Underwater explosion shock wave load probability characterization method based on Bayesian reasoning

Abstract

The invention discloses an underwater explosion shock wave load probability characterization method based on Bayesian inference, and relates to the field of underwater explosion load calculation, and the method comprises the steps: obtaining previous test data, carrying out the analysis processing of the test data, obtaining sample data, taking a Cole empirical model as a prior model of Bayesian inference, and carrying out the calculation of the probability of the underwater explosion shock wave load. Performing uncertainty analysis on load characterization parameters and calculation errors of the prior model in combination with the sample data to obtain target probability distribution of the load characterization parameters and the calculation errors; and by taking the target probability distribution as priori knowledge, carrying out updating calculation on the load characterization parameters and calculation errors by adopting a Bayesian inference method to obtain a Bayesian probability model of the underwater explosion shock wave load. And carrying out probability representation on the underwater explosion shock wave load based on the Bayesian probability model of the underwater explosion shock wave load. According to the method, the uncertainty of the underwater explosion shock wave load is effectively represented, and random input considering the load variability is provided for the anti-explosion reliability design of the underwater structure.

Description

Technical Field

[0001] This invention relates to the field of underwater explosion load calculation, and in particular to a method for probabilistic characterization of underwater explosion shock wave load based on Bayesian inference. Background Technology

[0002] Underwater explosions have always been a serious threat to important infrastructure such as cross-river and cross-sea bridges, underwater tunnels and pipelines, reservoir dams and port terminals. The core premise for analyzing and calculating the dynamic response and damage characteristics of underwater explosions is to establish a reasonable and accurate explosion load model.

[0003] Underwater explosion processes are extremely complex. The propagation patterns and characteristics of explosive loads are influenced by a variety of factors, including the type and composition of explosives, the method and size of the charge, the explosive yield and location, the properties of the medium itself, the characteristics of the underwater environment and boundary conditions, and the size and accuracy of sensors. The inherent randomness of these factors leads to significant uncertainties in the intensity and duration of underwater explosive loads. These uncertainties are often the key reason for significant deviations between actual loads and calculations based on empirical formulas.

[0004] Current structural blast-resistant design codes treat explosion loads as fixed and employ conservative design and evaluation methods based on maximum explosive charges to ensure structural safety. However, they cannot quantify the guarantee rate of a structure meeting its blast-resistant protection function under a specific explosion condition. Reliability-based blast-resistant design is an important direction for the development of structural engineering. Considering the uncertainties in structural blast-resistant design from a probabilistic perspective, research on the variability of structural explosion loads is essential.

[0005] Therefore, there is an urgent need for a probabilistic characterization method for underwater explosion shock wave loads based on Bayesian inference. Summary of the Invention

[0006] To address the aforementioned issues, this application proposes a probabilistic characterization method for underwater explosion shock wave loads based on Bayesian inference, comprising the following steps:

[0007] S1. Obtain past experimental data, analyze and process the experimental data to obtain sample data;

[0008] S2. Using the Cole empirical model as the prior model for Bayesian inference, and combining sample data, uncertainty analysis is performed on the load characterization parameters and calculation errors of the prior model to obtain the target probability distribution of the load characterization parameters and calculation errors.

[0009] S3. Using the target probability distribution as prior knowledge, Bayesian inference method is used to update the load characterization parameters and calculation error to obtain the Bayesian probability model of underwater explosion shock wave load.

[0010] S4. The underwater explosion shock wave load is predicted and calculated based on the Bayesian probability model.

[0011] Preferably, the specific content of the prior model in S2, which uses the Cole empirical model as the basis for Bayesian inference, includes:

[0012] Based on Cole's empirical formula, the time history expression of the underwater explosion shock wave pressure is obtained, and then the load characterization parameters of the underwater explosion shock wave are derived.

[0013] The load characterization parameters include the peak pressure p of the free-field shock wave. m Time constant θ, impulse I, and shock wave specific energy density e s ;

[0014] A dimensionless empirical model is established for the load characterization parameters.

[0015] Preferably, the time history expression for the pressure of the underwater explosion shock wave is:

[0016]

[0017] Where p(t) is the pressure time history of the underwater explosion shock wave; p m θ represents the peak pressure of the free-field shock wave; θ is the time constant, i.e., the peak pressure of the shock wave starts from p. m Decay to p m / e is the time used, where t is time, m is the explosive equivalent, and e is the natural constant;

[0018] Based on the time history expression of underwater explosion shock wave pressure, the water density ρ is introduced. w With the speed of sound propagation in water c w The impulse I and the shock wave specific energy density e are derived. s :

[0019] The preferred expression for the dimensionless empirical model is:

[0020]

[0021] In the formula, the response variable y = (p m ,θ / m 1 / 3 ,I / m 1 / 3 ,e s / m 1 / 3 ), where m 1 / 3 This represents the 1 / 3 power of the explosive equivalent.

[0022] Load characterization parameter k d =(k p ,k θ ,k I ,k e ), α d=(α p ,α θ ,α I ,α e ), where k d α d These represent the first and second computational parameters of the dimensionless empirical model expression, respectively, including the computational parameter k of the pressure peak prior model. p α p The computational parameter k of the time constant prior model θ α θ The calculation parameter k of the impulse prior model I α I The calculation parameter k of the prior model for shock wave specific energy density e α e ;

[0023] Wherein, the proportional detonation distance Z can be expressed as:

[0024]

[0025] In the formula, R is the detonation distance; m is the mass of the explosive; and NEQ is the TNT equivalent.

[0026] Preferably, in S2, the specific content of obtaining the target probability distribution of the load characterization parameters and calculation errors by combining sample data and performing uncertainty analysis on the load characterization parameters and calculation errors includes:

[0027] There are preset probability distribution types available, including but not limited to Normal, Lognormal, Weibull, and Gamma distributions.

[0028] By fitting the load characterization parameters with sample data, the power-law relationship between the load characterization parameters and the proportional blast distance Z is obtained.

[0029] The goodness-of-fit test results were obtained by using the Anderson-Darling statistic to test the goodness-of-fit between the load characterization parameters and the available probability distribution types.

