Prefabricated polyurethane directly-buried thermal insulation pipe limit state prediction method based on depth operator network architecture

Abstract

The invention provides a method for predicting the limit state of a prefabricated polyurethane directly-buried thermal insulation pipe based on a depth operator network architecture, and the method comprises the steps: firstly constructing a simulated use environment, carrying out the multi-batch limit stress test of polyurethane directly-buried pipes of different specifications under a multi-environment condition, recording the corresponding specifications and environment data as the input, and taking the maximum stress as the output, and carrying out the prediction of the limit state of the polyurethane directly-buried thermal insulation pipe. Constructing a data set; designing a feature screening module to remove features with low relevancy; thirdly, designing and constructing a limit state prediction model based on a depth operator network architecture; carrying out optimization training on the model by applying reinforcement learning strategy design; and deploying the trained prediction model to a related environment, and inputting the specification and environmental parameters of the thermal insulation pipe in a computer terminal to obtain the limit stress condition of the thermal insulation pipe. According to the method, redundant features introduced into the initial features can be eliminated, the model architecture is further optimized, the complexity of model training is reduced, and the design of the prediction model ensures that the ultimate stress of the polyurethane directly-buried thermal insulation pipe can be accurately predicted under various working conditions.

Description

Technical Field

[0001] The present invention belongs to the technical field of predicting the ultimate state of prefabricated polyurethane directly buried insulation pipes based on machine learning, and particularly relates to a method for predicting the ultimate state of prefabricated polyurethane directly buried insulation pipes based on a deep operator network architecture. Background Art

[0002] Polyurethane directly buried insulation pipes have excellent heat insulation performance, which can greatly reduce the heat loss during transportation, effectively save energy, and can be achieved whether it is for hot water, steam transportation or maintaining a low-temperature environment. Their characteristics of waterproofing, moisture-proofing and anti-corrosion protection can protect the pipeline from external erosion, extend its service life and reduce maintenance costs. At the same time, it is convenient for construction, adapts to complex terrains, can be installed quickly, and also has a certain structural support capacity to withstand various external forces. In addition, the flame retardant performance and alarm function of some products provide guarantee for safe operation, meet the requirements of the era of energy conservation and environmental protection, and play an irreplaceable role in the pipeline transportation systems in many fields. However, during the actual operation of polyurethane directly buried insulation pipes, they are affected by external forces such as soil pressure and vehicle loads, as well as thermal stresses caused by the transportation of hot media. Stress analysis can accurately calculate the stress distribution, avoid structural failures such as pipeline rupture, deformation and interface loosening caused by excessive stress, and ensure the structural safety of the pipeline; at the same time, it can also prevent the stress from being transmitted to the insulation layer, avoid damage to the insulation layer, and maintain the insulation performance and structural integrity. In addition, through stress analysis, design parameters can be optimized, fatigue life can be predicted, the service life of the pipeline can be extended, maintenance and replacement costs can be reduced, and the requirements of engineering specifications and standards can be met, ensuring public safety and ensuring the smooth acceptance and commissioning of the project. Therefore, ultimate stress analysis is of great significance in the specific use process of polyurethane insulation pipes.

[0003] At present, the analysis and prediction of the ultimate state of polyurethane directly buried pipes mainly focus on the following methods:

[0004] (1) Analytical method: The analytical method is a method based on theories such as elastic mechanics and material mechanics. By establishing an accurate mathematical model and using mathematical formulas and derivations to solve for stress. For some polyurethane directly buried insulation pipe structures with simple geometric shapes and clear boundary conditions, this method can be used to directly obtain the analytical expression of stress. For example, for a straight pipe with a constant cross-section under uniform temperature change and axial force, according to the theory of thermoelasticity, by establishing equilibrium equations, geometric equations and physical equations, the stress distribution formula inside the pipe can be derived, which can accurately give the stress values at each point of the pipe and provide a theoretical basis for stress analysis. However, for cases with complex shapes and boundary conditions, it is difficult or even impossible to solve by the analytical method;

[0005] (2) Numerical analysis method: Numerical analysis method is to use computer technology to discretize the actual structure of polyurethane direct-buried thermal insulation pipe into a finite number of units and nodes, perform mechanical analysis on each unit, and then solve the stress by assembling the overall equation set. Among them, the finite element method is one of the most commonly used numerical analysis methods. It divides the pipeline structure into a large number of small units, such as tetrahedral units, hexahedral units, etc., establishes a stiffness matrix for each unit, and then solves the displacement field of the entire structure according to the boundary conditions and load conditions, and then calculates the stress field through physical equations. This method can handle various complex geometric shapes, material properties and boundary conditions, and can accurately simulate the stress distribution of pipelines under different working conditions. It is widely used in stress analysis of polyurethane direct-buried thermal insulation pipes, but it requires professional software and certain computing resources, and has high requirements for model establishment and parameter setting.

[0006] (3) Experimental stress analysis method: The experimental stress analysis method is a method of determining stress distribution by directly measuring stress or physical quantities related to stress through experiments on actual polyurethane direct buried insulation pipes or their models. The most common method is the resistance strain gauge method, in which the resistance strain gauge is pasted on the surface of the pipe. When the pipe is deformed by force, the resistance value of the strain gauge will change. By measuring the resistance change and calculating the strain on the pipe surface based on the relationship between strain and resistance, the stress is obtained based on the constitutive relationship of the material. There is also the photoelastic method, in which a pipe model is made of photoelastic material. Under loading conditions, the stress distribution in the model is analyzed and calculated by observing the stripe pattern that appears in the photoelastic model, and the stress state of the actual pipe is inferred. The experimental stress analysis method can directly obtain the stress data of the pipe. The results are intuitive and reliable, and can be used to verify the results of numerical analysis and analytical methods. However, the experimental cost is high, and due to the limitations of experimental conditions and measurement technology, there are certain difficulties in measuring stress under some internal stresses and complex working conditions. Summary of the invention

[0007] In view of the above problems, the first aspect of the present invention proposes a method for constructing a limit state prediction model of a prefabricated polyurethane direct-buried thermal insulation pipe based on a deep operator network architecture, comprising the following process:

[0008] Step 1: Based on the use environment simulation, carry out the ultimate stress test on the direct buried pipes of different specifications to collect the maximum stress they can withstand under such conditions, record the maximum stress and the corresponding specification data and environmental data, and construct the initial data set;

[0009] Step 2: Based on the feature screening module of the relieff algorithm, the features with low correlation with the ultimate stress of the polyurethane direct-buried thermal insulation pipe are eliminated, the specification data and environmental data with the highest feature importance are obtained, and the training data set is constructed in combination with the corresponding maximum stress;

[0010] Step 3: Construct a limit state prediction model based on the deep operator network architecture, including a backbone network architecture, a branch network architecture, and a fusion output module. The backbone network is used to extract the physical property features of the directly buried insulation pipe. The branch network is used to extract the environmental features of the directly buried insulation pipe and perform weighted processing on the feature channels in the environmental features based on the self-attention mechanism to obtain self-attention environmental features. The fusion output module is used to adaptively fuse the two types of features output by the two network architectures and output the maximum stress prediction result based on the fused features.

