A method for calculating the dynamic response of magneto-electroelastic structures based on overlapping triangular elements

Abstract

The present invention discloses a method for calculating the dynamic response of a magneto-electroelastic structure based on overlapping triangular elements, which relates to the technical field of dynamic response prediction. For each overlapping triangular element, the node degrees of freedom are locally numerically approximated by a local enhanced interpolation basis function and a coefficient to be determined. During the construction of the local numerical approximation, a weighted function is constructed by introducing mid-edge nodes in each overlapping triangular element. Numerical interpolation is performed while satisfying the element decomposition characteristics. Based on this numerical interpolation, the governing matrix equations for the dynamics of the magneto-electroelastic structure are established, resulting in an overlapping finite element numerical model of the solver's dynamic response. After applying appropriate boundary conditions and external excitation, a high-precision prediction of the dynamic response of the magneto-electroelastic structure can be achieved. The present invention utilizes the above-mentioned method for calculating the dynamic response of a magneto-electroelastic structure based on overlapping triangular elements, improving the computational efficiency and accuracy of the prior art while also having strong mesh adaptability.

Description

Technical Field

[0001] The present invention relates to the technical field of dynamic response prediction, and in particular to a method for calculating the dynamic response of a magneto-electroelastic structure based on overlapping triangular units. Background Art

[0002] Magneto-electro-elastic materials are a typical type of multifunctional smart materials. Due to their simultaneous possession of multiple physical effects such as piezoelectric, piezo-magnetic and magneto-electric, magneto-electro-elastic structures can realize the mutual conversion between electrical energy, magnetic energy and mechanical energy under external interference. Through carefully designed magneto-electro-elastic structures, they can achieve exotic properties that conventional materials do not have, such as band gap, negative mass, negative modulus, negative refraction, acoustic focusing and acoustic stealth, etc. Common magneto-electro-elastic smart structures mainly include magneto-electro-elastic multilayer structures and magneto-electro-elastic functional gradient structures. Due to the unique mechanical properties of magneto-electro-elastic smart structures in certain specific fields, they have broad application prospects in mechanical engineering, transportation, electronic communications, aerospace and 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. Although accurate solutions to magneto-electro-elastic structural dynamic problems can be obtained through analytical or semi-analytical and semi-numerical methods, they are generally only effective for relatively simple geometric shapes (rectangular beams and rectangular plates) and boundary conditions (simply supported and free). 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 structures are a complex engineering problem involving multiple physical fields. Their calculation and analysis often involve the interaction and entanglement of different physical fields such as electric fields, magnetic fields, and force fields. Their dynamic characteristics are far more complex than those of pure elastic mechanics problems. The calculation accuracy of the standard finite element method is often affected by the quality of the mesh. When calculating and analyzing complex engineering problems such as magneto-electro-elastic multi-physics fields, the calculation accuracy and efficiency of the traditional finite element method are relatively low.

[0005] The dynamics of magneto-electro-elastic multilayer smart structures is a typical and complex multi-physics coupling problem (involving electric fields, magnetic fields, force fields, etc.), and the physical fields of different properties influence and couple with each other. Although it can be solved using the standard finite element method, magneto-electro-elastic multilayer structures usually contain a large number of artificially designed discontinuous material interfaces. At these material interfaces, the material properties will suddenly change, causing the corresponding physical fields to be distorted. Conventional numerical methods often fail to deal with these strong gradient changes or discontinuities due to relatively low computational accuracy or lack of sufficient computational stability. Therefore, there is an urgent need to develop efficient, stable and accurate advanced numerical techniques to study the dynamics of magneto-electro-elastic multilayer smart structures. Summary of the Invention

[0006] The purpose of the present invention is to provide a method for calculating the dynamic response of a magneto-electroelastic structure based on overlapping triangular units, thereby improving the calculation efficiency and accuracy of the prior art.

