Optimization Method for Forging Process Parameters of TC11 Titanium Alloy Thick-Walled Pipe

Abstract

This invention discloses a method for optimizing the forging process parameters of TC11 titanium alloy thick-walled tubes, belonging to the field of metal plastic forming process optimization technology. The invention includes: clearly defining the microstructure defects caused by uneven deformation energy in the radial forging of a specific tube blank and the optimization objectives; constructing a simulation model to calculate the forming dimensions and internal deformation energy distribution under different process parameters; training a neural network surrogate model with an embedded deformation energy uniformity optimization objective; using this model to drive a multi-objective optimization algorithm to automatically search for the optimal combination of process parameters; and selecting and outputting the final optimal process parameters from the optimization results. By embedding a quantitative index reflecting the uniformity of deformation energy distribution into the process optimization model, this invention can simultaneously optimize the internal deformation energy distribution of the material while optimizing macroscopic dimensions, solving the problem that existing methods struggle to effectively assess and actively control this implicit factor affecting the final microstructure and properties of components during the process design stage.

Description

Technical Field

[0001] This invention belongs to the field of metal plastic forming process optimization technology, and in particular relates to a method for optimizing the forging process parameters of TC11 titanium alloy thick-walled pipe diameter. Background Technology

[0002] TC11 titanium alloy is a martensitic α+β two-phase titanium alloy with good comprehensive performance. It is alloyed by adding elements such as aluminum, tin, zirconium, and molybdenum. It has high specific strength and creep strength at high temperatures, as well as good thermal stability and creep resistance. It is a key material for manufacturing hot-end components such as compressor disks and blades of aero engines. However, it has the characteristics of large deformation resistance and relatively narrow plastic forming window during hot working.

[0003] Radial forging, also known as radial forging, is a precision forming process that uses high-speed hammers to simultaneously or alternately forge a rotating or stationary billet in multiple directions (usually two or four symmetrical directions). This process can effectively improve the internal stress state of the material and increase the forging compression ratio through local continuous radial compression deformation. It is especially suitable for manufacturing thick-walled pipes, stepped shafts, and difficult-to-deform metal materials. Its multi-directional forging characteristics are conducive to obtaining a denser and more uniform internal structure.

[0004] Existing techniques for optimizing radial forging processes primarily focus on controlling the macroscopic dimensional accuracy, surface quality, or overall mechanical properties of forgings by adjusting process parameters such as temperature and deformation. This type of optimization typically relies on a combination of finite element simulation and optimization algorithms, with optimization objectives often aimed at minimizing directly measurable geometric deviations such as outer and inner diameters, or controlling the extrema of single physical fields such as equivalent stress and strain fields after forging. However, for critical components like aero-engine rotors, which have extremely high requirements for the uniformity of the material's internal microstructure, the quality of radial forging depends not only on macroscopic dimensions but also, and more importantly, on the uniformity of the material's internal microstructure, which is closely related to the distribution characteristics of deformation energy within the material during the forming process.

[0005] Currently, in the process design stage, there is a lack of effective means to assess and actively control the uniformity of deformation energy distribution within the material during forming. Deformation energy distribution, as a process state quantity, is difficult to measure directly online. Traditional post-processing analyses based on finite element simulation are mostly limited to observing distribution cloud maps or calculating statistical variance, failing to extract it as a quantitative indicator that can be parallel to macroscopic dimensional targets and embedded in automated optimization processes. Therefore, when facing the problem of deformation energy concentration in the core of thick-walled tube blanks of specific specifications during radial forging due to metal flow characteristics, existing optimization methods struggle to proactively prevent and suppress such latent defects during parameter optimization.

[0006] This limitation means that process development still relies to some extent on trial and error. Although existing optimization methods can effectively converge to a parameter solution that meets macroscopic dimensional requirements, this solution may not be optimal in terms of deformation energy distribution uniformity, and may even fall within the parameter range that easily induces abnormal microstructures in subsequent heat treatment (such as localized grain coarsening). Therefore, to address these issues, the following solution is proposed. Summary of the Invention

[0007] The purpose of this invention is to provide a method for optimizing the forging process parameters of TC11 titanium alloy thick-walled tubes. By embedding a quantitative index reflecting the uniformity of deformation energy distribution into the process optimization model, it is possible to optimize the internal deformation energy distribution of the material while optimizing the macroscopic dimensions. This solves the problem that existing methods are unable to effectively evaluate and actively control this hidden factor affecting the final microstructure and properties of the component during the process design stage.

