Hydrostatic spindle parameter optimization method based on multi-objective heterogeneous evolutionary algorithm

Abstract

The application discloses a kind of liquid static pressure spindle parameter optimization method based on multi-objective heterogeneous evolutionary algorithm, for static pressure spindle oil film stiffness and temperature rise collaborative optimization problem, with oil film thickness, orifice diameter, orifice depth as core design variable, MO-HAEA multi-objective optimization model is constructed, three kinds of heterogeneous sub-algorithms of MOPSO, MODE, NSGA-II are fused, UCB adaptive scheduling mechanism is introduced to dynamically allocate sub-algorithm execution period and resource, the dynamic balance of global exploration and local mining is realized through the advantage complementation between sub-algorithms, pareto frontier solution set is solved efficiently, after TOPSIS decision screening, the optimal structure parameter combination of high stiffness, low temperature rise is obtained, realize the precision promotion of static pressure spindle performance optimal design parameter solution.The method of the application discards fixed search mode, integrates the advantages of multiple heterogeneous sub-algorithms, significantly improves the performance of static pressure spindle, is suitable for static pressure spindle design optimization and related fields, solves the technical problem that oil film stiffness and temperature rise are difficult to collaborative optimization.

Description

Technical Field

[0001] This invention belongs to the field of hydrostatic spindle structure optimization technology, specifically relating to a method for optimizing the parameters of a hydrostatic spindle based on a multi-objective heterogeneous evolutionary algorithm. Background Technology

[0002] The rapid development of aerospace, precision molds, and optical devices has placed higher demands on the machining accuracy of CNC machine tools. As a core component of precision CNC machine tools, the spindle system's performance directly affects machining quality. Studies show that thermal errors account for 40% to 70% of machining errors in precision machine tools, making it a major factor affecting accuracy. Hydrostatic spindles, which form a stable oil film through an external oil supply system, offer advantages such as high rigidity, high damping, and high rotational accuracy, and are widely used in precision machining. However, existing hydrostatic spindle designs still have the following technical problems:

[0003] (1) It is difficult to optimize oil film stiffness and temperature rise in a coordinated manner;

[0004] Under high-speed and variable-load conditions, oil film stiffness is significantly affected by oil supply pressure, throttling method, and oil cavity structure. Simultaneously, the lubricating oil experiences a significant temperature rise due to shearing during flow, leading to a decrease in viscosity and weakening its load-bearing capacity. Stiffness and temperature rise are mutually restrictive, and existing structures struggle to simultaneously meet the requirements for high stiffness and low temperature rise.

[0005] (2) Existing optimization methods are mostly limited to single parameters or single objectives;

[0006] The paper "Lim H, Kim K, Ryu K. Effects of Recess Depth and Orifice Diameter on Pneumatic Hammer Instability in Hydrostatic Thrust Bearings: Analysis and Experimental Validation. Turbo Expo: Power for Land, Sea, and Air. American Society of Mechanical Engineers, 2024, 88025: V10AT22A020" studies the influence of the orifice diameter and oil chamber depth on the performance of a hydrostatic spindle. This paper analyzes the effects of oil supply pressure, orifice diameter, and oil chamber depth on the spindle's load-bearing capacity, stiffness, and damping characteristics using a combination of theoretical analysis and experimental verification. The results show that reducing the oil chamber depth helps suppress air hammer instability, and appropriately selecting the orifice diameter can improve spindle performance. However, changes in the orifice diameter may lead to a decrease in load-bearing capacity and stiffness. This method can effectively reveal the influence of orifice diameter and oil chamber depth on spindle performance, but it mainly focuses on the stability of thrust hydrostatic spindles and does not consider the influence of temperature rise. The paper “Gao S, Shang Y, Gao Q, et al. CFD-based investigation oneffects of orifice length–diameter ratio for the design of hydrostatic thrustbearings. Applied Sciences, 2021, 11(3): 959” studies the influence of orifice structural parameters on the performance of hydrostatic thrustbearings. The paper establishes a computational fluid dynamics (CFD) model to analyze the influence of the orifice length-diameter ratio (i.e., the orifice geometric parameters) on pressure distribution, flow characteristics and oil film stiffness. The results show that the orifice geometric parameters have a significant impact on the load capacity, stiffness and flow resistance of the spindle. This method can reflect the effect of orifice size parameters on spindle performance relatively accurately. However, its research mainly focuses on the single structural parameter of the orifice and does not conduct a combined analysis of parameters such as oil film thickness and oil cavity depth.The paper “Michalec M, Ondra M, Svoboda M, et al. Anovel geometry optimization approach for multi-recess hydrostatic bearing padoperating in static and low-speed conditions using CFD simulation. Tribology Letters, 2023, 71(2): 52” proposes a CFD simulation-based method for optimizing the structure of a hydrostatic spindle. This paper establishes a multi-cavity hydrostatic spindle model, optimizes the size and layout parameters of the cavities, and analyzes the influence of different structural parameters on the load-bearing capacity and energy loss. The results show that reasonable optimization of the cavity geometry parameters can effectively reduce energy loss and improve spindle performance. This method can achieve multi-parameter optimization design, but its optimization objects are mainly focused on the cavity area and position parameters, and insufficient attention is paid to key parameters such as the diameter of the throttling orifice and the thickness of the oil film.

[0007] Therefore, it is necessary to study a new method for optimizing the structural parameters of a hydrostatic spindle to solve the technical problem that it is difficult to optimize oil film stiffness and temperature rise in the existing technology. Summary of the Invention

[0008] To address the aforementioned technical problems, this invention provides a method for optimizing the parameters of a hydrostatic spindle based on a multi-objective heterogeneous evolutionary algorithm. Through precise optimization design, the performance of the hydrostatic spindle is significantly improved.

[0009] The technical solution adopted in this invention is: a method for optimizing the parameters of a hydrostatic master shaft based on a multi-objective heterogeneous evolutionary algorithm, the specific steps of which are as follows:

[0010] S1. Establish a three-dimensional oil film model that considers viscosity-temperature effect and fluid-thermal-solid coupling;

[0011] S11. Construct the viscosity-temperature coupling equation;

[0012] The temperature-viscosity-pressure coupling feedback mechanism is represented by an exponential model, as shown in the following expression:

[0013] ;

[0014] in, Indicates the temperature of the lubricating oil The dynamic viscosity below, Indicates the lubricating oil at the reference temperature The dynamic viscosity below, Indicates the viscosity-temperature coefficient of lubricating oil. This indicates the real-time temperature of the oil film. Indicates the reference temperature.

[0015] S12. The separation coupling method is used to solve the fluid-thermal-structure coupling problem.

[0016] First, create a 3D geometric model containing the fluid domain and the solid domain in 3D CAD, paying attention to the accurate handling of the coupling surface. Then, start the simulation software, import the geometry or mesh, perform basic solver settings, and generate fluid domain mesh and solid domain mesh in the mesh generation software respectively.

[0017] Then, in the simulation software, a fluid-thermal-solid coupled boundary is set, the mesh is generated and exported, and numerical simulation is performed to solve the continuity, momentum, and energy equations to obtain the temperature and pressure fields of the fluid domain. Next, it is determined whether the energy equation has converged. If it has converged, the temperature and pressure field data of the fluid domain are transferred to the solid domain to obtain the temperature and pressure fields. If it has not converged, the modeling stage is returned to adjust the model or mesh, and the fluid domain simulation is re-executed until convergence.

[0018] The material properties of the solid domain are redefined, and mechanical and thermal boundary constraints are applied. Using the transferred fluid temperature boundary data, the temperature field of the solid domain is solved in the simulation software. Based on the solid temperature field and the pressure load transferred by the fluid, the solid deformation field is solved. Finally, the results are output, including temperature, pressure, deformation, and stress, and the simulation ends.

[0019] S2. Based on step S1, select design variables and determine the parameter combination scheme through Latin hypercube sampling;

[0020] S21. Select design variables;

[0021] Based on the structural characteristics and optimization objectives of the hydrostatic spindle, and according to the actual situation, select... Key structural parameters are used as design variables.

