A method and system for predicting the strength of porous aluminum based on physical constraint symbolic regression

Abstract

This invention discloses a method and system for predicting the strength of porous aluminum based on physical constraint symbolic regression, belonging to the field of material property prediction technology. The method includes: conducting tensile tests on porous aluminum materials over a wide temperature range to obtain stress-strain data at different temperatures and porosities; constructing a physical constraint symbolic regression model based on the data, embedding fundamental principles of material mechanics into the symbolic regression process in the form of hard constraints; performing evolutionary calculations using the physical constraint symbolic regression model to automatically discover elastoplastic constitutive equations that satisfy the hard constraints; and then constructing a tensile strength theoretical characterization model based on the relationship between the elastoplastic constitutive equations and the failure strain to predict the strength of porous aluminum at any temperature within a wide temperature range. The tensile strength theoretical characterization model of this invention has a clear structure, explicit physical meaning, and few fitting parameters, overcoming the shortcomings of traditional empirical models that lack physical content and rely on numerous full-temperature-range destructive experiments.

Description

Technical Field

[0001] This invention relates to the field of material property prediction technology, and in particular to a method and system for predicting the strength of porous aluminum based on physical constraint symbolic regression. Background Technology

[0002] Porous aluminum, as a core lightweight structural material in high-end equipment fields such as aerospace and rail transportation, often faces severe challenges in extreme wide-temperature environments during actual service. With the continuous improvement of the operating temperature requirements of advanced equipment, the study of the wide-temperature mechanical properties of porous aluminum has become an important research direction in the scientific community, especially the strength and failure mechanism of the material under wide temperature range.

[0003] Currently, the mechanical properties of porous aluminum at different temperatures are typically obtained experimentally. Based on the experimental results, corresponding constitutive models or strength prediction models are fitted. However, these models do not fully consider the influence of temperature on material failure mechanisms and are only applicable to specific material systems or temperature ranges, lacking universality. Furthermore, wide-temperature-range mechanical experiments require multiple temperature points, constituting a series of destructive tests that require a large number of specimens and significant time investment, making them quite time-consuming and labor-intensive.

[0004] Therefore, there is an urgent need for a strength prediction method that can self-consistently describe the continuous evolution of porous aluminum from low-temperature brittleness to high-temperature toughness, reduce reliance on destructive experiments across the entire temperature range, and has a clear physical meaning. Summary of the Invention

[0005] The purpose of this invention is to provide a method and system for predicting the strength of porous aluminum based on physical constraint symbolic regression, which solves the problems in the current technology where constitutive models are difficult to self-consistently describe the continuous evolution of porous aluminum from low-temperature brittleness to high-temperature toughness, and where strength prediction relies on a large number of destructive experiments across the entire temperature range and lacks physical connotation.

[0006] To achieve the above objectives, this invention provides a method for predicting the strength of porous aluminum based on physical constraint symbolic regression, comprising the following steps: Tensile tests were conducted on porous aluminum materials at multiple temperature points and under different porosities over a wide temperature range to obtain stress-strain data corresponding to each temperature and porosity. Based on stress-strain data, a physical constraint symbolic regression model is constructed. The physical constraint symbolic regression model embeds the basic principles of mechanics of materials into the symbolic regression process in the form of hard constraints. The hard constraints include the monotonically increasing stress-strain curve constraint, the decreasing elastic modulus with increasing porosity constraint, and the non-negative plastic work constraint. The physical constraint symbolic regression model is used to perform evolution calculations on stress-strain data, automatically discovering and outputting elastoplastic constitutive equations that satisfy hard constraints; Based on the direct numerical relationship between the elastoplastic constitutive equation and the material failure strain, a temperature-dependent tensile strength theoretical characterization model is constructed. Based on the tensile strength theoretical characterization model, the tensile strength of porous aluminum at any temperature over a wide temperature range is predicted.