[0007] To achieve the above object, the present invention provides a method for calculating the dynamic response of a magneto-electroelastic structure based on overlapping triangular elements, comprising the following steps:

[0008] S1. Based on the constitutive equation of the magneto-electroelastic structure, the dynamic control equation of its discrete form is derived;

[0009] S2. Geometric modeling of the magneto-electroelastic structure and meshing using a standard triangular unit grid;

[0010] S3, setting the corresponding physical parameters of the material and medium for the magneto-electroelastic structure grid obtained in S2;

[0011] S4. Select an appropriate local enhanced interpolation basis function, construct a triangle unit node interpolation function based on the unit decomposition property, introduce virtual triangle unit mid-edge nodes in the process of constructing the interpolation function, and complete the node shape function and local high-order numerical approximation of the overlapping triangle units;

[0012] S5. Based on the standard "Galerkin" weighted residual method, a complete interpolation format of overlapping triangular elements is used to establish a numerical discrete model of the magneto-electroelastic smart structure dynamics problem and obtain the matrix equation of the system;

[0013] S6. Apply specific boundary conditions and solve the established matrix equations for the magneto-electro-elastic structure dynamics problem without considering external forces, obtain the natural frequencies of each order of the magneto-electro-elastic structure and the corresponding natural vibration modes, and complete the free vibration analysis of the magneto-electro-elastic structure;

[0014] S7. Apply specific boundary conditions, consider external excitation, and within a certain frequency range, solve the established matrix equations for the magneto-electroelastic structure dynamics problem to complete the harmonic response analysis of the magneto-electroelastic structure.

[0015] S8. Based on the node coefficients and the numerical approximation of the three-phase displacement field of the step magneto-electroelastic structure, the three-phase physical field response of any point in the calculation domain as the calculation frequency changes is obtained by interpolation.

[0016] Preferably, in S2, the magneto-electroelastic structure is divided into a standard triangular unit grid, each node is a central node of a polygonal unit composed of all surrounding single triangular units related to it, and the overlapping parts of the polygonal units are constructed overlapping triangular units.

[0017] Preferably, S4 comprises the following steps:

[0018] S41. 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] S42. Weighting function The shape function in the standard 6-node triangle element is interpolated with the nodal value of the shape function at each node; the nodal value of the shape function Constructed by Shepard function;

[0020] S43. 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;

[0021] S44, 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).

[0022] Preferably, in S4, the basis function of the Lagrange polynomial strengthening function is used to perform dimensionless operation, and the specific operation method is:

[0023]

[0024] Among them, (xJ ,y J ) is the spatial coordinate of node J, and h is the characteristic size of the overlapping triangular elements under consideration.

[0025] Preferably, the local field of node I in S43 is approximately ψ I (x) can be expressed as:

[0026]

[0027] Among them, u J is the degree of freedom at the corresponding node, The weight function corresponding to node I, J = I, L, M are the nodes of the triangular unit, and x = (x, y) are the spatial coordinates;

[0028] Degree of freedom u at node J J It can usually be expressed as

[0029] u J =p1a J,1 +p2a J,2 +p3a J,3 +p4a J,4 +…+p n a J,n =p n a J,n ;

[0030] Among them, p n represents a suitable reinforcement function basis function, a J,n represents the corresponding solution coefficients.

[0031] Preferably, the complete interpolation format for overlapping triangle elements has the following form:

[0032]

[0033] in,

[0034]

[0035] 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, 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, Represents the node value of the weight function at node i.

[0036] Preferably, the matrix form of the magneto-electroelastic structural dynamics control equation in S5 is as follows:

[0037]

[0038] in,

[0039] M=∫ Ω N T ρNdΩ;

[0040]

[0041] 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, and ρ represents the density of the magneto-electroelastic structure, MTU.

[0042] Preferably, S6 comprises the following steps:

[0043] S61. According to the matrix condensation technology, the size of the matrix equation is reduced and simplified to

[0044]

[0045] Among them, Kεq represents the equivalent stiffness matrix of the system;

[0046] S62. Introduce corresponding boundary conditions to obtain the node displacement vector, node electric potential vector, and node magnetic potential vector of the magneto-electroelastic structure;

[0047] S63. For the free vibration problem of magneto-electroelastic structure, the above equation has the following general solution:

[0048]

[0049] 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;

[0050] S64. The natural frequency and natural mode model of the magneto-electroelastic structure during free vibration are:

[0051]

[0052] Solve this equation to complete the free vibration analysis of the magneto-electroelastic structure.