[0022] The key structural parameters include: oil film thickness. Orifice diameter Orifice depth .

[0023] S22. Generate parameter combination schemes through Latin hypercube sampling (LHS);

[0024] The Latin hypercube sampling method is used for step S21. Sample each design variable to generate Group parameter combination schemes, each scheme corresponds to a set of input parameters The data is directly input into the three-dimensional oil film model for simulation calculation, and the corresponding performance indicators are obtained to construct the input-output sample set.

[0025] The performance indicators include: average temperature of the rotating surface of the oil film and oil film stiffness.

[0026] S3. Based on step S2, construct a high-precision prediction agent model based on the Kriging model;

[0027] The Kriging model consists of two parts: a regression function and a stochastic process. Therefore, a high-precision predictive surrogate model based on the Kriging model is constructed, expressed as follows:

[0028] ;

[0029] in, This represents the model's predicted value. Represents the global trend function. Let represent a random process with a mean of zero. The covariance function of this random process is expressed as follows:

[0030] ;

[0031] in, Represents the variance of a random process. Represents the relevant function, Let represent two different spatial sample points of a random process. The formal expression of the Gaussian correlation function is as follows:

[0032] ;

[0033] in, Indicates the relevant parameters, Indicates the number of design variables. They represent the first The and the first The first spatial sample point 3D coordinate components.

[0034] S4. Based on step S3, establish a multi-objective optimization model using the improved heterogeneous alternation evolutionary algorithm based on UCB scheduling;

[0035] S41, Algorithm initialization;

[0036] 1) Initialize parameters;

[0037] Set the maximum number of iterations. Population size UCB scheduling coefficient Scheduling cycle Exceeding the volume threshold Initialize the primary population. Establish external archives It is used to store non-dominated solutions generated during the iteration process.

[0038] 2) Construct a heterogeneous sub-algorithm pool;

[0039] The sub-algorithm pool includes three multi-objective evolutionary algorithms: Multi-objective Particle Swarm Optimization (MOPSO), Multi-objective Differential Evolutionary Algorithm (MODE), and Non-dominated Sorting Genetic Algorithm (NSGA-II). The initial parameters of each sub-algorithm are set uniformly.

[0040] 3) Sub-algorithm warm-up evaluation;

[0041] Perform short-time iterations on each sub-algorithm and record the initial performance metrics of each algorithm, including: hypervolume. Number of non-dominated solutions .

[0042] S42, UCB scheduler and selection mechanism;

[0043] Introducing a scheduler based on confidence upper bound (UCB), after a preset scheduling period... The algorithm dynamically selects the current optimal sub-algorithm for iteration, and its core mathematical model is as follows:

[0044] 1) Calculation of the overall score of the sub-algorithm;

[0045] For the first in the sub-algorithm pool Each sub-algorithm, its overall score The calculation expression is as follows:

[0046] ;

[0047] in, =1,2,3, corresponding to MOPSO, MODE, and NSGA-II; Sub-algorithm The cumulative reward value, Represents the UCB scheduling coefficient. Indicates the total number of scheduling operations. Sub-algorithm The number of times it was selected.

[0048] 2) Scheduling rules;

[0049] Select the comprehensive score for each scheduling. The highest-performing sub-algorithm is used as the execution algorithm for the current iteration cycle. After the cycle is completed, the scheduling count is updated. AND sub-algorithm selected times Then proceed to the next round of scheduling.

[0050] S43, Iteration and Population Update;

[0051] Iterative execution: Following the sub-algorithm selected by the UCB scheduler, the main population is... Perform the corresponding evolutionary operations to generate a progeny population.

[0052] Among them, MOPSO performs particle updates and velocity iterations; MODE performs mutation and crossover operations; and NSGA-II performs non-dominated sorting, crowding calculation, and selection operations.

[0053] Population interaction and selection: The offspring population is merged with the parent population, and the best individuals are retained through non-dominant ranking and crowding selection to update the main population. At the same time, the non-dominated solution is stored in an external archive. If the number of solutions in the external archives exceeds a preset threshold, redundant solutions are deleted by sorting by congestion.

[0054] S44, Reward Mechanism and Performance Feedback;

[0055] After each selected sub-algorithm is executed and the population is updated, the performance improvement of that sub-algorithm is calculated and used as a reward value. Update cumulative reward value The details are as follows:

[0056] 1) Super-volume increase : ;

[0057] in, This indicates the increased size of the external file after the update. This indicates the external file size exceeded the limit before the update.

[0058] 2) Increase in the number of non-dominated solutions: ;

[0059] in, , These represent the number of non-dominated solutions in the external archives before and after the update, respectively.

[0060] 3) Cumulative reward value update: ;

[0061] in, This represents the weighting coefficient.

[0062] S45. Update statistical information;

[0063] After completing the calculation of sub-algorithm reward values ​​and updating the cumulative reward values, various statistical information from the algorithm iteration process is updated synchronously. The specific update content and rules are as follows:

[0064] 1) Sub-algorithm performance statistics update;

[0065] Record the single-iteration performance metrics of the current sub-algorithm, including: hypervolume. Number of non-dominated solutions The iteration time is calculated, and then it is summarized with the historical iteration performance data of this sub-algorithm to calculate its average. Increase, average Improve the quantity and average iteration efficiency, and form a statistical ledger of sub-algorithm performance.

[0066] 2) Update scheduling-related parameters;

[0067] Update total scheduling count Increment it by 1, and simultaneously update the selection count of the corresponding sub-algorithm. It also increments by 1.

[0068] S46. Convergence judgment;

[0069] After each iteration, calculate the external archive. Super volume increase Stop iteration if any of the following conditions are met:

[0070] 1) Number of iterations Reaching the maximum number of iterations ;2) Super-volume increase .

[0071] Among them, external files after stopping iteration All non-dominated solutions constitute the Pareto front solution set, which includes multiple combinations of structural parameters capable of achieving synergistic optimization of principal shaft stiffness and temperature rise. .

[0072] If none of the conditions are met, return to S42 to continue iterative calculation.

[0073] S47. Establish a multi-objective optimization model;

[0074] A multi-objective optimization model is constructed with oil film temperature and oil film stiffness as optimization indices. First, the average temperature of the rotating surface of the oil film is selected as the thermal performance evaluation index, and the temperature objective function expression is constructed as follows:

[0075] ;

[0076] in, Represents a design variable vector. This indicates the transpose operation. This indicates the average temperature of the rotating surface of the oil film.

[0077] The stiffness objective function expression is reconstructed as follows:

[0078] ;

[0079] in, This indicates the stiffness of the oil film.

[0080] The multi-objective optimization model expression for the main shaft structure is as follows:

[0081] ;

[0082] in, This indicates the lower limit of the oil film thickness design. This indicates the upper limit of the oil film thickness design. This indicates the lower limit of the design depth for the throttle orifice. This indicates the upper limit of the design depth of the throttle orifice. This indicates the lower limit of the design diameter for the throttling orifice. This indicates the upper limit of the design diameter of the throttling orifice.

[0083] S5. Based on step S4, determine the optimal compromise solution using the TOPSIS method to optimize the parameters of the hydrostatic spindle.

[0084] The Pareto front solution set obtained by optimization using the MO-HAEA (Multi-Objective Heterogeneous Alternating Evolutionary Algorithm) algorithm includes multiple sets of non-dominated solutions, each set corresponding to a set of structural parameter combinations. Then, the Top-Ideal Solution Ranking Method (TOPSIS) is used for multi-objective decision-making. By calculating the relative proximity of each alternative solution to the positive and negative ideal solutions, all non-dominated solutions are ranked, and the solution with the highest proximity is the optimal compromise solution.

[0085] Furthermore, step S22 is specifically as follows:

[0086] S221. Selection of the number of sampling points;

[0087] First, based on the accuracy requirements of training a high-precision prediction surrogate model based on the Kriging model, the number of sampling points is selected. Then, the sampled data is divided into training set and validation set according to the actual situation.

