Quartz crucible material proportioning optimization method based on genetic algorithm

Abstract

The application belongs to the technical field of crucible material proportioning optimization, more particularly to a quartz crucible material proportioning optimization method based on a genetic algorithm; the present application collects and collates crucible performance data under different proportions, uses data cleaning and standardization to ensure data quality, and then divides the data into a training set and a test set for subsequent model training and evaluation, uses a regression model to establish a mathematical relationship between the proportion and the crucible performance, and ensures the prediction accuracy of the model through training and testing, thereby providing an accurate fitness function for the genetic algorithm; the introduction of the genetic algorithm enables the present method to efficiently search for the optimal proportion in the global range and avoid falling into a local optimal solution; the present method not only ensures the effectiveness of the optimized proportion, but also greatly shortens the research and development cycle and reduces the cost, and through systematic data processing, accurate model construction and efficient algorithm application, the global optimal proportion is quickly determined.

Description

Technical Field

[0001] This application belongs to the field of crucible material ratio optimization technology, and more specifically, relates to a method for optimizing the ratio of quartz crucible materials based on genetic algorithms. Background Technology

[0002] Quartz crucibles are commonly used containers in high-temperature experiments and industrial production, primarily for melting, smelting, and heat-treating metals, alloys, and glass. The performance of the crucible directly affects the accuracy of experimental results and the smooth progress of the production process. In the production of quartz crucibles, the material ratio is one of the key factors determining its performance.

[0003] Background of ingredient optimization: The main ingredients of quartz crucibles are quartz sand and alumina quartz sand. Quartz sand has good high-temperature resistance but low strength; while alumina quartz sand has high strength but poor high-temperature resistance. To balance the strength and high-temperature resistance of the crucible, the ratio of quartz sand to alumina quartz sand needs to be optimized.

[0004] Currently, the optimization of quartz crucible formulation mainly relies on empirical methods and trial-and-error methods. Both methods require continuous experimentation and iteration to find a relatively satisfactory formulation scheme. This requires a large number of experiments to verify the formulation, which may lead to local optima and make it impossible to guarantee finding the global optimum. Furthermore, obtaining the optimal formulation through continuous experimentation will significantly increase costs and lengthen the optimization cycle. Summary of the Invention

[0005] This invention provides a method for optimizing the material ratio of quartz crucibles based on genetic algorithms, which aims to solve the technical problems of the current empirical and trial-and-error methods, which cannot guarantee finding the global optimal solution and are costly and time-consuming.

[0006] The method for optimizing the material ratio of quartz crucibles based on genetic algorithms includes the following steps:

[0007] Step 1: Collect data on crucible strength and high-temperature resistance under different ratios of quartz sand and aluminum quartz sand. Each set of data includes ratio data and performance data. The ratio data is the proportion of quartz sand and aluminum quartz sand, and the performance data is the crucible strength and high-temperature resistance data under the corresponding ratio.

[0008] Step 2: Based on the collected data, perform data cleaning and standardization, and then divide the standardized data into training set and test set;

[0009] Step 3: Establish regression models for crucible strength and high-temperature resistance under different ratios of quartz sand and aluminosilicate sand, respectively;

[0010] Step 4: Train the established crucible strength regression model and high-temperature resistance performance regression model using the training set data, evaluate the prediction accuracy of the trained model using the test set, if the accuracy meets the requirements, output the trained crucible strength regression model and high-temperature resistance performance regression model, if the accuracy does not meet the requirements, continue to train the model until the accuracy meets the requirements;

[0011] Step 5: Use the trained crucible strength regression model and high-temperature resistance performance regression model as the fitness function of genetic algorithm, and iteratively solve the optimal ratio based on genetic algorithm.

[0012] The present application collects and organizes a large amount of crucible performance data under different ratios, uses data cleaning and standardization to ensure data quality, and then divides it into training set and test set for subsequent model training and evaluation. Then, a regression model is used to establish the mathematical relationship between the ratio and the performance of the crucible, and the prediction accuracy of the model is ensured through training and testing, thereby providing an accurate fitness function for genetic algorithm. The introduction of genetic algorithm enables the present method to efficiently search for the optimal ratio in the global range, avoiding local optimal solution. The present application greatly shortens the research and development cycle and reduces the cost. Through systematic data processing, accurate model construction and efficient algorithm application, the present application realizes the rapid determination of the global optimal ratio, improves the production efficiency and product quality, and achieves significant economic benefits and technical progress.

