A Simulation Method for the Hysteresis Characteristics of a Soft Magnetic Material under Stress

Abstract

The present invention discloses a method for simulating the hysteresis characteristics of soft magnetic materials under stress. First, it is assumed that the soft magnetic material is composed of a large number of magnetic domains. The icosahedron is subjected to six times of refined triangulation to determine the initial magnetization intensity directions of various magnetic domains. Based on the mesoscopic magnetization theory and Hooke's law respectively, mathematical models of magnetic domain magnetization intensity and strain are constructed. The expression of Gibbs free energy considering the second-order stress is derived to obtain the expression of the non-hysteretic volume fraction of magnetic domains. According to the homogenization theory, a mathematical model of the non-hysteretic magnetization intensity of the overall soft magnetic material is constructed. By applying the mesoscopic energy conservation principle, the hysteresis model of the soft magnetic material is derived and solved based on the fourth-order Runge-Kutta method to simulate the hysteresis loop of the soft magnetic material under different stresses. The present invention not only accurately and quickly simulates the hysteresis characteristics of soft magnetic materials under stress, but also lays a theoretical and technical foundation for research such as the accurate simulation of the electromagnetic-mechanical comprehensive characteristics and the global structural optimization design of electrical equipment containing soft magnetic materials.

Description

Technical Field

[0001] The present invention belongs to the field of analysis of the hysteresis characteristics of soft magnetic materials, and particularly relates to a method for simulating the hysteresis characteristics of soft magnetic materials under stress. Background Art

[0002] Soft magnetic materials have been widely used in the preparation of iron cores for electrical equipment such as transformers and motors. It should be noted that the hysteresis characteristics of such materials, as their inherent properties, will have an important impact on their own and other characteristics of the electrical equipment to which they belong (such as exciting current, voltage, energy loss, etc.). During the actual service process of electrical equipment, its own structure and the bending effect inside itself will cause the iron core to be continuously subjected to different degrees of compressive or tensile stress, which will cause the hysteresis characteristics of the soft magnetic material of the iron core to be greatly affected. Therefore, inventing an accurate, fast, and practical method for simulating the hysteresis characteristics of soft magnetic materials under stress has an important supporting role in research such as accurate evaluation of the performance of electrical equipment and its global optimal design.

[0003] However, the existing hysteresis simulation methods considering stress mainly start from the perspective of macroscopic modeling, and a large amount of experimental data is required to identify the model parameters, so the practicality is not high, and the magnetization mechanism of soft magnetic materials under stress cannot be revealed. For example:

[0004] 1) Literature 1: Li Yiling, Li Lin, Liu Ren. Research on the simulation method of the static hysteresis characteristics of electrical steel sheets under mechanical stress [J]. Electric Power, 2020, 53(10): 10 - 18. Aiming at the problem that the traditional Preisach hysteresis model cannot consider stress, this literature uses the measured hysteresis loop data under different stresses to identify its Everett function, and thus proposes a hysteresis simulation method under stress based on the Preisach hysteresis model. However, the Preisach hysteresis model belongs to a macroscopic hysteresis model, so this method necessarily requires a large amount of experimental data to identify its parameters in the stress state, and it cannot be used for the analysis of the magnetization mechanism of soft magnetic materials under stress.

[0005] 2) Literature 2: Luo Xu, Zhu Haiyan, Ding Yaping. Modified magnetization model of ferromagnetic materials based on the force - magnetic coupling effect [J]. Acta Physica Sinica, 2019, 68(18): 295 - 306. This literature combines the nonlinear magnetostrictive strain relationship in the Zheng Xiao - Jing - Liu Xing - En (Z - L) hysteresis model with the hysteresis theory of the Jiles - Atherton (J - A) hysteresis model, considers the influence of stress on the model parameters, and establishes a modified hysteresis model for the influence of elastic - plastic stress on the hysteresis curve of soft magnetic materials. However, this modified model still belongs to a macroscopic hysteresis model, introduces multiple additional parameters, requires a large amount of experimental data and more calculation time, and it cannot essentially analyze the magnetization mechanism of soft magnetic materials under stress.