[0030] The target probability distribution of the load characterization parameters is determined based on the power-law relationship characteristics and the goodness-of-fit test results.

[0031] Using root mean square error (RMSE) and coefficient of determination (R²) 2 The coefficient of variation (Cov) is used to evaluate the dimensionless empirical model of Cole, and then the bias of the dimensionless empirical model is quantified to obtain the model error (ME).

[0032] The proportional burst distance Z is segmented and subjected to Anderson-Darling goodness-of-fit test on the interval characteristics of ME, and then the target probability distribution of the calculation error is determined from the available probability distribution types.

[0033] Preferably, the expression for the model error ME is:

[0034]

[0035] In the formula, y test For experimental test values; y cal Calculate values ​​for the model.

[0036] Preferably, the expression for the Bayesian probability model of underwater explosion shock wave load includes:

[0037] (1) The parameter k is characterized by the load of the prior model. s =(k p ,k θ ,k I ,k e ), α s =(α p ,α θ ,α I ,α e The first Bayesian probability model, whose parameters are the Bayesian probability model parameters and the model error σ.

[0038] y k (x,Θ k )=y k,d (x,θ k )+σ k ε k ;

[0039] (2) Using the model correction term parameter β k And the second Bayesian probability model, where the model error σ is the parameter of the Bayesian probability model, is determined by the model correction term η. k (x,β k Correct the calculation results of the prior model;

[0040] y k (x,Θ k )=y k,d (x,θ k )+η k (x,β k )+σ k ε k ;

[0041] In the formula, y k (x,Θ k ) represents the probabilistic model of underwater explosive load characterization parameters, where the subscript k represents different characterization parameters; y k,d(x,θ k ), η k (x,β k ) represent the empirical model and its correction terms; x represents the observable input random variable; Θ represents the θ. k =(θ k ,β k ,σ k ) represents the unknown parameters of the probability model; σ k ε k σ represents the computational error of the corrected probability model; where σ k ε represents the model standard deviation. k Let θ be a random variable that follows a standard normal distribution. k Representing the empirical model y k,d The calculation parameter k d α d ;β k η represents the correction term in the second Bayesian probability model. k Calculation parameters;

[0042] Define the model variance σ k 2 The load is independent and linearly unrelated to the variable x, that is, for a given load characterization parameter Θ... k The variance of the probability model is Var[P(x,Θ]]. k )]=σ k 2 ;

[0043] The model correction term η k (x,β k The expression for ) is:

[0044]

[0045] Where, β k,i η represents the correction term in the second Bayesian probability model. k The i-th calculated coefficient of the calculated parameter, h k,i is the "interpretation" function for the correction terms of the shock wave load-related parameters, where x is the variable of the "interpretation" function.

[0046] Preferably, based on the observation dataset y k The posterior distribution of the Bayesian probability model parameters is updated using Bayes' theorem. The expression for the posterior distribution of the Bayesian probability model parameters is:

[0047]

[0048] In the formula, π(Θ) k ) represents the prior distribution of parameters in the Bayesian probability model; y k y represents the actual obtained evidence sample vector. k=[y1,y2,…,y n ];Θ k For unknown parameters in the probabilistic model; π(Θ) k |y k L(y) represents the posterior distribution of the parameters of the Bayesian probability model; k |Θ k L(y) is the likelihood function, used to quantify the consistency between the model and the data. k |Θ k )=L(y1,y2,…,y n ,Θ k ); c is the definite integral constant, representing the regularization factor;

[0049] Preferably, when the evidence sample information comes from n independent trial sets, the posterior distribution expression of the Bayesian probability model parameters can be further expressed as:

[0050]

[0051] Where i is the i-th data sample point of the n-th independent trial dataset, n is the number of independent trial datasets, and π is the posterior distribution function.

[0052] Based on the Bayesian probability model of underwater explosion shock wave load, the model calculation error is defined, and its expression is as follows:

[0053] r k (x,Θ k )=y k (x,Θ k )-y k,d (x,θ k )-η k (x,β k );

[0054] Where, r k For model calculation error, y k,d An empirical model for underwater explosion shock wave load.

[0055] In summary, the underwater explosion shock wave load probability characterization method based on Bayesian inference of the present invention has the following advantages compared with traditional technologies:

[0056] 1. The coefficient of variation of the load characterization parameters in this application is 0.03-0.48, and the coefficient of variation of the model error is 0.19-0.38, revealing the significant uncertainty characteristics of underwater explosion shock wave load;

[0057] 2. Bayesian inference methods effectively improve the accuracy of parameter estimation. By sampling parameters posteriorly, the RMSE is significantly reduced under limited sample conditions. Compared with traditional deterministic methods, Bayesian probability not only provides point estimates but also quantifies the uncertainty of the model, providing more comprehensive information for engineering risk assessment.

[0058] 3. The Bayesian probability model established in this application can effectively characterize the uncertainty of underwater explosion shock wave load, providing a stochastic input field that considers load variability for the explosion-resistant reliability design of underwater structures. This model framework can be extended to reliability analysis of other explosion load scenarios, providing a new tool for risk assessment and uncertainty propagation research.

