Method and system for analyzing diffusion coefficient of chloride ion in concrete in natural exposure environment

Abstract

The application provides a method and system for analyzing a chloride ion diffusion coefficient of concrete in a natural exposure environment, and relates to the technical field of material durability evaluation and testing. The method comprises the following steps: obtaining chloride ion content data at different depths and initial chloride ion content; constructing a nonlinear mathematical model containing an error function based on a chloride ion diffusion theory; linearly approximating the model by using a series expansion of the error function to generate a linear expression with to-be-optimized parameters as coefficients; calculating residual sum of squares under different parameter values by changing the to-be-optimized parameter values and solving corresponding linear equations; determining optimal parameter values by taking the minimum residual sum of squares; and inversely calculating the chloride ion diffusion coefficient and surface chloride ion content. The application converts a traditional nonlinear fitting problem into single-parameter global optimization under a linear framework, reduces the calculation complexity while ensuring the accuracy, and is convenient for engineering practical application.

Description

Technical Field

[0001] This invention relates to the field of material durability assessment and testing technology, and in particular to a method and system for analyzing the chloride ion diffusion coefficient of concrete in naturally exposed environments. Background Technology

[0002] In naturally exposed environments (especially marine or salt spray environments), concrete structures in water transport engineering are subject to long-term chloride ion corrosion, leading to internal steel reinforcement corrosion and severely impacting structural durability. To assess the corrosion resistance of concrete, it is typically necessary to calculate the chloride ion diffusion coefficient using measured chloride ion content data. Currently, engineering projects mainly employ a nonlinear mathematical model based on Fick's second law for fitting and solving this model. This model includes an error function, requiring the simultaneous inversion of two unknown parameters—the diffusion coefficient and the surface chloride ion content—using a nonlinear fitting method. However, the nonlinear fitting process is computationally complex, iteratively difficult, and demands high data processing and programming skills from engineering technicians. Furthermore, it is susceptible to the influence of initial value selection, making it difficult to guarantee convergence and stability, thus limiting the field application of this method.

[0003] Therefore, there is an urgent need for a method to determine the chloride ion diffusion coefficient that can ensure calculation accuracy, significantly reduce calculation complexity, and facilitate practical engineering applications. Summary of the Invention

[0004] To address the aforementioned problems in existing technologies, the first aspect of this invention proposes a method for analyzing the chloride ion diffusion coefficient of concrete in naturally exposed environments, comprising: S1: Obtain chloride ion content data of concrete samples at multiple different depths and the initial chloride ion content of the concrete as the first dataset; S2: Based on the first dataset and chloride ion diffusion theory, a nonlinear mathematical model is constructed that includes an error function and uses the diffusion coefficient and the chloride ion content on the surface to be determined as unknown parameters. S3: Based on the nonlinear mathematical model, the nonlinear mathematical model is linearized by using the series expansion of the error function to generate a linear expression. The linear expression represents the combination of the surface chloride ion content to be determined and the initial chloride ion content as a linear relationship with the diffusion coefficient to be determined as a function. The diffusion coefficient to be determined is then transformed into a parameter to be optimized. S4: Based on the first dataset and the linear expression, by changing the values ​​of the parameters to be optimized and solving the corresponding system of linear equations, the sum of squared residuals under different values ​​of the parameters to be optimized is calculated to obtain the optimal parameter values ​​that minimize the sum of squared residuals; wherein, S4 includes: S41: Set the numerical search range for the parameters to be optimized, and generate a parameter search sequence based on the first dataset; S42: Based on each parameter value to be optimized and the linear expression in the parameter search sequence, a set of temporary surface chloride ion content and temporary combined parameter values ​​are obtained by using the linear regression method; S43: Based on the temporary surface chloride ion content, temporary combined parameter values, linear expression, and the first dataset, calculate the sum of squared residuals corresponding to each parameter value to be optimized; S44: Compare all residual sums of squares, select the residual sum of squares with the smallest value, and determine the parameter value to be optimized corresponding to the residual sum of squares with the smallest value as the optimal parameter value; S5: Based on the optimal parameter values, the final results of the chloride ion diffusion coefficient and surface chloride ion content of concrete are obtained by back-calculation through preset mathematical relationships.

[0005] Compared with the prior art, the beneficial effects of the present invention are as follows: By constructing a complete process of "data acquisition, model building, linearization, parameter optimization, and result back-calculation," an efficient and stable solution for the chloride ion diffusion coefficient was achieved. First, chloride ion content data and initial chloride ion content at different depths were acquired to provide a measured basis for the model. Second, a nonlinear mathematical model incorporating an error function was constructed based on chloride ion diffusion theory to ensure the model has a clear physical basis. Subsequently, the nonlinear model was linearized using the series expansion of the error function, generating a linear expression with the parameters to be optimized as coefficients. This transformed the originally complex nonlinear fitting problem into a problem of solving a system of linear equations, significantly reducing computational complexity. Based on this, by changing the values ​​of the parameters to be optimized and solving the corresponding system of linear equations, the sum of squared residuals under different parameter values ​​was calculated. The optimal parameter values ​​were then selected based on minimizing the sum of squared residuals. Finally, the chloride ion diffusion coefficient and surface chloride ion content were back-calculated based on the optimal parameter values. The steps work synergistically: the linearization step achieves the crucial transformation from nonlinear to linear, allowing the model to be directly solved using linear regression; the parameter optimization step avoids local optima through a global search strategy, ensuring the stability and accuracy of the results. Overall, this method significantly reduces computational difficulty and technical requirements for users while maintaining the physical meaning and computational accuracy of the model, making it more suitable for rapid application in engineering fields. Attached Figure Description

[0006] To more clearly illustrate the specific embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the specific embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained from these drawings without creative effort.

[0007] Figure 1 The diagram shown is a flowchart illustrating a method for analyzing the chloride ion diffusion coefficient of concrete in a naturally exposed environment, according to an embodiment of the present invention. Figure 2 The diagram shown is a structural schematic of a concrete chloride ion diffusion coefficient analysis system provided in a naturally exposed environment according to an embodiment of the present invention. Detailed Implementation

[0008] To make the objectives, technical solutions, and advantages of this invention clearer, the technical solutions of this invention will be clearly and completely described below. Obviously, the described embodiments are only a part of the embodiments of this invention, and not all of them. Based on the embodiments of this invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this invention.

