A method of uncertainty analysis for predicting hydrogen redistribution in zirconium alloys

Abstract

The application discloses a kind of uncertainty analysis methods for predicting redistribution of hydrogen in zirconium alloy, comprising: determining the input parameters to be studied and the corresponding prior distribution range;MOAT method is used to analyze the importance degree of input parameter;Based on prior distribution, input parameter sampling is carried out, the input parameter obtained by sampling is input into simulation model, output parameter is obtained, training data set for training surrogate model is constructed, fitting is carried out on multiple groups of input parameters and output parameters, and surrogate model is obtained;The input parameters and the corresponding output parameters filtered by the MOAT method are obtained through the surrogate model, the Sobol index of the input parameters to the key performance indicators of nuclear fuel is calculated, and the key uncertainty parameters are identified;Based on the prior distribution of key uncertainty parameters, the posterior distribution after correction is obtained by using MCMC method.The application can effectively support the safety evaluation, life prediction and optimization design of nuclear fuel, and is suitable for nuclear fuel performance analysis scenarios of various types of nuclear reactors, such as pressurized water reactor and fast reactor.

Description

Technical Field

[0001] This invention relates to the field of uncertainty analysis of nuclear fuels, and more particularly to an uncertainty analysis method for predicting the redistribution of hydrogen in zirconium alloys. Background Technology

[0002] Nuclear fuel cladding is the primary safety barrier of the reactor core, its main function being to isolate radioactive fuel and its fission products from the cooling system. Zirconium is the preferred fuel cladding material for nuclear reactors due to its excellent nuclear properties, including a low thermal neutron absorption cross section, excellent mechanical properties, and strong corrosion resistance in high-temperature, high-pressure water and steam environments. However, under standard operating conditions of pressurized water reactors (PWRs), the outer surface of the zirconium alloy cladding is susceptible to water-side corrosion, which leads to a reduction-oxidation chemical reaction between the cladding and the coolant. This reaction forms a zirconium dioxide (ZrO2) oxide layer on the zirconium alloy surface and allows some free hydrogen to enter the cladding. When hydrogen is trapped and accumulates inside the zirconium cladding, it first dissolves in the solid solution.

[0003] In cladding materials, hydrogen diffusion is influenced by concentration gradients, temperature gradients, and hydrostatic stress. The diffusion behavior of hydrogen atoms in solid solutions is closely related to parameters of hydrogen atoms in zirconium alloys, such as diffusion coefficient, solubility, and various gradient conditions. Specifically, in the presence of a concentration gradient, hydrogen atoms migrate from regions of higher concentration to regions of lower concentration according to Fick's law. Furthermore, under the influence of a temperature gradient, hydrogen diffuses from regions of higher temperature to regions of lower temperature according to the Soray effect, and the diffusion rate of hydrogen in the cladding increases significantly with increasing temperature. Kammenzind's experiments show that stress has almost no effect on the chemical potential of hydrogen in the solid solution or the chemical potential of the hydrogen solution. The observed net transfer of hydrogen cannot be clearly attributed to the effect of stress on the dissolving solvent or the effect of stress on the chemical potential of dissolved hydrogen in the zirconium alloy lattice. Compared to the effect of temperature, the effect of applied stress on hydrogen migration is very small. Hydrogen generates additional tensile stress, which, combined with external loads, can enhance plastic deformation. Once the hydrogen concentration in the solid solution exceeds its limit, hydrides form. These hydrides have extremely low elongation and exhibit a brittle phase morphology, which leads to significant lattice deformation around the cladding and generates a strain field in the surrounding region. This change directly results in a significant reduction in the toughness of the zirconium alloy cladding material. Due to hydrogen formation, the zirconium matrix is ​​more prone to cracking, which can further develop into delayed hydride cracks (DHC). This phenomenon greatly increases the failure risk of the zirconium alloy cladding and poses a serious threat to the safe and stable operation of nuclear reactors. Therefore, it is necessary to propose a method for predicting the redistribution of hydrogen in zirconium alloys. Summary of the Invention

[0004] To address the technical problem of the difficulty in comprehensively and accurately quantifying the sources and extent of uncertainty in existing nuclear fuel performance analysis, which in turn affects nuclear fuel safety assessment and optimization design, this invention provides an uncertainty analysis method for predicting the redistribution of hydrogen in zirconium alloys. This invention provides two methods of sensitivity analysis to accurately identify and quantify the impact of key uncertainty parameters that have the most significant influence on nuclear fuel performance on quantities of interest, thus clarifying the main sources of uncertainty. This invention also provides an inverse uncertainty method based on Bayesian inference, which, combined with nuclear fuel performance test data or operational measurement data, reversely corrects the probability distribution of key uncertainty parameters.