[0006] 3) Literature 3: Wu Jiaqi. Simulation and Application of Magnetostriction Characteristics of Electrical Steel Sheets Considering Stress Influence [D]. Shenyang University of Technology, 2021. The magnetic hysteresis models based on the hyperbolic tangent function and the Helmholtz free energy were respectively derived, and the influences of tensile stress and compressive stress on the magnetic properties of magnetic materials were successfully simulated. However, this method is based on empirical formulas and requires a large amount of experimental data to fit its parameters, making it difficult to be applied in practice. Moreover, this method still only starts from a macroscopic perspective and does not analyze the magnetization mechanism of magnetic materials under stress.

[0007] 4) Literature 4: Chen Hao, Li Lin, Liu Yang. Simulation of Magnetic Hysteresis Characteristics of Electrical Steel Sheets under Mechanical Stress Based on the Energetic Model [J]. Proceedings of the CSEE, 2022: 1-12. This literature introduced the energy density additional term caused by stress into the total energy density and considered the dependence of each parameter in the model on stress to construct a magnetic hysteresis model of electrical steel sheets under stress. However, the magnetic hysteresis model constructed by introducing the stress additional term into the total energy density is an equivalent macroscopic magnetic hysteresis model, which cannot reveal the magnetization mechanism of soft magnetic materials under stress and requires a large amount of experimental data to identify its parameters.

[0008] Therefore, it is necessary to propose a new method for simulating the magnetic hysteresis characteristics of soft magnetic materials under stress starting from the multi-scale magnetization mechanism and the law of energy conservation of soft magnetic materials and combining with the J-A magnetic hysteresis model. Summary of the Invention

[0009] The purpose of the present invention is to provide a new method for simulating the magnetic hysteresis characteristics of soft magnetic materials under stress in view of the deficiencies of the above existing magnetic hysteresis models and methods, so as to solve the problems raised in the background technology.

[0010] To achieve the above purpose, the present invention provides the following technical solutions:

[0011] A method for simulating the magnetic hysteresis characteristics of soft magnetic materials under stress, comprising the following steps:

[0012] Step 1: Assume that the soft magnetic material is composed of a large number of magnetic domains with randomly distributed directions and a saturation magnetization intensity of M s , and perform six-time refined triangulation on the icosahedron to simulate the initial magnetization direction of the magnetic domain α in the soft magnetic material;

[0013] Step 2: Use the mesoscopic magnetization theory and Hooke's law to construct a mathematical model of the magnetization intensity M α , magnetostrictive strain ε α within a single magnetic domain α;

[0014] Step 3: According to the definition of the Gibbs free energy W α , divide it into the anisotropy energy Wα an 、The static magnetic energy \(W\) α mag and the magnetoelastic energy \(W\) α σ The sum of the three items, and introducing the second-order stress into the magnetoelastic energy, a new expression of Gibbs free energy is derived;

[0015] Step 4: According to the calculation principle of Boltzmann distribution, calculate the anhysteretic volume fraction \(f\) of each magnetic domain \(\alpha\) an in turn; Then, based on the homogenization theory, establish the anhysteretic magnetization intensity \(M\) of the soft magnetic material an with respect to the magnetization intensity \(M\) of the magnetic domain α , the magnetostrictive strain \(\varepsilon\) α and the anhysteretic volume fraction \(f\) corresponding to the magnetic domain an and other parametric mathematical models;

[0016] Step 5: Through the mesoscopic energy conservation principle, and the theory that the volume fraction \(f\) of the magnetic domain α causes the change of magnetization intensity, combined with the anhysteretic magnetization intensity \(M\) obtained in the above steps an , establish a differential equation of the magnetic domain volume fraction \(f\) α , then use the fourth-order Runge-Kutta method to solve the differential equation, and finally, based on the homogenization theory, obtain the overall magnetization intensity of the soft magnetic material under different stresses and simulate its hysteresis loop.

[0017] Preferably, in Step 1, it is assumed that the soft magnetic material is composed of a large number of magnetic domains at the mesoscopic scale between the macroscopic and microscopic scales. Under the influence of no external magnetic field, stress and other factors, the initial magnetization directions of the magnetic domains are randomly distributed, and the soft magnetic material as a whole does not show magnetism externally. Therefore, by performing six-time triangulation on the icosahedron, a sphere with 10242 coordinate points is obtained to simulate the initial magnetization directions of the magnetic domains in the soft magnetic material.

