A method for calculating the harmonic response of magneto-electroelastic structures using enhanced overlapping triangular elements

Abstract

The present invention discloses a method for calculating the harmonic response of a magneto-electro-elastic structure using enhanced overlapping triangular units, which relates to the technical field of magneto-electro-elastic intelligent materials. In the process of constructing a local enhanced interpolation basis function, a composite basis function consisting of a Lagrange polynomial basis function and a plane wave basis function is used for construction; in the process of constructing a local numerical approximation, a weighted function is constructed by introducing a mid-edge node in each enhanced overlapping triangular unit, and numerical interpolation is completed under the premise of satisfying the unit decomposition characteristics. Based on the numerical interpolation, a dynamic matrix control equation of the magneto-electro-elastic structure is established, and an enhanced overlapping finite element numerical model of the solver dynamic response is obtained. After applying appropriate boundary conditions and external excitations, a high-precision prediction of the harmonic response of the magneto-electro-elastic structure can be completed.

Description

Technical Field

[0001] The present invention relates to the technical field of magneto-electro-elastic smart materials, and in particular to a method for calculating the harmonic response of a magneto-electro-elastic structure by strengthening overlapping triangular units. Background Art

[0002] Magneto-electro-elastic materials are a typical type of multifunctional smart materials. Because magneto-electro-elastic smart structures have unique mechanical properties in certain specific fields, they have broad application prospects in mechanical engineering, transportation, electronic communications, aerospace, navigation, and national defense and military fields.

[0003] At present, analytical or semi-analytical methods are mainly used in engineering practice to study the dynamic performance of magneto-electro-elastic structures. For magneto-electro-elastic structures with complex geometric shapes in engineering practice, analytical methods are often not applicable and can only be solved by numerical methods.

[0004] The current standard finite element method is a commonly used numerical method for studying and analyzing the dynamic characteristics of magneto-electro-elastic structures. However, magneto-electro-elastic multilayer structures usually contain a large number of artificially designed discontinuous material interfaces. The material properties at these material interfaces will suddenly change, causing the corresponding physical fields to be distorted. Conventional numerical methods often fail when dealing with these strong gradient changes or discontinuities due to relatively low calculation accuracy or lack of sufficient calculation stability. Summary of the Invention

[0005] The purpose of the present invention is to provide a method for calculating the harmonic response of a magneto-electroelastic structure using enhanced overlapping triangular elements, which has the advantages of high efficiency, stability and accuracy.

[0006] To achieve the above object, the present invention provides a method for calculating the harmonic response of a magneto-electroelastic structure using enhanced overlapping triangular elements, comprising the following steps:

[0007] S1. Based on the constitutive equation of the magneto-electroelastic structure, the dynamic control equation of its discrete form is derived;

[0008] S2. Geometric modeling of the magneto-electroelastic structure is performed, and meshing is completed using a standard triangular unit mesh. Interpolation functions based on enhanced overlapping triangular finite element nodes and local high-order numerical approximations are constructed.

[0009] S3. The dynamic control equations of the magneto-electroelastic structure are derived in discrete form from the constitutive equations. Based on the control equations and the node shape functions of the reinforced overlapping triangles and the local high-order numerical approximation of the elements, a complete interpolation format of the reinforced overlapping triangle elements is obtained.

[0010] S4. Based on the standard Galerkin weighted residual method, an overlapping triangle finite element numerical discretization model for the magneto-electro-elastic smart structure dynamics problem is established and the matrix equation of the system is obtained;

[0011] S5. Apply specific boundary conditions, consider external simple harmonic excitation, and solve the established matrix equations for the magneto-electroelastic structure dynamics problem within a certain frequency range to complete the harmonic response analysis of the magneto-electroelastic structure.

[0012] S6. Based on the numerical approximation of the node coefficients and the three-phase displacement field of the magneto-electro-elastic structure obtained from the matrix equation of the magneto-electro-elastic structure dynamics problem, the three-phase physical field response at any point in the calculation domain as the calculation frequency changes is obtained by interpolation.

