Concrete nonlinear creep explicit modeling method and system

Abstract

The invention discloses a concrete nonlinear creep explicit modeling method and system, and relates to the technical field of concrete structures and materials.The concrete nonlinear creep explicit modeling method comprises the steps that concrete creep test data under all conditions are collected, and a test data set is constructed; preprocessing the concrete creep test data of the test data set; a DBO algorithm is adopted to optimize hyper-parameters of the LGBM model, and the LGBM model is constructed and trained through the test data set and the optimal hyper-parameters; the input features of the trained LGBM model are screened through an SHAP additive interpretation method, and key feature parameters are obtained; and establishing a concrete nonlinear creep explicit calculation formula by using a PySR symbol regression tool and taking the key characteristic parameters as input and the creep strain as output. The concrete nonlinear creep explicit modeling method provided by the invention has both calculation precision and engineering practical value.

Description

Technical Field

[0001] This invention relates to the field of concrete structure and materials technology, and more specifically to an explicit modeling method and system for nonlinear creep of concrete. Background Technology

[0002] Creep, a time-varying characteristic of concrete under sustained loads, adversely affects concrete structures, causing crack propagation, increased deformation, and prestress loss. The creep deformation of concrete varies under different stress levels (stress-to-strength ratio). When the stress level is less than 0.4, the creep deformation exhibits a linear relationship with the stress-to-strength ratio, known as linear creep. When the stress level is greater than 0.4, the creep deformation exhibits a non-linear relationship with the stress-to-strength ratio, known as non-linear creep. Under high stress levels, the creep rate of concrete accelerates significantly, leading to rapid propagation of microcracks and ultimately material failure, severely threatening the structural safety. Since long-span bridges, super high-rise buildings, and high arch dams, which are primarily load-bearing structures, endure high levels of sustained loads over long periods, accurate calculation of the non-linear creep of concrete is crucial for structural design and safety assessment.

[0003] Existing technologies mainly include: GL 2000 creep model, fib MC2010 creep model, ACI 209 creep model, AASHTO creep model, B4 creep model, JTG D62-2012 creep model, TB 1002.3-2005 creep model, and China Academy of Building Research-1986 creep model, etc. The main shortcomings of existing technologies are: (1) Insufficient accuracy of nonlinear creep calculation under high stress level: Most models are based on linear relationship and superposition principle and are established using experimental data under low stress level. They do not adequately consider the coupling mechanism between the development of microcracks inside concrete and nonlinear creep under high stress state, and it is difficult to accurately reflect the nonlinear characteristics of the rapid increase in creep rate after the stress level exceeds 0.4, resulting in a large deviation in nonlinear creep calculation under high stress level.

[0004] (2) The difficulty in balancing model complexity and engineering practicality: These models all rely on statistical regression on a large amount of experimental data to obtain empirical or semi-empirical formulas for creep. The establishment mechanisms of different models are not entirely the same, and the selection of parameters also has its own characteristics. Some high-precision models (such as the B4 model) have numerous parameters and complex calculations, and some parameters lack simple experimental measurement methods, making it difficult to promote their application in actual engineering design and rapid safety assessment. On the contrary, some simplified models (such as the ACI 209 model) ignore the effects of nonlinear development, loading age effect, or long-term environmental changes, making it difficult to meet the needs of large-span, super high-rise and other major projects for detailed analysis of long-term performance.

[0005] Therefore, how to provide an explicit modeling method for nonlinear creep of concrete that combines computational accuracy with practical engineering value is a problem that urgently needs to be solved by those skilled in the art. Summary of the Invention

[0006] In view of this, the present invention provides an explicit modeling method and system for nonlinear creep of concrete. The method utilizes the Dung Beetle Optimization (DBO) algorithm to optimize the hyperparameters of the Lightweight Gradient Boosting Machine (LGBM) model, accurately establishing the mapping relationship between different influencing factors and nonlinear creep of concrete. Furthermore, SHAP interpretable analysis is introduced to reveal the specific contribution of each input feature to the creep calculation results, clarifying the key characteristic parameters affecting nonlinear creep of concrete. After obtaining the key characteristic parameters affecting nonlinear creep of concrete, the method combines PySR symbolic regression to mine potential physical laws from the data, constructing a concise and highly accurate explicit calculation formula for nonlinear creep of concrete.

[0007] To achieve the above objectives, the present invention adopts the following technical solution: An explicit modeling method for nonlinear creep of concrete includes: Collect concrete creep test data under various conditions and construct an experimental dataset; The concrete creep test data of the experimental dataset were preprocessed and divided into training set and test set; The hyperparameters of the LGBM model are optimized using the DBO algorithm. The LGBM model is constructed and trained using the training set and the optimal hyperparameters. The trained LGBM model is then validated using the test set. The SHAP additive interpretation method is used to filter the input features of the trained LGBM model and obtain key feature parameters. Using the PySR symbolic regression tool, with the key feature parameters as input and creep strain as output, an explicit calculation formula for nonlinear creep of concrete is established.