[0053] Preferably, the external excitation in S7 includes mechanical excitation, electric field excitation or magnetic field excitation.

[0054] Preferably, in S7, the response of the magneto-electroelastic structure under external excitation (such as mechanical excitation, electric field excitation or magnetic field excitation) is considered, and the formula is:

[0055]

[0056] Where F represents the external excitation force vector acting on the magneto-electroelastic structure.

[0057] Therefore, the present invention adopts the above-mentioned method for calculating the dynamic response of magneto-electro-elastic structures based on overlapping triangular units. This method can realize high-order numerical interpolation without increasing unit nodes, and can use different enhanced interpolation basis functions to interpolate different physical fields according to the characteristics of different physical fields. Ultimately, even when using a coarse grid, very high-precision calculation results can still be obtained; in addition, this method also has strong grid adaptability, and even if an irregular calculation grid is used, relatively reliable calculation results can still be obtained. The method of the present invention can be used to realize high-precision numerical prediction of the dynamic response problem of magneto-electro-elastic intelligent elastic structures.

[0058] The technical solution of the present invention is further described in detail below through the accompanying drawings and embodiments. BRIEF DESCRIPTION OF THE DRAWINGS

[0059] Figure 1 Flow chart of the method of the present invention;

[0060] Figure 2 Schematic diagram of the magneto-electroelastic beam structure with two ends fixed in an embodiment of the present invention;

[0061] Figure 3 The model is obtained by discretizing the magneto-electroelastic beam structure with two ends clamped by the present invention using a triangular unit grid;

[0062] Figure 4 Schematic diagram of the construction process of overlapping triangle units based on a triangle unit grid in a Cartesian coordinate system of the present invention;

[0063] Figure 5 A relationship diagram between edge nodes and physical nodes in a triangle overlapping finite element in a Cartesian coordinate system of the present invention;

[0064] Figure 6 is the weight function W of the present invention J (x) and its defined circular support domain;

[0065] Figure 7 Schematic diagram of the first 10 modal vibration shapes of the magneto-electroelastic beam structure with two ends fixed

[0066] Figure 8Response curves of the magneto-electroelastic beam structure with two ends fixed at the present invention at the measuring point under the action of simple harmonic excitation force, (a) is the displacement response, (b) is the electric field response, and (c) is the magnetic field response. DETAILED DESCRIPTION

[0067] The technical solution of the present invention is further described below with reference to the accompanying drawings and embodiments.

[0068] 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" and the like 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" and the like 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.

[0069] Example

[0070] In the present invention, a magneto-electroelastic beam structure with two ends clamped in a plane stress state is used. 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 smart beam structure with two ends clamped are as follows: Figure 2 The present invention can effectively predict the dynamic response (including free vibration response and harmonic response under external excitation) of magneto-electroelastic structures of various complex geometric shapes.

[0071] Reference Figure 1 As shown, the present invention provides a method for predicting the dynamic response of a magneto-electroelastic structure based on an overlapping triangle finite element method, comprising the following steps:

[0072] S1: Based on the constitutive equation of the magneto-electroelastic structure, the dynamic control equation of its discrete form is derived.

[0073] S11: For an ideal three-dimensional linear magneto-electroelastic structure, which has obvious magneto-electro-elastic coupling characteristics, the corresponding constitutive governing equation is:

[0074]

[0075] Among them, σ k is the stress component, S k is the strain component, C jkare 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 m lm are the corresponding dielectric constant, permeability constant, piezoelectric constant, piezomagnetic constant and magnetoelectric constant respectively.

[0076] The strain-displacement relationship of the three-dimensional magneto-electroelastic structure is expressed as:

[0077]

[0078] Here, u, v, and w represent the displacements along the x, y, and z directions in the Cartesian coordinate system.