[0005] To achieve the objective of this invention, a method for uncertainty analysis in predicting the redistribution of hydrogen in zirconium alloys is provided, specifically comprising the following steps: A simulation model for predicting hydrogen redistribution in zirconium alloys was constructed. Based on the simulation model, the input parameters to be studied were determined, and the prior distribution range corresponding to each input parameter was determined. Based on the prior distribution, the MOAT method is used to analyze the importance of the input parameters and to perform preliminary screening of the input parameters; Input parameters are sampled based on the prior distribution, and the sampled input parameters are input into the simulation model to obtain the corresponding output parameters. A training dataset for training the surrogate model is constructed based on the sampled input parameters and the simulation output parameters. The surrogate model is obtained by fitting multiple sets of input parameters and output parameters. The input parameters and corresponding output parameters after being filtered by the MOAT method are obtained by using a proxy model. The Sobol index sensitivity analysis method is used to calculate the Sobol index of each input parameter on the key performance indicators of nuclear fuel. Based on the Sobol index, the key uncertainty parameters that have the most significant impact on the performance of nuclear fuel are identified. By combining nuclear fuel test or measured data, and based on the prior distribution of key uncertainty parameters, the MCMC method is used to obtain the corrected posterior distribution.

[0006] As a preferred technical solution, the Morris basic effects method, also known as the Morris one-time method (MOAT), is a global extension of the one-time method. In the one-time method, the model input parameters are varied one by one while keeping other parameters constant. MOAT, however, performs a computer experiment with a single randomized design, changing only one input parameter at a time to generate its basic effects (EE) sample. Given a model y(x), where It is dimensional input vector, EE corresponds to in The first reference point Input parameters and define them as follows:

[0007] in, Indicates in Dimensional grid jumps. MOAT for each input Perform N samplings, while randomly selecting reference points from the entire input space. For each input parameter... The mean, corrected mean, and standard deviation are defined as follows:

[0008]

[0009]

[0010] The Sobol exponential method is based on the principle of variance decomposition, which decomposes the total variance of the output into the contribution of each input parameter and their interactions. This method assumes that the model output can be represented as the sum of functions of the input parameters, and accordingly decomposes and quantifies the variance of the model output. For a given expression... The amount of attention, based on the first-order Sobol exponent of variance, is defined as:

[0011] in, yes One input parameter, and Indicates except All other parameters besides the inner expected value. The meaning of the inner expected value is that, at a fixed... In this case, for All possible values The average value. Then, the outer variance is... The calculation is performed over all possible values. The total Sobol exponent is defined as:

[0012] This indicator measures the first The sum of the first-order effects of the input parameters and their interactions with other parameters. The second term in the equation can be seen as the sum of the first-order effects of the input parameters and their interactions with other parameters. The first-order effect of all parameters other than the first parameter. Therefore, 1 minus the second term necessarily gives the first-order effect of the first parameter. The contribution of all relevant items. It is important to note that for uncertainty quantification studies with multiple points of interest, the Sobol index is calculated for each individual point of interest.

[0013] In the Bayesian inference framework, the product of the prior distribution and the likelihood function is used to construct the target posterior distribution:

[0014] To obtain the posterior distribution, this invention uses the Markov Chain Monte Carlo (MCMC) method from... Sampling is performed within the target posterior distribution. This sampling-based method repeatedly generates random samples to approximate the target posterior distribution. The core of this method lies in constructing a Markov chain, a stochastic process where the next state depends only on the current state, not on previous states. The key to MCMC is designing such Markov chains so that their steady-state distribution is equal to the target distribution we want to sample from. Regarding the processing of experimental data, if sufficient experimental data is available, it can be treated as univariate data, with other variables used as additional evaluation parameters. When experimental data is insufficient, data noise can be added to the high-fidelity model to simulate real experimental data.

[0015] Given the probability distribution of uncertain parameters, Latin hypercube sampling (LHS) can be used to construct computer experiments. LHS generates random samples of parameter values ​​from a multidimensional distribution, effectively reflecting real-world variability and has been widely used in Monte Carlo simulations to save computational costs. LHS is built upon the Latin square design, which uses a square grid containing one sample per row / column. The Latin hypercube generalizes from the Latin square design to accommodate arbitrarily high dimensions. Similarly, each axis-aligned hypercube contains only one random sample. In each input dimension, the intervals constituting the hypercube have the same probability, and the number of equally probable intervals is the same for each input dimension (variable). Thus, the parameter space of interest is efficiently filled using LHS samples. Furthermore, samples are generated sequentially, making it easy to record the samples generated so far. It's important to note that the LHS design is not unique, as the position of the sample within each hypercube remains random. To ensure that the generated samples represent the true distribution, methods such as maximin-LHS are often used to improve space-filling properties, maximizing the minimum distance between samples. Furthermore, it is worth noting that one of the assumptions underlying Latin hypercube sampling (LHS) is that the random variables are independent of each other. This assumption of independence is reasonable for the application of this study, as such information is difficult to obtain in kernel applications due to the typically sparse data.

