CN116386783A

CN116386783A - Degradation model confirmation with interval uncertainty and test design method thereof

Info

Publication number: CN116386783A
Application number: CN202310326610.XA
Authority: CN
Inventors: 孙东阳; 梁雪峰
Original assignee: Jiangsu University of Science and Technology
Current assignee: Jiangsu University of Science and Technology
Priority date: 2023-03-30
Filing date: 2023-03-30
Publication date: 2023-07-04

Abstract

The invention discloses a degradation model confirmation with interval uncertainty and a test design method thereof in the field of material test design, which are characterized in that an unbiased estimation quantity of a first moment average value and an unbiased estimation quantity of a second moment variance are calculated based on a test data sample set, an upper limit value and a lower limit value of unbiased estimation are respectively carried out linear interpolation on an upper boundary and a lower boundary of an interval variable in an interval variable set to obtain an upper boundary and a lower boundary of a test observation model in an interval process, a test observation model is formed, a model confirmation index is compared with a set model confirmation factor, and whether a prediction model is a real degradation model is judged; according to the invention, the interval limit of test data is quantitatively calculated by using an unbiased estimation method, the requirement on the number of test samples is reduced, the number and distribution of the observation time and the number of the test samples are fully considered, the optimal distribution of the test sample number and the observation time is obtained by adopting a collaborative optimization algorithm, and the cost of a model confirmation test is reduced.

Description

Degradation model confirmation with interval uncertainty and test design method thereof

Technical Field

The invention relates to a material test design technology, in particular to a material degradation model confirmation method and a material degradation test design method, so as to reduce the cost of a material degradation confirmation test.

Background

In general, the performance of a material may degrade due to storage, normal operation, or extreme conditions, and one effective method of reliability assessment is to use degradation data (e.g., stiffness or strength) to reflect the health of the material. Mathematical models describing the process of degradation of material properties are called degradation models, which are broadly divided into two categories, namely abrupt (impact) degradation and gradual (progressive) degradation. In sudden degradation, the material properties will instantaneously decrease to zero or a constant proportion of the previous properties, and the material is simply considered undamaged and completely damaged; in gradual degradation, the material properties change continuously as damage accumulates, and each increment requires a balancing iteration. In addition, materials often exhibit significant uncertainty in the degradation process due to various uncertainty factors from the manufacturing, molding process, and external environment. Thus, the degradation process is generally regarded as a random process with a specific probability density function, as it can effectively capture the time variability and uncertainty of the degradation process. Currently, common stochastic processes such as Wiener and Gamma processes are widely used to model material degradation processes, however, these probabilistic methods require complete stochastic information and knowledge, and obtaining accurate probability distributions for multidimensional stochastic processes is often difficult and expensive in most applications.

The interval theory utilizes the interval range to quantify uncertainty, and is more convenient for processing incomplete information with limited data. For these simulation models proposed based on interval theory, it is also necessary to evaluate the validity of the simulation results by model validation, i.e. to determine whether the simulation model accurately represents the authenticity within its intended application range. In general, model validation can be performed by qualitative (graphical comparison) or quantitative (validation indicators), such as methods based on hypothesis testing, area indicators, and distance indicators.

In theory, model validation of the degradation model is not a problem in the case of unlimited resources. However, in the real world, there is always a need to minimize the cost of validation experiments, one of the biggest challenges in validating experimental design is balancing the two conflicting goals of confidence and cost.

Disclosure of Invention

The invention provides a method for confirming a material degradation model with interval uncertainty and a test design method aiming at the uncertainty of a material in a degradation process, which can provide a confirmation test scheme and effectively reduce test cost.

The invention relates to a degradation model confirmation method with interval uncertainty, which is realized by the following steps:

s1: taking n material test pieces, respectively at m observation moments t _j Measuring degradation index of material to obtain n×m test data and single observation time t _j A measured test data sample set;

s2: calculating an unbiased estimation amount of a first moment average value and an unbiased estimation amount of a second moment variance based on the test data sample set, and an upper bound value and a lower bound value of unbiased estimation to obtain m unbiased estimated interval variables and interval variable sets of m observation moments;

s3: respectively carrying out linear interpolation on the upper boundaries and the lower boundaries of m interval variables in the interval variable set to obtain the upper boundaries and the lower boundaries of the experimental observation model of the interval process, so as to form the experimental observation model;

s4: respectively taking one interval variable of the experimental observation model and one interval variable of the prediction model, and obtaining the similarity of the two and the model confirmation index on the whole time domain;

s5: and comparing the model confirmation index with a set model confirmation factor, and judging whether the prediction model is a real degradation model.