[0007] Preferably, the specific content of constructing a physical constraint symbolic regression model based on stress-strain data includes: The stress-strain data are divided into a training set and a validation set, and a set of basic variables is determined; the basic variables include at least strain and porosity. Initialize the population and generate multiple candidate equations. Each candidate equation is represented by a tree structure. Leaf nodes include basic variables and constants within a preset numerical range, while internal nodes include addition, subtraction, multiplication, division operators, and exponentiation operators. Based on the constraints of monotonically increasing stress-strain curve, decreasing elastic modulus with increasing porosity, and non-negative plastic work, corresponding constraint functions are constructed. Each constraint function is used to calculate the quantitative degree of violation of the candidate equation for this specific constraint. The fitting error of each candidate equation is calculated based on the training set, and the fitness value of each candidate equation is calculated based on the quantitative violation value. For candidate equations that violate any hard constraint, an adaptive penalty factor is applied to the fitness value to penalize them. Based on the fitness value, the parent equation is selected using a selection operator, and the offspring equation is generated through crossover and mutation operations to form a new generation of population. Repeat the calculation of fitness value, penalty, selection, crossover, and mutation operations until the change in the optimal fitness value of the population is less than the preset threshold or the maximum number of iterations is reached over multiple generations. A candidate equation is selected from the final Pareto front, and the prediction accuracy and physical consistency are verified based on the validation set. The candidate equation that passes the verification is determined as the physically constrained symbolic regression model.

[0008] Preferably, the elastoplastic constitutive equation is in additive form, consisting of a superposition of a linear elastic response term and a nonlinear hardening term; the linear elastic response term includes a product coupling relationship between porosity and elastic modulus; and the nonlinear hardening term includes a product coupling relationship between porosity and the material's hardening ability.

[0009] Preferably, the expression for the elastoplastic constitutive equation is: ; in, For tensile strength, For modulus, Porosity ε In response, For yield strength, and For adjustable fitting parameters, n This is the hardening index.

[0010] Preferably, the theoretical characterization model of tensile strength is directly derived from the elastoplastic constitutive equation and the material failure strain, and the expression is: ; in, T For temperature, For temperature T Tensile strength at that time For temperature T Modulus at time, For temperature T Response in time, For temperature T Yield strength at that time Porosity and For adjustable fitting parameters, n This is the hardening index.

[0011] Preferably, the prediction of tensile strength is carried out according to the principle of temperature-specific calibration and temperature-specific calculation. Specifically, the parameters in the elastoplastic constitutive equation are calibrated for different temperatures, and the tensile strength at the corresponding temperature is calculated.

[0012] Preferably, the wide temperature range is -70℃ to 350℃; the porosity range is 0.6 to 0.9.

[0013] This invention provides a porous aluminum strength prediction system based on physical constraint symbolic regression, used to implement the aforementioned porous aluminum strength prediction method based on physical constraint symbolic regression, comprising: The data acquisition module is used to conduct tensile tests on porous aluminum materials at multiple temperature points and under different porosities within a wide temperature range, and to obtain stress-strain data corresponding to each temperature and porosity. The regression model construction module is used to construct a physical constraint symbolic regression model based on stress-strain data. The physical constraint symbolic regression model embeds the basic principles of mechanics of materials into the symbolic regression process in the form of hard constraints. The hard constraints include the monotonically increasing stress-strain curve constraint, the decreasing elastic modulus with increasing porosity constraint, and the non-negative plastic work constraint. The equation building module is used to perform evolution calculations on stress-strain data using a physical constraint symbolic regression model, and automatically discover and output elastoplastic constitutive equations that satisfy hard constraints. The characterization model construction module is used to construct a temperature-dependent tensile strength theoretical characterization model based on the direct numerical relationship between the elastoplastic constitutive equation and the material failure strain. The tensile strength prediction module is used to predict the tensile strength of porous aluminum at any temperature over a wide temperature range, based on the tensile strength theoretical characterization model.

[0014] The present invention provides a computer device comprising: a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program to implement the above-described method for predicting the strength of porous aluminum based on physical constraint symbolic regression.

[0015] The present invention provides a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the above-described method for predicting the strength of porous aluminum based on physical constraint symbolic regression.