[0008] To solve the above-mentioned technical problems, the present invention is achieved through the following technical solution:

[0009] This invention provides a method for optimizing forging process parameters of TC11 titanium alloy thick-walled tubes, comprising the following steps:

[0010] The specific application scenarios and corresponding microstructure defects of the TC11 titanium alloy thick-walled tube forging process are determined, and an optimization problem including process parameter variables, optimization objectives and constraints is defined accordingly.

[0011] A high-fidelity multiphysics finite element simulation model of the radial forging process under the application scenario is established, and simulation result data under different combinations of process parameters is collected based on the model. The simulation result data includes at least the final forging outer diameter and the historical data of the equivalent variable energy density of multiple integration points of the tube blank cross section distributed radially at the time of final forging.

[0012] A physically enhanced neural network surrogate model is constructed. This model takes a combination of process parameters as input and outputs the predicted final forging outer diameter and the predicted radial deformation energy density distribution. In the construction process of the neural network surrogate model, a deformation energy distribution uniformity calculation layer is embedded. This layer automatically calculates a uniformity index for quantitatively evaluating the radial energy distribution uniformity based on the predicted radial deformation energy density distribution. This uniformity index is incorporated into the training loss function of the neural network as one of the optimization objectives to guide the model to learn the inherent laws of process parameters that can improve the uniformity index.

[0013] Based on the improved multi-objective optimization algorithm, the trained neural network surrogate model is used as the fitness evaluation function to iteratively optimize the process parameter variables. In the process of evolution, the optimization algorithm applies adaptive selection pressure to the optimization objective dimension corresponding to the balance index in order to obtain the Pareto optimal solution set on the two objectives of final forging dimensional accuracy and deformation energy distribution balance.

[0014] Select the final optimal combination of process parameters from the Pareto optimal solution set and output it.

[0015] Furthermore, the specific application scenario is the radial forging of TC11 titanium alloy thick-walled tubes of specific specifications for high-pressure compressor rotors of aero engines. The microstructure defect problem refers to the abnormal grain growth in the core area of ​​the tube blank during the radial forging process due to excessive concentration of deformation energy.

[0016] Furthermore, the calculation process of the equilibrium index calculated by the deformation energy distribution equilibrium calculation layer includes: calculating the average value of the predicted radial deformation energy density distribution, normalizing the distribution based on the average value, and evaluating the dispersion of the normalized distribution using higher-order statistics, wherein the higher-order statistics are used to amplify the contribution of local sharp peaks in the distribution to enhance the sensitivity of the index to deformation energy concentration phenomena.

[0017] Furthermore, the physically enhanced neural network proxy model is a fully connected neural network with a shared hidden layer and a dual-output branch structure. The first output branch is used to predict the final forging outer diameter, and the second output branch is used to predict the radial deformation energy density distribution vector. The training loss function consists of three parts: the prediction error term of the final outer diameter, the overall prediction error term of the deformation energy density distribution vector, and the optimization term directly based on the balance index.

[0018] Furthermore, the improved multi-objective optimization algorithm adopts a non-dominated sorting genetic algorithm framework with an elitist strategy. The improvement lies in that, when calculating the individual crowding distance, an adaptively changing weight factor is assigned to the target dimension component corresponding to the equilibrium index. This weight factor increases with the number of iterations of the algorithm, so as to strengthen the search for the equilibrium of deformation energy distribution in the later stage of optimization.

[0019] The present invention has the following beneficial effects:

[0020] 1. This invention can transform the uniformity of internal deformation energy distribution, which is difficult to directly observe and control during radial forging, into a quantifiable and optimizable objective. By constructing a uniformity index that is highly sensitive to local energy concentration and making it one of the core objectives in multi-objective optimization, the optimization process can actively guide the process parameters to be adjusted in a direction that is conducive to the uniform dissipation of internal energy in the material.

