Modeling method of stress-electromagnetic coupling effect of amorphous wire based on micromagnetism theory

Abstract

The present invention discloses a method for modeling the stress-electromagnetic coupling effect of amorphous wires based on micromagnetism theory, which belongs to the field of magnetic material technology and electromagnetic property regulation technology. The present invention constructs a three-dimensional simulation model of amorphous wire micromagnetism, solves it through the finite element method, realizes the cross-scale simulation of the evolution of the magnetic domain structure of amorphous wire under external stress, further calculates the change of the effective magnetic permeability of amorphous wire under different stress fields, and integrates the skin effect and magneto-impedance theory to establish an amorphous wire electromagnetic impedance spectrum model. The present invention establishes a quantitative mapping relationship between effective magnetic permeability, electromagnetic impedance characteristics and external stress, which can deduce the magnetic domain structure under different stress fields, and output key parameters such as effective magnetic permeability and electromagnetic impedance spectrum, providing theoretical guidance for the wide application of amorphous wire in the fields of high-sensitivity stress sensor device design, broadband adjustable electromagnetic wave absorption / shielding, and self-sensing composite material structure health monitoring.

Description

Technical Field

[0001] The present invention belongs to the field of magnetic material technology and electromagnetic property regulation technology, relates to the stress-electromagnetic coupling effect of amorphous wires, and particularly relates to a modeling method of the stress-electromagnetic coupling effect of amorphous wires based on micromagnetism theory. Background Art

[0002] As a new type of intelligent fiber material developed in recent years, amorphous wire exhibits unique physical properties such as giant magneto-impedance effect and Barkhausen effect due to its unique magnetic domain structure characteristics and controllable magnetic anisotropy. This type of material has both micron-scale geometric characteristics and excellent mechanical properties, and has important application value in flexible stress sensing, biomedical testing, intelligent electromagnetic regulation and in-situ monitoring of composite materials. However, current research focuses on the magneto-impedance effect, and there are still deficiencies in the study of the coupling mechanism between stress field and electromagnetic parameters. In particular, due to the micron-scale size characteristics of fiber materials, traditional physical testing methods are difficult to achieve in-situ observation of the dynamic transformation of magnetic domains and electromagnetic parameter responses inside the material under the action of stress field. This technical bottleneck restricts the design and engineering application of amorphous wire functional devices.

[0003] As the core theoretical framework for describing the magnetization behavior of magnetic materials, micromagnetism theory effectively achieves a cross-scale correlation between the evolution of atomic-scale magnetic moments and macroscopic magnetic properties through a continuous field representation system for the magnetization intensity vector. By solving the Landau-Lifshitz-Gilbert equation, this theory can accurately reconstruct microscopic dynamic processes such as domain wall motion and magnetic moment rotation, providing a physical basis for revealing the magnetization mechanism under complex external field coupling. Currently, this theory has been successfully applied to the study of material systems such as nanomagnetic devices, two-dimensional heterojunctions, and amorphous alloy ribbons. However, in the field of amorphous wire materials with strong shape anisotropy, the magnetoelastic coupling effect caused by their unique quasi-one-dimensional structure lacks a systematic theoretical modeling method. Summary of the Invention

[0004] Based on the above-mentioned deficiencies in the research on stress-electromagnetic coupling effect of amorphous wire, the present invention proposes a modeling method for stress-electromagnetic coupling effect of amorphous wire based on micromagnetism theory, which includes the following steps:

[0005] S1: Based on the geometric morphology, magnetic parameters and intrinsic magnetoelastic anisotropy of the amorphous wire, a three-dimensional micromagnetic simulation model of the amorphous wire is constructed. The initial magnetic domain structure is obtained by solving the time-domain Landau-Lifshitz-Gilbert equation based on the amorphous wire using the finite element method.

[0006] S2: Based on the magnetoelastic coupling relationship of amorphous wires, the dynamic evolution of the magnetic domain structure of amorphous wires under the action of stress fields is solved;

[0007] S3: The effective permeability parameters are obtained by solving the frequency-domain Landau-Lifshitz-Gilbert equation based on amorphous wires using the finite element method;

[0008] S4: Integrate the skin effect and magneto-impedance theory to establish an electromagnetic impedance spectrum model for amorphous wires.

