An exhaust manifold thermal fatigue life multi-field coupling optimization method based on response surface method and back propagation neural network

Abstract

This invention relates to a multi-field coupled optimization method for the thermal fatigue life of exhaust manifolds based on response surface methodology and backpropagation neural networks. The method includes: conducting single-factor "thermal-fluid-solid" multi-field coupled finite element analysis on stiffener height, peripheral fin thickness, and inlet flange slot depth; designing a sample set and constructing a response surface model; expanding the sample dataset using two methods based on the experimental design of the response surface methodology; training and evaluating the backpropagation neural network model; introducing the Shapley additivity interpretation method to enhance the interpretability of the neural network model; and using the backpropagation neural network as a probabilistic surrogate model for a Bayesian optimization algorithm to search for the optimal parameter combination. This invention achieves multi-parameter collaborative optimization, systematically improving the thermal fatigue life of exhaust manifolds.

Description

Technical Field

[0001] This invention relates to the field of exhaust manifold thermal fatigue life optimization technology, specifically to a multi-field coupled optimization method for exhaust manifold thermal fatigue life based on response surface methodology and backpropagation neural network. Background Technology

[0002] The exhaust manifold has a large market size and wide range of applications, serving as a key component of the engine exhaust system. In actual service, the exhaust manifold is subjected to harsh conditions of alternating hot and cold temperatures, frequently leading to thermal fatigue cracking. Repair methods primarily involve welding to repair cracks or even replacing the entire exhaust manifold. This results in a surge in after-sales maintenance costs for manufacturers and intensifies the pressure on recycling and disposing of scrapped parts. Simultaneously, it increases vehicle maintenance costs for consumers, and the potential for crack propagation to cause exhaust leaks makes it neither economical nor safe. To alleviate these problems, it is essential to improve the thermal fatigue life of the exhaust manifold through structural design. Therefore, exploring a more effective methodological framework to enhance the scientific rigor of exhaust manifold structural design has become a key focus for the industry.

[0003] While adding stiffening ribs, fins, and hollow flanges are novel and easily implemented methods, they only optimize a single location on the exhaust manifold. They lack analysis of the overall effect when optimizing multiple locations simultaneously. This method of finding a single location for fixed-point structural optimization through simulation analysis often has limited impact on improving the thermal fatigue life of the exhaust manifold. Currently, there is a lack of a methodological framework for simultaneously optimizing the structure at multiple thermal stress concentration locations on the exhaust manifold, studying interaction effects, and exploring optimal parameter combinations to comprehensively improve thermal fatigue life. Summary of the Invention

[0004] This invention addresses the problems of existing exhaust manifold optimization methods, which only target a single location, lack multi-parameter collaborative optimization and interaction effect analysis, have low optimization efficiency, and offer limited improvement in thermal fatigue life. It provides a multi-field coupled optimization method for exhaust manifold thermal fatigue life based on response surface methodology and backpropagation neural networks. The method involves conducting single-factor thermo-fluid-structure interaction multi-field coupled finite element analysis on stiffener height, peripheral fin thickness, and hollow flange slot depth. Based on this, response surface analysis is implemented, and a preliminary data foundation is built by fitting the correlation between each parameter and thermal fatigue life. Simultaneously, sample data is supplemented and the dataset size is expanded based on the analysis results. A BPNN prediction model is constructed to establish a nonlinear mapping relationship between structural parameters and thermal fatigue life characteristics. The SHAP method is introduced to enhance the interpretability of the neural network model to address the "black box" problem. Finally, the BPNN is used as a probabilistic surrogate model for Bayesian optimization algorithms, aiming to minimize the equivalent plastic strain increment. This efficiently searches and determines the optimal combination of structural parameters within the parameter space, achieving multi-parameter collaborative optimization and systematically improving the thermal fatigue life of the exhaust manifold.

[0005] To achieve the above objectives, this invention proposes a multi-field coupled optimization method for the thermal fatigue life of exhaust manifolds based on the response surface methodology and backpropagation neural network, comprising the following steps:

[0006] S1: Select the height of the reinforcing ribs, the thickness of the peripheral fins, and the groove depth of the hollow flange of the exhaust manifold as optimization parameters, construct a three-dimensional geometric model of the exhaust manifold, and carry out single-factor thermo-fluid-solid multi-field coupled finite element analysis to obtain the influence law of individual changes of each parameter on the thermal fatigue life characterization quantity.

[0007] S2: Determine the range and level of parameter values ​​based on the results of single-factor analysis, construct sample groups using the Box-Behnken experimental design method, conduct multi-field coupled finite element simulations on each sample group, construct a response surface model based on the least squares method using a multivariate quadratic polynomial as the response function, and complete the preliminary fitting of parameters and response quantities.

[0008] S3: Using the core samples obtained from the response surface methodology experimental design as the base, the sample dataset is expanded by combining two methods: completing the fully factorial simulation data and generating expanded samples using the virtual sample method, thereby improving the parameter space coverage and sample size.

[0009] S4: Normalize and randomly partition the expanded dataset, construct a backpropagation neural network model, train the network using the Levenberg-Marquardt algorithm, and evaluate the model's prediction accuracy and generalization ability using multiple metrics.

[0010] S5: Introducing the Shapley additivity interpretation method to calculate the SHAP value of each input parameter, quantify the marginal contribution, influence direction and interaction of each parameter to the prediction result, and improve the interpretability of the backpropagation neural network model;

[0011] S6: The trained backpropagation neural network model is used as a probabilistic surrogate model for the Bayesian optimization algorithm. The optimization objective is to minimize the equivalent plastic strain increment. The optimal parameter combination is searched within the range of feasible engineering parameters, and the optimization results are verified by finite element simulation.

[0012] Furthermore, the specific details of the thermal-fluid-solid multi-field coupled finite element analysis described in step S1 include:

[0013] S11: Construct a three-dimensional geometric model of the exhaust manifold, apply stiffening ribs, peripheral fins and hollow flanges separately, and perform single-factor multi-field coupled simulation on the height of stiffening ribs, thickness of peripheral fins and groove depth of hollow flanges. The coupled simulation puts the thermal field, flow field and solid mechanical field in the same equation matrix for solution, and uses steady-state simulation for analysis.

