n₂ are the indices of two parents randomly selected from the triangular distribution [49]; θ_k is a random number chosen in the range [0, 1]. Through the population update procedure UPDATE.POP, a set of constitutive model parameter populations with lower cost and higher fitness can be generated.

4. Analysis of Intelligent Calibration Results

This section presents a comprehensive evaluation of the proposed intelligent calibration method for the CycLiq constitutive model. The analysis focuses on four key aspects: calibration accuracy, convergence behavior, repeatability, and uncertainty, to rigorously assess the method’s reliability and computational efficiency.

4.1. Calibration Accuracy

The intelligent calibration method was applied to a comprehensive dataset of 20 sets to test its calibration accuracy. The dataset incorporated the three types of numerical unit tests (conventional triaxial compression drainage tests under three confining pressures, undrained cyclic torsional shear test, and drained cyclic torsional shear test) with initial conditions as summarized in Table 1. The search ranges P*_min and P*_max for the CycLiq parameters in the GA model are provided in Table 2.

Table 2. Search ranges P*_min and P*_max for CycLiq model parameters in the GA model.

CycLiq Parameters	G0	κ	h	M	dre,1	dir	λc	e0	np	nd
maximum value	400	0.03	3.0	1.9	3.0	3.5	0.04	1.00	3	11
minimum value	100	0.005	0.5	1.0	0.5	0.5	0.01	0.85	0.5	1

The calibration accuracy is quantified by the relative error between the calibrated parameters and their reference values. Figure 3 presents the dispersion of calibration errors for all CycLiq constitutive model parameters across the comprehensive dataset.

Figure 3. The dispersion of calibration errors of the CycLiq constitutive model parameters.

The results demonstrate that the GA model achieves a high overall accuracy, with a maximum relative error of less than 20% for all parameters in the comprehensive dataset. However, the dispersion of errors varies among parameters. Specifically, parameters d_ir and n^d exhibit a relatively larger scatter in their calibration errors. This suggests that these parameters are either less sensitive to the cost function or have stronger interdependencies with other parameters, making them more challenging to identify uniquely. Despite this, the overall average relative error is 8.16%, corresponding to a calibration accuracy of 91.84%. This high accuracy demonstrates the parameter identifiability and inversion robustness of the proposed framework in a controlled numerical setting, confirming its capability to consistently recover a known parameter set from the corresponding system response. It should be noted, however, that such a metric primarily validates the internal consistency of the calibration procedure. The physical predictive validity of the calibrated parameters must be further assessed through their performance in simulating independent physical tests, as carried out in Section 5.

4.2. Convergence Behavior

This subsection analyzes the convergence process of the intelligent calibration method to elucidate its computational efficiency and the factors influencing it.

The selection of the initial population size is a critical parameter affecting the convergence speed of the GA. Figure 4 illustrates the relationship between the initial population size and the number of iterations required for convergence. As the initial population size increases from 50 to 250, the number of iterations required for convergence gradually decreases. This is because a larger initial population provides a better coverage of the parameter space, increasing the probability of including individuals close to the global optimum early in the process. When the initial population size reaches 250, the GA model converges after 210 iterations. Further increases in population size lead to a stabilized iteration count, indicating a point of diminishing returns. To achieve the target calibration accuracy (10⁻³), the GA model required 210 iterations within a computationally manageable timeframe. This cost stems primarily from the necessary, repeated calls to the high-fidelity numerical solver to maintain physical consistency, yet it represents a significant efficiency gain over traditional manual methods that can take weeks or months. The result demonstrates the feasibility of automated, globally optimal calibration. The inherently parallelizable structure of the GA also offers clear pathways for significant runtime reduction in future implementations.

Figure 4. Influence of initial population size on the number of convergence iterations.

For a given set of parameters P, numerical integration generates the time evolution of the solution vector X_k on a uniform time grid [28,50]. By recording the distribution of each constitutive model parameter during each iteration, the convergence evolution process of each constitutive model parameter for several different intelligent calibration methods can be obtained as shown in Figure 5. For the GA calibration process, the initial population’s values for each constitutive model parameter have a wide distribution range and low concentration. Parameters G₀, h, M, λ_c, and e₀ show a wide initial distribution but converge steadily and rapidly to the optimal solution after approximately 100–150 iterations. This indicates that these parameters are highly influential and well-constrained by the cost function. Parameters κ, d_re,1, d_ir, n^p, and n^d converge at a slower pace. Their initial distributions are wide, and the populations take longer to stabilize around the mean value. This slower convergence is consistent with the larger calibration errors observed in Figure 3, underscoring the challenge in precisely identifying these parameters due to potential lower sensitivity or higher interdependence.