[0059] The technical method of the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. Attached Figure Description

[0060] Figure 1 The figures shown are bar charts and corresponding probability distribution diagrams of various load characterization parameters in the embodiments of the present invention, wherein, Figure 1 (a) in the figure represents the calculation parameter k. θ Probability distribution diagram Figure 1 (b) in the figure represents the calculation parameter α. θ Probability distribution diagram Figure 1 (c) in the equation represents the calculation parameter k. p Probability distribution diagram Figure 1 In this context, (d) represents the calculation parameter α. p Probability distribution diagram Figure 1 (e) in the equation represents the calculation parameter k. I Probability distribution diagram Figure 1 (f) in the equation represents the calculation parameter α. I Probability distribution diagram Figure 1 In this context, (g) represents the calculation parameter k. e Probability distribution diagram Figure 1 (h) in the calculation parameter α e Probability distribution diagram;

[0061] Figure 2 This invention provides a comparison between experimental and predicted values ​​of underwater explosion load characterization parameters in an embodiment of the invention. Figure 2 (a) in the figure compares the experimental value and the predicted value of the time constant θ. Figure 2 (b) represents the peak pressure p. m Comparison of experimental values ​​and predicted values Figure 2 (c) in the figure shows a comparison between the experimental value and the predicted value of impulse I. Figure 2 In the equation (d), the specific energy density e of the shock wave is... s Comparison of experimental values ​​and predicted values;

[0062] Figure 3This is the error distribution of different ratio detonation distance models in the embodiments of the present invention, wherein, Figure 3 In the diagram, (a) represents the error distribution of the time constant θ model. Figure 3 (b) represents the peak pressure p. m Model error distribution Figure 3 In the diagram, (c) represents the error distribution of the impulse I model. Figure 3 In the equation (d), the specific energy density e of the shock wave is... s Model error distribution;

[0063] Figure 4 These are the mean and coefficient of variation of the explosion distance models with different ratios in the embodiments of the present invention;

[0064] Figure 5 This is the hierarchical structure of the Bayesian probability model in this embodiment of the invention;

[0065] Figure 6 The following is the posterior density distribution of the time constants of each model in Model I in this embodiment of the invention, wherein... Figure 6 (a) in the figure represents the calculation parameter α. θ The posterior distribution, Figure 6 (b) in the equation represents the calculation parameter k. θ The posterior distribution, Figure 6 (c) in the figure represents the calculation parameter σ. θ The posterior distribution, Figure 6 In the figure, (d) represents the variation of the standard deviation of the calculation parameter with the sample size;

[0066] Figure 7 The following is the posterior density distribution of peak pressure for each model in Model I of this embodiment of the invention, wherein... Figure 7 (a) in the figure represents the calculation parameter α. p The posterior distribution, Figure 7 (b) in the equation represents the calculation parameter k. p The posterior distribution, Figure 7 (c) in the figure represents the calculation parameter σ. p The posterior distribution, Figure 7 In the figure, (d) represents the variation of the standard deviation of the calculation parameter with the sample size;

[0067] Figure 8 The following is the impulse posterior density distribution of each model in Model I in this embodiment of the invention, wherein, Figure 8 (a) in the figure represents the calculation parameter α. I The posterior distribution, Figure 8 (b) in the equation represents the calculation parameter k. I The posterior distribution, Figure 8 (c) in the figure represents the calculation parameter σ. I The posterior distribution, Figure 8 In the figure, (d) represents the variation of the standard deviation of the calculation parameter with the sample size;

[0068] Figure 9 The following is the posterior density distribution of the shock wave density for each model in Model I of this embodiment of the invention, wherein... Figure 9 (a) in the figure represents the calculation parameter α. e The posterior distribution, Figure 9 (b) in the equation represents the calculation parameter k. e The posterior distribution, Figure 9 (c) in the figure represents the calculation parameter σ. e The posterior distribution, Figure 9 In the figure, (d) represents the variation of the standard deviation of the calculation parameter with the sample size;

[0069] Figure 10 The following is the posterior density distribution of the time constants of each model in Model II of this invention, wherein... Figure 10 (a) in the figure represents the calculation parameter α. θ The posterior distribution, Figure 10 (b) in the equation represents the calculation parameter k. θ The posterior distribution, Figure 10 (c) in the figure represents the calculation parameter σ. θ The posterior distribution, Figure 10 In the figure, (d) represents the variation of the standard deviation of the calculation parameter with the sample size;

[0070] Figure 11 The following is the posterior density distribution of peak pressure for each model in Model II of this invention, wherein... Figure 11 (a) in the figure represents the calculation parameter α. p The posterior distribution, Figure 11 (b) in the equation represents the calculation parameter k. p The posterior distribution, Figure 11 (c) in the figure represents the calculation parameter σ. p The posterior distribution, Figure 11 In the figure, (d) represents the variation of the standard deviation of the calculation parameter with the sample size;

[0071] Figure 12 The following is the impulse posterior density distribution of each model in Model II of this invention embodiment, wherein, Figure 12 (a) in the figure represents the calculation parameter α. I The posterior distribution, Figure 12 (b) in the equation represents the calculation parameter k. I The posterior distribution, Figure 12 (c) in the figure represents the calculation parameter σ. I The posterior distribution, Figure 12 In the figure, (d) represents the variation of the standard deviation of the calculation parameter with the sample size;

[0072] Figure 13 The following is the posterior density distribution of the shock wave density for each model in Model II of this invention, wherein... Figure 13 (a) in the figure represents the calculation parameter α.e The posterior distribution, Figure 13 (b) in the equation represents the calculation parameter k. e The posterior distribution, Figure 13 (c) in the figure represents the calculation parameter σ. e The posterior distribution, Figure 13 In the figure, (d) represents the variation of the standard deviation of the calculation parameter with the sample size;

[0073] Figure 14 This is a comparison of the measured values ​​and the calculated values ​​of each model in the embodiments of the present invention, wherein, Figure 14 (a) in the figure compares the experimental value and the predicted value of the time constant θ. Figure 14 (b) represents the peak pressure p. m Comparison of experimental values ​​and predicted values Figure 14 (c) in the figure shows a comparison between the experimental value and the predicted value of impulse I. Figure 14 In the equation (d), the specific energy density e of the shock wave is... s Comparison of experimental values ​​and predicted values;

[0074] Figure 15 This is a flowchart illustrating the steps of a method for probabilistic characterization of underwater explosion shock wave load based on Bayesian inference, as described in this invention. Detailed Implementation

[0075] The technical method of the present invention will be further described below with reference to the accompanying drawings and embodiments. It should be noted that, unless otherwise specifically stated, the relative arrangement, numerical expressions, and values ​​of the components and steps described in these embodiments do not limit the scope of this application.

[0076] The following description of at least one exemplary embodiment is merely illustrative and is in no way intended to limit the scope of this application and its application or use.