[0079] The relationship between the electric potential φ and the electric field vector E of the three-dimensional magneto-electroelastic structure can be expressed as

[0080]

[0081] The relationship between the magnetic potential ψ and the electric field vector H of the three-dimensional magneto-electroelastic structure can be expressed as

[0082]

[0083] S12: Substituting formulas (2)-(4) into formula (1), the constitutive equation of the magneto-electroelastic structure can be expressed in the following matrix form:

[0084]

[0085] The standard finite element interpolation of the three-phase different physical fields of the three-dimensional magneto-electroelastic structure can be expressed as

[0086] u=[N u ]{U}, φ=[N φ ]{Φ},ψ=[N ψ ]{Ψ} (8)

[0087] 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.

[0088] 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

[0089]

[0090] Among them, B u , B φ and B ψ are the gradient matrices of the displacement field, electric field, and magnetic field, respectively.

[0091] For magneto-electroelastic materials, since they involve three physical fields: displacement field, electric field, and magnetic field, their energy functional can be expressed as a function of these three physical fields, namely,

[0092]

[0093] Among them, S is the strain vector, E is the electric field vector, H is the magnetic field vector, 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.

[0094] According to different physical fields, appropriate interpolation functions are selected, and based on the Galerkin weighted residual method, the magneto-electroelastic structural dynamics governing equations can be written in the following matrix form:

[0095]

[0096] in

[0097]

[0098] where ρ represents the density of the magneto-electroelastic structure.

[0099] S2: Geometric modeling of the magneto-electroelastic smart beam structure with two ends clamped is performed, and the grid division is completed using the standard triangular unit grid, such as Figure 3 shown.

[0100] 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 overlapping triangle units to be constructed. For example, Figure 4 The ILM region composed of nodes I, L and M is an overlapping triangle unit to be discussed.

[0101] S3: Setting the corresponding physical parameters of the material and medium for the magneto-electroelastic structure grid obtained in step S2, as shown in Table 1.

[0102] Table 1 Material constants of CoFe2O4-BaTiO3 magneto-electroelastic structure

[0103] Material properties Material constants <![CDATA[Density (kg / m 3 )]]> ρ=5730 <![CDATA[Magnetic-electric constant (10 -12 Ns / VC)]]> m11=m22=6.0,m33=2500 Piezomagnetic constant (N / Am) q31=200,q33=260,q15=180 <![CDATA[Dielectric constant (10 -9 C / Vm)]]> ε11=ε22=0.9,ε33=7.5 <![CDATA[Permeability constant (10 -4 Ns 2 / C 2 )]]> μ11=μ22=-1.5,μ33=0.75 <![CDATA[Piezoelectric constant (C / m 2 )]]> e31=-3.5,e33=11,e15=0 Elastic constant (GPa) <![CDATA[C 11 =C 22 =200,C 12 =110,C 13 =C 23 =110,C 33 =190,C 44 =45]]>

[0104] S4: Based on the computational grid obtained in step S2 and the physical parameters given in step S3, select the appropriate local enhanced interpolation basis function, construct the triangle unit node interpolation function based on the unit decomposition property, introduce the virtual triangle unit mid-edge node in the process of constructing the interpolation function, and complete the node shape function and local high-order numerical approximation of the overlapping triangle unit.

[0105] S41: For overlapping triangular unit regions, the internal physical field variables can be expressed as the following interpolation format:

[0106] u(x)=h I (x)ψ I (x)+h L (x)ψ L (x)+h M (x)ψ M (x)(15)

[0107] 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.

[0108] The local numerical approximation at node I can be expressed as

[0109]

[0110] Among them, u J is the degree of freedom at the corresponding node, The weight function corresponding to node I.

[0111] Degree of freedom u at node J J It can usually be expressed as

[0112] u J =p1a J,1 +p2a J,2 +p3aJ,3 +p4a J,4 +…+p n a J,n =p n a J,n (17)

[0113] Among them, p n represents a suitable basis function, a J,n represents the corresponding solution coefficients.

[0114] Here, Lagrange polynomials are used as basis functions to interpolate the displacement field.

[0115] p=[1 xyx 2 xy y 2 …] (18)

[0116] Here, trigonometric functions are used as basis functions to interpolate the electric and magnetic fields.