[0016] In summary, the present invention provides a method and system for predicting the strength of porous aluminum based on physical constraint symbolic regression. Compared with traditional technologies, the advantages are as follows: The physical constraint symbolic regression method automatically discovers and constructs a theoretical characterization model for tensile strength, obtaining a mathematical expression for predicting the strength of porous aluminum over a wide temperature range. The structure is clear, each physical term has a distinct physical meaning, and the fitting parameters are few, overcoming the shortcomings of traditional empirical models that rely on a large number of fitting parameters and lack physical meaning. By utilizing stress-strain data from multiple temperature points and under different porosity conditions within a wide temperature range, the elastoplastic constitutive equation is parameter-fitted, enabling the prediction of the strength of porous aluminum at any temperature within a wide temperature range. This avoids the time-consuming and labor-intensive problem of traditional methods requiring numerous specimens for destructive experiments across the entire temperature range.

[0017] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. Attached Figure Description

[0018] Figure 1 This is a flowchart of a method for predicting the strength of porous aluminum based on physical constraint symbolic regression in this invention; Figure 2 This is a comparison of the elastoplastic constitutive equation and the stress-strain curves of porous aluminum materials at different temperatures (-70℃, 25℃, 150℃, 350℃) and porosities (0.6, 0.8, and 0.9) in this invention. Figure 2 (a) in the figure is a comparison of the elastoplastic constitutive equation and the stress-strain curve of a porous aluminum material with a porosity of 0.6 at a temperature of 350℃. Figure 2 (b) in the figure is a comparison of the elastoplastic constitutive equation and the stress-strain curve of a porous aluminum material with a porosity of 0.6 at a temperature of 150℃. Figure 2 (c) in the figure is a comparison of the elastoplastic constitutive equation and the stress-strain curve of a porous aluminum material with a porosity of 0.6 at a temperature of 25℃. Figure 2(d) in the figure is a comparison of the elastoplastic constitutive equation and the stress-strain curve of a porous aluminum material with a porosity of 0.6 at a temperature of -70℃. Figure 2 (e) in the figure is a comparison of the elastoplastic constitutive equation and the stress-strain curve of a porous aluminum material with a porosity of 0.8 at a temperature of 350℃. Figure 2 (f) in the figure is a comparison of the elastoplastic constitutive equation and the stress-strain curve of a porous aluminum material with a porosity of 0.8 at a temperature of 150℃. Figure 2 In the figure, (g) is a comparison of the elastoplastic constitutive equation and the stress-strain curve of a porous aluminum material with a porosity of 0.8 at a temperature of 25℃. Figure 2 In the figure, (h) is a comparison diagram of the elastoplastic constitutive equation and the stress-strain curve of a porous aluminum material with a temperature of -70℃ and a porosity of 0.8. Figure 2 In the figure, (i) is a comparison of the elastoplastic constitutive equation and the stress-strain curve of a porous aluminum material with a porosity of 0.9 at a temperature of 350℃. Figure 2 In the figure, (j) is a comparison diagram of the stress-strain curve of the elastoplastic constitutive equation and the porous aluminum material with a porosity of 0.9 at a temperature of 150℃. Figure 2 In the figure, (k) represents the elastoplastic constitutive equation and a comparison of the stress-strain curves of porous aluminum material with a porosity of 0.9 at a temperature of 25℃. Figure 2 (l) is a comparison diagram of the elastoplastic constitutive equation and the stress-strain curve of a porous aluminum material with a temperature of -70℃ and a porosity of 0.9; Figure 3 This is a comparison graph of the experimental and theoretical tensile strengths of porous aluminum material with a porosity of 0.6 at the target temperature in an embodiment of the present invention. Figure 4 This is a comparison graph of the experimental and theoretical tensile strengths of porous aluminum material with a porosity of 0.8 at the target temperature in an embodiment of the present invention. Figure 5 This is a comparison graph of the experimental and theoretical tensile strengths of porous aluminum material with a porosity of 0.9 at the target temperature in an embodiment of the present invention. Figure 6 This is a block diagram of a porous aluminum strength prediction system based on physical constraint symbolic regression in this invention. Detailed Implementation

[0019] The technical method of the present invention will be further described below with reference to the accompanying drawings and embodiments. It should be noted that, unless otherwise specifically stated, the relative arrangement, numerical expressions, and values ​​of the components and steps described in these embodiments do not limit the scope of this application.

[0020] The following description of at least one exemplary embodiment is merely illustrative and is not intended to limit the scope of this application or its application or use.

[0021] Techniques, systems, and equipment known to those skilled in the art may not be discussed in detail, but where appropriate, they should be considered part of the instruction manual.