[0077] Techniques, systems, and equipment known to those skilled in the art may not be discussed in detail, but where appropriate, they should be considered part of the instruction manual.

[0078] In all the examples shown and discussed herein, any specific values ​​should be interpreted as merely exemplary and not as limitations. Therefore, other examples of exemplary embodiments may have different values.

[0079] Unless otherwise defined, the technical or scientific terms used in this invention shall have the ordinary meaning as understood by one of ordinary skill in the art to which this invention pertains.

[0080] Underwater explosion shock wave loads exhibit significant variability and uncertainty. Classical deterministic empirical models, by neglecting this characteristic, lead to discrepancies between calculated results and measured data. This invention utilizes 682 sets of underwater explosion test data to analyze various parameters of the explosion load (peak pressure p). m Time constant θ, impulse I, and shock wave specific energy density e s Uncertainty analysis was performed, and a Bayesian probabilistic model integrating the Cole empirical model and physical correction terms was constructed. Bayesian inference was used to calibrate parameters, achieving a probabilistic characterization of the shock wave load. Specific steps are as follows: Figure 15 As shown.

[0081] Example

[0082] S1. Obtain past experimental data, analyze and process the experimental data to obtain sample data experimental dataset.

[0083] The main characterization parameters of underwater explosion shock wave load include pressure time history p(t) and peak pressure p. m Time constant θ, impulse I, and shock wave specific energy density e s This invention collected and screened a total of 682 sets of underwater explosion test data, including 677 sets of p m Data, 593 sets of θ data, 593 sets of I data, and 553 sets of e data s Data. The collected data primarily characterizes low-yield, shallow-water, and mid-to-long-field underwater explosion conditions. Given the scarcity of underwater explosion test data, this statistical analysis integrates field tests and indoor model chamber tests, covering various explosive types including TNT, RDX, HMX, CL-20, and emulsion explosives. Test design parameters include charge density, sensor depth measurement, and distance measurement. Explosion load test parameters mainly include time constant, peak pressure, impulse, and specific energy density. The statistical results are shown in Table 1.

[0084] Table 1. Statistical Analysis of Underwater Explosion Test Design Parameters and Test Data

[0085]

[0086] S2. Using the Cole empirical model in the literature “COLE RH, WELLER R. Underwater explosions[J]. Physics Today,1948,1(6):35” as the prior model for Bayesian inference, uncertainty analysis was performed on the load characterization parameters and calculation errors of the prior model in combination with sample data to obtain the target probability distribution of the load characterization parameters and calculation errors.

[0087] According to Cole's empirical formula, the pressure time history of an underwater explosion shock wave can be expressed as: p(t) = p m e -tθ (1).

[0088] In the formula, p(t) is the pressure time history of the underwater explosion shock wave; p m θ represents the peak pressure of the free-field shock wave; θ is the time constant, i.e., the peak pressure of the shock wave starts from p. m Decay to p m / e is the time used, where t is time, m is the explosive equivalent, and e is the natural constant.

[0089] Based on this pressure-time history curve, the water density ρ is known. w With the speed of sound propagation in water c w Under the given conditions, derive the impulse I and the specific energy density e of the shock wave. s :

[0090]

[0091] Where, ρ w Let c be the density of the water medium. w Let p be the sound velocity in water and p be the shock wave pressure. For the above-mentioned characterization parameter p... m , θ, I and e s Based on a large amount of experimental data, the following dimensionless empirical model was established: (4).

[0092] In the formula, the response variable y = (p m ,θ / m 1 / 3 ,I / m 1 / 3 ,e s / m 1 / 3 ), where m 1 / 3 The explosive equivalent is represented by a power of 1 / 3; the load characterization parameter k d =(k p ,k θ ,k I ,k e ) and α d =(α p ,α θ ,α I ,α e ), where k d α d These represent the first and second computational parameters of the dimensionless empirical model expression, respectively, including the computational parameter k of the pressure peak prior model. p α p The computational parameter k of the time constant prior model θ α θ The calculation parameter k of the impulse prior model I αI The calculation parameter k of the prior model for shock wave specific energy density e α e Its value depends on the properties of the explosive and the test scenario, and the load characterization parameter (k) d ,α d The calibration of the explosive directly determines the model's accuracy. Considering different types of explosives, the proportional detonation distance Z can be expressed as:

[0093]

[0094] In the formula, R is the detonation distance; m is the mass of the explosive; and NEQ is the TNT equivalent.

[0095] Taking the logarithm of equation (4) yields a linear relationship: ln y = ln k d -α d ln Z(6), where y is the response variable of the dimensionless empirical model expression, k d α d These are the first and second computational parameters of the dimensionless empirical model expression.

[0096] The load characterization parameters were fitted using sample data, and the Anderson-Darling goodness-of-fit test was used to determine the probability distribution type of the load characterization parameters.

[0097] The probability distribution types include, but are not limited to, Normal distribution, Lognormal distribution, Weibull distribution, and Gamma distribution.

[0098] The underwater explosion load characterization parameters were fitted using experimental data, and the fitting results are shown in Table 2. It can be seen that the load characterization parameters (θ, p) in the Cole empirical model... m 、I、e s Both θ and the proportional explosion distance Z exhibit a significant power-law relationship, but show varying degrees of dispersion. The greater dispersion of θ indicates that this parameter is most sensitive to changes in environmental factors.

[0099] Table 3 shows the statistical characteristics of the underwater explosion load characterization parameters. The results indicate that the values ​​of the empirical load characterization parameters vary significantly under different scenarios, with coefficients of variation ranging from 0.03 to 0.48. The coefficient of variation for the load characterization parameter in the time constant θ model is as high as 0.48. To determine the optimal probability distribution type for each load characterization parameter, the Anderson-Darling goodness-of-fit test was conducted using the Normal, Lognormal, Weibull, and Gamma distributions to examine the consistency between the sample data and the selected distribution models. The Anderson-Darling statistic represents the weighted squared distance from a point in the probability graph to the fitted line; a smaller value indicates a better fit. The goodness-of-fit test results for each load characterization parameter are shown in Table 4, with probability distribution comparisons as follows. Figure 1 As shown in Table 4, the results indicate that the Anderson-Darling statistics for each load characterization parameter are generally small, suggesting that the selected probability distributions can effectively describe parameter variability. However, significant differences exist in the probability density curves at the tails of the distributions (e.g., ...). Figure 1 As shown in Tables 2 and 4, it is advisable to choose a model with better tail-fit performance. Figure 1 Analysis shows that the load characterization parameters of TNT explosive and the fitting results of this invention closely approximate the statistical mean of the parameters. Therefore, the probabilistic calibration of the load characterization parameters can be achieved by evaluating the positional characteristics of the parameters in the probability distribution, such as the degree of deviation from the probability mean and median.

