Piezoelectric material characteristic value solving method and device based on spectral element method

Abstract

The invention provides a method and a device for solving a characteristic value of a piezoelectric material based on a spectral element method, and the method comprises the steps: building an efficient solving model of the spectral element method: building a piezoelectric model based on a solid-state multi-physical effect, and determining an internal relation equation of mechanical strain and an electric field; integrating by using the high-order polynomial interpolation function of the spectral element method to obtain a quality matrix, and enabling the quality matrix to meet a preset condition; and converting and optimizing the generalized eigenvalue problem: converting the standard form of the generalized eigenvalue problem into a new matrix form, and converting the generalized eigenvalue problem into a common eigenvalue problem through a new matrix rapid processing algorithm and keeping the symmetry of the matrix. The traditional method for solving the generalized eigenvalue problem of the piezoelectric material has the problems of high calculation complexity and low efficiency, the generalized eigenvalue problem is converted into a common eigenvalue problem by utilizing the advantages of high precision and high efficiency of the spectral element method, the calculation process is simplified, and the calculation efficiency is improved.

Description

Technical Field

[0001] The present invention belongs to the technical field of calculating the characteristic frequency of piezoelectric materials, and particularly relates to a method and device for solving the eigenvalues of piezoelectric materials based on the spectral element method. Background Art

[0002] In the modern technological system, piezoelectric materials are widely used in many key fields such as sensors, actuators, filters, and gyroscopes due to their unique electro-mechanical coupling characteristics. Taking sensors as an example, piezoelectric materials can efficiently convert physical quantities such as pressure and vibration into electrical signals, achieving precise perception of external environmental changes; in filters, their inherent frequency characteristics directly determine the signal screening and transmission effects, playing a decisive role in the performance of the device. Thus, the performance and stability of piezoelectric materials are crucial for the normal operation of related devices, and their natural frequency, as the core factor affecting dynamic characteristics, has become the key focus of research.

[0003] In traditional methods for calculating the eigenfrequency of piezoelectric materials, the finite element method (FEM), boundary element method (BEM), and finite difference method (FDM) are relatively commonly used. However, these traditional methods have many limitations that cannot be ignored in the actual application process. Taking the finite element method as an example, when dealing with complex boundary conditions, large-scale problems, and complex geometries, it is often necessary to solve the generalized eigenvalue problem. In the high-frequency band, this process not only has a huge computational cost but is also extremely prone to numerical instability, resulting in a significant reduction in the accuracy of the final solution. For example, when analyzing a piezoelectric sensor with a complex shape, the finite element method needs to divide a large number of small elements to approximate the real structure, which not only significantly increases the computational cost but also easily leads to the accumulation of numerical errors, making the calculation results deviate greatly from the actual situation.

[0004] The spectral element method (SEM), as an emerging numerical calculation method, has emerged in recent years in the research of structural dynamics and wave propagation problems, showing significant advantages. It takes advantage of the orthogonality of high-order interpolation polynomials and the unique characteristics of Gauss-Lobatto-Legendre (GLL) integration points to obtain a diagonal mass matrix in the frequency domain analysis, thus greatly simplifying the calculation process and effectively improving the calculation efficiency. Nevertheless, the application of the spectral element method in solving the eigenfrequency of piezoelectric materials is still in the exploratory stage, and the relevant research is not yet deep and perfect. Therefore, introducing the spectral element method into the field of solving the eigenfrequency of piezoelectric materials and developing an efficient algorithm adapted to it have important research value and practical significance. Summary of the Invention

[0005] In the prior art, piezoelectric materials are widely used in fields such as sensors, actuators, filters, and gyroscopes, and their performance and stability are significantly affected by the intrinsic frequency of the materials. Therefore, accurately solving the intrinsic frequency of piezoelectric materials is the key to designing and optimizing piezoelectric devices. However, traditional methods such as the Finite Element Method (FEM), Boundary Element Method (BEM), and Finite Difference Method (FDM) face problems of large computational amount and low efficiency when dealing with the generalized eigenvalue problem of piezoelectric materials, especially when dealing with complex boundary conditions and large-scale structures. The present invention provides a method and device for solving the eigenvalue of piezoelectric materials based on the spectral element method to solve the above-mentioned existing technical defect problems.

[0006] In a first aspect, the present invention proposes a method for solving the eigenvalue of piezoelectric materials based on the spectral element method, and the method includes the following steps:

[0007] Construct an efficient solution model of the spectral element method: establish a piezoelectric model based on solid-state multi-physical effects, and determine the internal relationship equation between mechanical strain and electric field; use the high-order polynomial interpolation function of the spectral element method to integrate to obtain a mass matrix, and make the mass matrix satisfy preset conditions;

[0008] Perform the conversion and optimization of the generalized eigenvalue problem: convert the standard form of the generalized eigenvalue problem into a new matrix form, and then through a new matrix fast processing algorithm and maintaining the symmetry of the matrix, convert the generalized eigenvalue problem into an ordinary eigenvalue problem;

[0009] Conduct application verification in piezoelectric devices: further apply the above method steps to the complex structure analysis of piezoelectric materials, including piezoelectric devices such as low-frequency filters, surface acoustic wave filters, and piezoelectric gyroscopes.