[0008] Preferably, the concrete creep test data includes aggregate-cement ratio, cement content, 28-day compressive strength, 28-day elastic modulus, volume ratio, loading age, temperature, humidity, stress level, loading stress, loading time, and creep flexibility.

[0009] Preferably, the optimization of the hyperparameters of the LGBM model using the DBO algorithm specifically includes: selecting the coefficient of determination as the objective function, setting the number of iterations of the DBO algorithm to 100 and the population size to 50, and defining the LGBM hyperparameters to be optimized and their search range; calculating the current optimal function value and the corresponding individual position according to the mechanism of the DBO algorithm, and selecting the current optimal individual; iterating the position of each individual in the population through the four behaviors of the dung beetle, i.e., updating the hyperparameters; and outputting the optimal hyperparameters when the dung beetle criterion or the set number of iterations is reached.

[0010] Preferably, the LGBM model learns the nonlinear relationship between each input feature and the creep compliance through a gradient boosting framework, and uses a decision tree ensemble mechanism to capture the interaction between variables and complex response patterns, thereby establishing a high-precision nonlinear mapping relationship. The entire process is as follows: The training set for concrete is: ; in, It is the sequence length; It is a factor that affects concrete creep; It refers to the creep flexibility of concrete; It is the number of features in each sample; It is the set of real numbers; The calculation formula for the initial model is as follows: ; in, It is a loss function; It is a constant that minimizes the loss function; The formula for calculating the residual is as follows: ; in, It is the first In the nth iteration The residuals of each sample; yes The predicted value; Fitting To obtain the leaf node region of the tree Optimal constant The expression is as follows: ; in, To determine the current model's response to the input after the (m-1)th iteration. The predicted value; The regression tree is updated as follows: ; in, These are the parameters involved in the iteration. It is the number of node regions. It is the prediction value of the current model for the input x after the m-th and m-1-th iterations; The resulting trained LGBM model is as follows: ; in, It is the final promotion tree.

[0011] Preferably, obtaining the key feature parameters specifically includes: ; in, Indicates the first The contribution of each feature; Represents the set of all features; Represents a subset of a given predictable feature; and Representing whether or not the first SHAP approximates complex models by superimposing multiple linear models, focusing on models with specific characteristics. Output model of each feature Defined as the linear sum of the input variables: ; in, Prediction results without eigenvalues; Indicates when the first The value is 1 when a feature is selected, otherwise it is 0.

[0012] Preferably, the explicit calculation formula for nonlinear creep of concrete is established using a frequency metric based on complexity to calculate the error, for the dataset <X, y> and the expression to be evaluated. The calculation formula is: ; in, This is the original expression currently undergoing mutation; It is an expression error; It is an expression The complexity is denoted by ; frecency is the number of times a specific complexity occurs within a time window; During the mutation process, the parent expression selects a mutation operation according to a preset probability distribution. PySR, combined with the simulated annealing algorithm, dynamically decides whether to accept the mutation based on the error change and complexity difference between the mutated expression and the parent expression. The probability of accepting the mutation is calculated by the following function: ; in, It is the new expression generated after the mutation; This is the annealing factor, used to control the algorithm's tolerance to error degradation; To simulate the temperature parameters in the annealing algorithm; The complexity of the original expression; The complexity of the new expression; After the population evolution stage is completed, the expression in the population is replaced with its algebraic equivalent to reduce the search space; then, the constants in the expression are fine-tuned through numerical optimization algorithm. The optimization goal is to minimize the error between the predicted value of the expression and the real data, and improve the fitting accuracy. Each population executes the asynchronous migration strategy in sequence, stores the optimal expression for different complexities of the current population, and maintains a set of optimal solutions shared across populations. After the migration is completed, the error of the expression is continuously reduced by iterating the above process until the termination condition is met, and finally the expression with the smallest error in the set is output.

[0013] Preferably, the method further includes verifying the explicit calculation formula for nonlinear creep of concrete using a test set, wherein the formula is: ; in, It's the loading time; For loading age; Humidity; Stress level; The compressive strength is measured over 28 days.