[0014] S12: Through internal fluid analysis of the exhaust manifold, the temperature and pressure fields of the fluid and solid domains are obtained. Then, the temperature obtained from the fluid analysis is placed in the thermal analysis module to solve the temperature field of the exhaust manifold, completing the fluid-thermal coupling process. Finally, the pressure and temperature fields obtained from the fluid and thermal analysis are used as boundary conditions and mapped to the static solid structure module to solve the thermal stress distribution and equivalent plastic strain distribution of the exhaust manifold, completing the fluid-thermal-solid three-field coupling analysis.

[0015] S13: The relationship between thermal fatigue life N and plastic strain range is expressed as follows:

[0016] (1);

[0017] in, Where N is the plastic strain range, C is the crack initiation life, and α is a material constant.

[0018] The equivalent plastic strain increment (ΔPEEQ) of the exhaust manifold within a specified operating cycle is used as... The value of is used for thermal fatigue life estimation, and and this As the core characterization parameter for the thermal fatigue life of the exhaust manifold.

[0019] A three-field steady-state coupled simulation of fluid-thermal-solid is adopted to realistically reproduce the actual service conditions of the exhaust manifold, significantly improving the calculation accuracy of thermal stress and equivalent plastic strain. ΔPEEQ is used as the thermal fatigue life characterization quantity to achieve rapid and accurate quantitative assessment of life, providing a reliable data foundation for subsequent optimization.

[0020] Furthermore, the reinforcing ribs, peripheral fins, and hollow flanges mentioned in step S1, wherein the reinforcing ribs are arranged between adjacent manifolds to improve structural rigidity and disperse thermal stress;

[0021] The peripheral fins are arranged around the exhaust manifold outlet to enhance heat dissipation and reduce local stress concentration.

[0022] The hollow flange is formed by milling a groove in the middle, which is used to reduce the flange stiffness and reduce the stress effect of external constraints on the manifold.

[0023] Furthermore, the response surface model construction described in step S2 specifically includes:

[0024] S21: Based on the single-factor analysis results of step S1, determine the value range of the three optimization parameters: stiffener height, peripheral fin thickness, and hollow flange slot depth. Each parameter is set to three levels: low, medium, and high.

[0025] S22: A sample group was constructed using a three-factor, three-level Box-Behnken experimental design, and multi-field coupled finite element simulations were performed on each sample group.

[0026] S23: The expression for a multivariate quadratic polynomial as a response function is:

[0027] (2);

[0028] in, For specific response quantities, The fitting parameters are to be determined. The levels of the m-th and n-th factors are respectively (where the levels can be set). k represents the total number of factors;

[0029] S24: Test the significance of the model through analysis of variance, requiring model P<0.05 and lack-of-fit P>0.05, to verify the effectiveness and reliability of the model fit.

[0030] The Box-Behnken experimental design enables efficient modeling with a small sample size. The multivariate quadratic polynomial accurately reflects the parameter coupling relationship, and the significance is rigorously verified by analysis of variance, ensuring reliable model fitting and accurate prediction, and providing a standard benchmark for subsequent data expansion and neural network training.

[0031] Furthermore, the sample dataset expansion described in step S3 specifically involves:

[0032] Based on the response surface methodology experimental design in step S2, the sample dataset is expanded in two ways:

[0033] The first approach is to conduct supplementary simulations on the parameter combinations that were not extracted by Box-Behnken in the fully factorial design, forming a fully factorial simulation dataset that covers the complete parameter space.

[0034] The second approach is to introduce a Gaussian noise virtual sample method based on a fully factorial dataset. Controllable random noise is added to the input parameters and output response respectively, which greatly expands the number of samples without changing the data distribution pattern.

[0035] Using the core samples of the response surface methodology as a base, the method significantly improves the parameter space coverage and data diversity by supplementing the complete factorial data and using virtual samples. This solves the problems of insufficient neural network training and weak generalization ability caused by insufficient sample size in traditional methods, and significantly improves the stability and accuracy of the model.

[0036] Furthermore, the training and evaluation of the backpropagation neural network model in step S4 specifically includes:

[0037] S41: After data augmentation, use the randperm function to generate random indices to shuffle the augmented data and avoid model training bias caused by sample order.

[0038] S42: The augmented dataset is divided into training, validation and test sets using a random partitioning strategy;

[0039] S43: The Levenberg-Marquardt algorithm is used to optimize network weights, and multi-dimensional training termination conditions are set to prevent overfitting;

[0040] S44: Select the mean squared error (MSE), mean relative error (MRE), mean absolute error percentage (MAPE), and coefficient of determination (R²) to evaluate the model performance, as shown in formula (3):

[0041] (3);

[0042] Where n is the number of samples, For the true value, For predicted values, It is the average of the true values.

[0043] The Levenberg-Marquardt algorithm is used for fast convergence and high fitting accuracy. Overfitting is effectively prevented by shuffling the data, splitting the dataset, and using early stopping under multiple conditions. MSE, MRE, MAPE, and R² are used for comprehensive evaluation to measure model performance in a comprehensive and objective manner, ensuring the reliability of prediction results.

[0044] Furthermore, the introduction of the Shapley additivity interpretation method in step S5 to enhance the interpretability of the neural network model specifically includes:

[0045] S51: Extract the total sample, which includes both the experimental sample and the fitted sample;

[0046] S52: Calculate the SHAP value of each feature for the corresponding sample to characterize the marginal contribution of the feature to the prediction result;

[0047] S53: Draw a SHAP summary diagram, a feature importance ranking diagram, and a dependency diagram to reveal the contribution direction, intensity, and interaction mechanism of each parameter to the equivalent plastic strain increment.

[0048] The SHAP method is introduced to crack the "black box" problem of neural networks, quantify the marginal contribution, influence direction and interaction of each parameter to the prediction results, clearly reveal the optimization mechanism, make the results theoretically supported and convincing, and facilitate engineering debugging and scheme improvement.