[0010] Preferably, in the step of constructing the efficient solution model of the spectral element method, the spectral element method uses high-order interpolation polynomials and Gauss-Lobatto–Legendre integration points, so that the mass matrix presents a diagonalized characteristic in frequency-domain analysis, and the mass matrix satisfies preset conditions.

[0011] The present invention proposes to use the spectral element method to solve the solid-electrical coupling eigenvalue problem of piezoelectric materials. Compared with the traditional Finite Element Method (FEM), the spectral element method uses high-order interpolation polynomials and Gauss-Lobatto-Legendre (GLL) integration points, so that the mass matrix presents a diagonalized characteristic in frequency-domain analysis. Through this innovative design, the calculation process is simplified, the complex matrix inversion operation is avoided, and the calculation efficiency is greatly improved.

[0012] Further preferably, in the step of constructing an efficient solution model of the spectral element method, the intrinsic relationship equation between mechanical strain and electric field is where τ is the stress tensor, ρ and D are the mass density and electric displacement vector of the piezoelectric material respectively, and u is the displacement vector at the location where deformation occurs;

[0013] The intrinsic relationship coupling between the mechanical strain tensor S and the electric field is given by the piezoelectric effect equation D = ∈E + e:S and the inverse piezoelectric effect equation T = c:S - e T E, where e, S, ε, E, and c are the piezoelectric coupling coefficient, mechanical strain tensor, dielectric constant, electric field strength, and elastic coefficient matrix respectively, T represents the stress tensor, and e T is the transpose matrix of the piezoelectric coupling coefficient matrix e.

[0014] Preferably, in the step of performing the transformation and optimization of the generalized eigenvalue problem, it specifically includes: transforming the standard form of the generalized eigenvalue problem into a new matrix form ω 2 MU + ωCU + KU = 0 into Let to obtain K g U t＝ wM g U t , where U is the displacement vector describing the displacement of nodes in the system, M is the diagonal mass matrix, C is the symmetric damping matrix, and K is the symmetric stiffness matrix; then, through a new matrix fast processing algorithm, it is transformed into an ordinary eigenvalue problem while ensuring the symmetry of the matrix, that is K n U n = wU n , and the new stiffness matrix has the same sparsity as the original stiffness matrix.

[0015] The present invention proposes a method for transforming a generalized eigenvalue problem. Traditional generalized eigenvalue solutions usually require dealing with non-symmetric matrices, and numerical instability is prone to occur during the calculation process. The present invention transforms the generalized eigenvalue problem into an ordinary eigenvalue problem by introducing a new matrix fast processing algorithm and maintains the symmetry of the matrix. This transformation method not only improves the calculation stability but also significantly reduces the storage and calculation costs.

[0016] Preferably, the specific steps for solving the eigenvalue include: data preparation, exporting the Comosl mesh file ".mphbin", and recording material parameters and solution parameters through a csv file; matrix assembly, processing the mesh data and material parameters, and assembling the mass matrix and stiffness matrix; matrix format processing, using the Intel MKL library and Eigen library to process the matrix format; first attempt to use the conventional finite element method. First, attempt to solve using the conventional finite element method. If it fails, adjust the solution parameters and use the ARPACK library to calculate the generalized eigenvalue; if successful, directly obtain the result; then use the optimized spectral element method to transform the generalized eigenvalue into an ordinary eigenvalue, and then adjust the parameters and use the ARPACK library to calculate the ordinary eigenvalue, and finally obtain the result.

[0017] Preferably, in the application verification step in the piezoelectric device, when the method is used for the analysis of complex structures of piezoelectric materials, it can maintain high precision at different degrees of freedom, and the calculation error in the high-degree-of-freedom model is controlled within 1e-5.

[0018] More preferably, by utilizing the orthogonality of the high-order interpolation function in the spectral element method and the selection of GLL integration points, a sparse and diagonalized mass matrix is constructed. Compared with the traditional finite element method, the calculation time is reduced by 40%, and the memory occupancy is reduced by 30%.

[0019] The calculation method of the present invention is applicable to the analysis of complex structures of various piezoelectric materials, including application scenarios such as low-frequency filters (such as LMR filters), surface acoustic wave filters (SAW filters), and piezoelectric gyroscopes. Tests and verifications show that the present invention can maintain high precision at different degrees of freedom. Especially in the high-degree-of-freedom model, the calculation error can be controlled within 1e-5, demonstrating strong adaptability and robustness.