[0022] In all the examples shown and discussed herein, any specific values ​​should be interpreted as merely exemplary and not as limitations. Therefore, other examples of exemplary embodiments may have different values.

[0023] Unless otherwise defined, the technical or scientific terms used in this invention shall have the ordinary meaning as understood by one of ordinary skill in the art to which this invention pertains.

[0024] This invention provides a method for predicting the strength of porous aluminum based on physical constraint symbolic regression, such as... Figure 1 As shown, it includes the following steps: S1. Tensile tests were conducted on porous aluminum materials at multiple temperature points and under different porosities within a wide temperature range to obtain stress-strain data corresponding to each temperature and porosity. The wide temperature range was -70℃ to 350℃, and the porosity range was 0.6 to 0.9.

[0025] S2. Based on the stress-strain data, construct a physical constraint symbolic regression model. This model embeds fundamental principles of materials mechanics into the symbolic regression process as hard constraints. These hard constraints include the monotonically increasing stress-strain curve constraint, the decreasing elastic modulus with increasing porosity constraint, and the non-negative plastic work constraint.

[0026] Furthermore, step S2 specifically includes the following steps: S201. Divide the stress-strain data into a training set and a validation set, and determine the set of basic variables. The basic variables include at least strain and porosity.

[0027] S202. Initialize the population and generate multiple candidate equations. Each candidate equation is represented by a tree structure. Leaf nodes include basic variables and constants within a preset numerical range. Internal nodes include addition, subtraction, multiplication, division operators, and exponentiation operators.

[0028] S203. Based on the constraints of monotonically increasing stress-strain curve, decreasing elastic modulus with increasing porosity, and non-negative plastic work, corresponding constraint functions are constructed respectively. Each constraint function is used to calculate the quantitative degree of violation of the candidate equation for this specific constraint.

[0029] S204. Calculate the fitting error of each candidate equation based on the training set, and calculate the fitness value of each candidate equation based on the quantitative violation degree value. Apply an adaptive penalty factor to the fitness value for candidate equations that violate any hard constraint.

[0030] S205. Based on the fitness value, the parent equation is selected using the selection operator, and the offspring equation is generated through crossover and mutation operations to form a new generation population.

[0031] S206. Repeat the calculation of fitness value, penalty, selection, crossover, and mutation operations until the change in the optimal fitness value of the population is less than the preset threshold or the maximum number of iterations is reached over multiple generations.

[0032] S207. Select a candidate equation from the final Pareto front and verify the prediction accuracy and physical consistency based on the validation set. The candidate equation that passes the validation is determined as the physically constrained symbolic regression model.

[0033] S3. Using a physical constraint symbolic regression model, the stress-strain data are evolved to automatically identify and output the elastoplastic constitutive equations that satisfy the hard constraints. The elastoplastic constitutive equations are in additive form, consisting of a superposition of linear elastic response terms and nonlinear hardening terms. The linear elastic response term includes the product coupling relationship between porosity and elastic modulus. The nonlinear hardening term includes the product coupling relationship between porosity and the material's hardening ability.

[0034] The expression for the elastoplastic constitutive equation is: ; in, For tensile strength, For modulus, Porosity ε In response, For yield strength, and For adjustable fitting parameters, n This is the hardening index.

[0035] S4. Based on the direct numerical relationship between the elastoplastic constitutive equation and the material failure strain, a temperature-dependent theoretical characterization model for tensile strength is constructed. Specifically, the tensile strength theoretical characterization model is directly derived from the elastoplastic constitutive equation and the material failure strain, and its expression is: ; in, T For temperature, For temperature T Tensile strength at that time For temperature T Modulus at time, For temperature T Response in time, For temperature T Yield strength at that time Porosity and For adjustable fitting parameters, n The hardening index is used. Physically constrained symbolic regression is a machine learning method that derives mathematical expressions, specifically elastoplastic constitutive equations. It combines the fundamental principles of materials mechanics with data-driven modeling, relying on an evolutionary algorithm framework. A fitness function is used to improve the accuracy and physical plausibility of the mathematical expression. A tree structure is used to represent candidate equations, with leaf nodes representing basic variables (strain and porosity) and constant terms, and internal nodes representing mathematical operators (addition, multiplication, and exponentiation). During the evolutionary process, new expressions are generated through crossover and mutation, finding the Pareto optimal solution between accuracy (generally measured by mean squared error) and complexity (measured by equation terms). Compared to traditional symbolic regression methods, the unique characteristic of physically constrained symbolic regression is its physical constraint mechanism: utilizing the monotonicity requirement given on the stress-strain curve of the instantaneously obtained candidate equations (i.e.,... σ / ε >0); through the sign constraint of the coefficients of the porosity coupling term, the elastic modulus decreases with increasing porosity (i.e., E / P <0), and the non-negativity of plastic work was verified by energy integration.