[0013] Preferably, S2 comprises the following steps:

[0014] S21. Divide the magneto-electroelastic structure into a standard triangular unit grid, where each node is the center node of a polygonal unit composed of all surrounding single triangular units associated with it, and each polygon overlaps with each other to form a reinforced overlapping triangular unit;

[0015] S22, overlapping triangular unit area, its internal position variables are expressed as the following interpolation format

[0016] u(x)=h I (x)ψ I (x)+h L (x)ψ L (x)+h M (x)ψ M (x);

[0017] Among them, h I (x),h L (x) and h M (x) are the shape functions of the standard linear triangle element at nodes I, L, and M, respectively, and ψ I (x),ψ L (x) and ψ M (x) are local numerical approximations at nodes I, L, and M, respectively;

[0018] S23. Generate virtual mid-edge nodes at the midpoints of each edge of each overlapping triangular element, and construct a weighting function in the local field approximation based on the shape function in the standard 6-node triangular element.

[0019] S23. Assign a set of high-order complete Langrange polynomial basis functions to the local analysis domain of each field node to construct the local field approximation ψ I (x), ψ L (x) and ψ M(x), and normalize it with the unit size to improve the stability of the system matrix;

[0020] S24, based on the unit decomposition characteristics and local field approximation ψ I (x), ψ L (x) and ψ M (x) Construct the unit interpolation function ρ in the overlapping triangle finite element I (x),ρ L (x) and ρ M (x).

[0021] Preferably, in S22, a composite basis function consisting of a Lagrange polynomial basis function and a plane wave basis function is selected as the basis function, and the enhanced interpolation basis function is:

[0022]

[0023] Among them, (x J ,y J ) is the spatial coordinate of node J, and h is the characteristic size of the overlapping triangular elements under consideration.

[0024] Preferably, the complete interpolation format of the enhanced overlapping triangle unit in S24 has the following form:

[0025]

[0026] in,

[0027]

[0028] is the weight function corresponding to node I, is the weight function corresponding to node M, is the weight function corresponding to node L, is the shape function of the 6-node quadratic triangular element, Indicates the node value of the weight function in the six nodes of the quadratic triangle element, u J is the degree of freedom at the corresponding node.

[0029] Preferably, the standard finite element interpolation expression of the three-phase different physical fields of the S3 three-dimensional magneto-electroelastic structure is:

[0030] u=[N u ]{U},φ=[N φ ]{Φ},ψ=[N ψ ]{Ψ};

[0031] Among them, U, Φ and Ψ are the node variables of displacement field, electric field and magnetic field respectively, N u,N φ and N ψ is the corresponding interpolation function.

[0032] Preferably, when analyzing the harmonic response of the magneto-electroelastic structure in S3, the boundary conditions are:

[0033]

[0034] Among them, B ψ , They represent the generalized displacements of three different physical fields, namely displacement field, electric field and magnetic field, corresponding to the intrinsic boundary conditions. They represent the surface force, voltage intensity and magnetic induction intensity corresponding to the three different physical fields of displacement field, electric field and magnetic field under natural boundary conditions respectively.

[0035] Preferably, the matrix form of the magneto-electroelastic structural dynamics control equation in S4 is as follows:

[0036]

[0037] in

[0038] M=∫ Ω N T ρNdΩ

[0039]

[0040] Among them, B u , B φ and B ψ are the gradient matrices of the displacement field, electric field and magnetic field respectively, U, Φ and Ψ are the node variables of the displacement field, electric field and magnetic field respectively, ρ represents the density of the magneto-electroelastic structure, C, q, e, m, ε and μ are related material constants.

[0041] Preferably, S7 comprises the following steps:

[0042] S71. Considering the response of the magneto-electroelastic structure under external excitation, using the matrix condensation technology, formula (24) is expressed as:

[0043]

[0044] Among them, Kεq represents the system equivalent stiffness matrix, is the amplitude of the solution, which is generally a function of the spatial position, is the imaginary unit, ω is the circular frequency, and F represents the external excitation force vector on the magneto-electroelastic structure;

[0045] S72. Given a specific frequency range, solve the above equation to obtain a response curve of the mechanical displacement response of the magneto-electroelastic structure to arbitrary frequency changes;

[0046] S73. After obtaining the response curve of the mechanical displacement response of the magneto-electroelastic structure to arbitrary displacement as the frequency changes, the response curves of the electric potential and magnetic potential response of the magneto-electroelastic structure as the frequency changes are obtained based on the formula, and finally the harmonic response analysis of the magneto-electroelastic structure under external excitation is completed.

[0047] Preferably, the external excitation in S7 includes mechanical excitation, electric field excitation or magnetic field excitation.