[0014] An explicit modeling system for nonlinear creep of concrete includes: The dataset construction unit collects concrete creep test data under various conditions and constructs the test dataset; The preprocessing unit preprocesses the concrete creep test data of the experimental dataset, dividing it into a training set and a test set; The optimization unit uses the DBO algorithm to optimize the hyperparameters of the LGBM model, constructs and trains the LGBM model using the training set and the optimal hyperparameters, and verifies the trained LGBM model using the test set. The feature selection unit uses the SHAP additive interpretation method to select the input features of the trained LGBM model and obtain key feature parameters. The formula construction unit uses the PySR symbolic regression tool, with the key feature parameters as input and creep strain as output, to establish an explicit calculation formula for nonlinear creep of concrete.

[0015] As can be seen from the above technical solution, compared with the prior art, this invention discloses an explicit modeling method and system for nonlinear creep of concrete. Through parameter sensitivity analysis of the constructed symbolic regression model, the influence of various key features on the nonlinear creep behavior of concrete is systematically revealed, and the results are highly consistent with the basic theories of concrete materials science. This indicates that the proposed computational model has good physical interpretability and can provide strong theoretical support and quantitative basis for the long-term performance evaluation and durability design of concrete structures. Attached Figure Description

[0016] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the provided drawings without creative effort.

[0017] Figure 1 This is a schematic diagram of the method flow provided by the present invention; Figure 2(a) shows the performance of the DBO-LGBM machine learning model provided by this invention; Figure 2(b) shows the performance of the LGBM machine learning model provided by this invention; Figure 2(c) shows the performance of the XGBoost machine learning model provided by this invention; Figure 2(d) shows the performance of the BPNN machine learning model provided by this invention; Figure 2(e) shows the performance of the ANN machine learning model provided by the present invention; Figure 2(f) shows the performance of the LR machine learning model provided by this invention; Figure 3 This is a summary diagram of SHAP provided by the present invention; Figure 4 The flowchart of the PySR symbolic regression method provided by this invention; Figure 5(a) is a comparison chart of the calculated and measured values ​​of the PySR model provided by the present invention; Figure 5(b) is a comparison chart of the calculated and measured values ​​of the CEB-FIP90 model provided by the present invention; Figure 5(c) is a comparison chart of the calculated and measured values ​​of the ACI209-92 model provided by the present invention; Figure 5(d) is a comparison chart of the calculated and measured values ​​of the ASSHTO model provided by the present invention; Figure 6(a) is a loading time sensitivity analysis diagram of the PySR symbolic regression model provided by the present invention; Figure 6(b) is a sensitivity analysis diagram of the loading age of the PySR symbolic regression model provided by the present invention; Figure 6(c) is a humidity sensitivity analysis diagram of the PySR symbolic regression model provided by the present invention; Figure 6(d) is a stress level sensitivity analysis diagram of the PySR symbolic regression model provided by the present invention; Figure 6(e) is a sensitivity analysis diagram of the 28-day compressive strength of the PySR symbolic regression model provided by the present invention. Detailed Implementation

[0018] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0019] This invention discloses an explicit modeling method and system for nonlinear creep of concrete, such as... Figure 1 As shown, the steps are as follows: Step 1: Construct a dataset for nonlinear creep tests on concrete. The system collects concrete creep test data covering various mix proportions, environmental conditions, and loading conditions to construct a high-quality dataset. The test data should include concrete raw material parameters (such as water-cement ratio, cement content, and aggregate type), curing conditions (such as relative humidity and temperature), loading age, stress level, and the corresponding measured values ​​of creep strain over time. Data sources may include laboratory tests, literature, or engineering monitoring data.

[0020] Step 2: Data Preprocessing. The raw data needs to be cleaned, normalized, and feature-engineered to remove outliers and fill in missing values, ensuring the integrity and consistency of the input features. The preprocessed data is then randomly divided into an 80% training set and a 20% test set for subsequent model training and evaluation.

[0021] Step 3: Optimize the hyperparameters of the LGBM model using the DBO algorithm. The DBO algorithm efficiently searches for the optimal parameter combination in the solution space by simulating four behaviors of dung beetles: rolling a ball, reproduction, foraging, and stealing, and selects the coefficient of determination (R²) from the training set. 2 Using the fitness function, the hyperparameters of the LGBM model, such as the learning rate, number of network layers, and number of neurons per layer, are optimized. Through continuous iteration, the globally optimal or near-optimal hyperparameter configuration is finally obtained, thereby significantly enhancing the model's nonlinear fitting ability and generalization performance.

[0022] Step 4: Establish the mapping relationship between different influencing factors and nonlinear creep of concrete. Based on the DBO-optimized LGBM model, a high-precision nonlinear mapping relationship is constructed between input features (such as material composition, environmental and mechanical conditions) and output target (creep strain). The DBO-LGBM model can capture the complex nonlinear, time-varying, and coupling effects in concrete creep behavior, achieving accurate calculation of nonlinear creep of concrete and providing a reliable foundation for subsequent interpretability analysis.