[0018] Preferably, the magnetization intensity \(M\) of the magnetic domain in Step 2 α can be expressed as:

[0019] \(M\) α = \(M\) s \(\alpha\) = \(M\) s [\(\alpha_1\alpha_2\alpha_3\)] t

[0020] where \(\alpha\) = [\(\alpha_1\alpha_2\alpha_3\)] t is the initial magnetization direction of the magnetization intensity \(M\) α of the magnetic domain \(\alpha\).

[0021] Preferably, the magnetostrictive strain \(\varepsilon\) in Step 2 α is:

[0022]

[0023] where λ 100 and λ 111 are the magnetostrictive strain constants when the cubic crystal is saturated magnetized along the <100> and <111> directions; when the magnetic material is an isotropic material, it is considered that λ 100 = λ 111 = λ m , and λ m is the maximum magnetostrictive strain constant of the magnetic material.

[0024] Preferably, in step three, the Gibbs free energy W α , the magnetocrystalline anisotropy energy , the magnetostatic energy and the magnetoelastic energy and other expressions are as follows:

[0025]

[0026]

[0027]

[0028]

[0029] where represents the first-order component of the magnetoelastic energy, represents the second-order component of the magnetoelastic energy, K1 and K2 are the anisotropy constants of the soft magnetic material, μ0 is the vacuum permeability constant, H is the applied magnetic field, σ α is the stress tensor, E α is the fourth-order magnetostriction tensor, R α is the second-order tensor formed by the magnetization direction cosines, and N is the sixth-order magnetostriction tensor.

[0030] Preferably, in step four, according to the Boltzmann distribution principle, the non-hysteretic volume fraction f an expression is:

[0031]

[0032] where is the material adjustable parameter, determined by the non-hysteretic initial permeability χ0 and the saturation magnetization M s . Preferably, the non-hysteretic magnetization M an of the magnetic material obtained by the homogenization theory in step four is:

[0033] M an = M α = ∑ α f an M α

[0034] Preferably, the mesoscopic energy conservation equation in step five is as follows:

[0035]

[0036] where k is the restraint coefficient between magnetic domains, and δ M is a coefficient introduced to prevent non - physical solutions, δ is the direction coefficient, M irr is the irreversible magnetization component, and M is the magnetization intensity;

[0037] According to the theory that the change in magnetization intensity is caused by the change in the volume fraction of magnetic domains, we can obtain:

[0038] dM ani = M s df ani

[0039] dM i = M s df αi

[0040] where M ani , f ani , M i , f αi represent the anhysteretic magnetization intensity, anhysteretic volume fraction, magnetization intensity, and volume fraction of the i - th magnetic domain respectively, and a new model for simulating the magnetic properties of soft magnetic materials under stress is derived:

[0041]

[0042] where f αi is the volume fraction of the i - th magnetic domain, al is the average internal coupling field parameter of the magnetic domain, c is the reversible magnetization coefficient, and α i represents the initial magnetization direction of the i - th magnetic domain. Finally, the fourth - order Runge - Kutta method is used to solve the differential equation to obtain the volume fraction f αi of each magnetic domain;

[0043]

[0044] where f αi (i + 1) is the volume fraction of the next magnetic domain, f αi (i) is the volume fraction of the current magnetic domain, and k1, k2, k3, k4 are the intermediate quantities of the Runge - Kutta method;

[0045] Then, through the homogenization theory in step four, the overall magnetization intensity M of the soft magnetic material under different stress conditions in the multi - scale case is obtained:

[0046] M = M α = ∑ α fa M α

[0047] Advantages of the present invention:

[0048] A simulation method for the magnetic properties of soft magnetic materials under stress proposed by the present invention combines a simplified multi-scale theory based on magnetic domain angles with the principle of mesoscopic energy conservation, fundamentally revealing the magnetization mechanism of soft magnetic materials. At the same time, a second-order stress tensor is introduced to correctly simulate the non-monotonic effect of the magnetic properties of soft magnetic materials under tensile stress. This method belongs to a physical model and does not require a large amount of experimental data to identify its parameters, so it has high practicability and can reveal the magnetization mechanism of soft magnetic materials under different stresses. Its physical meaning is clear, uses less experimental data and parameters, reduces the calculation amount on the premise of ensuring the simulation accuracy, and improves the operation speed, which is of great significance for the structural optimization of the iron core of actual power equipment and the improvement of equipment operation efficiency. Brief description of the drawings