[0016] To address the high computational costs associated with complex simulations or physical experiments, a response surface model can be constructed to replace the actual physical model. Then, a Monte Carlo-based Sobel method and response curves are applied to this surrogate model. This study uses a Gaussian process (GP) as the surrogate model. A Gaussian process is a nonparametric probabilistic model based on a Bayesian framework, adept at flexibly modeling and predicting unknown functions. Its core principle lies in treating the objective function as a single instance of a stochastic process defined in the input space, while assuming that the joint distribution of this process follows a multivariate Gaussian distribution.

[0017] The present invention also provides an uncertainty analysis system for predicting the redistribution of hydrogen in zirconium alloys.

[0018] The present invention also provides a computer device.

[0019] The present invention also provides a computer-readable storage medium.

[0020] Compared with the prior art, the present invention can achieve at least the following beneficial effects: (1) Based on the preliminary sensitivity screening of the MOAT method, this invention quickly and cost-effectively identifies a few key parameters that have a significant impact on the output results and selects a subset of parameters that need to be focused on from a large number of candidate parameters; (2) This invention uses global sensitivity quantitative analysis based on the Sobol index to conduct more accurate and comprehensive quantitative sensitivity analysis on the basis of the key parameters screened by MOAT, and to distinguish the main effects of each parameter and the interaction effects with other parameters. (3) The MCMC method used in this invention does not provide a single optimal parameter value, but a complete posterior probability distribution of the parameters. It can directly quantify the uncertainty of the model parameters and pass this uncertainty to the final prediction, thereby giving a prediction result with a confidence interval, which greatly improves the reliability of the model prediction and the decision support value.

[0021] (4) The uncertainty method used in this invention has a certain degree of universality and can be used to analyze parameters in different stages of nuclear fuel service performance. Attached Figure Description

[0022] Figure 1 This is a flowchart illustrating an uncertainty analysis method based on Bayesian methods in an embodiment of the present invention.

[0023] Figure 2 This is a schematic diagram of the MOAT calculation results for predicting the relevant model parameters of hydrogen redistribution in zirconium alloy under temperature gradient in an experiment based on Sawatzky, as shown in the embodiment of the present invention.

[0024] Figure 3This is a schematic diagram illustrating the Sobol index calculation results for predicting hydrogen redistribution in zirconium alloys under temperature gradients in an experiment based on Sawatzky, as shown in this embodiment of the invention.

[0025] Figure 4 This is a schematic diagram of the posterior distribution of important parameters obtained using the MCMC method in an embodiment of the present invention. Detailed Implementation

[0026] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0027] This invention provides an uncertainty analysis method for predicting hydrogen redistribution in zirconium alloys based on a Bayesian framework. It utilizes methods such as the MOAT method, the Sobol exponent method, and Markov chain Monte Carlo to address the technical problem that existing nuclear fuel performance analysis cannot comprehensively and accurately quantify the sources and extent of uncertainty, thereby affecting nuclear fuel safety assessment and optimization design.

[0028] Please see Figure 1 The present invention provides an uncertainty analysis method for predicting hydrogen redistribution in zirconium alloys based on a Bayesian framework, comprising the following steps: Step S1: Construct a simulation model to predict the redistribution of hydrogen in zirconium alloy. Based on the simulation model, determine the input parameters for the study using existing knowledge or expert advice, and determine the prior distribution range corresponding to each input parameter.

[0029] In this step, the simulation model for predicting hydrogen redistribution in zirconium alloys includes: Hydrogen migration in zirconium solid solutions is driven by both the concentration gradient and the temperature gradient, and the specific calculation formula is as follows:

[0030] In the formula, It is the concentration of dissolved hydrogen in the matrix. It is diffuse heat. It is the cladding temperature. It is the ideal gas constant. The gradient represents the hydrogen diffusion coefficient in the fitted zirconium alloy, which mainly depends on temperature, and is calculated as follows:

[0031] Indicates activation energy. It is the total hydrogen flux. The diffusion coefficient inside the zirconium alloy. The diffusion coefficient is the exponential factor before the diffusion coefficient.

[0032] When the hydrogen concentration in the solid solution is higher than When hydrogen in a solid solution is converted into precipitated hydrides, the hydrogen solubility in the solid solution will be lower than that of the precipitated hydrides. If precipitated hydrides are present, they will dissolve in the solid solution. When the hydrogen concentration in the solid solution is between the two solubility limits, neither precipitation nor dissolution occurs. The interconversion between hydrogen in the solid solution and precipitated hydrides depends on the following two equations:

[0033]

[0034] It's TSS D Pre-exponential factor, It's TSS P Pre-exponential factor, and This represents the corresponding activation energy.