[0011] Step 4: Optimize and train the constructed limit state prediction model based on the reinforcement learning strategy to obtain the final limit state prediction model.

[0012] Preferably, the specific data processing process of the feature screening module based on the relieff algorithm in Step 2 is as follows:

[0013] S21: Data selection and classification: Based on the initial dataset Data of the limit state of the polyurethane directly buried insulation pipe, randomly select M sample data from it, and group the M sample data according to the data similarity SDU. The number of groups is L.

[0014] S22: Feature difference calculation: Calculate the feature difference TZC corresponding to each feature in the same group of samples and the feature difference BZC in different groups of samples respectively.

[0015] S23: Feature correlation calculation: Calculate the feature correlation Tmax based on the calculated difference TZC and BZC of each feature.

[0016] S24: Repeatedly calculate the feature correlation: Repeat the process from S21 to S23 for a total of B iterations. Average and weight the feature correlation t∈[1, B], and define the result TO as the final feature importance. The higher the feature importance, the stronger the connection degree between this feature and the ultimate stress of the polyurethane directly buried insulation pipe.

[0017] S25: Feature screening process: Based on the calculated feature importance of each feature, discard the redundant features with the feature importance in the last 40%.

[0018] Preferably, in S21,

[0019]

[0020] where Datayz j represents the j-th parameter value of a piece of data in the dataset Data, It represents the j-th parameter value of another piece of data in the dataset Data, where j ∈ [1, C]; C is the total number of environmental data, specification data, and maximum stress data.

[0021] The specific process of grouping is as follows: Randomly select one piece of data from M pieces of sample data to start grouping. Then, calculate the SDU values of all the remaining data and this piece of data in turn. After the calculation, combine the piece of data with the largest SDU value with this piece of data to form a data group; After the grouped data is removed, continue to perform the above-mentioned data classification on the remaining data until the data grouping is completed.

[0022] In S22, calculate the feature difference degree TZC corresponding to each feature in the same-group samples and the feature difference degree BZC in different-group samples respectively. The formulas are as follows:

[0023]

[0024] Among them, It is the parameter value of the k-th feature of the w-th and m-th data in the same-group sample data, where T is the sum of environmental data and specification data; and They respectively represent the parameter values of the k-th feature of the g-th and l-th data in different-group sample data, |*| represents the absolute value operation function.

[0025] Preferably, the backbone network architecture consists of three parts: an input layer, an intermediate layer, and an output layer; Select the specification feature parameters of the polyurethane directly buried insulation pipe as the input of this backbone network architecture. After feature screening, the number of retained polyurethane directly buried insulation pipe specification feature parameters is p. Therefore, the number of neurons in the input layer is p; The network architecture of the intermediate layer is mainly composed of three parts: a fully connected layer, a Relu activation function, and a Silu activation function. In the intermediate layer, there are first Wq fully connected layers, and each fully connected layer has Ws neurons. Then, it is activated by the Relu function to enhance the non-linear expression ability of the model. Finally, there are Wh fully connected layers, and each fully connected layer has Wf neurons in each layer and the Silu activation function; The designed output layer contains 4 different neurons. Then, through the spatial attention layer, the outputs of the 4 neurons in the hidden layer are adaptively fused to obtain the output Feaz of the backbone network architecture.

[0026] Preferably, a stacking structure is adopted in the branch network architecture; the stacking structure is composed of two parallel network paths, and each network path is composed of an input layer, an intermediate layer, and an output layer; the environmental characteristic parameters are used as the input of this branch network architecture. There are a total of q neurons in the input layer, and q is the total number of environmental characteristic data retained after screening; the structure of the intermediate layer is as follows: first, there are three sequentially connected fully connected layers, and then the data is non-linearly activated through the sigmoid activation function, and then there are four sequentially connected fully connected layers. In these four fully connected layers, finally, non-linear normalization processing is performed through the Softmax activation function; the number of neurons in the output layer of the network path is set to 1; the output FZS of the output layer in each network path is sent to the self-attention mechanism unit for processing to obtain the output result Fe, and then a fully connected layer is used to further integrate the outputs of the two network paths to obtain the output Feaf of the branch network architecture.

[0027] Preferably, the self-attention mechanism unit maps the output FZS of each network path in the branch network architecture to multiple feature spaces. The self-attention mechanism includes L parallel self-attention taps, and then the output of the L attention taps is integrated through the channel attention layer to obtain the output result Fe of the final self-attention mechanism; the feature processing process of each self-attention tap is as follows:

[0028]

[0029] Among them, Softmax(*) represents the Softmax linear normalization function, and Fea is the output result of the self-attention tap; the matrices Q, K, and V are the query, key, and value matrices mapped after the feature FZS is processed by three different matrices. Specifically:

[0030] Q = W Q *FZS

[0031] K = W K *FZS

[0032] V = W V *FZS

[0033] Among them, W Q ,W K ,W V are three different feature mapping matrices to process the feature FZS in three different feature spaces.

[0034] Preferably, the fusion output module further fuses the output Feaz obtained from the backbone network architecture and the output Feaf obtained from the branch network architecture to obtain the predicted result Yout of the ultimate stress of the prefabricated polyurethane directly buried thermal insulation pipe. First, the BN normalization layer is used to normalize Feaz and Feaf respectively to optimize the distribution of the feature channels and enhance the robustness of the model when outputting the predicted result of the ultimate stress of the prefabricated polyurethane directly buried thermal insulation pipe, obtaining and Secondly, the bilinear interpolation upsampling layer is used to upsample to the same size; Thirdly, the dot product layer is used to perform a dot product operation on the upsampled and to obtain the fused feature Ffz; Thirdly, the fused feature is input into the Dropout layer to optimize the feature channel parameters in Ffz to obtain the feature Ffz* with frozen parameters, reducing the risk of model overfitting; Finally, Ffz* is input into the L2 regularization layer and then into the Relu activation function to obtain the final predicted result Yout of the ultimate stress of the prefabricated polyurethane directly buried thermal insulation pipe, where the L2 regularization layer constrains the model weights and enhances the robustness of the model during prediction.