[0049] Figure 1 A three-dimensional perspective view after triangulating an icosahedron;

[0050] Figure 2 A flow chart of a new method for simulating the hysteresis characteristics of a soft magnetic material under stress;

[0051] Figure 3 A graph showing the variation trend of non-hysteretic magnetization with stress and magnetic field;

[0052] Figure 4 Experimental data and simulation of the hysteresis loop of a soft magnetic material under a 50 MPa tensile stress. Detailed implementation manners

[0053] The present invention will be further described in detail below with reference to the drawings and specific embodiments:

[0054] As shown in the Figure 2 flow chart of a new method for simulating the hysteresis characteristics of a soft magnetic material under stress, to establish a new hysteresis model, the following steps are specifically included:

[0055] Step 1: Assume that the soft magnetic material is composed of a large number of magnetic domains with randomly distributed directions and a saturation magnetization of M s , perform six refinements of triangulation on the icosahedron and remove the duplicates. Finally, a sphere with 10242 coordinate points is obtained, and each coordinate point is used to simulate the initial magnetization direction of a certain magnetic domain α i in the soft magnetic material;

[0056] As Figure 1The figure shows a three-dimensional perspective view after the icosahedron is triangulated six times. It obtains 10,242 coordinate points to simulate the initial magnetization direction of the soft magnetic material, avoiding the complex operation of taking the derivative of the Gibbs free energy and greatly reducing the operation time.

[0057] Step 2: Use the mesoscopic magnetization theory and Hooke's law to construct the mathematical models of the magnetization intensity M α and magnetostrictive strain ε α within a single magnetic domain α.

[0058] First, assume from Step 1 that the soft magnetic material consists of a large number of magnetic domains, and its initial magnetization direction is also as Figure 1 shown. Therefore, the magnetization intensity M α and magnetostrictive strain ε α of a single magnetic domain can be analyzed first, and their mathematical models are as follows:

[0059] M α = M s α = M s [α1α2α3] t

[0060]

[0061] where α = [α1α2α3] t is the initial magnetization direction of the magnetization intensity M α of magnetic domain α; λ 100 , λ 111 are the magnetostrictive strain constants when the cubic crystal is magnetized to saturation along the <100> and <111> directions. When the material is an isotropic material, it is considered that λ 100 = λ 111 = λ m , and λ m is the maximum magnetostrictive strain constant of the material.

[0062] Step 3: Determine the definition of the Gibbs free energy W α and divide it into the sum of the anisotropy energy W α an , the magnetostatic energy W α mag and the magnetoelastic energy W α σ to list the relevant expressions. Considering from the mesoscopic level, introduce the second-order stress into the magnetoelastic energy and derive a new expression for the Gibbs free energy.

[0063] Since it involves the calculation of a sixth-order tensor and its expression is extremely complex, only the simplification work of the multi-scale theory is considered under a relatively ideal condition, ensuring that the operation can be carried out at the fastest speed without losing accuracy.

[0064]

[0065]

[0066] Wherein, K1 and K2 are the anisotropy constants of the soft magnetic material. When the material is an isotropic material, the parameter value is 0.

[0067]

[0068] Wherein, H is the applied magnetic field, and H1, H2, and H3 are the magnetic field components of the applied magnetic field projected onto the three direction axes of the coordinate system respectively; μ0 is the vacuum permeability constant.

[0069] In actual situations, the stress applied to the material is extremely complex. When the stress is small, relevant simplifications can be made, that is, only the first-order stress parameter is considered because the magnitude of the stress is not yet sufficient to change the trend of the magnetic properties of the material at this time. However, when the stress is relatively large, due to factors such as the influence of magnetic domains, the trend of the magnetic properties of the soft magnetic material has changed to a certain extent, and this change is sufficient to affect the operation of relevant machines macroscopically. Therefore, at this time, the second-order stress factor must be considered to accurately predict the relevant changes.