[0020] The invention further optimizes the calculation efficiency. By utilizing the orthogonality of the high-order interpolation function in the spectral element method and the selection of GLL integration points, a sparse and diagonalized mass matrix is constructed. In frequency-domain analysis, this matrix form can significantly reduce the calculation complexity, making the algorithm of the present invention reduce the calculation time by about 40% compared with the traditional finite element method and reduce the memory occupancy by about 30%. It is particularly suitable for the eigenfrequency analysis of large-scale multi-scale problems.

[0021] In a second aspect, an embodiment of the present invention provides a device for solving the eigenvalue of a piezoelectric material based on the spectral element method. The device includes:

[0022] A solution model construction module, configured to construct an efficient solution model of the spectral element method, establish a piezoelectric model based on solid-state multi-physical effects, and determine the internal relationship equation between mechanical strain and electric field; use the high-order polynomial interpolation function of the spectral element method to integrate to obtain a mass matrix, and make the mass matrix meet the preset conditions;

[0023] An eigenvalue conversion module is used to perform the conversion and optimization of the generalized eigenvalue problem, convert the standard form of the generalized eigenvalue problem into a new matrix form, and then through a new matrix fast processing algorithm while maintaining the symmetry of the matrix, convert the generalized eigenvalue problem into an ordinary eigenvalue problem.

[0024] Preferably, in the solution model construction module, the spectral element method adopts high-order interpolation polynomials and Gauss-Lobatto–Legendre integration points, so that the mass matrix shows diagonalization characteristics in frequency domain analysis. Compared with the traditional finite element method, the calculation time is reduced by 40%, and the memory occupancy is reduced by 30%. The mass matrix satisfies M ij = 0 for i≠j.

[0025] Preferably, it further includes: an application verification module, which is used to apply the above modules to the complex structure analysis of piezoelectric materials for application verification in piezoelectric devices. The piezoelectric devices include low-frequency filters, surface acoustic wave filters, and piezoelectric gyroscopes. When the application verification module is used for the complex structure analysis of piezoelectric materials, it can maintain high precision under different degrees of freedom, and the calculation error in the high-degree-of-freedom model is controlled within 1e-5.

[0026] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0027] The present invention first introduces the spectral element method into the solution of the eigenfrequency of piezoelectric materials and proposes a new matrix fast processing method. This method can convert the generalized eigenvalue problem into an ordinary eigenvalue problem, thereby simplifying the calculation process and improving the calculation efficiency; the present invention proposes a new matrix fast processing method that can convert the generalized eigenvalue problem into an ordinary eigenvalue problem; this method is significantly superior to the traditional method in terms of calculation accuracy and efficiency, especially when dealing with complex structures and boundary conditions, showing strong adaptability and stability; the feasibility and superiority of this method are verified through two example tests of LMR filters and piezoelectric rate gyroscopes, providing a novel and effective solution for the eigenfrequency analysis of piezoelectric materials; in summary, the research in this paper provides a new and efficient calculation tool for the eigenfrequency analysis of piezoelectric materials, with broad application prospects and potential value. Description of the Drawings

[0028] The drawings are included to provide a further understanding of the embodiments and are incorporated into and constitute a part of this specification. The drawings illustrate the embodiments and are used together with the description to explain the principles of the present invention. Other embodiments and many of the intended advantages of the embodiments will be readily recognized, as they become better understood by reference to the following detailed description. The elements of the drawings are not necessarily to scale with each other. The same reference numerals refer to corresponding like parts.

[0029] Figure 1 is a schematic flow chart of a method for solving the eigenvalue of piezoelectric materials based on the spectral element method according to an embodiment of the present invention;

[0030] Figure 2 is a schematic diagram of the specific operation process of the eigenvalue solving step in the embodiment of the present invention;

[0031] Figure 3 is a schematic diagram of the 3D model of the LMR filter according to a specific embodiment of the present invention;

[0032] Figure 4 is a comparison of the solving time of the LMR filter according to a specific embodiment of the present invention;

[0033] Figure 5 is a schematic diagram of the 3D model of the SAW filter according to a specific embodiment of the present invention;

[0034] Figure 6 is a comparison of the solving time of the SAW filter according to a specific embodiment of the present invention;

[0035] Figure 7 is a schematic diagram of the error of different orders of the spectral element method in a specific embodiment of the present invention;

[0036] Figure 8 is a schematic diagram of the 3D model of the piston model according to a specific embodiment of the present invention;

[0037] Figure 9 is a comparison of the solving time of the piston model according to a specific embodiment of the present invention;

[0038] Figure 10 is a schematic diagram of the architecture of a method for solving the eigenvalue of piezoelectric materials based on the spectral element method according to an embodiment of the present invention. Specific Embodiments

[0039] The present invention will be further described in detail below with reference to the drawings and embodiments. It can be understood that the specific embodiments described herein are only used to explain the related invention, rather than limiting the invention. Additionally, it should be noted that for the sake of description, only parts related to the relevant invention are shown in the drawings.