[0048] Therefore, the present invention adopts the above-mentioned method of calculating the harmonic response of a magneto-electroelastic structure using enhanced overlapping triangular elements. The node degrees of freedom of each constructed overlapping triangular element are not node displacement variables in the traditional finite element method, but local numerical approximations composed of local enhanced interpolation basis functions and coefficients to be calculated. In the process of constructing the local enhanced interpolation basis functions, a composite basis function composed of Lagrange polynomial basis functions and plane wave basis functions is used for construction. In the process of constructing the local numerical approximation, a weighting function is constructed by introducing mid-edge nodes in each overlapping triangular element. Numerical interpolation is completed under the premise of satisfying the element decomposition characteristics. Based on this numerical interpolation, the dynamic matrix governing equations of the magneto-electroelastic structure are established, and an enhanced overlapping finite element numerical model of the solver dynamic response is obtained. After applying appropriate boundary conditions and external excitations, a high-precision prediction of the dynamic response of the magneto-electroelastic structure can be completed. Since high-order interpolation is achieved without increasing the number of unit nodes, the computational efficiency and accuracy of the existing technology are improved, while having strong grid adaptability.

[0049] The technical solution of the present invention is further described in detail below through the accompanying drawings and embodiments. BRIEF DESCRIPTION OF THE DRAWINGS

[0050] Figure 1 Flow chart of the method of the present invention;

[0051] Figure 2 This is a schematic diagram of the magneto-electroelastic cantilever beam structure with holes of the present invention;

[0052] Figure 3 The model is obtained by discretizing the magneto-electroelastic cantilever beam structure with holes using a triangular unit grid;

[0053] Figure 4 Schematic diagram of the construction process of the reinforced overlapping triangular units based on the triangular unit grid of the present invention;

[0054] Figure 5 A relationship diagram between edge nodes and physical nodes in a triangle reinforced overlapping finite element of the present invention;

[0055] Figure 6 is the weight function W of the present invention J (x) and its defined circular support domain;

[0056] Figure 7 These are the harmonic response results of the displacement field and electric field at the measuring point of the magneto-electroelastic cantilever beam structure with a hole under the action of a simple harmonic excitation force. (a) is the displacement field, and (b) is the electric field. DETAILED DESCRIPTION

[0057] The technical solution of the present invention is further described below with reference to the accompanying drawings and embodiments.

[0058] Unless otherwise defined, the technical or scientific terms used in the present invention shall have the usual meanings understood by persons of ordinary skill in the field to which the present invention belongs. The words "first", "second" and similar terms used in the present invention do not indicate any order, quantity or importance, but are only used to distinguish different components. Words such as "include" or "comprise" mean that the elements or objects preceding the word include the elements or objects listed after the word and their equivalents, without excluding other elements or objects. Words such as "connect" or "connected" are not limited to physical or mechanical connections, but may include electrical connections, whether direct or indirect. "Up", "down", "left", "right" and the like are only used to indicate relative positional relationships. When the absolute position of the object being described changes, the relative positional relationship may also change accordingly.

[0059] Example

[0060] In a specific embodiment of the present invention, a magneto-electroelastic cantilever beam structure with a hole is used in a plane stress state. The magneto-electroelastic structure is composed of BaTiO3 with a volume fraction of 60% and CoFe2O4 with a volume fraction of 40%. The geometric dimensions of the magneto-electroelastic cantilever beam structure with a hole are as follows: Figure 2 The present invention can effectively predict the dynamic response of magneto-electroelastic structures of various complex geometric shapes under the action of simple harmonic excitation forces.

[0061] Reference Figure 1 As shown, the present invention provides a prediction method for harmonic response analysis of magneto-electro-elastic smart material structures using reinforced overlapping triangular units, comprising the following steps:

[0062] S1: Based on the constitutive equation of the magneto-electroelastic structure, the dynamic control equation of its discrete form is derived;

[0063] S2: Geometric modeling of the magneto-electroelastic smart beam structure with two ends fixed is performed, and the grid division is completed using the standard triangular unit grid (such as Figure 3 and setting the corresponding physical parameters of the material and medium for the magneto-electroelastic structure grid;

[0064] A composite basis function consisting of Lagrange polynomials and plane wave functions is selected as the local enhanced interpolation basis function. The triangle unit node interpolation function is constructed based on the unit decomposition property. In the process of constructing the interpolation function, virtual triangle unit mid-edge nodes are introduced to complete the node shape function and local high-order numerical approximation of the enhanced overlapping triangle units.