[0013] Preferably, the crucible strength regression model and high-temperature resistance performance regression model mentioned in step 3 are as follows:

[0014] Crucible strength regression model:

[0015]

[0016] In the formula, x1 and x2 represent the proportion of quartz sand and aluminum quartz sand respectively, satisfying x1+x2=1; β0 is the bias term of the crucible strength regression model, β1 and β2 represent the linear influence of quartz sand and aluminum quartz sand on the strength of the crucible; β3 and β4 represent the quadratic term coefficients, representing the nonlinear influence of the square term of the ratio on the strength of the crucible; β5 represents the interaction term coefficient, representing the interaction between quartz sand and aluminum quartz sand;

[0017] High-temperature resistance performance regression model:

[0018]

[0019] In the formula, γ0, γ1, …, γ5 represent the coefficients of the high-temperature resistance regression model; γ0 represents the bias term of the high-temperature resistance regression model; γ1 and γ2 represent the linear effects of quartz sand and aluminum quartz sand on the high-temperature resistance, respectively; γ3 and γ4 represent the quadratic coefficients, which are the nonlinear effects of the matching square terms on the high-temperature resistance; and γ5 represents the interaction term coefficient of the high-temperature resistance regression model, which is the interaction between the quartz sand and the aluminum quartz sand;

[0020] Preferably, the crucible strength regression model and the high-temperature resistance regression model are respectively established with objective functions, and the model parameters are optimized by using an Adam optimizer.

[0021] The objective function of the crucible strength regression model is as follows:

[0022]

[0023] In the formula, represents the i-th prediction of the crucible strength by the model; represents the quartz sand ratio of the i-th sample; represents the aluminum quartz sand ratio of the i-th sample; represents the real crucible strength corresponding to the i-th sample; N represents the number of samples; L S represents the optimization objective function for the crucible strength;

[0024] The objective function of the high-temperature resistance regression model is as follows:

[0025]

[0026] In the formula, L T represents the objective function for the crucible high-temperature resistance; represents the i-th prediction of the crucible high-temperature resistance by the model; represents the real crucible high-temperature resistance corresponding to the i-th sample.

[0027] Preferably, the specific steps of optimizing the model parameters by using the Adam optimizer are as follows:

[0028] First-order moment estimation and second-order moment estimation:

[0029] m t = α1m t-1 + (1-α1)g t ;

[0030]

[0031] In the formula, g t represents the gradient at the current time t; and m t-1 represents the momentum at the last time, i.e., the first-order moment estimation; and vt-1 denotes the second moment estimation at the last time; and a1 and a2 denote hyperparameters;

[0032] Bias correction:

[0033]

[0034] where: denotes the corrected momentum estimation; denotes the corrected second moment estimation;

[0035] Update parameters: use the corrected and to update each parameter t :

[0036]

[0037] where: t+1 denotes the updated model parameter; and t denotes the model parameter before update; η denotes the learning rate; and ∈ denotes a constant to prevent the denominator from being zero;

[0038] where the learning rate η is updated every predetermined epoch:

[0039] η t+1 = η t × γ;

[0040] where: γ is a decay factor; η t+1 denotes the updated learning rate; and η t denotes the learning rate before update.

[0041] Preferably, the fitness function and the constraint condition of the genetic algorithm are as follows:

[0042] Fitness function:

[0043] F(x1, x2) = w1·S(x1, x2) + w2·T(x1, x2);

[0044] where: S(x1, x2) denotes the output of the crucible strength regression model; T(x1, x2) denotes the output of the high-temperature resistance performance regression model; w1 denotes the weight coefficient of the strength; w2 denotes the weight coefficient of the high-temperature resistance performance; and w1 + w2 = 1;

[0045] Constraint condition:

[0046] Proportion constraint of quartz sand and aluminum quartz sand:

[0047] x1 + x2 = 1;

[0048] 0 < x1 < 1, 0 < x2 < 1.

[0049] Crucible strength constraint:

[0050] S(x1, x2) ≥ min_strength;

[0051] wherein: min_strength represents a preset minimum strength requirement of the crucible;

[0052] Crucible high temperature performance constraint:

[0053] T(x1, x2) ≥ min_temperature;

[0054] wherein: min_temperature represents a preset minimum high temperature performance constraint of the crucible.

[0055] Preferably, the specific steps for iteratively solving the optimal proportion based on the genetic algorithm are as follows:

[0056] Initialize the population: set the population size to N, indicating that there are N individuals, randomly generate the proportion x1 and x2 of each individual, wherein the proportion of each individual randomly generated satisfies the quartz sand and aluminum quartz sand ratio constraint;

[0057] Selection operation: calculate the fitness F(x1, x2) of each individual according to the fitness function, calculate the selection probability according to the fitness, and then select the individual according to the calculated probability through random selection operation, wherein the probability calculation is as follows:

[0058]

[0059] wherein: F penalized,i represents the fitness of individual i after punishment; N represents the population size; P i represents the probability of individual i being selected;

