Machine learning based method for dual-target optimization of resistivity and magnetostriction properties of tb-dy-fe alloys

Abstract

The present application relates to a Tb-Dy-Fe alloy resistivity and magnetostriction performance double-target collaborative optimization method based on machine learning, belongs to the cross technical field of material informatics and rare earth magnetostrictive functional materials, and solves the problem that magnetostrictive performance and resistivity are difficult to consider in the prior art. The method comprises the following steps: S1, constructing a Tb-Dy-Fe alloy multi-performance database; S2, constructing a magnetostrictive performance special feature descriptor and constructing a resistivity special feature descriptor; S3, training a magnetostriction coefficient prediction model and training a resistivity prediction model; S4, constructing a double-target optimization framework: taking the maximization of resistivity and the maximization of the saturation magnetostriction coefficient as the optimization target, setting the constraint condition, adopting a multi-target optimization algorithm for solution, and outputting a Pareto optimal solution set; S5, Pareto front analysis and scheme recommendation: performing cluster analysis on the Pareto optimal solution set, identifying a typical composition interval, selecting an optimal scheme according to an application scenario, and outputting a recommended composition and predicted performance.

Description

Technical Field

[0001] This invention relates to the interdisciplinary field of materials informatics and rare earth magnetostrictive functional materials, and in particular to a dual-objective synergistic optimization method for resistivity and magnetostrictive properties of Tb-Dy-Fe alloys based on machine learning. Background Technology

[0002] Tb-Dy-Fe alloy (typical composition Tb) 0.3 Dy 0.7 Fe 1.95 Terfenol-D (trade name) is currently the material with the highest magnetostriction coefficient at room temperature, with a saturation magnetostriction coefficient of 1500-2000 ppm, far exceeding that of traditional nickel-based magnetostrictive materials (approximately -35 ppm). This material has broad application prospects in fields such as sonar transducers, precision positioning actuators, active vibration control, magnetostrictive delay lines, and energy harvesting devices.

[0003] However, Tb-Dy-Fe alloys face the following technical bottlenecks in practical applications: First, high magnetostriction coefficients typically require high rare earth content and a specific Tb / Dy ratio, which often leads to a decrease in alloy resistivity. Low resistivity generates significant eddy current losses under alternating magnetic field conditions, reducing device efficiency and limiting the upper limit of operating frequency. For example, in sonar transducer applications (operating frequency 1-20kHz), eddy current losses can lead to 10-30% energy loss, significantly reducing the device's electroacoustic conversion efficiency. Second, the performance of Tb-Dy-Fe based alloys is affected by multiple factors such as the ratio of main elements, the type and content of dopants, and the preparation process, resulting in a large composition-process space. Taking a system containing four independent variables (such as Tb / Dy ratio, total rare earth content, dopant content, and annealing temperature) as an example, if each variable has 10 levels, then 10... 4 With 10,000 sets of experiments, the traditional "trial and error" method is insufficient to exhaustively enumerate them, significantly extending the experimental cycle and causing costs to skyrocket. Furthermore, existing research primarily focuses on a single objective (maximizing the magnetostriction coefficient), lacking a systematic exploration of synergistic improvements in multi-objective performance. Optimization strategies and technical pathways that consider multiple objective performances, such as resistivity and magnetostriction, have not yet been established, making it difficult to simultaneously meet the dual requirements of high magnetostriction effect and low eddy current loss in devices. To address these issues, machine learning-driven material optimization methods, due to their ability to efficiently explore high-dimensional composition spaces and establish complex composition-performance mapping relationships, have become an important approach to overcoming this technological bottleneck.

[0004] In recent years, machine learning methods have made significant progress in optimizing material composition and have been initially applied in the field of magnetostrictive materials. However, the synergistic optimization of magnetostrictive properties and resistivity has not yet been achieved. Summary of the Invention

[0005] In view of the above analysis, the present invention aims to provide a dual-objective synergistic optimization method for resistivity and magnetostrictive properties of Tb-Dy-Fe alloy based on machine learning, in order to solve at least one of the following technical problems: (1) lack of a dedicated feature descriptor system that can effectively characterize the magnetoelastic coupling mechanism of rare earth ions 4f electrons, and general physical parameters are difficult to accurately capture the intrinsic mapping relationship between composition and magnetostrictive properties; (2) no dedicated feature descriptor for resistivity has been established, and synergistic optimization of magnetostrictive properties and resistivity cannot be achieved; (3) lack of a multi-objective optimization framework for different application scenarios, and it is difficult to provide customized composition design schemes for specific applications such as sonar transducers and precision actuators.

[0006] The objective of this invention is mainly achieved through the following technical solutions: This invention provides a dual-objective synergistic optimization method for the resistivity and magnetostrictive properties of Tb-Dy-Fe alloys based on machine learning, comprising the following steps: S1. Construct a multi-performance database for Tb-Dy-Fe alloys; S2. Construct a dedicated feature descriptor for magnetostrictive properties and a dedicated feature descriptor for resistivity; S3. Train the magnetostriction coefficient prediction model and train the resistivity prediction model; S4. With maximizing resistivity and maximizing saturation magnetostriction as optimization objectives, a dual-objective optimization framework is constructed, constraints are set, and a multi-objective optimization algorithm is used to solve the problem, outputting the Pareto optimal solution set. S5. Perform cluster analysis on the Pareto optimal solution set, identify typical component intervals, select the optimal solution according to the application scenario, and output recommended components and prediction performance.

[0007] Optionally, in S2, the magnetostrictive property-specific feature descriptors include: a magnetoelastic coupling descriptor group based on the Stevens factor of rare earth ions, a spin-orbit coupling descriptor group based on the 4f electronic properties of rare earth ions, and a crystal structure descriptor group.

[0008] Optionally, the spin-orbit coupling descriptor set based on the 4f electronic properties of rare earth ions includes the equivalent spin-orbit coupling energy and the orbital angular momentum contribution factor. Equivalent spin-orbit coupling energy The calculation is expressed using the following formula:

[0009] Where, x Tb Indicates the percentage of Tb atoms. x Dy Indicates the percentage of Dy atoms. ; Orbital angular momentum contribution factor The calculation is expressed using the following formula:

[0010] Where, x Tb L represents the percentage of Tb atoms. Tb =3, x Dy L represents the percentage of Dy atoms. Dy =5.