[0035] Total hydrogen concentration in zirconium alloy cladding It is the hydrogen concentration in the solid solution. (in ppm) and hydrogen concentration in hydrides The sum:

[0036] The equilibrium equation between hydrogen in the solid solution and the precipitated hydride can be expressed as:

[0037] in It is the source term of hydrides, which is represented in the HNGD model as the hydride nucleation rate ( ), growth rate ( ) and dissolution rate ( The sum of ). This refers to the reaction run time. When the hydrogen concentration in the solid solution is higher than... When the precipitated hydride continues to grow, this is called the hydride growth process. However, if there is no hydride at this time, the hydrogen in the solid solution will only diffuse into the zirconium alloy. , , The expression is:

[0038]

[0039]

[0040] in, , and These represent the kinetic coefficients for nucleation, growth, and dissolution, respectively. This represents the growth dimension, and can take a value of 2.5. It represents the degree to which the reaction is completed. The kinetic coefficient equation can be summarized as follows:

[0041]

[0042]

[0043] in , , and These are the exponential constants of Arrhenius's laws for dissolution, nucleation, diffusion-controlled growth, and reaction-controlled growth, respectively. Hydrogen diffusion activation energy, Hydride formation energy, Volume fraction of zirconium alloy cladding Hydrogen solubility atomic fraction The activation energy required for hydride growth is indicated by the number of mobs ("mob") and the reaction rate at the hydride-matrix interface ("th"), both of which are incorporated into the overall growth kinetics parameter K. G middle; These are reaction-driven growth kinetic parameters. These are diffusion-driven growth kinetic parameters.

[0044] In this step, it is necessary to determine the input parameters to be analyzed and their corresponding prior distributions. For general input parameters, a roughly reasonable range of prior distributions can be determined under different circumstances through historical data and expert opinions. In one embodiment, the input parameters involved are shown in Table 1. Through a review of the literature, the default values ​​of different input parameters were varied by 20%-30% to serve as the upper and lower limits of the prior distribution.

[0045] In one embodiment, during sensitivity analysis, the relative root mean square error (RMSE) is used as the measure of interest to compare the differences between the diffusion-reaction model values ​​simulated in COMSOL (i.e., the simulation model) and Sawazky's experimental data (an existing known experiment, which will not be elaborated upon here). Its definition is:

[0046] in, The initial total hydrogen concentration under different experimental data, This indicates the number of sample slices used to measure hydrogen content in the experiment. These are the predicted values ​​for hydrogen concentration at different locations. The hydrogen concentration was measured at different locations during the experiment.

[0047] Table 1. Prior distribution range of input parameters

[0048] S2: Based on the prior distribution, the MOAT method is used to analyze the importance of the input parameters, and the input parameters are initially screened to provide a data foundation for subsequent performance model verification and analysis.

[0049] To assess the impact of input parameters on the uncertainty of model output, the Morris basic effects method (also known as the MOAT method) was used for parameter selection in the global sensitivity analysis.

[0050] The MOAT method executes computer experiments with individual randomized designs, changing only one input parameter at a time to generate a sample of its base effect (EE). Given a model... , where input parameters It is 3D input vector, Representing the The last input parameter (i.e., the dimension). Basic effect sample. Corresponding to the reference point The first There are 1 input parameter, defined as:

[0051] Indicates the first Reference points At this point, the first Each input parameter is increased by a perturbation step size. The model's output value when all other parameters remain constant; Indicates the first Reference points At this point, the first Reduce the number of input parameters The model's output value when all other parameters remain constant; Indicates the first A reference point, i.e., a specific input vector in the parameter space, is used as a benchmark for calculating the basic effects.

[0052] The MOAT method provides basic effect samples for each input parameter. conduct Secondary sampling involves randomly selecting reference points from the entire input parameter space. Basic effect samples for each input parameter are then taken. mean Corrected mean and standard deviation Defined as:

[0053]

[0054]

[0055] The mean and corrected mean reflect the overall impact of input parameters on the output. The standard deviation indicates the presence of nonlinear effects or interactions between parameters. The corrected mean is calculated because for some input parameters, the error value may change sign at the output, leading to a canceling effect. Examining the mean, corrected mean, and standard deviation of an input parameter provides a qualitative assessment of its importance, thus identifying a few key parameters that significantly influence the output. Larger... or A value indicates that the input parameter is more important, while a larger value indicates that the input parameter is more important. The MOAT value indicates that the input parameter has a nonlinear and / or interactive effect on the model output. The MOAT mean estimates the overall impact of the input parameter on the amount of attention, while the MOAT standard deviation measures the nonlinear effect of the input parameter and its interaction with other input parameters. Parameters with high MOAT mean and MOAT variance significantly affect the amount of attention, while parameters with large standard deviations usually indicate strong parameter interactions, nonlinear effects, or both.

[0056] The MOAT method excels in cost-effectiveness because it requires relatively few model evaluations. However, a drawback is its reliance on basic effects, which are computed using finite differences. The final metric, however, is obtained by averaging multiple basic effects at different points in the input parameter space, thus eliminating dependence on specific computational points. Therefore, this method can be considered a global selection approach. It's important to note that MOAT provides a qualitative metric because it prioritizes input parameters according to their importance.

