A method and system for predicting the vibration characteristics of a magneto-electro-elastic structure based on an overlapping finite element method

Abstract

The application belongs to the field of magneto-electro-elastic structure vibration characteristics, and particularly discloses a magneto-electro-elastic structure vibration characteristic prediction method and system based on an overlapping finite element method, which comprises the following steps: discretizing physical field variables of a magneto-electro-elastic structure based on the overlapping finite element method, wherein the physical field variables include a displacement field, an electric potential field and a magnetic potential field; deriving a Galerkin weak form of a dynamic control equation of the magneto-electro-elastic structure by using a weighted residual method, and combining the discretized physical field variables to obtain an overlapping finite element dynamic discrete linear equation set of the magneto-electro-elastic structure; reducing the degrees of freedom of the overlapping finite element dynamic discrete linear equation set by using a static condensation method to obtain an overlapping finite element control equation, and further realizing the vibration characteristic prediction of the magneto-electro-elastic structure. The application realizes accurate prediction of the structure natural frequency, modal shape and multi-physical field harmonic response under external excitation, and has significant advantages in calculation accuracy and efficiency compared with the traditional finite element method.

Description

Technical Field

[0001] This invention belongs to the field of vibration characteristics of magneto-electro-elastic structures, and more specifically, relates to a method and system for predicting the vibration characteristics of magneto-electro-elastic structures based on the overlapping finite element method. Background Technology

[0002] Magnetoelastic materials are advanced smart materials made from piezoelectric and piezomagnetic materials. These materials integrate the interactive conversion of electric fields, magnetic fields, and mechanical stress, exhibiting remarkable multi-physics coupling properties. They can not only generate magnetic responses to changes in electric fields or induce electrical signals to changes in magnetic fields, but also modulate electromagnetic properties through mechanical stress, and vice versa. This unique physical property makes magnetoelastic materials of immeasurable importance in many fields. Currently, researchers have proposed many magnetoelastic smart devices such as sensors and energy harvesters, and magnetoelastic materials have become one of the hot topics in current materials science research.

[0003] In early research on magnetoelastic materials, researchers often used analytical methods to study magnetoelastic structures with simple geometric features. These methods, based on the fundamental governing equations of magnetoelastic materials, could yield highly reliable results through calculation. However, when solving the dynamic problems of complex magnetoelastic structures, analytical methods often became unsuitable. In such cases, researchers preferred to use numerical methods for in-depth analysis.

[0004] The finite element method (FEM) is a stable and efficient numerical method widely used in engineering practice. It can handle various complex physical fields and coupling problems, and therefore, the FEM is also used to predict the dynamic characteristics of magneto-electro-elastic structures. Many scholars have used the FEM to study magneto-electro-elastic structures and achieved significant results. However, it is worth noting that although the FEM is a powerful analytical tool, its solution accuracy is limited to some extent by the mesh refinement. Therefore, to ensure the reliability and accuracy of the results, a fine mesh generation strategy is usually required. This means that when dealing with complex structures, researchers need to spend a lot of effort to obtain a satisfactory mesh. Therefore, it is necessary to study more accurate and efficient dynamic prediction models for magneto-electro-elastic structures to provide technical support for theoretical research and engineering practice. Summary of the Invention

[0005] In view of the above-mentioned defects or improvement needs of the existing technology, the present invention provides a method and system for predicting the vibration characteristics of magneto-electro-elastic structures based on the overlapping finite element method. The purpose is to achieve high-precision and high-efficiency prediction of the vibration characteristics of magneto-electro-elastic structures.

[0006] To achieve the above objectives, according to one aspect of the present invention, a method for predicting the vibration characteristics of magnetoelectroelastic structures based on the overlapping finite element method is proposed, comprising the following steps:

[0007] The physical field variables of the magnetoelectroelastic structure are discretized based on the overlapping finite element method. The physical field variables include displacement field, electric potential field and magnetic potential field.