[0117] p=[1 cos(πx) sin(πx) … cos(nπx-nπy) sin(nπx-nπy)] (19)

[0118] 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 first-order Lagrange polynomial basis functions and first-order trigonometric basis functions as basis functions of the enhanced interpolation function, but in fact, when calculating specific engineering problems, the order of the basis function 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 basis function of the selected Lagrange polynomial enhancement function must be dimensionless. Therefore, the enhanced interpolation basis function selected when interpolating the displacement field, electric field and magnetic field is

[0119]

[0120] 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.

[0121] S42: 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

[0122]

[0123] Where, is the shape function of the 6-node quadratic triangular element, Represents the node value of node i.

[0124] Based on the finite sphere meshless technology, the weight function node value of node i It can be expressed as

[0125]

[0126] Among them, W J (x) is the weight function defined as follows

[0127]

[0128] Among them, the parameter s = d J / αr J ,

[0129] d J represents the distance from node J to the field point of interest x, 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).

[0130] In order to meet the unit decomposition properties and fragmentation test, the weight function node value The values of are shown in Table 2.

[0131] Table 2 Node values of weight function in nodal triangle elements The value of

[0132]

[0133]

[0134] Substituting formula (21) into formula (16), we can get

[0135]

[0136] Similarly, for nodes L and M, we can get

[0137]

[0138] Substituting equations (24)-(26) into equation (15), we can obtain

[0139]

[0140] in,

[0141]

[0142] S5: Based on the magneto-electroelastic structural dynamics control equations in step S1 and the node shape functions and local high-order numerical approximations of the elements constructed in step S4, a complete interpolation format of overlapping triangular elements can be obtained. Then, based on the standard "Galerkin" weighted residual method, an overlapping triangular finite element numerical discrete model of the magneto-electroelastic intelligent structural dynamics problem can be established and the matrix equation of the system can be obtained.

[0143] Based on the interpolation format in Equation (27), the dynamic matrix control equation of the magneto-electroelastic structure based on overlapping triangular units in Equation (13) can be obtained, and then the free vibration analysis of the magneto-electroelastic structure can be completed.

[0144] S6: Apply specific boundary conditions and solve the established matrix equation of the magneto-electroelastic structure dynamics problem without considering external forces to obtain the natural frequencies of each order of the magneto-electroelastic structure and the corresponding natural vibration modes of each order, thus completing the free vibration analysis of the magneto-electroelastic structure.

[0145] 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.

[0146] Table 3 Expressions of various material matrices in the magneto-electroelastic structure

[0147]

[0148]

[0149] S61: In order to reduce the size of the matrix equation, according to the matrix condensation technology, method (22) can be simplified to

[0150]

[0151] in,

[0152]

[0153] S62: By introducing the corresponding boundary conditions and solving the matrix equation (16), the node displacement vector of the magneto-electroelastic structure can be obtained. The node electric potential vector and the node magnetic potential vector can be obtained by the following equations:

[0154]

[0155] For the free vibration problem of magneto-electroelastic structure, Equation (16) has the following general solution:

[0156]

[0157] 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.

[0158] Substituting the above formula into formula (16), we can get

[0159]

[0160] Solving this equation can obtain the natural frequency and natural vibration mode of the magneto-electroelastic structure during free vibration, and complete the free vibration analysis of the magneto-electroelastic structure. The results of the first 10 natural frequencies are shown in Table 4, and the corresponding natural vibration modes are shown in Figure 7 shown.

[0161] S7: Apply specific boundary conditions, consider external excitations (including mechanical excitation, electric field excitation, and magnetic field excitation), and within a certain frequency range, solve the established matrix equations of the magneto-electroelastic structure dynamics problem to complete the harmonic response analysis (forced vibration analysis) of the magneto-electroelastic structure.

[0162] S71: Considering the response of the magneto-electroelastic structure under external excitation (such as mechanical excitation, electric field excitation or magnetic field excitation), formula (24) can be expressed as

[0163]

[0164] Where F represents the external excitation force vector acting on the magneto-electroelastic structure.

[0165] S72: Given a specific frequency range, solving formula (29) can obtain the response curve of the mechanical displacement response of the magneto-electroelastic structure with arbitrary displacement and arbitrary frequency change.