[0023] Step 5: Obtain key feature parameters through SHAP interpretability analysis. Based on the trained LGBM model, the SHAP method is introduced to perform local and global interpretability analysis on the model calculation results. The SHAP method quantifies the contribution of each input feature to a single calculation result by calculating the SHAP value of each sample feature, revealing the direction and intensity of the influence of each factor on creep. By analyzing the distribution and ranking of SHAP values, key parameters that play a dominant role in concrete creep (such as water-cement ratio, relative humidity, stress level, etc.) are identified, thereby enhancing the physical interpretability of the model and providing a basis for feature selection for subsequent symbolic regression.

[0024] Step 6: Establish an explicit calculation formula for nonlinear creep of concrete based on the PySR symbolic regression method. After identifying the key influencing factors, the PySR symbolic regression tool is used, with these key features as input and creep strain as output, to automatically search for physically meaningful mathematical expressions from the data. PySR combines genetic algorithms and sparse regression to evolve a concise, highly generalizable, and accurate explicit formula from a large space of candidate functions. The resulting analytical expression not only facilitates engineering applications and code embedding but also reveals the underlying physical mechanisms, realizing the transformation from a "black box model" to a "white box formula," thus completing the interpretable explicit modeling of nonlinear creep of concrete.

[0025] In the specific embodiment 1 provided by the present invention: (1) Constructing a dataset for nonlinear creep tests of concrete. The Northwest University Creep Database collected 29,177 observations of creep compliance under different compressive strengths and loading conditions (such as stress level, age, temperature, and humidity), providing a solid data foundation for studying the creep characteristics of concrete. The data used in this embodiment comes from this database. Considering existing models and practical experience, this embodiment initially selected 11 characteristic parameters as model inputs: aggregate-cement ratio, cement content, 28-day compressive strength, 28-day elastic modulus, volume ratio, loading age, temperature, humidity, stress level, loading stress, and loading time, with creep compliance as the output.

[0026] (2) Data preprocessing. Creep test data with stress levels greater than 0.4 were selected from the Northwest University creep database, and samples lacking any of the above input parameters were removed. After the above preprocessing, 1169 data points were finally obtained. The preprocessed data were randomly divided into 80% training set and 20% test set for subsequent model training and evaluation. The input and output parameters in the dataset and their value ranges are detailed in Table 1.

[0027] Table 1 Input and output parameters and their range of variation

[0028] (3) Optimize the hyperparameters of the LGBM model using the DBO algorithm. DBO is a novel intelligent optimization algorithm, which mainly updates and optimizes the position based on four behaviors of dung beetles: rolling, reproduction, foraging, and stealing. This algorithm considers both global exploration and local exploitation, and has the characteristics of fast convergence speed and high accuracy. It can be used for hyperparameter tuning of the LGBM model to further improve the prediction accuracy of the model, thereby better serving the accurate modeling and mechanism analysis of the nonlinear creep behavior of concrete. The specific process is as follows: (1) Select the coefficient of determination (R 2 (1) As the objective function, the number of iterations of the DBO algorithm is set to 100 and the population size to 50, and the hyperparameters of the LGBM to be optimized and their search range are defined; (2) According to the mechanism of the DBO algorithm, the current optimal function value and the corresponding individual position are calculated, and the current optimal individual is selected; (3) The position of each individual in the population is iterated through the four behaviors of the dung beetle, that is, the hyperparameter combination is updated; (4) When the sword-collecting criterion or the set number of iterations is reached, the optimal hyperparameter combination is output. The DBO optimization algorithm is used to search for the best hyperparameter combination of the LGBM model, and the optimization results are shown in Table 2.

[0029] Table 2 LGBM Hyperparameter Optimization Results

[0030] (4) Establish the mapping relationship between different influencing factors and nonlinear creep of concrete. Based on the completion of data preprocessing and model hyperparameter optimization, the LGBM model is constructed and trained using the 80% training set data obtained in step (2) and the optimal hyperparameter combination of the LGBM model obtained by DBO algorithm in step (3) (such as num_leaves=5, min_child_samples=15, max_depth=47, learning_rate=0.01, n_estimators=1791). The input consists of 11 influencing factors, including aggregate-cement ratio, cement content, 28-day compressive strength, 28-day elastic modulus, volume ratio, loading age, temperature, humidity, stress level, loading stress, and loading time. The output is the creep flexibility. During the training process, the LGBM model learns the nonlinear relationship between each input feature and the creep flexibility through the gradient boosting framework, and uses the decision tree ensemble mechanism to capture the interaction between variables and complex response patterns, thereby establishing a high-precision nonlinear mapping relationship. The whole process is as follows: The dataset for nonlinear creep of concrete is as follows: (1) in It is the sequence length; It is a factor that affects concrete creep; It refers to the creep flexibility of concrete; It is the number of features in each sample; It is the set of real numbers.