[0035] Preferably, the reinforcement learning strategy comprehensively optimizes the number of fully connected layers in the intermediate layer of the model backbone network and the number of neurons Ws, Wf contained in each fully connected layer Wq, Wh to find the specific number of fully connected layers and the number of neurons in each layer corresponding to the best model performance;

[0036] The vector composed of the four quantity combinations of the number of fully connected layers Wq, Wh and the corresponding number of neurons Ws, Wf is used as the state space corresponding to this reinforcement learning; in its action space, it adaptively optimizes and changes the parameters Wq, Wh, Ws, Wf to find the model parameters corresponding to the best model performance; the design of its reward factor is realized by comparing the magnitudes of the loss functions. Using the constructed training dataset, based on the model parameters corresponding in the state space, the loss function value Lossq of the model at this time is obtained; when the state is updated, that is, an action to update the model parameters is taken, and then the loss function value Lossh corresponding to the model after the parameter update is calculated; if the value of Qjl is greater than 0, a positive reward value of 2*Qjl is given to this action, and if the value of Qjl is less than 0, a negative reward value of 2*Qjl is given to this action:

[0037] Qjl = Lossh - Lossq

[0038] Meanwhile, a reinforcement learning network is constructed. A policy network is constructed to optimize the parameters Wq, Wh, Ws, and Wf, and a value network is constructed to adaptively judge the value corresponding to this set of model parameters. The designed policy network mainly consists of 4 interconnected fully connected layers and a Relu activation function, and the designed value network mainly consists of 6 interconnected fully connected layers and a Silu activation function. The input of its network architecture is a set of four-dimensional vectors, representing a corresponding situation of Wq, Wh, Ws, and Wf. The constructed dataset is used and the prediction model constructed with the input of this architecture at this time is trained, and the loss function after the training process is recorded to calculate the reward factor.

[0039] In the second aspect of the present invention, a method for predicting the ultimate state of a prefabricated polyurethane directly buried insulation pipe based on a deep operator network architecture is proposed. The ultimate state prediction model constructed by the construction method described in the first aspect is deployed in a computer terminal, and the following process is included:

[0040] The specification data and environmental data of the polyurethane directly buried insulation pipe are obtained in real time and standardized into model input data;

[0041] The standardized specification data and environmental data are input into the ultimate state prediction model; the ultimate stress result of the polyurethane directly buried insulation pipe of this specification under this environmental condition is output.

[0042] Preferably, the specification characteristic parameters of the polyurethane directly buried insulation pipe reserved for prediction after screening are: the steel pipe layer thickness GT of the insulation pipe, the insulation layer thickness WT of the insulation pipe, the diameter BD of the insulation pipe, the compressive strength BQ of the insulation pipe, the thermal conductivity BR of the insulation pipe, and the water absorption rate BH of the insulation pipe; the reserved environmental characteristic parameters are the environmental temperature TM, the buried depth SD of the directly buried pipe, the surrounding soil density TD, and the soil friction coefficient TS.

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

[0044] (1) Feature screening module based on the relieff algorithm: A feature screening module is designed based on the relieff algorithm. By screening the importance of features in the dataset, only the features with a relatively high correlation with the ultimate stress are retained. According to the design of this feature screening module, the redundant features introduced in the initial features can be eliminated, thereby further optimizing the model architecture, reducing the complexity of model training, enhancing the deep fitting ability of the model, optimizing the complexity of feature selection when predicting the ultimate state of the polyurethane directly buried insulation pipe, and enhancing the prediction ability of the model;

[0045] (2) Establishing a limit state prediction model for directly buried polyurethane insulation pipes based on the deep operator network architecture: Based on the deep fitting ability of the deep operator network architecture, the present invention designs a limit state prediction model for directly buried polyurethane insulation pipes based on this architecture, and introduces a self-attention mechanism therein to further improve the generalization ability and deep fitting ability of the model. Ensure that accurate prediction of the ultimate stress of directly buried polyurethane insulation pipes can be achieved under various working conditions;

[0046] (3) Construction of a model optimization module based on reinforcement learning: Based on the reinforcement learning strategy, a model parameter architecture optimization module is constructed, and a loss function, state, action, and reward factor are designed to find the model architecture parameters corresponding to the best model performance. Based on the design of this module, the fitting performance of the model can be further optimized, and the accuracy and robustness of the prediction of the ultimate stress of directly buried polyurethane insulation pipes can be enhanced. Brief Description of the Drawings

[0047] In order to more clearly illustrate the technical solutions of the present invention or the prior art, the following will briefly introduce the drawings required for the description of the embodiments or the prior art. Obviously, the following description is only one embodiment of the present invention. For those of ordinary skill in the art, other drawings can be obtained based on these drawings without creative efforts.

[0048] Figure 1 It is the overall process logic block diagram of the present invention;

[0049] Figure 2 It is the operation flow chart of feature screening of the present invention;

[0050] Figure 3 It is the network structure diagram of the limit state prediction model of the present invention;

[0051] Figure 4 It is the network structure diagram of the fusion output module of the present invention;

[0052] Figure 5 It is the comparison diagram of the experimental results of the model performance of the present invention. Detailed Embodiments

[0053] The overall process of the present invention is as Figure 1 shown:

[0054] Construction of the ultimate state prediction data set for polyurethane direct-buried thermal insulation pipes: A simulated use environment for prefabricated polyurethane direct-buried pipes is constructed in a laboratory scenario, and ultimate stress tests are carried out on polyurethane direct-buried pipes of different specifications under different environmental conditions to collect the maximum stress that the polyurethane direct-buried pipes can withstand under such conditions. The corresponding polyurethane direct-buried pipe specification data and the environmental data during the test are recorded as the input data of the model, and the maximum stress it can withstand is recorded as the output data of the model. In order to ensure the comprehensiveness of data collection, multiple batches of ultimate stress tests under multiple specifications and multiple environmental conditions are carried out to complete the construction of the ultimate state prediction data set for polyurethane direct-buried thermal insulation pipes;