[0057] S3: Sample input parameters based on the prior distribution, input the sampled input parameters into the simulation model to obtain the corresponding output parameters, and construct a training dataset for training the surrogate model based on the sampled input parameters and the simulation output parameters. The surrogate model is obtained by fitting multiple sets of input and output parameters.

[0058] This step employs Latin hypercube sampling. The accuracy of uncertainty quantification analysis and the approximation effect of the surrogate model are directly related to the data sampling used in the simulation model evaluation. Latin hypercube sampling (LHS) is a method consisting of a... The design given by the matrix, where each column is random arrangement, Indicates the number of model evaluations. This indicates the number of input parameters. Latin hypercube sampling (LHS) has good projection properties in any single dimension, meaning that the sampled data always has good space-filling properties in any one-dimensional input parameter space. Uncertainty analysis of input parameter sampling using Latin hypercube sampling (LHS) will... In the hypercube The model evaluation generates an optimal Latin hypercube sample, which is then mapped onto the probability distribution of the input parameters. The optimal Latin hypercube sample exhibits spatial filling, meaning that sample points should be distributed as uniformly as possible throughout the input parameter space. Simultaneously, it avoids duplication and sample point clustering on any subspace projection of the input parameter space. The process of constructing the optimal Latin hypercube (LHS) is formulated as a global optimization problem. It is important to note that the optimal Latin hypercube obtained from the optimization may be of lower quality depending on the initial design. Therefore, global optimization is repeated starting from different initial LHS designs.

[0059] In one embodiment, to address the computational time issue of the simulation model, the type of surrogate model is determined based on the input-output mapping relationship of the simulation model. The types of surrogate models include response surface models, kriging models, etc.

[0060] In one embodiment, to address the high computational costs associated with complex simulations or physical experiments, a response surface model can be constructed to replace the real physical model, and then a Monte Carlo-based Sobol method and response curves can be applied to this replacement model.

[0061] In one embodiment, a Gaussian process-based surrogate model, also known as a Kriging model, is employed as a unified, computationally efficient alternative to simultaneously and efficiently support Sobol global sensitivity analysis and inverse uncertainty analysis. A Gaussian process is a nonparametric probabilistic model based on a Bayesian framework, adept at flexibly modeling and predicting unknown functions. Its core principle lies in treating the objective function as a single realization of a stochastic process defined on the input space, while assuming that the joint distribution of this process follows a multivariate Gaussian distribution. A Gaussian process is defined by its mean and covariance functions. When training the Gaussian process-based surrogate model, the hyperparameters of the mean and covariance functions are optimized. Consider the input parameters... and attention The surrogate model based on the Gaussian process obtained by fitting is expressed as:

[0062] in, It is the mean function. It has a mean of zero and a covariance of Gaussian process, The hyperparameters representing the covariance function, the covariance function Used to describe the correlation between input points Represents input parameters The other input parameter is different; both are points in the model input space.

[0063] Based on training dataset Train a surrogate model based on a Gaussian process, which is applicable to arbitrary input parameters. The predictions follow a Gaussian distribution:

[0064] Among them, the predicted mean and prediction variance It is given by the following formula:

[0065]

[0066] Here, Test point The covariance vector between all training points, It is the variance of observation noise. It is the covariance matrix between training points; Indicates that given an input vector Given the training dataset D, output the number of followers. The conditional probability distribution; The mean is variance is Gaussian distribution; superscript Represents the transpose of a vector or matrix; Indicates test point Its own prior covariance, Represents the identity matrix.

[0067] A key innovation of this invention lies in fully utilizing the predictive uncertainty provided by the Gaussian process. An adaptive update process is used to update the training dataset and the surrogate model. This adaptive update process employs an uncertainty-based sampling strategy, such as expectation enhancement or upper confidence bound strategies. This uncertainty-based sampling strategy intelligently selects the next most "informative" sample point by balancing the predicted mean and predicted variance, which is then used by the simulation model for calculation, and this updated sample point updates the training dataset and the Gaussian process-based surrogate model itself. This closed-loop process ensures rapid convergence to the global optimum or a high-precision approximation of the entire design space with the highest sampling efficiency.

[0068] Step S4: Obtain the input parameters and corresponding output parameters after filtering by the MOAT method through the surrogate model, and calculate the Sobol index of each input parameter on the key performance indicators of nuclear fuel using the Sobol index sensitivity analysis method; by sorting the Sobol index by size, accurately identify the key uncertainty parameters that have the most significant impact on nuclear fuel performance, clarify the main sources of uncertainty, and provide direction for subsequent targeted optimization.

[0069] In one embodiment, when there are few original input parameters, step S2 can be omitted. This step obtains all input parameters and corresponding output parameters through a proxy model.