[0009] The specific embodiments of the present invention will be described below.

[0010] Example 1 like Figure 1 As shown, the first aspect of this invention proposes a method for analyzing the chloride ion diffusion coefficient of concrete in a naturally exposed environment, comprising: S1: Obtain chloride ion content data of concrete samples at multiple different depths and the initial chloride ion content of the concrete as the first dataset; S2: Based on the first dataset and chloride ion diffusion theory, a nonlinear mathematical model is constructed that includes an error function and uses the diffusion coefficient and the chloride ion content on the surface to be determined as unknown parameters. S3: Based on the nonlinear mathematical model, the nonlinear mathematical model is linearized by using the series expansion of the error function to generate a linear expression. The linear expression represents the combination of the surface chloride ion content to be determined and the initial chloride ion content as a linear relationship with the diffusion coefficient to be determined as a function. The diffusion coefficient to be determined is then transformed into a parameter to be optimized. S4: Based on the first dataset and the linear expression, by changing the values ​​of the parameters to be optimized and solving the corresponding system of linear equations, the sum of squared residuals under different values ​​of the parameters to be optimized is calculated to obtain the optimal parameter values ​​that minimize the sum of squared residuals; wherein, S4 includes: S41: Set the numerical search range for the parameters to be optimized, and generate a parameter search sequence based on the first dataset; S42: Based on each parameter value to be optimized and the linear expression in the parameter search sequence, a set of temporary surface chloride ion content and temporary combined parameter values ​​are obtained by using the linear regression method; S43: Based on the temporary surface chloride ion content, temporary combined parameter values, linear expression, and the first dataset, calculate the sum of squared residuals corresponding to each parameter value to be optimized; S44: Compare all residual sums of squares, select the residual sum of squares with the smallest value, and determine the parameter value to be optimized corresponding to the residual sum of squares with the smallest value as the optimal parameter value; S5: Based on the optimal parameter values, the final results of the chloride ion diffusion coefficient and surface chloride ion content of concrete are obtained by back-calculation through preset mathematical relationships.

[0011] The analysis of the chloride ion diffusion coefficient in concrete under natural exposure aims to assess the rate at which chloride ions penetrate the interior of concrete, a key parameter for predicting structural durability. This method provides an efficient and stable computational pathway for this analysis. The method begins with data collection, specifically obtaining chloride ion content data from concrete samples at multiple depths, as well as the initial chloride ion content of the concrete itself. These measured values ​​form the basis for all subsequent calculations and are referred to as the first dataset. Here, "depth" refers to the distance perpendicularly inward from the exposed concrete surface, "chloride ion content" is typically expressed as a percentage of the concrete's mass or the mass of the cementitious material, and "initial chloride ion content" refers to the inherent chloride ion concentration of the concrete before service or in areas unaffected by external chloride ion erosion.

[0012] Based on these data and analytical objectives, a matching theoretical model needs to be established. The diffusion process of chloride ions in concrete is typically described using a model based on Fick's second law. Under certain initial conditions (such as uniform initial chloride ion distribution inside) and boundary conditions (such as constant surface chloride ion concentration), the solution of this partial differential equation contains a special mathematical function—the Gaussian error function. Therefore, the constructed nonlinear mathematical model naturally includes this error function, and its core unknowns are the "diffusion coefficient to be determined" (a physical quantity characterizing the diffusion rate) and the "surface chloride ion content to be determined" (characterizing the stable value of chloride ion concentration at the exposed surface). Directly performing a two-parameter nonlinear fitting on this complex expression containing the error function is the key to the computational complexity of traditional methods.

[0013] To address this challenge, this invention introduces a core element: linearization approximation. The principle is that a complex error function can be approximated with high precision using a simpler polynomial function derived from the Taylor expansion of the error function. Substituting the Taylor expansion (e.g., a finite-term polynomial consisting of the first few terms) into the original mathematical model, replacing the original error function, yields an "intermediate approximation model." Although the model still contains unknown parameters, its mathematical form has changed from including transcendental functions to only including power functions, multiplication, and other elementary operations. A further ingenious aspect lies in reorganizing the model by introducing a new variable, the "parameter to be optimized." For example, let the parameter to be optimized, k, be equal to the square root of the diffusion coefficient D_t. Substituting this relationship into the intermediate approximation model, and through a series of algebraic rearrangements and recombinations, the model can be rewritten in a special form. This form reveals that for each sampling depth x_i, the measured chloride ion content can be approximately represented as the sum of two terms. The first term is directly related to the chloride ion content C_s on the surface to be determined. The second term is the product of (C_s - initial chloride ion content C_0) and a coefficient φ(x_i, k), where φ is only a function of the depth x_i and the parameter k to be optimized. If we consider C_s and (C_s - C_0) as two unknowns to be solved, then for all sampled depth data points, the entire model constitutes a system of linear equations with φ(x_i, k) as the coefficient. Thus, we have successfully "reconstructed" the original nonlinear model into a "linear expression".

[0014] After the model is reconstructed into a linear form, the solution strategy also changes. The diffusion coefficient D_t is now implicitly included in the parameter k to be optimized. The solution process is transformed into trying a series of possible values ​​of k within a reasonable preset range. For each given value of k, all coefficients φ(x_i, k) in the linear expression become known, and a set of "temporary" C_s and temporary combined parameter values ​​can be quickly and stably solved using linear regression methods (such as least squares). Then, this set of solutions is used to predict the chloride ion content at each depth, and the sum of squares of the differences between the solutions and the measured values ​​is calculated, i.e., the "residual sum of squares". By traversing the search range of k, the k value that minimizes the residual sum of squares can be found, which is the "optimal parameter value". This process essentially decomposes a complex nonlinear minimization problem into a combined problem of performing a one-dimensional search on the single variable k at the outer layer and repeatedly solving a system of simple linear equations at the inner layer. Finally, using the pre-defined simple mathematical relationship between the optimal parameter value k and the diffusion coefficient D_t (such as the square relationship in the previous example), the final result of the diffusion coefficient is calculated, and the temporary C_s corresponding to the optimal k value also becomes the final result of the surface chloride ion content.

[0015] The temporary combination parameter value is numerically equal to the temporary surface chloride ion content minus the initial chloride ion content.