[0008] Using the weighted residual method, the Galerkin weak form of the governing equations of the magneto-electro-elastic structure is derived. Combined with the discretized physical field variables, the overlapping finite element dynamic discrete linear equations of the magneto-electro-elastic structure are obtained.

[0009] The degrees of freedom of the overlapping finite element dynamic discrete linear equation system are reduced by using the static condensation method, and the overlapping finite element control equations are obtained.

[0010] Dynamic analysis was performed based on the overlapping finite element control equations to obtain the vibration characteristic parameters of the magneto-electro-elastic structure.

[0011] As a further preferred embodiment, the physical field variables of the magneto-electro-elastic structure are discretized based on the overlapping finite element method, and expressed as follows:

[0012]

[0013] Among them, u d φ and ψ represent the displacement field, electric potential field, and magnetic potential field of the magneto-electro-elastic structure, respectively; Nn represents the total number of nodes in the magneto-electro-elastic structure after meshing. γ represents the node values, namely the displacement, electric potential, and magnetopotential corresponding to node J; J This represents the overlapping finite element interpolation function corresponding to node J.

[0014] As a further preferred method, the overlapping finite element method interpolation function is constructed as follows:

[0015]

[0016] Where Ne represents the number of nodes in an overlapping finite element, h I Let I be the shape function of the first-order element in the finite element method corresponding to node I; virtual nodes are introduced at the midpoint of each edge of the overlapping finite element to form a second-order element. Parameters are constructed for the virtual nodes; Nh is the number of nodes in a quadratic element. Let i be the shape function of the quadratic element in the finite element method corresponding to node i. yes The corresponding weighting coefficients.

[0017] As a further preferred option, the overlapping finite element method interpolation function γ corresponding to node J is... J Specifically:

[0018]

[0019] Where K is the node directly connected to node J in the overlapping finite element, and β represents an adjustable coefficient. h represents the quadratic element shape function corresponding to the virtual node between nodes J and K. J h K Let J and K represent the linear unit shape functions corresponding to nodes J and K, respectively.

[0020] As a further preferred embodiment, the node value is based on the basis function p. J and unknown variable a J This indicates that the basis functions are first-order complete polynomial basis functions.

[0021] As a further preferred approach, the discretized physical field variables are represented in matrix form:

[0022] u d =N u u,φ=N φ φ, ψ = N ψ ψ

[0023] N u =[N1 N1x N1z]

[0024] N φ =[N2 N2x N2z]

[0025] N ψ =[N2 N2x N2z]

[0026] N1=[γ1I e γ2I e ...γ Nn I e ],

[0027] N2=[γ1γ2...γ Nn ]

[0028] Where u, φ, and ψ represent the responses of the displacement field, electric potential field, and magnetic potential field, respectively, each containing the unknown variable a. J The vector; construct a Cartesian coordinate system with the direction perpendicular to the plane of the magneto-electro-elastic structure as the y-axis, where x and z represent coordinate variables; γ1, γ2…γ Nn These represent the overlapping finite element interpolation functions corresponding to nodes 1, 2, ..., Nn, respectively.

[0029] As a further preferred embodiment, the overlapping finite element dynamic discrete linear equations of the magneto-electro-elastic structure are determined, including:

[0030] The weighted residual method is adopted, and the weighted residual formula is determined based on the governing equations of the magneto-electro-elastic structure. Then, Green's divergence theorem is applied, considering pre-set boundary conditions, and combined with the matrix form of the discretized physical field variables, to obtain the overlapping finite element dynamic discrete linear equations of the magneto-electro-elastic structure, expressed as:

[0031]

[0032] in:

[0033]

[0034] Gradient matrix B u B φ B ψ Represented as:

[0035] B u =L u N u B φ =L φ N φ B ψ =L ψ N ψ

[0036]

[0037] in, Let Ω and Γ represent the second derivative of u with respect to time, respectively, and let Ω and Γ represent the computational domain and boundary of the magneto-electro-elastic structure. These represent the applied traction force, charge, and magnetic induction boundary conditions, respectively. ρ represents density. The matrices C, e, q, m, ε, and μ correspond to elasticity, piezoelectricity, piezomagneticity, dielectricity, magnetoelectricity, and permeability constant, respectively. x, y, and z represent coordinate variables.