[0065] S3: Based on the governing equations of magneto-electroelastic structural dynamics, node shape functions, and local high-order numerical approximations of the elements, a complete interpolation format for reinforced overlapping triangular elements can be obtained;

[0066] S4: Based on the standard Galerkin weighted residual method, an enhanced overlapping triangle finite element numerical discretization model for the dynamics problem of magneto-electroelastic smart structures is established and the matrix equation of the system is obtained;

[0067] S5: Apply specific boundary conditions, consider external excitations (including mechanical excitation, electric field excitation and magnetic field excitation), give a certain frequency range, solve the established matrix equations of the magneto-electroelastic structure dynamics problem, and complete the harmonic response analysis (forced vibration analysis) of the magneto-electroelastic structure.

[0068] S6: Based on the node coefficients obtained by solving the matrix equations of the magneto-electroelastic structure dynamics problem and the numerical approximation of the three-phase displacement field of the magneto-electroelastic structure, the three-phase physical field response (including mechanical displacement response, electric field response and magnetic field response) at any point in the calculation domain as the calculation frequency changes can be obtained by interpolation.

[0069] The following is a detailed description of each of the above steps:

[0070] Based on the constitutive equation of the magneto-electroelastic structure, the discrete form of the dynamic control equation S11 is derived: For an ideal three-dimensional linear magneto-electroelastic structure, which has obvious magneto-electro-elastic coupling characteristics, the corresponding constitutive control equation is:

[0071]

[0072] Among them, σ k is the stress component, S k is the strain component, C jk are the corresponding elastic constants (j, k = 1, 2, 3, ..., 6), D l represents the generalized displacement component of the electric field, E l represents the electric field intensity component (l=1,2,3), B m represents the generalized displacement component of the magnetic field, H m represents the magnetic field intensity component (m=1,2,3), ε lm , μ ml , e kl ,q km and mlm are the corresponding dielectric constant, permeability constant, piezoelectric constant, piezomagnetic constant and magnetoelectric constant respectively.

[0073] The strain-displacement relationship of the three-dimensional magneto-electroelastic structure is expressed as:

[0074]

[0075] Here, u, v, and w represent the displacements along the x, y, and z directions in the Cartesian coordinate system.

[0076] The relationship between the electric potential φ and the electric field vector E of the three-dimensional magneto-electroelastic structure can be expressed as

[0077]

[0078] The relationship between the magnetic potential ψ and the electric field vector H of the three-dimensional magneto-electroelastic structure can be expressed as

[0079]

[0080] S12: Substituting formulas (2)-(4) into formula (1), the constitutive equation of the magneto-electroelastic structure can be expressed in the following matrix form:

[0081]

[0082]

[0083] S13: The standard finite element interpolation of the three-phase different physical fields of the three-dimensional magneto-electroelastic structure can be expressed as

[0084] u=[N u ]{U},φ=[N φ ]{Φ},ψ=[N ψ ]{Ψ}(8)

[0085] Among them, U, Φ and Ψ are the node variables of displacement field, electric field and magnetic field respectively, N u ,N φ and N ψ is the corresponding interpolation function when interpolating different physical fields.

[0086] For a two-dimensional structure under plane stress state, using standard linear triangular elements, the gradient matrix for different physical fields can be expressed as

[0087]

[0088] Among them, B u , B φ and B ψ are the gradient matrices of the displacement field, electric field, and magnetic field, respectively.

[0089] When performing harmonic response analysis of magneto-electroelastic structures, corresponding boundary conditions must be applied. Common boundary conditions in engineering practice are:

[0090]

[0091] Where B ψ , They represent the generalized displacements of three different physical fields, namely displacement field, electric field and magnetic field, corresponding to the intrinsic boundary conditions. They represent the surface force, voltage intensity and magnetic induction intensity corresponding to the three different physical fields of displacement field, electric field and magnetic field under natural boundary conditions respectively.

[0092] When the magneto-electroelastic structure is not subject to surface forces, external electric fields, and external magnetic fields, Equation (13) can be expressed as

[0093]

[0094] Where, Represents the second derivative of the displacement vector with respect to time.

[0095] Apply the Galerkin weighted residual method, multiply Equation (14) by a test function, and integrate it over the entire problem domain Ω

[0096]

[0097] Where t represents the test function.