[0011] Optionally, the magnetoelastic coupling descriptor set based on rare-earth ion Stevens factor includes an equivalent second-order Stevens factor, an equivalent fourth-order Stevens factor, and a magnetostriction driving factor. Equivalent second-order Stevens factor The calculation is expressed using the following formula: Where, x Tb x represents the percentage of Tb atoms. Dy Indicates the percentage of atoms in Dy, x Mi This represents the atomic percentage of the i-th dopant element. , , Indicates the first i The second-order Stevens factor of a doped element; wherein, when the doped element M... i When it is a non-rare earth element, When the doping element Mi is a rare earth element, Take the theoretical value of the second-order Stevens factor corresponding to the rare earth ion; Equivalent fourth-order Stevens factor The calculation is expressed using the following formula:

[0012] Where, x Tb Indicates the percentage of Tb atoms x Dy Indicates the percentage of Dy atoms. x Mi This represents the atomic percentage of the i-th dopant element. Denotes the fourth-order Stevens factor of the i-th doped element; wherein, when the doped element M i When it is a non-rare earth element, When the doping element M i When it is a rare earth element, Take the theoretical value of the fourth-order Stevens factor corresponding to the rare earth ion; Magnetostriction driving factor The calculation is expressed using the following formula: in, J(J+1) eff This is the weighted average of the total angular momentum quantum numbers of rare earth ions. .

[0013] Optionally, in S2, the resistivity-specific feature descriptors include: an electronic scattering descriptor group based on rare-earth ion magnetic scattering, an electronic structure descriptor group, and a defect scattering descriptor group.

[0014] Optionally, the set of electronic scattering descriptors based on rare-earth ion magnetic scattering includes: spin disorder scattering factor and composition disorder scattering factor; Spin disorder scattering factor The calculation is expressed using the following formula:

[0015] Where, x Tb Indicates the percentage of Tb atoms, g J Tb = 3 / 2, which is Tb 3+ Lande g factor, J Tb =6, for Tb 3+ The total angular momentum quantum number, x Dy G represents the percentage of Dy atoms. J Dy = 4 / 3, for Dy 3+ Lande g factor, J Dy =15 / 2 is Dy 3+ The total angular momentum quantum number; Component disorder scattering factor The calculation is expressed using the following formula:

[0016] in, The percentage of atoms of the i-th element. Z represents the atomic percentage of the j-th element. i Let be the atomic number of the i-th element. Let be the atomic number of the j-th element.

[0017] Optionally, in S4, the multi-objective optimization algorithm is the multi-objective evolutionary algorithm NSGA-III or the multi-objective Bayesian optimization method.

[0018] Optionally, in S1, constructing a multi-performance database of Tb-Dy-Fe alloys includes: collecting Tb-Dy-Fe-based alloy data, with each data record containing composition parameters, process parameters, magnetostrictive performance parameters, resistivity parameters, and auxiliary performance parameters, and preprocessing the data.

[0019] Optionally, the composition parameters include: Tb atomic percentage x Tb , Dy atomic percentage x Dy Fe atomic percentage x Fe Dopant element M i and the atomic percentage x of the i-th dopant element Mi (i=1,2,...,n).

[0020] The present invention also provides the application of the above-described method in the composition design of Tb-Dy-Fe alloys, which are used in sonar transducers, precision positioning actuators, vibration energy harvesters, or magnetostrictive delay lines.

[0021] Compared with the prior art, the present invention can achieve at least one of the following beneficial effects: (1) To address the problem that general physical parameters are difficult to accurately capture the intrinsic mapping relationship between composition and magnetostrictive properties, this invention constructs a dedicated descriptor system for magnetostrictive properties. For the first time, it systematically uses parameters with clear physical meanings, such as Stevens factor, spin-orbit coupling energy, and magnetostrictive driving factor, as machine learning features. This effectively characterizes the physical essence of the 4f electron magnetoelastic coupling mechanism of rare earth ions and accurately establishes the intrinsic mapping relationship between composition and magnetostrictive properties, thereby achieving high-precision performance prediction (the examples show that the relative error of the magnetostrictive coefficient prediction is only 2.2%-3.2%).

[0022] (2) This invention constructs a dedicated resistivity feature descriptor, realizing dual-objective collaborative optimization of magnetostrictive performance and resistivity. Addressing the trade-off between performance and loss in magnetostrictive device applications, this invention employs a multi-objective optimization algorithm, establishing a multi-objective optimization framework and outputting a Pareto optimal solution set. This provides customized component schemes for different application scenarios such as sonar transducers and precision actuators, overcoming the limitations of existing single-objective optimization methods.

[0023] (3) The present invention significantly shortens the material development cycle. Traditional trial and error methods require 100-500 experiments to screen the optimal components. The present invention uses machine learning prediction combined with multi-objective optimization, which can greatly reduce the number of experiments, significantly shorten the development cycle and reduce costs.

[0024] (4) This invention supports active learning and experimental closed loop. The multi-objective Bayesian optimization method provided can be combined with small sample data. Through the closed loop iteration of "prediction-experiment-feedback", it can efficiently converge to the Pareto front in the exploration of new component systems.

[0025] (5) The method model of the present invention has high prediction accuracy. Specifically, the relative error in predicting the magnetostrictive properties of the alloy is 2.2%-3.2%, and the relative error in predicting the resistivity of the alloy is 1.4%-3.2%.

[0026] In this invention, the above-described technical solutions can be combined with each other to achieve more preferred combinations. Other features and advantages of this invention will be set forth in the following description, and some advantages will become apparent from the description or may be learned by practicing the invention. Attached Figure Description

[0027] The accompanying drawings are for illustrative purposes only and are not intended to limit the invention. Throughout the drawings, the same reference numerals denote the same parts.

[0028] Figure 1 A schematic diagram of the overall process of the dual-objective collaborative optimization method for resistivity and magnetostrictive properties of Tb-Dy-Fe alloy based on machine learning provided in an embodiment of the present invention. Figure 2 This is a schematic diagram of the integrated learning and training framework for the magnetostriction coefficient prediction model and resistivity prediction model provided in an embodiment of the present invention. Detailed Implementation

[0029] The preferred embodiments of the present invention will now be described in detail with reference to the accompanying drawings, which constitute a part of the present invention and are used together with the embodiments of the present invention to illustrate the principles of the present invention, but are not intended to limit the scope of the present invention.