[0038] As a further preferred embodiment, when the external excitation force is not considered and a free vibration analysis is performed on the magneto-electroelastic structure, the overlapping finite element governing equation is expressed as:

[0039]

[0040] When an external excitation force is applied and harmonic response analysis is performed on the magneto-electro-elastic structure, the overlapping finite element control equation is expressed as:

[0041]

[0042] in:

[0043]

[0044] As a further preferred approach, dynamic analysis based on overlapping finite element control equations is performed to obtain the vibration characteristic parameters of the magneto-electro-elastic structure, including:

[0045] Based on the overlapping finite element control equations, the free vibration analysis of the magneto-electro-elastic structure is carried out to obtain the natural frequencies and mode shapes of the magneto-electro-elastic structure.

[0046] Based on the overlapping finite element control equations, harmonic response analysis is performed on the magneto-electro-elastic structure to obtain the displacement harmonic response of the magneto-electro-elastic structure. Then, the electric potential and magnetomotive force responses of the magneto-electro-elastic structure are obtained based on the displacement harmonic response.

[0047] According to another aspect of the present invention, a system for predicting the vibration characteristics of a magneto-electro-elastic structure based on the overlapping finite element method is provided, comprising a processor for executing the above-described method for predicting the vibration characteristics of a magneto-electro-elastic structure based on the overlapping finite element method.

[0048] In summary, compared with the prior art, the above-described technical solutions conceived by this invention mainly possess the following technical advantages:

[0049] 1. This invention uses overlapping finite element methods to discretize the magneto-electro-elastic structure and combines it with the structural dynamics control equations to construct a magneto-electro-elastic structure dynamics calculation model based on overlapping finite element methods. This enables accurate prediction of the natural frequencies, mode shapes, and multi-physics harmonic responses of the magneto-electro-elastic structure under external excitation. Compared with the traditional finite element method, it has significant advantages in both calculation accuracy and efficiency, and can provide technical support for the theoretical research and engineering application of magneto-electro-elastic composite materials.

[0050] 2. This invention simulates the magnetic field, electric field, and displacement field of a magneto-electro-elastic structure by interpolation using the overlapping finite element method. It also employs first-order complete polynomial basis functions as local field approximations to enhance the performance of the overlapping finite element method. This allows for accurate simulation of the structural vibration characteristics even with relatively coarse meshing, further improving the efficiency of dynamic simulation of magneto-electro-elastic materials.

[0051] 3. The method of this invention can be calculated based on the finite element mesh commonly used in engineering practice, and can be easily integrated into existing finite element analysis software, thus reducing the technical threshold and promotion cost. Attached Figure Description

[0052] Figure 1 This is a flowchart of the method for predicting the vibration characteristics of magnetoelectroelastic structures based on the overlapping finite element method according to an embodiment of the present invention;

[0053] Figure 2 This is a schematic diagram of a magneto-electro-elastic cantilever beam model according to an embodiment of the present invention;

[0054] Figure 3This is a schematic diagram of the finite element mesh generation of the magneto-electro-elastic cantilever beam model according to an embodiment of the present invention, where (a) is mesh 1 and (b) is mesh 2;

[0055] Figure 4 The graph shows the multi-physics harmonic response curves at the observation point of the magneto-electro-elastic cantilever beam structure under unit harmonic force excitation in an embodiment of the present invention, where (a) is the displacement harmonic response, (b) is the electric potential harmonic response, and (c) is the magnetopotential harmonic response. Detailed Implementation

[0056] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention. Furthermore, the technical features involved in the various embodiments of this invention described below can be combined with each other as long as they do not conflict with each other.

[0057] This invention provides a method for predicting the vibration characteristics of magnetoelectroelastic structures based on the overlapping finite element method, such as... Figure 1 As shown, it includes the following steps:

[0058] S1: Obtain the mesh data of the magneto-electro-elastic structure and set the corresponding material parameters; based on the mesh data, use the overlapping finite element method to discretize the magnetic field, electric field and displacement field of the magneto-electro-elastic structure, and combine the control equations of magneto-electro-elastic material dynamics to derive the overlapping finite element dynamic discrete linear equations of the structure, and set the boundary conditions according to the model.