[0070] The Sobol index sensitivity analysis method is based on the principle of variance decomposition, which considers the amount of attention... Total variance The method decomposes the variance of the surrogate model into the contribution of each input parameter and their interactions. It assumes that the output of the surrogate model can be represented as the sum of functions of the input parameters, and accordingly decomposes and quantifies the variance of the surrogate model's output.

[0071] For defined as Attention volume, first-order Sobol exponent based on variance Defined as:

[0072] Represents the conditional expectation function about Find the variance of the distribution. Indicates when When a certain fixed value is taken, the amount of attention output by the model Compared to all other uncertain parameters The average value, It is the first One input parameter, and Indicates except All other parameters besides; This represents the functional relationship between the amount of attention and the input parameters. This indicates the number of input parameters.

[0073] The meaning of the inner expected value is that, under a fixed... In this case, for All possible values ​​are considered in the attention count. The average value. The outer variance is the mean value. The calculation is performed over all possible values. Total Sobol exponent Defined as:

[0074] Represents the conditional expectation function with respect to Find the variance of the distribution. Indicates when excluding All parameters except When fixed, the amount of attention output by the model Compared to The average value.

[0075] The Sobol index The measure is the first The sum of the first-order effects of the input parameters and their interactions with other input parameters. The second term in the equation can be seen as the sum of the first-order effects of the input parameters and their interactions with other input parameters. The first-order effects of all parameters other than the input parameters. Therefore, Subtracting the second term will necessarily give the result of the subtraction of the second term. The contribution of all relevant items. It is important to note that for uncertainty quantification studies with multiple concerns, the Sobol index is calculated for each concern individually.

[0076] By sorting the Sobol index by magnitude, we can accurately identify the key uncertainty parameters that have the most significant impact on nuclear fuel performance, clarify the main sources of uncertainty, and provide direction for subsequent targeted optimization.

[0077] S5: Combining nuclear fuel test / measurement data, and based on the prior distribution of key uncertainty parameters, the MCMC method is used to obtain a corrected posterior distribution, reducing parameter uncertainty. In this step, hydrogen concentration is used as the basis for the calculation. As a measure of attention.

[0078] In the Bayesian inference framework, the product of the prior distribution and the likelihood function is used to construct the target posterior distribution:

[0079] Indicates that in a given and Under the condition of hyperparameters The posterior probability distribution; It is the prior distribution of the calibration parameters, where and These represent the attention value output by the proxy model and the experimental data, respectively. It is the likelihood function. Here, the prior and posterior distributions reflect the hyperparameters before and after considering the experimental data. The degree of belief in possible values. The likelihood function based on all experimental data is defined as:

[0080] This represents each dimension of the value being monitored. It is variance. Representing the One experimental measurement value, It is the first The attention value output by the proxy model, superscript Represents the transpose of a vector or matrix.

[0081] As can be seen from formula (12), in order to obtain the posterior distribution, this invention uses the Markov Chain Monte Carlo (MCMC) method from... Sampling is performed in the process. This sampling-based method repeatedly generates random samples to approximate the target posterior distribution. The core of this method lies in constructing a Markov chain, a stochastic process in which the next state depends only on the current state and not on previous states. The key to the MCMC method is designing such Markov chains so that their steady-state distribution is equal to the target distribution to be sampled. Existing technologies only have two measured hydrogen concentration datasets available. This limited sample size introduces significant uncertainty and potential bias, thus affecting the MCMC posterior estimate. To overcome this deficiency, this invention develops a diffusion-reaction model in COMSOL Multiphysics. By systematically varying the initial hydrogen concentration from low to high at 3 wt.ppm increments from 64 wt.ppm, 22 detailed concentration distributions are generated. To better simulate real measurement noise, this invention adds zero-mean Gaussian noise with a mean of zero and a standard deviation of 5% to each simulation curve (the simulation model is calculated every 3 wt.ppm for initial concentrations from 64-130, and zero-mean Gaussian noise with a mean of zero and a standard deviation of 5% is added to the generated hydrogen content distribution curve to simulate experimental data), thus obtaining a synthetic experimental dataset that includes statistical uncertainty while maintaining physical consistency. This expanded dataset not only greatly expands the observation space but also provides sufficient and reliable observational support for subsequent Bayesian parameter inversion, ensuring robustness and reasonable confidence intervals for post-convergence. 10% of the samples extracted using the Markov chain Monte Carlo method are designated as additional burnt samples. The role of these additional burnt samples in MCMC estimation is to eliminate biases generated during the pre-convergence phase of the chain. During the initial iterations, the Markov chain is typically far from the target posterior, and the parameter values ​​are mainly determined by the starting point and transient dynamics; these values ​​neither conform to a steady-state distribution nor are they merely weakly correlated. By discarding the top 10% of samples, it is possible to ensure that only samples from well-mixed, steady-state states are used to estimate the posterior mean, confidence intervals, and model evidence, thereby avoiding systematic biases introduced by initial transient behavior. In Monte Carlo Romanov chain (MCMC) processes, using a surrogate model can significantly improve computational efficiency, especially when the computational cost of the simulation model is high, and the surrogate model is constructed in a data-driven manner to capture the behavioral characteristics of the real system or model.