[0031] The calculation formula for the initial model is as follows: (2) in It is a loss function; It is a constant that minimizes the loss function.

[0032] The formula for calculating the residual is as follows: (3) in, It is the first In the nth iteration The residuals of each sample; yes The predicted value; This represents the current model's prediction of the input x after the (m-1)th iteration.

[0033] Fitting To obtain the leaf node region of the tree Optimal constant The expression is as follows: (4) in, To determine the current model's response to the input after the (m-1)th iteration. The predicted value; The regression tree is updated as follows: (5) in, These are the parameters involved in the iteration. It is the number of node regions. It is the prediction value of the current model for the input x after the m-th and m-1-th iterations.

[0034] The final model obtained is as follows: (6) in It is the final promotion tree.

[0035] To quantitatively evaluate the computational performance of the model, three evaluation metrics were used: coefficient of determination (R²), root mean square error (RMSE), and mean absolute error (MAE). Among these, R²... 2 The higher the value, the lower the RMSE and MAE values, indicating higher accuracy of the model calculation. The expressions for these indicators are as follows: (7) (8) (9) in For the true value, For calculated values, This is the average value. To calculate the sample size.

[0036] To demonstrate the superiority of the DBO-LGBM model, its prediction performance was compared with that of LGBM, XGBoost, BPNN, ANN, and LR machine learning models. The results of different machine learning models calculating the nonlinear creep of concrete are shown in Figures 2(a)-2(f), and Table 3 lists the specific computational performance indicators. As can be seen from Figures 2(a)-2(f) and Table 3, the overall difference between the predicted and actual values ​​of the DBO-LGBM model is smaller, exhibiting the best prediction performance. The MAE value of the DBO-LGBM model for predicting the nonlinear creep of concrete is reduced by 42.6%, 51.7%, 56.6%, 55.7%, and 66.3% compared to LGBM, XGBoost, BPNN, ANN, and LR, respectively. The results indicate that the DBO-LGBM model accurately establishes the mapping relationship between different influencing factors and the nonlinear creep of concrete.

[0037] Table 3 Performance Comparison of Different Machine Learning Models

[0038] (5) Key feature parameters were obtained through SHAP interpretability analysis. Although the DBO-LGBM machine learning model has high prediction accuracy, its internal mechanism is not transparent, making it difficult to reveal the specific influence of each input variable on the output result. In order to improve the interpretability of the model and clarify the important feature parameters affecting the nonlinear creep of concrete, the SHAP additive interpretation method was used to perform interpretability analysis on the above DBO-LGBM machine learning model.

[0039] SHAP is a method based on game theory for interpreting machine learning models. It approximates complex machine learning models using multiple linear models and calculates the SHAP value for each linear model, thus indicating the contribution of each feature (parameter) to the prediction process. The formula for calculating the SHAP value is as follows: (10) In the formula: Indicates the first The contribution of each feature; Represents the set of all features; Represents a subset of a given predictable feature; and Representing whether or not the first The model results for each characteristic. SHAP approximates complex models by superimposing multiple linear models, for models with... Output model of each feature Defined as the linear sum of the input variables: (11) in Prediction results without eigenvalues; Indicates when the first The value is 1 when a feature is selected, otherwise it is 0.

[0040] Figure 3 The SHAP summary plot arranges features from highest to lowest importance on the vertical axis, while the horizontal axis represents the contribution of each feature to the prediction result. Color (e.g., red for high feature values, blue for low feature values) visually illustrates the relationship between different features and the target variable. Figure 3 It can be seen that the top 7 input features that have the greatest impact on the prediction of nonlinear creep of concrete are stress level, loading stress, loading time, humidity, 28-day compressive strength of concrete, loading age, and 28-day elastic modulus of concrete.

[0041] (6) An explicit calculation formula for nonlinear creep of concrete was established based on the PySR symbolic regression method.

[0042] PySR is a Python-based symbolic regression tool designed to automatically discover equations that balance accuracy and conciseness from data. This method does not require pre-defining the specific form of the function expression; instead, it leverages the search capabilities of a genetic algorithm to automatically select a suitable function expression for a specific task. The method initializes multiple populations, each consisting of a number of randomly generated expressions. These populations evolve independently at different stages, achieving large-scale parallelism, allowing different populations to evolve at different speeds without affecting each other. In each generation, each population first independently executes two phases: evolution and optimization. During the evolution phase, each population undergoes mutation and crossover operations by selecting expressions based on error magnitude through a tournament selection process. The algorithm uses a frequency-based metric of complexity to calculate the error for the dataset <X, y> and the expression to be evaluated. The error calculation formula for PySR is: (12) In the formula: This is the original expression currently undergoing mutation; It is an expression error; It is an expression The complexity of an expression is the sum of the complexities of all its operators, variables, and constants; frecency is the number of times a specific complexity occurs within a time window.

