Polyurethane creep property prediction method based on relevance vector machine

Abstract

The invention relates to the field of performance prediction of materials, and discloses a polyurethane creep performance prediction method based on a relevance vector machine. Firstly, experimental data are obtained, and data abnormal value processing is carried out; secondly, cutting a demarcation point; then, expanding the data according to the demarcation point, and carrying out normalization processing; then, initializing each parameter of the relevance vector machine, training the model by the relevance vector machine, and carrying out iterative optimization; and finally, carrying out creep property prediction by using the trained relevance vector machine model. Experiments prove that the method can accurately reflect that the relevance vector machine model has good prediction performance, and can effectively solve the problem that the creep performance of the rigid polyurethane foam material is difficult to predict under the coupling of multiple factors.

Description

Technical Field

[0001] The invention relates to the field of material performance prediction, and specifically discloses a polyurethane creep performance prediction method based on a correlation vector machine. Background Art

[0002] Rigid polyurethane foam is a polymer material. Due to its high rigidity, low density, good heat insulation, sound absorption and vibration reduction performance, it has been widely used in aviation, national defense, automobile and other fields in recent years. In military applications, rigid polyurethane foam has a long storage time and is often subjected to large loads. Creep as the main factor will cause changes in the structural properties of the material, affecting its long-term service reliability. In practical applications, factors such as temperature, humidity, and pressure have a greater impact on rigid polyurethane foam. However, most of the current creep analysis and research on rigid polyurethane foam considers a single factor, and the failure mechanism is difficult to reflect under the combined action of multiple factors. The document with application number "201310237622.1" discloses "a method for determining and predicting the creep failure life of polymer materials". The creep strain-time relationship of the polymer material is determined by creep experiments under at least three groups of different test stresses to obtain the corresponding creep strain-time curve, thereby determining the creep failure life of the polymer material. However, the invention does not consider the creep of polyurethane under the influence of multiple factors. The document with application number "202111048278.2" discloses a quantitative analysis method for the compressive strength of submerged polyurethane based on strength prediction, determines the influence of water and air on the compressive strength of polyurethane in different stages of immersion, establishes a compressive strength prediction model for polyurethane in different stages of immersion, quantitatively analyzes the change in the compressive strength of submerged polyurethane when the water and air content changes, calculates the compressive strength of polyurethane in different stages of immersion, and thus predicts the service life of polyurethane. However, under the condition that the creep mechanism of rigid polyurethane foam materials is unknown, the above methods only consider the influence of a single factor on the life of polyurethane, and do not consider the important factor-temperature changes. The model established to describe the creep process of the material cannot analyze the creep performance of rigid polyurethane foam materials under the coupling conditions of multiple factors (such as temperature, humidity, pressure, etc.), and it is difficult to regress and predict the creep performance of polyurethane materials. Summary of the invention

[0003] In view of the above analysis, the present invention proposes a polyurethane creep performance prediction method based on a correlation vector machine. The model describing the material creep process established under the condition that the creep mechanism of rigid polyurethane foam materials is unclear is not comprehensive enough, and the creep performance prediction of rigid polyurethane foam materials under multi-factor coupling conditions is difficult.

[0004] To achieve the above object, the technical solution of the present invention is as follows:

[0005] A polyurethane creep performance prediction method based on a relevance vector machine comprises the following steps:

[0006] Step 1, obtaining experimental data of rigid polyurethane under different temperature, humidity and pressure conditions, and processing data outliers;

[0007] Step 2: Determine the cutoff point, expand the data according to the cutoff point, and then perform normalization;

[0008] Step 3, initializing the correlation vector machine regression prediction model;

[0009] Step 4: Determine the hyperparameters, use the RVM training model, iteratively optimize the hyperparameters, and obtain a trained RVM model;

[0010] Step 5: Use the trained RVM model to predict the creep properties of polyurethane.

