Degradation model identification with interval uncertainty and its test design method

Abstract

The application discloses a degradation model confirmation and test design method with interval uncertainty in the field of material test design. The first moment average unbiased estimator and the second moment variance unbiased estimator are calculated based on a test data sample set, and the upper limit value and the lower limit value of the unbiased estimation are calculated. The upper limit and the lower limit of the interval variables in the interval variable set are linearly interpolated to obtain the upper limit and the lower limit of the test observation model of the interval process, and a test observation model is formed. The model confirmation index is compared with the set model confirmation factor to determine whether the prediction model is a real degradation model. The interval limit of the test data is quantitatively calculated by using the unbiased estimation method, the demand for the number of test samples is reduced, the number and distribution of observation time and the number of test samples are fully considered, the collaborative optimization algorithm is adopted to obtain the optimal distribution of the test sample quantity and the observation time, and the cost of the model confirmation test is reduced.

Description

Technical Field

[0001] This invention relates to materials test design technology, specifically to a method for confirming materials degradation models and its test design method, thereby reducing the cost of materials degradation confirmation tests. Background Technology

[0002] Typically, material properties degrade due to storage, normal operation, or extreme conditions, and an effective method for reliability assessment is to utilize degradation data (such as stiffness or strength) to reflect the material's health status. Mathematical models describing the degradation process of material properties are called degradation models, which are broadly classified into two categories: abrupt (impact) degradation and gradual (progressive) degradation. In abrupt degradation, material properties decrease instantaneously to zero or a constant proportion of their previous values, and the material is simply considered undamaged and completely damaged; in gradual degradation, material properties change continuously with the accumulation of damage, and each increment requires balancing iterations. Furthermore, materials often exhibit significant uncertainty during degradation due to various uncertainties arising from manufacturing, forming processes, and the external environment. Therefore, the degradation process is often viewed as a stochastic process with a specific probability density function, as this effectively captures the time-varying and uncertainties of the degradation process. Currently, common stochastic processes such as the Wiener process and the Gamma process are widely used to simulate the degradation process of materials. However, these probabilistic methods require complete stochastic information and knowledge, and obtaining the accurate probability distribution of multidimensional stochastic processes is usually difficult and expensive in most applications.

[0003] Interval theory utilizes interval ranges to quantify uncertainty, making it more suitable for handling incomplete information with limited data. For simulation models based on interval theory, model validation is still needed to evaluate the validity of the simulation results, i.e., to determine whether the simulation model accurately represents reality within its intended application range. Generally, model validation can be conducted qualitatively (graphical comparison) or quantitatively (validation indicators), such as hypothesis testing methods, area indicators, and distance indicators.

[0004] In theory, with unlimited resources, model validation of a degradation model is not a problem. However, in the real world, it is always necessary to minimize the cost of validation experiments, and one of the biggest challenges in designing validation experiments is balancing the conflicting objectives of confidence and cost. Summary of the Invention

[0005] This invention addresses the uncertainty exhibited by materials during degradation by providing a method for validating a material degradation model with range uncertainty, as well as an experimental design method. This method can provide a validating experimental scheme and effectively reduce experimental costs.

[0006] The technical solution of the present invention for verifying a degradation model with interval uncertainty is implemented according to the following steps:

[0007] S1: Take n material specimens and observe them at m observation times t. j The degradation index of the material was measured, n×m experimental data points were obtained, and a single observation time t was obtained. j A sample set of measured experimental data;

[0008] S2: Based on the experimental data sample set, calculate the unbiased estimator of the first moment mean and the unbiased estimator of the second moment variance, as well as the upper and lower bounds of the unbiased estimators, to obtain the interval variables and the interval variable set of m unbiased estimators for m observation times.

[0009] S3: Perform linear interpolation on the upper and lower boundaries of the m interval variables in the interval variable set to obtain the upper and lower boundaries of the experimental observation model of the interval process, thus forming the experimental observation model;

[0010] S4: Take one interval variable from the experimental observation model and one interval variable from the prediction model respectively, and obtain the similarity between the two and the model validation index over the entire time domain;

[0011] S5: Compare the model validation index with the set model validation factor to determine whether the prediction model is a true degradation model.

[0012] The present invention provides an experimental design method for a degradation model with interval uncertainty: the design variables include the aforementioned n, m, and the vector t at the observation time. obs ={t1,t2,...,t m The experimental design optimization model is as follows:

[0013]

[0014] Where, n min and n max Let m be the minimum and maximum number of test samples, respectively. min and m max T represents the minimum and maximum number of observation times, respectively. max This is the longest experimental observation time. It is the prediction model interval. η is the experimental observation model interval; η is the model validation factor; and C is the total cost. E (n,m,t obs ) = C s ·n+C T ·n·m,C s and C T These are the costs of a single experimental sample and a single observation in an experiment, respectively.