[0016] The pre-defined mathematical relationship is D_t=k 2 The chloride ion diffusion coefficient is calculated based on the optimal parameter values, and the corresponding temporary surface chloride ion content is used as the final surface chloride ion content.

[0017] This method achieves a fundamental shift in computational paradigm through a series of tightly linked steps: mathematical model construction, linearization approximation of the error function, algebraic reconstruction of the model into a linear expression, and coordinated optimization of the outer single-parameter scan and the inner linear solution. It avoids the iterations, differentiations, initial guesses, and potential convergence risks inherent in traditional nonlinear fitting algorithms. Mature, stable, and fast algorithms exist for solving linear equations, while the outer single-parameter global scan ensures the discovery of the global optimum rather than getting trapped in local optima. This synergistic effect significantly reduces the complexity of the entire computational process, the technical requirements for users, and the dependence on computational software, while ensuring the accuracy and stability of the solution. Thus, it successfully transforms a theoretical model originally confined to complex laboratory analysis into a highly efficient tool that can be quickly applied in engineering fields or routine testing.

[0018] In some implementations, S1 includes: S11: Drill concrete core samples from concrete structures in naturally exposed environments; S12: Perform layered powder sampling on the concrete core sample to obtain powder samples corresponding to different depths; S13: Perform chemical analysis on the powder sample and determine the chloride ion content data of the powder sample.

[0019] This embodiment specifies the method for obtaining the "first dataset," which is a crucial operational link in anchoring theoretical analysis methods to the real physical world, ensuring the authenticity and representativeness of the input data. The specific implementation begins with sample collection, specifically drilling complete "concrete core samples" from actual concrete structures (such as wharf decks, piers, and beams) under natural exposure conditions such as ocean tidal zones, splash zones, or salt spray environments, using hydraulic or mechanical core drilling rigs. This step is critical because laboratory-cast test blocks differ from actual structures in terms of material homogeneity, curing conditions, and loading history. Core sampling preserves the original structure's material properties, microstructure, and true chloride ion erosion state to the greatest extent possible, allowing subsequent analytical conclusions to directly correspond to the target physical structure.

[0020] After obtaining the core sample, the core procedure is "layered powder sampling." While the chloride ion concentration in concrete changes continuously with depth, the measurement must be discrete. In practice, the cylindrical core sample is fixed, and starting from one end (the end face simulating the exposed surface of the structure), material is removed layer by layer along the axial direction (depth direction) of the core sample using mechanical cutting or low-speed grinding at pre-set, equally spaced "depth intervals" (e.g., every 5 or 10 millimeters). The powder generated from each layer of grinding or drilling must be collected independently and completely, and immediately placed in a clean, sealed container with a clear depth label. This process must be extremely careful to prevent mixing or contamination between powders from different depth layers, as this would severely distort the judgment of the true chloride ion concentration gradient. The purpose of layered sampling is to discretize the continuous concentration distribution function into a series of data points (depth, chloride ion content) corresponding to specific depths, providing the necessary input for the mathematical model.

[0021] The final step is to determine the "chloride ion content data." Each collected powder sample must be tested according to the chemical analysis methods specified in national or industry standards. Commonly used methods include silver nitrate titration or chloride-selective electrode method. These methods, through specific chemical reactions or potential measurements, can accurately determine the percentage of water-soluble chloride ions in the sample mass. Simultaneously, the "initial chloride ion content of the concrete" also needs to be determined using similar methods. This is typically obtained by analyzing powder obtained from the deepest part of the same core sample (the part confirmed to be unaffected by external erosion), or by referring to the original formula data of the same batch of concrete. All the measurement results—the percentage of chloride ion content at each depth and the initial chloride ion content percentage—together constitute the first dataset used for subsequent mathematical calculations.

[0022] Therefore, this embodiment constructs a standardized data link from macroscopic structure to microscopic chemical composition. Core sampling ensures the authenticity of the sample source, layered powder sampling technology enables refined and discretized capture of the spatial distribution gradient of chloride ions, and standardized chemical analysis provides accurate quantitative evidence. These three sub-steps are interconnected and indispensable. Together, they fundamentally guarantee the quality of the data input to the mathematical model, avoiding systematic or random errors introduced by irregular sampling, sample contamination, or inaccurate testing. This provides a solid physical foundation and practical significance for subsequent complex mathematical analyses based on this data, and is the fundamental prerequisite for the reliable application of the entire analytical method in engineering practice.

[0023] In some implementations, S12 includes: S121: Starting from the surface of the concrete core sample, drill in layers at predetermined intervals along the depth direction; S122: When drilling at each depth layer, control the lateral dimension of the drilling area and make the lateral dimension gradually decrease with increasing depth, while ensuring that the lateral dimension is greater than a predetermined multiple of the maximum particle size of coarse aggregate in the concrete.

[0024] This embodiment proposes more refined technical constraints on the key operation of "layered powder sampling," focusing particularly on controlling the "lateral dimensions of the drilling area." This requirement deeply considers the inherent characteristics of concrete as a multiphase composite heterogeneous material, aiming to maximize the representativeness and reliability of the data from the sampling stage. Concrete is not a homogeneous medium, but a composite material composed of coarse aggregates, fine aggregates, cement paste, and interfacial transition zones of varying sizes. If the size of the sampling area is inappropriate, especially if the sampling volume is too small at a certain depth, the obtained powder may accidentally originate mainly from a large aggregate, a mass of cement paste, or a pore, failing to represent the average chemical composition of the concrete at that depth. This can lead to abnormal fluctuations or deviations in the measured chloride ion content data. This "sampling representativeness error" directly interferes with the accurate fitting of the diffusion law by the subsequent mathematical model.

[0025] In practice, the operation requires a dynamic adjustment mindset. When starting to drill or grind concrete core samples in layers, the "lateral dimension" of the drilling area (e.g., the diameter of a circular drill bit, or the side length of a square sampling area) should be set to a relatively large initial value for the first depth layer (the outermost layer). For example, grinding can be performed within a large initial radius, using the geometric center of the core sample cross-section as a reference. The purpose is to obtain a sufficiently large volume of material in the initial stage, so that the aggregates, pastes, and other phases contained therein tend to be statistically homogeneous, thereby obtaining a sample that can represent the "macroscopic average" chloride ion content of that depth layer.