[0055] Construction of feature screening module based on relieff algorithm: In the data set construction stage, in order not to miss out the features with high correlation with the ultimate stress of polyurethane direct-buried thermal insulation pipe, some redundant features will inevitably be introduced. These redundant features will increase the complexity of the model and the difficulty of model training, making it difficult to successfully fit the model. In order to eliminate the adverse effects of redundant features, the present invention designs a feature screening model based on relieff algorithm to eliminate features with low correlation with the ultimate stress of polyurethane direct-buried thermal insulation pipe;

[0056] Construction of the limit state prediction model of prefabricated polyurethane direct buried thermal insulation pipe based on the deep operator network architecture: Since the deep operator network architecture can approximate nonlinear continuous operators with high accuracy, it can maintain a small prediction error in the field of limit state prediction. Therefore, the present invention constructs a limit state prediction model of prefabricated polyurethane direct buried thermal insulation pipe based on the deep operator network architecture. In order to further improve the performance of the model, a self-attention mechanism is designed inside the model;

[0057] Construction of model architecture optimization module based on reinforcement learning: Based on the reinforcement learning strategy, a parameter optimization module for the model parameters of the limit state prediction model of prefabricated polyurethane direct buried thermal insulation pipe is designed. The network architecture, state, action and reward factor used in reinforcement learning are designed in detail to obtain the optimized limit state prediction model of polyurethane direct buried thermal insulation pipe;

[0058] Model deployment and use: The hardware equipment required by the present invention and the corresponding polyurethane direct buried thermal insulation pipe limit state prediction model are deployed in the relevant use environment. The specification parameter conditions of the thermal insulation pipe and the relevant environmental specification parameter conditions are input on the computer terminal to obtain the limit stress of the polyurethane prefabricated direct buried thermal insulation pipe under such environmental conditions, so as to achieve the effect of improving the quality of the project.

[0059] The method of the present invention is further described below with reference to specific embodiments.

[0060] 1. Dataset Construction

[0061] The present invention aims to predict the ultimate stress of directly buried polyurethane insulation pipes under real usage scenarios. Accurate training data is a prerequisite for ensuring the prediction accuracy. To this end, the present invention constructs an experimental scenario in the laboratory to simulate the actual use of directly buried polyurethane insulation pipes, and conducts multiple batches of ultimate stress tests on directly buried polyurethane insulation pipes of different specifications under different experimental conditions to complete the construction of the ultimate state prediction data set of directly buried polyurethane insulation pipes.

[0062] Selection of specifications of directly buried polyurethane insulation pipes: To ensure the comprehensiveness of data collection and not miss the features highly correlated with the ultimate stress of polyurethane insulation pipes. The present invention selects a variety of directly buried polyurethane insulation pipes of different specifications for experiments. The specification parameters selected in this embodiment include: the thickness GT of the steel pipe layer of the insulation pipe, the thickness WT of the insulation layer of the insulation pipe, the diameter BD of the insulation pipe, the width and length BW, BC of the insulation pipe, the average density BM of the insulation pipe, the compressive strength BQ of the insulation pipe, the thermal conductivity BR of the insulation pipe, and the water absorption rate BH of the insulation pipe;

[0063] Selection of experimental environment parameters: The ultimate stress of directly buried polyurethane insulation pipes is not only related to the self-specification parameters of the insulation pipes, but also greatly affected by their usage environment. Therefore, the experimental environment parameters selected in this embodiment are the environmental temperature TM, the buried depth SD of the directly buried pipe, the surrounding soil density TD, the soil friction coefficient TS, the soil moisture TC, the soil pH value TJ, the soil porosity TK, and the soil bulk density TR.

[0064] Collection of ultimate stress data: Based on the described specification parameters and experimental environment parameters of directly buried polyurethane insulation pipes. Conduct ultimate stress tests on directly buried polyurethane insulation pipes under a specific set of insulation pipe specification parameters and experimental environment parameters, and record their ultimate stress value JY. Record the specification parameters GE of the directly buried polyurethane insulation pipe used in this set of tests and the environmental parameters HC. According to the above data records, obtain a complete set of data Datayz;

[0065] GE = [GT, WT, BD, BW, BC, BM, BQ, BR, BH]

[0066] HC = [TM, SD, TD, TS, TC, TJ, TK, TR]

[0067] Datayz = [GE, HC, JY]

[0068] Construction of the initial data set for predicting the ultimate state of polyurethane directly buried thermal insulation pipes: To ensure the comprehensiveness of data collection, a total of N groups of tests on the ultimate state of thermal insulation pipes were carried out to cover all the specification parameters of polyurethane directly buried thermal insulation pipes and the experimental environment parameters. The data collected in each test were combined to obtain the initial data set Datay for predicting the ultimate state of polyurethane directly buried thermal insulation pipes;

[0069] Datay = [Datayz 1 ,Datayz 2 ,Datayz 3 ,...,Datayz N

[0070] Data standardization processing: To eliminate the dimensional difference and scale difference existing in the initial data, the data in the data set Datay were subjected to unfolding standardization processing. Take a piece of data Datayz i in the initial prediction data set Datay of the ultimate state of polyurethane directly buried thermal insulation pipes, where i ∈ [1, N]. Calculate the average value μ and the standard deviation α of all the specific parameter values α contained in this piece of data.

[0071]

[0072] Among them, α* represents the parameter value after the standardization operation, and α represents the initial parameter value in the data.

[0073] Construction of the standard prediction data set for the ultimate state of polyurethane directly buried thermal insulation pipes: Based on the described data standardization operation, each piece of data in the data set Datay was standardized in turn. After the standardization operation was completed, the standard prediction data set Data for the ultimate state of polyurethane directly buried thermal insulation pipes was obtained.

[0074] II. Feature screening based on the relieff algorithm

[0075] In the data set construction stage, to ensure the comprehensiveness of feature collection, some redundant features will inevitably be introduced into the data set. These redundant features will increase the difficulty of model training and at the same time reduce the generalization ability of the model. To eliminate the influence of redundant features and enhance the model performance, the present invention constructs a feature screening module based on the relieff algorithm, which only retains the features with a high degree of correlation with the ultimate stress of polyurethane directly buried thermal insulation pipes, so as to achieve the purpose of enhancing the generalization ability of the model. The process is as Figure 2 shown.