[0060] Crossover operation: adopt single-point crossover method to perform crossover operation based on a preset crossover probability, for two parent individuals and randomly select a crossover point k, and exchange the genes of the parents after the crossover point to obtain two crossed offspring and

[0061] Mutation operation: based on a predetermined mutation probability to control the frequency of mutation, change the genes in the individual to obtain a new individual;

[0062] Constraint correction: in the crossover and mutation operations, each generated individual needs to satisfy the quartz sand and aluminum quartz sand ratio constraint, the crucible strength constraint, and the crucible high temperature performance constraint;

[0063] If the ratio of quartz sand and aluminum quartz sand does not meet the constraint, it is adjusted by normalization:

[0064]

[0065] Based on this, the adjusted ratio of quartz sand and aluminum quartz sand is obtained;

[0066] If the crucible strength constraint is not met, a penalty is imposed on the fitness, and the specific penalty is as follows:

[0067] F penalized (x1,x2)=F(x1,x2)-λ1·max(0,min_strength-S(x1,x2));

[0068] In the formula: λ1 represents a penalty coefficient; F penalized (x1,x2) represents the fitness function value after adjustment based on the penalty;

[0069] If the crucible high temperature resistance constraint is not met, a penalty is imposed on the fitness, and the specific penalty is as follows:

[0070] F penalized (x1,x2)=F(x1,x2)-λ2·max(0,min_temperature-T(x1,x2));

[0071] In the formula: λ2 represents a penalty coefficient;

[0072] Termination: In each iteration, new individuals are generated and fitness is evaluated through selection, crossover, mutation and constraint conditions, and the iteration stops when the predetermined condition is met, and the value obtained that meets the constraint condition and has the highest fitness is taken as the optimal solution.

[0073] Preferably, an elite reservation strategy is introduced in the genetic algorithm, and the fitness of each iteration is sorted from large to small, and the individuals with high fitness value are selected based on the elite reservation ratio to enter the next iteration;

[0074] Wherein the elite reservation ratio is dynamically adjusted based on the fitness change amount, and the specific adjustment is as follows:

[0075]

[0076] In the formula: Ft represents the fitness value of the optimal individual of the tth iteration; Ft-1 represents the fitness value of the optimal individual of the (t-1)th iteration; ΔF represents the fitness change amount of two adjacent iterations;

[0077] Based on the calculated fitness change amount and the pre-set fitness threshold, if the fitness change amount is less than the fitness threshold, the proportion of elite individuals is increased:

[0078] p elite ′=min(p elite +δ,1);

[0079] In the formula: p elite 'Modified elite retention ratio; δ represents the predefined adjustment amount; min(p elite +δ,1) is used to ensure that the adjusted elite retention rate does not exceed 100%; p elite This indicates the percentage of elites retained before the modification.

[0080] If the change in fitness exceeds a threshold, then reduce the proportion of elites retained:

[0081] p elite =max(p) elite -δ,p min );

[0082] In the formula: p min This indicates the pre-set minimum elite ratio.

[0083] The beneficial effects of this invention include:

[0084] This invention collects and organizes a large amount of crucible performance data under different formulations, ensuring data quality through data cleaning and standardization. The data is then divided into training and testing sets for subsequent model training and evaluation. Next, a regression model is used to establish the mathematical relationship between the formulation and crucible performance. Training and testing ensure the model's prediction accuracy, thus providing an accurate fitness function for the genetic algorithm. The introduction of the genetic algorithm enables this method to efficiently search for the optimal formulation globally, avoiding getting trapped in local optima. This invention significantly shortens the development cycle and reduces costs. Through systematic data processing, precise model construction, and efficient algorithm application, it achieves rapid determination of the globally optimal formulation, improving production efficiency and product quality, resulting in significant economic benefits and technological progress. Attached Figure Description

[0085] To more clearly illustrate the technical solutions in the embodiments of this application, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0086] Figure 1 This is an overall step diagram provided for an embodiment of the present invention.

[0087] Figure 2 A flowchart illustrating the steps of iterative solution using a genetic algorithm provided in an embodiment of the present invention. DETAILED DESCRIPTION

[0088] In order to make the technical problems, technical solutions and beneficial effects to be solved in the present application more clear, the present application will be further described in detail below in combination with the drawings and examples. It should be understood that the specific examples described herein are only used to explain the present application and do not limit the present application.

[0089] Referring to Figure 1 As shown in the figure, the quartz crucible material ratio optimization method based on genetic algorithm includes the following steps:

[0090] Step 1: Collect crucible strength and high temperature resistance performance data under different quartz sand and aluminum quartz sand ratios, each set of data including ratio data and performance data, the ratio data being the quartz sand and aluminum quartz sand ratio, and the performance data being the crucible strength and high temperature resistance performance data under the corresponding ratio;

[0091] According to the set ratio range, experiments are carried out, for example, once every 10% to obtain multiple ratio points (such as Table 1):

[0092] Table 1 shows an example of ratio experimental data

[0093] Quartz sand ratio Aluminum quartz sand ratio Crucible strength High temperature resistance 90％ 10％ 200 MPa 1500℃ 80％ 20％ 220 MPa 1550℃ ... ... ... ...