[0049] Furthermore, step S6, which describes using a backpropagation neural network (BPNN) as a probabilistic surrogate model for the Bayesian optimization algorithm to search for the optimal parameter combination, specifically includes:

[0050] S61: Using a backpropagation neural network as a probabilistic proxy model to proxy an unknown objective function;

[0051] S62: Exploration and development in the optimization process using expected improvement criteria;

[0052] S63: The optimization objective is to minimize the equivalent plastic strain increment, and the range of variable values ​​is limited during the optimization process;

[0053] S64: The optimal parameter combination obtained by optimization is verified by finite element simulation to confirm that it can effectively reduce the equivalent plastic strain increment and improve the thermal fatigue life, thus ensuring the engineering applicability of the optimization results.

[0054] By using BPNN as a surrogate model to replace a large number of repetitive finite element calculations, the computational load is significantly reduced and the optimization efficiency is improved. The EI criterion is adopted to balance exploration and development and avoid local optima. By combining engineering feasibility constraints and simulation verification, the optimal combination is ensured to be real, effective and feasible for engineering implementation.

[0055] The beneficial effects of the present invention through the above technical solution are as follows:

[0056] (1) Significantly improve the accuracy and targeting of exhaust manifold structure optimization. Through "thermal-fluid-solid" multi-field coupling simulation, it fits the actual high-temperature service conditions, effectively solving the problem of large deviation between traditional simulation and actual conditions. Combined with sample expansion and BPNN model, it accurately captures the complex nonlinear relationship between the three parameters and ΔPEEQ, fundamentally improving thermal stress concentration, significantly improving thermal fatigue life, and reducing cracking and repair / replacement costs.

[0057] (2) The optimization efficiency is significantly improved. The BPNN model is used as a proxy model for Bayesian optimization to replace the traditional time-consuming finite element simulation optimization, which greatly reduces the amount of computation and solves the problem of large amount of computation and long cycle of traditional optimization methods. Combined with the efficient search capability of the EI criterion, the global optimal parameters are quickly converged, overcoming the defect of being prone to getting trapped in local optima.

[0058] (3) The model is highly interpretable. The SHAP method is introduced to quantify the contribution and interaction effect of each parameter, solve the problem of the "black box" of neural networks being uninterpretable and the optimization mechanism being unclear, clarify the multi-parameter synergistic mechanism, make the optimization results more theoretically supported and convincing, and facilitate parameter adjustment and scheme improvement in engineering applications.

[0059] (4) It is highly practical for engineering. In response to the shortcomings of existing technologies that can only optimize a single position and lack multi-position collaborative design, it realizes the synchronous optimization of multiple structural parameters, takes into account both the improvement of thermal fatigue life and the feasibility of actual processing, and provides a general method reference for the multi-position collaborative optimization of high-temperature load-bearing components of similar power machinery. Attached Figure Description

[0060] Figure 1This is a schematic diagram of the methodological framework of the present invention;

[0061] Figure 2 This is a schematic diagram of the exhaust manifold geometric model used in this invention;

[0062] Figure 3 A schematic diagram of the exhaust manifold geometric model with three optimized structures applied as used in this invention;

[0063] Figure 4 The response surface and contour plot of the interaction of various factors on ΔPEEQ in this invention are shown.

[0064] Figure 5 This is a diagram of the backpropagation neural network structure constructed in this invention.

[0065] Figure 6 This is a comparison chart showing the fitting effects of the BPNN model constructed in this invention on the training, validation, test sets, and all data.

[0066] Figure 7 This paper presents the performance evaluation and prediction results analysis of the backpropagation neural network model constructed in this invention.

[0067] Figure 8 This is a graph showing the feature importance and interaction effect analysis of the BPNN model based on SHAP values ​​in this invention.

[0068] Figure 9 This is the iterative convergence curve of the Bayesian optimization algorithm used in this invention.

[0069] Figure 10 This is a flowchart of the steps of the present invention.

[0070] The attached diagrams are numbered as follows: 1 is the exhaust manifold, 2 is the peripheral fin, 3 is the hollow flange, and 4 is the reinforcing rib. Detailed Implementation

[0071] The present invention will be further described below with reference to the accompanying drawings and specific embodiments:

[0072] Example 1

[0073] like Figures 1-10 As shown, a multi-field coupled optimization method for the thermal fatigue life of an exhaust manifold based on response surface methodology and backpropagation neural network includes the following steps:

[0074] S1: Select the height of the reinforcing rib plate 4, the thickness of the peripheral fin 2, and the groove depth of the hollow flange 3 as optimization parameters to construct a three-dimensional geometric model of the exhaust manifold 1, and carry out single-factor thermal-fluid-solid multi-field coupled finite element analysis to obtain the influence law of individual changes of each parameter on the thermal fatigue life characterization quantity.

[0075] S2: Determine the range and level of parameter values ​​based on the results of single-factor analysis, construct sample groups using the Box-Behnken experimental design method, conduct multi-field coupled finite element simulations on each sample group, construct a response surface model based on the least squares method using a multivariate quadratic polynomial as the response function, and complete the preliminary fitting of parameters and response quantities.

[0076] S3: Using the core samples obtained from the response surface methodology experimental design as the base, the sample dataset is expanded by combining two methods: completing the fully factorial simulation data and generating expanded samples using the virtual sample method, thereby improving the parameter space coverage and sample size.

[0077] S4: Normalize and randomly partition the expanded dataset, construct a backpropagation neural network model, train the network using the Levenberg-Marquardt algorithm, and evaluate the model's prediction accuracy and generalization ability using multiple metrics.

[0078] S5: Introducing the Shapley additivity interpretation method to calculate the SHAP value of each input parameter, quantify the marginal contribution, influence direction and interaction of each parameter to the prediction result, and improve the interpretability of the backpropagation neural network model;

[0079] S6: The trained backpropagation neural network model is used as a probabilistic surrogate model for the Bayesian optimization algorithm. The optimization objective is to minimize the equivalent plastic strain increment. The optimal parameter combination is searched within the range of feasible engineering parameters, and the optimization results are verified by finite element simulation.

[0080] The specific components of the thermo-fluid-solid multi-field coupled finite element analysis described in step S1 include:

[0081] S11: Construct a three-dimensional geometric model of exhaust manifold 1, apply stiffening ribs 4, peripheral fins 2 and hollow flanges 3 separately, and perform single-factor multi-field coupled simulation on the height of stiffening ribs 4, the thickness of peripheral fins 2 and the slot depth of hollow flanges 3. The coupled simulation puts the thermal field, flow field and solid mechanical field in the same equation matrix for solution, and uses steady-state simulation for analysis.