[0030] Machine learning methods have made significant progress in material composition optimization and have been initially applied in the field of magnetostrictive materials. However, the synergistic optimization of magnetostrictive properties and resistivity has not yet been achieved. For example, a machine learning database method for RFe2-type magnetostrictive materials uses compound composition content and physical parameter data (including atomic radius difference, melting point temperature, specific heat, thermal conductivity, electronegativity, valence electron concentration, bulk modulus, shear modulus, Poisson's ratio, and volume magnetic susceptibility) as feature operators, and magnetostrictive values ​​as target operators to construct the database. The study found that this method mainly relies on composition-weighted average physical parameters and fails to specifically characterize the spin-orbit coupling strength and magnetoelastic coupling effect of rare earth ions in Tb-Dy-Fe alloys, making it difficult to accurately predict the magnetostrictive properties of complex multi-component alloy systems. More importantly, this method does not involve the construction of a resistivity prediction model, thus failing to achieve synergistic optimization of magnetostrictive properties and resistivity.

[0031] To achieve synergistic optimization of magnetostrictive properties and resistivity, the following technical challenges need to be overcome: First, establish a dedicated feature descriptor system that can effectively characterize the 4f electron magnetoelastic coupling mechanism of rare-earth ions, avoiding the difficulty of accurately capturing the intrinsic mapping relationship between composition and magnetostrictive properties using general physical parameters; Second, establish a dedicated resistivity feature descriptor to achieve synergistic optimization of magnetostrictive properties and resistivity; Third, develop a multi-objective optimization framework for different application scenarios to provide customized composition design schemes for specific applications such as sonar transducers and precision actuators.

[0032] like Figure 1 As shown, in a first aspect, the present invention provides a dual-objective synergistic optimization method for the resistivity and magnetostrictive properties of Tb-Dy-Fe alloys based on machine learning, comprising the following steps: S1. Construct a multi-performance database for Tb-Dy-Fe alloys; S2. Construct a dedicated feature descriptor for magnetostrictive properties and a dedicated feature descriptor for resistivity; S3. Train the magnetostriction coefficient prediction model and train the resistivity prediction model; S4. Construct a dual-objective optimization framework: With maximizing resistivity and maximizing saturation magnetostriction as optimization objectives, set constraints, use a multi-objective optimization algorithm to solve the problem, and output the Pareto optimal solution set. S5, Pareto Front Analysis and Solution Recommendation: Perform cluster analysis on the Pareto optimal solution set, identify typical component intervals, select the optimal solution based on the application scenario, and output recommended components and prediction performance.

[0033] Specifically, in S1, constructing a multi-performance database for Tb-Dy-Fe alloys includes: collecting Tb-Dy-Fe-based alloy data, with each data record containing composition parameters, process parameters, magnetostrictive property parameters, resistivity parameters, and auxiliary performance parameters, and preprocessing the data. The magnetostrictive property parameters are measured using the resistance strain gauge method, and the resistivity parameters are measured using the four-probe method.

[0034] Composition parameters include: Tb atomic percentage x Tb , Dy atomic percentage x Dy Fe atomic percentage x Fe Dopant element M i and its content x Mi (i=1,2,...,n).

[0035] Among them, dopant element M i It is selected from one or more of the following elements: Al, Ga, Nb, Si, Co, Mn, Pr, Ho, Er, B, C, Zr.

[0036] Process parameters include: directional solidification casting speed v (mm / min), annealing temperature T. a (°C), Annealing time t a (h).

[0037] Magnetostrictive performance parameters include: saturation magnetostriction coefficient λ s (ppm), low-field magnetostriction coefficient λ 33 (ppm, test magnetic field H=1kOe).

[0038] Resistivity parameters include: room temperature resistivity ρ (μΩ·cm).

[0039] Auxiliary performance parameters: Curie temperature T c (°C).

[0040] Specifically, the data on the aforementioned Tb-Dy-Fe based alloys can be collected from publicly available literature, patents, and laboratory data.

[0041] Data preprocessing includes: imputation of missing values ​​using the K-Nearest Neighbors (KNN) algorithm, outlier detection and removal based on box plots, and data standardization using Min-Max normalization or Z-score standardization. The database size should be no less than 200 records if sufficient data is available.

[0042] In S2, the dedicated feature descriptors for magnetostrictive properties include: a magnetoelastic coupling descriptor group based on the Stevens factor of rare earth ions, a spin-orbit coupling descriptor group based on the 4f electronic properties of rare earth ions, and a crystal structure descriptor group.

[0043] The magnetoelastic coupling descriptor set based on rare-earth ion Stevens factors includes an equivalent second-order Stevens factor, an equivalent fourth-order Stevens factor, and a magnetostriction driving factor.

[0044] Specifically, the equivalent second-order Stevens factor The calculation is expressed using the following formula: Where, x Tb x represents the percentage of Tb atoms. Dy Indicates the percentage of atoms in Dy, x Mi This represents the atomic percentage of the i-th dopant element. , , Denotes the second-order Stevens factor of the i-th doped element; wherein, when the doped element M i When it is a non-rare earth element, When the dopant element Mi is a rare earth element (a rare earth element other than the main elements Tb and Dy), Take the theoretical value of the second-order Stevens factor corresponding to the rare earth ion.

[0045] Equivalent fourth-order Stevens factor The calculation is expressed using the following formula:

[0046] Where, x Tb Indicates the percentage of Tb atoms x Dy Indicates the percentage of Dy atoms. x Mi This represents the atomic percentage of the i-th dopant element. Let Mi represent the fourth-order Stevens factor of the i-th dopant element; where, when the dopant element Mi is a non-rare earth element, When the dopant element Mi is a rare earth element (a rare earth element other than the main elements Tb and Dy), Take the theoretical value of the fourth-order Stevens factor corresponding to the rare earth ion.

[0047] When the dopant element is Pr , ; When the dopant element is Ho , ; When the dopant element is Er , .