[0088] Then the value range of each design variable is divided into... Given several equally probable subintervals, and randomly select one sample point from any subinterval of each variable, ensuring that each subinterval is selected only once, finally randomly combine the sample points of each variable to generate... Different parameter combination schemes, i.e., sampling sample matrix The expression is as follows:

[0089] ;

[0090] in, , , They represent the first The values ​​of oil film thickness, orifice diameter, and orifice depth in the group sampling scheme are determined. .

[0091] S222, Sample homogeneity verification;

[0092] First, the sample homogeneity is verified using the maximum-minimum distance criterion or correlation coefficient analysis, i.e., the sampling sample matrix is ​​calculated. The Euclidean distance between any two sets of parameters is calculated, ensuring that the minimum distance is greater than a threshold; and the calculation... Pearson correlation coefficient among design variables .

[0093] in, <0.1.

[0094] S223, Simulation Input Scheme;

[0095] The product generated in step S221 The parameter combinations are sequentially input into the three-dimensional oil film model considering viscosity-temperature effect and fluid-structure interaction constructed in step S1, and CFD simulation calculations are performed to obtain the average temperature of the oil film rotation surface and the oil film stiffness corresponding to each set of parameters, and an input-output sample set is constructed.

[0096] Furthermore, in step S3, after the Kriging proxy model is established, the prediction accuracy of the model is verified, as follows:

[0097] By comparing the simulation results with the prediction results of the Kriging surrogate model, and using the coefficient of determination... The root mean square error (RMSE) and mean relative error (MARE) are used to evaluate the model accuracy. Let the true sample value be... The surrogate model predicts the value. The sample size is The sample mean is .

[0098] (1) The expression for the coefficient of determination is as follows:

[0099] ;

[0100] (2) The root mean square error expression is as follows:

[0101] ;

[0102] (3) The expression for the average relative error is as follows:

[0103] .

[0104] Furthermore, step S5 is specifically as follows:

[0105] S51. Construct the decision matrix;

[0106] Suppose that the Pareto front solution set output by the MO-HAEA algorithm contains There are 1 non-dominated solution, and each solution corresponds to 2 optimization objectives, namely oil film stiffness. Average temperature of oil film rotating surface Then construct the decision matrix. Matrix elements Indicates the first The nondominated solution is at the th... The response values ​​under each optimization objective are expressed by the following decision matrix:

[0107] ;

[0108] in, , This represents the number of non-dominated solutions in the Pareto front solution set. This corresponds to optimization objective 1, namely oil film stiffness. Maximize the goal; This corresponds to optimization objective 2, namely the average temperature of the oil film rotating surface. Minimize the target.

[0109] S52, Decision matrix normalization processing;

[0110] For decision matrix Normalization is performed to eliminate the influence of dimensions, then when Maximize the target oil film stiffness The normalization formula is as follows:

[0111] ;

[0112] when Minimize the average temperature of the target oil film's rotating surface. The normalization formula is as follows:

[0113] ;

[0114] in, This represents the elements of the normalized decision matrix.

[0115] S53. Determine the weighting coefficients and construct the weighted normalization matrix;

[0116] Based on the actual usage requirements of the hydrostatic spindle, the weighting coefficients for the two optimization objectives are determined, with the stiffness weighting coefficient set to... The weighting factor for temperature is set to And satisfy .

[0117] Then construct a weighted normalized matrix based on the weight coefficients. Matrix elements The weighted normalized matrix expression is as follows:

[0118] ;

[0119] S54. Determine the positive ideal solution and the negative ideal solution;

[0120] Positive Ideal Solution It is a vector composed of the optimal values ​​of each objective in the weighted normalized matrix, representing the negative ideal solution. It is a vector composed of the worst-case values ​​of each objective, and its calculation expression is as follows:

[0121] ;

[0122] ;

[0123] in, Let represent the maximum values ​​of all non-dominated solutions (or alternatives) on the first and second objectives, respectively, i.e., the components of the positive ideal solution. These represent the minimum values ​​of all non-dominated solutions on the first and second objectives, respectively, i.e., the components of the negative ideal solution.

[0124] S55. Calculate the Euclidean distance between each non-dominated solution and the ideal solution;

[0125] The first step is to calculate the Euclidean distance. A nondominated solution and a positive ideal solution distance and negative ideal solutions - distance The calculation expressions are as follows:

[0126] ;

[0127] in, The smaller the value, the closer the non-dominated solution is to the optimal objective; The larger the value, the further the non-dominated solution is from the worst objective.

[0128] S56. Calculate the relative proximity and determine the optimal compromise solution;

[0129] Definition of the first Relative closeness of non-dominated solutions This is used to measure how close the solution is to the ideal solution, and its calculation expression is as follows:

[0130] ;

[0131] in, .

[0132] Then, the relative proximity of all non-dominated solutions is calculated. Sort and select The largest nondominated solution is taken as the optimal compromise solution, and the corresponding combination of structural parameters is... This refers to the optimal structural parameters of the hydrostatic spindle, and finally the optimized oil film structural parameters and calculated and predicted performance.

[0133] The beneficial effects of this invention are as follows: This invention addresses the challenge of coordinating the optimization of oil film stiffness and temperature rise in hydrostatic spindles. Using oil film thickness, orifice diameter, and orifice depth as core design variables, it constructs a three-dimensional oil film model considering viscosity-temperature effects and fluid-structure interaction to obtain accurate simulation data. Latin hypercube sampling is then used to determine the parameter combinations for oil film thickness, orifice diameter, and orifice depth. A high-precision predictive surrogate model is constructed based on the Kriging model to replace traditional time-consuming CFD simulations. Furthermore, an improved heterogeneous alternating evolutionary algorithm (MO-HAEA) based on UCB scheduling is employed to perform Pareto optimization with the goal of maximizing stiffness and minimizing temperature. Finally, the optimal structural parameters are determined from the Pareto front solution set using the TOPSIS method, achieving a significant improvement in the accuracy of solving the optimal design parameters for hydrostatic spindle performance. Compared to traditional single multi-objective evolutionary algorithms, the MO-HAEA proposed in this invention, combined with the UCB scheduling strategy, abandons the fixed search mode and integrates the advantages of multiple heterogeneous sub-algorithms. Through UCB adaptive dynamic optimization scheduling, it intelligently balances global exploration and local mining, avoiding premature convergence and blind search. It can efficiently adapt to complex optimization problems with multiple variables and strong coupling in hydrostatic principal shafts, obtain Pareto solution sets with better distribution and faster convergence, and improve the overall accuracy, efficiency and robustness of multi-objective optimization. Attached Figure Description

[0134] Figure 1 This is a flowchart of a method for optimizing the parameters of a hydrostatic spindle based on a multi-objective heterogeneous evolutionary algorithm, according to the present invention.

[0135] Figure 2 This is a flowchart of the overall solution process for fluid-thermal-solid coupling in an embodiment of the present invention.

[0136] Figure 3 This is a flowchart of the improved heterogeneous alternation evolution algorithm (MO-HAEA) based on UCB scheduling in an embodiment of the present invention.

[0137] Figure 4 This is a schematic diagram showing the comparison between the predicted value and the actual value of the Kriging proxy model described in this embodiment of the invention.

[0138] Figure 5This is a schematic diagram of the temperature field comparison before and after radial oil film optimization in an embodiment of the present invention.

[0139] Figure 6 This is a schematic diagram of the pressure field comparison before and after radial oil film optimization in an embodiment of the present invention.

[0140] Figure 7 This is a schematic diagram comparing the spindle temperature field before and after optimization in an embodiment of the present invention.

[0141] Figure 8 This is a schematic diagram of the deformation field of the main shaft before optimization in an embodiment of the present invention.

[0142] Figure 9 This is a schematic diagram of the optimized spindle deformation field in an embodiment of the present invention. Detailed Implementation

[0143] The method of the present invention will be further described below with reference to the accompanying drawings and embodiments.

[0144] like Figure 1 The flowchart of a method for optimizing the parameters of a hydrostatic master shaft based on a multi-objective heterogeneous evolutionary algorithm, as shown in the figure, includes the following specific steps:

[0145] S1. Establish a three-dimensional oil film model that considers viscosity-temperature effect and fluid-thermal-solid coupling;

[0146] S11. Construct the viscosity-temperature coupling equation;

[0147] The viscosity of lubricating oil changes with temperature. As the temperature increases, the viscosity decreases, which in turn changes the pressure distribution in the ReynoLds equation, forming a coupled feedback mechanism of temperature-viscosity-pressure.