[0009] Compared with the prior art, the innovative advantages of the present invention are:

[0010] 1) Based on the geometric morphology, magnetic parameters, and intrinsic magnetoelastic anisotropy of amorphous wires, this paper constructs a three-dimensional micromagnetic simulation model of amorphous wires. Through micromagnetic theory, it reveals the dynamic evolution of the magnetic domain structure of amorphous wires under the action of an external stress field, breaking through the limitations of traditional methods in characterizing microstructural transformations.

[0011] 2) The present invention adopts the frequency domain perturbation method to solve the effective magnetic permeability parameters of amorphous wires, establishes the stress-effective magnetic permeability mapping relationship, and clarifies the response mechanism of the effective magnetic permeability of amorphous wires with changes in stress field.

[0012] 3) This invention integrates the skin effect and magneto-impedance theory to establish an electromagnetic impedance spectrum model for amorphous wires, enabling quantitative characterization of the dynamic response of the electromagnetic impedance characteristics of amorphous wires to stress loading. The entire method achieves a quantitative mapping of "stress loading-magnetic domain reconstruction-permeability change-impedance response";

[0013] 4) Based on the method of the present invention, a visual representation of the dynamic transformation of magnetic domains and the response of electromagnetic parameters inside amorphous wires under the action of stress fields can be obtained, which helps to solve the difficult problems of difficulty in obtaining electromagnetic parameters and insufficient coupling of multi-physical fields in amorphous wire research.

[0014] This method can be used to deduce the magnetic domain structure of amorphous wires under different stress fields, and output key parameters such as effective magnetic permeability and electromagnetic impedance spectrum, providing support for the widespread application of amorphous wires in the design of high-sensitivity stress sensor devices, broadband adjustable electromagnetic wave absorption / shielding, and self-sensing composite material structure health monitoring. BRIEF DESCRIPTION OF THE DRAWINGS

[0015] Figure 1 Schematic diagram of the framework of the modeling method for the stress-electromagnetic coupling effect of amorphous wire;

[0016] Figure 2 Schematic diagram for setting the initial state structural parameters of the amorphous wire micromagnetic model;

[0017] Figure 3 Schematic diagram of the initial magnetic domain distribution characteristics of the amorphous wire micromagnetic model;

[0018] Figure 4 Schematic diagram of the magnetic domain structure evolution under axial tensile stress of 100 MPa;

[0019] Figure 5 Schematic diagram of the magnetic domain structure evolution under axial tensile stress of 200 MPa;

[0020] Figure 6 This is the numerical simulation diagram of the frequency response characteristics of the effective magnetic permeability of the initial amorphous wire;

[0021] Figure 7 This is a comparison chart of the frequency response characteristics of the effective magnetic permeability of amorphous wire under different axial stresses;

[0022] Figure 8 This is the numerical simulation diagram of the impedance spectrum characteristics of the initial amorphous wire;

[0023] Figure 9 A comparison of the impedance spectrum characteristics of amorphous wires under different axial stresses. DETAILED DESCRIPTION

[0024] The technical solution of the present invention is further described below with reference to the accompanying drawings and embodiments. The present invention includes but is not limited to the described embodiments.

[0025] In response to the problems of limited physical characterization methods and lack of theoretical models in the existing research on the stress-electromagnetic coupling effect of amorphous wires, the present invention proposes a modeling method for the stress-electromagnetic coupling effect of amorphous wires based on micromagnetic theory. This method aims to achieve a cross-scale simulation of the evolution of the magnetic domain structure of amorphous wires under external stress, and establishes a quantitative mapping relationship between effective magnetic permeability, electromagnetic impedance characteristics and external stress. This method uses micromagnetic simulation technology to achieve a three-dimensional reconstruction of the magnetic domain structure of amorphous wires under the action of an external stress field and a visual representation of the evolution law. The present invention establishes an effective magnetic permeability calculation model for the stress-electromagnetic coupling effect, and quantitatively characterizes the dynamic response characteristics of the frequency-varying magnetic permeability of amorphous wires under different stress fields.

[0026] The present invention proposes a modeling method for stress-electromagnetic coupling effect of amorphous wire based on micromagnetic theory, and its theoretical framework diagram is shown in the figure: Figure 1 As shown, the method includes the following steps:

[0027] S1: Based on the geometric morphology, magnetic parameters and intrinsic magnetoelastic anisotropy of the amorphous wire, a three-dimensional micromagnetic simulation model of the amorphous wire is constructed. The initial magnetic domain structure is obtained by solving the time-domain Landau-Lifshitz-Gilbert equation based on the amorphous wire using the finite element method.