[0048] Magnetostriction driving factor The calculation is expressed using the following formula: in, J(J+1) eff This is the weighted average of the total angular momentum quantum numbers of rare earth ions. .

[0049] The spin-orbit coupling descriptor set based on the 4f electronic properties of rare earth ions includes the equivalent spin-orbit coupling energy and the orbital angular momentum contribution factor.

[0050] Equivalent spin-orbit coupling energy The calculation is expressed using the following formula:

[0051] Where, x Tb Indicates the percentage of Tb atoms. x Dy Indicates the percentage of Dy atoms. (Based on Hartree-Fock calculations).

[0052] Orbital angular momentum contribution factor The calculation is expressed using the following formula:

[0053] Where, x Tb L represents the percentage of Tb atoms. Tb =3, x Dy L represents the percentage of Dy atoms. Dy =5.

[0054] The crystal structure descriptor set includes the Laves phase stability factor and lattice distortion degree.

[0055] Laves phase stability factor The calculation is expressed using the following formula:

[0056] Where, r RE r is the equivalent radius of a rare earth atom. Fe Where is the radius of an iron atom.

[0057] Lattice distortion The calculation is expressed using the following formula:

[0058] in, r represents the atomic percentage of the i-th element. i Let be the radius of the i-th atom. is the average atomic radius.

[0059] Constructing a dedicated feature descriptor for magnetostrictive properties involves the following steps: 1. Composition analysis and parameter extraction: Analyze the composition parameters of each sample in the multi-performance database, extract the atomic percentages of Tb, Dy, Fe and each dopant element Mi; and extract the theoretical values ​​of the second / fourth order Stevens factor, spin-orbit coupling energy constant, orbital angular momentum quantum number and atomic radius of the corresponding elements from the built-in physical parameter dictionary; 2. Calculation of magnetoelasticity and spin-orbit coupling characteristics: Based on the aforementioned formulas, combined with the extracted atomic percentages and physical parameters, the type of dopant element Mi (rare earth or non-rare earth) is determined, and the equivalent second-order Stevens factor is calculated accordingly. Equivalent fourth-order Stevens factor Magnetostriction driving factor Equivalent spin-orbit coupling energy and orbital angular momentum contribution factor The magnetoelastic coupling descriptor set and the spin-orbit coupling descriptor set were obtained respectively. 3. Calculation of crystal structure characteristics: Based on the aforementioned formula and the extracted atomic radii, calculate the Laves phase stability factor. With lattice distortion This yields a set of crystal structure descriptors, which in turn leads to a dedicated feature descriptor for magnetostrictive properties. 4. Feature Vector Concatenation: The magnetoelastic coupling descriptor set, spin-orbit coupling descriptor set, and crystal structure descriptor set obtained in the above steps are concatenated with the original composition parameters (elemental content) and process parameters (annealing temperature, annealing time, pulling speed) to form a complete magnetostrictive input feature vector X. λ .

[0060] In S2, the resistivity-specific feature descriptors include: an electronic scattering descriptor group based on rare-earth ion magnetic scattering, an electronic structure descriptor group, and a defect scattering descriptor group.

[0061] The electronic scattering descriptor set based on rare-earth ion magnetic scattering includes: spin disorder scattering factor and composition disorder scattering factor.

[0062] Spin disorder scattering factor The calculation is expressed using the following formula:

[0063] Where, x Tb Indicates the percentage of Tb atoms, g J Tb = 3 / 2, which is Tb 3+ Lande g factor, J Tb =6, for Tb 3+ The total angular momentum quantum number, x Dy G represents the percentage of Dy atoms. J Dy = 4 / 3, for Dy 3+ Lande g factor, J Dy =15 / 2 is Dy 3+ The total angular momentum quantum number.

[0064] Component disorder scattering factor The calculation is expressed using the following formula:

[0065] in, The percentage of atoms of the i-th element. Z represents the atomic percentage of the j-th element. i Let be the atomic number of the i-th element. Let be the atomic number of the j-th element.

[0066] The electronic structure descriptor set includes valence electron concentration, df hybridization strength estimator, and Fermi level density of states estimator.

[0067] Valence electron concentration The calculation is expressed using the following formula:

[0068] in, VEC represents the atomic percentage of the i-th element. i Let be the number of valence electrons of the i-th element.

[0069] df hybridization strength estimator The calculation is expressed using the following formula:

[0070] in, n represents the percentage of Fe atoms. dFe This represents the number of 3d electrons in Fe. n represents the atomic percentage of rare earth elements. fRE This represents the number of 4f electrons in rare earth elements.

[0071] Fermi level density of states estimation factor N(E) F The calculation of ) is expressed by the following formula:

[0072] in, N represents the atomic percentage of the i-th element. i (E F ) represents the theoretical value of the Fermi level density of states of the i-th pure element.

[0073] The defect scattering descriptor set includes: atomic size mismatch parameter and electronegativity mismatch parameter.

[0074] Atomic size mismatch parameters The calculation is expressed using the following formula:

[0075] in, r represents the atomic percentage of the i-th element. i Let be the radius of the i-th atom. is the average atomic radius.

[0076] Electronegativity mismatch parameter The calculation is expressed using the following formula;

[0077] in, χ represents the atomic percentage of the i-th element. i Let the Pauling electronegativity of the i-th element be... The weighted average electronegativity of all elements in an alloy system is expressed by the following formula: ,in, χ represents the atomic percentage of the i-th element. i Let be the Pauling electronegativity of the i-th element.

[0078] Constructing a resistivity-specific feature descriptor involves the following steps: 1. Composition analysis and parameter extraction: Analyze the composition parameters of the sample and extract the atomic percentage of each element; and extract the corresponding Landé g-factor, total angular momentum quantum number, atomic number, valence electron number, atomic radius and Pauling electronegativity from the physical parameter dictionary; 2. Calculation of electron scattering characteristics: Based on the aforementioned formulas, calculate the spin disorder scattering factor and the composition disorder scattering factor respectively to obtain the electron scattering descriptor set; 3. Calculation of electronic structure and defect scattering characteristics: Based on the aforementioned formulas, the valence electron concentration, df hybridization intensity estimation factor, Fermi level density of states estimation factor, atomic size mismatch parameter and electronegativity mismatch parameter are calculated respectively to obtain the electronic structure descriptor set and the defect scattering descriptor set, and then the resistivity-specific characteristic descriptor is obtained. 4. Feature vector concatenation: The resistivity-specific feature descriptor calculated above is concatenated with the original component parameters and process parameters to form a complete resistivity input feature vector.