The invention relates to a test design method of a degradation model with interval uncertainty, which comprises the following steps: the design variables include the vectors t of n, m and the observation time _obs ＝{t ₁ ,t ₂ ,...,t _m The experimental design optimization model is:

wherein n is _min And n _max Respectively the minimum value and the maximum value of the number of test samples, m _min And m _max Respectively represent the minimum and maximum observation time quantity, T _max Is the maximum experimental observation time that can be achieved,

is a predictive model interval, +.>

Is a test observation model interval; eta is the model validation factor, total cost C _E (n,m,t _obs )＝C _s ·n+C _T ·n·m，C _s And C _T The cost of one test sample and one test observation, respectively.

The invention has the outstanding advantages that after the technical scheme is adopted:

1. the interval method is used as a non-probability method and is used as a beneficial supplement for the uncertainty analysis of the limited information, the consistency between the prediction model and the test data is accurately represented, and the reliability of the model is improved.

2. According to the invention, a novel measurement standard based on interval process overlapping is established, the difference between the prediction model and the test data is effectively measured, the index based on interval process overlapping is applied to confirm the degradation model, and the interval limit of the test data is quantitatively calculated by using an unbiased estimation method, so that the requirement on the number of test samples is reduced.

3. The invention fully considers the design variables of the number and the distribution of the observation time and the number of the test samples, and improves the accuracy of the confirmation experiment.

4. The invention provides a confirmation test optimization model, which adopts a collaborative optimization algorithm to obtain the optimal distribution of test sample size and observation time, and reduces the cost of the model confirmation test.

Drawings

FIG. 1 is a schematic representation of interval variables of a trial observation and prediction model;

fig. 2 is a schematic diagram of a relationship between interval similarity and position of test data and interval process of a prediction model:

FIG. 3 is a schematic illustration of the steps of a proof test design;

FIG. 4 is a schematic representation of loading of a composite sample in an application example;

FIG. 5 is a graph of fatigue damage for a composite material at two stress levels in an application example;

FIG. 6 is a graph of the trend of the composite material validation index over the number of observation times at two stress levels;

FIG. 7 is an optimized profile of a composite material at a first stress level confirming the index observation time distribution;

FIG. 8 is an optimized profile of a composite material at a second stress level confirming the index observation time distribution;

FIG. 9 is an optimal test protocol for a model validation test at two pressure levels;

fig. 10 is a graph showing the observation time profiles in two optimal test protocols.

Detailed Description

Taking n material test pieces, wherein the material of the test pieces can be metal, nonmetal, composite material and the like. At m observation times t respectively _j Measuring degradation indexes of the material, wherein the degradation indexes comprise elastic modulus, yield strength, tensile strength, rigidity, plasticity, toughness and the like, and therefore, the vector of the observation time is t _obs ＝{t ₁ ,t ₂ ,...,t _m N×m test data are obtained after measurement, wherein a single observation time t _j The measured test data sample set is { alpha } ₁ ,α ₂ ,...,α _n }。

For a single observation time t _j Measured test data sample set { alpha } ₁ ,α ₂ ,...,α _n The existing methods are: by minimum value in sample set

Maximum->

Interval variable for obtaining test data ∈ ->

α _i Is the sample set { alpha } ₁ ,α ₂ ,...,α _n Single data in }, i=1, 2, n. this regionThe boundary of the inter-variable alpha' "contains the sample set { alpha } ₁ ,α ₂ ,...,α _n Intervals of all data in }. Interval variable according to test data->

The interval variable can be calculated by>

The first moment average μ '"(α) and the second moment variance Var'" (α) of the test data of (a) are respectively:

the upper limit value of the boundary alpha ' of the single time interval is calculated from the first moment average value mu ' (alpha) and the second moment variance Var ' (alpha) of the test data

And lower boundary valueα′：

Thereby obtaining a single time interval boundary

However, when the number of test pieces is too small, the interval cannot accurately reflect the variable limit, so the invention adopts an unbiased estimation method for obtaining the interval boundary alpha at an accurate single moment ^I . The method specifically comprises the following steps:

for a single observation time t _j Measured test data sample set { alpha } ₁ ,α ₂ ,...,α _n First calculate the average value alpha ^* Then, the first-order moment average value unbiased estimation amount μ (α) and the second-order moment variance unbiased estimation amount Var (α) are calculated according to the following formulas:

wherein alpha is _i Is the sample set { alpha } ₁ ,α ₂ ,...,α _n Single data in i=1, 2,..n.