[0148] The temperature-viscosity-pressure coupling feedback mechanism is represented by an exponential model, as shown in the following expression:

[0149] ;

[0150] in, Indicates the temperature of the lubricating oil The dynamic viscosity below, Indicates the lubricating oil at the reference temperature The dynamic viscosity below, Indicates the viscosity-temperature coefficient of lubricating oil. This indicates the real-time temperature of the oil film. Indicates the reference temperature.

[0151] S12. The separation coupling method is used to solve the fluid-thermal-structure coupling problem.

[0152] First, create a 3D geometric model containing both fluid and solid domains in 3D CAD (SOLIDWORKS), paying attention to the accurate handling of coupling surfaces. Then, launch the simulation software, import the geometry or mesh, and perform basic solver settings. Finally, generate separate meshes for the fluid and solid domains in the mesh generation software.

[0153] Then, in the simulation software, a fluid-thermal-solid coupled boundary is set, the mesh is generated and exported, and numerical simulation is performed to solve the continuity, momentum, and energy equations to obtain the temperature and pressure fields of the fluid domain. Next, it is determined whether the energy equation has converged. If it has converged, the temperature and pressure field data of the fluid domain are transferred to the solid domain to obtain the temperature and pressure fields. If it has not converged, the modeling stage is returned to adjust the model or mesh, and the fluid domain simulation is re-executed until convergence.

[0154] The material properties of the solid domain are redefined, and mechanical and thermal boundary constraints are applied. Using the transferred fluid temperature boundary data, the temperature field of the solid domain is solved in the simulation software. Based on the solid temperature field and the pressure load transferred by the fluid, the solid deformation field is solved. Finally, the results are output, including temperature, pressure, deformation, and stress, and the simulation ends.

[0155] In this embodiment, the coupled analysis of the temperature field, based on the calculated boundary conditions in Fluent, is performed using Fluent and Mechanical simulation software in ANSYS. A unidirectional fluid-thermal-structure interaction (FTE) strategy is employed to obtain the corresponding field distribution and structural response. The overall FTE-thermal-structure interaction solution process is as follows: Figure 2 As shown.

[0156] S2. Based on step S1, select design variables and determine the parameter combination scheme through Latin hypercube sampling;

[0157] S21. Select design variables;

[0158] Based on the structural characteristics and optimization objectives of the hydrostatic spindle, and according to the actual situation, select... In this embodiment, key structural parameters are used as design variables. =3.

[0159] The key structural parameters include:

[0160] Oil film thickness : Directly affects the stiffness and load-bearing capacity of the oil film, with a value range of [10μm, 40μm];

[0161] Orifice diameter This determines the oil supply flow rate and pressure regulation characteristics, with a value range of [0.4mm, 1mm].

[0162] Orifice depth : Affects the throttling damping characteristics and oil film flow state, with a value range of [0.2mm, 0.8mm].

[0163] The unoptimized oil film structure parameters and simulation performance of this embodiment are shown in Table 1.

[0164] Table 1

[0165]

[0166] S22. Generate parameter combination schemes through Latin hypercube sampling (LHS);

[0167] Latin hypercube sampling is a multidimensional random sampling method based on stratified sampling. It can achieve uniform sampling of the design variable space with the fewest sampling points while ensuring sample coverage.

[0168] This embodiment uses the Latin hypercube sampling method for step S21. Sample each design variable to generate Group parameter combination schemes, each scheme corresponds to a set of input parameters The data is directly input into the three-dimensional oil film model for simulation calculation, and the corresponding performance indicators are obtained to construct the input-output sample set.

[0169] The performance indicators include: average temperature of the rotating surface of the oil film and oil film stiffness.

[0170] S3. Based on step S2, construct a high-precision prediction agent model based on the Kriging model;

[0171] The Kriging model consists of two parts: a regression function and a stochastic process. Therefore, a high-precision predictive surrogate model based on the Kriging model is constructed, expressed as follows:

[0172] ;

[0173] in, This represents the model's predicted value. Represents the global trend function. Let represent a random process with a mean of zero. The covariance function of this random process is expressed as follows:

[0174] ;

[0175] in, Represents the variance of a random process. Represents the relevant function, Let represent two different spatial sample points of a random process. The formal expression of the Gaussian correlation function is as follows:

[0176] ;

[0177] in, Indicates the relevant parameters, Indicates the number of design variables. They represent the first The and the first The first spatial sample point 3D coordinate components.

[0178] After the proxy model trains its parameters using the sample data in the training set obtained in step S22, it can establish an approximate mapping relationship between the design variables and the objective function, thereby enabling rapid prediction of oil film temperature and oil film stiffness.

[0179] S4. Based on step S3, establish a multi-objective optimization model using the improved heterogeneous alternation evolutionary algorithm based on UCB scheduling;

[0180] S41, Algorithm initialization;

[0181] 1) Initialize parameters;

[0182] Set the maximum number of iterations. Population size UCB scheduling coefficient Scheduling cycle Exceeding the volume threshold Initialize the primary population. Establish external archives It is used to store non-dominated solutions generated during the iteration process.

[0183] 2) Construct a heterogeneous sub-algorithm pool;

[0184] The sub-algorithm pool includes three multi-objective evolutionary algorithms that complement each other: MOPSO (Multi-Objective Particle Swarm Optimization), MODE (Multi-Objective Differential Evolutionary Algorithm), and NSGA-II (Non-Dominated Sorting Genetic Algorithm). The initial parameters of each sub-algorithm are uniformly set to ensure fair comparison.

[0185] 3) Sub-algorithm warm-up evaluation;

[0186] Each sub-algorithm is iterated over a short period of time, and the initial performance indicators (hypervolume HV, number of non-dominated solutions ND) of each algorithm are recorded as the initial basis for subsequent UCB scheduling.

[0187] S42, UCB scheduler and selection mechanism;

[0188] Introduce a scheduler based on upper confidence limit (UCB), which operates every preset scheduling period. The algorithm dynamically selects the current best sub-algorithm for iteration, balancing "utilization" (selecting the best-performing algorithm) with "exploration" (trying algorithms that have not been fully run). Its core mathematical model is as follows:

[0189] 1) Calculation of the overall score of the sub-algorithm;

[0190] For the first in the sub-algorithm pool Each sub-algorithm, its overall score The calculation expression is as follows:

[0191] ;

[0192] in, =1,2,3, corresponding to MOPSO, MODE, and NSGA-II; Sub-algorithm The cumulative reward value, Represents the UCB scheduling coefficient. Indicates the total number of scheduling operations. Sub-algorithm The number of times it was selected.

[0193] 2) Scheduling rules;

[0194] Select the comprehensive score for each scheduling. The highest-performing sub-algorithm is used as the execution algorithm for the current iteration cycle. After the cycle is completed, the scheduling count is updated. AND sub-algorithm selected times Then proceed to the next round of scheduling.

[0195] S43, Iteration and Population Update;

[0196] Iterative execution: Following the sub-algorithm selected by the UCB scheduler, the main population is... Perform the corresponding evolutionary operations to generate a progeny population.

[0197] Among them, MOPSO performs particle updates and velocity iterations; MODE performs mutation and crossover operations; and NSGA-II performs non-dominated sorting, crowding calculation, and selection operations.

[0198] Population interaction and selection: The offspring population is merged with the parent population, and the best individuals are retained through non-dominant ranking and crowding selection to update the main population. At the same time, the non-dominated solution is stored in an external archive. If the number of solutions in the external archives exceeds a preset threshold, redundant solutions are deleted by sorting by congestion to ensure the diversity of the solution set.

[0199] S44, Reward Mechanism and Performance Feedback;

[0200] After each selected sub-algorithm is executed and the population is updated, the performance improvement of that sub-algorithm is calculated and used as a reward value. Update cumulative reward value This enables dynamic feedback on the performance of the sub-algorithm, as detailed below:

[0201] 1) Super-volume increase (Reflecting the convergence and distribution of the solution set): ;

[0202] in, This indicates the increased size of the external file after the update. This indicates the external file size exceeded the limit before the update.