[0082] S12: Through internal fluid analysis of exhaust manifold 1, the temperature and pressure fields of the fluid and solid domains are obtained. Then, the temperature obtained from the fluid analysis is placed in the thermal analysis module to solve the temperature field of exhaust manifold 1, completing the fluid-thermal coupling process. Finally, the pressure and temperature fields obtained from the fluid and thermal analysis are used as boundary conditions and mapped to the static solid structure module to solve the thermal stress distribution and equivalent plastic strain distribution of exhaust manifold 1, completing the fluid-thermal-solid three-field coupling analysis.

[0083] S13: The relationship between thermal fatigue life N and plastic strain range is expressed as follows:

[0084] (1);

[0085] in, Where α is the plastic strain range, N is the crack initiation life, and C and α are material constants.

[0086] The equivalent plastic strain increment of exhaust manifold 1 within a specified working cycle is used as... The value of is used for thermal fatigue life estimation, and and This serves as the core characterization parameter for the thermal fatigue life of the exhaust manifold 1.

[0087] The reinforcing rib 4, peripheral fins 2, and hollow flange 3 mentioned in step S1, wherein the reinforcing rib 4 is arranged between adjacent manifolds 1 to improve structural rigidity and disperse thermal stress;

[0088] The peripheral fins 2 are arranged around the outlet of the exhaust manifold 1 to enhance heat dissipation and reduce local stress concentration.

[0089] The hollow flange 3 is formed by milling a groove in the middle, which is used to reduce the stiffness of the flange 3 and reduce the stress effect of external constraints on the manifold 1.

[0090] The response surface model construction described in step S2 specifically includes:

[0091] S21: Based on the single-factor analysis results of step S1, determine the value range of the three optimization parameters: height of stiffening rib plate 4, thickness of peripheral fin 2, and groove depth of hollow flange 3. Each parameter is set to three levels: low, medium, and high.

[0092] S22: A sample group was constructed using a three-factor, three-level Box-Behnken experimental design, and multi-field coupled finite element simulations were performed on each sample group.

[0093] S23: The expression for a multivariate quadratic polynomial as a response function is:

[0094] (2);

[0095] in, For specific response quantities, The fitting parameters are to be determined. The levels of the m-th and n-th factors are respectively (where the levels can be set). k represents the total number of factors;

[0096] S24: Test the significance of the model through analysis of variance, requiring model P<0.05 and lack-of-fit P>0.05, to verify the effectiveness and reliability of the model fit.

[0097] The sample dataset expansion mentioned in step S3 specifically involves:

[0098] Based on the response surface methodology experimental design in step S2, the sample dataset is expanded in two ways:

[0099] The first approach is to conduct supplementary simulations on the parameter combinations that were not extracted by Box-Behnken in the fully factorial design, forming a fully factorial simulation dataset that covers the complete parameter space.

[0100] The second approach is to introduce a Gaussian noise virtual sample method based on a fully factorial dataset. Controllable random noise is added to the input parameters and output response respectively, which greatly expands the number of samples without changing the data distribution pattern.

[0101] The training and evaluation of the backpropagation neural network model in step S4 specifically includes:

[0102] S41: After data augmentation, use the randperm function to generate random indices to shuffle the augmented data and avoid model training bias caused by sample order.

[0103] S42: The augmented dataset is divided into training, validation and test sets using a random partitioning strategy;

[0104] S43: The Levenberg-Marquardt algorithm is used to optimize network weights, and multi-dimensional training termination conditions are set to prevent overfitting;

[0105] S44: Select the mean squared error (MSE), mean relative error (MRE), mean absolute error percentage (MAPE), and coefficient of determination (R²) to evaluate the model performance, as shown in formula (3):

[0106] (3);

[0107] Where n is the number of samples, For the true value, For predicted values, It is the average of the true values.

[0108] Step S5, which introduces the Shapley additivity interpretation method to enhance the interpretability of the neural network model, specifically includes:

[0109] S51: Extract the total sample, which includes both the experimental sample and the fitted sample;

[0110] S52: Calculate the SHAP value of each feature for the corresponding sample to characterize the marginal contribution of the feature to the prediction result;

[0111] S53: Draw a SHAP summary diagram, a feature importance ranking diagram, and a dependency diagram to reveal the contribution direction, intensity, and interaction mechanism of each parameter to the equivalent plastic strain increment.

[0112] Step S6, which describes using the backpropagation neural network as a probabilistic surrogate model for the Bayesian optimization algorithm to search for the optimal parameter combination, specifically includes:

[0113] S61: Using a backpropagation neural network as a probabilistic proxy model to proxy an unknown objective function;

[0114] S62: Exploration and development in the optimization process using expected improvement criteria;

[0115] S63: The optimization objective is to minimize the equivalent plastic strain increment, and the range of variable values ​​is limited during the optimization process;

[0116] S64: The optimal parameter combination obtained by optimization is verified by finite element simulation to confirm that it can effectively reduce the equivalent plastic strain increment and improve the thermal fatigue life, thus ensuring the engineering applicability of the optimization results.

[0117] This method takes the exhaust manifold 1 of a small-sized V6 "3-2" type engine as an example. Figure 2 As shown, the specific implementation steps are as follows:

[0118] like Figure 3 The exhaust manifold 1 constructed in this invention is divided into two groups, left and right, each containing three cylinders. The three intake channels of the two groups of sub-manifolds 1 merge into one in-group outlet. The two in-group outlets serve as independent exhaust outlets, directly connected to two parallel exhaust systems downstream. The three-dimensional model of exhaust manifold 1 is shown below. Figure 2 As shown. When constructing the 3D model, some structures that have little impact on the simulation analysis but would significantly increase the simulation time were ignored, and the left and right structures of exhaust manifold 1 were made symmetrical.

[0119] Reinforcing rib 4: Designed between adjacent manifolds 1 at both ends of the left and right sides to increase wall thickness and distribute stress;

[0120] Peripheral fins 2: Fins 2 surround the outlet to increase heat dissipation and reduce stress concentration;

[0121] Hollow flange 3: The middle part of the material is milled, leaving only the inner and outer edges (supports) to reduce the stiffness of flange 3, thereby reducing the external constraints of exhaust manifold 1.