[0079] In S3, the magnetostriction coefficient prediction model adopts the existing model. Training the magnetostriction coefficient prediction model includes: using the magnetostriction performance-specific feature descriptor constructed in S2, along with the original composition parameters (content of each element) and process parameters (annealing temperature, annealing time, pulling speed) as the magnetostriction input feature vector X. λ With the saturation magnetostriction coefficient λ s Or the low-field magnetostriction coefficient λ 33 y, as the target variable for magnetostriction λ A machine learning approach was used to train a model for predicting the magnetostriction coefficient.

[0080] Specifically, combined Figure 2 As shown, the machine learning method adopts an ensemble learning strategy, and the base learners are selected from two or more of the following models: random forest model, gradient boosting regression model, XGBoost model, support vector regression model, Gaussian process regression model, and multilayer perceptron model; the ensemble strategy adopts a stacking strategy or a weighted average strategy.

[0081] The selection of the base learner is determined based on the data distribution characteristics of the target variable and the prediction requirements: for target variables with sufficient data and complex nonlinear relationships, a combination of tree models (random forest model, XGBoost model, gradient boosting model) and kernel methods (support vector regression model) is preferred; for target variables that require prediction of uncertainty estimation, a Gaussian process regression model is preferred.

[0082] In a preferred embodiment, when training the magnetostriction coefficient prediction model, the base learner is selected from the random forest model (MRF). λ (n) estimators =200, max depth =15), XGBoost model MXGB λ (learning) rate =0.05,n estimators =300) and the support vector regression model MSVR λ (kernel='rbf', C=10); Stacking ensemble strategy is used, and Ridge Regression is employed as the meta-learner. Five-fold cross-validation is used to evaluate model performance, requiring a test set R... 2 ≥0.85, RMSE≤80ppm.

[0083] The selection of the above base learner combinations is based on the following technical considerations: (1) Random forest models have good anti-overfitting ability and are suitable for handling high-dimensional feature inputs; (2) XGBoost models perform well in capturing feature interaction effects and have strong fitting ability for complex nonlinear relationships between components and performance; (3) Support vector regression models use kernel function mapping, which can effectively handle nonlinear regression problems in small sample situations. The prediction mechanisms of the three models are complementary, and the bias of a single model can be effectively reduced by integrating them through the Stacking strategy. Those skilled in the art can select other base learner combinations according to the actual data characteristics, such as replacing the XGBoost model with the LightGBM model, or replacing the support vector regression model with the Gaussian process regression model, all of which can achieve similar technical effects.

[0084] The resistivity prediction model adopts an existing model. Training the resistivity prediction model includes: using the resistivity-specific feature descriptor constructed by S2 along with the original component parameters and process parameters as the resistivity input feature vector X. ρ Using room temperature resistivity ρ as the target resistivity variable y ρ A resistivity prediction model was trained using machine learning methods.

[0085] Specifically, the machine learning method adopts an ensemble learning strategy, with the base learners selected from two or more of the following models: random forest, gradient boosting regression, XGBoost, support vector regression, Gaussian process regression, and multilayer perceptron; the ensemble strategy adopts either a stacking strategy or a weighted average strategy.

[0086] The selection of the base learner is determined based on the data distribution characteristics of the target variable and the prediction requirements: for target variables with sufficient data and complex nonlinear relationships, a combination of tree models (random forest model, XGBoost model, gradient boosting model) and kernel methods (support vector regression model) is preferred; for target variables that require prediction of uncertainty estimation, a Gaussian process regression model is preferred.

[0087] In a preferred embodiment, the base learners are selected from a gradient boosting regression model MGBRρ, a Gaussian process regression model MGPRρ, and a multilayer perceptron model MMLPρ (hidden_layers=(64,32,16)); a weighted average strategy is used for ensemble integration: Mρ = w1·MGBRρ + w2·MGPRρ + w3·MMLPρ, where Mρ represents the final resistivity prediction value output by the ensemble model; MGBRρ, MGPRρ, and MMLPρ represent the resistivity prediction values ​​output by the gradient boosting regression model, the Gaussian process regression model, and the multilayer perceptron model, respectively; w1, w2, and w3 are the weight coefficients of the corresponding base learners, and satisfy w1+w2+w3=1. Five-fold cross-validation is used to evaluate the model performance, requiring R²≥0.80 and MAPE≤10% on the test set.

[0088] The selection of the above base learner combination is based on the following technical considerations: (1) Gaussian process regression model can provide uncertainty estimation (confidence interval) of the predicted value, which is crucial for the calculation of the acquisition function in subsequent multi-objective Bayesian optimization; (2) Gradient boosting regression model has high prediction accuracy on structured data; (3) Multilayer perceptron model can capture deep nonlinear features. The weighted average strategy instead of the stacking strategy is adopted to preserve the probability output characteristics of the Gaussian process regression model, so that the ensemble model can output the predicted mean and the predicted variance at the same time. Those skilled in the art can also adjust the base learner combination according to actual needs. For example, in application scenarios where uncertainty estimation is not required, the Gaussian process regression model can be replaced with a random forest model.

[0089] In addition, S3 includes feature selection before training the magnetostriction coefficient prediction model and the resistivity prediction model. This process selects the features most useful for the target performance and trains the model using the most effective features, which can improve the model's accuracy. Feature selection employs one or more combinations of the following strategies: feature correlation analysis based on mutual information, feature importance ranking based on recursive feature elimination, and feature contribution analysis based on SHAP values.

[0090] In S4, the multi-objective optimization algorithm is either the multi-objective evolutionary algorithm NSGA-III or the multi-objective Bayesian optimization method.

[0091] When using a multi-objective Bayesian optimization method to solve the problem, the following steps are included: Construct Gaussian process surrogate models to fit the saturation magnetostriction coefficient λ respectively. s and resistivity; The desired hypervolume improvement is used as the acquisition function to select the next experimental point; After obtaining the measured performance of new data points through experiments, the surrogate model is updated, and the iteration continues until the Pareto front converges.