[0026] As sampling depth increases, layers of material are removed, naturally reducing the cross-sectional area of ​​the remaining core sample. This is a physical constraint. In this process, it's not feasible to passively use a fixed, small-sized tool to drill to the bottom; instead, a strategy is needed to "control the lateral dimension of the drilling area and ensure it gradually decreases with depth." This means that when drilling deeper layers, it may be necessary to use a smaller diameter drill bit or adjust the positioning of the grinding jig to reduce the sampling area. However, this reduction is not arbitrary; the core control principle is that at each drilling depth, regardless of the adjusted lateral dimension of that layer, it must be ensured that the dimension is "greater than a predetermined multiple of the maximum coarse aggregate size in the concrete." This "predetermined multiple" is a threshold based on statistics and engineering experience, typically set at 3 times or more. The purpose of setting this multiple is very clear: to ensure that at any depth, the lateral dimension of the sampling area is large enough on a microscopic scale to cover multiple coarse aggregate particles and the mortar matrix filling them. Only in this way can the obtained powder sample be the product of mixing various components of concrete at that depth in volume proportion, and its chloride ion content can truly reflect the macroscopic average concentration at that depth, rather than a special value of a certain local micro-region.

[0027] By implementing this active control strategy of "initial large range, dynamically shrinking with depth, but always ensuring the size is much larger than the characteristic aggregate particle size," sampling noise caused by the microscopic inhomogeneity of concrete materials can be systematically suppressed. This ensures that powder samples obtained from different depths, despite differences in absolute volume, can effectively represent the overall material condition of their respective depth layers. The resulting chloride ion content data sequence will exhibit smoother and more stable changes with depth, more clearly revealing the macroscopic statistical laws of chloride ion diffusion. This provides high-quality, low-noise input for subsequent mathematical models, greatly enhancing the adaptability of the entire analytical method to complex, heterogeneous concrete materials in practical engineering and the confidence level of the final analytical results.

[0028] In some implementations, S3 includes: S31: Obtain the Taylor expansion of the error function as an approximate expression; S32: Substitute the approximate expression into the nonlinear mathematical model, replace the error function part in the nonlinear mathematical model, and obtain the intermediate approximate model; S33: Based on the intermediate approximation model, parameters to be optimized are introduced to replace the diffusion coefficient to be determined, and the intermediate approximation model is reconstructed into a linear expression through algebraic transformation.

[0029] This embodiment breaks down and refines the core step of "linearization," clearly revealing the specific mathematical operation process for transforming a nonlinear mathematical model containing a complex error function into a linearly solvable expression. This process begins with the approximation of the error function, a special mathematical tool. The Gaussian error function is mathematically an integral form, lacking an elementary function expression; its direct inclusion in the model makes linearization of the equations with respect to parameters impossible. Therefore, the first step is to "obtain the Taylor expansion of the error function as an approximate expression." Taylor expansion is a method of infinitely approximating a smooth function using a polynomial. For the error function, performing a Taylor expansion at the point where the independent variable is 0 (i.e., the origin) yields an infinite series. In actual calculations, it's unnecessary to take infinitely many terms; simply truncating the first few terms of the series (such as the first five terms, i.e., the "fifth-order Taylor expansion") allows for a highly accurate approximation of the original error function within an interval where the absolute value of the independent variable is not too large. This truncated polynomial is the "approximate expression" used to replace the complex error function in the original model.

[0030] After obtaining the approximate expression, the crucial substitution operation is performed: "substituting the approximate expression into the nonlinear mathematical model, replacing the error function part in the nonlinear mathematical model." Assume the original model is of the form C(x) = C_s - (C_s - C_0) × erf(φ(x, D_t)), where erf is the error function. Replacing erf(φ) with its fifth-order Taylor expansion polynomial P(φ) yields a new equation: C(x) ≈ C_s - (C_s - C_0) × P(φ(x, D_t)). This new equation is the "intermediate approximation model." Although the right-hand side of the equation still contains the parameters C_s, C_0, and D_t, the mathematical form has changed from containing the transcendental function erf to a polynomial P(φ) containing only φ, which is typically some elementary function of the depth x and the diffusion coefficient D_t. This transformation eliminates the main mathematical obstacle to solving the original model.

[0031] Finally, "based on the intermediate approximation model, parameters to be optimized are introduced to replace the diffusion coefficient to be determined, and the intermediate approximation model is reconstructed into a linear expression through algebraic transformations." This involves setting D_t = k... 2The relationship is substituted into the intermediate approximation model. At this point, φ(x, D_t) in the model is transformed into φ'(x, k), becoming a function of depth x and parameter k. Next, an "algebraic transformation" is performed: the polynomial P(φ'(x, k)) is expanded, and then like terms containing C_s and C_0 are combined. After simplification, the entire equation can be rearranged into the following form: for each depth x_i, we have C(x_i) ≈ β0 + β1 × ψ(x_i, k). Here, β0 and β1 are two new unknowns, which have a simple linear relationship with C_s and C_0 (usually β0 = C_s, β1 = C_s - C_0 or similar forms). And ψ(x_i, k) is a coefficient function completely calculated from depth x_i and the parameter k to be optimized, which comes from the previous polynomial expansion and combination. The new expression C(x_i) ≈ β0 + β1 × ψ(x_i, k) is the final "linear expression". In this linear expression, for a given set of depth data {x_i} and any fixed value of k, the coefficient ψ(x_i, k) is a computable constant. Thus, the entire model is represented as a standard system of linear equations with β0 and β1 as unknowns.

[0032] This embodiment first uses Taylor expansion to approximate the error function in the model using a polynomial. Then, through variable substitution, the key parameter D_t is transformed into a more manageable outer parameter k. Finally, through algebraic restructuring, the linear structure of the model with respect to variables β0 and β1 is uncovered and made explicit. This series of operations is based on a deep understanding of the mathematical properties of the model, and its final effect is to construct a completely new solution framework: transforming the problem of finding C_s and D_t into the problem of finding the optimal value of k and the corresponding optimal linear values ​​of β0 and β1, thus laying a solid mathematical foundation for the global optimization strategy.

[0033] In some implementations, the Taylor expansion in S31 uses a fifth-order Taylor expansion.