[0036] Tensile experiments yielded complete stress-strain curves for porous aluminum at different temperatures and porosities, resulting in a multiphysics dataset encompassing thermomechanical response and pore parameters. Symbolic regression was performed using this hybrid dataset. The algorithm explored the space of mathematical expressions through an iterative process of initialization, evaluation, selection, crossover, and mutation within a genetic programming framework. The core principle is to treat physical priors as hard constraints, ensuring that candidate expressions conform to macroscopic trends such as dimensionality consistency, zero-strain initial conditions, and a decreasing elastic modulus with increasing porosity. These physical constraints are determined by an adaptive penalty function and data fit goodness of fit to guide the search. Within this hybrid physical information framework, the physical constraint symbolic regression algorithm not only effectively identified the additive combination structure of elastic and nonlinear hardening terms but also accurately recognized the product coupling relationship between each term and porosity.

[0037] For the linear elastic response term, the elastic modulus is not a constant, but a product term. Depends on porosity P In other words, the linear elastic response term exhibits the following characteristics: , and The product of and physically characterizes the linear weakening effect of porosity on the initial stiffness of a material. For the nonlinear hardening term, the product form is also derived. The reference yield strength Multiply The porosity is used to characterize the hardening capacity of porous aluminum. Finally, a physically constrained symbolic regression algorithm combines these two physically understandable parts: the linear elastic response corrected for porosity and the nonlinear power-law hardening regulated by porosity. The superposition relationship between the two is expressed in additive form, thus automatically generating a unified elastoplastic constitutive equation. This equation not only combines different deformation mechanisms through additive structures but also describes the quantitative influence of porosity on each mechanism through embedded multiplicative relationships, thereby achieving a unification of mathematical form and physical mechanism.

[0038] The advantage of the elastoplastic constitutive equation is that it can use a data-driven approach to give a mathematical expression for the nonlinear mechanical response of high-porosity (0.6~0.9) porous aluminum materials as a function of temperature, as shown in the stress-strain curves. Figure 2 As shown. Figure 2 (a) Figure 2 (b) Figure 2 (c) Figure 2 (d) Figure 2 (e) Figure 2 (f) in Figure 2 (g) in Figure 2 (h) in Figure 2 (i) Figure 2 (j) in Figure 2 (k) and Figure 2 The goodness-of-fit values ​​of (l) in the equations were 0.9836, 0.9414, 0.7788, 0.9955, 0.9494, 0.8474, 0.9930, 0.9812, 0.9297, 0.9964, 0.9713, and 0.9336, respectively. The elastoplastic constitutive equations accurately reproduced the key phenomena observed in the experiment: as the temperature gradually increased from a low temperature (-70℃) to a high temperature (350℃), the material's mechanical behavior changed from a brittle property with high strength and low plasticity to a ductile property with low strength and high plasticity. The elastoplastic constitutive equations are suitable for predicting several important mechanical parameters, reflecting the relationship between porosity, ultimate strength, and temperature, and also demonstrating the sensitivity of ultimate strain to temperature and the thermal softening of the elastic modulus. The deviations between these predicted results and the experimental data were generally within the error range of ±10%. Its inherent nonlinear characteristics in its mathematical structure can well reflect the unique deformation laws of porous materials, namely the process of slow buckling and collapse of pore walls in the low-temperature region, and the large-scale deformation manifestation dominated by diffusion creep in the high-temperature region. The ability to describe across scales from microscopic mechanisms to macroscopic reactions demonstrates the unique advantage of physical constraint symbolic regression in discovering physical laws from data.