[0015] The outstanding advantages of this invention after adopting the above technical solution are:

[0016] 1. This invention uses the interval method as a non-probabilistic method as a useful supplement to the uncertainty analysis of limited information, accurately characterizing the consistency between the prediction model and the experimental data, and improving the reliability of the model.

[0017] 2. This invention establishes a novel metric based on interval process overlap, which effectively measures the difference between the prediction model and the experimental data. The index based on interval process overlap is applied to confirm the degradation model. By using an unbiased estimation method to quantify the interval boundaries of the experimental data, the requirement for the number of experimental samples is reduced.

[0018] 3. This invention fully considers design variables such as the number and distribution of observation times and the number of test samples, thereby improving the accuracy of the validation experiment.

[0019] 4. This invention proposes an optimized model for validation experiments, employing a collaborative optimization algorithm to obtain the optimal distribution of experimental sample size and observation time, thereby reducing the cost of model validation experiments. Attached Figure Description

[0020] Figure 1 This is a schematic diagram of interval variables in the experimental observation and prediction model;

[0021] Figure 2 This is a schematic diagram illustrating the positional relationship between the interval process of the prediction model and the interval similarity of the experimental data:

[0022] Figure 3 This is a schematic diagram illustrating the steps of optimal experimental design;

[0023] Figure 4 This is a schematic diagram of the loading of the composite material sample in the application example;

[0024] Figure 5 These are fatigue damage curves of the composite material under two stress levels in an application example;

[0025] Figure 6 This is a graph showing the trend of composite material confirmation indicators under two stress levels as a function of the number of observation times;

[0026] Figure 7 It is an optimized distribution diagram of the observation time distribution of the confirmation index of composite materials at the first stress level;

[0027] Figure 8 This is an optimized distribution diagram of the observation time distribution of the confirmation index of composite materials under the second stress level;

[0028] Figure 9It is the optimal test scheme for model validation tests under two pressure levels;

[0029] Figure 10 This is a distribution diagram of observation times in two optimal experimental schemes. Detailed Implementation

[0030] Take n material specimens, the materials of which can be metals, non-metals, composite materials, etc. Observe them at m observation times t. j The degradation indices of materials are measured, including elastic modulus, yield strength, tensile strength, stiffness, plasticity, toughness, etc. Therefore, the vector at the observation time is t. obs ={t1,t2,...,t m After measurement, n×m experimental data points are obtained, where a single observation time t j The experimental data sample set for measurement is {α1, α2, ..., α...} n}

[0031] For a single observation time t j The measured experimental data sample set {α1,α2,...,α n The existing method is to use the minimum value in the sample set. and maximum value Interval variables for obtaining experimental data α i It is the sample set {α1, α2, ..., α} n For a single data point in the interval {i = 1, 2, ..., n}, the boundary of the interval variable α″′ contains the sample set {α1, α2, ..., α″′}. n The interval of all data in}. Based on the interval variable of the experimental data. The interval variable can be calculated using the following formula. The mean of the first moment μ″′(α) and the variance of the second moment Var″′(α) of the experimental data are as follows:

[0032]

[0033] The upper bound of the boundary α′ of a single time interval is calculated from the mean of the first moment μ″′(α) and the variance of the second moment Var″′(α) of the experimental data. and lower bound value α ′:

[0034]

[0035] This yields the boundary of a single time interval.

[0036] However, when the number of samples is too small, this interval cannot accurately reflect the variable boundaries. Therefore, this invention employs an unbiased estimation method to obtain the interval boundary α at an accurate single time point. I Specifically:

[0037] For a single observation time t j The measured experimental data sample set {α1,α2,...,α n First, calculate its average value α. * Then, the unbiased estimator of the first moment mean μ(α) and the unbiased estimator of the second moment variance Var(α) are calculated according to the following formulas, respectively:

[0038]

[0039] Where α i It is the sample set {α1, α2, ..., α} n In the context of a single data point, i = 1, 2, ..., n.