[0028] The amorphous wire of the present invention is an amorphous ferromagnetic alloy fiber. In the experiment of the present invention, the three-dimensional simulation model of the amorphous wire micromagnetism is a concentric cylindrical model equivalent to the amorphous wire core-shell double-layer magnetic domain structure; the establishment steps include:

[0029] S11: setting model structure parameters according to the geometric shape of the amorphous wire, wherein the model structure parameters include the inner diameter, outer diameter and height of the concentric cylinders;

[0030] S12: setting magnetic parameters of the amorphous wire, wherein the magnetic parameters include saturation magnetization, exchange interaction constant, and magnetostriction coefficient;

[0031] S13: constructing an effective field according to the intrinsic magnetoelastic anisotropy, wherein the effective field includes an exchange field and a magnetoelastic anisotropy field;

[0032] S2: Based on the magnetoelastic coupling relationship of amorphous wires, the dynamic evolution of the magnetic domain structure of amorphous wires under the action of stress fields is solved;

[0033] In S2, the external stress field affects the stress on the amorphous wire. Then, the magnetic domain structure is changed, and the S2 is specifically:

[0034] First, based on the external stress of amorphous wire and intrinsic internal stress of amorphous wire The stress of amorphous wire is obtained ;

[0035] Then the stress of the amorphous wire Substituting it into the time-domain Landau-Lifshitz-Gilbert equation of S1 to solve the magnetic domain structure, the dynamic evolution process of the magnetic domain structure of the amorphous wire under the action of the stress field is obtained.

[0036] S3: The effective permeability parameters are obtained by solving the frequency-domain Landau-Lifshitz-Gilbert equation based on amorphous wires using the finite element method;

[0037] S4: By integrating the skin effect and magneto-impedance theory, the impedance of amorphous wire at different frequencies (the value of the amorphous wire impedance spectrum) is obtained, thereby establishing an amorphous wire electromagnetic impedance spectrum model.

[0038] From the above introduction to the method, it can be seen that this method can be used to deduce the magnetic domain structure of amorphous wires under different stress fields, and output key parameters such as effective magnetic permeability and electromagnetic impedance characteristics, providing theoretical guidance for the wide application of amorphous wires in the design of high-sensitivity stress sensor devices, broadband adjustable electromagnetic wave absorption / shielding, and self-sensing composite material structure health monitoring.

[0039] In a specific embodiment of the present invention, solving the time-domain Landau-Lifshitz-Gilbert equation based on the amorphous wire by the finite element method to obtain the initial magnetic domain structure specifically includes the following steps:

[0040] The time-domain Landau-Lifshitz-Gilbert equation based on amorphous wire is:

[0041]

[0042] Where: is the unit magnetic moment, is the gyromagnetic ratio, is the effective field, is the Gilbert damping factor, is the exchange interaction coefficient, is the magnetostriction coefficient, is stress, is the vacuum permeability, is the saturation magnetization strength. Convert the equation into a weak equation form:

[0043]

[0044] The Galerkin method based on the finite element method can be used to obtain the numerical solution of the equation and obtain the unit magnetic moment The distribution of . is a second-order Lagrangian test function, The model domain representing the three-dimensional simulation model of amorphous wire micromagnetism, is the spatial position, is the spatial coordinate component. In the time domain solution , 、 They are time, The unit magnetic moment at time, is the time increment; it can be further substituted into the weak equation:

[0045]

[0046] The Galerkin method based on the finite element method is used to obtain the equation, and the distribution of unit magnetic moment in the three-dimensional simulation model of amorphous wire micromagnetism is obtained, which is the initial magnetic domain structure of the amorphous wire.

[0047] In a specific embodiment of the present invention, the S3 solves the frequency domain Landau-Lifshitz-Gilbert equation based on the amorphous wire by the finite element method to obtain the effective permeability parameter, which specifically includes:

[0048] S31: Due to the skin effect, the coupling with the demagnetization field must be further considered in the frequency domain solution based on the S13 effective field. The time domain Landau-Lifshitz-Gilbert equation is transformed into the frequency domain Landau-Lifshitz-Gilbert equation. The transformation process is as follows:

[0049]

[0050]

[0051] The unit magnetic moment is decomposed into the static stable magnetic moment and dynamic magnetic moment , is the angular frequency, and the effective field is decomposed into the static stable effective field and the dynamic perturbation effective field , the weak form of the frequency domain Landau-Lifshitz-Gilbert equation is:

[0052]

[0053] The perturbation field is introduced, and the dynamic magnetic moment under different frequency perturbation fields can be obtained by solving the weak equation with finite elements. changes.