[0039] For the failure modes of porous aluminum materials, a direct calculation method for tensile strength based on the elastoplastic constitutive equation is further proposed. The core idea is that, given the known failure strain of the porous aluminum material, the tensile strength can be directly calculated from the elastoplastic constitutive equation, i.e., the expression of the theoretical characterization model of tensile strength is as follows: .

[0040] The entire calculation process follows the principle of temperature-specific calibration and temperature-specific calculation, fully taking into account the objective laws of material parameter changes with temperature. Consistent modeling from constitutive relations to strength prediction can be achieved without adding other fitting parameters.

[0041] S5. Based on the tensile strength theoretical characterization model, the tensile strength of porous aluminum at any temperature within a wide temperature range is predicted. The prediction of tensile strength is carried out according to the principle of temperature-specific calibration and calculation. Specifically, the parameters in the elastoplastic constitutive equation are calibrated for different temperatures, and the tensile strength at the corresponding temperature is calculated.

[0042] In an exemplary embodiment of the present invention, based on the present invention, the tensile strength of porous aluminum materials with porosities of 0.6, 0.8, and 0.9 is predicted in the temperature range of -70℃ to 350℃ according to the tensile strength theoretical characterization model. The specific data are as follows: Tensile fracture tests were conducted on porous aluminum materials with a porosity of 0.6 at different temperatures to obtain the Young's modulus of the porous aluminum materials at different target temperatures (-70℃, 25℃, 150℃, and 350℃). E Fracture strain ε f Data such as yield strength of the material at the reference temperature σ The parameters of the temperature-dependent tensile strength of porous aluminum materials with a porosity of 0.6 are shown in Table 1.

[0043] Table 1. Tensile test parameters of porous aluminum materials with a porosity of 0.6 (hardening index) n =0.6)

[0044] Substituting the experimental data from Table 1 into the tensile strength theoretical characterization model, the predicted results of the tensile strength theoretical characterization model were obtained. The tensile strength of the porous aluminum material with a porosity of 0.6 at the target temperature was calculated, and the correlation between the tensile strength and temperature was obtained as follows: Figure 3 As shown, and compared with experimental measurements, from Figure 3 It can be seen that the calculated values ​​and the experimental test values ​​are in good agreement.

[0045] Similarly, porous aluminum materials with porosities of 0.8 and 0.9 were selected for tensile fracture tests at different temperatures to obtain the Young's modulus of the materials at different target temperatures (-70℃, 25℃, 150℃, 350℃). E Fracture strain ε f Data such as yield strength of the material at the reference temperature σ The parameters of the temperature-dependent tensile strength of porous aluminum materials with porosities of 0.8 and 0.9 are shown in Tables 2 and 3, respectively.

[0046] Table 2 Tensile test parameters for porous aluminum materials with a porosity of 0.8 (hardening index) n =0.6)

[0047] Table 3 Tensile test parameters of porous aluminum materials with a porosity of 0.9 (hardening index) n =0.6)

[0048] Substituting the experimental data from Tables 2 and 3 into the tensile strength theoretical characterization model, the predicted results of the model were obtained. The tensile strengths of porous aluminum with porosities of 0.8 and 0.9 at the target temperatures were calculated, and the correlation between tensile strength and temperature was shown below. Figure 4 and Figure 5 As shown, and compared with experimental measurements, from Figure 4 and Figure 5 It can be seen that the calculated values ​​and the experimental test values ​​are in good agreement.

[0049] This invention provides a reliable method for calculating the tensile strength of porous aluminum at various temperatures and porosity levels. It can calculate the tensile strength not only at different temperatures but also at different porosity levels, and takes into account the influence of temperature on the failure mechanism. This establishes a new method for predicting the tensile strength of porous aluminum over a wide temperature range, reducing the need for destructive experiments across the entire temperature range.

[0050] This invention takes porous aluminum as the research object. Based on the stress-strain data obtained from tensile tests over a wide temperature range (-70℃~350℃) under different temperatures and porosities, the invention uses a physical constraint symbolic regression method. By embedding the basic principles of materials mechanics as physical constraints into the symbolic regression process, the invention automatically discovers elastoplastic constitutive equations that satisfy physical consistency. Based on the strain-strength relationship of the elastoplastic constitutive equations, a temperature-dependent tensile strength theoretical characterization model is constructed to predict the strength of porous aluminum at different temperatures.