[0094] As described above, the ratio data is obtained through experiments, and it should be noted that in the experiments, we only focus on the relationship between the ratio of quartz sand and aluminum quartz sand and the strength and high temperature resistance performance of the crucible, so when producing the crucible corresponding to the ratio of quartz sand and aluminum quartz sand in the experiment, other production control parameters need to be ensured to be the same; that is, the crucible is produced by quartz sand and aluminum quartz sand with different ratios in the same production environment; and how to ensure other production control parameters is a conventional technical means in the art, so it will not be described in detail in the present application.

[0095] Step 2: Based on the collected data, data cleaning and data standardization are performed, and then the standardized data is divided into a training set and a test set;

[0096] The data cleaning includes missing value processing and outlier processing, and we delete the data with missing values and outliers. The problem with missing values and outliers is that we also obtain historical data. As mentioned in the background art, the current common practice is to perform crucible ratio experiments by experience or trial and error, so these historical data can be fully utilized. The reason for deleting the data with missing values and outliers here is that the data input to the model is less, and the historical data and experimental data can fully meet the training of the regression model; secondly, we delete the missing data and outliers to ensure the authenticity of the data.

[0097] Where the detection of outliers we adopt the interquartile range method to detect outliers, the steps are as follows:

[0098] Sort the crucible strength and high temperature performance data;

[0099] Calculate the first quartile data (Q1): the 25th position of the sorted data;

[0100] Calculate the third quartile data (Q3): the 75th position of the sorted data;

[0101] Calculate the interquartile range (IQR): IQR = Q3-Q1;

[0102] Suppose the crucible strength data is as follows:

[0103] The sorted data is: [200, 220, 230, 240, 250, 260, 270, 350]

[0104] Calculate Q1 (the first quartile): the first quartile position is the (8+1) / 4 = 2.25 position, take the average of the 2nd and 3rd data points: Q1 = 220+0.25×(230-220) = 220+2.5 = 222.5;

[0105] Calculate Q3 (the third quartile): the third quartile position is the 3(8+1) / 4 = 6.75 position, take the average of the 6th and 7th data points: Q3 = 260+0.75×(270-260) = 260+7.5 = 267.5;

[0106] Calculate IQR: IQR = Q3-Q1 = 267.5-222.5 = 45;

[0107] Determine the boundary of outliers:

[0108] Calculate the upper and lower boundaries: lower boundary = Q1-1.5×IQR;

[0109] Upper boundary = Q3+1.5×IQR;

[0110] Example calculation:

[0111] Lower boundary = 222.5-1.5×45 = 222.5-67.5 = 155;

[0112] Upper boundary = 267.5+1.5×45 = 267.5+67.5 = 335;

[0113] Any data point below the lower boundary or above the upper boundary is considered an outlier:

[0114] For example, data point 350, which is above the upper boundary, is determined as an outlier and is thus removed. When removing outliers, the corresponding data needs to be extracted. For example, if we remove the crucible strength data, the corresponding high-temperature resistance performance data and the matching data also need to be removed.

[0115] The standardization adopts a z-score standardization method to convert the data into a standard normal distribution with a mean of 0 and a standard deviation of 1.

[0116] Step 3: Establish a crucible strength regression model and a high-temperature resistance performance regression model under different quartz sand and aluminum quartz sand matching ratios using a regression model.

[0117] The crucible strength regression model and the high-temperature resistance performance regression model mentioned in step 3 are as follows:

[0118] Crucible strength regression model:

[0119]

[0120] In the formula, x1 and x2 represent the proportions of quartz sand and aluminum quartz sand, respectively, and satisfy x1+x2=1; β0 is the bias term of the crucible strength regression model; β1 and β2 represent the linear effects of quartz sand and aluminum quartz sand on crucible strength; β3 and β4 represent the quadratic coefficients, representing the nonlinear effects of the square terms of the matching ratios on the crucible strength; and β5 represents the interaction term coefficient, representing the interaction between quartz sand and aluminum quartz sand.

[0121] High-temperature resistance performance regression model:

[0122]

[0123] In the formula, γ0, γ1, …, γ5 represent the coefficients of the high-temperature resistance performance regression model; γ0 represents the bias term of the high-temperature resistance performance regression model; γ1 and γ2 represent the linear effects of quartz sand and aluminum quartz sand on high-temperature resistance performance; γ3 and γ4 represent the quadratic coefficients, representing the nonlinear effects of the square terms of the matching ratios on high-temperature resistance performance; and γ5 represents the interaction term coefficient of the high-temperature resistance performance regression model, representing the interaction between quartz sand and aluminum quartz sand.