[0043] During the mutation process, the parent expression selects various mutation operations according to a preset probability distribution, such as inserting nodes, deleting nodes, mutating nodes, or perturbing constant values. PySR, combined with the simulated annealing algorithm, dynamically decides whether to accept the mutation based on the error change and complexity difference between the mutated expression and the parent expression. The probability of accepting the mutation is calculated by the following function: (13) in, It is the new expression generated after the mutation; This is the annealing factor, used to control the algorithm's tolerance to error degradation; To simulate the temperature parameters in the annealing algorithm; The complexity of the original expression; The complexity of the new expression; After the population evolution phase, the optimization phase begins. This phase first replaces the expressions in the population with their algebraic equivalents to reduce the search space. Then, numerical optimization algorithms are used to fine-tune the constants in the expressions. The optimization objective is to minimize the error between the predicted values ​​and the actual data, thereby improving the fitting accuracy.

[0044] After completing the "evolution to optimization" phase, each population enters the asynchronous migration phase. Each population sequentially executes the asynchronous migration strategy, storing the optimal expressions for different complexities for the current population, while maintaining a shared set of optimal solutions across populations. After migration, by iterating through the above process, the error of the expression is continuously reduced until a termination condition is met, ultimately outputting the expression with the smallest error in the set. Figure 4 The PySR workflow is demonstrated.

[0045] Different parameters affect concrete creep. To select suitable parameters for formula derivation, the top 7 factors with the highest mean of SHAP feature importance ranking were chosen as important feature parameters. These feature parameters were then applied as input variables in PySR, and operators such as addition, subtraction, multiplication, division, exponentiation, logarithm, square root, and square were selected for symbolic regression.

[0046] The symbol search process in PySR is controlled collaboratively by multiple sets of hyperparameters. The population size (population_size) determines the number of candidate expressions retained within each independent evolutionary population, while the number of iterations (niterations) and the number of iteration cycles per iteration (ncycles_per_iteration) jointly limit the depth and breadth of the algorithm's exploration before termination. The values ​​of these parameters need to be dynamically adjusted based on the dataset size, feature dimensions, and the nonlinearity of the underlying patterns. The complexity of the expression structure is constrained by the maximum number of nodes (maxsize) and the maximum tree depth (maxdepth), which respectively limit the total number of operators and the nesting depth of the expression. To balance model complexity and accuracy, maxsize and maxdepth are set to 20 and 10, respectively. Table 4 shows the parameter settings used for the PySR algorithm in this embodiment.

[0047] Table 4 Parameter settings for the PySR algorithm

[0048] Similar to previous machine learning models, the model is trained using a training set and validated using a test set. The final fitting formula is: (14) in, It's the loading time; For loading age; Humidity; Stress level; The compressive strength is measured over 28 days.

[0049] The PySR symbolic regression model has a fairly simple and compact formula. Among them... and There is a significant positive correlation between them, which can usually be approximated by empirical formulas, while Variable ratio The variables contain more information, so formula (14) only includes 5 of the 7 input variables used in the model development process.

[0050] To further verify the effectiveness of the PySR symbolic regression model, Table 5 lists commonly used creep empirical models such as CEB-FIP90, ACI209-92, and ASSHTO. Figures 5(a)-5(d) show a comparison between the calculated values ​​of different models and the measured creep values. In the PySR model, most of the red dots are concentrated on or near the black line, and the calculated values ​​show good consistency with the experimental values, indicating that the PySR model can predict the nonlinear creep of concrete materials well. The red dots of the CEB-FIP90 model are more dispersed in the figure, and some points are far from the black line, indicating that the predictive performance of this model is relatively poor. The ACI209-92 model tends to provide underestimated predictions for high creep values, while the ASSHTO model tends to provide overestimated predictions for low creep values; both have poor accuracy.

[0051] Table 6 shows the performance indicators of each calculation model. It can be seen that the PySR symbolic regression model has the lowest RMSE and MAE values ​​and the highest R² value, showing the best predictive performance in the calculation of nonlinear creep of concrete. Compared with the ACI209-92, CEB-FIP90 and ASSHTO models, the R² values ​​are improved by 35.0%, 35.0% and 39.7%, respectively.

[0052] Table 5 Common Creep Model Formulas

[0053] Note: The creep coefficient is . Slump The content of fine aggregate, For air content, The theoretical thickness of the component.