[0076] S1. Data selection and classification: Based on the obtained initial data set Data of the ultimate state of polyurethane directly buried thermal insulation pipes, M sample data are randomly selected from it. To facilitate the subsequent calculation of feature importance, after the data selection is completed, the M sample data are grouped according to the data similarity SD U.​

[0077]

[0078] Among them, Datayz j represents the j-th parameter value of a piece of data in the data set Data, represents the j-th parameter value of another piece of data in the data set Data, where j ∈ [1, 18]. In this embodiment, the number of groups is 8, and the specific grouping process is as follows. Start grouping from a randomly selected piece of data among the M pieces of sample data, and calculate the SDU values of all the remaining data and this piece of data in turn. After the calculation, combine the piece of data with the largest SDU value and this piece of data to form a data group. After the grouped data is removed, continue to perform the above-mentioned data classification on the remaining data until the data grouping is completed;

[0079] S2, Feature difference degree calculation: Based on the described parameter selection, a total of 9 different specification characteristic parameters of polyurethane directly buried thermal insulation pipes and 8 different environmental characteristic parameters are selected in this embodiment. Based on the sample grouping situation after the S1 process operation, calculate the feature difference degree TZC corresponding to each feature in the same group of samples and the feature difference degree BZC in different groups of samples respectively;

[0080]

[0081] Among them, the parameter values of the k-th feature of the w-th and m-th pieces of data in the same group of sample data, and respectively represent the parameter values of the k-th feature of the g-th and l-th pieces of data in different groups of sample data, |*| represents the absolute value operation function;

[0082] S3, Feature correlation degree calculation: Based on the difference degrees TZC and BZC of each feature calculated in the S2 process, calculate the feature correlation degree Tmax:

[0083]

[0084] In the same group of samples, the smaller the feature difference degree, the stronger the correlation degree of this feature. While in different groups of samples, the larger the feature difference degree, the weaker the correlation degree of this feature;

[0085] S4, Multiple iteration feature correlation degree calculation: To avoid the calculation error caused by a single data sampling, in the present invention, the processes S1 to S3 are repeated, and the number of iterations is 6. The feature correlation degree calculated each time Perform average weighting processing for \(t\in[1,6]\), and define the result \(TO\) as the final feature importance. The higher the feature importance, the stronger the connection between this feature and the ultimate stress of the polyurethane directly buried insulation pipe;

[0086]

[0087] Among them, represents the feature correlation degree calculated for the \(k\)-th feature in the \(t\)-th calculation;

[0088] S5. Feature screening process: Based on the calculated feature importance of each feature, it is found that the feature importance of the features in the last 40% among the 17 features has a significant decrease compared to the features in the first 60%. Therefore, in the present invention, the redundant features with feature importance in the last 40% will be discarded. After the feature screening is completed, the retained specification feature parameters of the polyurethane directly buried insulation pipe are: the steel pipe layer thickness \(GT\) of the insulation pipe, the insulation layer thickness \(WT\) of the insulation pipe, the diameter \(BD\) of the insulation pipe, the compressive strength \(BQ\) of the insulation pipe, the thermal conductivity \(BR\) of the insulation pipe, and the water absorption rate \(BH\) of the insulation pipe. The retained environmental feature parameters are the environmental temperature \(TM\), the buried depth \(SD\) of the directly buried pipe, the surrounding soil density \(TD\), and the soil friction coefficient \(TS\).

[0089] III. Construction of the ultimate state prediction model for prefabricated polyurethane directly buried insulation pipes based on the deep operator network architecture

[0090] The ultimate state prediction model for prefabricated polyurethane directly buried insulation pipes constructed in the present invention mainly consists of three parts, namely the backbone network structure, the branch network architecture, and the fusion output module, as Figure 3 shown.

[0091] Backbone network architecture design: The backbone network architecture consists of an input layer, an intermediate layer, and an output layer. In the present invention, the feature parameters of the polyurethane directly buried insulation pipe are selected as the input of this backbone network architecture. After feature screening, the number of retained feature parameters of the polyurethane directly buried insulation pipe is 6. Therefore, the number of neurons in the input layer is 6. The network architecture of the intermediate layer mainly consists of a fully connected layer, a Relu activation function, and a Silu activation function. In the intermediate layer, first there are \(Wq\) fully connected layers, each fully connected layer has \(Ws\) neurons, and then it is activated by the Relu function to enhance the non-linear expression ability of the model. Finally, there are \(Wh\) fully connected layers, each fully connected layer has \(Wf\) neurons in each layer and the Silu activation function. The designed output layer contains 4 different neurons, and then the outputs of the 4 neurons in the hidden layer are adaptively fused through the spatial attention layer to obtain the output \(Feaz\) of the backbone network architecture.

[0092] Branch Network Architecture Design: In the designed branch network architecture, a stacking structure is adopted to enhance the overall fitting ability of the model. The stacking structure consists of 2 parallel network paths, and each network path still consists of an input layer, an intermediate layer, and an output layer. In the branch network architecture, the environmental feature parameters are used as the input of this network architecture. Since the number of environmental feature parameters after screening is 4, a total of 4 neurons are included in the input layer. The structure design of the intermediate layer is as follows: First, there are 3 sequentially connected fully connected layers, and each fully connected layer contains 300 neurons. Subsequently, the data is non-linearly activated through the sigmoid activation function. Then, there are 4 sequentially connected fully connected layers, and in these 4 fully connected layers, each layer has 400 neurons. Finally, non-linear normalization processing is performed through the Softmax activation function. The number of neurons in the output layer of this network path is set to 1. The output FZS of the output layer in each network path is sent into the self-attention mechanism for further processing. Subsequently, a fully connected layer is used to further integrate the outputs of the 2 network paths to obtain the output Feaf of the branch network architecture.

[0093] Self-attention Mechanism Design: This self-attention mechanism is used to map the output FZS of each network path in the branch network architecture into multiple feature spaces to further enhance the overall fitting ability of the model. The feature processing process of each self-attention head is as follows:

[0094]

[0095] Among them, Softmax(*) represents the Softmax linear normalization function, and Fea is the output result of the self-attention head. The matrices Q, K, and V are the query, key, and value matrices mapped after processing the feature FZS through three different matrices. Specifically:

[0096] Q = W Q *FZS

[0097] K = W K *FZS

[0098] V = W V *FZS

[0099] Among them, W Q ,W K ,W V are three different feature mapping matrices to send the feature FZS into three different feature spaces for processing and enhance the deep data fitting ability of the model.