Figure 5. Convergence evolution process of each parameter during the GA calibration process.

4.3. Repeatability and Uncertainty

To investigate the repeatability and uncertainty of the method, the typical constitutive model parameter calibration process was repeated 100 times. The distribution of the calibrated parameters from these 100 independent runs is shown in Figure 6, and the corresponding statistical results are summarized in Table 3.

Figure 6. Distribution of calibrated parameters for 100 repeated calibration runs.

Table 3. Statistical results of the calibrated parameter distribution from 100 repeated calibrations.

CycLiq Parameters	Minimum Value	Maximum Value	Average Value μ	σ/μ (%)
G0	112.23	121.20	116.80	1.574
κ	0.0050	0.0063	0.0056	4.467
h	0.77	0.84	0.81	1.879
M	1.21	1.25	1.23	0.475
dre1	0.50	0.64	0.56	6.588
dir	0.87	1.04	0.98	3.426
λc	0.0161	0.0236	0.0211	5.886
e0	0.8208	0.8526	0.8378	0.791
np	0.82	1.08	0.93	4.851
nd	5.73	7.61	6.60	5.942

From Figure 6 and the corresponding statistical analysis in Table 3, the intelligent calibration method demonstrates good repeatability while also quantifying the inherent uncertainty in parameter identification. Most parameters exhibit a normalized standard deviation σ/μ below 5%. The critical state stress ratio M and critical void ratio e₀ show notably low variability (σ/μ < 1%), confirming their role as well-constrained, key parameters of the constitutive model. In contrast, parameters n^d, d_re1, and λ_c present higher dispersion. This stems from the specific information content of the calibration test suite: n^d, governing state-dependent stiffness, would benefit from a broader range of initial density conditions; d_re1, associated with reversible dilatancy, is less directly constrained by the undrained cyclic tests that dominate the calibration; and λ_c, controlling compressive behavior, is primarily constrained by volumetric strain during monotonic shearing, which was not exhaustively characterized across all states in the current test combination. Importantly, this parameter-level variability does not compromise the model’s predictive accuracy at the macroscopic scale, as all calibrated sets achieve consistently high overall fit to the experimental curves, demonstrating that the framework reliably delivers robust and precise simulations even within quantified uncertainty bounds. Thus, the method not only provides optimized parameter values but also transparently assesses their reliability, advancing beyond conventional calibration practices that offer no such uncertainty quantification.

5. Verification of the Effectiveness of the Intelligent Calibration Method

This section validates the effectiveness of the proposed intelligent calibration method through a series of laboratory tests on engineering materials from an extremely high earth-rock dam project. The calibrated CycLiq model parameters are rigorously evaluated by comparing numerical predictions against experimental results, both for tests used in the calibration and for independent tests under different loading conditions.

5.1. Laboratory Testing Program

The validation material is a specific sand from an extremely high earth-rock dam project. The sand has a density ranging from 1508 to 1526 kg/m³, a specific gravity of 2.7, a coefficient of uniformity C_u = 5.16, and a coefficient of curvature Cc = 1.57. According to these gradation parameters (C_u ≥ 5 and Cc = 1–3), the soil is classified as well-graded. A series of laboratory unit tests was conducted. The test types and initial conditions are summarized in Table 4.

Table 4. Test types and initial conditions for a specific engineering sand.

Type of Laboratory Test	Confining Pressure (kPa)	Dynamic Shear Stress τd (kPa)	Density (kg/m3)
Conventional triaxial compression drainage test	200/400/600	-	1512/1517/1526
Undrained cyclic torsional shear test	200	20/30/40	1508/1514/1513
Drained cyclic torsional shear test	200	20/30/40	1514/1515/1524

The test equipment used in this paper is the hollow cylinder torsional shear apparatus (Figure 7a). The sample was prepared using the standardized layered sand pluviation method [51]. The sample is a hollow cylinder specimen with an inner diameter of 3 cm, outer diameter of 5 cm, and height of 20 cm (Figure 7b). Careful saturation control achieved B-values consistently above 96%. The testing procedures adhered to established protocols [51]. While minor experimental variability, such as slight fluctuations in initial density, is inherent in physical testing, the dataset represents high-quality, realistic measurements suitable for engineering calibration. The subsequent analysis demonstrates that the proposed intelligent calibration framework is robust to such controlled, yet realistic, data conditions.