[0124] The crucible strength regression model and the high-temperature resistance performance regression model each have an objective function, and the Adam optimizer is used to optimize the model parameters.

[0125] In this embodiment, the influence of the quartz sand and aluminum quartz sand ratio on the crucible strength and high temperature resistance performance is accurately described by the crucible strength regression model and the high temperature resistance performance regression model. Not only the linear influence of individual components is considered, but also the non-linear influence and interaction is included, providing a multi-dimensional performance prediction model. Secondly, the trained crucible strength regression model and high temperature resistance performance regression model are used as the fitness function of the genetic algorithm, which can effectively explore the optimal solution in the multi-dimensional variable space. The global search ability of the genetic algorithm and the accurate prediction of the regression model make the material ratio optimization process more efficient and accurate.

[0126] The objective function of the crucible strength regression model is as follows:

[0127]

[0128] In the formula: represents the i-th prediction of the crucible strength of the model; represents the quartz sand ratio of the i-th sample; represents the aluminum quartz sand ratio of the i-th sample; represents the true crucible strength corresponding to the i-th sample; N represents the number of samples; L S represents the optimization objective function for the crucible strength;

[0129] The objective function of the high temperature resistance performance regression model is as follows:

[0130]

[0131] In the formula: L T represents the objective function for the crucible high temperature resistance performance; represents the i-th prediction of the crucible high temperature resistance performance of the model; represents the true crucible high temperature resistance performance corresponding to the i-th sample.

[0132] The specific steps of using the Adam optimizer to optimize the model parameters are as follows:

[0133] First-order moment estimation and second-order moment estimation:

[0134] m t = α1m t-1 + (1-α1)g t ;

[0135]

[0136] In the formula: g t represents the gradient at the current time t; m t-1 represents the momentum at the last time, i.e. the first-order moment estimation; v t-1denotes the second moment estimation at the last time step; and a1 and a2 denote hyperparameters;

[0137] Bias correction:

[0138]

[0139] where: denotes the corrected momentum estimation; denotes the corrected second moment estimation;

[0140] Update parameters: the corrected and are used to update each parameter t :

[0141]

[0142] where: t+1 denotes the updated model parameter; and t denotes the model parameter before update; denotes the learning rate; and denotes a constant to prevent the denominator from being zero;

[0143] where the learning rate is updated every predetermined epoch:

[0144] η t+1 = η t * gamma;

[0145] where: gamma is a decay factor; and t+1 denotes the updated learning rate; and t denotes the learning rate before update.

[0146] In this embodiment, the Adam optimizer is used to optimize the model parameters. The Adam optimizer has the advantages of high computational efficiency, low memory occupation, and strong adaptability to sparse gradient data. It can dynamically adjust the learning rate and improve the convergence speed, so as to find the optimal model parameters faster.

[0147] Step 4: The established crucible strength regression model and high-temperature resistance regression model are trained using the training set data, and the prediction accuracy of the trained model is evaluated using the test set. If the accuracy meets the requirements, the trained crucible strength regression model and high-temperature resistance regression model are output. If the accuracy does not meet the requirements, the model is continuously trained until the accuracy meets the requirements. The accuracy of the model is evaluated using the mean square error, root mean square error, or mean absolute error. If the error is higher than the set threshold, it means that the requirements are not met and the model needs to be continuously trained. Otherwise, the trained model is obtained.

[0148] Step 5: The trained crucible strength regression model and high-temperature resistance regression model are used as the fitness function of the genetic algorithm, and the optimal ratio is iteratively solved based on the genetic algorithm;

[0149] The fitness function and constraint conditions of the genetic algorithm are as follows:

[0150] Fitness function:

[0151] F(x1, x2) = w1·S(x1, x2) + w2·T(x1, x2);

[0152] In the formula: S(x1, x2) represents the output of the crucible strength regression model; T(x1, x2) represents the output of the high-temperature resistance regression model; w1 represents the weight coefficient of strength; w2 represents the weight coefficient of high-temperature resistance; wherein w1 + w2 = 1; The setting of the weight coefficient allows the user to adjust the optimization focus according to actual needs, for example, giving priority to strength in some applications, and paying more attention to high-temperature resistance in other applications.