[0054] S32: Use perturbation method to solve the frequency domain magnetic susceptibility tensor: ,

[0055] S33: Obtain the effective magnetic permeability of the amorphous wire: .

[0056] In a specific embodiment of the present invention, the electromagnetic impedance calculation in S4 adopts a hierarchical progressive algorithm:

[0057] S41: Calculate the skin depth of amorphous wire under electromagnetic waves of different frequencies:

[0058] Where: is the skin depth, is the electromagnetic wave frequency, is the vacuum permeability, is the conductivity;

[0059] S42: Solving the propagation constant of electromagnetic waves in amorphous wires :

[0060]

[0061] S43: Calculate the impedance of amorphous wire at different frequencies to obtain the electromagnetic impedance spectrum model of amorphous wire; for:

[0062]

[0063] Where: is the impedance, is the DC resistance, is the radius of the amorphous wire, is the Bessel function of the first kind of order 0 and is a Bessel function of the first kind of order 1.

[0064] The present embodiment is described and illustrated below through preferred embodiments:

[0065] Based on the geometric shape, magnetic parameters and intrinsic magnetoelastic anisotropy of the amorphous wire, an amorphous wire model with a core-shell structure was constructed in the micromagnetic module of COMSOL software. The height of the concentric cylinder is 5 μm, the inner diameter of the outer circle is 3 μm, and the inner diameter of the inner circle is 0.6 μm. The magnetostriction coefficient of the amorphous wire ( ) is -2×10 -7 , saturation magnetization ( ) is 5×10 5 A / m, vacuum permeability ( ) is 4π×10 -7 H / m, and the gyromagnetic ratio is 2.21×10 5 m / (s·A), and the exchange coefficient is 3.8×10 -11 A·m. The effective magnetoelastic field in the core of the amorphous wire is 0, and the circumferential direction of the shell ( ) is subjected to a shrinkage stress of 600 MPa, the axial direction ( ) is subjected to a contraction stress of 300 MPa, then the effective field of the amorphous wire shell in the X, Y, and Z directions in the Cartesian coordinate system is 、 、 They are:

[0066] + A▽ 2

[0067] + A▽ 2

[0068] + A▽ 2

[0069]

[0070]

[0071]

[0072] 、 、 are the unit magnetic moments in the X, Y, and Z directions respectively, 、 、 are the stresses in the X, Y, and Z directions respectively, and x, y, and z are the coordinate values ​​in the X, Y, and Z directions respectively. .

[0073] Substituting the above parameters into the Landau–Lifshitz–Gilbert equation:

[0074] Convert the time-domain Landau-Lifshitz-Gilbert equation for amorphous wires into a weak equation form:

[0075]

[0076] Where, is a second-order Lagrangian test function, is the model domain of the amorphous wire micromagnetism three-dimensional simulation model, is the spatial position, is the spatial coordinate component. In order to speed up the solution, the Gilbert damping factor Take it as 0.5. The initial magnetic moments in the X, Y, and Z directions are set as:

[0077]

[0078]

[0079]

[0080] In the time domain solution , which can be further substituted into the weak equation:

[0081]

[0082] The Galerkin method based on the finite element method is used to obtain the equation, obtain the distribution of the unit magnetic moment, and complete the solution of the initial magnetic domain structure. In this embodiment, the initial state structure parameters of the amorphous wire micromagnetism three-dimensional simulation model are set as follows Figure 2 As shown, the solution is then performed after the default grid division. After the solution is completed, the magnetic moment distribution in the model is as follows Figure 3 As shown, it shows an obvious spiral magnetic domain structure.

[0083] This embodiment takes the tensile stress in the axial direction as an example of the external stress field. When the amorphous wire is subjected to the external tensile stress in the axial direction, due to the initial The shrinkage stress is 300 MPa (the intrinsic internal stress of amorphous wire ), under external tensile stresses of 100 MPa and 200 MPa ( ) under the action of The value of will decrease, which will reduce the axial component of the amorphous wire magnetic moment and gradually transform the initial spiral magnetic domain structure into a bamboo-like magnetic domain structure, such as Figure 4 and Figure 5 shown.