Figure 7. Laboratory unit test equipment and specimen dimensions. (a) Hollow cylinder torsional shear apparatus (b) Specimen dimensions.

To verify the effectiveness of the intelligent calibration method for CycLiq constitutive model parameters, conventional triaxial compression tests under different confining pressures (200 kPa/400 kPa/600 kPa), undrained cyclic torsional shear tests with dynamic shear stress level τ_d = 20 kPa, and drained cyclic torsional shear tests with dynamic shear stress level τ_d = 20 kPa were used as calibration tests to determine the CycLiq constitutive model parameters for the certain sand.

5.2. Calibration and Predictive Performance of the CycLiq Model

5.2.1. Parameter Calibration and Comparative Analysis

The intelligent GA model was used to determine the CycLiq parameters for the engineering sand. The calibration utilized the conventional triaxial compression tests under different confining pressures (200 kPa/400 kPa/600 kPa), undrained cyclic torsional shear tests with dynamic shear stress level τ_d = 20 kPa, and drained cyclic torsional shear tests with dynamic shear stress level τ_d = 20 kPa to determine the CycLiq constitutive model parameters. For comparison, the calibration results from Zhou et al. [41] for the same sand are also presented in Table 5.

Table 5. Comparison of calibrated CycLiq parameters for an engineering sand.

Parameters	G0	κ	h	M	dre,1	dir	λc	e0	np	nd
Zhou 2019 [41]	106	0.0052	0.70	1.12	0.52	0.74	0.0181	0.882	0.74	2.75
GA calibration	112	0.006	0.82	1.43	0.12	2.14	0.016	0.88	2.2	7.2

The GA calibration yields parameter values that are different from those obtained by Zhou’s method [41]. For instance, the critical state stress ratio M is calibrated as 1.43 by the GA, which is significantly higher and likely more representative of the sand’s true critical state compared to the value of 1.12 from Zhou’s method. Similarly, the plastic modulus h and the dilatancy-related parameters (d_re,1, d_ir, n^p, n^d) show substantial differences. This quantitative comparison highlights that the proposed physics-embedded GA framework identifies a different and, as will be shown, mechanically more consistent parameter set.

5.2.2. Performance on Calibration Tests

The predictive capability of the calibrated model is first verified against the tests used in the calibration process. Figure 8, Figure 9 and Figure 10 show the comparison between experimental results and numerical predictions for the conventional triaxial compression drainage test (200 kPa), the undrained cyclic torsional shear test (CSR = 0.1), and the drained cyclic torsional shear test (CSR = 0.1), while Table 6, Table 7 and Table 8 list the differences in characteristic point, respectively. For comparison with existing research findings, the corresponding simulation results obtained by calibrating the CycLiq constitutive model parameters are also provided.

Figure 8. Comparison of predicted (GA model and RBF model from Zhou et al. [41]) and experimental results for the conventional triaxial compression drainage test. The stress–strain curves (Left column) and volumetric deformation–strain curves (Right column) under pressures of (a,b) 200 kPa; (c,d) 400 kPa; (e,f) 600 kPa.

Figure 9. Comparison of predicted and experimental results for the undrained cyclic torsional shear test (CSR = 0.1). Shown are the effective stress path (Left column) and stress–strain curve (Right column) from: (a,b) Test; (c,d) RBF (Zhou 2019 [41]); (e,f) GA model.

Figure 10. Comparison of predicted and experimental results for the drained cyclic torsional shear test (CSR = 0.1). Shown are the stress–volumetric strain (left column) and stress–strain curves (right column) from: (a,b) Test; (c,d) RBF (Zhou 2019 [41]); (e,f) GA model.

Table 6. Differences in characteristic points for simulating the conventional triaxial drainage test (200 kPa).