[0100] Table 2 Fitting results of characteristic parameters of underwater explosion load.

[0101]

[0102] Table 3. Statistics of parameter values ​​for empirical formulas for underwater explosion loads.

[0103]

[0104]

[0105] Table 4. Results of Anderson-Darling goodness-of-fit test for each load characterization parameter.

[0106]

[0107] Using root mean square error (RMSE) and coefficient of determination (R²) 2 The coefficient of variation (Cov) is used to evaluate the Cole dimensionless empirical model, and then the bias of the Cole dimensionless empirical model is quantified to obtain the model error (ME).

[0108]

[0109] In the formula, y e For experimental test results; y pre The result is the result of the prior model calculation; N is the number of samples.

[0110] The smaller the RMSE and Cov values, and the higher the R... 2 A value closer to 1 indicates higher model prediction accuracy. The evaluation results for each empirical model are shown in Table 5. The results show that, except for the θ model, the R-value... 2 <0.8, p m 、I、e s R of the model 2 A value >0.9 indicates relatively good deterministic accuracy, but its coefficient of variation is between 0.297 and 0.580, revealing significant random uncertainty.

[0111] Figure 2 The predictive performance of the empirical model for various characterization parameters of underwater explosion loads is demonstrated. Overall, the load characterization parameters θ and p... m , I and e s When the values ​​are relatively small, they are mainly distributed near the contour lines (red diagonal lines) between measured and calculated values. For conditions with larger θ (θ>200μs), the model calculation results are generally lower than expected; for p... m Larger operating conditions (p) m >60MPa), the model calculation results are on average too high. I and e s The model calculation results show a systematic underestimation, and the dispersion of the model calculation results gradually increases with the increase of impulse I. This is because, under the condition of small proportional distance, on the one hand, the ionization effect caused by high temperature and high pressure increases the uncertainty of various parameters of the explosion load; on the other hand, when the explosive mass is small, although the energy contained in the detonator is small, it has a large impact on the test results.

[0112] Table 5. Evaluation Results of Empirical Model for Underwater Explosion Load

[0113] Computational model θ <![CDATA[p m ]]> I <![CDATA[e s ]]> RMSE 41.13 5.37 504.92 6.86 <![CDATA[R 2 ]]> 0.75 0.84 0.92 0.95 Cov 0.42 0.22 0.57 1.03

[0114] Considering the influence of random noise, the model error between the test and calculated values ​​should exhibit a better normal distribution with a non-zero mean. To quantify the systematic bias of the model, the model error (ME) is defined as:

[0115]

[0116] In the formula, y test For experimental test values; y calThe values ​​are calculated for the model. The statistical results of ME are shown in Table 6. The results show that the mean and median of ME for each model range from 0.93 to 1.13 and 0.91 to 1.07, respectively, with coefficients of variation ranging from 0.19 to 0.38, indicating significant variability. The goodness of fit of the probability distribution of ME, evaluated by the Anderson-Darling test, is shown in Table 7. The results indicate that the load characterization parameters θ and p... m , I and e s The optimal distributions of ME are, in order, log-normal, normal, Weibull, and Gamma distributions, among which only p m The model error fits the normal distribution best, reflecting p m The model's prediction accuracy is relatively high; I and e s The ME distribution of the model shows a slight right skewness, while the ME distribution of the θ model shows a significant right skewness, meaning that the proportion of samples with ME(θ)<1 is higher, indicating that the θ model generally provides higher predicted values.

[0117] Table 6. Statistical Description of Model Errors

[0118] Model error total mean Standard deviation coefficient of variation Minimum value median Maximum value ME(θ) 593 1.08 0.40 0.38 0.29 0.99 2.71 <![CDATA[ME(p m )]]> 677 1.02 0.19 0.19 0.46 1.02 1.55 ME(I) 593 1.06 0.33 0.31 0.20 1.06 2.12 <![CDATA[ME(e s )]]> 553 1.07 0.36 0.34 0.11 1.02 2.49

[0119] Table 7. Model Error and Anderson-Darling Goodness-of-Fit Test Results

[0120]

[0121]

[0122] The proportional burst distance Z is segmented and subjected to Anderson-Darling goodness-of-fit test on the interval characteristics of ME, thereby determining the random frame in the probability distribution type.

[0123] Given the varying sample sizes for different detonation distances, Z is segmented to analyze the interval characteristics of ME. Based on the sample mean estimation theory, the minimum effective sample size is determined using the following formula:

[0124]

[0125] In the formula, n is the sample size; z α / 2 Let z be the standard differentiation number at a confidence level of 1-α. Taking a confidence level of 95%, then z α / 2 The value is taken as 1.96; σ is the sample standard deviation, which is used instead of the population sample standard value since the value of σ is unknown; E is the allowable mean estimation error at a given confidence level, which is 10% in this blast load analysis. Therefore, it can be inferred that θ and p within each proportional blast distance interval... m 、I、e sThe minimum sample sizes are 56, 14, 37, and 45, respectively. The distribution of model error ME for each load characterization parameter under different proportional blast distances is shown in [reference needed]. Figure 3 , Figure 4 As can be seen from the figure, p m The mean error of the model is closer to 1 than other models, and the coefficient of variation is relatively low, indicating that the peak pressure model has relatively high accuracy. For other parameter calculation models, the mean errors of most explosion parameters at different proportional explosion distances are greater than 1, indicating that θ, I, and e... s The model calculation results were too large, and all three models showed significant variability. However, as the proportional detonation distance increased, the mean and variability of the three models converged to a stable value.