[0098] Based on Green's theorem and using integration by parts, we can get

[0099]

[0100] Substituting the constitutive relation shown in formula (1) and the boundary conditions in formula (12) and formula (13) into formula (16), we can obtain

[0101]

[0102] Substituting the interpolation format shown in formula (8) into formula (17), we can get

[0103]

[0104] Based on formula (16)-formula (18), we can get

[0105]

[0106] Where C, ε, μ, e, q and m are the elastic constant matrix, dielectric constant matrix, permeability constant matrix, piezoelectric constant matrix, piezomagnetic constant matrix and magnetoelectric constant matrix, respectively.

[0107] Based on formula (19), the governing equations of magneto-electroelastic structural dynamics can be written in the following matrix form:

[0108]

[0109] in

[0110]

[0111] where ρ represents the density of the magneto-electroelastic structure.

[0112] S2 is discretized using standard three-node triangular elements, such as Figure 4 The magneto-electroelastic structure used in this specific embodiment is composed of 60% by volume of BaTiO3 and 40% by volume of CoFe2O4. The corresponding material constant matrix is shown in Table 1.

[0113] Table 1 Material constants of CoFe2O4-BaTiO3 magneto-electroelastic structure

[0114]

[0115]

[0116] The magneto-electroelastic structure is divided into standard triangular unit grids (such as Figure 4 As shown), for node I, the 5 triangle units associated with node I together form a 6-node polygon unit centered on node I; similarly, the 6 triangle units associated with node L together form a 7-node polygon unit centered on node L; and the 7 triangle units associated with node M together form an 8-node polygon unit centered on node M. In a standard triangular mesh, each node can be regarded as the central node of a polygon unit composed of all the surrounding single triangle units associated with it. Here we define the overlapping parts of each polygon as the reinforced overlapping triangle unit to be constructed. For example, Figure 4 The ILM region composed of nodes I, L and M is an enhanced overlapping triangle unit to be discussed.

[0117] For overlapping triangular element regions, the internal physical field variables can be expressed as the following interpolation format:

[0118] u(x)=h I (x)ψ I (x)+h L (x)ψ L (x)+h M (x)ψ M (x)(22)

[0119] Among them, h I (x),h L (x) and h M (x) is the shape function in the standard linear triangle element, ψ I (x),ψ L (x) and ψ M (x) is a local numerical approximation.

[0120] The local numerical approximation at node I can be expressed as

[0121]

[0122] Among them, u J is the degree of freedom at the corresponding node, The weight function corresponding to node I.

[0123] Degree of freedom u at node J J It can usually be expressed as

[0124] u J =p1a J,1 +p2a J,2 +p3a J,3 +p4a J,4 +…+p n a J,n =p n a J,n (twenty four)

[0125] Among them, p n represents a suitable basis function, a J,n represents the corresponding solution coefficients.

[0126] When interpolating different physical fields, different basis functions can be used, or the same basis function can be used. The computational efficiency and accuracy are different when using different basis functions. A composite basis function composed of Lagrange polynomials and plane wave functions is selected as the basis function of the enhancement function, as follows:

[0127]

[0128] in, is the imaginary unit, k is the wave number, q represents the number of selected plane wave basis functions propagating in different directions, n = 0, 1, 2, …, q-1.

[0129] In actual numerical calculations, the higher the order of the selected basis function, the higher the calculation accuracy obtained, but the scale of the obtained system matrix and the amount of calculation required for related matrix operations will be larger. This specific embodiment only uses the first-order Lagrange polynomial basis function and the plane wave function (q=8) containing 8 propagating directions as the basis functions of the enhanced interpolation function, but in fact, when calculating specific engineering problems, the order of the Lagrange polynomial and the number of plane wave functions can be flexibly selected according to the accuracy requirements; in addition, in order to reduce the condition number of the obtained system matrix and improve the stability of the numerical calculation, the composite basis function composed of the Lagrange polynomial and the plane wave function must be dimensionlessly operated. Therefore, the enhanced interpolation basis function selected when interpolating the displacement field, electric field and magnetic field is

[0130]

[0131] Among them, (x J ,y J ) is the spatial coordinate of node J, and h is the characteristic size of the overlapping triangular elements under consideration.