[0166] S73: After obtaining the response curve of the mechanical displacement response of the magneto-electroelastic structure with the frequency change, the response curves of the electric potential and magnetic potential response of the magneto-electroelastic structure with the frequency change can be obtained at once based on formula (18), and finally the harmonic response analysis of the magneto-electroelastic structure under external excitation is completed. The displacement response, electric potential response and magnetic field response at the measuring point are obtained as follows: Figure 8 shown.

[0167] S8: Based on the node coefficients obtained from solving the matrix equations for the magneto-electroelastic structure dynamics problem in step S7 and the numerical approximation of the three-phase displacement field of the magneto-electroelastic structure obtained in step 4, the three-phase physical field response (including mechanical displacement response, electric field response, and magnetic field response) at any point in the computational domain as the computational frequency changes can be obtained through interpolation. The obtained response results include, but are not limited to, mechanical displacement response, electric field response, magnetic field response, stress response contour map, strain response contour map, etc.

[0168] Table 4 shows the calculation results of the first 10 natural frequencies of the magneto-electroelastic beam with two ends clamped using the triangular elements, quadrilateral elements in the standard finite element method and the overlapping finite element method provided by the present invention under the same mesh and number of nodes (305 nodes).

[0169] Table 4 Calculation results of the natural frequencies of the first 10 sections of the magneto-electroelastic structure

[0170]

[0171] Figure 8 Given are curves of displacement response, electric field response and magnetic field response at the measuring point as the calculation frequency changes under the action of unit simple harmonic excitation force. The reference solution is the calculation result using high-order finite elements in an extremely fine grid (24641 nodes). It can be seen that the overlapping finite element method given in the present invention can calculate results that are closer to the reference solution and more accurate than the triangular elements and quadrilateral elements in the standard finite element method under the same grid node distribution. This shows that the overlapping finite element method given in the present invention has higher solution accuracy than the standard finite element method when calculating the dynamic response of the magneto-electroelastic structure using the same grid, and has great potential application value in engineering practice.

[0172] Therefore, the present invention adopts the above-mentioned method for calculating the dynamic response of a magneto-electroelastic structure based on overlapping triangular elements. For each constructed overlapping triangular element, the nodal degrees of freedom are not the nodal displacement variables used in the traditional finite element method, but rather a local numerical approximation consisting of a local enhanced interpolation basis function and the coefficients to be calculated. 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 performed while satisfying the element decomposition characteristics. Based on this numerical interpolation, the governing matrix equations for the dynamics of the magneto-electroelastic structure are established, resulting in an overlapping finite element numerical model of the solver's dynamic response. After applying appropriate boundary conditions and external excitations, a high-precision prediction of the dynamic response of the magneto-electroelastic structure can be achieved. Because high-order interpolation is achieved without increasing the number of element nodes, the computational efficiency and accuracy of the existing technology are improved.

[0173] 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 dynamic response of magneto-electroelastic structures based on 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 and meshing using a standard triangular unit grid; S3, setting the corresponding physical parameters of the material and medium for the magneto-electroelastic structure grid obtained in S2; S4. Selecting a local enhanced interpolation basis function, constructing a triangle unit node interpolation function based on the unit decomposition property, introducing virtual triangle unit mid-edge nodes in the process of constructing the interpolation function, and completing the node shape function and local high-order numerical approximation of the overlapping triangle units; S5. Based on the standard Galerkin weighted residual method, a complete interpolation format of overlapping triangular elements is used to establish a numerical discrete model of the magneto-electroelastic smart structure dynamics problem and obtain the matrix equation of the system. S6. Apply boundary conditions and solve the established matrix equations for the magneto-electro-elastic structure dynamics problem without considering external forces, obtain the natural frequencies of each order of the magneto-electro-elastic structure and the corresponding natural vibration modes, and complete the free vibration analysis of the magneto-electro-elastic structure; S7. Apply boundary conditions, consider external excitation, and solve the established matrix equations for the magneto-electroelastic structure dynamics problem within a given frequency range to complete the harmonic response analysis of the magneto-electroelastic structure. S8. Based on the node coefficients and the numerical approximation of the three-phase displacement field of the step magneto-electroelastic structure, the three-phase physical field response of any point in the calculation domain as the calculation frequency changes is obtained by interpolation.