[0126] Table 8 shows the Anderson-Darling goodness-of-fit test results for the model errors of each load under different proportional blast distances. The results show that the Anderson-Darling statistics for the model errors under normal, lognormal, Weibull, and Gamma distributions are all relatively small, indicating that each probability density function can well describe the probability distribution of each model error. Among them, the normal and Gamma distributions are more obvious. Based on the optimal probability distribution fitting results, the model error can be evaluated by judging the position of the model error in the probability distribution and its closeness to the mean and median of the probability model.

[0127] Table 8. Results of Anderson-Darling goodness-of-fit test for model errors under different proportional burst distances.

[0128]

[0129]

[0130] S3. Using the target probability distribution as prior knowledge, the Bayesian inference method is employed to update the load characterization parameters and calculation errors, resulting in a Bayesian probabilistic model of the underwater explosion shock wave load, as follows: Figure 5 As shown, the expression for the Bayesian probability model of underwater explosion shock wave load includes:

[0131] (1) The parameter k is characterized by the load of the prior model. s =(k p ,k θ ,k I ,k e ), α s =(α p ,α θ ,α I ,α eThe first Bayesian probability model is defined by the parameters of the Bayesian probability model, including the model error σ.

[0132] y k (x,Θ k )=y k,d (x,θ k )+σ k ε k .

[0133] (2) Using the model correction term parameter β k And the second Bayesian probability model, where the model error σ is the parameter of the Bayesian probability model, is obtained through the model correction term ηk(x,β). k The calculation results of the prior model are corrected.

[0134] y k (x,Θ k )=y k,d (x,θ k )+η k (x,β k )+σ k ε k (12).

[0135] In the formula, y k (x,Θ k ) represents the probabilistic model of underwater explosive load characterization parameters, where the subscript k represents different characterization parameters; y k,d (x,θ k ), η k (x,β k ) represents the empirical model and its correction terms, reflecting the model's uncertainty. The empirical model, through correction terms, can further expand and combine various factors affecting the underwater explosion load variation to increase the model's accuracy; x represents observable input random variables, such as explosive mass, explosion location information, explosive properties, and measurement errors, reflecting the uncertainty of the model's input parameters; Θ k =(θ k ,β k ,σ k ) represents the unknown parameters of the probability model; σ k ε k This represents the computational error of the corrected probability model, comprehensively reflecting the model's uncertainty from the perspectives of computational model, input parameters, and experimental testing. σ represents this error. k ε represents the model standard deviation. k Let θ be a random variable that follows a standard normal distribution. k Representing the empirical model y k,d The calculation parameter k d α d ;β k η represents the correction term in the second Bayesian probability model.k The calculation parameters.

[0136] To simplify the subsequent analysis, we assume here that the model variance σ k 2 The variables x and Θ are independent and linearly unrelated, meaning that for a given Bayesian probability model parameter Θ... k The variance of the probability model is Var[P(x,Θ]]. k )]=σ k 2 .

[0137] For the model correction term η k (x,β k The determination of ) can be based on engineering experience and mechanism analysis, using a series of basic elementary functions h. i It can be represented by a linear combination of (x):

[0138]

[0139] Where, β k,i η represents the correction term in the second Bayesian probability model. k The i-th calculated coefficient of the calculated parameter, h k,i Here, x is the "interpretation" function for the correction terms of the shock wave load-related parameters, and x is the variable of the "interpretation" function.

[0140] Bayesian probabilistic model parameter update: based on the observed dataset y k Update the posterior distribution of the parameters of the Bayesian probability model using Bayes' theorem:

[0141]

[0142] In the formula, π(Θ) k The load characterization parameters are represented by y, reflecting the prior knowledge of the overall parameter distribution; k y represents the actual obtained evidence sample vector. k =[y1,y2,…,y n ];Θ k For unknown parameters in the probabilistic model; π(Θ) k |y k L(y) represents the posterior distribution of the parameters of the Bayesian probability model, reflecting the update of the prior after obtaining evidence sample information; k |Θ k L(y) is the likelihood function, used to quantify the consistency between the model and the data. k |Θ k )=L(y1,y2,…,y n ,Θ k); c is the definite integral constant, representing the regularization factor. When the evidence sample information comes from n independent trial sets, equation (14) can be further expressed as:

[0143]

[0144] Where i is the i-th data sample point of the n-th independent trial dataset, n is the number of independent trial datasets, and π is the posterior distribution function.

[0145] Based on equation (12), the model calculation error is defined as follows:

[0146] r k (x,Θ k )=y k (x,Θ k )-y k,d (x,θ k )-η k (x,β k (16).

[0147] Where, r k For model calculation error, y k,d An empirical model for underwater explosion shock wave load.

[0148] Assuming the model's computational error follows a Gaussian normal distribution, the likelihood function is:

[0149]

[0150] In the formula, Σ k Calculate the covariance matrix of the error for the model. Assuming the parameters of the Bayesian probabilistic model are independent, the prior distribution of the Bayesian probabilistic model parameters can adopt a non-information prior distribution, i.e., taking π(θ). k )∝1,π(β k )∝1,π(σ k )∝1 / σ k The prior distribution can be expressed as:

[0151]

[0152] Because high-dimensional integrals with a definite integral constant *c* are difficult to solve analytically, this invention utilizes Markov Chain Monte Carlo (MCMC) conditional sampling algorithms to calculate the posterior distribution of Bayesian probability model parameters, such as Gibbs sampling, MH sampling, and delayed rejection adaptive MH sampling. After updating the Bayesian probability model parameters, a feature of the posterior distribution of the parameters is typically selected as its estimate, i.e.:

[0153]

[0154] Or further, select a posterior distribution sample Θk,i ~π(Θ) k |y k The mean and variance of the shock wave load can be expressed as follows:

[0155]

[0156] Bayesian probability model verification and analysis:

[0157] (1) Selection of parameters for the probabilistic Bayesian probability model.