[0092] When using the multi-objective evolutionary algorithm NSGA-III, the algorithm parameters are set as follows: population size of 200, maximum number of iterations of 500 generations, crossover probability of 0.9, and mutation probability of 0.1. The algorithm process includes the following steps: first, initialize the population; then, calculate the fitness of individuals; perform non-dominated sorting; adopt a reference point association selection strategy; perform SBX crossover and polynomial mutation operations; and repeat the iteration until convergence. The output is the Pareto optimal solution set P* = {x1*, x2*, ..., x...}. k *}, where P* represents the Pareto optimal solution set, x1*, x2*, ..., x k * represents the 1st to the kth Pareto optimal solution in the solution set, and each optimal solution x* corresponds to a specific combination of alloy composition and process parameters.

[0093] In S4, maximizing resistivity f 1(x) = ρ pred (x), maximizing the saturation magnetostriction coefficient f 2(x) = λ s,pred (x). Where x represents an input decision variable vector containing alloy composition and process parameters; f 1(x) and f 2(x) represent the first and second optimization objective functions, respectively; ρ pred (x) represents the predicted room temperature resistivity for input x, output by the resistivity prediction model; λ s,pred(x) represents the predicted value of the saturated magnetostriction coefficient for input x, output by the magnetostriction coefficient prediction model.

[0094] Constraints include composition constraints, performance constraints, and process constraints.

[0095] Compositional constraints: The alloy chemical formula adopts (Tb x Dy (1-x) )Fe (2-y) M y Let x represent the atomic percentage of Tb in rare earth elements, x ∈ [0.20, 0.55]; 1-x represent the atomic percentage of Dy in rare earth elements; 2-y represent the stoichiometric coefficient of Fe, (2-y) ∈ [1.90, 2.00]; and y represent the stoichiometric coefficient of dopant element M, y ∈ [0, 0.10].

[0096] Process constraints: v is the directional solidification casting speed, v ∈ [0.5, 10] mm / min, T a T is the annealing temperature. a ∈ [800,1100]℃, t a Insulation time, t a ∈ [1, 3]h.

[0097] Cost constraints: Where cost(x) represents the estimated total raw material cost for the corresponding composition scheme x; x i Price represents the atomic percentage of the i-th element; i Cost represents the unit price of the raw material corresponding to the i-th element. max This indicates the maximum cost limit threshold set based on actual engineering needs.

[0098] Optional auxiliary performance screening constraint: Performance constraint. When the database contains Curie temperature or hardness data, HV(x) ≤ HV can be set. max and T c (x) ≥ T c min (e.g., 350℃), λ s (x) ≥ λ min (e.g., 800ppm) is used as a screening condition, where HV(x) and T c (x) and λ s (x) represent the predicted hardness, Curie temperature, and saturation magnetostriction coefficient for the corresponding composition scheme x, respectively; HV max Indicates the maximum allowable hardness for the application scenario; T c min Indicates the minimum Curie temperature limit required by the application scenario; λ minThis represents the lower limit of the minimum saturation magnetostriction coefficient required for the application scenario.

[0099] In S5, the typical component range includes the high magnetostriction region (saturation magnetostriction coefficient λ). s >1200ppm, relatively low room temperature resistivity ρ), high resistivity region (room temperature resistivity ρ>80μΩ·cm, saturation magnetostriction coefficient λ) s (relatively low) and equilibrium zone (λ) s Both ρ and ρ are at a moderate level.

[0100] The optimal solution is selected based on the application scenario, including: (a) For low-frequency large displacement applications (such as precision positioners): the high magnetostriction region solution is preferred, and the constraint λ is recommended. s (a) For high-frequency transducer applications (such as sonar): the high resistivity region scheme is preferred, and λ constraint is recommended. s ≥ 700ppm and ρ ≥ 100μΩ·cm; (c) For broadband applications: select a balanced region scheme, and constrain λ is recommended. s ≥ 1000ppm and ρ ≥ 80μΩ·cm. Output recommended composition scheme and predicted performance, including a 95% confidence interval.

[0101] The method of the present invention will be described in detail below with reference to specific embodiments.

[0102] Example 1: Optimization of High Resistivity Tb-Dy-Fe Alloy for Sonar Transducers Background: Sonar transducers operate at frequencies of 1k-15kHz. Under high-frequency operating conditions, eddy current loss is the main factor limiting device efficiency. Therefore, the material is required to have the highest possible resistivity (preferably ρ ≥ 100μΩ·cm), while maintaining a saturation magnetostriction coefficient (λ) that meets application requirements. s ≥ 700ppm).

[0103] Step 1: Data Preparation. 256 data points on Tb-Dy-Fe based alloys were collected from literature and experiments. According to (Tb... x Dy (1-x) )Fe (2-y) M y Chemical formula representation, composition range as follows: atomic percentage of Tb in rare earth elements x ∈ [0.27, 0.55], stoichiometric number (2-y) of Fe ∈ [1.90, 1.98], doping element M includes Co (stoichiometric number 0-0.05), Al (stoichiometric number 0-0.03), Nb (stoichiometric number 0-0.02), Mn (stoichiometric number 0-0.05), and Si (stoichiometric number 0-0.03). Performance range: saturation magnetostriction coefficient λ s700-1800 ppm, room temperature resistivity ρ: 100-160 μΩ·cm. Data preprocessing employed the K-nearest neighbor algorithm for missing value imputation, box plot method for outlier detection and removal, and Z-score standardization for data standardization.

[0104] Step 2: Construct dedicated feature descriptors for magnetostrictive properties and resistivity. Calculate the feature descriptors according to the aforementioned formulas, including a magnetoelastic coupling descriptor set based on rare-earth ion Stevens factors (equivalent to a second-order Stevens factor α). J eff Equivalent fourth-order Stevens factor β J eff Magnetostriction driving factor Λ drive Spin-orbit coupling descriptor set based on the 4f electronic properties of rare earth ions (equivalent spin-orbit coupling energy ξ) eff Orbital angular momentum contribution factor L eff Crystal structure descriptor group (Laves phase stability factor Φ) Laves lattice distortion δ r ), Electron scattering descriptor set (spin disorder scattering factor S) spin Component disorder scattering factor S comp ), Electronic structure descriptor set (valence electron concentration VEC, df hybridization strength estimator H) df ), Defect scattering descriptor set (atomic size mismatch parameter δ) size Electronegativity mismatch parameter δ χ The original composition parameters and process parameters were also included. After recursive feature elimination (RFE) screening, 12 key features were retained for modeling, among which the key features retained for the magnetostrictive model included α. J eff β J eff Λ drive ξ eff L eff Φ Laves and the Tb / Dy ratio, etc.; key features preserved by the resistivity model include S spin S comp ,VEC,δ size And the content of doping elements, etc.