[0034] This embodiment specifies the mathematical approximation used in the linearization process, explicitly stating that "the Taylor expansion used is a fifth-order Taylor expansion." The core of the linearization process is to approximate the Gaussian error function in the nonlinear mathematical model of chloride ion diffusion with a simple polynomial function. The Gaussian error function is a special function in integral form, lacking an elementary closed-form expression, making direct parameter solving in the model very difficult. Taylor expansion is a standard function approximation method that expands a smooth function at a point (usually the zero point for the error function) into an infinite series. This series is determined by the values ​​of its derivatives at that point, exhibiting a power series form. In practical calculations, the first finite number of terms of this infinite series must be truncated to construct a practical approximate polynomial. "Fifth-order" indicates the truncation order, meaning that all terms from the zeroth to the fifth power of the independent variable in the power series are used to construct the approximate polynomial.

[0035] The fifth-order Taylor expansion is specifically erf(z)≈ ,in Substitute this expansion into the nonlinear mathematical model and introduce the parameters to be optimized. After algebraic merging, ψ(x_i,k) is represented as containing A polynomial function.

[0036] Choosing the fifth-order Taylor expansion as the approximation tool is an optimized result after a thorough trade-off between computational accuracy and implementation complexity. In terms of accuracy, a higher-order Taylor expansion generally results in a better fit between the polynomial and the original function near the origin, and also a wider applicable range of independent variables. For the error function, if only a first-order (linear) expansion is used, the approximation error increases rapidly when the independent variable is slightly large, leading to unacceptable deviations in the final inverted chloride ion diffusion coefficient. This high accuracy ensures that the subsequent linearization reconstruction and parameter solving process based on this approximation maintains a high degree of consistency with the results of traditional nonlinear fitting methods using the original error function model, meeting the error requirements of engineering analysis. Regarding complexity, although seventh-order or higher-order expansions can provide a wider effective range or slightly higher accuracy, the increase in the number of polynomial terms significantly increases the complexity of subsequent algebraic reconstruction. Substituting higher-order polynomials into the model and rearranging them into a linear expression introduces more higher-order terms and cross terms, making the final expression for the coefficients extremely lengthy and complex, increasing the computational load, and thus deviating from the original intention of this method to simplify calculations. The fifth-order expansion achieves the best balance between accuracy and complexity. It provides sufficient accuracy to ensure the reliability of engineering applications, while its polynomial form is sufficiently concise, making subsequent algebraic operations (such as combining like terms and extracting coefficients) clear and straightforward. The coefficient function in the final linear expression has a simple form, making it easy to calculate in programming or implement in a spreadsheet.

[0037] Therefore, explicitly limiting the Taylor expansion to fifth order provides implementers with a deterministic, validated, and high-precision approximation scheme. It is not a vague, arbitrarily chosen concept, but a specific technical instruction. This limitation ensures the stability and repeatability of the linearization process; all users will work based on a mathematical approximation of the same precision, thus guaranteeing the consistency of the method's output. It controls systematic errors introduced by inappropriate approximation tool selection at the source, allowing the entire quasi-linearization method to be built on a robust and optimal mathematical approximation foundation. This is one of the key mathematical guarantees that the method can reduce computational complexity without sacrificing computational accuracy, enabling engineers to use this simplified method to replace traditional complex nonlinear fitting.

[0038] In some implementations, S42 includes: S421: Given the parameter values ​​to be optimized, based on the linear expression and the first dataset, list a system of linear equations with the temporary surface chloride ion content and the temporary combined parameter values ​​as unknowns; S422: The linear equation system is solved using the least squares method to obtain the temporary surface chloride ion content and temporary combination parameter values.

[0039] This embodiment specifies the mathematical method for solving the linear equation system during parameter optimization: "the least squares method is used to solve the linear equation system." In the parameter optimization stage, for each assumed value of the parameter k to be optimized, the original chloride ion diffusion problem is transformed into a linear regression problem. Specifically, for a set of measured data points, where each point contains a depth x_i and a corresponding chloride ion content C_i, there exists a linear relationship: C_i ≈ β0 + β1 × ψ(x_i, k). Here, β0 and β1 are the temporary surface chloride ion content and temporary combined parameter values ​​to be solved, and ψ(x_i, k) is a known coefficient calculated from the depth x_i and the current parameter k. When there are multiple data points (usually more than two), the above formula constitutes a linear equation system containing multiple equations. Due to the inherent measurement errors and material discreteness in the measured data, these equations are usually mathematically "overdetermined," meaning that no single set of (β0, β1) can simultaneously and precisely satisfy all the equations. Therefore, an optimal criterion is needed to find the most suitable solution.

[0040] The least squares method is the most classic and well-defined method for solving such overdetermined linear equation systems. Its principle is to find a set of parameter estimates (β0, β1) such that the sum of squares of the differences (called residuals) between all predicted values ​​calculated by the model (i.e., β0 + β1 × ψ(x_i, k)) and the corresponding measured values ​​C_i is minimized. This "residual sum of squares" is a holistic quantitative indicator of the goodness of fit. The implementation process follows standard mathematical derivation: First, based on the above linear model form, the residual sum of squares S(β0, β1) = Σ [ C_i - (β0 + β1 × ψ_i) ] is constructed. 2 The summation applies to all data points. To find β0 and β1 that minimize S, we need to apply the extremum condition from calculus, i.e., take the partial derivatives of β0 and β1 respectively and set them equal to zero. This operation transforms the minimization problem into solving a system of equations consisting of two linear equations, which is called the "normal equations" or "normal equations" in statistics. By solving this normal equations, we can directly obtain the unique values ​​of β0 and β1 that minimize the sum of squared residuals.

[0041] The least squares method offers significant technical advantages and ease of implementation. First, it is based on a clear mathematical optimization criterion (minimizing the squared error), possessing a solid statistical theoretical foundation. Under the assumption that measurement errors follow a normal distribution, the obtained parameter estimates are optimal linear unbiased estimates. Second, the solution process is deterministic; the result can be obtained by solving a simple system of two linear equations, eliminating the need for complex iterative searches, resulting in high computational efficiency and eliminating the risk of non-convergence. Third, the least squares method is extremely universal and standardized. Almost all data processing software, programming languages ​​(such as Python's NumPy / SciPy libraries, MATLAB), and even spreadsheet software (such as Microsoft Excel) have built-in functions or tools (such as the LINEST function) for easy implementation. This greatly reduces the requirement for users to develop complex algorithms, allowing engineers to quickly learn and apply it. Finally, the calculated residual sum of squares, S, itself is the direct criterion used in subsequent steps to compare the merits of different k values ​​and thus find the global optimal solution. Therefore, this embodiment concretizes the crucial computational step of solving a system of linear equations from a possible, fuzzy operation into a standard, optimal mathematical procedure. It ensures that the solution process is stable, efficient, and the results are comparable, regardless of the number of data points. It provides a reliable and unified inner-layer calculation module for the entire global parameter optimization process and is an important computational foundation for the method to achieve convenient and reliable engineering applications.