[0100] In the self-attention mechanism designed in the present invention, a total of 8 parallel self-attention taps are designed as described above. Then, the channel attention layer is used to integrate the outputs of the 8 attention taps to obtain the output result Fe of the final self-attention mechanism:

[0101]

[0102] Among them, zyl(*) represents the operation function of the channel attention layer.

[0103] Fusion output module: The fusion output module further fuses the output Feaz obtained from the backbone network architecture and the output Feaf obtained from the branch network architecture to obtain the predicted result Yout of the ultimate stress of the prefabricated polyurethane directly buried insulation pipe. First, the BN normalization layer is used to normalize Feaz and Feaf respectively to optimize the distribution of the feature channels to enhance the robustness of the model when outputting the predicted result of the ultimate stress of the prefabricated polyurethane directly buried insulation pipe, and obtain and Secondly, the bilinear interpolation upsampling layer is used to upsample to the same size; Thirdly, the dot product layer is used to perform a dot product operation on the upsampled and to obtain the fusion feature Ffz; Thirdly, the fusion feature is input into the Dropout layer to optimize the feature channel parameters in Ffz to obtain the feature Ffz* with frozen parameters, reducing the risk of model overfitting; Finally, Ffz* is input into the L2 regularization layer and then into the Relu activation function to obtain the final predicted result Your of the ultimate stress of the prefabricated polyurethane directly buried insulation pipe, where the L2 regularization layer constrains the model weights and enhances the robustness of the model during prediction. The structure is as Figure 4 shown.

[0104] Yout = zrh(Feaf, Feaz)

[0105] Among them, zrh(*, *) represents the fusion output module.

[0106] IV. Model Optimization Training Based on Reinforcement Learning

[0107] In the present invention, in order to further improve the overall performance of the prefabricated polyurethane directly buried insulation pipe ultimate state prediction model and enhance the deep fitting ability of the model. After the overall architecture of the model is constructed, a model optimization module is constructed based on reinforcement learning technology to comprehensively optimize the number of fully connected layers in the intermediate layer of the model backbone network and the number of neurons Ws, Wf contained in each fully connected layer Wq, Wh, so as to find the specific number of fully connected layers and the number of neurons in each layer corresponding to the best model performance;

[0108] Loss function construction: It is clear that the loss function used by the model during the training phase is reinforcement learning to help construct the reward factor used in reinforcement learning. The present invention constructs the loss function Loss used in the model based on the mean square error and the absolute value error:

[0109]

[0110] Among them, Yout o , o ∈ [1, N] represents the ultimate stress test result of the polyurethane directly buried thermal insulation pipe predicted by the model, and JY o represents the ultimate stress test result of the polyurethane directly buried thermal insulation pipe obtained from the actual test.

[0111] State, action, and reward factor construction: The reinforcement learning strategy used in the present invention is to optimize the model parameters in the ultimate state prediction model of the polyurethane directly buried thermal insulation pipe. Therefore, a vector composed of the number of fully connected layers Wq, Wh and the corresponding number of neurons Ws, Wf, a total of 4 numerical combinations, constitutes the state space corresponding to this reinforcement learning. In its action space, it is the adaptive optimization and change of the parameters Wq, Wh, Ws, Wf to find the model parameters corresponding to the best model performance. The design of its reward factor is achieved by comparing the magnitudes of the loss functions. Using the data set constructed in process S1, based on the model parameters corresponding in the state space, the loss function value Lossq of the model at this time is obtained. When the state is updated, that is, an action to update the model parameters is taken. Then calculate the loss function value Lossh corresponding to the model after the parameter update. If the value of Qjl is greater than 0, a positive reward value of 2 * Qjl is given to this action. If the value of Qjl is less than 0, a negative reward value of 2 * Qjl is given to this action;

[0112] Qjl = Lossh - Lossq

[0113] Reinforcement learning network construction: In the present invention, a policy network is constructed to optimize the parameters Wq, Wh, Ws, Wf to find the network model parameters that make the model performance the best. A value network is constructed to adaptively judge the value corresponding to adopting this set of model parameters. In the present invention, the optimized parameters are relatively simple. Therefore, the designed policy network mainly consists of 4 interconnected fully connected layers and the Relu activation function. The designed value network mainly consists of 6 interconnected fully connected layers and the Silu activation function. The input of its network architecture is a four-dimensional vector group, representing a corresponding situation of Wq, Wh, Ws, Wf. Using the data set constructed in process S1 and placing the prediction model constructed by the input of this architecture at this time for training, and recording the loss function after the training process to calculate the reward factor.

[0114] Reinforcement learning network training: Randomly initialize a set of network parameters \(W_q\), \(W_h\), \(W_s\), \(W_f\), and send them into the reinforcement learning network for optimization training. Adopt a stochastic gradient descent training strategy and combine the reward factor to adaptively update the parameters \(W_q\), \(W_h\), \(W_s\), \(W_f\) using the policy network and the value network. Continuously update the model parameters. When the loss function of the model is less than the set threshold \(Th\), stop the training of the model. And set the obtained network parameters \(W_q\), \(W_h\), \(W_s\), \(W_f\) as the number of fully connected layers finally used in the middle layer of the model and the number of neurons used therein. Obtain the finally optimized limit state prediction model of the polyurethane directly buried heat-insulated pipe.

[0115] V. Model Deployment and Application

[0116] Model deployment: Deploy the finally optimized limit state prediction model of the polyurethane directly buried heat-insulated pipe and the parameter standardization program to the computer terminal to ensure that this prediction model can function smoothly and guarantee the smooth realization of its prediction function.

[0117] Development of parameter input interface: Develop a visual interface for inputting the specification parameters and usage environment parameter conditions of the polyurethane directly buried heat-insulated pipe, and deploy this interface and the software in the backend to the computer terminal, and adaptively match it with the prediction model to ensure that its parameters can be smoothly transmitted into the limit state prediction model of the polyurethane directly buried heat-insulated pipe.

[0118] Ultimate stress prediction: Send various specification parameters input through the parameter input interface into the limit state prediction model of the polyurethane directly buried heat-insulated pipe after being processed by the data standardization program to obtain the ultimate stress result of this specification of polyurethane directly buried heat-insulated pipe under this environmental condition.