[0153] Constraint conditions:

[0154] Quartz sand and aluminum quartz sand ratio constraint:

[0155] x1 + x2 = 1;

[0156] 0 < x1 < 1, 0 < x2 < 1;

[0157] Crucible strength constraint:

[0158] S(x1, x2) ≥ min_strength;

[0159] In the formula: min_strength represents the preset minimum crucible strength requirement;

[0160] Crucible high-temperature resistance constraint:

[0161] T(x1, x2) ≥ min_temperature;

[0162] In the formula: min_temperature represents the preset minimum high-temperature performance constraint of the crucible.

[0163] Referring to Figure 2 The specific steps of iteratively solving the optimal ratio based on the genetic algorithm are as follows:

[0164] Initialize the population: set the population size to N, indicating that there are N individuals, randomly generate the ratio x1 and x2 of each individual, and each randomly generated individual ratio satisfies the quartz sand and aluminum quartz sand ratio constraint;

[0165] Selection operation: according to the fitness function, the fitness F(x1, x2) of each individual is calculated, the selection probability is calculated according to the fitness, and then the individual is selected according to the calculated probability through a random selection operation, wherein the probability is calculated as follows:

[0166]

[0167] In the formula, F penalized,i represents the fitness of the individual i after punishment; N represents the population size; P i represents the probability of selection of the individual i;

[0168] Crossover operation: a single-point crossover method is used to perform a crossover operation based on a preset crossover probability, and for two parent individuals and a crossover point k is randomly selected, and the genes of the parents after the crossover point are exchanged to obtain two crossed offspring and

[0169] Mutation operation: the frequency of mutation is controlled based on a predetermined mutation probability, the genes in the individual are changed, and a new individual is obtained;

[0170] Constraint correction: in the crossover and mutation operations, each generated individual needs to satisfy the quartz sand and aluminum quartz sand ratio constraint, the crucible strength constraint, and the crucible high-temperature performance constraint;

[0171] If the quartz sand and aluminum quartz sand ratio constraint is not satisfied, normalization is performed for adjustment:

[0172]

[0173] Based on this, the adjusted quartz sand and aluminum quartz sand ratio is obtained;

[0174] If the crucible strength constraint is not satisfied, a penalty is applied to the fitness, and the specific penalty is as follows:

[0175] F penalized (x1, x2) = F(x1, x2) - λ1·max(0, min_strength - S(x1, x2));

[0176] In the formula, λ1 represents a penalty coefficient; F penalized (x1, x2) represents the fitness function value after adjustment based on the penalty;

[0177] If the crucible high-temperature performance constraint is not satisfied, a penalty is applied to the fitness, and the specific penalty is as follows:

[0178] F penalized(x1,x2) = F(x1,x2) - λ2·max(0,min_temperature - T(x1,x2));

[0179] wherein λ2 represents a penalty coefficient;

[0180] For individuals that do not meet the strength and high-temperature resistance performance constraints, the fitness of the individuals is reduced by a penalty function. This method effectively reduces the influence of unqualified individuals in the population and gradually guides the population to evolve in the direction of meeting the constraint conditions. It should be noted that if any individual does not meet the above three constraints at the same time, the above three constraint correction steps need to be performed in turn. If any individual does not meet any of the above constraints, only the corresponding constraint correction needs to be performed. If any individual meets any of the above two constraints at the same time, only the corresponding two constraint corrections need to be performed in turn.

[0181] Termination: In each iteration, new individuals are continuously generated and fitness is evaluated through selection, crossover, mutation and constraint conditions, and the iteration is stopped until a predetermined condition is reached. The value obtained that meets the constraint conditions and has the highest fitness is taken as the optimal solution.

[0182] An elite reservation strategy is introduced in the genetic algorithm. The fitness of each iteration is sorted from large to small, and the individuals with high fitness values are selected based on the elite reservation ratio to enter the next iteration;

[0183] wherein the elite reservation ratio is dynamically adjusted based on the fitness change amount, and the specific adjustment is as follows:

[0184]

[0185] wherein: Ft represents the fitness value of the optimal individual in the tth iteration; Ft-1 represents the fitness value of the optimal individual in the (t-1)th iteration; and ΔF represents the fitness change amount of two adjacent iterations;

[0186] The calculated fitness change amount is compared with a pre-set fitness threshold. If the fitness change amount is less than the fitness threshold, the proportion of elite individuals is increased:

[0187] p elite ′ = min(p elite + δ, 1);

[0188] wherein p elite ′ represents the modified elite reservation ratio; δ represents a pre-defined adjustment amount; min(p elite + δ, 1) is used to ensure that the adjusted elite reservation ratio does not exceed 100%; p elite represents the elite reservation ratio before modification; and p

[0189] If the fitness change amount is greater than the threshold, then the elite reservation ratio is reduced:

[0190] p elite ′=max(p elite -δ,p min );

[0191] In the formula, p min represents a pre-set minimum elite ratio.