[0040] Substituting the accurate first-order moment mean μ(α) and second-order moment variance Var(α) into the single-time interval boundary of the existing method, respectively... upper bound value and lower bound value α The calculation formula for ' is used to replace the first moment mean μ″′(α) and second moment variance Var″′(α) of the experimental data, respectively, to obtain the single observation time t of this invention. j The upper bound of the unbiased estimate and lower bound value α for:

[0041]

[0042] Therefore, in this invention, at a single observation time t j The interval variable α of the unbiased estimator I Represented as:

[0043]

[0044] For n×m experimental data, according to the unbiased estimation method described above, the experimental data are quantified at each observation time, and t is calculated for each observation time. j The interval variable α of the unbiased estimator I This allows us to obtain m unbiased estimates of the interval variable α for m observation times. I (t1),α I (t2),...,α I (t m The interval variable α, which is an unbiased estimate of m observation times, is... I (t1),αI (t2),...,α I (t m ) to form a set of interval variables based on experimental data α I (t j ) is the tth j An interval variable for an unbiased estimate of the observation time.

[0045] With the prediction model to be confirmed In contrast, interval variable sets based on experimental data Since these are interval variables observed only at discrete times, comparing them and establishing confirmation factors is difficult. Therefore, this invention employs linear interpolation to obtain the interval variable set from the interval variable set. Constructing an experimental observation model for continuous interval processes Among them, for the set of interval variables Linear interpolation is performed on the upper boundaries of the m interval variables to obtain the upper boundary of the experimental observation model of the interval process. For the set of interval variables The lower boundary of the experimental observation model of the interval process is obtained by linear interpolation of the lower boundaries of the m interval variables. Finally, an experimental observation model based on experimental data was obtained.

[0046] Figure 1 The diagram illustrates the interval process of the prediction model and experimental observations. According to the diagram, if the experimental observation model... and prediction models The higher the overlap, the closer the prediction model is to the experimental observation model, resulting in better model confirmation. However, directly calculating the overlap can lead to the overlap being more important for time periods with wider boundaries than for time periods with narrower boundaries. To consider the overlap across the entire time domain on an average basis, this invention integrates the overlap over the time domain.

[0047] Take the experimental observation model An interval variable and prediction models An interval variable Based on the interval sorting strategy, the six different positional relationships between the two interval variables are as follows: Figure 2 As shown, the mathematical expression for similarity is defined as:

[0048]

[0049] Among them, the similarity Deg(a I ,b I ) depends on a I and b IDifferent positional relationships, for Figure 2 The similarity Deg(a) is calculated for six different positions within the same data set. I ,b I The calculation is as follows:

[0050]

[0051] Similarity Deg(a I ,b I The value range of ) is [0,1], and the similarity Deg(a) is... I ,b I The larger the value of Deg(a), the more similar the two interval variables are; while Deg(a) I ,b I =1 means that the two interval variables are the same.

[0052] Considering the entire time domain, based on similarity Deg(a I ,b I ), to obtain the prediction model Interval process and experimental observation model The model validation metrics over the entire time domain are:

[0053]

[0054] Among them, model validation indicators Representative prediction model and experimental observation model The degree of overlap is used to determine the predictive model based on the model confirmation index. and experimental observation model The degree of overlap, when the model confirms the indicators At that time, the prediction model The interval process does not coincide with the interval process observed in the experiment, and the prediction model... Completely unreliable. And for... Higher means better predictive models Compared with actual experimental observation model The closer, especially when If the interval process of the prediction model is exactly the same as the interval process of the experimental data, then the prediction model is valid. This is a degradation model for materials.

[0055] A preferred embodiment of the present invention is: due to the limited number of experimental samples, the model validation index... Due to the uncertainty, the model validation index is modified based on the expected value. The model validation index representing the degree of overlap is modified as follows:

[0056]

[0057] in It confirms the expected value of the indicator, with a range of [0,1].

[0058] Confirmation indicators for the modified model Compared with the pre-defined model validation index η, if: If the inequality indicating the goodness of fit to the interval process is satisfied, then the prediction model... A true degradation model is defined as η, and vice versa. A larger η value indicates a more stringent requirement for model validation.

[0059] Model validation metrics are the core of validation experiments. This invention utilizes metrics derived from predictive models. Based on the obtained prior information, a validation design optimization (VEDO) model is established. Following the process of establishing validation indices described above, the design variables in the validation experiment include the number of material times (n), the number of observation times (m), and a vector t representing the distribution of observation times. obs ={t1,t2,...,t m To minimize the cost of the model validation experiment, the optimized model for the validation experiment design is as follows:

[0060]

[0061] Where, n min and n max Let m be the minimum and maximum number of test samples, respectively. min and m max T represents the minimum and maximum number of observation times, respectively. max If the observation time is the maximum, then the total cost C of the experiment can be calculated using the following formula. E for:

[0062] C E (n,m,t obs ) = C s ·n+C T ·n·m,

[0063] Where C s and C T These represent the costs of a single experimental sample and a single observation. Based on the VEDO model, the cost of the confirmation experiment is minimized.