[0042] In some implementations, S422 includes: S4221: Construct a normal system of equations based on a system of linear equations; S4222: The system of equations is solved using Cramer's rule to obtain the temporary surface chloride ion content and temporary combination parameter values.

[0043] This embodiment further refines the solution process of the least squares method, explicitly specifying that "Crem's rule is used to solve the system of normal equations." After the linear fitting problem is transformed into solving the system of normal equations using the least squares method, this system of normal equations is a system of two linear equations in two variables, β0 and β1. Its standard form can be expressed as: n×β0 + (Σψ_i)×β1 = ΣC_i and (Σψ_i)×β0 + (Σ(ψ_i)×β1 = ΣC_i) 2 _i))×β1 = Σ(ψ_i × C_i). Where n is the number of data points, ψ_i is short for ψ(x_i, k), and Σ represents the summation over all data points.

[0044] Cramer's rule is a classic analytical method for solving systems of linear equations in linear algebra, particularly suitable for systems with a small number of unknowns. First, we need to calculate the determinant of the coefficients, called the coefficient determinant D. For the system of equations above, D = n × Σ(ψ) 2 _i) - (Σψ_i) 2 Next, two new determinants need to be calculated: D0 and D1. D0 is the determinant calculated by replacing the first column of the coefficient determinant (i.e., the coefficient column of β0) with the constant terms of the system of equations (i.e., the values ​​on the right side of the equals sign), that is, D0 = (ΣC_i) × Σ(ψ 2 _i) - (Σψ_i) × Σ(ψ_i × C_i). D1 is the determinant calculated by replacing the second column of the coefficient determinant (i.e., the coefficient column of β1) with the constant term column, i.e., D1 = n × Σ(ψ_i × C_i) - (Σψ_i) × (ΣC_i). Finally, the solution to the system of equations is given by the following formulas: β0 = D0 / D, β1 = D1 / D. As long as the coefficient determinant D is not zero (which holds true under the normal condition that the data points are not collinear), the solution is unique and definite.

[0045] Using Cramer's rule offers several practical benefits. First, it avoids matrix inversion. For cases with only two unknowns, the determinant calculation is very simple, involving only basic multiplication, addition, and subtraction, resulting in minimal computation and a low probability of error. Second, its solution process is a fully step-by-step analytical formula with a clear and intuitive logical chain. This allows technicians to easily complete the entire solution manually, even in environments lacking advanced mathematical software, such as on-site using only calculators or simple spreadsheets. This is invaluable for understanding the algorithm's principles or for rapid verification. Third, the rule is easily implemented on any tool with basic computational capabilities. For example, in Microsoft Excel, simply calculating a few summation terms and the determinant value sequentially according to the formula yields the result, without needing to call a dedicated solver module. This significantly reduces reliance on software functionality and enhances the method's adaptability to various application scenarios. Furthermore, during the calculation, the value of the coefficient determinant D itself can serve as a simple indicator of numerical stability. If the absolute value of D is very close to zero, it suggests that the current data point or the ψ_i sequence calculated from the k value may lead to an "illness" in the equation system, and the solution is sensitive to small fluctuations in the data. This provides an immediate, built-in checkpoint for evaluating the reliability of the solution under the current k value. Therefore, this embodiment concretizes the solution of the least squares method from a relatively macroscopic "solving equation system" instruction into a series of explicit and sequentially executable arithmetic operation steps. It completely opens the "black box" of the core calculation process, giving the implementer the highest level of operational transparency and controllability, ensuring that even under the most basic calculation conditions, the key calculation links of this method can be executed accurately and reliably. This is a very important detail design for achieving the widespread applicability of this method in engineering settings.

[0046] In some implementations, S5 is followed by: S6: Based on the final results of chloride ion diffusion coefficient and surface chloride ion content, combined with a nonlinear mathematical model, calculate the theoretical predicted values ​​of chloride ion content at multiple different depths. S7: Based on theoretical predictions and chloride ion content data, the coefficient of determination is calculated to obtain the goodness-of-fit evaluation results of the nonlinear mathematical model.

[0047] This embodiment introduces an extension step for result verification and model evaluation after calculating the chloride ion diffusion coefficient and surface chloride ion content. While not a necessary step for core parameter inversion, this step provides crucial quantitative evidence for quality control and conclusion reliability throughout the analysis process. To implement this step, the "final results of the chloride ion diffusion coefficient and surface chloride ion content" are first used; these two parameter values ​​represent the optimal model parameters obtained through this method. Then, these two parameters are used as known quantities and substituted back into the "nonlinear mathematical model containing error functions" initially constructed in step S2. This model is the original theoretical equation based on Fick's diffusion law without any linearization approximations. Next, for each actual sampling depth x_i in the first dataset, a "theoretical prediction value," denoted as C_calc(x_i), is calculated using this complete theoretical model and the determined parameters (C_s, D_t). This process is a deterministic forward calculation; that is, the predicted chloride ion content is directly obtained from the input parameters and depth through the error function table or calculation library functions.

[0048] After obtaining the theoretical predicted value sequence for all depth points, it is necessary to perform a systematic quantitative comparison with the corresponding measured "chloride ion content data" (i.e., C_i) in the first dataset. To use a comprehensive index to characterize the overall goodness of fit, the "coefficient of determination R" is introduced here. 2 "As an evaluation statistic, the coefficient of determination is calculated following standard statistical methods: First, the total variation between the measured values ​​C_i and their mean C_avg is calculated, i.e., the total sum of squared deviations SST = Σ (C_i - C_avg)." 2 Secondly, the unexplained variation between the theoretical predictions and the observed values ​​is calculated, i.e., the residual sum of squares SSE = Σ [C_i - C_calc(x_i)]. 2 Finally, the coefficient of determination is defined as R0. 2 = 1 - (SSE / SST). R 2 The value range of R is between 0 and 1, which intuitively explains the proportion of measured data variability that the theoretical model can "capture" or "explain". 2 The closer R is to 1, the better the model's predicted values ​​match the actual values, indicating that almost all data variation can be explained by the model, and the goodness of fit is high; conversely, R... 2 The closer a value is to 0, the weaker the model's predictive ability, and the more data variation is that the model cannot explain.