[0070]

[0071]

[0072] Wherein, is the first-order component of the magnetoelastic energy; the operator ‘:’ is the tensor scalar product operation; σ α is the second-order stress tensor; λ 100 , λ 111 are responsible for the first-order magnetoelastic effect, is the second-order component of the magnetoelastic energy, and the coefficients λ s , λ‘’ s are responsible for the second-order magnetoelastic effect component; tr(σ α ) represents the trace of the second-order stress tensor matrix

[0073]

[0074] Wherein, σ α is the stress tensor, E α is the fourth-order magnetostriction tensor, R α is the second-order tensor formed by the magnetization direction cosine, and N is the sixth-order magnetostriction tensor.

[0075] When there is no external magnetic field or stress, the magnetization direction of the soft magnetic material is determined by its own magnetocrystalline anisotropy energy, which can be calculated by minimizing the Gibbs free energy. However, usually, the calculation of the minimum Gibbs free energy often involves derivative operations, which are extremely complex. Therefore, to avoid this complex operation, in step one, it has been assumed that the magnetic material consists of a large number of magnetic domains with randomly distributed directions. After multiple refinement operations on the icosahedron and normalization, in the resulting three-dimensional map, the coordinates of each point can represent the initial magnetization direction of a certain magnetic domain α in the crystal coordinate system.

[0076] Step 4: Establish a mathematical model for the non-hysteretic volume fraction f of the magnetic domains of the soft magnetic material according to the Boltzmann distribution law, and then obtain the non-hysteretic magnetization M of the soft magnetic material based on the homogenization principle. an an

[0077] After determining the initial magnetization direction of the magnetic domains in step one, the non-hysteretic volume fraction f of the magnetic domains can be calculated through the Boltzmann distribution function. an :

[0078]

[0079]

[0080] where A s is an adjustable material parameter; χ0 is the initial magnetic permeability of the non-hysteretic magnetization; μ0 is the magnetic permeability of vacuum.

[0081] The resulting non-hysteretic volume fraction is much less complex than before simplification, making the model more efficient in operation.

[0082] After that, the overall non-hysteretic magnetization M of the soft magnetic material an can be obtained by vector multiplying the magnetization M α of a single magnetic domain α and its non-hysteretic volume fraction f an , and then summing up the calculation results of all magnetic domains, we can get:

[0083] M an = M α = ∑ α f an M α = M α1 · f an1 + M α2 · f an2 + … + M αi · f ani + …

[0084] where i represents the i-th magnetic domain, and the resulting non-hysteretic magnetization Man along with the externally applied magnetic field H and stress σ α the change trends are as Figure 3 shown.

[0085] Step Five: On the basis of the above simplified multi-scale theory, combined with the mesoscopic energy conservation principle, a new method for modeling the hysteresis model is derived.

[0086] According to the mesoscopic energy conservation principle, the following energy conservation equation is obtained:

[0087]

[0088] In the formula, k is the restraint coefficient between magnetic domains; δ is the direction coefficient, δ = 1 when dH / dt > 0, and δ = -1 when dH / dt < 0; M irr is the irreversible magnetization component; δ M is a coefficient introduced to prevent non-physical solutions, and the expression is as follows:

[0089]

[0090] The relationship between the magnetization intensity M and the irreversible magnetization intensity M irr is as follows:

[0091] M = M rev + M irr

[0092] M rev = c(M an - M irr )

[0093] In the formula, M rev is the reversible magnetization component; M an is the non-hysteretic magnetization intensity of the soft magnetic material derived above; c is the reversible magnetization coefficient;

[0094] The differential equation of the irreversible magnetization intensity M irr is:

[0095]

[0096] Combining the above formulas and the mesoscopic energy conservation equation, the macroscopic model can be derived as:

[0097]

[0098] In the formula, al and a are the average field parameter of the internal coupling of magnetic domains and the shape parameter of the non-hysteretic magnetization curve respectively;

[0099] However, according to the simplified multi-scale theory, the magnetization of soft magnetic materials is caused by the change in the volume fraction of magnetic domains. Therefore, it is necessary to calculate the volume fraction f of a single magnetic domain in soft magnetic materials according to the homogenization theory. α , and to simplify the calculation and improve the operation speed, the anhysteretic magnetization M an is split. First, calculate dM ani and dM i on a single magnetic domain, as shown in the following formula:

[0100] dM ani = M s df ani

[0101] dM i = M s df αi

[0102] f ani and f αi represent the anhysteretic volume fraction and volume fraction of the i-th magnetic domain.