[0011] Furthermore, in step 2, the maximum slope x′ of the data curve is determined. max Half of (t) is used as the cutoff point for data expansion;

[0012] The data expansion uses interpolation, assuming that the interval between each adjacent data point is Δx i , where i represents the i-th interval, and sets the total number of data points to be expanded. The specific expansion method is:

[0013] For slopes greater than The number of expansions between each point is

[0014]

[0015] For slopes less than The number of expansions between each point is

[0016]

[0017] Furthermore, the regression prediction model of the correlation vector machine in step 3 is specifically described as follows:

[0018] The regression model of the relevance vector machine based on Bayesian theory is:

[0019] t=Φω+ε (3)

[0020] Where ω=(ω 0 ,...,ω n ) T is the weight of the model, which is an N+1-dimensional column vector; Φ is the kernel function matrix, and Φ=(Φ 1 ,Φ 2 ...,Φ n )T , Φ i (x i )=[1,K(x i ,x 1 ),...,K(x i ,x n )], i = 1, 2, ..., N, K(·) is the kernel function; ε is independent and identically distributed Gaussian noise, and ε ~ N(0,σ 2 ), σ 2 is the noise variance of the model output data t;

[0021] RVM is based on the Bayesian framework and introduces a hyperparameter α, which makes ω obey a Gaussian distribution with a mean of zero. The weight prior is in the form of

[0022]

[0023] In the formula, α=(α 0 ,...,α N ) T , is an N+1-dimensional hyperparameter, α i is the weight ω i Hyperparameters of

[0024] Set {α i}=0.1,i=0,1,...,n-1,σ 2 =var(t)*0.1, set the maximum number of iterations, and perform convergence training.

[0025] Furthermore, the process of determining the hyperparameters in step 4 is as follows:

[0026] First, calculate the covariance matrix Σ and mean μ of the posterior distribution of ω:

[0027] Based on the prior distribution and likelihood distribution, the posterior distribution of ω also satisfies the Gaussian distribution:

[0028]

[0029] The posterior variance and mean of the weights are obtained as follows:

[0030] Σ=(σ -2 Φ T Φ+A) -1 (6)

[0031] μ=σ -2 ΣΦ T t (7)

[0032] Where A = diag(α 0 ,α 1 ,...,α N), that is, A is a diagonal matrix with α i is the weight ω i Hyperparameters of

[0033] Then, we use the iterative estimation method to update α i and σ 2 :

[0034] Obtained by marginal integration of the parameters:

[0035] p(tx,α,σ 2 )=∫p(tx,ω,σ 2 )p(ωα)dω (8)

[0036] Get the marginal likelihood of the hyperparameters:

[0037] p(tx,α,σ 2 )=N(0,C) (9)

[0038] Where C = σ 2 I+ΦA -1 Φ T

[0039] Take the logarithm of both sides and then calculate α i and σ 2 Taking the partial derivative and making it equal to zero, we get

[0040]

[0041] where γ i =1-α i Σii, Σii is the element on the i-th diagonal of the covariance Σ;

[0042] Finally, determine whether the iteration reaches the maximum number of iterations or meets the convergence condition. If so, end the iteration, otherwise repeat step 4.

[0043] Compared with the prior art, the present invention has the following beneficial effects:

[0044] (1) Different from the previous method that considers the influence of a single factor on the creep performance of polyurethane, this method considers the creep of polyurethane under the combined action of multiple factors such as temperature, pressure, and humidity. Among them, the role of temperature in the whole process is very important. By establishing a model that can reflect the comprehensive effect of multiple factors, the creep performance of polyurethane under complex environmental conditions can be predicted more accurately.

[0045] (2) When expanding the data, the principle of "dense at the front and sparse at the back" is used. The data in the front part with a larger slope is expanded more densely, while the data in the back part with a smaller slope is expanded more sparsely. By carefully capturing the changing trends of the data in different slope sections, the accuracy of creep performance prediction can be improved.

[0046] (3) The method of the present invention can accurately reflect that the correlation vector machine model has good prediction performance and can effectively solve the problem of difficulty in predicting the creep performance of rigid polyurethane foam materials under the coupling of multiple factors. BRIEF DESCRIPTION OF THE DRAWINGS

[0047] Figure 1 It is the overall flow chart of creep prediction method based on correlation vector machine;

[0048] Figure 2 This is a comparison chart of the original data and the expanded data;

[0049] Figure 3 This is a comparison chart of the actual data and the predicted data results. DETAILED DESCRIPTION

[0050] like Figure 1 The overall flow chart of the present invention is shown. First, the experimental data is obtained and the data outliers are processed; second, the demarcation point is cut; then, the data is expanded according to the demarcation point and normalized; then, the parameters of the correlation vector machine are initialized, the correlation vector machine training model is trained, and iterative optimization is performed; finally, the trained correlation vector machine model is used to predict the creep performance. The specific implementation steps are shown below.