[0122] The material used for the exhaust manifold 1 housing needs to be specified according to national standards and relevant literature. It is worth noting that the physical and chemical properties of the material change with temperature. This invention takes this into account during simulation and provides the relevant properties of the material at different temperatures to increase the reliability of the simulation. The exhaust manifold 1 housing material is ductile iron, and its relevant properties are shown in Tables 1 and 2 below:

[0123]

[0124]

[0125] Considering both the accuracy and computational efficiency of complex flow simulations, it is assumed that the relevant parameters of the exhaust gas fluid in exhaust manifold 1 are steady, and that the fluid flows in three dimensions and is incompressible. Specific data are shown in Table 3 below:

[0126]

[0127] The SSTk-ω turbulence model was selected. The SSTk-ω turbulence model is a mathematical model used in computational fluid dynamics (CFD) to simulate turbulent flow. It involves two equations to solve for two turbulence-related variables: k (turbulent energy) and ω (specific dissipation rate).

[0128] The turbulent kinetic energy (k) is shown in equation (4):

[0129] (4);

[0130] The specific dissipation ratio (ω) is shown in formula (5):

[0131] (5);

[0132] In the formula: This represents the average turbulent velocity. These are the coordinate components; Indicates the generation of ω. This represents k generated due to the average velocity gradient. and ω and k represent the effective diffusion coefficients, respectively. This represents the dissipation of ω and k in turbulence. This represents the cross-diffusion term. Represents user-defined source terms.

[0133] The SST k-ω turbulence model combines the high accuracy of the k-ω model in the near-wall region with the robustness of the k-ε model in free shear flow, optimizes the calculation of turbulent viscosity, and can accurately calculate the heat transfer coefficient, making it suitable for studying the internal flow field characteristics of exhaust manifold 1.

[0134] When determining the boundary conditions, the actual operating conditions of exhaust manifold 1 need to be considered. The six inlet values ​​of exhaust manifold 1 are set to the same value, the two outlet values ​​are set to the same value, and the wall heat transfer coefficient is 50. The inlet uses a velocity inlet boundary, with a set velocity of 10 m / s, turbulence intensity of 10%, and inlet temperature of 1023.15 K. The outlet uses a pressure outlet boundary, with a set gauge pressure of 0 kPa and recirculation turbulence intensity of 10%. Steady-state thermal analysis of the solid domain applies a third type of boundary condition, and circumferential constraints are applied to the bolt hole section.

[0135] Taking the original model of exhaust manifold 1 as the object, the mesh independence of the fluid part was first verified. Using the average static temperature as the index, five groups of meshes with different sizes were set. The size of group 1 was increased by 20% to obtain group 2, and the size of group 2 was increased by 30%, 40%, and 100% respectively to obtain groups 3, 4, and 5. When generating the volume mesh, the mesh growth rate was set to 1.15 to achieve a smooth transition of mesh size. The Curvature & proximity size function was used to adaptively refine the mesh based on geometric curvature and feature proximity. The number of gap filling element layers was set to 3 to ensure accurate capture of near-wall flow and heat transfer characteristics. For the steady-state thermal and static structural modules, five groups of tetrahedral meshes with different sizes were set using the overall average temperature of the exhaust manifold 1 shell as the index. The mesh generation data are shown in Tables 4 and 5.

[0136]

[0137]

[0138] The results show that in the fluid domain, the mesh scheme with a minimum mesh size of 0.39 mm and a maximum mesh size of 7.9 mm can maximize the reduction of computation time while meeting the accuracy requirement of relative deviation of core indicators ≤2%. In the steady-state thermal and static structure modules, the overall average temperature relative deviation is only 1.31% when the mesh size is 1.5 mm, which meets the irrelevance judgment criterion and minimizes the consumption of time and resources while ensuring simulation accuracy.

[0139] According to formula (1), α = 0.4, C = 0.25, Take ΔPEEQ.

[0140] The height of the reinforcing rib 4, the thickness of the peripheral fins 2, and the groove depth of the hollow flange 3 were selected as factors in the response surface methodology (RSM) fitting analysis. Before constructing the RSM system, single-factor simulation experiments were conducted on these three factors. Based on the single-factor simulation results, it was found that when only the reinforcing ribs were applied between manifolds 1, and the height of the reinforcing ribs was 16 mm, the exhaust manifold 1 exhibited the smallest ΔPEEQ and the largest thermal fatigue life. Similarly, when only the peripheral fins 2 were added at the outlet, and the thickness of the fins 2 was 2 mm, the exhaust manifold 1 exhibited the smallest ΔPEEQ and the largest thermal fatigue life. Finally, when only the inlet flange 3 was grooved, and the groove depth was 1 mm, the exhaust manifold 1 exhibited the smallest ΔPEEQ and the largest thermal fatigue life.

[0141] This invention employs the Box-Behnken design method for sample design. The Box-Behnken design method selects parameter combinations at the midpoint of the sample space edge and the center of the sample space as samples. Each factor always has three levels: the maximum, minimum, and median of the factor's value range. Table 6 shows the factor and level design table. A response surface methodology system is constructed using three factors and three levels, totaling 17 data sets, including 12 factorial points and 5 replicated zero points. The Box-Behnken design and results are shown in Table 7.

[0142]

[0143]

[0144] A multinomial regression fitting analysis was performed, and the regression equation of ΔPEEQ with the three parameters was obtained as follows:

[0145]

[0146] Based on the above equation and combined with equation (1), the regression equation for the thermal fatigue life of the exhaust manifold can be derived:

[0147]

[0148] Table 8 shows the results of the analysis of variance for the multiple regression model. A represents height, B represents depth, C represents thickness, and the experimental index is ΔPEEQ. A p-value < 0.0001 and an F-value of 40.51 indicate that the model term is significant. For the model test terms, a p-value < 0.05 indicates that the model term is significant, and a p-value < 0.01 indicates that the model term is highly significant. In this case, the p-values ​​for A, B, and C are all 0.0001, and the p-value for BC is 0.0016. 2 The p-value is 0.0474, C 2 The p-value is 0.0002, therefore we know that A, B, C, BC, B 2 C 2 This is an important salient model term. The F-value for the lack-of-fit term is 1.92, the P-value is 0.2678, and the R-value is... 2 The value is 0.9812, adjust R. 2 The value is 0.9569, and the predicted R is... 2 The value is 0.8100. All data indicators in the experiment are within a reasonable range. The model with the missing term is not significant. The model fit is good and the data is relatively reliable.