Feature	q_0.5	q_1	q_2	q_5	q_10	q_15	εv_0.5	εv_1	εv _2
Test	164.66	256.91	381.99	554.46	618.01	618.01	−0.37	−0.65	−1.03
Zhou 2019 [41]	478.39	587.20	668.98	672.27	633.88	615.02	−0.75	−0.75	−0.45
GA calibration	251.09	311.27	395.46	536.20	630.11	630.11	−0.80	−1.03	−1.13
Feature	εv _5	εv _10	εv _15	εvmax	εvmax _ε1	φmax	q/p	Dilatation rate_max	MSE (%)
Test	−1.37	−0.94	−0.94	−1.37	4.91	37.44	2.76	−0.02	0.00
Zhou 2019 [41]	0.81	2.32	3.10	−0.78	0.72	39.27	2.68	−0.25	2.59
GA calibration	−1.21	−1.16	−1.16	−1.21	5.29	38.41	2.72	−0.02	0.22

Table 7. Differences in characteristic points for simulating the undrained cyclic torsional shear test (CSR = 0.1).

Feature	u_40	u_80	u_120	u_160	Liq1st	γ1st	γ1+	γ1−	γ2+
Test	10.03	32.73	64.04	84.13	88.50	0.74	1.02	−1.21	1.87
Zhou 2019 [41]	0.17	0.21	0.19	0.14	0.50	0.06	0.01	−0.11	0.04
GA calibration	4.06	7.22	9.69	11.40	12.50	0.42	0.65	−0.98	1.15
Feature	γ2−	γ5+	γ5−	γ10+	γ10−	λ	Gd	-	MSE (%)
Test	−1.91	4.58	−4.98	-	-	0.02	321.76	-	0.00
Zhou 2019 [41]	−0.14	0.10	−0.19	0.15	−0.24	-	-	-	5.62
GA calibration	−1.51	2.64	−2.93	4.35	−4.52	0.03	367.24	-	0.87

Table 8. Differences in characteristic points for simulating the drained cyclic torsional shear test (CSR = 0.1).

Feature	εv_20(%)	e_20	MSE (%)
Test	0.17	0.45	0.00
GA calibration	0.17	0.46	0.17

As shown in Figure 8, the GA-calibrated model demonstrates a superior fit to the experimental stress–strain curve and volumetric strain behavior compared to Zhou’s method. The predicted deviatoric stress q at various strain levels and the dilation characteristics are captured with significantly higher accuracy. This is quantitatively confirmed by the Mean Squared Error (MSE), which is 0.22% for the GA model, an order of magnitude lower than the 2.59% MSE of the Zhou model (Table 6). The GA model successfully replicates the strain-softening and dilatant behavior post-peak strength, which the Zhou model fails to capture.

For the undrained cyclic test, the GA-calibrated model effectively simulates the key liquefaction phenomena. It accurately captures the cumulative buildup of pore water pressure, predicting liquefaction (Figure 9f) at the 12.5th cycle, which is close to the experimental observation of the 8.5th cycle (Figure 9b). While the Zhou’s model fails entirely to predict pore pressure ratio and grossly underestimates shear strain amplitudes, the GA model’s predictions, though not perfect, are quantitatively and qualitatively in much closer agreement with the experimental data (MSE: GA 0.87% vs. Zhou 5.62%) as shown in Table 7.

In the drained test, the model’s ability to capture cyclic densification is evaluated. The GA-calibrated model excellently predicts the accumulation of volumetric strain over cycles. After 20 cycles, the predicted volumetric strain ε_v is 0.17%, matching the experimental value of 0.17% exactly, and the corresponding void ratio change is also accurately captured as in Table 8. This demonstrates the successful calibration of the reversible dilatancy parameters.

5.3. Model Generalization and Predictive Validation

To evaluate the predictive robustness and extrapolation capacity of the calibrated CycLiq model, its parameter set, determined from tests at CSR = 0.1, was used to simulate the behavior of the same sand under elevated CSR levels of 0.15 and 0.20. This procedure is properly characterized as an extrapolation test within an expanded loading range of the same material, rather than a full external validation involving different soil types or constitutive frameworks. The stringency of this test arises from the requirement to accurately predict responses under both undrained and drained boundary conditions, which represent two distinct physical regimes governed, respectively, by pore pressure evolution and volumetric strain accumulation.