[0064] Based on the total cost C of the experiment E By minimizing the number of observation times *n* and the number of samples *m*, the lowest experimental cost can be obtained. The VEDO model can be transformed into a multi-objective optimization problem:

[0065]

[0066] The optimal solution set for multi-objective optimization is obtained by solving the formula, and the lowest cost test scheme is obtained from the optimal solution set.

[0067] For the VEDO model, n and m are discrete design variables, t obs It is an N T The presence of continuous design variables in the dimensional model leads to a mixed-variable optimization problem (MVOP). Generally, mixed variables increase the complexity of the search space, making the optimization problem more difficult to solve.

[0068] This invention takes into account that both the number of experimental samples and the observation time will affect the model validation index. Therefore, it first considers the observation time t... obs Perform separate optimizations, and then determine the number of experimental samples n. In other words, the optimization process is divided into two parts: optimizing the observation time and the number of experimental samples separately. A schematic diagram of the steps is shown below. Figure 3 As shown, specifically:

[0069] Step 1: Set the initial parameters for the experimental design, including n min ,n max ,m min ,m max , and η;

[0070] Step 2: Set up the initial experimental plan, where the number of observation times m = m min and the number of test samples n = n max Generate the current test plan.

[0071] Step 3: In the previous step, the current experimental plan was determined, including the number of experimental samples n and the number of time points m. This step will optimize the distribution of the m observation time points t. obs ={t1,t2,...,t m The optimized model is obtained as follows:

[0072]

[0073] The above optimization model is a continuous optimization problem, which is solved using a genetic algorithm.

[0074] Step 4: If the confirmation factor of the current optimal experimental scheme is greater than η, then record the observation time distribution t of the current optimal experimental scheme. obs Proceed to step 5, optimize the number of experimental samples n. If not, increase the number of observation times and return to step 3.

[0075] Step 5: In the previous step, the distribution of observation times was determined. The number of experimental samples n is the optimization objective. To obtain the minimum number of experimental samples n that satisfies the requirement η, the optimization model is:

[0076]

[0077] This optimization model is an optimization problem with a single discrete variable, which can be easily solved using the enumeration method.

[0078] Step 6: Record the current optimal experimental design, including the observation time t. obs And the number of experimental samples n. Continue to increase the number of observation times m = m + 1, and return to step 3.

[0079] Step 7: Obtain all recorded test protocols and select the one with the lowest cost.

[0080] The following is an application example of the present invention:

[0081] Taking the fatigue test of composite laminate at two stress levels (40% and 60% of its tensile strength, i.e., 40%σ0 and 60%σ0) as an example, where the stacking sequence is [0 / 90]7, a schematic diagram of the specimen is shown below. Figure 4 The stiffness degradation of composite materials under fatigue load in the loading direction is modeled as an interval process, where the mean function Eij m (c) A general stiffness degradation model is adopted:

[0082]

[0083] Where D(c) is the fatigue failure index, and E0 and E f These represent the material stiffness corresponding to the initial cycle and the final stable cycle, respectively, where c represents the normalized fatigue cycle parameter, c = n / N. f , where n and N f These represent the number of application cycles and the final stabilization period, respectively. p and q represent relevant parameters. The fatigue test parameters under the two stress levels are shown in Table 1 below.

[0084] Table 1. Fatigue test parameters under two stress levels.

[0085]

[0086] like Figure 5 As shown, under two different stress levels, damage accumulates nonlinearly in the hemp / epoxy composite material. These processes are divided into three stages: I is the region with a significant initial slope at the start of the curve; II is the region with a constant slope; and III is the region where the slope gradually increases before the final stabilization period.

[0087] For hemp / epoxy resin composite laminates, the degradation process is dispersed due to material defects and fluctuations in fatigue loads, and this dispersion gradually increases over time. Therefore, the radius function is defined as...

[0088]

[0089] Among them, b1 and b2 are related to fatigue testing, as shown in Table 1.

[0090] Figure 6 The diagram shows the trend of the degradation model confirmation index as a function of the number of observation times *m* at two stress levels, where each observation time is uniformly distributed. It can be seen that the degradation model confirmation index gradually increases with the increase of the number of observation times. Clearly, more observation times represent a more detailed description of the entire degradation process, but this also leads to high observation costs; therefore, it is necessary to optimize the distribution of observation times.