[0054] Table 6 Performance metrics of each computational model

[0055] To intuitively analyze the influence of various characteristic parameters on the nonlinear creep of concrete in the symbolic regression model, a parameter sensitivity analysis was performed on the PySR symbolic regression model formula. When performing parameter analysis on a certain parameter, the average value of the other characteristic parameters in the dataset was taken, and then the values ​​of the parameter analysis variables were changed one by one to obtain the changing trend and magnitude of the model output.

[0056] As shown in Figure 6(a), loading time has a significant positive promoting effect on the creep flexibility of concrete. In the initial stage of loading, creep deformation develops rapidly, and as the loading duration increases, the creep growth rate gradually decreases. This trend is consistent with the typical logarithmic time law of concrete creep. As shown in Figure 6(b), the effect of loading age on the creep flexibility of concrete shows a trend of first decreasing rapidly and then gradually leveling off, especially around 28 days, where the rate of change decreases significantly. This is because when load is applied to concrete at an early age, the hydration reaction of cement particles is not yet fully completed, so it is more prone to viscoelastic flow under external stress, resulting in a higher initial creep flexibility. When the loading age approaches 28 days, the hydration reaction of most silicate cement has entered the middle and late stages, the microstructure tends to stabilize, and its ability to resist creep also stabilizes. This phenomenon is not only highly consistent with the physicochemical evolution process of cement-based materials, but also consistent with the international standard practice of using 28 days as the reference age for evaluating the strength of concrete. Figure 6(c) characterizes the relationship between ambient relative humidity and creep flexibility. There is a negative correlation between humidity and the creep flexibility of concrete. In high humidity environments, the higher water content in concrete can reduce volume shrinkage caused by drying, thus helping to slow down creep. Figure 6(d) reflects the effect of stress level on creep flexibility. The curve shows a clear positive relationship, meaning that the higher the stress level, the greater the creep flexibility. Figure 6(e) shows a significant negative correlation between 28-day compressive strength and creep flexibility. Creep flexibility decreases monotonically with the increase of 28-day concrete strength. When the 28-day compressive strength of concrete exceeds 60 MPa, the concrete has entered the category of high-strength concrete, with a relatively dense microstructure. Therefore, further increasing the concrete strength gradually reduces the contribution to inhibiting creep, and the downward trend of the curve tends to flatten out.

[0057] This invention proposes an explicit modeling method for nonlinear creep of concrete that combines high accuracy with engineering practicality by integrating the DBO-LGBM-SHAP-PySR method. This effectively solves the technical problems of large calculation deviations and an imbalance between model complexity and practicality in existing models under high stress levels. Compared with traditional empirical models such as ACI209-92, CEB-FIP90, and ASSHTO, the calculation accuracy is improved by 35.0%, 35.0%, and 39.7%, respectively. In terms of engineering applications, the established explicit formula contains only five input parameters, is concise, and the parameters are easy to obtain, significantly reducing model complexity and computational cost. It can be directly embedded into design specifications and safety assessment systems for large-span structures, providing a reliable tool for long-term deformation prediction and life assessment for major projects such as long-span bridges and super high-rise buildings, and has significant socio-economic value.

[0058] The various embodiments in this specification are described in a progressive manner, with each embodiment focusing on its differences from other embodiments. Similar or identical parts between embodiments can be referred to interchangeably. For the apparatus disclosed in the embodiments, since they correspond to the methods disclosed in the embodiments, the description is relatively simple; relevant parts can be referred to the method section.

[0059] The above description of the disclosed embodiments enables those skilled in the art to make or use the invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of the invention. Therefore, the invention is not to be limited to the embodiments shown herein, but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. An explicit modeling method for nonlinear creep of concrete, characterized in that, include: Collect concrete creep test data under various conditions and construct an experimental dataset; The concrete creep test data of the experimental dataset were preprocessed and divided into training set and test set; The hyperparameters of the LGBM model are optimized using the DBO algorithm. The LGBM model is constructed and trained using the training set and the optimal hyperparameters. The trained LGBM model is then validated using the test set. The SHAP additive interpretation method is used to filter the input features of the trained LGBM model and obtain key feature parameters. Using the PySR symbolic regression tool, with the key feature parameters as input and creep strain as output, an explicit calculation formula for nonlinear creep of concrete is established.

2. The explicit modeling method for nonlinear creep of concrete according to claim 1, characterized in that, The concrete creep test data includes aggregate-cement ratio, cement content, 28-day compressive strength, 28-day elastic modulus, volume ratio, loading age, temperature, humidity, stress level, loading stress, loading time, and creep flexibility.