[0103] In the original macroscopic model, since the influence of factors such as magnetic domains at the mesoscopic scale could not be fully simulated, the effective magnetic field H e was used to equivalently represent the actual magnetic field. After introducing the simplified multi-scale theory, since the model itself starts from the mesoscopic magnetic domain perspective, only the externally applied magnetic field H originally applied needs to be considered.

[0104] After combining the simplified multi-scale theory with the mesoscopic energy conservation principle, the differential equation expression of the volume fraction f αi on the i-th magnetic domain of soft magnetic materials is derived:

[0105]

[0106] where α i represents the initial magnetization direction of the i-th magnetic domain.

[0107] Since the saturation magnetization M s and the saturation magnetostriction coefficient λ m of the same material are fixed, after determining the mathematical model, the experimental data measured under the saturation magnetic induction intensity and different stress conditions of the material can be used to fit the data with the Matlab toolbox to obtain the saturation magnetization M s and the saturation magnetostriction coefficient λ m . Then, combined with the algorithm, the specific values of the parameters al, a, c, and k of the model can be extracted.

[0108] Step 6: Solve the above differential equation according to the fourth-order Runge-Kutta method, and then, based on the homogenization theory, calculate the magnetization intensity of the entire soft magnetic material to obtain the hysteresis loop diagram of the soft magnetic material.

[0109] The fourth-order Runge-Kutta method is as follows:

[0110] f αi (i + 1) = f ai (i) + 16(k1 + 2k2 + 2k3 + k4)

[0111] k1 = df α [H(i), f α (i), d, d M

[0112]

[0113]

[0114] k4 = df a [H(i) + h, f α (i) + hk3, d, d M

[0115] In the formula, h is the step size, h = H(i + 1) - H(i), f αi (i) is the volume fraction of the current magnetic domain, f αi (i + 1) is the volume fraction of the next magnetic domain, and k1, k2, k3, and k4 are intermediate calculation quantities of the Runge-Kutta method.

[0116] Finally, in combination with the homogenization theory, sum the vector products of the volume fraction f αi of each magnetic domain and the magnetization intensity M α of the corresponding magnetic domain:

[0117] M = M α = ∑ α f a M α

[0118] Finally, the magnetization intensity of the soft magnetic material under relevant stress conditions and the hysteresis curve diagram that changes with the applied magnetic field are obtained. As Figure 4 shown are the curves of the experimental values and simulation values of electrical steel sheets under a maximum magnetic density of 1.7 T and a tensile stress of 50 MPa. Its global average error can be reduced to about 7%, proving the correctness of this method.​​

Claims

1. A method for simulating the hysteresis characteristics of a soft magnetic material under stress, characterized in that: It includes the following steps: Step 1: Assume that the soft magnetic material is composed of a large number of magnetic domains with randomly distributed directions and a saturation magnetization of M s , and perform six times of refined triangulation on the icosahedron to simulate the initial magnetization direction of the magnetic domain α in the soft magnetic material; Step 2: Construct a mathematical model of the magnetization intensity M α and magnetostrictive strain ε α within a single magnetic domain α using mesoscopic magnetization theory and Hooke's law; Step 3: According to the definition of Gibbs free energy W α it is divided into the anisotropy energy the magnetostatic energy and the magnetoelastic energy which are the sum of three terms. And the second-order stress is introduced into the magnetoelastic energy to derive a new expression of Gibbs free energy; Step 4: According to the calculation principle of Boltzmann distribution, calculate the anhysteretic volume fraction f of each magnetic domain α in turn an ; Then, based on the homogenization theory, establish the mathematical model of the anhysteretic magnetization M of the soft magnetic material an with respect to the magnetization M α of the magnetic domain, the magnetostrictive strain ε α and the anhysteretic volume fraction f corresponding to the magnetic domain an parameters Step 5: By means of the mesoscopic energy conservation principle and the theory of the change in magnetization intensity caused by the volume fraction f of magnetic domains, combined with the anhysteretic magnetization intensity M obtained in the above steps α a differential equation of the magnetic domain volume fraction f is established, and then the fourth-order Runge-Kutta method is used to solve the differential equation. Finally, based on the homogenization theory, the overall magnetization intensity of the soft magnetic material under different stresses is obtained, and its hysteresis loop is simulated. an α ​​ 2. A method for simulating the hysteresis characteristics of a soft magnetic material under stress according to claim 1, characterized in that: In step one, it is assumed that the soft magnetic material is composed of a large number of magnetic domains at the mesoscopic scale, which is between the macroscopic and microscopic scales. Without the influence of external magnetic fields and stress factors, the initial magnetization directions of the magnetic domains are randomly distributed, and the soft magnetic material as a whole does not show magnetism externally. Therefore, by performing six triangulations on the icosahedron, a sphere with 10242 coordinate points is obtained to simulate the initial magnetization directions of the magnetic domains in the soft magnetic material.