[0091] To optimize the distribution of observation times, the experimental sample size n = 100 and the number of observation times m = 11 were set, where c = [c1, c2, ..., c...]. 11 ], c1=0 and c 11 =1. A genetic algorithm is used to obtain the optimal distribution of observation times. In the initial scheme, the observation times are uniformly distributed under both stress levels. Figure 7 and Figure 8 The results show the confirmation of the initial and optimized observation times under two stress levels. It can be seen that the confirmation results of the optimized observation points are better than those of the initial observation points. The model confirmation index for 40% σ0 improved from 0.8676 to 0.9166, and the model confirmation index for 60% σ0 improved from 0.8062 to 8725. Therefore, optimizing the distribution of observation points can effectively improve the confirmation results.

[0092] Table 2 lists the relevant parameters for the model validation experimental design, according to the present invention. Figure 9 The optimal solution sets for the confirmation tests under two stress levels are presented.

[0093] Table 2. Relevant parameters of the experimental design

[0094]

[0095] For 40% σ0, the minimum number of observations is 9, and at least 50 test cases are required. When the number of observations is 20, the number of test cases can be reduced to 7. Similarly, for 60% σ0, the minimum number of observations is 10, and at least 37 test cases are required. When the number of observations is 20, the number of test cases can be reduced to 9. The cost of all test plans in the optimal solution set was calculated according to Table 2. Figure 10The distribution of observation times in two optimal experimental schemes is shown. The lowest experimental cost for 40%σ0 is 425 yuan (n=11, m=13); the lowest experimental cost for 60%σ0 is 480 yuan (n=12, m=15). The scheme effectively reduces the cost of the confirmation experiment.

Claims

1. A method for verifying a degenerate model with interval uncertainty, characterized in that: Includes the following steps: S1: Take n material specimens and observe them at m observation times t. j The degradation index of the material was measured, n×m experimental data points were obtained, and a single observation time t was obtained. j A sample set of measured experimental data; S2: Based on the experimental data sample set, calculate the unbiased estimator of the first moment mean and the unbiased estimator of the second moment variance, as well as the upper and lower bounds of the unbiased estimators, to obtain the interval variables and the interval variable set of m unbiased estimators for m observation times. The first-order moment average unbiased estimator Unbiased estimator of the variance of the second moment Upper bound of unbiased estimator lower bound value Obtain m unbiased estimates of the interval variable α at m observation times. I (t1),α I (t2),...,α I (t m ) and interval variable set α * It is the average of the sample set, α i It is the sample set {α1, α2, ..., α} n In the context of a single data point, i = 1, 2, ..., n; S3: Perform linear interpolation on the upper and lower boundaries of the m interval variables in the interval variable set to obtain the upper and lower boundaries of the experimental observation model of the interval process, thus forming the experimental observation model; S4: Take one interval variable from the experimental observation model and one interval variable from the prediction model respectively, and obtain the similarity between the two and the model validation index over the entire time domain. It is a predictive model It is an experimental observation model; S5: Compare the model validation index with the set model validation factor to determine whether the prediction model is a true degradation model.

2. The method for verifying a degenerate model with interval uncertainty according to claim 1, characterized in that: Modify the model validation index based on the expected value; the modified model validation index. These are the model confirmation indicators before modification. It is the expected value, and its range is [0,1].

3. The validation method for a degradation model with interval uncertainty according to claim 1, characterized in that: The aforementioned similarity a I b is an interval variable in the experimental observation model. I It is an interval variable in the prediction model.

4. An experimental design method for validating the degradation model according to claim 1, characterized in that: Includes the following steps: The design variables include n, m, and the vector t at the observation time. obs ={t1,t2,...,t m The experimental design optimization model is as follows: Where, n min and n max Let m be the minimum and maximum number of test samples, respectively. min and m max T represents the minimum and maximum number of observation times, respectively. max This is the longest experimental observation time. It is the prediction model interval. It is the experimental observation model interval; η is the model validation factor, and the total cost C E (n,m,t obs ) = C s ·n+C T ·n·m,C s and C T These are the costs of a single experimental sample and a single observation in an experiment, respectively.

5. The experimental design method according to claim 4, characterized in that: The experimental design optimization model described is a multi-objective optimization model:

6. The experimental design method according to claim 5, characterized in that: First optimize the vector t at the observation time. obs Optimize separately, and then determine the number of material specimens n.

7. The experimental design method according to claim 6, characterized in that: The vector t at the observation time obs The optimization model is as follows: If the current model validation factor is greater than η, then the optimization model that minimizes the number of material specimens n while satisfying the η requirement is:

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