[0192] In this embodiment, the elite reservation ratio is dynamically adjusted according to the fitness change amount, effectively solving the problems that may be caused by a fixed elite reservation ratio; the elite ratio is increased when the fitness change amount is small, ensuring that excellent individuals will not be eliminated; the elite ratio is reduced when the fitness change amount is large, enhancing the diversity and exploration ability of the population; by dynamically adjusting the elite ratio, the algorithm is prevented from converging to a local optimal solution too early, the diversity of the population is maintained, and the global search ability is improved

[0193] The above merely describes a preferred embodiment of the present application and is not used to limit the present application, and any modification, equivalent replacement, and improvement made within the spirit and principle of the present application shall be included in the protection scope of the present application.

Claims

1. A method for optimizing the material ratio of quartz crucibles based on genetic algorithms, characterized in that, Includes the following steps: Step 1: Collect data on crucible strength and high-temperature resistance under different ratios of quartz sand and aluminum quartz sand. Each set of data includes ratio data and performance data. The ratio data is the proportion of quartz sand and aluminum quartz sand, and the performance data is the crucible strength and high-temperature resistance data under the corresponding ratio. Step 2: Based on the collected data, perform data cleaning and standardization, and then divide the standardized data into training set and test set; Step 3: Establish regression models for crucible strength and high-temperature resistance under different ratios of quartz sand and aluminosilicate sand, respectively; Step 4: Use the training set data to train the established crucible strength regression model and high temperature resistance performance regression model, and use the test set to evaluate the prediction accuracy of the trained model. If the accuracy meets the requirements, output the trained crucible strength regression model and high temperature resistance performance regression model. If the accuracy does not meet the requirements, continue training the model until the accuracy meets the requirements. Step 5: Use the trained crucible strength regression model and high temperature resistance performance regression model as the fitness function of the genetic algorithm, and iteratively solve the optimal ratio based on the genetic algorithm; An elite retention strategy is introduced into the genetic algorithm. Individuals with higher fitness values ​​are selected for the next iteration based on the fitness ratio of each iteration. The elite retention ratio is dynamically adjusted based on the change in fitness, as detailed below: In the formula: This represents the fitness value of the best individual in the t-th iteration; ΔF represents the fitness value of the optimal individual in the (t-1)th iteration; ΔF represents the change in fitness between two adjacent iterations. The calculated change in fitness is compared with a pre-set fitness threshold. If the change in fitness is less than the fitness threshold, the proportion of elite individuals is increased. p elite '=min(p elite +d,1); In the formula: p elite 'Modified elite retention ratio; δ represents the predefined adjustment amount; min(p elite +δ,1) is used to ensure that the adjusted elite retention rate does not exceed 100%; p elite This indicates the percentage of elites retained before the modification. If the change in fitness exceeds a threshold, then reduce the proportion of elites retained: p elite '=max(p elite -δ,p min ); In the formula: p min This indicates the pre-set minimum elite ratio.

2. The method for optimizing the material ratio of quartz crucibles based on genetic algorithms according to claim 1, characterized in that, The crucible strength regression model and high-temperature resistance performance regression model mentioned in step 3 are as follows: Crucible strength regression model: In the formula: x1 and x2 represent the proportions of quartz sand and aluminum quartz sand, respectively, satisfying x1 + x2 = 1; β0 is the bias term of the crucible strength regression model, β1 and β2 represent the linear effects of quartz sand and aluminum quartz sand on the crucible strength, respectively; β3 and β4 represent the quadratic coefficients, representing the nonlinear effects of the square of the proportions on the crucible strength; β5 represents the interaction coefficient, representing the interaction between quartz sand and aluminum quartz sand. High-temperature resistance performance regression model: In the formula: γ0, γ1, ..., γ5 represent the coefficients of the high-temperature resistance performance regression model; γ0 represents the bias term of the high-temperature resistance performance regression model; γ1 and γ2 represent the linear effects of quartz sand and aluminosilicate sand on the high-temperature resistance performance, respectively; γ3 and γ4 represent the coefficients of the quadratic term, which are the nonlinear effects of the matched square term on the high-temperature resistance performance; γ5 represents the coefficient of the interaction term of the high-temperature resistance performance regression model, which is the interaction between quartz sand and aluminosilicate sand. The crucible strength regression model and the high temperature resistance performance regression model are respectively established with objective functions, and the Adam optimizer is used to optimize the model parameters.

3. The method for optimizing the material ratio of quartz crucibles based on genetic algorithms according to claim 2, characterized in that, The objective function of the crucible strength regression model is as follows: In the formula: This represents the crucible strength predicted by the model in the i-th iteration; This represents the quartz sand ratio of the i-th sample; This represents the aluminum silica sand ratio of the i-th sample; L represents the actual crucible strength corresponding to the i-th sample; N represents the number of samples; L S This represents the objective function for optimizing the crucible strength. The objective function of the high-temperature resistance performance regression model is as follows: In the formula: L T Represent the objective function regarding the high-temperature resistance performance of the crucible; This represents the high-temperature resistance performance of the crucible predicted by the model in the i-th iteration. This represents the actual high-temperature resistance performance of the crucible corresponding to the i-th sample.