[0051] Example:

[0052] Step 1: obtaining experimental data of rigid polyurethane under different temperature, humidity and pressure conditions and performing data preprocessing;

[0053] After the data is obtained from the experiment, there may be various problems in the sampled data, such as missing values, outliers, etc. These problems will affect the training and prediction performance of the model. Therefore, after obtaining the sampled data, it is necessary to preprocess the missing values ​​and outliers (delete or correct) to improve the quality and credibility of the data.

[0054] In the experiment, the creep data of polyurethane material were obtained under the experimental conditions of humidity 75% RH, temperature 65°C, pressures 0.1MPa, 0.2MPa, 0.4MPa, and temperature 75°C, pressures 0.1MPa, 0.2MPa, 0.4MPa, respectively;

[0055] Step 2: Cutting at the cutoff point, expanding the data according to the cutoff point, and normalizing it;

[0056] Since the sampling data is limited, in order to ensure the integrity and reliability of the data, the present invention uses data expansion technology to increase the size and diversity of the data set. And because the data is nonlinear, in order to improve the performance and generalization ability of the model, the slope of the data curve is used here to determine the density or sparsity of the data expansion. Since the creep variable changes rapidly in the early stage and slowly in the later stage, it is decided to adopt the principle of "dense at the beginning and sparse at the end" for data expansion, that is, the data is densely expanded in the part that changes rapidly in the early stage, and the data is sparsely expanded in the part that changes slowly in the later stage. First, calculate the slope of the data curve, that is, the derivative x′(t), and determine the maximum value x′ of the slope max (t), and with the maximum slope x′ max Half of (t) is used as the dividing point, that is, for slopes greater than The part where the data changes more dramatically needs more intensive expansion; for the part where the slope change rate is less than The part where the data changes more slowly needs to be expanded more sparsely.

[0057] After confirming the dividing point, use interpolation to expand the data, assuming that the interval between each adjacent data point is Δx i , where i represents the i-th interval, and the total number of data points to be expanded is set to 1000. The rate of change of the slope of the curve is greater than The points between them are densely expanded, and the number of expansions required is D 前 , the rate of change of the curve slope is less than The expansion between the points is sparse, and the number of expansions required is D 后 , assuming the slope of the curve reaches The corresponding time point is k, and then the number of expansions required for each interval is determined based on the length of each interval. Specifically:

[0058] For slopes greater than The number of expansions between each point is

[0059]

[0060] For slopes less than The number of expansions between each point is

[0061]

[0062] Afterwards, the expanded data is normalized.

[0063] Step 3 initializes the parameters of the correlation vector machine model;

[0064] The regression model of the relevance vector machine based on Bayesian theory is:

[0065] t=Φω+ε (3)

[0066] Where ω=(ω 0 ,...,ω n ) T is the weight of the model, which is an N+1-dimensional column vector; Φ is the kernel function matrix, and Φ=(Φ 1 ,Φ 2 ...,Φ n ) T , Φ i (x i )=[1,K(x i ,x 1 ),...,K(x i ,x n )], i = 1, 2, ..., N, K(·) is the kernel function; ε is independent and identically distributed Gaussian noise, and ε ~ N(0,σ 2 ), σ 2 is the noise variance of the model output data t.

[0067] RVM is based on the Bayesian framework and introduces a hyperparameter α, which makes ω obey a Gaussian distribution with a mean of zero. The weight prior is in the form of

[0068]

[0069] In the formula, α=(α 0 ,...,α N ) T , is an N+1-dimensional hyperparameter, α i is the weight ω i Hyperparameters of .

[0070] Set {α i}=0.1,i=0,1,...,n-1,σ 2 =var(t)*0.1, set the maximum number of iterations to 1000. During RVM training, alphaTolerance is a threshold to measure the change of the alpha parameter. When the change of the alpha parameter is less than this threshold, the algorithm considers that the convergence condition has been reached and the training process can be terminated. Here, the convergence condition alphaTolerance is set to 10 -6 .