[0049] The calculated coefficient of determination R 2As an output of "Goodness-of-Fit Evaluation Results for Nonlinear Mathematical Models," it has profound practical significance. Firstly, it provides users with an objective and standardized quality benchmark. Users do not need to rely on subjective observation of charts; they can simply use R... 2 A high R value can quickly assess the overall effect of this parameter inversion and the reliability of parameters such as the obtained diffusion coefficient. 2 Values ​​(e.g., greater than 0.85 or 0.9) significantly increase confidence in the analytical conclusions, supporting their use in durability assessments or life predictions. Conversely, a lower R-value... 2 The value serves as a clear warning signal, prompting users to review data quality (e.g., sampling or testing errors), check the model's applicability premises (e.g., whether Fick diffusion is dominant), or consider whether the heterogeneity of concrete materials is too significant. Secondly, this step logically constitutes a complete analytical loop from "theoretical model assumptions" to "parameter estimation" and then to "hypothesis testing," reflecting the rigor of scientific methods. It doesn't end with simply outputting parameters in a "black box," but rather uses the original data to retrospectively verify the theoretical model based on these parameters. Finally, through long-term engineering applications, accumulating R values ​​from different cases... 2 The evaluation value helps to assess the general applicability of the diffusion model to a certain type of structure or environment from a macroscopic perspective, providing data support for the continuous improvement of the method and the definition of its application scope. Therefore, the evaluation step added in this embodiment greatly enriches the information dimensions of the method output, upgrading this analysis method from a parameter calculation tool to a complete analysis system with self-diagnostic capabilities, assisting engineers in making more scientific and reliable judgments and decisions.

[0050] Example 2 like Figure 2 As shown, in a second aspect, the present invention proposes a system for analyzing the chloride ion diffusion coefficient of concrete in a naturally exposed environment. The system employs the method for analyzing the chloride ion diffusion coefficient of concrete in a naturally exposed environment proposed in any of the above embodiments. The system includes: The data acquisition module is used to perform step S1: acquire chloride ion content data of concrete samples at multiple different depths and the initial chloride ion content of the concrete as the first dataset; The model building module is used to perform step S2: based on the first dataset and chloride ion diffusion theory, construct a nonlinear mathematical model that includes an error function and uses the diffusion coefficient and the chloride ion content on the surface to be determined as unknown parameters; The linearization module is used to execute step S3: Based on the nonlinear mathematical model, the nonlinear mathematical model is linearized by using the series expansion of the error function to generate a linear expression. The linear expression represents the combination of the surface chloride ion content to be determined and the initial chloride ion content as a linear relationship with the diffusion coefficient to be determined as a function of the diffusion coefficient to be determined, and the diffusion coefficient to be determined is transformed into the parameter to be optimized. The parameter optimization module is used to execute step S4: based on the first dataset and the linear expression, by changing the values ​​of the parameters to be optimized and solving the corresponding system of linear equations, the sum of squared residuals under different values ​​of the parameters to be optimized is calculated to obtain the optimal parameter values ​​that minimize the sum of squared residuals; wherein, S4 includes: S41: Set the numerical search range for the parameters to be optimized, and generate a parameter search sequence based on the first dataset; S42: Based on each parameter value to be optimized and the linear expression in the parameter search sequence, a set of temporary surface chloride ion content and temporary combined parameter values ​​are obtained by using the linear regression method; S43: Based on the temporary surface chloride ion content, temporary combined parameter values, linear expression, and the first dataset, calculate the sum of squared residuals corresponding to each parameter value to be optimized; S44: Compare all residual sums of squares, select the residual sum of squares with the smallest value, and determine the parameter value to be optimized corresponding to the residual sum of squares with the smallest value as the optimal parameter value; The result inverse calculation module is used to execute step S5: based on the optimal parameter values, inverse calculation is performed through preset mathematical relationships to obtain the final results of the chloride ion diffusion coefficient and surface chloride ion content of the concrete.

[0051] This system corresponds to the method proposed in Example 1, and will not be described in detail here.

[0052] 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 the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the technical solutions of the embodiments of the present invention.