3. A method for simulating the hysteresis characteristics of a soft magnetic material under stress according to claim 1, characterized in that: The magnetization intensity M of the magnetic domain in Step 2 α can be expressed as: M α = M s α = M s [α1 α2 α3] t where α = [α1 α2 α3] t is the magnetization M α of the magnetic domain α in the initial magnetization direction.

4. A method for simulating the hysteresis characteristics of a soft magnetic material under stress according to claim 1 or 3, characterized in that: The magnetostrictive strain ε in Step 2 α is as follows: where λ 100 , λ 111 are the magnetostrictive strain constants when the cubic crystal is saturated magnetized along the <100> and <111> directions; when the magnetic material is an isotropic material, it is considered that λ 100 = λ 111 = λ m , and λ m is the maximum magnetostrictive strain constant of the magnetic material.

5. A method for simulating the hysteresis characteristics of a soft magnetic material under stress according to claim 1, characterized in that: In step 3, the Gibbs free energy $W$ α , the magnetocrystalline anisotropy energy $W$ α an , the magnetostatic energy $W$ α mag and the magnetoelastic energy $W$ α σ are expressed as follows: Among them, W α σ1 represents the first-order component of the magnetoelastic property, and W α σ2 represents the second-order component of the magnetoelastic property. K1 and K2 are the anisotropy constants of the soft magnetic material, μ0 is the vacuum permeability constant, H is the applied magnetic field, and σ α is the stress tensor, E α is the fourth-order magnetostriction tensor, R α is the second-order tensor formed by the magnetization direction cosines, and N is the sixth-order magnetostriction tensor.

6. A method for simulating the hysteresis characteristics of a soft magnetic material under stress according to claim 1, characterized in that: In step four, according to the Boltzmann distribution principle, the non-hysteretic volume fraction f an The expression is: Among them, is a material adjustable parameter, which is determined by the non-hysteretic initial magnetic permeability χ0 and the saturation magnetization M s determined.

7. A method for simulating the hysteresis characteristics of a soft magnetic material under stress according to claim 1, characterized in that: The non-hysteretic magnetization intensity M of the magnetic material obtained from the homogenization theory in Step 4 an is as follows: M an = <M α >= ∑ α f an M α 。 8. A method for simulating the hysteresis characteristics of a soft magnetic material under stress according to claim 1, characterized in that: The mesoscopic energy conservation equation in step five is: where k is the pinning coefficient between magnetic domains, and δ M is a coefficient introduced to prevent non - physical solutions, δ is the direction coefficient, and M irr is the irreversible magnetization component, and M is the magnetization intensity; According to the theory that the change in magnetization intensity is caused by the change in the volume fraction of magnetic domains, it can be obtained that: dM ani = M s df ani dM i = M s df αi where M ani , f ani , M i , f αi respectively represent the anhysteretic magnetization intensity, anhysteretic volume fraction, magnetization intensity and volume fraction of the i-th magnetic domain, and a new model for simulating the magnetic properties of soft magnetic materials under stress is derived: where, f αi is the volume fraction of the i-th magnetic domain, al is the average internal coupling field parameter of the magnetic domain, c is the reversible magnetization coefficient, α i represents the initial magnetization direction of the i-th magnetic domain, and finally, the fourth-order Runge-Kutta method is used to solve the differential equation to obtain the volume fraction f of each magnetic domain αi ; where f αi (i + 1) is the volume fraction of the next magnetic domain, and f αi (i) is the volume fraction of the current magnetic domain. k1, k2, k3, and k4 are intermediate quantities of the Runge-Kutta method; Then, through the homogenization theory in step four, the magnetization intensity M of the whole soft magnetic material under different stress conditions in the multi-scale case is obtained: M = <M α > = ∑ α f a M α 。

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