3. The explicit modeling method for nonlinear creep of concrete according to claim 1, characterized in that, The specific steps for optimizing the hyperparameters of the LGBM model using the DBO algorithm include: selecting the coefficient of determination as the objective function, setting the number of iterations of the DBO algorithm to 100 and the population size to 50, and defining the LGBM hyperparameters to be optimized and their search range; calculating the current optimal function value and the corresponding individual position according to the mechanism of the DBO algorithm, and selecting the current optimal individual; iterating the position of each individual in the population through the four behaviors of the dung beetle, i.e., updating the hyperparameters; and outputting the optimal hyperparameters when the dung beetle criterion or the set number of iterations is reached.

4. The explicit modeling method for nonlinear creep of concrete according to claim 1, characterized in that, The LGBM model learns the nonlinear relationship between each input feature and the creep compliance through a gradient boosting framework, and uses a decision tree ensemble mechanism to capture the interactions and complex response patterns between variables, thereby establishing a high-precision nonlinear mapping relationship. The entire process is as follows: The training set for concrete is: ； in, It is the sequence length; These are factors that affect concrete creep; It refers to the creep flexibility of concrete; It is the number of features in each sample; It is the set of real numbers; The calculation formula for the initial model is as follows: ； in, It is a loss function; It is a constant that minimizes the loss function; The formula for calculating the residual is as follows: ； in, It is the first In the nth iteration The residuals of each sample; yes The predicted value; Fitting To obtain the leaf node region of the tree Optimal constant The expression is as follows: ； in, To determine the current model's response to the input after the (m-1)th iteration. The predicted value; The regression tree is updated as follows: ； in, These are the parameters involved in the iteration. It is the number of node regions. It is the prediction value of the current model for the input x after the m-th and m-1-th iterations; The resulting trained LGBM model is as follows: ； in, It is the final promotion tree.

5. The explicit modeling method for nonlinear creep of concrete according to claim 1, characterized in that, The acquisition of key feature parameters specifically includes: ； in, Indicates the first The contribution of each feature; Represents the set of all features; Represents a subset of a given predictable feature; and Representing whether or not the first SHAP approximates complex models by superimposing multiple linear models, focusing on models with specific characteristics. Output model of each feature Defined as the linear sum of the input variables: ； in, Prediction results without eigenvalues; Indicates when the first The value is 1 when a feature is selected, otherwise it is 0.

6. The explicit modeling method for nonlinear creep of concrete according to claim 1, characterized in that, The established explicit calculation formula for nonlinear creep of concrete uses a frequency metric based on complexity to calculate the error, for the dataset <X, y> and the expression to be evaluated. The calculation formula is: ； in, This is the original expression currently undergoing mutation; It is an expression error; It is an expression The complexity is denoted by ; frecency is the number of times a specific complexity occurs within a time window; During the mutation process, the parent expression selects a mutation operation according to a preset probability distribution. PySR, combined with the simulated annealing algorithm, dynamically decides whether to accept the mutation based on the error change and complexity difference between the mutated expression and the parent expression. The probability of accepting the mutation is calculated by the following function: ； in, It is the new expression generated after the mutation; This is the annealing factor, used to control the algorithm's tolerance to error degradation; To simulate the temperature parameters in the annealing algorithm; The complexity of the original expression; The complexity of the new expression; After the population evolution stage is completed, the expression in the population is replaced with its algebraic equivalent to reduce the search space; then, the constants in the expression are fine-tuned through numerical optimization algorithm. The optimization goal is to minimize the error between the predicted value of the expression and the real data, and improve the fitting accuracy. Each population executes the asynchronous migration strategy in sequence, stores the optimal expression for different complexities of the current population, and maintains a set of optimal solutions shared across populations. After the migration is completed, the error of the expression is continuously reduced by iterating the above process until the termination condition is met, and finally the expression with the smallest error in the set is output.

7. The explicit modeling method for nonlinear creep of concrete according to claim 1, characterized in that, It also includes verifying the explicit calculation formula for nonlinear creep of concrete using a test set. The formula is as follows: ； in, It's the loading time; For loading age; Humidity; Stress level; The compressive strength is measured over 28 days.

8. A concrete nonlinear creep explicit modeling system, employing the concrete nonlinear creep explicit modeling method according to any one of claims 1-7, characterized in that, include: The dataset construction unit collects concrete creep test data under various conditions and constructs the test dataset; The preprocessing unit preprocesses the concrete creep test data of the experimental dataset, dividing it into a training set and a test set; The optimization unit uses the DBO algorithm to optimize the hyperparameters of the LGBM model, constructs and trains the LGBM model using the training set and the optimal hyperparameters, and verifies the trained LGBM model using the test set. The feature selection unit uses the SHAP additive interpretation method to select the input features of the trained LGBM model and obtain key feature parameters. The formula construction unit uses the PySR symbolic regression tool, with the key feature parameters as input and creep strain as output, to establish an explicit calculation formula for nonlinear creep of concrete.

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