[0149]

[0150] Based on the Box-Behnken experiment results, the following diagram was drawn: Figure 4 The three-dimensional response surface plot and contour plot are shown. According to... Figure 4 (a) It can be seen that under the interaction of height and depth, ΔPEEQ gradually increases with increasing height, and the change pattern is close to linear. It also gradually increases with increasing depth. The overall change of ΔPEEQ is similar to that of both factors, but within the horizontal change range of depth, the tangent plane of the response surface on the side of its single factor gradually becomes steeper and the slope of the tangent gradually increases, and the response surface is concave overall.

[0151] according to Figure 4 (b) It can be seen that under the interactive influence of thickness and height, ΔPEEQ approaches its minimum state when the thickness value reaches approximately 2. As the thickness increases, ΔPEEQ exhibits a change state of first decreasing and then increasing. Specifically, within the range of the thickness range, the increase value of ΔPEEQ is less than the decrease value. The tangent plane of the response surface on the side of the single factor first gradually becomes gentler and then gradually becomes steeper, and the tangent slope first gradually decreases and then gradually increases. According to Figure 4 (c) It can be seen that, under the interaction of thickness, ΔPEEQ is more affected by depth than by thickness, and the interaction effect between the two is significant. According to the contour map, the response point is located in the blue-green area of ​​the map, and the minimum value of PEEQ can be predicted.

[0152] According to the software calculation results, when the height is 14.509mm, the depth is 0.173mm, and the thickness is 2.937mm, the minimum value of ΔPEEQ is 0.00870744.

[0153] The constructed neural network model adopts a 3-10-8-1 hierarchical structure, such as... Figure 5 The input layer contains 3 neurons to correspond to the 3 input parameters. There are 2 hidden layers: the first layer contains 10 neurons and uses the tansig activation function to enhance the model's ability to fit nonlinear relationships; the second layer contains 8 neurons and uses the purelin activation function to avoid generalization errors caused by excessive nonlinearity. The number of neurons in the hidden layers was determined based on empirical formulas and repeated adjustments to balance the model's fitting ability and generalization ability. The output layer contains 1 neuron to output the predicted ΔPEEQ value.

[0154] The overall process includes three core steps: dataset preprocessing and augmentation, network construction and training, and performance evaluation and visualization. At the same time, special handling strategies are implemented to address key issues such as insufficient sample size, overfitting risk, and training stability.

[0155] To address the overfitting problem caused by insufficient original sample size, a Gaussian noise enhancement method is used to generate virtual samples.

[0156] After completing the construction and significance verification of the response surface model based on the simulation results of the 17 BBD test points in Table 7, in order to further improve the coverage and representativeness of the dataset to the design space, 12 supplementary test points that were not included in the full factorial design were selected for additional simulation. Combined with the original 17 test points of BBD, the full factorial simulation dataset is finally formed as shown in Table 9.

[0157] Add normally distributed noise to the input features of each set of original samples, with the noise intensity being 2% of the standard deviation of the input features.

[0158] A weak noise is added to the target output, with the noise intensity being 1% of the output standard deviation, to simulate the measurement error of the experiment.

[0159] The dataset was eventually expanded from 29 groups to 609 groups, significantly improving sample diversity.

[0160]

[0161] After data augmentation, the `randperm` function was used to generate random indices, shuffling the 609 augmented datasets to avoid training bias caused by sample order. Min-Max linear normalization was implemented using the `mapminmax` function, mapping the input features and target output to the [-1,1] interval to eliminate dimensional differences and optimize model training. To further control overfitting, a random partitioning strategy was used to divide the augmented dataset into training, validation, and test sets, with proportions of 70%, 15%, and 15%, respectively. The training set was used for iterative updates of network parameters, the validation set for real-time monitoring of model generalization performance (early stopping was triggered when validation performance showed no improvement for several consecutive rounds), and the test set for final evaluation of the model's actual prediction performance. The Levenberg-Marquardt algorithm corresponding to the `trainlm` function was used for model training. This algorithm combines the advantages of gradient descent and Gauss-Newton methods, offering both fast convergence speed and high fitting accuracy.

[0162] To ensure model training efficiency and generalization ability, multi-dimensional training termination conditions were set:

[0163] (1) Maximum number of rounds: 1000: to avoid training time being too long.

[0164] (2) Target error 1e-7: to ensure the accuracy of model training.

[0165] (3) Early stopping parameter: Training is terminated when the validation set error does not decrease for 60 consecutive rounds, which effectively suppresses overfitting.

[0166] After the BP neural network is constructed, the model is evaluated using the four metrics in formula (4). Figure 6 ,7 As can be seen, the neural network model constructed in this invention exhibits excellent prediction performance and good generalization ability. The prediction errors of the training set, validation set, and test set in the error distribution histogram are all closely concentrated near the zero error line, showing an approximately symmetrical normal distribution, indicating that the model has small prediction bias, high stability, and no obvious systematic bias. The mean squared error (MSE) convergence curve shows that after about 20 iterations, the MSE of the training set, validation set, and test set all decrease rapidly and tend to stabilize. Finally, the curves of the three sets are highly consistent, with no obvious overfitting or underfitting, verifying the rationality of the model structure and the effectiveness of the training process. The comparison curve between the actual value and the predicted value of the test set further confirms that the model prediction results are highly consistent with the actual data, with a mean absolute percentage error (MAPE) of only 0.39%, indicating that the model can accurately capture the mapping relationship between structural parameters and target response, and has the high precision and reliability required for engineering applications.