[0071] Step 4: determine the hyperparameters, use the RVM to train the model, iteratively optimize the hyperparameters, and obtain a trained RVM model;

[0072] In the experiment, time, temperature, humidity and pressure were used as independent variables, and five sets of data were used as training, including humidity 75% RH, temperature 65°C, pressures 0.1MPa, 0.2MPa, 0.4MPa, and temperature 75°C, pressures 0.1MPa, 0.2MPa;

[0073] Step 4.1 Determine hyperparameters: Calculate the covariance matrix Σ and mean μ of the posterior distribution of ω:

[0074] Based on the prior distribution and likelihood distribution, the posterior distribution of ω also satisfies the Gaussian distribution:

[0075]

[0076] The posterior variance and mean of the weights are obtained as follows:

[0077] Σ=(σ -2 Φ T Φ+A) -1 (6)

[0078] μ=σ -2 ΣΦ T t (7)

[0079] Where A = diag(α 0 ,α 1 ,...,α N ), that is, A is a diagonal matrix with α i is the weight ω i Hyperparameters of .

[0080] Step 4.2 iteratively optimizes the hyperparameters. The process is:

[0081] Update α using iterative estimation i and σ 2 :

[0082] Obtained by marginal integration of the parameters:

[0083] p(tx,α,σ 2 )=∫p(tx,ω,σ 2 )p(ωα)dω (8)

[0084] This gives the marginal likelihood of the hyperparameters:

[0085] p(tx,α,σ 2 )=N(0,C) (9)

[0086] Where C = σ 2 I+ΦA -1 Φ T

[0087] Take the logarithm of both sides and then calculate α i and σ 2 Taking the partial derivative and making it equal to zero, we get

[0088]

[0089] where γ i =1-α i Σii, Σii is the element on the i-th diagonal of covariance Σ.

[0090] Step 4.3 determines whether the iteration reaches the maximum number of iterations or meets the convergence condition. If so, the iteration ends, otherwise go to 4.1.

[0091] Step 5 uses the trained RVM model to predict the creep properties of polyurethane.

[0092] In this experiment, time, temperature, humidity and pressure were used as independent variables, and the creep amount at humidity 75% RH, temperature 75°C and pressure 0.4 MPa was studied as the prediction output.

[0093] Tables 1 to 5 below are the original experimental data, where time, temperature, humidity and pressure are used as inputs, and the creep amount under the corresponding conditions is used as output:

[0094] Table 1:

[0095] Time / Day Temperature / ℃ Humidity / %RH Pressure / MPa Creep amount / mm 0 65 75 0.1 0.0000 3 65 75 0.1 0.0018 7 65 75 0.1 0.0052 10 65 75 0.1 0.0065 15 65 75 0.1 0.0084 30 65 75 0.1 0.0105 45 65 75 0.1 0.0113 59 65 75 0.1 0.0090

[0096] Table 2:

[0097] Time / Day Temperature / ℃ Humidity / %RH Pressure / MPa Creep amount / mm 0 65 75 0.2 0.0000 3 65 75 0.2 0.0043 7 65 75 0.2 0.0063 10 65 75 0.2 0.0097 15 65 75 0.2 0.0111 30 65 75 0.2 0.0124 45 65 75 0.2 0.0136 59 65 75 0.2 0.0133

[0098] Table 3:

[0099]

[0100]

[0101] Table 4:

[0102] Time / Day Temperature / ℃ Humidity / %RH Pressure / MPa Creep amount / mm 0 75 75 0.1 0.0000 3 75 75 0.1 0.0043 7 75 75 0.1 0.0058 10 75 75 0.1 0.0072 15 75 75 0.1 0.0085 30 75 75 0.1 0.0101 45 75 75 0.1 0.0125 59 75 75 0.1 0.0148 90 75 75 0.1 0.0138

[0103] Table 5:

[0104] Time / Day Temperature / ℃ Humidity / %RH Pressure / MPa Creep amount / mm 0 75 75 0.2 0.0000 3 75 75 0.2 0.0054 7 75 75 0.2 0.0106 10 75 75 0.2 0.0131 15 75 75 0.2 0.0151 30 75 75 0.2 0.0183 45 75 75 0.2 0.0205 59 75 75 0.2 0.0242 90 75 75 0.2 0.0257

[0105] Table 6:

[0106]

[0107]

[0108] Experimental results:

[0109] RMSE MAE MAPE 0.0029 0.0019 4.83％

[0110] RMSE: Root mean square error, which refers to the square root of the mean of the square of the difference between the predicted value and the true value.

[0111]

[0112] Among them, y represents the predicted result and x represents the true value. The smaller the RMSE value, the smaller the difference between the model's predicted result and the true value.

[0113] MAE: Mean absolute error, which refers to the average of the absolute values ​​of the differences between the predicted values ​​and the true values.