[0132] Generate virtual mid-edge nodes (such as Figure 5 As shown), based on the shape function in the standard 6-node triangular element, the weight function in the local field approximation is constructed Weighting function It can be obtained by interpolating the shape function in the standard 6-node triangle element and the node value of the shape function in each node. By introducing three virtual mid-edge nodes, the weight function It can be expressed as

[0133]

[0134] Where, is the shape function of the 6-node quadratic triangular element, Represents the node value of node i.

[0135] Based on the finite sphere meshless technology, the weight function node value of node i It can be expressed as

[0136]

[0137] Among them, W J (x) is the weight function defined as follows

[0138]

[0139] Among them, the parameter s = d J / αr J, d J represents the distance from node J to any point x in the computational domain, α is a constant and 1≤α≤2, r J Indicates the radius of the circle centered at node J. In actual calculations, r J .It must be set large enough to ensure that the circle with node J as the center can cover the overlapping unit ILM (such as Figure 6 shown).

[0140] In order to meet the unit decomposition properties and fragmentation test, the weight function node value The values of are shown in Table 2.

[0141] Table 2 Node values of weight function in nodal triangle elements The value of

[0142]

[0143] Substituting formula (27) into formula (23), we can get

[0144]

[0145] Similarly, for nodes L and M, we can get

[0146]

[0147]

[0148] Substituting equations (30)-(32) into equation (22), we can obtain

[0149]

[0150] in,

[0151]

[0152] Taking the magneto-electroelastic structure in a plane stress state as an example, the expression of the corresponding material constant matrix is shown in Table 3.

[0153]

[0154]

[0155] In order to reduce the size of the matrix equation, according to the matrix condensation technology, method (20) can be simplified to

[0156]

[0157] in,

[0158]

[0159] Introducing the corresponding boundary conditions, the node displacement vector of the magneto-electroelastic structure can be obtained by solving the matrix equation (35). The node electric potential vector and node magnetic potential vector can be obtained by the following equations:

[0160]

[0161] For the free vibration problem of magneto-electroelastic structure, Equation (35) has the following general solution:

[0162]

[0163] in, is the amplitude of the solution, which is generally a function of the spatial position, is the imaginary unit, ω is the circular frequency, and t is the time.

[0164] Considering the response of the magneto-electroelastic structure under external excitation (such as mechanical excitation, electric field excitation or magnetic field excitation), formula (35) can be expressed as

[0165]

[0166] Where Kεq represents the equivalent stiffness matrix of the system, and F represents the external excitation force vector acting on the magneto-electroelastic structure.

[0167] Given a specific frequency range, solving formula (39) can obtain the response curve of the mechanical displacement response of the magneto-electroelastic structure with arbitrary displacement and arbitrary frequency change.

[0168] After obtaining the response curve of the mechanical displacement response of the magneto-electroelastic structure with frequency changes for any displacement, the response curves of the electric potential and magnetic potential responses of the magneto-electroelastic structure with frequency changes can be obtained at once based on formula (37), and finally the harmonic response analysis of the magneto-electroelastic structure under external excitation is completed, in which the displacement response, electric potential response and magnetic field response at the measuring point are obtained.

[0169] Based on the node coefficients obtained by solving the matrix equations of the magneto-electroelastic structural dynamics problem and the numerical approximation of the three-phase displacement field of the magneto-electroelastic structure, the three-phase physical field response of any point in the calculation domain as the calculation frequency changes can be obtained by interpolation. The response results obtained include but are not limited to mechanical displacement response, electric field response and magnetic field response, stress response cloud map, strain response cloud map, etc.

[0170] Figure 7The present invention provides the calculation results of the harmonic response of the displacement field and electric field of the magneto-electroelastic cantilever beam structure with a hole under the action of a unit simple harmonic excitation force, using the overlapping finite element of the triangular element in the standard finite element method and only using the first-order Lagrange polynomial as the basis function, and the enhanced overlapping finite element method given in the present invention using the composite function composed of Lagrange polynomials and plane wave functions as the basis function under exactly the same grid and number of nodes (546 nodes). The reference solution is the calculation result using high-order finite elements under extremely fine grid (11191 nodes). It can be seen that the enhanced overlapping finite element method given in the present invention is Under the same grid node distribution, the finite element method can not only calculate results that are closer to the reference solution and more accurate than the triangular elements in the standard finite element method, but also obtain harmonic response analysis results with better accuracy than those obtained by the general overlapping finite element method. The main reason for this is that when constructing the enhanced overlapping finite element based on the triangular grid provided by the present invention, a composite function consisting of Lagrange polynomials and plane wave functions is used as the basis function. The plane wave function added to the basis function can significantly improve the solution accuracy of the enhanced overlapping finite element provided by the present invention when performing harmonic response analysis of magneto-electroelastic structures; Figure 7 The harmonic response analysis results given also show that the enhanced overlapping finite element method given in the present invention has higher solution accuracy than the standard finite element method and the general overlapping finite element method when using the same grid to calculate the harmonic response of the magneto-electroelastic structure under the action of simple harmonic excitation force, and has great potential application value in engineering practice.