[0084] After obtaining the magnetic domain structure of the amorphous wire under different stress fields through the above steps, we can further calculate the change in the effective magnetic permeability of the amorphous wire under different stress fields. Due to the skin effect, electromagnetic waves propagate along the near-surface of the amorphous wire shell. When only considering the propagation area, it is necessary to further introduce a demagnetization field on top of the magnetoelastic effective field. To facilitate calculations, it is also necessary to convert the above-mentioned time-domain Landau-Lifshitz-Gilbert equation into the frequency-domain Landau-Lifshitz-Gilbert equation. Let:

[0085]

[0086]

[0087] The frequency domain Landau-Lifshitz-Gilbert equation is:

[0088]

[0089] When solving in the frequency domain, the Gilbert damping factor = 0.02. The weak form of the frequency-domain Landau-Lifshitz-Gilbert equation is:

[0090]

[0091] At this time, a perturbation field with different frequencies is applied in a certain direction to calculate the corresponding direction. The change of is given by the formula:

[0092]

[0093] The magnetic susceptibility can be calculated, and further the conversion relationship between magnetic permeability and magnetic susceptibility is:

[0094]

[0095] The effective magnetic permeability of the amorphous wire in this direction can be calculated. In the specific application of amorphous wire, the main focus is on its effective magnetic permeability in the circumferential direction. Therefore, we applied a magnetic field at different frequencies in the circumferential direction. The perturbation magnetic field is calculated, and then its effective magnetic permeability in the circumferential direction is calculated. Figure 6 As shown in Figure 2, there is an obvious ferromagnetic resonance at around 700 MHz. Similarly, the change in the effective magnetic permeability of the amorphous wire under axial tensile stress can be calculated, as shown in Figure 2. Figure 7 As shown in the figure, with the increase of axial tensile stress, the ferromagnetic resonance frequency gradually moves to low frequency, and the effective magnetic permeability value at the ferromagnetic resonance frequency also gradually decreases.

[0096] Based on the circumferential effective magnetic permeability of the amorphous wire, the skin depth of electromagnetic wave propagation can be further calculated:

[0097]

[0098] in is the conductivity, with a value of 1×10 6 S / m. Next, the impedance of a cylindrical magnetic conductor is calculated using the equation:

[0099]

[0100]

[0101] where R dc =75 Ω, R=15 μm, from which the electromagnetic impedance spectrum value of the amorphous wire can be calculated, such as Figure 8 As shown in Figure 2, the impedance is highest at the ferromagnetic resonance frequency of 700 MHz. Similarly, the change in the impedance spectrum characteristics of the amorphous wire under axial tensile stress can be calculated, as shown in Figure 2. Figure 9 As shown in Figure 3, as the axial tensile stress increases, the ferromagnetic resonance frequency gradually moves to a lower frequency, and the impedance at the ferromagnetic resonance frequency also gradually decreases.

[0102] In summary, based on the method of the present invention, a visual representation of the dynamic transformation of magnetic domains and electromagnetic parameter responses inside amorphous wires under stress fields can be obtained. This method can be used to deduce the magnetic domain structure of amorphous wires under different stress fields, and output key parameters such as effective magnetic permeability and electromagnetic impedance characteristics.

[0103] The above-described embodiments merely represent several implementation methods of the present invention. While the descriptions are relatively specific and detailed, they should not be construed as limiting the scope of the present invention. It should be noted that a person skilled in the art would be able to make various modifications and improvements without departing from the spirit of the present invention, all of which fall within the scope of protection of the present invention. Therefore, the scope of protection of the present invention shall be determined by the appended claims.