The accurate first moment average value mu (alpha) and the second moment variance Var (alpha) are respectively substituted into the single time interval boundary of the prior method

Upper bound of>

And lower boundary valueαThe ' calculation formula is used for replacing the first moment average value mu ' (alpha) and the second moment variance Var ' (alpha) of the test data respectively to obtain the single observation time t of the invention _j Upper bound of unbiased estimation of (c)

And lower boundary valueαThe method comprises the following steps:

the invention is at a single observation time t _j Is an unbiased estimated interval variable alpha ^I Expressed as:

for n×m test data, according to the unbiased estimation method described above, the test data are quantized at each observation time, and each observation time t is calculated one by one _j Is an unbiased estimated interval variable alpha ^I Obtaining m unbiased estimated interval variables alpha at m observation moments ^I (t ₁ ),α ^I (t ₂ ),...,α ^I (t _m ). Interval variable alpha of m unbiased estimates of m observation moments ^I (t ₁ ),α ^I (t ₂ ),...,α ^I (t _m ) Composition of interval variable set based on test data

α ^I (t _j ) Is the t _j An unbiased estimated interval variable of observation time.

Prediction model to be confirmed

In contrast, interval variable set based on test data +.>

Is only an interval variable at discrete observation times, so it is difficult to compare the two and establish a validation factor. Therefore, the invention adopts linear interpolation method to add/drop from interval variable set>

Constructing a test observation model of the continuous interval process +.>

Wherein, for interval variable set->

Linear interpolation is carried out on the upper boundaries of the m interval variables to obtain the upper boundary +.>

For interval variable set->

Linear interpolation is carried out on the lower boundaries of m interval variables in the interval process to obtain the lower boundary of the experimental observation model of the interval process

Finally obtaining a test observation model based on test data>

FIG. 1 shows a schematic diagram of the interval course of a predictive model and experimental observations, from which it can be seen that if the experimental observations model

And predictive model->

The higher the overlap ratio of (2), the closer the prediction model is to the experimental observation model, i.e. the better the model confirmation result. However, if the calculation of the degree of coincidence is directly performed, the degree of coincidence in the period of the wider interval boundary is more important than the degree of coincidence in the period of the narrower interval boundary, and in order to take the degree of coincidence over the whole time domain into consideration on average, the present invention performs integration of the degree of coincidence over the time domain.

Taking test observation model

Is +.>

And predictive model->

Is a variable of an interval

Based on the interval ordering strategy, six different positional relationships of two interval variables are shown in fig. 2, and the mathematical expression of the similarity is defined as:

wherein, similarity Deg (a ^I ,b ^I ) Depending on a ^I And b ^I For the six different position cases in fig. 2, the similarity Deg (a ^I ,b ^I ) The calculation is as follows:

similarity Deg (a) ^I ,b ^I ) The value range of (2) is [0,1 ]]Similarity Deg (a) ^I ,b ^I ) The larger the size of the container,the more similar the two interval variables are; and Deg (a) ^I ,b ^I ) =1 means that the two interval variables are identical.

Based on the similarity Deg (a ^I ,b ^I ) Obtaining a predictive model

Interval course and experimental observation model->

Model validation metrics over the entire time domain are:

wherein the model confirms the index

Representing predictive model +.>

Is>

Is used for judging the prediction model according to the model confirmation index>

Is>

Is determined by the degree of overlap of the model +.>

Prediction model +.>

The interval course of (2) is not coincident with the interval course of experimental observation, predictive model->

Is totally unreliable. But for->

Higher means predictive model +.>

Observation model +.>

The closer, in particular when->

If the interval course of the prediction model is identical to the interval course of the test data, the prediction model is +.>

Is a degradation model of the material.

The preferred scheme of the invention is as follows: due to the limited number of test samples, a model validation index is caused

Therefore, the model confirmation index is modified according to the expected value, and the model confirmation index representing the overlapping degree is modified as follows:

wherein the method comprises the steps of

Is the expected value of the confirmation index, and the value range is [0,1]。

Validating the modified model against an index

Compared with a preset model validation index η, if: />

Inequality representing the degree of fit with respect to the interval course is satisfied, then the predictive model +.>

Is a true degradation model, and vice versa. Where a larger value of η means a more stringent requirement for model validation.