[0171] Therefore, the present invention adopts the above-mentioned method of calculating the harmonic response of a magneto-electroelastic structure using enhanced overlapping triangular elements. The node degrees of freedom of each constructed overlapping triangular element are not node displacement variables in the traditional finite element method, but local numerical approximations composed of local enhanced interpolation basis functions and coefficients to be calculated. In the process of constructing the local enhanced interpolation basis functions, a composite basis function composed of Lagrange polynomial basis functions and plane wave basis functions is used for construction. In the process of constructing the local numerical approximation, a weighting function is constructed by introducing mid-edge nodes in each overlapping triangular element. Numerical interpolation is completed under the premise of satisfying the element decomposition characteristics. Based on this numerical interpolation, the dynamic matrix governing equations of the magneto-electroelastic structure are established, and an enhanced overlapping finite element numerical model of the solver dynamic response is obtained. After applying appropriate boundary conditions and external excitations, a high-precision prediction of the dynamic response of the magneto-electroelastic structure can be completed. Since high-order interpolation is achieved without increasing the number of unit nodes, the computational efficiency and accuracy of the existing technology are improved, while having strong grid adaptability.

[0172] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention rather than to limit the same. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that they can still modify or replace the technical solutions of the present invention with equivalents, and these modifications or equivalent replacements cannot cause the modified technical solutions to deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. A method for calculating the harmonic response of magneto-electroelastic structures using enhanced overlapping triangular elements, characterized in that: The following steps are involved: S1. Based on the constitutive equation of the magneto-electroelastic structure, the dynamic control equation of its discrete form is derived; S2. Geometric modeling of the magneto-electroelastic structure is performed, and meshing is completed using a standard triangular unit mesh. Interpolation functions based on enhanced overlapping triangular finite element nodes and local high-order numerical approximations are constructed. S3. The dynamic control equations of the magneto-electroelastic structure are derived in discrete form from the constitutive equations. Based on the control equations and the node shape functions of the reinforced overlapping triangles and the local high-order numerical approximation of the elements, a complete interpolation format of the reinforced overlapping triangle elements is obtained. S4. Based on the standard Galerkin weighted residual method, an overlapping triangle finite element numerical discretization model for the magneto-electro-elastic smart structure dynamics problem is established and the matrix equation of the system is obtained; S5. Apply specific boundary conditions, consider external simple harmonic excitation, and solve the established matrix equations for the magneto-electroelastic structure dynamics problem within a certain frequency range to complete the harmonic response analysis of the magneto-electroelastic structure. S6. Based on the numerical approximation of the node coefficients and the three-phase displacement field of the magneto-electro-elastic structure obtained from the matrix equation of the magneto-electro-elastic structure dynamics problem, the three-phase physical field response at any point in the calculation domain as the calculation frequency changes is obtained by interpolation.

2. The method for calculating the harmonic response of a magneto-electroelastic structure using enhanced overlapping triangular elements according to claim 1, characterized in that: S2 includes the following steps: S21. Divide the magneto-electroelastic structure into a standard triangular unit grid, where each node is the center node of a polygonal unit composed of all surrounding single triangular units associated with it, and each polygon overlaps with each other to form a reinforced overlapping triangular unit; S22, overlapping triangular unit area, its internal position variables are expressed as the following interpolation format u(x)=h I (x)ψ I (x)+h L (x)ψ L (x)+h M (x)ψ M (x); Among them, h I (x),h L (x) and h M (x) are the shape functions of the standard linear triangle element at nodes I, L, and M, respectively, and ψ I (x),ψ L (x) and ψ M (x) are local numerical approximations at nodes I, L, and M, respectively; S23. Generate virtual mid-edge nodes at the midpoints of each edge of each overlapping triangular element, and construct a weighting function in the local field approximation based on the shape function in the standard 6-node triangular element. S23. Assign a set of high-order complete Langrange polynomial basis functions to the local analysis domain of each field node to construct the local field approximation ψ I (x), ψ L (x) and ψ M (x), and normalize it with the unit size to improve the stability of the system matrix; S24, based on the unit decomposition characteristics and local field approximation ψ I (x), ψ L (x) and ψ M (x) Construct the unit interpolation function ρ in the overlapping triangle finite element I (x),ρ L (x) and ρ M (x).