Claims

1. A modeling method for stress-electromagnetic coupling effect of amorphous wire based on micromagnetism theory, characterized in that: The method comprises the following steps: S1: Based on the geometric morphology, magnetic parameters and intrinsic magnetoelastic anisotropy of the amorphous wire, a three-dimensional micromagnetic simulation model of the amorphous wire is constructed. The initial magnetic domain structure is obtained by solving the time-domain Landau-Lifshitz-Gilbert equation based on the amorphous wire using the finite element method. S2: Based on the magnetoelastic coupling relationship of amorphous wires, the dynamic evolution of the magnetic domain structure of amorphous wires under the action of stress fields is solved; S3: The effective permeability parameters are obtained by solving the frequency-domain Landau-Lifshitz-Gilbert equation based on amorphous wires using the finite element method; S4: Integrate the skin effect and magneto-impedance theory to establish an electromagnetic impedance spectrum model for amorphous wires; The construction of the electromagnetic impedance spectrum model of the amorphous wire in S4 adopts a hierarchical progressive algorithm, and S4 includes: S41: Calculate the skin depth of amorphous wire under electromagnetic waves of different frequencies: ； Where: is the skin depth, is the electromagnetic wave frequency, is the vacuum permeability, is the conductivity; S42: Solving the propagation constant of electromagnetic waves in amorphous wires : ； S43: Calculate the impedance of the amorphous wire at different frequencies to obtain an electromagnetic impedance spectrum model of the amorphous wire; wherein the impedance Z of the amorphous wire at different frequencies is: ； Where: Z is impedance, is the DC resistance, is the radius of the amorphous wire, is the Bessel function of the first kind of order 0 and is a Bessel function of the first kind of order 1.

2. The amorphous wire stress-electromagnetic coupling effect modeling method based on micromagnetism theory according to claim 1, characterized in that: The amorphous wire in S1 is an amorphous ferromagnetic alloy fiber.

3. The amorphous wire stress-electromagnetic coupling effect modeling method based on micromagnetism theory according to claim 1, characterized in that: The amorphous wire micromagnetism three-dimensional simulation model in S1 is a concentric cylindrical model equivalent to the amorphous wire core-shell double-layer magnetic domain structure, and its establishment steps include: S11: setting model structure parameters according to the geometric shape of the amorphous wire, wherein the model structure parameters include the inner diameter, outer diameter and height of the concentric cylinders; S12: setting magnetic parameters of the amorphous wire, wherein the magnetic parameters include saturation magnetization, exchange interaction constant, and magnetostriction coefficient; S13: constructing an effective field according to the intrinsic magnetoelastic anisotropy, wherein the effective field includes an exchange field and a magnetoelastic anisotropy field.

4. The method for modeling stress-electromagnetic coupling effect of amorphous wires based on micromagnetism theory according to claim 1, characterized in that: In S1, the initial magnetic domain structure is obtained by solving the time-domain Landau-Lifshitz-Gilbert equation based on the amorphous wire using the finite element method, including: Convert the time-domain Landau-Lifshitz-Gilbert equation for amorphous wires into a weak equation form: ； Where, is a second-order Lagrangian test function, is the model domain of the amorphous wire micromagnetism three-dimensional simulation model, is the unit magnetic moment, is the Gilbert damping factor, is the gyromagnetic ratio, is the exchange interaction coefficient, is the spatial position, is the spatial coordinate component, is the magnetostriction coefficient, is stress, is the vacuum permeability, is the saturation magnetization; In the time domain solution ,in, 、 They are time t, The unit magnetic moment at time, is the time increment; further substituted into the weak equation: ； The Galerkin method based on the finite element method is used to obtain the equation, and the distribution of unit magnetic moment in the three-dimensional simulation model of amorphous wire micromagnetism is obtained, which is the initial magnetic domain structure of the amorphous wire.

5. The method for modeling stress-electromagnetic coupling effect of amorphous wire based on micromagnetism theory according to claim 1, characterized in that: In S2, the external stress field affects the stress on the amorphous wire. Then change the magnetic domain structure; first based on the external stress on the amorphous wire and intrinsic internal stress of amorphous wire The stress of amorphous wire is obtained ; Then the stress of the amorphous wire Substituting it into the time-domain Landau-Lifshitz-Gilbert equation of S1 to solve the magnetic domain structure, the dynamic evolution process of the magnetic domain structure of the amorphous wire under the action of the stress field is obtained.

6. The method for modeling stress-electromagnetic coupling effect of amorphous wires based on micromagnetism theory according to claim 4, characterized in that: The S3 specifically includes: S31: Transform the time-domain Landau-Lifshitz-Gilbert equation into the frequency-domain Landau-Lifshitz-Gilbert equation. The weak equation form of the frequency-domain Landau-Lifshitz-Gilbert equation is: ； The unit magnetic moment Disassembled into static stable magnetic moment and dynamic magnetic moment , is the angular frequency, The dynamic magnetic moment under different frequency perturbation fields is obtained by finite element solution of the weak equation. changes; S32: Solve the frequency domain magnetic susceptibility tensor: , S33: Obtain the effective magnetic permeability of the amorphous wire: .

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