4. The method for optimizing the material ratio of quartz crucibles based on genetic algorithms according to claim 3, characterized in that, The specific steps for optimizing the model parameters using the Adam optimizer are as follows: First-order moment estimate and second-order moment estimate: m t =α1m t-1 +(1-α1)g t ; Where: g t The gradient at time t is represented by m. t-1 This represents the momentum at the previous moment, i.e., the first moment estimate; v t-1 α1 represents the second moment estimate of the previous time step; α2 and α1 represent the hyperparameters. Deviation correction: In the formula: This indicates the revised momentum estimate; This represents the corrected second-order moment estimate; Update parameters: Use the corrected ones and To update each parameter θ t : In the formula: θ t+1 Represents the updated model parameters; θ t Represents the model parameters before the update; η represents the learning rate; ∈ represents a constant to prevent the denominator from being zero; The learning rate η is updated every predetermined epoch: or t+ 1=h t ×c; In the formula: γ is the attenuation factor; η t+1 η represents the updated learning rate. t This represents the learning rate before the update.

5. The method for optimizing the material ratio of quartz crucibles based on genetic algorithms according to claim 1, characterized in that, The fitness function and constraints of the genetic algorithm are as follows: Fitness function: F(x1,x2)=w1·S(x1,x2)+w2·T(x1,x2); In the formula: S(x1,x2) represents the output of the crucible strength regression model; T(x1,x2) represents the output of the high temperature resistance performance regression model; w1 represents the weighting coefficient of strength; w2 represents the weighting coefficient of high temperature resistance performance; where w1+w2=1; Constraints: The ratio constraints of quartz sand and aluminum quartz sand: x1 + x2 = 1; 0 <x1<1,0<x2<1; Crucible strength constraints: S(x1,x2)≥min_strength; In the formula: min_strength represents the preset minimum strength requirement of the crucible; High-temperature resistance constraints of crucibles: T(x1,x2)≥min_temperature; In the formula: min_temperature represents the preset minimum high-temperature performance constraint of the crucible.

6. The method for optimizing the material ratio of quartz crucibles based on genetic algorithms according to claim 5, characterized in that, The specific steps for iteratively solving the optimal ratio using a genetic algorithm are as follows: Population initialization: Let the population size be N, representing N individuals. Randomly generate the ratios x1 and x2 for each individual, where the randomly generated ratios of each individual satisfy the ratio constraints of quartz sand and aluminum quartz sand. Selection operation: Calculate the fitness F(x1,x2) of each individual based on the fitness function, calculate the selection probability according to the fitness, and then select an individual based on the calculated probability through a random selection operation. The probability calculation is as follows: In the formula: F penalized,i P represents the fitness of individual i after punishment; N represents the population size; P represents the population size after punishment. i This represents the probability that individual i is selected; Crossover operation: A single-point crossover method is used to perform the crossover operation based on a preset crossover probability for two parent individuals. and Randomly select a crossover point k, and swap the genes of the parent generation after the crossover point to obtain two offspring. and Mutation operation: Based on a predetermined mutation probability, the frequency of mutation is controlled to change the genes in an individual and obtain a new individual; Constraint Correction: In the aforementioned crossover and mutation operations, each generated individual needs to satisfy the constraints on the ratio of quartz sand and aluminum quartz sand, the crucible strength constraint, and the crucible's high-temperature resistance constraint. If the ratio constraints for quartz sand and aluminosilicate sand are not met, adjustments are made through normalization: Based on this, the adjusted ratio of quartz sand and aluminum quartz sand was obtained. If the crucible strength constraint is not met, a penalty is imposed on the fitness, as follows: F penalized (x1,x2)=F(x1,x2)-λ1·max(0,min_strength-S(x1,x2)); In the formula: λ1 represents the penalty coefficient; F penalized (x1, x2) represents the fitness function value after penalty adjustment; If the high-temperature resistance constraint of the crucible is not met, a penalty will be imposed on the adaptability, as follows: F penalized (x1,x2)=F(x1,x2)-λ2·max(0,min_temperature-T(x1,x2)); In the formula: λ2 represents the penalty coefficient; Termination: In each iteration, new individuals are continuously generated and their fitness is evaluated through selection, crossover, mutation, and constraints until the predetermined conditions are met, at which point the iteration stops, and the value that meets the constraints and has the highest fitness is taken as the optimal solution.

Images