[0105] Step 3: Model Training. Train the magnetostriction coefficient prediction model M. λ Employing an ensemble learning strategy: using a random forest model (n estimators =200, max depth =15), XGBoost model (learning rate =0.05, nestimators A base learner (kernel=300) and a support vector regression model (kernel='rbf', C=10) were used, ensembled using a stacking strategy, with ridge regression as the meta-learner. 5-fold cross-validation results: R 2 = 0.91, RMSE = 62 ppm. Training resistivity prediction model M ρ Gradient boosting regression model, Gaussian process regression model, and multilayer perceptron model (hidden) were employed. layers =(64,32,16)) was used as the base learner, and a weighted average strategy was employed for ensemble learning (the weights were determined through validation set performance optimization). 5-fold cross-validation results: R 2 = 0.87, MAPE = 7.2%. Both models meet the performance thresholds required by the original paper (magnetostrictive model R). 2 ≥0.85, resistivity model R 2 ≥0.80).

[0106] Step 4: Construct a dual-objective optimization framework. Maximizing room temperature resistivity and maximizing saturation magnetostriction coefficient are the dual optimization objectives, solved using the multi-objective evolutionary algorithm NSGA-III. Specific algorithm parameters are set as follows: population size 200, maximum number of iterations 500 generations, crossover probability 0.9, and mutation probability 0.1. Considering the application requirements of sonar transducers, specific performance constraints are set as follows: λ s The concentrations are ≥ 700 ppm and ρ ≥ 100 μΩ·cm; the composition and process constraints are set according to the boundary range in step 1. After the algorithm runs and solves the problem, it outputs the Pareto optimal solution set under these constraints.

[0107] Step 5: Pareto Front Analysis and Solution Recommendation. For sonar applications (high pG priority), the top five solutions selected from the Pareto front are shown in Table 1.

[0108] Table 1 Recommended Scheme for Tb-Dy-Fe Alloys in Sonar Transducers

[0109] Experimental verification: Scheme 4 was selected for experimental verification. The sample was prepared using the directional solidification method, with a pulling speed v = 3 mm / min. The annealing process involved holding at 1060℃ for 2 hours followed by furnace cooling. The measured composition was (Tb) 0.37 Dy 0.63 )Fe 1.97 Al 0.015 Si 0.015 Measured λ s=1070 ppm (predicted 1105±50, relative error 3.2%), measured ρ = 145 μΩ·cm (predicted 147±5, relative error 1.4%). The verification results show that the model prediction accuracy meets engineering requirements (engineering requirements for model prediction accuracy relative error within ±25%).

[0110] Example 2: Optimization of Tb-Dy-Fe Alloy with High Magnetostriction Coefficient for Precision Positioners Background: Precision positioners operate at low frequencies (<100Hz) and are insensitive to eddy current losses; therefore, a high magnetostriction coefficient is prioritized to achieve large displacement output. This example is based on the database and trained prediction model built in Example 1, adjusting and optimizing the constraints for the precision positioner application scenario. Since low-frequency applications have lower resistivity requirements, while precision positioners have requirements for machinability, a hardness constraint is added. The constraint is set as: λ s ≥ 1500 ppm (high magnetostriction preferred), ρ ≥ 60 μΩ·cm, HV ≤ 650 (ensuring machinability), T c ≥ 300℃ (to ensure operating temperature stability).

[0111] Optimization results: Pareto front with high λ s The recommended solutions are shown in Table 2.

[0112] Table 2 Recommended Scheme for Tb-Dy-Fe Alloy for Precision Positioners

[0113] Experimental Verification: Scheme 1 was selected for experimental verification. The sample was prepared using the directional solidification method, with a tensile speed v = 3 mm / min. The annealing process involved holding at 1060℃ for 2 hours followed by furnace cooling. The measured composition was (Tb) 0.50 Dy 0.50 )Fe 1.95 Co 0.05 Measured λ s = 1545ppm (predicted 1580±70 ppm, relative error 2.2%), measured ρ = 62 μΩ·cm (predicted 64±4 μΩ·cm, relative error 3.2%), measured HV = 620 (satisfies HV ≤ 650 constraint). The verification results show that the model's prediction accuracy in the high magnetostriction region also meets engineering requirements.

[0114] Example 3: Optimization of Broadband Tb-Dy-Fe Alloy for Vibration Energy Harvesters Background: Vibration energy harvesters need to operate efficiently over a wide frequency range (10Hz-1kHz). Sufficient magnetostriction coefficient is needed to ensure energy conversion efficiency, while moderate resistivity is required to control eddy current losses in the mid-to-high frequency range. Based on the broadband application scenario, the optimization objective is to select a scheme from the equilibrium region of the Pareto front, with the constraint λ. s ≥ 1000ppm and ρ ≥ 80μΩ·cm.

[0115] This embodiment is based on the database and trained prediction model constructed in Embodiment 1, and is optimized for broadband application scenarios of vibration energy harvesters. The component constraint is (Tb x Dy (1-x) )Fe (2-y) M y Where x ∈ [0.30, 0.50], (2-y) ∈ [1.92, 1.98], y ∈ [0.02, 0.08], and M is Al (stoichiometric coefficient 0-0.03), Ga (stoichiometric coefficient 0-0.06), or Nb (stoichiometric coefficient 0-0.02); the performance constraint is λ. s ≥ 1000 ppm, ρ ≥ 80 μΩ·cm, T c ≥ 350℃.

[0116] Optimization Results: The Pareto front was solved using the NSGA-III algorithm. The top five schemes meeting broadband application requirements were selected from the equilibrium region, as shown in Table 3. Table 3: Recommended Schemes for Tb-Dy-Fe Alloys in Vibration Energy Harvesters

[0117] All of the above solutions meet the performance constraints (λ) for broadband applications. s (≥ 1000ppm and ρ ≥ 80μΩ·cm), in the equilibrium region of the Pareto front.