[0114]

[0115] Among them, y represents the predicted result and x represents the true value. The smaller the MAE value, the more accurate the model's prediction is.

[0116] MAPE: Mean absolute percentage error, which refers to the ratio of the difference between the predicted value and the true value to the true value.

[0117]

[0118] Among them, y represents the predicted result and x represents the true value. The smaller the value of MAPE is, the more accurate the prediction of the model is.

[0119] like Figure 3 The figure shows the comparison results of the predicted data and the original data under experimental conditions. It can be seen from the figure that the predicted data is relatively close to the original data. The calculated MAPE is 4.83%, which can reflect the accuracy and reliability of the model prediction.

[0120] The above description is only a preferred embodiment of the present invention and does not limit the present invention in any form. Any simple modification, equivalent changes and modifications made to the above embodiments based on the technical essence of the present invention are still within the protection scope of the technical solution of the present invention.

Claims

1. A polyurethane creep performance prediction method based on a relevance vector machine, characterized in that: The following steps are involved: Step 1, obtaining experimental data of rigid polyurethane under different temperature, humidity and pressure conditions, and processing data outliers; Step 2: Determine the cutoff point, expand the data according to the cutoff point, and then perform normalization; Step 3, initializing the correlation vector machine regression prediction model; Step 4: Determine the hyperparameters, use the RVM training model, iteratively optimize the hyperparameters, and obtain a trained RVM model; Step 5: Use the trained RVM model to predict the creep properties of polyurethane.

2. The polyurethane creep performance prediction method based on relevance vector machine according to claim 1 is characterized in that: In step 2, the maximum slope x′ of the data curve is determined. max Half of (t) is used as the cutoff point for data expansion; The data expansion uses interpolation, assuming that the interval between each adjacent data point is Δx i , where i represents the i-th interval, and sets the total number of data points to be expanded. The specific expansion method is: For slopes greater than The number of expansions between each point is For slopes less than The number of expansions between each point is 3. The polyurethane creep performance prediction method based on relevance vector machine according to claim 2 is characterized in that: The regression prediction model of the relevant vector machine in step 3 is specifically described as follows: The regression model of the relevance vector machine based on Bayesian theory is: t=Φω+ε (3) where ω=(ω0,...,ω n ) T is the weight of the model, which is an N+1-dimensional column vector; Φ is the kernel function matrix, and Φ=(Φ1,Φ2...,Φ n ) T , Φ i (x i )=[1,K(x i ,x1),...,K(x i ,x n )], i = 1, 2, ..., N, K(·) is the kernel function; ε is independent and identically distributed Gaussian noise, and ε ~ N(0,σ 2 ), σ 2 is the noise variance of the model output data t; RVM is based on the Bayesian framework and introduces a hyperparameter α, which makes ω obey a Gaussian distribution with a mean of zero. The weight prior is in the form of In the formula, α=(α0,...,α N ) T , is an N+1-dimensional hyperparameter, α i is the weight ω i Hyperparameters of Set {α i }=0.1,i=0,1,...,n-1,σ 2 =var(t)*0.1, set the maximum number of iterations, and perform convergence training.

4. The polyurethane creep performance prediction method based on relevance vector machine according to claim 3 is characterized in that: The process of determining the hyperparameters in step 4 is as follows: First, calculate the covariance matrix Σ and mean μ of the posterior distribution of ω: Based on the prior distribution and likelihood distribution, the posterior distribution of ω also satisfies the Gaussian distribution: The posterior variance and mean of the weights are obtained as follows: S=(s -2 F T (F+A) -1 (6) μ = σ -2 SF T t (7) Among them, A=diag(α0,α1,...,α N ), that is, A is a diagonal matrix with α i is the weight ω i Hyperparameters of Then, we use the iterative estimation method to update α i and σ 2 : Obtained by marginal integration of the parameters: p(t|x,a,s 2 )=∫p(t|x,ω,σ 2 )p(ω|α)dω (8) Get the marginal likelihood of the hyperparameters: p(t|x,a,s 2 )=N(0,C) (9) where C = σ 2 I + ΦA -1 Φ T Take the logarithm of both sides and then calculate α i and σ 2 Taking the partial derivative and making it equal to zero, we get where γ i =1-α i Σii, Σii is the element on the i-th diagonal of the covariance Σ; Finally, determine whether the iteration reaches the maximum number of iterations or meets the convergence condition. If so, end the iteration, otherwise repeat step 4.

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