[0082] Figure 2 This is a MOAT plot. In the plot, parameters further right than D0 have a greater impact on the amount of attention. TSS D0 These three parameters; Figure 3 This demonstrates how the Sobol exponent method quantifies the magnitude of uncertainty (which can filter out D0). TSSD0 (Three parameters) Figure 4 For the four parameters (including D0, TSS D0 TSS p0 The accuracy of the Sobol index is higher than that of MOAT. In this embodiment, since there are not many input parameters, we do not rely entirely on MOAT for selection. We also refer to the results of the Sobol index to decide on the next step of analysis for the four parameters. The posterior probability distribution and joint probability distribution are obtained by inverse uncertainty analysis using the MCMC method.

[0083] The method provided in the foregoing embodiments of this invention first employs the MOAT (Morris One-At-a-Time) method to construct a multivariate input sample space for nuclear fuel performance analysis. This enables efficient sampling of key input parameters of nuclear fuel (such as fuel pellet thermal conductivity, cladding material mechanical properties, and operating condition parameters), ensuring that the samples fully cover the uncertainty range of each parameter and providing a comprehensive data foundation for subsequent uncertainty analysis. Next, the Sobol exponent sensitivity analysis method is used to calculate the Sobol exponent of each input parameter on key nuclear fuel performance indicators. The Sobol exponent accurately identifies the key uncertainty parameters that have the most significant impact on nuclear fuel performance, clarifying the main sources of uncertainty and providing direction for subsequent targeted optimization. Finally, the MCMC (Markov Chain Monte Carlo) inverse uncertainty analysis method is used, combined with nuclear fuel performance test data or operational measurement data, to reverse-correct the probability distribution of key uncertainty parameters. By constructing a likelihood function between the nuclear fuel performance prediction model and the measured data, the MCMC method is used to efficiently sample and obtain the posterior probability distribution of the corrected parameters, further reducing the uncertainty of key parameters and improving the accuracy and reliability of nuclear fuel performance prediction.

[0084] In one embodiment, an uncertainty analysis system for predicting the redistribution of hydrogen in zirconium alloys is provided for implementing the method described in Example 1. The system includes the following modules: The module for determining input parameters and prior distribution constructs a simulation model for predicting the redistribution of hydrogen in zirconium alloys. Based on the simulation model, it determines the input parameters to be studied and the prior distribution range corresponding to each input parameter. The MOAT initial screening module is used to analyze the importance of input parameters based on prior distribution using the MOAT method, and to perform preliminary screening of input parameters. The surrogate model construction module is used to sample input parameters based on a prior distribution, input the sampled input parameters into the simulation model to obtain the corresponding output parameters, construct a training dataset for training the surrogate model based on the sampled input parameters and the simulation output parameters, and fit multiple sets of input parameters and output parameters to obtain the surrogate model. The uncertainty parameter determination module is used to obtain the input parameters and corresponding output parameters after being filtered by the MOAT method through a surrogate model, calculate the Sobol index of each input parameter on the key performance indicators of nuclear fuel using the Sobol index sensitivity analysis method, and identify the key uncertainty parameters that have the most significant impact on nuclear fuel performance based on the Sobol index. The MCMC inverse uncertainty analysis module is used to correct the prior distribution of key uncertainty parameters by combining nuclear fuel test or measured data, and then using the MCMC method to obtain the corrected posterior distribution.

[0085] In one embodiment, a computer-readable storage medium is provided, which may be a ROM, RAM, disk, optical disk, or other storage medium. The storage medium stores one or more programs, which, when executed by a processor, implement the calculation method of the Bayesian method based on the above embodiment 1 for uncertainty analysis of hydrogen redistribution in zirconium alloy under predicted temperature gradient.

[0086] In one embodiment, a computer device is provided, which may be a desktop computer, laptop computer, smartphone, PDA handheld terminal, tablet computer or other terminal device with display function. The computing device includes a processor and a memory. The memory stores one or more programs. When the processor executes the program stored in the memory, it implements the calculation method of the Bayesian method based on the above embodiment 1 for uncertainty analysis of hydrogen redistribution in zirconium alloy under predicted temperature gradient.

[0087] The system, equipment, and medium described herein have the technical effects achieved by the method described in the foregoing embodiments.

[0088] The embodiments of the present invention can implement the above analysis method on an existing simulation model by using the uncertainty analysis module of COMSOL.

[0089] The above embodiments are preferred embodiments of the present invention, but the embodiments of the present invention are not limited to the above embodiments. Any changes, modifications, substitutions, combinations, or simplifications made without departing from the spirit and principle of the present invention shall be considered equivalent substitutions and shall be included within the protection scope of the present invention.