[0021] 2. This invention helps to improve the potential uniformity of the internal structure of the material while ensuring the macroscopic dimensional accuracy of the forging; by deeply embedding the uniformity index into the training loss function of the neural network surrogate model, the model inherently tends to those parameter combinations that can produce a more uniform deformation energy distribution when learning the mapping relationship between process parameters and results; in the multi-objective optimization stage, the improved algorithm further balances dimensional accuracy and distribution uniformity, and the final Pareto solution set provides a balance between the two, making the process design more comprehensive.

[0022] 3. This invention reduces the computational dependence on direct iterative optimization of high-fidelity simulation by constructing a physically enhanced neural network surrogate model. Once the surrogate model is trained, its evaluation speed of process parameter combinations is faster than that of a complete finite element simulation. This enables efficient exploration of a broad process parameter space and the running of multi-objective optimization algorithms for a sufficient number of algebras to approximate the ideal Pareto front. This architecture improves the overall optimization efficiency while ensuring the physical credibility of the optimization process.

[0023] Of course, any product implementing this invention does not necessarily need to achieve all of the advantages described above at the same time. Attached Figure Description

[0024] To more clearly illustrate the technical solutions of the embodiments of the present invention, the accompanying drawings used in the description of the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0025] Figure 1 This is a schematic flowchart of the method for optimizing the forging process parameters of TC11 titanium alloy thick-walled tubes according to the present invention. Detailed Implementation

[0026] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0027] Please see Figure 1 As shown, this invention provides a method for optimizing forging process parameters of TC11 titanium alloy thick-walled tubes, comprising the following steps:

[0028] The specific application scenarios and corresponding microstructure defects of the TC11 titanium alloy thick-walled tube forging process are determined, and an optimization problem including process parameter variables, optimization objectives and constraints is defined accordingly.

[0029] A high-fidelity multiphysics finite element simulation model of the radial forging process in the application scenario is established, and simulation result data under different process parameter combinations are collected based on the model. The simulation result data includes at least the final forging outer diameter and the historical data of the equivalent variable energy density of multiple integration points of the tube blank cross section along the radial direction at the final forging moment.

[0030] A physically enhanced neural network surrogate model is constructed. This model takes a combination of process parameters as input and outputs the predicted final forging outer diameter and the predicted radial deformation energy density distribution. In the process of constructing the neural network surrogate model, a deformation energy distribution uniformity calculation layer is embedded. This layer automatically calculates a uniformity index for quantitatively evaluating the radial energy distribution uniformity based on the predicted radial deformation energy density distribution. This uniformity index is incorporated into the training loss function of the neural network as one of the optimization objectives to guide the model to learn the inherent laws of process parameters that can improve the uniformity index.

[0031] Based on the improved multi-objective optimization algorithm, the trained neural network surrogate model is used as the fitness evaluation function to iteratively optimize the process parameter variables. In the process of evolution, the optimization algorithm applies adaptive selection pressure to the optimization objective dimension corresponding to the balance index in order to obtain the Pareto optimal solution set on the two objectives of final forging dimensional accuracy and deformation energy distribution balance.

[0032] Select the final optimal combination of process parameters from the Pareto optimal solution set and output it.

[0033] The specific application scenario is the radial forging of TC11 titanium alloy thick-walled tubes of specific specifications for high-pressure compressor rotors of aero engines. The microstructure defect problem refers to the abnormal grain growth in the core area of ​​the tube blank during the radial forging process due to excessive concentration of deformation energy.

[0034] The calculation process of the equilibrium index calculated by the deformation energy distribution equilibrium calculation layer includes: calculating the average value of the predicted radial deformation energy density distribution, normalizing the distribution based on the average value, and evaluating the dispersion of the normalized distribution using higher-order statistics. The higher-order statistics are used to amplify the contribution of local sharp peaks in the distribution to enhance the sensitivity of the index to deformation energy concentration phenomena.

[0035] The physically enhanced neural network proxy model is a fully connected neural network with a shared hidden layer and a dual-output branch structure. The first output branch is used to predict the final forging outer diameter, and the second output branch is used to predict the radial deformation energy density distribution vector. The training loss function consists of three parts: the prediction error term of the final outer diameter, the overall prediction error term of the deformation energy density distribution vector, and the optimization term directly based on the balance index.