[0158] Given the significant uncertainty in the parameters of the prior Bayesian probability model, using deterministic parameters for calculation and analysis would increase the bias of the results. Therefore, this invention employs two strategies for selecting Bayesian probability model parameters:

[0159] Model I: (1) The first Bayesian probability model with the prior model load characterization parameters ks=(kp,kθ,kI,ke), αs=(αp,αθ,αI,αe) and model error σ as Bayesian probability model parameters.

[0160] Model II: Using the model correction term parameter β k And the second Bayesian probability model, where the model error σ is the parameter of the Bayesian probability model, is obtained through the model correction term ηk(x,β). k The calculation results of the prior model are corrected.

[0161] For the model correction term η k (x,β k The determination of ) can be based on engineering experience and mechanism analysis, using a series of basic elementary functions h. i The effect of underwater explosions is represented by a linear combination of (x). Existing research indicates that factors influencing underwater explosion effects include: medium properties (atmospheric pressure, water density, sound velocity in water), explosive properties (charge shape, explosive mass, density, detonation velocity, heat of explosion), and explosion location (detonation distance, detonation depth). Based on this understanding, and combined with dimensional analysis, the "interpretation" function h(x) for the correction terms of shock wave load-related parameters can be selected as:

[0162] h(x) = [R / r0, ρ w / ρ e ,c w / c e Q e / Q TNT (19).

[0163] In the formula, ρ e c e Q e And r0 represents the explosive density, detonation velocity, heat of explosion, and equivalent radius; Q TNTp0 represents the thermal induction of TNT; p0 is atmospheric pressure. For specific explosives and fluid media, ρ... e c e Q e and Q TNT Since this has been determined, the "interpretation" function introduces a constant term, and h(x) is further simplified to:

[0164] h(x) = [1, R / r0] (20).

[0165] Considering the influence of higher-order terms, the model correction terms can be expressed as follows:

[0166]

[0167] (2) Parameter update results of Bayesian probability model.

[0168] Tables 9 and 10 compare the parameter estimates and performance evaluations of the probabilistic models under different sample sizes. Table 9 shows that Model I demonstrates that Bayesian update has a sample efficiency advantage; as the sample size gradually increases from 8% to 100%, θ and p... m 、I、e s The posterior variance of the Bayesian probability model parameters systematically shrinks, consistent with the theoretical expectation of Bayesian updates, θ, p m The model achieves excellent RMSE levels at both 16% and 60% sample sizes, a characteristic that offers significant engineering application value, as it balances experimental cost and efficiency while ensuring a certain level of model accuracy. (I, e) s The variance of the Bayesian probability model parameters converges the fastest, but the RMSE changes less, reflecting I, e s The parameters of the energy model are less affected by the sample size. Comparing the data in Tables 9 and 10, the calculation results of Model II show that θ, p m 、I、e s The posterior variance of the Bayesian probability model parameters also shows a systematic contraction, but convergence is slow, and the RMSE is not sensitive to changes in sample size. Therefore, it is reflected that Model I has lower parameter dispersion and is more suitable for working conditions where the explosive properties are known and the environment is simple, while Model II quantifies the influence of the "explanation" function through the β coefficient and is more suitable for more complex environments.

[0169] Figures 6-13 The evolution characteristics of the posterior distribution of the parameters in the probabilistic Bayesian model are presented. The evolution shows that with a small sample size (<20%), the posterior distribution exhibits a wide-tailed, multi-modal characteristic, while with a full sample size (100%), the distribution converges to a compact, unimodal distribution. This indicates that the degree of agreement between the posterior distribution and the actual situation increases with the increase in the measured sample size. Based on the data in Tables 9 and 10, in the Bayesian probabilistic modeling process, sampling can be stopped when the sample size reaches 40-60%, achieving the optimal cost point.

[0170] Table 11 Figure 14 A comprehensive comparison of the computational accuracy of the prior model and the probabilistic model reveals that both types of probabilistic models based on the Bayesian method exhibit high reliability. Specifically, as the sample size increases, the parameters of Model I show less dispersion compared to the uncertain parameters of Model II, indicating lower parameter uncertainty in Model I. However, Model 2, by introducing the interpretable term h(x), effectively considers the effects of explosive density, shape, detonation velocity, heat of explosion, and equivalent yield, factors not considered in the prior model and Model I, thus significantly improving the physical interpretability of the model.

[0171] Table 9. Mean, standard deviation, and model evaluation of each load characterization parameter in Model I.

[0172]

[0173] Table 10 Mean, standard deviation and model evaluation of each load characterization parameter in Model II

[0174]

[0175]

[0176] Table 11 Evaluation of Model Calculation Results

[0177]

[0178] Finally, it should be noted that the above embodiments are only used to illustrate the technical methods of the present invention and not to limit them. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the technical methods of the present invention, and these modifications or equivalent substitutions cannot cause the modified technical methods to deviate from the spirit and scope of the technical methods of the present invention.

Claims

1. A method for probabilistic characterization of underwater explosion shock wave loading based on Bayesian inference, characterized in that, The method comprises the following steps: S1, obtaining past test data, and analyzing and processing the experimental data to obtain sample data; S2, taking the Cole empirical model as a prior model for Bayesian inference, and combining the sample data to perform uncertainty analysis on the load characteristic parameters and the calculation error of the prior model to obtain target probability distributions of the load characteristic parameters and the calculation error; S3, taking the target probability distributions as prior knowledge, and performing updating calculation on the load characteristic parameters and the calculation error by using the Bayesian inference method to obtain a Bayesian probability model of the underwater explosion shock wave load; S4, performing prediction calculation on the underwater explosion shock wave load based on the Bayesian probability model of the underwater explosion shock wave load.

2. The method of claim 1, wherein, The specific content of taking the Cole empirical model as the prior model for Bayesian inference in S2 comprises: obtaining an underwater explosion shock wave pressure time history expression based on the Cole empirical formula, and further deriving load characteristic parameters of the underwater explosion shock wave; The load characterizing parameters include a free field shock wave pressure peak value p m , a time constant θ, an impulse I, and a shock wave specific energy density e s ; establishing a dimensionless empirical model for the load characteristic parameters.