[0051] This invention establishes a mathematical expression for predicting the strength of porous aluminum over a wide temperature range (i.e., a theoretical characterization model of tensile strength). This model is automatically discovered through a physical constraint symbolic regression method. The model structure is clear, each physical term has a definite physical meaning, and the number of fitting parameters is small, overcoming the shortcomings of traditional empirical models that rely on a large number of fitting parameters and lack physical meaning. After calibrating the theoretical characterization model of tensile strength using experimental data over a wide temperature range, the strength of porous aluminum can be predicted at any temperature within that range. This avoids the time-consuming and labor-intensive problem of traditional methods requiring numerous specimens for destructive experiments across the entire temperature range.

[0052] This invention provides a porous aluminum strength prediction system based on physical constraint symbolic regression, used to implement the aforementioned porous aluminum strength prediction method based on physical constraint symbolic regression, such as... Figure 6 As shown, it includes: The data acquisition module is used to conduct tensile tests on porous aluminum materials at multiple temperature points and under different porosities within a wide temperature range, and to obtain stress-strain data corresponding to each temperature and porosity.

[0053] The regression model construction module is used to build a physically constrained symbolic regression model based on stress-strain data. This model embeds fundamental principles of materials mechanics into the symbolic regression process as hard constraints. These hard constraints include a monotonically increasing stress-strain curve constraint, a decreasing elastic modulus with increasing porosity constraint, and a non-negative plastic work constraint.

[0054] The equation building module is used to perform evolution calculations on stress-strain data using a physical constraint symbolic regression model, and automatically discover and output elastoplastic constitutive equations that satisfy hard constraints.

[0055] The characterization model construction module is used to construct a temperature-dependent tensile strength theoretical characterization model based on the direct numerical relationship between the elastoplastic constitutive equation and the material failure strain.

[0056] The tensile strength prediction module is used to predict the tensile strength of porous aluminum at any temperature over a wide temperature range, based on the tensile strength theoretical characterization model.

[0057] The present invention provides a computer device comprising: a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program to implement the above-described method for predicting the strength of porous aluminum based on physical constraint symbolic regression.

[0058] The present invention provides a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the above-described method for predicting the strength of porous aluminum based on physical constraint symbolic regression.

[0059] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit them. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the technical solutions of the present invention, and these modifications or equivalent substitutions cannot cause the modified technical solutions to deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. A method for predicting the strength of porous aluminum based on physical constraint symbolic regression, characterized in that, Includes the following steps: Tensile tests were conducted on porous aluminum materials at multiple temperature points and under different porosities over a wide temperature range to obtain stress-strain data corresponding to each temperature and porosity. Based on stress-strain data, a physical constraint symbolic regression model is constructed. The physical constraint symbolic regression model embeds the basic principles of mechanics of materials into the symbolic regression process in the form of hard constraints. The hard constraints include the monotonically increasing stress-strain curve constraint, the decreasing elastic modulus with increasing porosity constraint, and the non-negative plastic work constraint. The physical constraint symbolic regression model is used to perform evolution calculations on stress-strain data, automatically discovering and outputting elastoplastic constitutive equations that satisfy hard constraints; Based on the direct numerical relationship between the elastoplastic constitutive equation and the material failure strain, a temperature-dependent tensile strength theoretical characterization model is constructed. Based on the tensile strength theoretical characterization model, the tensile strength of porous aluminum at any temperature over a wide temperature range is predicted.