[0036] The improved multi-objective optimization algorithm adopts a non-dominated sorting genetic algorithm framework with an elitist strategy. Its improvement lies in that, when calculating the crowding distance of individuals, an adaptively changing weight factor is assigned to the target dimension component corresponding to the equilibrium index. This weight factor increases with the number of iterations of the algorithm, so as to strengthen the search for the equilibrium of deformation energy distribution in the later stage of optimization.

[0037] One specific application of this embodiment is:

[0038] Step S1: Determine the specific application scenario and define the optimization problem.

[0039] The process involves radial forging of a TC11 titanium alloy thick-walled tube blank for the third-stage rotor of an engine's high-pressure compressor. The initial specifications of the tube blank are an outer diameter of Φ350±2mm, an inner diameter of Φ180±1.5mm, and a height of 450±3mm. The target forging specifications are an outer diameter of Φ320±1mm, an inner diameter of Φ185±1mm, and a height of 480±2mm. The specific problem is to avoid uneven distribution of radial deformation energy in the cross-section of the tube blank during radial forging, which could lead to abnormal local grain growth (grain size exceeding ASTM level 5) in the core region (radius r ranging from 0 to approximately 40% of the inner diameter) during subsequent solution aging heat treatment.

[0040] Based on this, the optimization variable is defined as: the initial temperature of the hammerhead. (°C), initial temperature of tube blank (°C), pass reduction sequence (mm) (4 rounds in total) and pause time after each round of forging (s); Define two core optimization objectives as: 1) Absolute value of the outer diameter error of the forging. Minimize; 2) Indicator of radial deformation energy distribution uniformity in the final forging cross section of the tube blank Maximize; while satisfying the constraints: the maximum equivalent stress after forging should not exceed 85% of the yield strength of the material at that temperature, and internal cracks should be avoided; the final forging temperature should not be lower than 30°C below the β phase transformation point of the titanium alloy.

[0041] Step S2: Construction of high-fidelity multiphysics finite element simulation model and data acquisition

[0042] A three-dimensional thermo-mechanical coupled finite element model of the forging process of the TC11 thick-walled tube of the specified diameter was established based on a commercial finite element software platform; the material model adopted the high-temperature rheological stress constitutive equation of TC11 titanium alloy fitted based on internal experimental data of the company. And considering the dynamic recrystallization softening effect, in the formula, For rheological stress; Symbols for functional relations; To be truly adaptable; Strain rate; For deformation temperature; boundary conditions are precisely set: including the friction model between the hammer and the tube blank (using a shear friction model, friction factor). The model calculates the radiation and convective heat transfer coefficients between the tube blank and the environment; and inputs a set of process parameters. Perform simulation calculations;

[0043] After the simulation calculation is completed, two key result data are extracted: 1) the outer diameter of the tube blank at the final forging time. 2) Historical equivalent variable energy density data of N (e.g., N=50) integration points uniformly selected radially from the inner wall to the outer wall on the cross-section of the tube blank at the final forging point; equivalent variable energy density Calculated by integration:

[0044]

[0045] In the formula, For the ultimate true response; For strain infinitesimal elements;

[0046] For each simulation case j, a radial deformation energy density distribution vector will be obtained. .

[0047] Step S3: Construct a physically enhanced neural network surrogate model and embed a deformation energy distribution uniformity calculation module.

[0048] The purpose of this step is to establish a rapid mapping relationship from process parameters to optimization objectives and to directly embed the balance evaluation.

[0049] Step S31: Prepare the training dataset: Within the feasible region of process parameters, generate M groups (e.g., M=500) of different combinations of process parameters using the Latin hypercube sampling method. For each set of parameters, perform the simulation in step S2 and collect the corresponding data. This forms the total dataset;

[0050] Step S32: Design and train a dual-output branch neural network: Construct a fully connected neural network with a shared hidden layer and two independent output branches; the shared layer is responsible for learning the higher-order correlation between process parameters and complex physical fields; the first output branch (size prediction branch) outputs a scalar, namely the predicted outer diameter. The second output branch (deformation energy distribution prediction branch) outputs an N-dimensional vector, which is the predicted radial deformation energy density distribution. ;

[0051] Step S33: In the network's loss function, design and embed the deformation energy distribution equilibrium index for the second output branch. The computational layer and related loss terms, specifically:

[0052] In network forward propagation After that, it is not directly compared with the simulation results. Instead of calculating the simple mean squared error, a custom, non-trainable layer is added, which is based on... Automatically calculate the radial deformation energy distribution uniformity index defined in this invention. ;

[0053] This indicator The calculation formula is:

[0054]

[0055] In the formula, This represents the number of radial sampling points; For the summation index, represents the first... One sampling point; For the first Predicted equivalent variable energy density at each point; This represents the average value of the predicted radial deformation energy density; It is a small constant (such as 1e-6) used to prevent division by zero errors and enhance numerical stability; As a normalization factor, it is usually taken from all samples in the training set. The approximate maximum value of the part inside the square root in the calculation formula is used to approximate the maximum value of the part inside the square root. It roughly maps to the (0,1] interval; the formula uses the fourth power instead of the quadratic power to amplify the contribution of sharp peaks in the distribution (i.e. deformation energy concentration areas), making it abnormally sensitive to local energy accumulation, thus better meeting the physical requirements of suppressing abnormal grain growth.

[0056] Constructing a composite loss function Used for network training:

[0057]

[0058] In the formula, This represents the total loss during neural network training. This represents the weighting coefficient for the dimensional error loss term; This is the actual final forging outer diameter; The final forging outer diameter predicted by the neural network; These are the weighting coefficients for the error loss term in the deformation energy distribution vector; This represents the true radial deformation energy distribution vector; This is the radial deformation energy distribution vector predicted by the neural network; The first term represents the absolute error of the size prediction, the second term represents the overall mean square error of the deformation energy distribution vector to ensure the macroscopic accuracy of the distribution shape, and the third term directly addresses the balance index. The optimization terms aim to guide the model to learn those features during agent training. The inherent laws governing the increase in process parameters; The weighting coefficients for each loss term are determined through validation set performance adjustments.

[0059] Step S4: Multi-objective parameter optimization based on improved non-dominated sorting genetic algorithm

[0060] Step S41, Initialization: Set the population size (e.g., 100), and use the physical augmentation neural network trained in step S3 as the fitness evaluation function; each individual represents a set of process parameters. ;

[0061] Step S42, Fitness Assessment: The process parameters of each individual in the population are encoded and input into the trained neural network; after forward propagation, two fitness values ​​are obtained simultaneously: (Outer diameter error, which needs to be minimized) and (Because it needs to be maximized) Therefore, taking the negative transforms the problem into a minimization problem.

[0062] Step S43, Improved Non-Dominated Sort and Selection: A non-dominated sorting genetic algorithm with elitist strategy (NSGA-II) framework is adopted; however, when calculating crowding, the target... (correspond Introducing adaptive weights: Among individuals with similar rankings, priority is given to those that enable... The value has increased significantly (i.e.) Individuals with significantly reduced (even if they are in) There is a slight disadvantage; this is achieved by modifying the way congestion distance is calculated. The distance component in each dimension is multiplied by a weighting factor greater than 1. , It increases slightly with the number of iterations, so as to place greater emphasis on optimizing the distribution balance of deformation energy in the later stages of optimization;

[0063] Step S44, Genetic Operations and Iteration: Perform simulated binary crossover and polynomial mutation on the selected individuals to generate offspring population; merge the parent and offspring generations, and repeat the evaluation, sorting, and selection process of steps S42 to S43 until the set maximum number of generations (e.g., 200 generations) is reached.

[0064] Step S45: Obtain the Pareto front: After the algorithm converges, it outputs a set of non-dominated solutions (Pareto optimal solution set); these solutions represent the best trade-off between the accuracy of the outer diameter dimensions and the uniformity of the deformation energy distribution.

[0065] Step S5: Optimal process parameter decision and output;

[0066] From the Pareto front, process experts select a final solution based on the specific priorities of the current production batch (e.g., whether to prioritize dimensional stability or microstructure uniformity); the system then combines the process parameters corresponding to this solution. Together with its predicted outer diameter and deformation energy distribution uniformity index It is output in a structured process card format to guide actual radial forging production.