[0118] Experimental Verification: Scheme 3 was selected for experimental verification. The sample was prepared using the directional solidification method, with a pulling speed v = 2.5 mm / min. The annealing process involved holding at 1060℃ for 2 hours followed by furnace cooling. The measured composition was (Tb) 0.40 Dy 0.60 )Fe 1.96 Ga 0.04 Measured λ s = 1125ppm (predicted 1150±60 ppm, relative error 2.2%), measured ρ = 90 μΩ·cm (predicted 88±4 μΩ·cm, relative error 2.3%). The validation results show that the model's prediction accuracy in the equilibrium region meets the requirements for engineering applications, and this composition scheme is suitable for broadband applications such as vibration energy harvesters.

[0119] The above description is only a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any changes or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in the present invention should be included within the scope of protection of the present invention.

Claims

1. A dual-objective synergistic optimization method for resistivity and magnetostrictive properties of Tb-Dy-Fe alloys based on machine learning, characterized in that, Includes the following steps: S1. Construct a multi-performance database for Tb-Dy-Fe alloys; S2. Construct a dedicated feature descriptor for magnetostrictive properties and a dedicated feature descriptor for resistivity; S3. Train the magnetostriction coefficient prediction model and train the resistivity prediction model; S4. With maximizing resistivity and maximizing saturation magnetostriction as optimization objectives, a dual-objective optimization framework is constructed, constraints are set, and a multi-objective optimization algorithm is used to solve the problem, outputting the Pareto optimal solution set. S5. Perform cluster analysis on the Pareto optimal solution set, identify typical component intervals, select the optimal solution according to the application scenario, and output recommended components and prediction performance.

2. The method according to claim 1, characterized in that, In S2, the dedicated feature descriptors for magnetostrictive properties include: a magnetoelastic coupling descriptor group based on the Stevens factor of rare earth ions, a spin-orbit coupling descriptor group based on the 4f electronic properties of rare earth ions, and a crystal structure descriptor group.

3. The method according to claim 2, characterized in that, The spin-orbit coupling descriptor set based on the 4f electronic properties of rare earth ions includes the equivalent spin-orbit coupling energy and the orbital angular momentum contribution factor. Equivalent spin-orbit coupling energy The calculation is expressed using the following formula: Where, x Tb Indicates the percentage of Tb atoms. x Dy Indicates the percentage of Dy atoms. ; Orbital angular momentum contribution factor The calculation is expressed using the following formula: Where, x Tb L represents the percentage of Tb atoms. Tb =3, x Dy L represents the percentage of Dy atoms. Dy =5.

4. The method according to claim 2, characterized in that, The magnetoelastic coupling descriptor set based on rare-earth ion Stevens factors includes an equivalent second-order Stevens factor, an equivalent fourth-order Stevens factor, and a magnetostriction driving factor. Equivalent second-order Stevens factor The calculation is expressed using the following formula: Where, x Tb x represents the percentage of Tb atoms. Dy Indicates the percentage of atoms in Dy, x Mi This represents the atomic percentage of the i-th dopant element. , , Indicates the first i The second-order Stevens factor of a doped element; wherein, when the doped element M... i When it is a non-rare earth element, When the doping element Mi is a rare earth element, Take the theoretical value of the second-order Stevens factor corresponding to the rare earth ion; Equivalent fourth-order Stevens factor The calculation is expressed using the following formula: Where, x Tb Indicates the percentage of Tb atoms x Dy Indicates the percentage of Dy atoms. x Mi This represents the atomic percentage of the i-th dopant element. Denotes the fourth-order Stevens factor of the i-th doped element; wherein, when the doped element M i When it is a non-rare earth element, When the doping element M i When it is a rare earth element, Take the theoretical value of the fourth-order Stevens factor corresponding to the rare earth ion; Magnetostriction driving factor The calculation is expressed using the following formula: in, J(J+1) eff This is the weighted average of the total angular momentum quantum numbers of rare earth ions. .

5. The method according to claim 1, characterized in that, In S2, the resistivity-specific feature descriptors include: an electronic scattering descriptor group based on rare-earth ion magnetic scattering, an electronic structure descriptor group, and a defect scattering descriptor group.

6. The method according to claim 5, characterized in that, The electronic scattering descriptor set based on rare-earth ion magnetic scattering includes: spin disorder scattering factor and composition disorder scattering factor; Spin disorder scattering factor The calculation is expressed using the following formula: Where, x Tb Indicates the percentage of Tb atoms, g J Tb = 3 / 2, which is Tb 3+ Lande g factor, J Tb =6, for Tb 3+ The total angular momentum quantum number, x Dy G represents the percentage of Dy atoms. J Dy = 4 / 3, for Dy 3+ Lande g factor, J Dy =15 / 2 is Dy 3+ The total angular momentum quantum number; Component disorder scattering factor The calculation is expressed using the following formula: in, The percentage of atoms of the i-th element. Z represents the atomic percentage of the j-th element. i Let be the atomic number of the i-th element. Let be the atomic number of the j-th element.

7. The method according to claim 1, characterized in that, In S4, the multi-objective optimization algorithm is either the multi-objective evolutionary algorithm NSGA-III or the multi-objective Bayesian optimization method.

8. The method according to claim 1, characterized in that, In S1, the construction of a multi-performance database for Tb-Dy-Fe alloys includes: collecting Tb-Dy-Fe-based alloy data, with each data record containing composition parameters, process parameters, magnetostrictive performance parameters, resistivity parameters, and auxiliary performance parameters, and preprocessing the data.

9. The method according to claim 8, characterized in that, Composition parameters include: Tb atomic percentage x Tb , Dy atomic percentage x Dy Fe atomic percentage x Fe Dopant element M i and the atomic percentage x of the i-th dopant element Mi (i=1,2,...,n).

10. The application of the method according to any one of claims 1-9 in the design of Tb-Dy-Fe alloy composition, wherein the Tb-Dy-Fe alloy is used in sonar transducers, precision positioning actuators, vibration energy harvesters, or magnetostrictive delay lines.

Images