2. The method for predicting the strength of porous aluminum based on physical constraint symbolic regression according to claim 1, characterized in that, The specific content of constructing a physical constraint symbolic regression model based on stress-strain data includes: The stress-strain data are divided into a training set and a validation set, and a set of basic variables is determined; the basic variables include at least strain and porosity. Initialize the population and generate multiple candidate equations. Each candidate equation is represented by a tree structure. Leaf nodes include basic variables and constants within a preset numerical range, while internal nodes include addition, subtraction, multiplication, division operators, and exponentiation operators. Based on the constraints of monotonically increasing stress-strain curve, decreasing elastic modulus with increasing porosity, and non-negative plastic work, corresponding constraint functions are constructed. Each constraint function is used to calculate the quantitative degree of violation of the candidate equation for this specific constraint. The fitting error of each candidate equation is calculated based on the training set, and the fitness value of each candidate equation is calculated based on the quantitative violation value. For candidate equations that violate any hard constraint, an adaptive penalty factor is applied to the fitness value to penalize them. Based on the fitness value, the parent equation is selected using a selection operator, and the offspring equation is generated through crossover and mutation operations to form a new generation of population. Repeat the calculation of fitness value, penalty, selection, crossover, and mutation operations until the change in the optimal fitness value of the population is less than the preset threshold or the maximum number of iterations is reached over multiple generations. A candidate equation is selected from the final Pareto front, and the prediction accuracy and physical consistency are verified based on the validation set. The candidate equation that passes the verification is determined as the physically constrained symbolic regression model.

3. The method for predicting the strength of porous aluminum based on physical constraint symbolic regression according to claim 1, characterized in that, The elastoplastic constitutive equation is in additive form, consisting of the superposition of a linear elastic response term and a nonlinear hardening term; the linear elastic response term includes the product coupling relationship between porosity and elastic modulus; the nonlinear hardening term includes the product coupling relationship between porosity and the material's hardening ability.

4. The method for predicting the strength of porous aluminum based on physical constraint symbolic regression according to claim 3, characterized in that, The expression for the elastoplastic constitutive equation is: ； in, For tensile strength, For modulus, Porosity ε In response, For yield strength, and For adjustable fitting parameters, n This is the hardening index.

5. The method for predicting the strength of porous aluminum based on physical constraint symbolic regression according to claim 1, characterized in that, The theoretical characterization model for tensile strength is derived directly from the elastoplastic constitutive equation and the material failure strain, and its expression is: ； in, T For temperature, The temperature is T Tensile strength at that time The temperature is T Modulus at time, For temperature T Response in time, For temperature T Yield strength at that time Porosity and For adjustable fitting parameters, n This is the hardening index.

6. The method for predicting the strength of porous aluminum based on physical constraint symbolic regression according to claim 1, characterized in that, The prediction of tensile strength is carried out according to the principle of temperature-specific calibration and temperature-specific calculation. Specifically, the parameters in the elastoplastic constitutive equation are calibrated for different temperatures, and the tensile strength at the corresponding temperature is calculated.

7. The method for predicting the strength of porous aluminum based on physical constraint symbolic regression according to claim 1, characterized in that, The wide temperature range is -70℃ to 350℃; the porosity range is 0.6 to 0.

9.

8. A porous aluminum strength prediction system based on physical constraint symbolic regression, characterized in that, The method for predicting the strength of porous aluminum based on physical constraint symbolic regression as described in any one of claims 1-7 includes: The data acquisition module is used to conduct tensile tests on porous aluminum materials at multiple temperature points and under different porosities within a wide temperature range, and to obtain stress-strain data corresponding to each temperature and porosity. The regression model construction module is used to construct a physical constraint symbolic regression model based on stress-strain data. The physical constraint symbolic regression model embeds the basic principles of mechanics of materials into the symbolic regression process in the form of hard constraints. The hard constraints include the monotonically increasing stress-strain curve constraint, the decreasing elastic modulus with increasing porosity constraint, and the non-negative plastic work constraint. The equation building module is used to perform evolution calculations on stress-strain data using a physical constraint symbolic regression model, and automatically discover and output elastoplastic constitutive equations that satisfy hard constraints. The characterization model construction module is used to construct a temperature-dependent tensile strength theoretical characterization model based on the direct numerical relationship between the elastoplastic constitutive equation and the material failure strain. The tensile strength prediction module is used to predict the tensile strength of porous aluminum at any temperature over a wide temperature range, based on the tensile strength theoretical characterization model.

9. A computer device, comprising: A memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that the processor executes the computer program to implement the porous aluminum strength prediction method based on physical constraint symbolic regression as described in any one of claims 1-7.

10. A computer-readable storage medium having a computer program stored thereon, characterized in that, When executed by a processor, the computer program implements the method for predicting the strength of porous aluminum based on physical constraint symbolic regression as described in any one of claims 1-7.

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