[0059] Furthermore, the specific steps of step S1 are as follows:

[0060] S11: The three physical field variables in the magneto-electro-elastic structure are discretized based on the overlapping finite element method, as detailed below:

[0061]

[0062] In the formula u d φ and ψ represent the displacement field, electric potential field, and magnetic potential field in the computational domain of the magneto-electro-elastic structure, respectively. For unknown nodes, γ represents the displacement, electric potential, and magnetic potential corresponding to mesh node J, respectively; Nn represents the total number of mesh nodes in the magneto-electro-elastic structure; J γ represents the overlapping finite element method interpolation function corresponding to mesh node J. J The expression is:

[0063]

[0064] In the formula h IThe interpolation function used in the traditional finite element method, i.e., the shape function of a first-order finite element; Ne represents the number of nodes in an overlapping finite element; in In its construction, virtual nodes are introduced, that is, virtual nodes are introduced at the midpoint of each edge of the overlapping finite element:

[0065]

[0066] In the formula It is the interpolation function after introducing virtual nodes, i.e., the quadratic finite element shape function; Nh is the number of nodes in a quadratic finite element; It is an interpolation function The corresponding weighting coefficients include an adjustable coefficient β for controlling higher-order terms; in this embodiment, β = 0.01. For example, if initially divided into quadrilateral meshes, each finite element element has 4 nodes, i.e., Ne = 4; after introducing virtual nodes, an 8-node quadrilateral element is formed, i.e., Nh = 8.

[0067] Combining formulas (2) and (3), the interpolation function can be obtained:

[0068]

[0069] Another part of equation (1) is that the node values ​​are determined by various basis functions p. n and unknown variable a Jn Composition; with For example, It can be represented as:

[0070]

[0071] Considering computational accuracy and efficiency, and taking into account practical engineering applications, first-order complete polynomial basis functions are selected as local approximations:

[0072] p J =[1,x,z] (6)

[0073] In this embodiment, a planar rectangular coordinate system is constructed with the plane where the magnetoelastic structure is located as the xoz plane and the direction perpendicular to the plane of the magnetoelastic structure as the y-axis; that is, (x,z) are coordinate variables.

[0074] Combining equations (2)-(6), equation (1) can be simplified into matrix form:

[0075]

[0076] In the formula, matrix N u N φ and N ψ By interpolation function and basis function p J constitute

[0077] N u =[N1 N1x N1y] (8)

[0078] N φ =[N2 N2x N2y] (9)

[0079] N ψ =[N2 N2x N2y] (10)

[0080]

[0081] N2=[γ1γ2γ3γ4] (12)

[0082] u, φ, and ψ are unknown variables included in the solution to the requirement. Jn The vectors describe the responses of the three physical fields, respectively.

[0083] S12: Using the weighted residual method, the Galerkin weak form of the governing equations for the magneto-electro-elastic material dynamics is derived. Combined with formula (7), an overlapping finite element dynamic discrete model for solving the vibration characteristics of the magneto-electro-elastic structure is obtained. The specific steps are as follows:

[0084] Two-dimensional magnetoelastic materials satisfy the following constitutive relations (i.e., the governing equations of dynamics):

[0085]

[0086] In the formula, σ, D, and B represent stress, electric displacement, and magnetic induction, respectively; S, E, and H represent the corresponding strain field, electric field, and magnetic field, respectively; and C, e, q, ε, m, and μ represent the elastic, piezoelectric, piezomagnetic, dielectric, magnetoelectric, and permeability constants (i.e., the magnetoelastic material parameters set in step S1), respectively. The subscripts i, j, k, and l correspond to the x or z axis. For example, when i and j both represent x, σ ij That is, σ xx , representing the stress in the x-direction. Additionally, in equation (15), D i,i D represents i Take the partial derivative with respect to i, B i,i B i Take the partial derivative with respect to i, where ρ is the density. It's acceleration.