[0067] In the description of this specification, references to terms such as "an embodiment," "example," "specific example," etc., indicate that a specific feature, structure, material, or characteristic described in connection with that embodiment or example is included in at least one embodiment or example of the invention. In this specification, illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in one or more embodiments or examples.

[0068] The preferred embodiments of the present invention disclosed above are merely illustrative of the invention. These preferred embodiments do not exhaustively describe all details, nor do they limit the invention to the specific implementations described. Clearly, many modifications and variations can be made based on the content of this specification. This specification selects and specifically describes these embodiments to better explain the principles and practical applications of the invention, thereby enabling those skilled in the art to better understand and utilize the invention. The invention is limited only by the claims and their full scope and equivalents.

Claims

1. A method for optimizing forging process parameters of TC11 titanium alloy thick-walled pipes, characterized in that, The optimization method includes the following steps: The specific application scenarios and corresponding microstructure defects of the TC11 titanium alloy thick-walled tube forging process are determined, and an optimization problem including process parameter variables, optimization objectives and constraints is defined accordingly. A high-fidelity multiphysics finite element simulation model of the radial forging process under the application scenario is established, and simulation result data under different combinations of process parameters is collected based on the model. The simulation result data includes at least the final forging outer diameter and the historical data of the equivalent variable energy density of multiple integration points of the tube blank cross section distributed radially at the time of final forging. A physically enhanced neural network surrogate model is constructed. This model takes a combination of process parameters as input and outputs the predicted final forging outer diameter and the predicted radial deformation energy density distribution. In the construction process of the neural network surrogate model, a deformation energy distribution uniformity calculation layer is embedded. This layer automatically calculates a uniformity index for quantitatively evaluating the radial energy distribution uniformity based on the predicted radial deformation energy density distribution. This uniformity index is incorporated into the training loss function of the neural network as one of the optimization objectives to guide the model to learn the inherent laws of process parameters that can improve the uniformity index. Based on the improved multi-objective optimization algorithm, the trained neural network surrogate model is used as the fitness evaluation function to iteratively optimize the process parameter variables. In the process of evolution, the optimization algorithm applies adaptive selection pressure to the optimization objective dimension corresponding to the balance index in order to obtain the Pareto optimal solution set on the two objectives of final forging dimensional accuracy and deformation energy distribution balance. Select the final optimal combination of process parameters from the Pareto optimal solution set and output it.

2. The method for optimizing forging process parameters of TC11 titanium alloy thick-walled pipes according to claim 1, characterized in that, The specific application scenario is the radial forging of TC11 titanium alloy thick-walled tubes of a specific specification for high-pressure compressor rotors of aero engines. The microstructure defect problem refers to the abnormal grain growth in the core area of ​​the tube blank during the radial forging process due to excessive concentration of deformation energy.

3. The method for optimizing forging process parameters of TC11 titanium alloy thick-walled pipes according to claim 1, characterized in that, The calculation process of the equilibrium index calculated by the deformation energy distribution equilibrium calculation layer includes: calculating the average value of the predicted radial deformation energy density distribution, normalizing the distribution based on the average value, and evaluating the dispersion of the normalized distribution using higher-order statistics, wherein the higher-order statistics are used to amplify the contribution of local sharp peaks in the distribution to enhance the sensitivity of the index to deformation energy concentration phenomena.

4. The method for optimizing forging process parameters of TC11 titanium alloy thick-walled pipes according to claim 1, characterized in that, The physically enhanced neural network proxy model is a fully connected neural network with a shared hidden layer and a dual-output branch structure. The first output branch is used to predict the final forging outer diameter, and the second output branch is used to predict the radial deformation energy density distribution vector. The training loss function consists of three parts: the prediction error term of the final outer diameter, the overall prediction error term of the deformation energy density distribution vector, and the optimization term directly based on the balance index.

5. The method for optimizing forging process parameters of TC11 titanium alloy thick-walled pipes according to claim 1, characterized in that, The improved multi-objective optimization algorithm adopts a non-dominated sorting genetic algorithm framework with an elitist strategy. The improvement lies in that, when calculating the crowding distance of individuals, an adaptively changing weight factor is assigned to the target dimension component corresponding to the equilibrium index. This weight factor increases with the increase of the algorithm iteration generation, so as to strengthen the search for the equilibrium of deformation energy distribution in the later stage of optimization.

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