To further examine the extrapolation capability and generality of the calibrated model, predictions were made for loading conditions not included in the calibration dataset, specifically, undrained and drained cyclic torsional shear tests under elevated cyclic stress ratios (CSR = 0.15 and CSR = 0.20, corresponding to τ_d = 30 kPa/40 kPa). These tests serve as independent checks of the model’s performance beyond its calibration range. It should be noted that for certain model parameters, e.g., ξ = 0.7, d_re,2 = 30, γ_dr = 0.05, and α = 30, were held constant. This practice is grounded in their well-constrained theoretical ranges; for instance, α and ξ are closely linked to critical-state properties and typically vary within narrow limits for a given soil type [21,52]. It also aligns with established calibration strategies, where such parameters are preset to stabilize the high-dimensional optimization, allowing the algorithm to focus on the remaining influential parameters [29]. The initial void ratio e_in was assigned according to the measured in situ density. The corresponding numerical predictions, derived from the intelligently calibrated parameter set, are systematically compared against experimental results in Figure 11, Figure 12, Figure 13 and Figure 14 and discussed in the following sections.

Figure 11. Prediction results for the undrained cyclic torsional shear test (CSR = 0.15). Shown are the stress–volumetric strain (left column) and stress–strain curves (right column) from: (a,b) Test; (c,d) GA model.

Figure 12. Prediction results for the undrained cyclic torsional shear test (CSR = 0.20). Shown are the stress–volumetric strain (left column) and stress–strain curves (right column) from: (a,b) Test; (c,d) GA model.

Figure 13. Prediction results for the drained cyclic torsional shear test (CSR = 0.15). Shown are the stress–volumetric strain (left column) and stress–strain curves (right column) from: (a,b) Test; (c,d) GA model.

Figure 14. Prediction results for the drained cyclic torsional shear test (CSR = 0.20). Shown are the stress–volumetric strain (left column) and stress–strain curves (right column) from: (a,b) Test; (c,d) GA model.

As shown in Figure 11, the model demonstrates excellent predictive capability for higher stress levels. For both CSR = 0.15 and CSR = 0.20, it accurately captures the accelerated pore pressure generation and the resulting larger shear strain amplitudes. The number of cycles to liquefaction decreases with increasing CSR, a trend correctly predicted by the model. This successful extrapolation confirms that the GA-calibrated parameters are not merely curve-fitted to a specific load but genuinely capture the underlying soil mechanics.

Similarly, for the drained condition at higher CSRs as in Figure 12, the model reliably predicts the increased rate of cyclic densification (volumetric strain accumulation) with increasing shear stress amplitude. The close agreement between the predicted and measured stress–strain hysteresis loops and volumetric strain paths across all verification tests (Figure 13 and Figure 14 further corroborate this) underscores the model’s robustness. The single set of parameters calibrated from the lower CSR tests successfully generalizes to predict the soil’s response under more severe loading, proving the method’s practical utility for engineering applications.

6. Conclusions

This study successfully developed and validated an intelligent calibration framework that integrates a numerical solver for fundamental unit tests with a genetically optimized algorithm to address the critical challenge of determining parameters for the sophisticated cyclic liquefaction constitutive model (CycLiq). The proposed methodology advances beyond purely data-driven mapping by embedding the physical model directly within the optimization loop, thereby promoting both data-fitting accuracy and mechanistic consistency. The framework’s performance was rigorously evaluated in terms of calibration accuracy, convergence behavior, repeatability, and computational efficiency for the tested materials and conditions. Its practical utility was demonstrated through a comprehensive series of laboratory tests on a specific sand from an extremely high earth-rock dam project. The principal achievements of this research are as follows:

• A robust, physics-embedded calibration framework was developed and validated, integrating a genetic algorithm with a numerical solver to automate parameter identification for the CycLiq model under the investigated conditions.
• The proposed method achieved a high calibration accuracy of 91.84% in parameter recovery and demonstrated superior performance in simulating key laboratory tests compared to an existing data-driven approach, suggesting its effectiveness for the calibration challenge at hand.
• The robustness of the calibrated model was supported by repeatability analysis, and its predictive capability was shown to extend to higher CSR levels for the same material, indicating good generalizability within the tested loading spectrum.

While the results are promising, several aspects warrant further investigation. The computational cost of the optimization, though acceptable for a one-time calibration, highlights a practical consideration for broader application. The identifiability of certain parameters remains partially dependent on the specific suite of calibration tests employed, and in some cases, parameter coupling can lead to solution non-uniqueness even with comprehensive data. Future work should therefore explore efficiency improvements through surrogate modeling or hybrid algorithms, seek to incorporate a wider variety of test types to better constrain all parameters, and validate the framework’s applicability to different soil types and more complex constitutive models.