3. The method for calculating the harmonic response of a magneto-electroelastic structure using enhanced overlapping triangular elements according to claim 2, characterized in that: In S22, a composite basis function consisting of Lagrange polynomial basis functions and plane wave basis functions is selected as the basis function, and the enhanced interpolation basis function is: Among them, (x J ,y J ) is the spatial coordinate of node J, and h is the characteristic size of the overlapping triangular elements under consideration.

4. The method for calculating the harmonic response of a magneto-electroelastic structure using enhanced overlapping triangular elements according to claim 3, characterized in that: The complete interpolation format for enhanced overlapping triangle elements in S24 has the following form: in, φ J I is the weight function corresponding to node I, is the weight function corresponding to node M, is the weight function corresponding to node L, is the shape function of the 6-node quadratic triangular element, Indicates the node value of the weight function in the six nodes of the quadratic triangle element, u J is the degree of freedom at the corresponding node.

5. The method for calculating the harmonic response of a magneto-electroelastic structure using enhanced overlapping triangular elements according to claim 4, characterized in that: The standard finite element interpolation expression of the three-phase different physical fields of the S3 three-dimensional magneto-electroelastic structure is: u=[N u ]{U},φ=[N φ ]{Φ},ψ=[N ψ ]{Ψ}; Among them, U, Φ and Ψ are the node variables of displacement field, electric field and magnetic field respectively, N u ,N φ and N ψ is the corresponding interpolation function.

6. The method for calculating the harmonic response of a magneto-electroelastic structure using enhanced overlapping triangular elements according to claim 5, characterized in that: When analyzing the harmonic response of the magneto-electroelastic structure in S3, the boundary conditions are: Among them, B ψ , They represent the generalized displacements of three different physical fields, namely displacement field, electric field and magnetic field, corresponding to the intrinsic boundary conditions. They represent the surface force, voltage intensity and magnetic induction intensity corresponding to the three different physical fields of displacement field, electric field and magnetic field under natural boundary conditions respectively.

7. The method for calculating the harmonic response of a magneto-electroelastic structure using enhanced overlapping triangular elements according to claim 6, characterized in that: The matrix form of the governing equations of magneto-electroelastic structural dynamics in S4 is as follows: in M=∫ Ω N T ρNdΩ; Among them, B u , B φ and B ψ are the gradient matrices of the displacement field, electric field and magnetic field respectively, U, Φ and Ψ are the node variables of the displacement field, electric field and magnetic field respectively, ρ represents the density of the magneto-electroelastic structure, C, q, e, m, ε and μ are related material constants.

8. The method for calculating the harmonic response of a magneto-electroelastic structure using enhanced overlapping triangular elements according to claim 7, characterized in that: S7 includes the following steps: S71. Considering the response of the magneto-electroelastic structure under external excitation, the matrix equation is expressed as follows after using the matrix condensation technique: Among them, Kεq represents the equivalent stiffness matrix of the system, is the amplitude of the solution, which is generally a function of the spatial position, is the imaginary unit, ω is the circular frequency, and F represents the external excitation force vector on the magneto-electroelastic structure; S72. Given a specific frequency range, solve the above equation to obtain a response curve of the mechanical displacement response of the magneto-electroelastic structure to arbitrary frequency changes; S73. After obtaining the response curve of the mechanical displacement response of the magneto-electroelastic structure to arbitrary displacement as the frequency changes, the response curves of the electric potential and magnetic potential response of the magneto-electroelastic structure as the frequency changes are obtained based on the formula, and finally the harmonic response analysis of the magneto-electroelastic structure under external excitation is completed.

9. The method for calculating the harmonic response of a magneto-electroelastic structure using enhanced overlapping triangular elements according to claim 8, characterized in that: The external excitations of the magneto-electroelastic structure considered in S7 include mechanical excitation, electric field excitation and magnetic field excitation.

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