Claims

1. An uncertainty analysis method for predicting the redistribution of hydrogen in zirconium alloys, characterized in that, Includes the following steps: A simulation model for predicting hydrogen redistribution in zirconium alloys was constructed. Based on the simulation model, the input parameters to be studied were determined, and the prior distribution range corresponding to each input parameter was determined. Based on the prior distribution, the MOAT method is used to analyze the importance of the input parameters and to perform preliminary screening of the input parameters; Input parameters are sampled based on the prior distribution, and the sampled input parameters are input into the simulation model to obtain the corresponding output parameters. A training dataset for training the surrogate model is constructed based on the sampled input parameters and the simulation output parameters. The surrogate model is obtained by fitting multiple sets of input parameters and output parameters. The input parameters and corresponding output parameters after being filtered by the MOAT method are obtained by using a proxy model. The Sobol index sensitivity analysis method is used to calculate the Sobol index of each input parameter on the key performance indicators of nuclear fuel. Based on the Sobol index, the key uncertainty parameters that have the most significant impact on the performance of nuclear fuel are identified. By combining nuclear fuel test or measured data, and based on the prior distribution of key uncertainty parameters, the MCMC method is used to obtain the corrected posterior distribution.

2. The uncertainty analysis method for predicting hydrogen redistribution in zirconium alloys according to claim 1, characterized in that, In the MOAT method, the "baseline effect" of each input parameter is a metric used to measure the impact of a single input parameter on the model output. The baseline effect is defined by calculating the change in the model output when a single input parameter changes: This represents the basic effect sample. Indicates the first Reference points At this point, the first Each input parameter is increased by a perturbation step size. The output value of the model at that time. Indicates the first Reference points At this point, the first Reduce the number of input parameters The output value of the model at that time; Calculate the basic effect sample for each input parameter mean Corrected mean and standard deviation A qualitative assessment of the importance of each input parameter.

3. The uncertainty analysis method for predicting hydrogen redistribution in zirconium alloys according to claim 1, characterized in that, When sampling input parameters based on prior distribution, Latin hypercube sampling is used.

4. The uncertainty analysis method for predicting hydrogen redistribution in zirconium alloys according to claim 1, characterized in that, The surrogate model can be either a response surface model or a kriging model. When the surrogate model is a kriging model, the input parameters are considered. and attention The fitted surrogate model is as follows: in, It is the mean function. It is a Gaussian process.

5. The uncertainty analysis method for predicting hydrogen redistribution in zirconium alloys according to claim 1, characterized in that, Based on training dataset The agent model is trained, and the training dataset and agent model are updated through an adaptive update process. The adaptive update process adopts an uncertainty-based sampling strategy, which intelligently selects the next most informative sample point by balancing the prediction mean and prediction variance, and submits it to the simulation model for calculation, thereby updating the training dataset and agent model.

6. The uncertainty analysis method for predicting hydrogen redistribution in zirconium alloys according to claim 1, characterized in that, When the number of input parameters is small, the initial screening step of input parameters using the MOAT method can be omitted.

7. The uncertainty analysis method for predicting the redistribution of hydrogen in zirconium alloys according to any one of claims 1-6, characterized in that, The expression for the posterior distribution is: in, Indicates that in a given and Under the condition of hyperparameters The posterior probability distribution; It is the prior distribution of the calibration parameters, where and These represent the attention value output by the proxy model and the experimental data, respectively. It is the likelihood function.

8. An uncertainty analysis system for predicting the redistribution of hydrogen in zirconium alloys, characterized in that, The system for implementing the method according to any one of claims 1-7 includes the following modules: The module for determining input parameters and prior distribution constructs a simulation model for predicting the redistribution of hydrogen in zirconium alloys. Based on the simulation model, it determines the input parameters to be studied and the prior distribution range corresponding to each input parameter. The MOAT initial screening module is used to analyze the importance of input parameters based on prior distribution using the MOAT method, and to perform preliminary screening of input parameters. The surrogate model construction module is used to sample input parameters based on a prior distribution, input the sampled input parameters into the simulation model to obtain the corresponding output parameters, construct a training dataset for training the surrogate model based on the sampled input parameters and the simulation output parameters, and fit multiple sets of input parameters and output parameters to obtain the surrogate model. The uncertainty parameter determination module is used to obtain the input parameters and corresponding output parameters after being filtered by the MOAT method through a surrogate model, calculate the Sobol index of each input parameter on the key performance indicators of nuclear fuel using the Sobol index sensitivity analysis method, and identify the key uncertainty parameters that have the most significant impact on nuclear fuel performance based on the Sobol index. The MCMC inverse uncertainty analysis module is used to correct the prior distribution of key uncertainty parameters by combining nuclear fuel test or measured data, and then using the MCMC method to obtain the corrected posterior distribution.

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, When the processor executes the computer program, it implements the method according to any one of claims 1-7.

10. A computer-readable storage medium storing a computer program, characterized in that, When the computer program is executed by a processor, it implements the method described in any one of claims 1-7.

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