3. The method of probabilistic characterization of underwater explosion shock wave load based on Bayesian inference according to claim 2, characterized in that, The underwater explosion shock wave pressure time history expression is: p(t) = p m e -tθ ; where p(t) is the underwater blast pressure time history; p m is the free-field blast pressure peak; θ is the time constant, i.e., the time it takes for the blast peak to decay from p m to p m / e, t is time, m is the explosive mass, and e is the natural constant. Based on the time history expression of underwater explosion shock wave pressure, the water density ρ is introduced. w With the speed of sound propagation in water c w The impulse I and the shock wave specific energy density e are derived. s .

4. The method according to claim 3, wherein, The expression of the dimensionless empirical model is: where the response variable y = (p m , θ / m 1 / 3 , I / m 1 / 3 , e s / m 1 / 3 ), where m 1 / 3 represents 1 / 3 power of the explosive equivalent; Load characterization parameter k d = (k p , k θ , k I , k e ), a d = (a p , a θ , a I , a e ), wherein k d , a d represent first and second calculation parameters of a dimensionless empirical model expression, respectively, including calculation parameters k p , a p of a pressure peak prior model, calculation parameters k θ , a θ of a time constant prior model, calculation parameters k I , a I of a momentum prior model, and calculation parameters k e , a e of a shock wave specific energy density prior model. wherein, the proportional stand-off distance Z is expressed as: In the formula, R is the stand-off distance; m is the mass of explosive; and NEQ is the TNT equivalent.

5. The method of claim 1, wherein, The specific content of combining the sample data to perform uncertainty analysis on the load characteristic parameters and the calculation error of the prior model to obtain target probability distributions of the load characteristic parameters and the calculation error in S2 comprises: presetting selectable probability distribution types, and the selectable probability distribution types comprise normal distribution Normal, lognormal distribution Lognormal, Weibull distribution and Gamma distribution; fitting the load characteristic parameters by using the sample data to obtain a power-law relationship characteristic of the load characteristic parameters and the proportional stand-off distance Z; performing goodness-of-fit test on the load characteristic parameters and the selectable probability distribution types by using Anderson-Darling statistical values to obtain goodness-of-fit test results; determining the target probability distribution of the load characteristic parameters according to the power-law relationship characteristic and the goodness-of-fit test results; The Cole dimensionless empirical model is evaluated by root mean square error (RMSE), determination coefficient (R 2 and coefficient of variation (Cov) to quantify the model error (ME) of the Cole dimensionless empirical model. dividing the proportional stand-off distance Z into segments to perform Anderson-Darling goodness-of-fit test on the interval characteristics of ME, and then determining the target probability distribution of the calculation error in the selectable probability distribution types.

6. The method of probabilistic characterization of underwater explosion shock wave load based on Bayesian inference according to claim 5, characterized in that, The expression of the model error ME is: where y test is the test value; y cal is the model calculated value.

7. The method of claim 1, wherein, The expression of the Bayesian probability model of the underwater explosion shock wave load comprises: (1) the prior model load table characteristic parameter k s = (k p , k θ , k I , k e ), a s = (a p , a θ , a I , a e ) and the model error σ are the first Bayesian probability model of the Bayesian probability model parameters; y k (x,Θ k )=y k,d (x,θ k )+σ k ε k ; (2) a second Bayesian probability model taking model correction term parameter β k and model error σ as Bayesian probability model parameters, and correcting the prior model calculation result by model correction term η k (x, β k ) y k (x,Θ k )=y k,d (x,θ k )+η k (x,β k )+σ k ε k ; where y k (x,Θ k ) is the probability model of the characteristic parameters of underwater explosion load, subscript k represents different characteristic parameters; y k,d (x,θ k ), η k (x,β k ) are the empirical model and its correction term, respectively; x is the observed input random variable; Θ k =(θ k ,β k ,σ k ) is the unknown parameter of the probability model; σ k ε k represents the calculation error of the corrected probability model; wherein σ k is the model standard deviation, ε k is a random variable obeying the standard normal distribution, θ k represents the calculation parameters k k,d , α d of the empirical model y d ; β k represents the calculation parameters of the correction term η k of the second Bayesian probability model. Definition of model variance σ k 2 are mutually independent and linearly uncorrelated, i.e. for a given load characterization parameter Θ k , the variance of the probability model is Var[P(x, Θ k )] = σ k 2 ; The model correction term η k (x, β k ) is expressed as: where β k,i represents the second Bayesian probability model correction term η k is the i-th calculation coefficient of the calculation parameter of the first Bayesian probability model correction term h k,i is the "explanation" function of the shock wave load related parameter correction term, and x is the variable of the "explanation" function.

8. The method of claim 7, wherein, Based on the observation dataset y k The posterior distribution of the Bayesian probability model parameters is updated by the Bayesian theorem, and the expression of the posterior distribution of the Bayesian probability model parameters is: where π(Θ k |y k ) is the Bayesian probability model parameter prior distribution; y k is the actual acquired evidence sample vector, y k = [y1, y2, …, y n ]; Θ k is the probability model unknown parameter; π(Θ k |y k ) is the Bayesian probability model parameter posterior distribution; L(y k |Θ k ) is the likelihood function, quantifying model consistency with data, L(y k |Θ k ) = L(y1, y2, …, y n , Θ k ); c is a definite integral constant, representing a regularization factor.

9. The method of claim 8, wherein, When the evidence sample information comes from n independent test sets, the expression of the posterior distribution of the Bayesian probability model parameters is: wherein, i is the i th data sample point of the n th independent test data set, n is the number of independent test data sets, and π is a posterior distribution function; according to the Bayesian probability model of the underwater explosion shock wave load, the model calculation error is defined, and the expression of the model calculation error is: r k (x, Θ k ) = y k (x, Θ k ) - y k,d (x, Θ k ) - η k (x, β k ); where r k is the model error, y k,d is the empirical model for underwater blast shock wave loading.

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