[0087] Furthermore, the boundary conditions applied to the magneto-electro-elastic structure are expressed by the following formula:

[0088]

[0089] In the formula, and Γ represents the applied boundary conditions of displacement, electric potential, magnetic potential, traction force, charge, and magnetic induction, respectively. u,Γ φ ,Γ ψ ,Γ T ,Γ E and Γ M For the corresponding boundary.

[0090] Consider a magnetoelectroelastic structure with a computational domain of Ω and a boundary of Γ. By combining the test function t with equation (15) and integrating over Ω, the weighted residual formula can be obtained:

[0091]

[0092] Applying Green's divergence theorem and considering the boundary conditions, we can obtain:

[0093]

[0094] Then, using equations (13), (14), and (19) and considering the overlapping finite element discretization scheme (7), the discrete Galerkin weak form of the governing equations for the two-dimensional magnetoelectroelastic dynamics problem based on the overlapping finite element method is obtained:

[0095]

[0096] In the formula

[0097] Matrices C, e, q, m, ε, and μ contain the material constants in equation (13), and the gradient matrix B u B φ and B ψ Represented as:

[0098] B u =L u N u B φ =L φ N φ B ψ =L ψ N ψ , (twenty one)

[0099]

[0100] Equation (20) above is the discrete linear equation set of overlapping finite element dynamics for the magneto-electro-elastic structure.

[0101] S2: For the derived overlapping finite element dynamic discrete linear equations, the static condensation method is used to reduce the degrees of freedom of the linear equations, resulting in an overlapping finite element control equation that is only related to the displacement field.

[0102] Furthermore, the specific steps of step S2 are as follows:

[0103] 1) In the free vibration analysis of the magneto-electro-elastic structure, the external excitation force is not considered, i.e., f u =0,f φ =0,f ψ =0, from the second and third equations in equation (20) we get:

[0104]

[0105] Substituting equation (24) into the second equation of equation (20), and equation (23) into the third equation of equation (20), we get:

[0106]

[0107] Substituting equations (25) and (26) into the first equation of equation (20), we obtain the overlapping finite element governing equations:

[0108]

[0109] In the formula

[0110]

[0111] 2) When an external mechanical excitation force is applied and harmonic response analysis is performed on the magneto-electro-elastic structure, f φ =0,f ψ If = 0, then the overlapping finite element governing equations are expressed as:

[0112]

[0113] Equations (27) and (33) above are the overlapping finite element control equations that are only related to the displacement field.

[0114] S3: Solve the overlapping finite element control equations according to the problem to obtain the natural frequency, mode shape, and displacement harmonic response of the magneto-electro-elastic structure; from the obtained displacement harmonic response results, inversely calculate the electromotive force and magnetomotive force response of the structure to obtain all the results related to the vibration characteristics of the magneto-electro-elastic structure.

[0115] Specifically, based on the overlapping finite element control equation (27), the free vibration analysis of the magneto-electro-elastic structure can be performed to obtain the natural frequency and mode shape; based on the overlapping finite element control equation (33), the harmonic response analysis of the magneto-electro-elastic structure can be performed to obtain the displacement harmonic response, and then the electric potential and magnetomotive force response can be obtained.

[0116] The following is a detailed description using specific embodiments:

[0117] For a magneto-electro-elastic cantilever beam fixed at the left end and subjected to a unit harmonic force at the top of the right end, such as Figure 2As shown, the beam is 0.3m long, 0.02m wide, and 0.001m thick. The magnetoelastic material used is BaTiO3-CoFe2O4, where BaTiO3 is a piezoelectric material with a volume ratio of 60%, and CoFe2O4 is a piezomagnetic material with a corresponding volume ratio of 40%. Although exemplary embodiments of this disclosure are shown in the accompanying drawings, it should be understood that the invention can be used for various two-dimensional magnetoelastic structures and should not be limited to the embodiments set forth herein. This invention can effectively predict the vibration characteristics of various two-dimensional magnetoelastic structures (such as beams, plates, etc., and different magnetoelastic materials).