[0203] 2) Increase in the number of non-dominated solutions: ;

[0204] in, , These represent the number of non-dominated solutions in the external archives before and after the update, respectively.

[0205] 3) Cumulative reward value update: ;

[0206] in, This represents the weighting coefficient.

[0207] S45. Update statistical information;

[0208] After completing the calculation of sub-algorithm reward values ​​and updating the cumulative reward values, various statistical information during the algorithm iteration process is updated synchronously to provide a comprehensive and accurate decision-making basis for the next round of UCB scheduler sub-algorithm selection, ensuring the scientific nature and adaptability of the scheduling strategy. The specific update content and rules are as follows:

[0209] 1) Sub-algorithm performance statistics update;

[0210] Record the single-iteration performance metrics of the current sub-algorithm, including: hypervolume. Number of non-dominated solutions The iteration time is calculated, and then it is summarized with the historical iteration performance data of this sub-algorithm to calculate its average. Increase, average Improve the quantity and average iteration efficiency, and create a statistical ledger of sub-algorithm performance for comparative analysis of subsequent sub-algorithm performance.

[0211] 2) Update scheduling-related parameters;

[0212] Update total scheduling count Increment it by 1, and simultaneously update the selection count of the corresponding sub-algorithm. Similarly, increment by 1 to ensure that the UCB composite score calculation formula is accurate. and The real-time performance and accuracy.

[0213] S46. Convergence judgment;

[0214] After each iteration, calculate the external archive. Super volume increase Stop iterating if any of the following conditions are met:

[0215] 1) Number of iterations Reaching the maximum number of iterations ;2) Super-volume increase .

[0216] Among them, external files after stopping iteration All non-dominated solutions constitute the Pareto front solution set, which includes multiple combinations of structural parameters capable of achieving synergistic optimization of principal shaft stiffness and temperature rise. .

[0217] If none of the conditions are met, return to S42 to continue iterative calculation.

[0218] The final Pareto front solution set obtained through iterative optimization using the MO-HAEA algorithm must satisfy the following conditions: uniform distribution of the solution set, no obvious clustering, and overvolume. The values ​​tend to stabilize, indicating that the algorithm has converged. The stiffness and temperature corresponding to the solution set are better than the initial parameter combination, verifying the effectiveness of the optimization effect. The algorithm flow of the improved heterogeneous alternation evolution algorithm (MO-HAEA) based on UCB scheduling is as follows: Figure 3 As shown.

[0219] S47. Establish a multi-objective optimization model;

[0220] Considering that hydrostatic milling spindles need to maintain a low oil film temperature to improve thermal stability and have a high oil film stiffness to ensure machining accuracy and load-bearing capacity during actual operation, a multi-objective optimization model is constructed with oil film temperature and oil film stiffness as optimization indicators.

[0221] Increased oil film temperature leads to decreased lubricating oil viscosity, alters pressure distribution characteristics, and may weaken the oil film's load-bearing capacity, thus affecting the spindle's operational stability. To mitigate the adverse effects of thermal effects on system performance, the average temperature of the oil film's rotating surface is selected as the thermal performance evaluation index, and the temperature objective function expression is constructed as follows:

[0222] ;

[0223] in, Represents a vector of design variables. This indicates the transpose operation. This indicates the average temperature of the rotating surface of the oil film.

[0224] Oil film stiffness is an important parameter for evaluating the load-carrying capacity and disturbance resistance of hydrostatic bearings. It is defined as the rate of change of oil film load-carrying capacity with respect to displacement disturbance. Therefore, the objective function expression for stiffness is constructed as follows:

[0225] ;

[0226] in, This indicates the stiffness of the oil film.

[0227] The multi-objective optimization model expression for the main shaft structure is as follows:

[0228] ;

[0229] in, This indicates the lower limit of the oil film thickness design. This indicates the upper limit of the oil film thickness design. This indicates the lower limit of the design depth for the throttle orifice. This indicates the upper limit of the design depth of the throttle orifice. This indicates the lower limit of the design diameter for the throttling orifice. This indicates the upper limit of the design diameter of the throttling orifice.

[0230] S5. Based on step S4, determine the optimal compromise solution using the TOPSIS method to optimize the parameters of the hydrostatic spindle.

[0231] The Pareto front solution set obtained by the MO-HAEA algorithm includes multiple sets of non-dominated solutions (each set of solutions corresponds to a set of structure parameter combinations). Each non-dominated solution can achieve different balances between principal shaft stiffness and temperature rise, with no absolute superiority or inferiority. To select the optimal compromise solution from the Pareto front solution set that balances maximum principal shaft stiffness and minimum temperature rise, TOPSIS (Topology for Approximating Ideal Solutions) is used for multi-objective decision-making. This method ranks all non-dominated solutions by calculating the relative closeness between each candidate solution and the positive and negative ideal solutions, and the solution with the highest closeness is the optimal compromise solution.

[0232] In this embodiment, step S22 is specifically as follows:

[0233] S221. Selection of the number of sampling points;

[0234] First, based on the accuracy requirements of training a high-precision prediction surrogate model based on the Kriging model, the number of sampling points is selected. The sampled data is then divided into a training set and a validation set based on the actual situation. In this embodiment, 80% of the samples are used as the training set and 20% as the validation set.

[0235] Then the value range of each design variable is divided into... Given several equally probable subintervals, and randomly select one sample point from any subinterval of each variable, ensuring that each subinterval is selected only once, finally randomly combine the sample points of each variable to generate... Different parameter combination schemes, i.e., sampling sample matrix The expression is as follows:

[0236] ;

[0237] in, , , They represent the first The values ​​of oil film thickness, orifice diameter, and orifice depth in the group sampling scheme are determined. .

[0238] S222, Sample homogeneity verification;

[0239] First, the sample homogeneity is verified using the maximum-minimum distance criterion or correlation coefficient analysis, i.e., the sampling sample matrix is ​​calculated. The Euclidean distance between any two sets of parameters is calculated to ensure that the minimum distance is greater than a threshold, thus avoiding sample clustering; and the calculation... Pearson correlation coefficient among design variables .

[0240] in, A value <0.1 ensures that there is no strong correlation between variables, satisfying the orthogonality requirement of Latin hypercube sampling.

[0241] S223, Simulation Input Scheme;

[0242] The product generated in step S221 The parameter combinations are sequentially input into the three-dimensional oil film model considering viscosity-temperature effect and fluid-structure interaction constructed in step S1, and CFD simulation calculations are performed to obtain the average temperature of the oil film rotation surface and the oil film stiffness corresponding to each set of parameters. An input-output sample set is constructed to provide a data foundation for subsequent Kriging surrogate model training.

[0243] In this embodiment, after the Kriging proxy model is established in step S3, the prediction accuracy of the model is verified, as follows:

[0244] After establishing the Kriging surrogate model, its prediction accuracy needs to be verified to ensure that the surrogate model accurately reflects the mapping relationship between design variables and oil film performance indicators, thus replacing the high-precision simulation model in multi-objective optimization calculations. This is achieved by comparing simulation results with the Kriging surrogate model predictions and using the coefficient of determination. The root mean square error (RMSE) and mean relative error (MARE) are used to evaluate the model accuracy. Let the true sample value be... The surrogate model predicts the value. The sample size is The sample mean is .

[0245] (1) Coefficient of determination;

[0246] The coefficient of determination is used to measure the goodness of fit of a model, and its expression is as follows:

[0247] ;

[0248] when The closer the value is to 1, the better the surrogate model fits the sample data.

[0249] (2) Root mean square error;

[0250] Root mean square error (RMSE) measures the deviation between predicted and actual values, and is expressed as follows:

[0251] ;

[0252] The smaller the RMSE, the smaller the prediction error of the surrogate model.