[0119] Data analysis: According to the analysis results of the predicted ultimate stress, help relevant staff understand the ultimate stress conditions of various specifications of polyurethane prefabricated directly buried heat-insulated pipes under different environmental conditions, so as to reasonably select the polyurethane prefabricated directly buried heat-insulated pipes used in the project according to the predicted ultimate stress conditions and the actual usage scenarios, in order to further improve the overall quality of the project.

[0120] VI. Experimental Results and Explanation

[0121] To verify that the method proposed in the present invention has a high detection accuracy, a performance comparison experiment is carried out with the existing limit state prediction algorithm for polyurethane pre-buried heat-insulated pipes. The results are as Figure 5As shown. The two existing algorithms are the TFRN algorithm and the BRTY algorithm respectively. Except for the environmental temperature parameter, keep the insulation pipe specification parameters and the remaining environmental parameters the same, and conduct comparative experiments at different operating temperatures (-20 degrees Celsius, 0 degrees Celsius, 10 degrees Celsius, 30 degrees Celsius). The pros and cons of the algorithm performance are realized by comparing its prediction error value (MAPE value). The larger the MAPE value, the greater the prediction error of this algorithm, that is, the overall performance of the model is worse.

[0122]

[0123] Among them, yi represents the true ultimate stress value of the polyurethane pre-embedded insulation pipe in the experiment, represents the predicted ultimate stress value of the polyurethane pre-embedded insulation pipe. The results show that at the four temperature points, the algorithm proposed in the present invention has a smaller prediction error, that is, it shows that the algorithm proposed in the present invention has better performance than the existing algorithms.

[0124] The above are only the preferred embodiments of the present application and are not used to limit the present application. For those skilled in the art, various changes and modifications can be made to the present application. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present application shall be included in the protection scope of the present application.

[0125] Although the specific implementation manners of the present invention have been described above, they are not limitations on the protection scope of the present invention. Those skilled in the art should understand that based on the technical solutions of the present invention, various modifications or deformations that can be made without creative efforts by those skilled in the art are still within the protection scope of the present invention.

Claims

1. A method for constructing a limit state prediction model for prefabricated polyurethane direct-buried thermal insulation pipe based on a deep operator network architecture, characterized in that: The process includes: Step 1: Based on the use environment simulation, carry out the ultimate stress test on the direct buried pipes of different specifications to collect the maximum stress they can withstand under such conditions, record the maximum stress and the corresponding specification data and environmental data, and construct the initial data set; Step 2: Based on the feature screening module of the relieff algorithm, the features with low correlation with the ultimate stress of the polyurethane direct-buried thermal insulation pipe are eliminated, the specification data and environmental data with the highest feature importance are obtained, and the training data set is constructed in combination with the corresponding maximum stress; Step 3, constructing a limit state prediction model based on a deep operator network architecture, including a trunk network architecture, a branch network architecture, and a fusion output module, wherein the trunk network is used to extract the physical property characteristics of the direct-buried thermal insulation pipe, the branch network is used to extract the environmental characteristics of the direct-buried thermal insulation pipe and perform weighted processing on the feature channels in the environmental characteristics based on the self-attention mechanism to obtain self-attention environmental characteristics, and the fusion output module is used to adaptively fuse the two features output by the two network architectures, and output the maximum stress prediction result based on the fusion feature; Step 4: Optimize and train the constructed limit state prediction model based on the reinforcement learning strategy to obtain the final limit state prediction model.

2. A method for constructing a limit state prediction model for prefabricated polyurethane direct-buried thermal insulation pipe based on a deep operator network architecture as claimed in claim 1, characterized in that: The specific data processing process of the feature screening module based on the relief algorithm in step 2 is as follows: S21, data selection and classification: Based on the initial data set Data of the limit state of polyurethane direct-buried thermal insulation pipe, M sample data are randomly selected from it, and the M sample data are grouped according to the data similarity SDU, and the number of groups is L; S22, feature difference calculation: calculate the feature difference TZC corresponding to each feature in the same group of samples and the feature difference BZC in different groups of samples respectively; S23, feature correlation calculation: calculating the feature correlation Tmax based on the calculated differences TZC and BZC of each feature; S24, multiple iterations of feature correlation calculation: repeat the process S21 to S23, a total of B iterations, and calculate the feature correlation obtained each time. Perform average weighted processing, and define the result TO as the final feature importance. The higher the feature importance, the stronger the connection between this feature and the ultimate stress of polyurethane direct-buried thermal insulation pipe. S25, feature screening process: based on the calculated feature importance of each feature, the redundant features whose feature importance is in the bottom 40% are discarded.

3. A method for constructing a limit state prediction model for prefabricated polyurethane direct-buried thermal insulation pipe based on a deep operator network architecture as claimed in claim 2, characterized in that: In S21, Among them, Datayz j Indicates the jth parameter value of a data set Data. Represents the jth parameter value of another data in the data set Data, j∈[1,C]; where C is the total number of environmental data, specification data and maximum stress data; The specific process of grouping is: randomly select one piece of data from M sample data to start grouping, calculate the SDU value of all other remaining data and this piece of data in turn, and after the calculation is completed, group the data with the largest SDU value. The data is combined with the data to form a data group; the grouped data is eliminated, and the remaining data is further classified as described above until the data grouping is completed; In S22, the feature difference TZC corresponding to each feature in the same group of samples and the feature difference BZC in different groups of samples are calculated respectively, and the formula is as follows: in, The parameter value of the kth feature of the wth and mth data in the same group of sample data, Where T is the sum of environmental data and specification data; and Respectively represent the parameter values ​​of the kth feature of the gth and lth data in different groups of sample data, |*| represents the absolute value operation function.

4. A method for constructing a limit state prediction model for prefabricated polyurethane direct-buried thermal insulation pipe based on a deep operator network architecture as claimed in claim 1, characterized in that: The backbone network architecture consists of three parts: an input layer, an intermediate layer, and an output layer; the specification characteristic parameters of the polyurethane direct-buried thermal insulation pipe are selected as the input of this backbone network architecture. After feature screening, the number of specification characteristic parameters of the polyurethane direct-buried thermal insulation pipe retained is p, so the number of neurons in the input layer is p; the network architecture of the intermediate layer is mainly composed of three parts: a fully connected layer, a Relu activation function, and a Silu activation function. In the intermediate layer, there are first Wq fully connected layers, each of which has Ws neurons, and then activated by the Relu function to improve the nonlinear expression ability of the model, and finally there are Wh fully connected layers, each of which has Wf neurons and a Silu activation function; the designed output layer contains 4 different neurons, and then the outputs of the 4 neurons in the hidden layer are adaptively fused through the spatial attention layer to obtain the output Feaz of the backbone network architecture.