Model validation index is the core of validation test, and the invention utilizes a slave prediction model

And (5) establishing a Validation Experiment Design Optimization (VEDO) model according to the obtained prior information. According to the above procedure of establishing the confirmation index, in the confirmation test, the design variables include the number n of material times, the number m of observation times, and the vector t representing the distribution of observation times _obs ＝{t ₁ ,t ₂ ,...,t _m }. To minimize the cost of the model validation test, the validation test design optimization model is:

wherein n is _min And n _max Respectively the minimum value and the maximum value of the number of test samples, m _min And m _max Respectively represent the minimum and maximum observation time quantity, T _max The maximum test observation time is calculated as the total cost C of the test by the following formula _E The method comprises the following steps:

C _E (n,m,t _obs )＝C _s ·n+C _T ·n·m，

wherein C is _s And C _T The cost of one test sample and one test observation, respectively. Based on the VEDO model, the cost of validation experiments is minimized.

According to the total cost of the test C _E By minimizing the number of observation times n and the number of samples m, the lowest test cost can be obtained. The VEDO model can be converted into a multi-objective optimization problem as follows:

the optimal solution set of the multi-objective optimization is obtained by solving a formula, and the test scheme with the lowest cost is obtained from the optimal solution set.

For the VEDO model, n and m are discrete design variables, t _obs Is N _T The continuous design of dimensions creates a Mixed Variable Optimization Problem (MVOP). Generally, the mixing variables increase the complexity of the search space, increasing the difficulty of solving this optimization problem.

The invention considers that the quantity of test samples and the observation time can influence the model confirmation index, and firstly, the observation time t is influenced _obs The individual optimizations are performed and then the number of test samples n is determined. That is, the optimization process is divided into two parts, and the observation time and the number of test samples are optimized respectively, and the step schematic diagram is shown in fig. 3, specifically:

step 1: setting initial parameters of experimental design including n _min ,n _max ,m _min ,m _max And eta;

step 2: setting an initial test plan, wherein the number of observation moments m=m _min And the number of test samples n=n _max A current test plan is generated.

Step 3: in the last step, a current test plan is determined, comprising the number of test samples n and the number of moments m. The present step will optimize the distribution of m observation moments for observation moment t _obs ＝{t ₁ ,t ₂ ,...,t _m Obtaining an optimization model as follows:

the optimization model is a continuous optimization problem, and the invention adopts a genetic algorithm to solve.

Step 4: if the model confirmation factor of the current optimal test scheme is larger than eta, recording the observation time distribution t of the current optimal test scheme _obs Step 5 is entered to optimize the number of test samples n, if not, the number of observation times is increased, and step 3 is returned to.

Step 5: in the last step, the distribution of observation instants is determined. The test sample number n is an optimization target, the minimum test sample number n meeting the eta requirement is obtained, and the optimization model is as follows:

the optimization model is an optimization problem of a single discrete variable, and can be simply solved by adopting an enumeration method.

Step 6: recording the current optimal test scheme including the observation time t _obs And the number of test samples n. The number of observation times m=m+1 continues to be increased and step 3 is returned.

Step 7: all recorded test protocols were obtained and the test protocol with the lowest cost was selected.

One application example of the present invention is provided below:

at two stress levels (40% and 60% of its tensile strength, i.e. 40% sigma) with a composite laminate ₀ And 60% sigma ₀ ) The following fatigue test is exemplified in which the stacking order is [0/90 ]] ₇ The schematic diagram of the sample is shown in FIG. 4. The stiffness degradation of the composite material under fatigue loading in the loading direction is modeled as an interval process, wherein the mean function E ^m (c) A general stiffness degradation model is adopted:

wherein D (c) is fatigue failure index, E ₀ And E is _f Material stiffness corresponding to initial and final stabilization cycles, respectively, c represents normalized fatigue cycle parameters, c=n/N _f Wherein N and N _f The cycle number and the final stabilization period, p and q, respectively, represent the relevant parameters, and the fatigue test parameters at two stress levels are shown in Table 1 below.

Table 1 fatigue test parameters at two stress levels.

As shown in fig. 5, at two different stress levels, the damage builds up non-linearly in the hemp/epoxy composite, and these processes are divided into three stages: i is the region with a significant initial slope at the beginning of the curve. II is a constant slope region. III is the region where the slope gradually increases before the final settling period.