2. The method for calculating the dynamic response of a magneto-electroelastic structure based on overlapping triangular elements according to claim 1, characterized in that: In S2, the magneto-electroelastic structure is divided into a standard triangular unit grid. Each node is the central node of a polygonal unit composed of all the surrounding single triangular units related to it. The overlapping parts of the polygonal units are constructed as overlapping triangular units.

3. The method for calculating the dynamic response of a magneto-electroelastic structure based on overlapping triangular elements according to claim 2, characterized in that: S4 includes the following steps: S41. 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. ; S42. Weighting function The shape function in the standard 6-node triangle element is interpolated with the nodal value of the shape function at each node; the nodal value of the shape function Constructed by Shepard function; S43. Assign a set of high-order complete Lagrange polynomial basis functions to the local analysis domain of each field node to construct a local field approximation. , and , and normalize it with the unit size to improve the stability of the system matrix; S44, based on unit decomposition characteristics and local field approximation , and Constructing element interpolation functions in overlapping triangular finite elements , and .

4. The method for calculating the dynamic response of a magneto-electroelastic structure based on overlapping triangular elements according to claim 3, characterized in that: In S4, the basis function of the Langrange polynomial strengthening function is used for dimensionless operation. The specific operation method is: ； in, is the spatial coordinate of node J, and , is the characteristic size of the overlapping triangular element under consideration.

5. The method for calculating the dynamic response of a magneto-electroelastic structure based on overlapping triangular elements according to claim 4, characterized in that: Local field approximation of node I in S43 Expressed as: ； in, is the degree of freedom at the corresponding node, The weight function corresponding to node I, are the nodes of the triangular element, is the spatial coordinate; Degrees of freedom at node J Expressed as ； in, represents a suitable reinforcement function basis function, represents the corresponding solution coefficients.

6. The method for calculating the dynamic response of a magneto-electroelastic structure based on overlapping triangular elements according to claim 5, characterized in that: The complete interpolation format for overlapping triangle elements has the following form: ； in, ； , and is the shape function in the standard linear triangle element, , and is a local numerical approximation, is the weight function corresponding to node M, is the weight function corresponding to node L, (i=1,2, …6) is the shape function of the 6-node quadratic triangular element, Represents the node value of the weight function at node i.

7. The method for calculating the dynamic response of a magneto-electroelastic structure based on overlapping triangular elements according to claim 6, characterized in that: The matrix form of the governing equations of magneto-electroelastic structural dynamics in S5 is as follows: ； in, ； ； ； ； ； ； ； , and are the gradient matrices of displacement field, electric field and magnetic field respectively, , and are the nodal variables of displacement field, electric field and magnetic field respectively, Represents the density of the magneto-electroelastic structure, MTU.

8. The method for calculating the dynamic response of a magneto-electroelastic structure based on overlapping triangular elements according to claim 7, characterized in that: S6 has the following steps: S61. According to the matrix condensation technology, the size of the matrix equation is reduced and simplified to ； in, Represents the equivalent stiffness matrix of the system; S62. Introduce corresponding boundary conditions to obtain the node displacement vector, node electric potential vector, and node magnetic potential vector of the magneto-electroelastic structure; S63. For the free vibration problem of magneto-electroelastic structure, the above equation has the following general solution: ； in, is the amplitude of the solution, which is generally a function of the spatial position, is the imaginary unit, is the circular frequency, t is the time; S64. The natural frequency and natural mode model of the magneto-electroelastic structure during free vibration are: ； Solve this equation to complete the free vibration analysis of the magneto-electroelastic structure.

9. The method for calculating the dynamic response of a magneto-electroelastic structure based on overlapping triangular elements according to claim 8, characterized in that: The external excitation in S7 includes mechanical excitation, electric field excitation or magnetic field excitation.

10. The method for calculating the dynamic response of a magneto-electroelastic structure based on overlapping triangular elements according to claim 9, characterized in that: In S7, the response of the magneto-electroelastic structure under external excitation is considered, and the formula is: ； Where, Represents the external excitation force vector acting on the magneto-electroelastic structure.

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