Claims

1. A method for analyzing the chloride ion diffusion coefficient of concrete in naturally exposed environments, characterized in that, include: S1: Obtain chloride ion content data of concrete samples at multiple different depths and the initial chloride ion content of the concrete as the first dataset; S2: Based on the first dataset and chloride ion diffusion theory, a nonlinear mathematical model is constructed that includes an error function and uses the diffusion coefficient and the chloride ion content on the surface to be determined as unknown parameters. S3: Based on the nonlinear mathematical model, the nonlinear mathematical model is linearized by using the series expansion of the error function to generate a linear expression. The linear expression represents the combination of the surface chloride ion content to be determined and the initial chloride ion content as a linear relationship with the diffusion coefficient to be determined as a function. The diffusion coefficient to be determined is then transformed into a parameter to be optimized. S4: Based on the first dataset and the linear expression, by changing the values ​​of the parameters to be optimized and solving the corresponding system of linear equations, the sum of squared residuals under different values ​​of the parameters to be optimized is calculated to obtain the optimal parameter values ​​that minimize the sum of squared residuals; wherein, S4 includes: S41: Set the numerical search range for the parameters to be optimized, and generate a parameter search sequence based on the first dataset; S42: Based on each parameter value to be optimized and the linear expression in the parameter search sequence, a set of temporary surface chloride ion content and temporary combined parameter values ​​are obtained by using the linear regression method; the temporary combined parameter value is numerically equal to the temporary surface chloride ion content minus the initial chloride ion content; S43: Based on the temporary surface chloride ion content, temporary combined parameter values, linear expression, and the first dataset, calculate the sum of squared residuals corresponding to each parameter value to be optimized; S44: Compare all residual sums of squares, select the residual sum of squares with the smallest value, and determine the parameter value to be optimized corresponding to the residual sum of squares with the smallest value as the optimal parameter value; S5: Based on the optimal parameter values, the final results of the chloride ion diffusion coefficient and surface chloride ion content of the concrete are obtained through back calculation using a preset mathematical relationship; the preset mathematical relationship is D_t=k 2 The chloride ion diffusion coefficient is calculated based on the optimal parameter value, and the chloride ion content of the corresponding temporary surface is used as the final surface chloride ion content. D_t is the diffusion coefficient, and k is the parameter to be optimized. S3 includes: S31: Obtain the Taylor expansion of the error function as an approximate expression; specifically: for the error function, perform a Taylor expansion at the point where its independent variable is 0 to obtain an infinite series, truncate the first five terms of the infinite series, and the resulting polynomial is the approximate expression; S32: Substitute the approximate expression into the nonlinear mathematical model and replace the error function part in the nonlinear mathematical model to obtain the intermediate approximate model; specifically: the original model is of the form C(x)=C_s-(C_s-C_0)×erf(φ(x,D_t)), where erf is the error function. Replace erf(φ) with its fifth-order Taylor expansion polynomial P(φ) to obtain a new equation: C(x)≈C_s-(C_s-C_0)×P(φ(x,D_t)), and the new equation is the intermediate approximate model; S33: Based on the intermediate approximation model, parameters to be optimized are introduced to replace the diffusion coefficient to be determined, and the intermediate approximation model is reconstructed into a linear expression through algebraic transformation; specifically: D_t=k 2 Substituting the relationship into the intermediate approximation model, φ(x, D_t) in the model is transformed into φ'(x, k), becoming a function of depth x and parameter k. Next, an algebraic transformation is performed: the polynomial P(φ'(x, k)) is expanded, and then the terms containing C_s and C_0 are combined with like terms. After simplification, the entire equation is rearranged into the following form: for each depth x_i, we have C(x_i)≈β0+β1×ψ(x_i, k), where β0 and β1 are two new unknowns, which have a simple linear relationship with C_s and C_0, β0=C_s, β1=C_s-C_0, and ψ(x_i, k) is a coefficient function calculated from depth x_i and the parameter k to be optimized. Due to the previous polynomial expansion and combination, this new expression C(x_i)≈β0+β1×ψ(x_i, k) is the final linear expression. S42 includes: S421: Given the parameter values ​​to be optimized, based on the linear expression and the first dataset, list a system of linear equations with the temporary surface chloride ion content and the temporary combined parameter values ​​as unknowns; S422: The linear equation system is solved by the least squares method to obtain the temporary surface chloride ion content and temporary combination parameter values; S422 includes: S4221: Construct a normal system of equations based on a system of linear equations; S4222: The system of equations is solved using Cramer's rule to obtain the temporary surface chloride ion content and temporary combination parameter values.

2. The method for analyzing the chloride ion diffusion coefficient of concrete in a naturally exposed environment according to claim 1, characterized in that, S1 includes: S11: Drill concrete core samples from concrete structures in naturally exposed environments; S12: Perform layered powder sampling on the concrete core sample to obtain powder samples corresponding to different depths; S13: Perform chemical analysis on the powder sample and determine the chloride ion content data of the powder sample.

3. The method for analyzing the chloride ion diffusion coefficient of concrete in a naturally exposed environment according to claim 2, characterized in that, S12 includes: S121: Starting from the surface of the concrete core sample, drill in layers at predetermined intervals along the depth direction; S122: When drilling at each depth layer, control the lateral dimension of the drilling area and make the lateral dimension gradually decrease with increasing depth, while ensuring that the lateral dimension is greater than a predetermined multiple of the maximum particle size of coarse aggregate in the concrete, specifically more than 3 times.

4. The method for analyzing the chloride ion diffusion coefficient of concrete in a naturally exposed environment according to claim 1, characterized in that, The Taylor expansion in S31 uses a fifth-order Taylor expansion.

5. The method for analyzing the chloride ion diffusion coefficient of concrete in a naturally exposed environment according to claim 1, characterized in that, Following S5 are: S6: Based on the final results of chloride ion diffusion coefficient and surface chloride ion content, combined with a nonlinear mathematical model, calculate the theoretical predicted values ​​of chloride ion content at multiple different depths. S7: Based on theoretical predictions and chloride ion content data, the coefficient of determination is calculated to obtain the goodness-of-fit evaluation results of the nonlinear mathematical model.

6. A system for analyzing the chloride ion diffusion coefficient of concrete in naturally exposed environments, characterized in that, The system employs the method for analyzing the chloride ion diffusion coefficient of concrete in a naturally exposed environment as described in any one of claims 1 to 5. The system comprises: The data acquisition module is used to perform step S1: acquire chloride ion content data of concrete samples at multiple different depths and the initial chloride ion content of the concrete as the first dataset; The model building module is used to perform step S2: based on the first dataset and chloride ion diffusion theory, construct a nonlinear mathematical model that includes an error function and uses the diffusion coefficient and the chloride ion content on the surface to be determined as unknown parameters; The linearization module is used to execute step S3: Based on the nonlinear mathematical model, the nonlinear mathematical model is linearized by using the series expansion of the error function to generate a linear expression. The linear expression represents the combination of the surface chloride ion content to be determined and the initial chloride ion content as a linear relationship with the diffusion coefficient to be determined as a function of the diffusion coefficient to be determined, and the diffusion coefficient to be determined is transformed into the parameter to be optimized. The parameter optimization module is used to execute step S4: based on the first dataset and the linear expression, by changing the values ​​of the parameters to be optimized and solving the corresponding system of linear equations, the sum of squared residuals under different values ​​of the parameters to be optimized is calculated to obtain the optimal parameter values ​​that minimize the sum of squared residuals; wherein, S4 includes: S41: Set the numerical search range for the parameters to be optimized, and generate a parameter search sequence based on the first dataset; S42: Based on each parameter value to be optimized and the linear expression in the parameter search sequence, a set of temporary surface chloride ion content and temporary combined parameter values ​​are obtained by using the linear regression method; S43: Based on the temporary surface chloride ion content, temporary combined parameter values, linear expression, and the first dataset, calculate the sum of squared residuals corresponding to each parameter value to be optimized; S44: Compare all residual sums of squares, select the residual sum of squares with the smallest value, and determine the parameter value to be optimized corresponding to the residual sum of squares with the smallest value as the optimal parameter value; The result inverse calculation module is used to execute step S5: based on the optimal parameter values, inverse calculation is performed through preset mathematical relationships to obtain the final results of the chloride ion diffusion coefficient and surface chloride ion content of the concrete.

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