[0167] 100 sets of sample data were extracted, including 29 experimental samples and 71 fitted samples. The SHAP value of each feature corresponding to the sample was calculated to characterize the marginal contribution of the feature to the prediction result. Figure 8 It can be seen that among the three key structural parameters of thickness, depth, and height, thickness has the largest average SHAP absolute value and the most significant impact on the model output, followed by depth, while height has the smallest impact. In terms of the direction of influence, the SHAP value of thickness is mostly positive, indicating that its increase contributes positively to the target response; the SHAP value of depth increases significantly with its own value, also showing a positive impact, and there is a clear interaction effect with thickness, with higher SHAP values ​​overall for high-thickness samples; while the SHAP value of height is mostly negative when its value is low, showing a negative response, and gradually turns positive as the height increases, with a more prominent positive impact in high-thickness and high-depth samples, reflecting the coupling effect between parameters.

[0168] The optimization objective is to minimize the ΔPEEQ value predicted by the BP neural network. The optimization variables are the stiffener height, slot depth, and fin thickness. Considering that excessively small milling slot depths may significantly reduce accuracy during actual processing, the variable value ranges are determined as follows: height ∈ [14, 18], slot depth ∈ [0.5, 2], and fin thickness ∈ [1, 3]. The BP neural network is encapsulated as the objective function. Normalization and denormalization are used to connect the optimization variables with the model input and output to ensure prediction accuracy. The acquisition function is the Expected Improvement Criterion (EI) built into the toolbox. The BPNN, as the posterior estimation carrier, provides a quantified decision input to the acquisition function by outputting the deterministic predicted value of ΔPEEQ. Based on this posterior estimate, the acquisition function quantifies and evaluates the expected improvement value of each candidate point in the variable search space to select the optimal sampling point. After 100 iterations, the posterior estimate converges to near the global optimum, as shown below. Figure 9 As shown, the optimal combination of height, depth, and thickness extracted represents the best result of the BPNN's posterior estimation of the true objective function. According to the Bayesian optimization results, the minimum value of ΔPEEQ is 0.008713 when the height is 14.0037 mm, the depth is 0.5066 mm, and the thickness is 2.0615 mm.

[0169] To evaluate the prediction performance, the optimal parameter combination optimization model derived using the response surface methodology was used and simulation analysis was performed to compare the actual ΔPEEQ with the predicted value. Similarly, the optimal parameter combination optimization model derived using this methodological framework was used and simulation analysis was performed to compare the actual ΔPEEQ with the predicted value. The comparison results are shown in Table 10. It can be seen that the ΔPEEQ obtained using this methodological framework is smaller and more accurate than the actual value.

[0170]

[0171] To further illustrate the optimization effect of this methodological framework, several optimization methods were compared, and the results are shown in Table 11. As can be seen from the table, while single-location optimization can improve the thermal fatigue life of exhaust manifold 1 to a certain extent, it has limitations; while multi-location collaborative optimization, through parameter combination control, increases the improvement. The optimal parameter combination provided by this methodological framework reduces ΔPEEQ to 0.00874 and increases the thermal fatigue life to 4373. The optimized exhaust manifold 1 shows a 43.47% improvement in thermal fatigue life compared to the original model, fully verifying the significant advantages of this methodological framework in accurately capturing the complex nonlinear coupling relationship between parameters and achieving globally optimal structural design, providing efficient and reliable methodological support for improving the thermal fatigue life of exhaust manifold 1.

[0172]

[0173] In summary, this paper proposes a multi-field coupled optimization method for the thermal fatigue life of exhaust manifolds based on response surface methodology (RSM) and backpropagation neural networks (BPNN). Taking a V6-cylinder 3-2 type exhaust manifold as the research object, and combining previous research, three optimized structures were designed: inter-manifold reinforcing ribs, outlet peripheral fins, and inlet slotted flanges. Thermal-fluid-structure multi-field coupled finite element analysis was conducted on each structure under different design parameters. Based on this, the Box-Behnken method was used to design sample groups and construct a response surface model to fit the correlation between parameters and thermal fatigue life. To address the "black box" problem of the BPNN model, the Shapley additivity interpretation method was introduced to enhance the model's interpretability and describe the interaction effects between parameters. Samples were supplemented and the dataset was expanded based on the response surface analysis results. To accurately establish the nonlinear mapping relationship between structural parameters and thermal fatigue life, a BPNN model was constructed. With maximizing thermal fatigue life as the ultimate goal, the BPNN was used as a probabilistic surrogate model for the Bayesian optimization algorithm to search for the optimal parameter combination. Finally, the optimal parameter combination was determined. This study provides a methodological framework with promotional value for effectively improving the thermal fatigue life of exhaust manifolds, and also provides a technical reference for multi-position collaborative optimization of high-temperature load-bearing components in similar power machinery.

[0174] The embodiments described above are merely preferred embodiments of the present invention and are not intended to limit the scope of the present invention. Therefore, all equivalent changes or modifications made to the structure, features and principles described in the claims of the present invention should be included within the scope of the present invention.

Claims

1. A multi-field coupled optimization method for the thermal fatigue life of an exhaust manifold based on response surface methodology and backpropagation neural network, characterized in that, Includes the following steps: S1: Select the height of the reinforcing ribs, the thickness of the peripheral fins, and the groove depth of the hollow flange of the exhaust manifold as optimization parameters, construct a three-dimensional geometric model of the exhaust manifold, and carry out single-factor thermo-fluid-solid multi-field coupled finite element analysis to obtain the influence law of individual changes of each parameter on the thermal fatigue life characterization quantity. S2: Determine the range and level of parameter values ​​based on the results of single-factor analysis, construct sample groups using the Box-Behnken experimental design method, conduct multi-field coupled finite element simulations on each sample group, construct a response surface model based on the least squares method using a multivariate quadratic polynomial as the response function, and complete the preliminary fitting of parameters and response quantities. S3: Using the core samples obtained from the response surface methodology experimental design as the base, the sample dataset is expanded by combining two methods: completing the fully factorial simulation data and generating expanded samples using the virtual sample method, thereby improving the parameter space coverage and sample size. S4: Normalize and randomly partition the expanded dataset, construct a backpropagation neural network model, train the network using the Levenberg-Marquardt algorithm, and evaluate the model's prediction accuracy and generalization ability using multiple metrics. S5: Introducing the Shapley additivity interpretation method to calculate the SHAP value of each input parameter, quantify the marginal contribution, influence direction and interaction of each parameter to the prediction result, and improve the interpretability of the backpropagation neural network model; S6: The trained backpropagation neural network model is used as a probabilistic surrogate model for the Bayesian optimization algorithm. The optimization objective is to minimize the equivalent plastic strain increment. The optimal parameter combination is searched within the range of feasible engineering parameters, and the optimization results are verified by finite element simulation.