5. The method for constructing a limit state prediction model for prefabricated polyurethane direct-buried thermal insulation pipe based on a deep operator network architecture according to claim 1, characterized in that: The branch network architecture adopts a stacking structure; the stacking structure is composed of two parallel network paths, each network path is composed of three parts: an input layer, an intermediate layer, and an output layer; the environmental feature parameters are used as the input of this branch network architecture, and the input layer contains a total of q neurons, q is the total number of environmental feature data retained after screening; the structure of the intermediate layer is: first, three fully connected layers are connected in sequence, and then the data is nonlinearly activated through the simgoid activation function, followed by four fully connected layers connected in sequence, and in these four fully connected layers, nonlinear normalization is finally performed through the Softmax activation function; the number of neurons in the output layer in the network path is set to 1; the output FZS of the output layer in each network path is sent to the self-attention mechanism unit for processing to obtain the output result Fe, and then a fully connected layer is used to further integrate the outputs of the two network paths to obtain the output Feaf of the branch network architecture.

6. A method for constructing a limit state prediction model for prefabricated polyurethane direct-buried thermal insulation pipe based on a deep operator network architecture as claimed in claim 5, characterized in that: The self-attention mechanism unit maps the output FZS of each network path in the branch network architecture to multiple feature spaces. The self-attention mechanism includes L parallel self-attention taps, and then the channel attention layer is used to integrate the outputs of the L attention taps to obtain the final output result Fe of the self-attention mechanism; the feature processing process of each self-attention tap is: Among them, Softmax(*) represents the Softmax linear normalization function, Fea is the output result of the self-attention tap; the matrices Q, K, and V are the query, key, and value matrices obtained by mapping the feature FZS after processing it with three different matrices. Specifically: Q=W Q *FZS W=W K *FZS V=W V *FZS Among them, W Q , W K , W V are three different feature mapping matrices to send the feature FZS into three different feature spaces for processing.

7. A method for constructing a limit state prediction model for prefabricated polyurethane direct-buried thermal insulation pipe based on a deep operator network architecture as claimed in claim 1, characterized in that: The fusion output module further fuses the output Feaz obtained by the trunk network architecture and the output Feaf obtained by the branch network architecture to obtain the ultimate stress prediction result Yout of the prefabricated polyurethane direct-buried thermal insulation pipe; firstly, the BN standardization layer is used to standardize Feaz and Feaf respectively to optimize the distribution degree of the characteristic channel, and obtain and Secondly, a bilinear interpolation upsampling layer is used to convert Upsample to Same size; again, use the dot product layer to upsample and A dot product operation is performed to obtain the fused feature Ffz; then, the fused feature is input into the Dropout layer to optimize the feature channel parameters in Ffz to obtain the feature Ffz* of the frozen parameter; finally, Ffz* is input into the L2 regularization layer and then into the Relu activation function to obtain the final prefabricated polyurethane direct-buried insulation pipe ultimate stress prediction result Yout, where the L2 regularization layer constrains the model weights.

8. The method for constructing a limit state prediction model for prefabricated polyurethane direct-buried thermal insulation pipe based on a deep operator network architecture according to claim 1, characterized in that: The reinforcement learning strategy is to comprehensively optimize the number of fully connected layers in the middle layer of the model backbone network and the number of neurons Ws, Wf contained in each fully connected layer Wq, Wh, so as to find the specific number of fully connected layers and the number of neurons in each layer corresponding to the best model performance; The vector composed of the number of fully connected layers Wq, Wh and the corresponding number of neurons Ws, Wf is used as the state space corresponding to this reinforcement learning; in its action space, it is the adaptive optimization and change of the parameters Wq, Wh, Ws, Wf to find the model parameters corresponding to the best model performance; the design of its reward factor is realized by comparing the size of the loss function, using the constructed training data set, based on the corresponding model parameters in the state space, to obtain the loss function value Lossq of the model at this time; When the state is updated, the action of updating the model parameters is taken, and then the loss function value Lossh corresponding to the model after the parameter update is calculated; if the value of Qjl is greater than 0, a positive reward value of 2*Qjl is given to this action, and if the value of Qjl is less than 0, a negative reward value of 2*Qjl is given to this action: Qjl=Lossh-Lossq At the same time, a reinforcement learning network is constructed, a policy network is constructed to optimize the parameters Wq, Wh, Ws, and Wf, and a value network is constructed to adaptively judge the value corresponding to this set of model parameters; the designed policy network is mainly composed of 4 interconnected fully connected layers and Relu activation functions, and the designed value network is mainly composed of 6 interconnected fully connected layers and Silu activation functions; the input of its network architecture is a set of four-dimensional vectors, representing a set of corresponding Wq, Wh, Ws, and Wf situations. The constructed data set is used to train the prediction model constructed by the architecture input at this time, and the loss function after the training process is recorded to calculate the reward factor.

9. A method for predicting the limit state of prefabricated polyurethane direct-buried thermal insulation pipe based on a deep operator network architecture, characterized in that: The limit state prediction model constructed by the construction method according to any one of claims 1 to 8 is deployed in a computer terminal, and includes the following process: Obtain specification data and environmental data of polyurethane direct buried thermal insulation pipe in real time, and standardize them into model input data; The standardized specification data and environmental data are input into the limit state prediction model; the ultimate stress results of the polyurethane direct-buried thermal insulation pipe of this specification under this environmental condition are output.

10. A method for predicting the limit state of a prefabricated polyurethane direct-buried thermal insulation pipe based on a deep operator network architecture as claimed in claim 9, characterized in that: After screening, the specification characteristic parameters of the polyurethane direct-buried insulated pipe retained for prediction are: the steel pipe layer thickness GT of the insulation pipe, the insulation layer thickness WT of the insulation pipe, the diameter BD of the insulation pipe, the compressive strength BQ of the insulation pipe, the thermal conductivity BR of the insulation pipe and the water absorption rate BH of the insulation pipe; the retained environmental characteristic parameters are the ambient temperature TM, the burial depth SD of the direct-buried pipe, the surrounding soil density TD, and the soil friction coefficient TS.

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