[0253] (3) Average relative error;

[0254] The mean relative error is used to evaluate the relative deviation between the predicted value and the actual value, and its expression is as follows:

[0255] ;

[0256] The established Kriging surrogate model was used to predict oil film temperature and oil film stiffness, and the prediction results were compared and analyzed with the simulation results. By calculating the coefficient of determination, root mean square error, and mean relative error, the predictive ability of the surrogate model can be quantitatively evaluated.

[0257] When the coefficient of determination When the value is close to 1 and the error index is small, it indicates that the Kriging surrogate model has high prediction accuracy and can well reflect the nonlinear relationship between design variables and oil film temperature and oil film stiffness. Under the condition of meeting accuracy requirements, the Kriging surrogate model can be used to replace the high-precision simulation model in subsequent multi-objective optimization calculations, thereby significantly reducing the number of simulation calls and improving the efficiency of structural parameter optimization.

[0258] Table 2 shows the accuracy verification of the Kriging surrogate model. Figure 3The results show a comparison between the predicted and actual values ​​of the Kriging surrogate model. The results demonstrate that the established Kriging surrogate model possesses high prediction accuracy and good reliability, and can replace computationally expensive fluid-thermal-structure interaction simulations for multi-objective optimization calculations, thereby effectively improving the efficiency of structural parameter optimization.

[0259] Table 2

[0260]

[0261] In this embodiment, step S5 is specifically as follows:

[0262] S51. Construct the decision matrix;

[0263] Suppose that the Pareto front solution set output by the MO-HAEA algorithm contains There are 1 non-dominated solution, and each solution corresponds to 2 optimization objectives, namely oil film stiffness. Average temperature of oil film rotating surface Then construct the decision matrix. Matrix elements Indicates the first The nondominated solution is at the th... The response values ​​under each optimization objective are expressed by the following decision matrix:

[0264] ;

[0265] in, , This represents the number of non-dominated solutions in the Pareto front solution set. This corresponds to optimization objective 1, namely oil film stiffness. Maximize the goal; This corresponds to optimization objective 2, namely the average temperature of the oil film rotating surface. Minimize the target.

[0266] S52, Decision matrix normalization processing;

[0267] Since the two optimization objectives have different dimensions (stiffness in N / μm, temperature in °C), to eliminate the influence of dimensions, the decision matrix needs to be adjusted. After normalization, when Maximize the target oil film stiffness The normalization formula is as follows:

[0268] ;

[0269] when Minimize the average temperature of the target oil film's rotating surface. The normalization formula is as follows:

[0270] ;

[0271] in, This represents the elements of the normalized decision matrix.

[0272] S53. Determine the weighting coefficients and construct the weighted normalization matrix;

[0273] Based on the actual usage requirements of the hydrostatic spindle, the weighting coefficients for the two optimization objectives are determined, with the stiffness weighting coefficient set to... The weighting factor for temperature is set to And satisfy .

[0274] Then construct a weighted normalized matrix based on the weight coefficients. Matrix elements The weighted normalized matrix expression is as follows:

[0275] ;

[0276] S54. Determine the positive ideal solution and the negative ideal solution;

[0277] Positive Ideal Solution It is a vector composed of the optimal values ​​of each objective in the weighted normalized matrix, representing the negative ideal solution. It is a vector composed of the worst-case values ​​of each objective, and its calculation expression is as follows:

[0278] ;

[0279] ;

[0280] in, Let represent the maximum values ​​of all non-dominated solutions (or alternatives) on the first and second objectives, respectively, i.e., the components of the positive ideal solution. These represent the minimum values ​​of all non-dominated solutions on the first and second objectives, respectively, i.e., the components of the negative ideal solution.

[0281] S55. Calculate the Euclidean distance between each non-dominated solution and the ideal solution;

[0282] The first step is to calculate the Euclidean distance. A nondominated solution and a positive ideal solution distance and negative ideal solutions - distance The calculation expressions are as follows:

[0283] ;

[0284] in, The smaller the value, the closer the non-dominated solution is to the optimal objective; The larger the value, the further the non-dominated solution is from the worst objective.

[0285] S56. Calculate the relative proximity and determine the optimal compromise solution;

[0286] Definition of the first Relative closeness of non-dominated solutions This is used to measure how close the solution is to the ideal solution, and its calculation expression is as follows:

[0287] ;

[0288] in, , The larger the value, the closer the non-dominated solution is to the ideal solution, and the better the overall performance.

[0289] Then, the relative proximity of all non-dominated solutions is calculated. Sort and select The largest nondominated solution is taken as the optimal compromise solution, and the corresponding combination of structural parameters is... The optimal structural parameters of the hydrostatic spindle are obtained, which can achieve the optimal balance between spindle stiffness and temperature rise, and meet the actual use requirements of high-precision equipment. The optimized oil film structure parameters and calculated predicted performance are shown in Table 3.

[0290] Table 3

[0291]

[0292] In this embodiment, step S6 is also included to verify the optimization results, as follows:

[0293] S61. Comparison of oil film optimization before and after based on Fluent;

[0294] Based on Fluent simulation, finite element simulation was performed on the structure with optimized parameters, and the results were compared with the structure with the original parameters. Table 4 shows the comparison results before and after oil film optimization. Figure 5 Comparison of temperature field before and after radial oil film optimization (cloud map) Figure 6 Comparison of pressure field contour plots before and after radial oil film optimization. Figure 5 (a) and (b) are temperature field cloud diagrams before and after radial oil film optimization, respectively. Figure 6 (a) and (b) are pressure field cloud diagrams before and after radial oil film optimization, respectively.

[0295] Table 4

[0296]

[0297] Therefore, the finite element simulation results of the optimized structure agree well with the predicted values ​​of the surrogate model (stiffness error 2.8%, temperature error 0.6%). Furthermore, the radial stiffness of the oil film increased by approximately 97% after optimization, demonstrating the significant effect of the optimized design in enhancing the radial load-bearing capacity of the oil film. The temperature decreased by approximately 1.95%, indicating that the optimization had a minimal impact on temperature rise while improving stiffness, and the thermal characteristics remained essentially stable. The optimized oil film temperature distribution is more uniform, and the local high-temperature areas are significantly reduced. The optimized oil film pressure value is significantly increased and more uniformly distributed, further verifying the effectiveness of the optimized design in terms of load-bearing capacity.

[0298] S62. Comparison of spindle optimization before and after Mechanical-based optimization;

[0299] The oil film temperature field results obtained from Fluent simulation are mapped and applied to the overall spindle system model as thermal boundary conditions through data exchange between Mechanical and Fluent. Simultaneously, the motor heating power and cooling water jacket cooling power are added to the spindle temperature field calculation to more completely and accurately solve for the overall temperature field distribution of the spindle system. Furthermore, after obtaining the temperature field distribution, it is applied as an external load to the structural deformation field simulation module to calculate the thermal deformation of the spindle under the action of the temperature field. Table 5 compares the highest temperature and deformation field results of the spindle system before and after optimization. Figure 7 To optimize the before-and-after spindle temperature field comparison contour map, Figure 8 To optimize the deformation field contour map of the front main shaft, Figure 9 This is the optimized principal shaft deformation field contour map. Among them, Figure 7 (a) and (b) are the spindle temperature field contour maps before and after optimization, respectively; Figure 8 (a), (b), (c), and (d) represent the total deformation, X-direction deformation, Y-direction deformation, and Z-direction deformation before optimization, respectively. Figure 9 (a), (b), (c), and (d) represent the total deformation, X-direction deformation, Y-direction deformation, and Z-direction deformation after optimization, respectively.

[0300] Table 5

[0301]

[0302] From the overall temperature field of the spindle, the peak temperature of the optimized spindle decreased from 26.09°C to 26.04°C, a decrease of approximately 0.19%. Although the temperature decrease is not significant, the temperature field distribution is more uniform, and the local high-temperature areas are significantly reduced, indicating that the optimized design improves the heat concentration phenomenon and is beneficial to the thermal stability and operational reliability of the spindle. From the overall deformation results, the peak total deformation of the optimized spindle decreased from 0.008995 mm to 0.008283 mm, a decrease of approximately 7.9%. The deformation in each direction is as follows: X-direction from 0.004452 mm to 0.004558 mm, a slight increase of approximately 2.3%; Y-direction from 0.004514 mm to 0.004513 mm; Z-direction from 0.008342 mm to 0.007679 mm, a decrease of approximately 7.9%. The contour plot shows that the thermal deformation distribution of the optimized spindle and spindle core is more uniform, and the high-deformation areas are reduced, especially with effective control of longitudinal thermal expansion in the Z-direction. This indicates that the optimized design reduced the heat concentration effect, thereby minimizing the impact of thermal expansion on the radial oil film and bearing clearance, and improving the overall thermodynamic stability of the system. After spindle optimization, both the total amount and distribution of thermal deformation were improved, with the total deformation decreasing by approximately 8% and the local deformation becoming more uniform.