[0118] In S1, the finite element mesh of the cantilever beam is first imported, using mesh 1: 2 × 30 = 60 elements, as follows. Figure 3 As shown in (a); in addition, the material parameters are determined. The magneto-electro-elastic material parameters used in this embodiment are shown in Table 1.

[0119] Table 1. Material parameters of BaTiO3 (60%)-CoFe2O4 (40%)

[0120]

[0121] Then, based on grid 1, the magnetic field, electric field and displacement field of the magneto-electro-elastic structure are discretized using overlapping finite element method. Combined with the control equation of magneto-electro-elastic material dynamics, the overlapping finite element dynamic discretization model of the structure is derived and constructed. The finite element stiffness and mass matrix of the cantilever beam model are obtained, and boundary conditions are set according to the model.

[0122] In S2, the static condensation method is used to eliminate the magnetic and electric terms in the equations, resulting in a magneto-electro-elastic structure control equation that is only related to displacement.

[0123] In S3, the natural frequencies of the cantilever beam made of magneto-electro-elastic material are first solved using the overlapping finite element method, as shown in Table 2. Table 2 also presents the numerical solution and reference solution obtained by the traditional finite element method. For the numerical solution, a mesh of 2: 8 × 120 = 960 elements is used. Figure 3 As shown in (b), the reference solution is obtained using an extremely fine mesh. As can be seen from Table 2, compared with the traditional finite element method, the overlapping finite element method only requires a coarser mesh and a shorter time to obtain a more accurate solution. Therefore, the overlapping finite element model proposed in this invention can achieve efficient analysis of the dynamic characteristics of magneto-electro-elastic structures.

[0124] Table 2 shows the natural frequencies of cantilever beams made of magneto-electro-elastic materials obtained by different methods and the required numerical costs.

[0125]

[0126] Furthermore, the displacement harmonic response of the magneto-electro-elastic cantilever beam under a unit harmonic force was calculated using the overlapping finite element method, and compared with that obtained using the traditional finite element method (both methods yielded results based on mesh 1). Figure 4 As shown in (a).

[0127] Then, by applying formulas (25) and (26), the harmonic responses of the two physical fields of magnetoelectricity at the observation point of the magnetoelectric elastic structure are calculated, such as Figure 4 As shown in (b) and (c), the curve comparison shows that the overlapping finite element method is more accurate than the traditional finite element method.

[0128] Those skilled in the art will readily understand that the above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for predicting the vibration characteristics of magnetoelectroelastic structures based on the overlapping finite element method, characterized in that, Includes the following steps: The physical field variables of the magnetoelectroelastic structure are discretized based on the overlapping finite element method. These physical field variables include displacement field, electric potential field, and magnetic potential field; expressed as: Among them, u d , , ψ These represent the displacement field, electric potential field, and magnetic potential field of the magneto-electro-elastic structure, respectively. Nn This represents the total number of nodes in the magneto-electro-elastic structure after mesh generation; , ), , The node values ​​represent the node values ​​respectively. J The corresponding displacement, electric potential, and magnetic potential; γ J Represents a node J The corresponding overlapping finite element method interpolation function is constructed as follows: in, Ne This represents the number of nodes in an overlapping finite element. h I For nodes I The corresponding shape function of a first-order element in the finite element method; virtual nodes are introduced at the midpoint of each edge of the overlapping finite element to form a second-order element. Construct parameters for the virtual nodes; Nh Let be the number of nodes in a quadratic element. For nodes i The corresponding finite element method quadratic element shape function, yes The corresponding weighting coefficients include adjustable coefficients for controlling higher-order terms. β ; node J Corresponding overlapping finite element method interpolation function γ J Specifically: in, K For overlapping finite elements and nodes J Directly connected nodes Represents a node J , K The quadratic unit shape function corresponding to the virtual nodes between them h J , h K Representing nodes respectively J , K The corresponding first-order unit shape function; Using the weighted residual method, the Galerkin weak form of the governing equations of the magneto-electro-elastic structure is derived. Combined with the discretized physical field variables, the overlapping finite element dynamic discrete linear equations of the magneto-electro-elastic structure are obtained. The degrees of freedom of the overlapping finite element dynamic discrete linear equation system are reduced by using the static condensation method, and the overlapping finite element control equations are obtained. Dynamic analysis was performed based on the overlapping finite element control equations to obtain the vibration characteristic parameters of the magneto-electro-elastic structure.