2. The multi-field coupled optimization method for thermal fatigue life of exhaust manifolds based on response surface methodology and backpropagation neural network as described in claim 1, characterized in that, The specific components of the thermo-fluid-solid multi-field coupled finite element analysis described in step S1 include: S11: Construct a three-dimensional geometric model of the exhaust manifold, apply stiffening ribs, peripheral fins and hollow flanges separately, and perform single-factor multi-field coupled simulation on the height of stiffening ribs, thickness of peripheral fins and groove depth of hollow flanges. The coupled simulation puts the thermal field, flow field and solid mechanical field in the same equation matrix for solution, and uses steady-state simulation for analysis. S12: Through internal fluid analysis of the exhaust manifold, the temperature and pressure fields of the fluid and solid domains are obtained. Then, the temperature obtained from the fluid analysis is placed in the thermal analysis module to solve the temperature field of the exhaust manifold, completing the fluid-thermal coupling process. Finally, the pressure and temperature fields obtained from the fluid and thermal analysis are used as boundary conditions and mapped to the static solid structure module to solve the thermal stress distribution and equivalent plastic strain distribution of the exhaust manifold, completing the fluid-thermal-solid three-field coupling analysis. S13: The relationship between thermal fatigue life N and plastic strain range is expressed as follows: （1）； in, Where N is the plastic strain range, C is the crack initiation life, and α is a material constant. The equivalent plastic strain increment of exhaust manifold 1 within a specified working cycle is used as... The value of is used for thermal fatigue life estimation, and and this This serves as the core characterization parameter for the thermal fatigue life of the exhaust manifold 1.

3. The multi-field coupled optimization method for thermal fatigue life of exhaust manifolds based on response surface methodology and backpropagation neural network as described in claim 1, characterized in that, The reinforcing ribs, peripheral fins, and hollow flanges mentioned in step S1, wherein the reinforcing ribs are arranged between adjacent manifolds to improve structural rigidity and disperse thermal stress; The peripheral fins are arranged around the exhaust manifold outlet to enhance heat dissipation and reduce local stress concentration. The hollow flange is formed by milling a groove in the middle, which is used to reduce the flange stiffness and reduce the stress effect of external constraints on the manifold.

4. The multi-field coupled optimization method for thermal fatigue life of exhaust manifolds based on response surface methodology and backpropagation neural network as described in claim 1, characterized in that, The response surface model construction described in step S2 specifically includes: S21: Based on the single-factor analysis results of step S1, determine the value range of the three optimization parameters: stiffener height, peripheral fin thickness, and hollow flange slot depth. Each parameter is set to three levels: low, medium, and high. S22: A sample group was constructed using a three-factor, three-level Box-Behnken experimental design, and multi-field coupled finite element simulations were performed on each sample group. S23: The expression for a multivariate quadratic polynomial as a response function is: （2）； in, For specific response quantities, The fitting parameters are to be determined. The levels of the m-th and n-th factors are respectively (where the levels can be set). k represents the total number of factors; S24: Test the significance of the model through analysis of variance, requiring model P<0.05 and lack-of-fit P>0.05, to verify the effectiveness and reliability of the model fit.

5. The multi-field coupled optimization method for thermal fatigue life of exhaust manifolds based on response surface methodology and backpropagation neural network as described in claim 1, characterized in that, The sample dataset expansion mentioned in step S3 specifically involves: Based on the response surface methodology experimental design in step S2, the sample dataset is expanded in two ways: The first approach is to conduct supplementary simulations on the parameter combinations that were not extracted by Box-Behnken in the fully factorial design, forming a fully factorial simulation dataset that covers the complete parameter space. The second approach is to introduce a Gaussian noise virtual sample method based on a fully factorial dataset. Controllable random noise is added to the input parameters and output response respectively, which greatly expands the number of samples without changing the data distribution pattern.

6. The multi-field coupled optimization method for thermal fatigue life of exhaust manifolds based on response surface methodology and backpropagation neural network as described in claim 1, characterized in that, The training and evaluation of the backpropagation neural network model in step S4 specifically includes: S41: After data augmentation, use the randperm function to generate random indices to shuffle the augmented data and avoid model training bias caused by sample order. S42: The augmented dataset is divided into training, validation and test sets using a random partitioning strategy; S43: The Levenberg-Marquardt algorithm is used to optimize network weights, and multi-dimensional training termination conditions are set to prevent overfitting; S44: Select the mean squared error, mean relative error, mean absolute error percentage and coefficient of determination to evaluate the model performance, as shown in formula (3): （3）； Where n is the number of samples, For the true value, For predicted values, It is the average of the true values.

7. The multi-field coupled optimization method for thermal fatigue life of exhaust manifolds based on response surface methodology and backpropagation neural network as described in claim 1, characterized in that, Step S5, which introduces the Shapley additivity interpretation method to enhance the interpretability of the neural network model, specifically includes: S51: Extract the total sample, which includes both the experimental sample and the fitted sample; S52: Calculate the SHAP value of each feature for the corresponding sample to characterize the marginal contribution of the feature to the prediction result; S53: Draw a SHAP summary diagram, a feature importance ranking diagram, and a dependency diagram to reveal the contribution direction, intensity, and interaction mechanism of each parameter to the equivalent plastic strain increment.

8. The multi-field coupled optimization method for thermal fatigue life of exhaust manifolds based on response surface methodology and backpropagation neural network as described in claim 1, characterized in that, Step S6, which describes using the backpropagation neural network as a probabilistic surrogate model for the Bayesian optimization algorithm to search for the optimal parameter combination, specifically includes: S61: Using a backpropagation neural network as a probabilistic proxy model to proxy an unknown objective function; S62: Exploration and development in the optimization process using expected improvement criteria; S63: The optimization objective is to minimize the equivalent plastic strain increment, and the range of variable values ​​is limited during the optimization process; S64: The optimal parameter combination obtained by optimization is verified by finite element simulation to confirm that it can effectively reduce the equivalent plastic strain increment and improve the thermal fatigue life, thus ensuring the engineering applicability of the optimization results.

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