[0303] S63. Comparison with existing optimization algorithms;

[0304] The simulation results of the structural parameters obtained by the method of this invention are compared with those obtained by the existing NSGA-II optimization algorithm. The optimized structural parameters and corresponding simulation results are shown in Table 6. It can be seen that the optimized results obtained by the existing NSGA-II optimization method have better performance in both temperature and stiffness indicators compared with the structural parameters obtained by this optimization method.

[0305] Table 6

[0306]

[0307] In summary, the method of this invention achieves optimized design of the structural parameters of a hydrostatic master shaft by integrating high-precision multiphysics modeling, efficient sampling, surrogate models, an improved heterogeneous alternating evolutionary algorithm, and a multi-objective decision-making method. Comparative verification using the same boundary conditions demonstrates that the optimized structural parameters effectively improve oil film stiffness and reduce oil film temperature compared to the original parameters. Furthermore, simulation results obtained based on the existing NSGA-II optimization algorithm are compared, verifying that the proposed optimization algorithm balances optimization efficiency and design accuracy.

[0308] Those skilled in the art will recognize that the embodiments described herein are intended to help the reader understand the implementation methods of the present invention, and should be understood that the scope of protection of the present invention is not limited to such specific statements and embodiments. Those skilled in the art can make various other specific modifications and combinations based on the technical teachings disclosed in this invention without departing from the spirit of the invention, and these modifications and combinations are still within the scope of protection of the present invention.

Claims

1. A method for optimizing the principal parameters of a hydrostatic spindle based on a multi-objective heterogeneous evolutionary algorithm, the specific steps of which are as follows: S1. Establish a three-dimensional oil film model that considers viscosity-temperature effect and fluid-thermal-solid coupling; S11. Construct the viscosity-temperature coupling equation; The temperature-viscosity-pressure coupling feedback mechanism is represented by an exponential model, as shown in the following expression: ； in, Indicates the temperature of the lubricating oil The dynamic viscosity below, Indicates the lubricating oil at the reference temperature The dynamic viscosity below, Indicates the viscosity-temperature coefficient of lubricating oil. This indicates the real-time temperature of the oil film. Indicates reference temperature; S12. The separation coupling method is used to solve the fluid-thermal-structure coupling problem. First, create a 3D geometric model containing the fluid domain and the solid domain in 3D CAD, paying attention to the accurate processing of the coupling surface. Then, start the simulation software, import the geometry or mesh, perform basic solver settings, and generate fluid domain mesh and solid domain mesh respectively in the mesh generation software. Then, in the simulation software, a fluid-thermal-solid coupled boundary is set, the mesh is generated and exported, and numerical simulation is performed to solve the continuity, momentum, and energy equations to obtain the temperature and pressure fields of the fluid domain. Next, it is determined whether the energy equation has converged. If it has converged, the temperature and pressure field data of the fluid domain are transferred to the solid domain to obtain the temperature and pressure fields. If it has not converged, the modeling stage is returned to adjust the model or mesh, and the fluid domain simulation is re-executed until convergence. The material properties of the solid domain are redefined, and mechanical and thermal boundary constraints are applied. The temperature field of the solid domain is solved in the simulation software using the transferred fluid temperature boundary data. Based on the solid temperature field and the pressure load transmitted by the fluid, the solid deformation field is solved, and the final output results include: temperature, pressure, deformation, and stress. The simulation ends. S2. Based on step S1, select design variables and determine the parameter combination scheme through Latin hypercube sampling; S21. Select design variables; Based on the structural characteristics and optimization objectives of the hydrostatic spindle, and according to the actual situation, select... One key structural parameter is used as a design variable; The key structural parameters include: oil film thickness. Orifice diameter Orifice depth ; S22. Generate parameter combination schemes through Latin hypercube sampling (LHS); The Latin hypercube sampling method is used for step S21. Sample each design variable to generate Group parameter combination schemes, each scheme corresponds to a set of input parameters The data is directly input into the three-dimensional oil film model for simulation calculation, and the corresponding performance indicators are obtained to construct an input-output sample set. The performance indicators include: average temperature of the oil film rotating surface and oil film stiffness; S3. Based on step S2, construct a high-precision prediction agent model based on the Kriging model; The Kriging model consists of two parts: a regression function and a stochastic process. Therefore, a high-precision predictive surrogate model based on the Kriging model is constructed, expressed as follows: ； in, This represents the model's predicted value. Represents the global trend function. Let represent a random process with a mean of zero; and the covariance function of the random process is expressed as follows: ； in, Represents the variance of a random process. Represents the relevant function, Let represent two different spatial sample points of a random process; then the formal expression of the Gaussian correlation function is as follows: ； in, Indicates the relevant parameters, Indicates the number of design variables. They represent the first The and the first The first spatial sample point 3D coordinate components; S4. Based on step S3, establish a multi-objective optimization model using the improved heterogeneous alternation evolutionary algorithm based on UCB scheduling; S41, Algorithm initialization; 1) Initialize parameters; Set the maximum number of iterations. Population size UCB scheduling coefficient Scheduling cycle Exceeding the volume threshold Initialize the primary population. Establish external archives This is used to store non-dominated solutions generated during the iteration process; 2) Construct a heterogeneous sub-algorithm pool; The sub-algorithm pool includes three multi-objective evolutionary algorithms: Multi-objective Particle Swarm Optimization (MOPSO), Multi-objective Differential Evolutionary Algorithm (MODE), and Non-dominated Sorting Genetic Algorithm (NSGA-II). The initial parameters of each sub-algorithm are set uniformly. 3) Sub-algorithm warm-up evaluation; Perform short-time iterations on each sub-algorithm and record the initial performance metrics of each algorithm, including: hypervolume. Number of non-dominated solutions ; S42, UCB scheduler and selection mechanism; Introducing a scheduler based on confidence upper bound (UCB), after a preset scheduling period... The algorithm dynamically selects the current optimal sub-algorithm for iteration, and its core mathematical model is as follows: 1) Calculation of the overall score of the sub-algorithm; For the first in the sub-algorithm pool Each sub-algorithm, its overall score The calculation expression is as follows: ； in, =1,2,3, corresponding to MOPSO, MODE, and NSGA-II; Sub-algorithm The cumulative reward value, Represents the UCB scheduling coefficient. Indicates the total number of scheduling operations. Sub-algorithm The number of times it was selected; 2) Scheduling rules; Select the comprehensive score for each scheduling. The highest-performing sub-algorithm is used as the execution algorithm for the current iteration cycle. After the cycle is completed, the scheduling count is updated. AND sub-algorithm selected times Proceed to the next round of scheduling; S43, Iteration and Population Update; Iterative execution: Following the sub-algorithm selected by the UCB scheduler, the main population is... Perform the corresponding evolutionary operations to generate a offspring population; Among them, MOPSO performs particle updates and velocity iterations; MODE performs mutation and crossover operations; and NSGA-II performs non-dominated sorting, crowding calculation, and selection operations. Population interaction and selection: The offspring population is merged with the parent population, and the best individuals are retained through non-dominant ranking and crowding selection to update the main population. At the same time, the non-dominated solution is stored in an external archive. If the number of solutions in the external archives exceeds a preset threshold, redundant solutions are deleted by sorting by congestion. S44, Reward Mechanism and Performance Feedback; After each selected sub-algorithm is executed and the population is updated, the performance improvement of that sub-algorithm is calculated and used as a reward value. Update cumulative reward value The details are as follows: 1) Super-volume increase : ; in, This indicates the increased size of the external file after the update. This indicates the external file size exceeded the limit before the update; 2) Increase in the number of non-dominated solutions: ; in, , These represent the number of non-dominated solutions in the external archive after the update and before the update, respectively. 3) Cumulative reward value update: ; in, Indicates the weighting coefficient; S45. Update statistical information; After completing the calculation of sub-algorithm reward values ​​and updating the cumulative reward values, various statistical information from the algorithm iteration process is updated synchronously. The specific update content and rules are as follows: 1) Sub-algorithm performance statistics update; Record the single-iteration performance metrics of the current sub-algorithm, including: hypervolume. Number of non-dominated solutions The iteration time is calculated, and then it is summarized with the historical iteration performance data of this sub-algorithm to calculate its average. Increase, average Improve the quantity and average iteration efficiency, and create a statistical ledger of sub-algorithm performance; 2) Update scheduling-related parameters; Update total scheduling count Increment it by 1, and simultaneously update the selection count of the corresponding sub-algorithm. It also increments by 1; S46. Convergence judgment; After each iteration, calculate the external archive. Super volume increase Stop iterating if any of the following conditions are met: 1) Number of iterations Reaching the maximum number of iterations ;2) Super-volume increase ; Among them, external files after stopping iteration All non-dominated solutions constitute the Pareto front solution set, which includes multiple combinations of structural parameters capable of achieving synergistic optimization of principal shaft stiffness and temperature rise. ; If none of the conditions are met, return to S42 and continue iterative calculation; S47. Establish a multi-objective optimization model; A multi-objective optimization model is constructed with oil film temperature and oil film stiffness as optimization indices. First, the average temperature of the rotating surface of the oil film is selected as the thermal performance evaluation index, and the temperature objective function expression is constructed as follows: ； in, Represents a vector of design variables. This indicates the transpose operation. This indicates the average temperature of the rotating surface of the oil film; The stiffness objective function expression is reconstructed as follows: ； in, Indicates oil film stiffness; The multi-objective optimization model expression for the main shaft structure is as follows: ； in, This indicates the lower limit of the oil film thickness design. This indicates the upper limit of the oil film thickness design. This indicates the lower limit of the design depth for the throttle orifice. This indicates the upper limit of the design depth of the throttle orifice. This indicates the lower limit of the design diameter for the throttling orifice. Indicates the upper limit of the design diameter of the throttle orifice; S5. Based on step S4, determine the optimal compromise solution using the TOPSIS method to optimize the parameters of the hydrostatic spindle. The Pareto front solution set obtained by the MO-HAEA algorithm includes multiple sets of non-dominated solutions, each set of solutions corresponding to a set of structure parameter combinations. Then, the Top-Ideal Solution Ranking Method (TOPSIS) is used for multi-objective decision-making. By calculating the relative proximity of each alternative solution to the positive and negative ideal solutions, all non-dominated solutions are ranked, and the solution with the highest proximity is the optimal compromise solution.