For the scrim/epoxy composite laminates, the degradation process is dispersed due to material imperfections and fluctuations in fatigue loading, and this dispersion increases gradually over time. Thus, the radius function is defined as

Wherein b ₁ And b ₂ Is related to fatigue test, see table 1.

Fig. 6 shows the trend of the degradation model validation index over the number m of observation instants at two pressure levels, wherein each observation instant is evenly distributed. It can be seen that the degradation model confirmation index gradually increases as the number of observation times increases. It is clear that more observation instants represent a more detailed description of the entire degradation process, but this also results in expensive observation costs, and therefore the distribution of observation instants is optimized.

In order to optimize the distribution of observation times, a trial number of samples n=100 and an observation time number of m=11 were set, where c= [ c ] ₁ ,c ₂ ,...,c ₁₁ ]，c ₁ =0 and c ₁₁ =1. And obtaining optimal distribution of observation time by adopting a genetic algorithm, wherein the observation time is uniformly distributed under two stress levels in an initial scheme. Fig. 7 and 8 are the results of confirmation of the initial observation time and the optimal observation time at two stress levels. It can be seen that the confirmation result of the optimized observation point is better than the confirmation result of the initial observation point. 40% sigma ₀ The model confirmation index of (a) is increased from 0.8676 to 0.9166, 60 percent sigma ₀ The model validation index of (2) is increased from 0.8062 to 8725. Thus, to the observation pointOptimization of the distribution can effectively improve the validation results.

Table 2 lists the relevant parameters of the model validation test design and figure 9 shows the optimal solution set for the validation test at two stress levels according to the present invention.

Table 2 parameters relating to the test design

For 40% sigma ₀ The minimum number of observation times is 9 and at least 50 test samples are required. When the number of observation times is 20, the number of test samples can be reduced to 7. Likewise, for 60% sigma ₀ The minimum number of observation times is 10, at least 37 test samples are required. When the number of observation times is 20, the number of test samples can be reduced to 9. The cost of all test plans in the optimal solution set is calculated according to table 2, and fig. 10 shows the distribution of observation times in two optimal test schemes. Of which 40% sigma ₀ The lowest cost of the test is 425 (n=11, m=13); 60% sigma ₀ The lowest test cost is 480 yuan ((n=12, m=15), and the scheme effectively reduces the cost of the confirmation test.

Claims

1. A degradation model confirmation method with interval uncertainty is characterized by comprising the following steps:

2. The degradation model validation method with interval uncertainty of claim 1, wherein: modifying the model confirmation index according to the expected value, wherein the modified model confirmation index

Is a model validation index before modification, +.>

Is an expected value with the value range of [0,1 ]]。

3. The degradation model validation method with interval uncertainty of claim 1, wherein: the first moment average value unbiased estimation quantity

Second moment variance unbiased estimation

Upper bound value of unbiased estimation +.>

Lower boundary value

Obtainingm unbiased estimated interval variables alpha for m observation times ^I (t ₁ ),α ^I (t ₂ ),...,α ^I (t _m ) Sum interval variable set +.>

α ^* Is the average value of the sample set, alpha _i Is the sample set { alpha } ₁ ,α ₂ ,...,α _n Single data in i=1, 2,..n.

4. A degradation model validation method with interval uncertainty as claimed in claim 1, wherein: the similarity is

a ^I Is an interval variable of the experimental observation model, b ^I Is an interval variable of the prediction model.

5. The degradation model validation method with interval uncertainty of claim 4, wherein: model validation index over the entire time domain

Is a predictive model

Is a test observation model.

6. A test design method based on the degradation model confirmation of claim 1, characterized by comprising the steps of: the design variables include the vectors t of n, m and the observation time _obs ＝{t ₁ ,t ₂ ,...,t _m The experimental design optimization model is:

is a predictive model interval, +.>

7. The test design method according to claim 6, wherein: the experimental design optimization model is a multi-objective optimization model:

8. the test design method according to claim 7, wherein: vector t of observation time is optimized firstly _obs And (3) independently optimizing, and determining the number n of the material test pieces.

9. The test design method according to claim 8, wherein: vector t of observation time _obs The optimization model of (2) is:

if the current model validation factor is larger than eta, obtaining an optimized model of the minimum number n of material test pieces meeting the eta requirement as follows: />