2. The method for predicting the vibration characteristics of magnetoelectroelastic structures based on the overlapping finite element method as described in claim 1, characterized in that, The node values ​​are based on basis functions. and unknown variables This indicates that the basis functions are first-order complete polynomial basis functions.

3. The method for predicting the vibration characteristics of magneto-electro-elastic structures based on the overlapping finite element method as described in claim 2, characterized in that, The discretized physical field variables are represented in matrix form: Among them, u, ψ and ψ represent the responses to the displacement field, electric potential field, and magnetic potential field, respectively, all of which contain unknown variables. The vector; with the direction perpendicular to the plane of the magneto-electro-elastic structure as y Construct a Cartesian coordinate system using axes. x , z Represents coordinate variables; , … Representing nodes 1, 2, ... respectively Nn The corresponding overlapping finite element method interpolation function.

4. The method for predicting the vibration characteristics of magnetoelectroelastic structures based on the overlapping finite element method as described in claim 3, characterized in that, The system of discrete linear equations for the overlapping finite element dynamics of the magneto-electro-elastic structure is determined, including: The weighted residual method is adopted, and the weighted residual formula is determined based on the governing equations of the magneto-electro-elastic structure. Then, Green's divergence theorem is applied, considering pre-set boundary conditions, and combined with the matrix form of the discretized physical field variables, to obtain the overlapping finite element dynamic discrete linear equations of the magneto-electro-elastic structure, expressed as: in: ， ， ， ， ， ， Gradient matrix B u B ϕ B ψ Represented as: in, Let Ω and Γ represent the second derivative of u with respect to time, respectively, and let Ω and Γ represent the computational domain and boundary of the magneto-electro-elastic structure. , , These represent the applied traction force, electric charge, and magnetic induction boundary conditions, respectively. ρ Density ， The matrices C, e, q, m, ε, and μ correspond to elasticity, piezoelectricity, piezomagneticity, dielectricity, magnetoelectricity, and permeability constant, respectively. x , y , z Represents coordinate variables.

5. The method for predicting the vibration characteristics of magnetoelectroelastic structures based on the overlapping finite element method as described in claim 4, characterized in that, When performing free vibration analysis on the magneto-electro-elastic structure without considering external excitation forces, the overlapping finite element governing equations are expressed as follows: When an external excitation force is applied and harmonic response analysis is performed on the magneto-electro-elastic structure, the overlapping finite element control equation is expressed as: in: 。 6. The method for predicting the vibration characteristics of magnetoelectroelastic structures based on the overlapping finite element method as described in any one of claims 1-5, characterized in that, Based on the overlapping finite element control equations, dynamic analysis was performed to obtain the vibration characteristic parameters of the magneto-electro-elastic structure, including: Based on the overlapping finite element control equations, the free vibration analysis of the magneto-electro-elastic structure is carried out to obtain the natural frequencies and mode shapes of the magneto-electro-elastic structure. Based on the overlapping finite element control equations, harmonic response analysis is performed on the magneto-electro-elastic structure to obtain the displacement harmonic response of the magneto-electro-elastic structure. Then, the electric potential and magnetomotive force responses of the magneto-electro-elastic structure are obtained based on the displacement harmonic response.

7. A system for predicting the vibration characteristics of magnetoelectroelastic structures based on the overlapping finite element method, characterized in that, Includes a processor, the processor being configured to execute the method for predicting the vibration characteristics of a magnetoelectroelastic structure based on the overlapping finite element method as described in any one of claims 1-6.

Images