2. The method for optimizing the parameters of a hydrostatic master shaft based on a multi-objective heterogeneous evolutionary algorithm according to claim 1, characterized in that, The specific steps of S22 are as follows: S221. Selection of the number of sampling points; First, based on the accuracy requirements of training a high-precision prediction surrogate model based on the Kriging model, the number of sampling points is selected. Then, the sampled data is divided into training set and validation set according to the actual situation; Then the value range of each design variable is divided into... Given several equally probable subintervals, and randomly select one sample point from any subinterval of each variable, ensuring that each subinterval is selected only once, finally randomly combine the sample points of each variable to generate... Different parameter combination schemes, i.e., sampling sample matrix The expression is as follows: ； in, , , They represent the first The values ​​of oil film thickness, orifice diameter, and orifice depth in the group sampling scheme are determined. ; S222, Sample homogeneity verification; First, the sample homogeneity is verified using the maximum-minimum distance criterion or correlation coefficient analysis, i.e., the sampling sample matrix is ​​calculated. The Euclidean distance between any two sets of parameters is calculated, ensuring that the minimum distance is greater than a threshold; and the calculation... Pearson correlation coefficient among design variables ; in, <0.1; S223, Simulation Input Scheme; The product generated in step S221 The parameter combinations are sequentially input into the three-dimensional oil film model considering viscosity-temperature effect and fluid-structure interaction constructed in step S1, and CFD simulation calculations are performed to obtain the average temperature of the oil film rotation surface and the oil film stiffness corresponding to each set of parameters, and an input-output sample set is constructed.

3. The method for optimizing the parameters of a hydrostatic master shaft based on a multi-objective heterogeneous evolutionary algorithm according to claim 1, characterized in that, In step S3, after the Kriging proxy model is established, the prediction accuracy of the model is verified, as follows: By comparing the simulation results with the prediction results of the Kriging surrogate model, and using the coefficient of determination... The root mean square error (RMSE) and mean relative error (MARE) are used to evaluate the model accuracy; let the true value of the sample be... The surrogate model predicts the value. The sample size is The sample mean is ; (1) The expression for the coefficient of determination is as follows: ； (2) The root mean square error expression is as follows: ； (3) The expression for the average relative error is as follows: 。 4. The method for optimizing the parameters of a hydrostatic master shaft based on a multi-objective heterogeneous evolutionary algorithm according to claim 1, characterized in that, Step S5 is as follows: S51. Construct the decision matrix; Suppose that the Pareto front solution set output by the MO-HAEA algorithm contains There are 1 non-dominated solution, and each solution corresponds to 2 optimization objectives, namely oil film stiffness. Average temperature of oil film rotating surface Then construct the decision matrix. Matrix elements Indicates the first The nondominated solution is at the th... The response values ​​under each optimization objective are expressed by the following decision matrix: ； in, , This represents the number of non-dominated solutions in the Pareto front solution set. This corresponds to optimization objective 1, namely oil film stiffness. Maximize the goal; This corresponds to optimization objective 2, namely the average temperature of the oil film rotating surface. Minimize the target; S52, Decision matrix normalization processing; For decision matrix Normalization is performed to eliminate the influence of dimensions, then when Maximize the target oil film stiffness The normalization formula is as follows: ； when Minimize the average temperature of the target oil film's rotating surface. The normalization formula is as follows: ； in, Represents the elements of the normalized decision matrix; S53. Determine the weighting coefficients and construct the weighted normalization matrix; Based on the actual usage requirements of the hydrostatic spindle, the weighting coefficients for the two optimization objectives are determined, with the stiffness weighting coefficient set to... The weighting factor for temperature is set to And satisfy ; Then construct a weighted normalized matrix based on the weight coefficients. Matrix elements The weighted normalized matrix expression is as follows: ； S54. Determine the positive ideal solution and the negative ideal solution; Positive Ideal Solution It is a vector composed of the optimal values ​​of each objective in the weighted normalized matrix, representing the negative ideal solution. It is a vector composed of the worst-case values ​​of each objective, and its calculation expression is as follows: ； ； in, Let represent the maximum values ​​of all non-dominated solutions on the first and second objectives, respectively, i.e., the components of the positive ideal solution. These represent the minimum values ​​of all non-dominated solutions on the first and second objectives, respectively, i.e., the components of the negative ideal solution; S55. Calculate the Euclidean distance between each non-dominated solution and the ideal solution; The first step is to calculate the Euclidean distance. A nondominated solution and a positive ideal solution distance and negative ideal solutions - distance The calculation expressions are as follows: ； in, The smaller the value, the closer the non-dominated solution is to the optimal objective; The larger the value, the further the non-dominated solution is from the worst objective; S56. Calculate the relative proximity and determine the optimal compromise solution; Definition of the first Relative closeness of non-dominated solutions This is used to measure how close the solution is to the ideal solution, and its calculation expression is as follows: ； in, ; Then, the relative proximity of all non-dominated solutions is calculated. Sort and select The largest nondominated solution is taken as the optimal compromise solution, and the corresponding combination of structural parameters is... This refers to the optimal structural parameters of the hydrostatic spindle, and finally the optimized oil film structural parameters and calculated and predicted performance.

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