[0040] It should be noted that, without conflict, the embodiments in the present invention and the features in the embodiments can be combined with each other. The present invention will be described in detail below with reference to the drawings and embodiments.

[0041] First, an embodiment of the present invention discloses a method for solving the eigenvalue of piezoelectric materials based on the spectral element method, as Figure 1 shown, the method includes the following steps:

[0042] S1. Construct an efficient solution model for the spectral element method: Establish a piezoelectric model based on solid-state multi-physical effects, and determine the internal relationship equation between mechanical strain and electric field; Use the high-order polynomial interpolation function of the spectral element method to integrate to obtain a mass matrix, and make the mass matrix meet the preset conditions;

[0043] Specifically, in this step, the internal relationship equation between mechanical strain and electric field is where τ is the stress tensor, ρ and D are the mass density and electric potential displacement vector of the piezoelectric material respectively, and u is the displacement vector at the location where deformation occurs.

[0044] The internal relationship coupling between the mechanical strain tensor S and the electric field is given by the piezoelectric effect equation D = ∈E + e:S and the inverse piezoelectric effect equation T = c:S - e T E, where e, S, ε, E, and c are the piezoelectric coupling coefficient, mechanical strain tensor, dielectric constant, electric field strength, and elastic coefficient matrix respectively, T represents the stress tensor, and e T is the transpose matrix of the piezoelectric coupling coefficient matrix e.

[0045] The spectral element method uses high-order interpolation polynomials and Gauss-Lobatto–Legendre integration points, so that the mass matrix shows a diagonalized characteristic in frequency-domain analysis, and the mass matrix meets the preset conditions, that is, the mass matrix satisfies M ij = 0 for i ≠ j.

[0046] By using the orthogonality of the high-order interpolation function in the spectral element method and the selection of GLL integration points, construct the sparse and diagonalized mass matrix. Compared with the traditional finite element method, the calculation time is reduced by 40% and the memory occupancy is reduced by 30%.

[0047] S2. Perform the transformation and optimization of the generalized eigenvalue problem: Transform the standard form of the generalized eigenvalue problem into a new matrix form, and then through a new matrix fast processing algorithm and maintaining the symmetry of the matrix, transform the generalized eigenvalue problem into an ordinary eigenvalue problem;

[0048] This step specifically includes: Transforming the standard form of the generalized eigenvalue problem into a new matrix form ω 2 MU + ωCU + KU = 0 is transformed into Let Get K g U t = wM g U t, where U is the displacement vector describing the displacement of nodes in the system, M is the diagonal mass matrix, C is the symmetric damping matrix, and K is the symmetric stiffness matrix; then, by applying a new matrix fast processing algorithm, it is transformed into a general eigenvalue problem while ensuring the symmetry of the matrix, that is K n U n = ωU n , and the new stiffness matrix has the same sparsity as the original stiffness matrix.

[0049] Furthermore, as Figure 2 shown, the specific steps for eigenvalue solving include:

[0050] S21. Data preparation: Export the comosl mesh file ".mphbin" and record the material parameters and solution parameters through a csv file;

[0051] S22. Matrix assembly: Process the mesh data and material parameters and assemble the mass matrix and stiffness matrix;

[0052] S23. Matrix format processing: Use the intel mkl library and eigen library to process the matrix format;

[0053] S231. First try with the conventional finite element method: First try to solve using the conventional finite element method. If it fails, adjust the solution parameters and use the arpack library to calculate the generalized eigenvalues; if it succeeds, directly obtain the result;

[0054] S232. Then use the optimized spectral element method to transform the generalized eigenvalues into general eigenvalues, and then adjust the parameters to use the arpack library to calculate the general eigenvalues, and finally obtain the result.

[0055] S3. Application verification in piezoelectric devices: Apply the above method steps to the complex structure analysis of piezoelectric materials, including piezoelectric devices such as low-frequency filters, surface acoustic wave filters, and piezoelectric gyroscopes.

[0056] Specifically, in this step, when the method disclosed in this embodiment is used for the complex structure analysis of piezoelectric materials, it can maintain high precision at different degrees of freedom, and the calculation error in the high-degree-of-freedom model is controlled within 1e-5.

[0057] The present invention first introduces the spectral element method into the solution of the intrinsic frequency of piezoelectric materials and proposes a new matrix fast processing method, which can transform the generalized eigenvalue problem into an ordinary eigenvalue problem, thereby simplifying the calculation process and improving the calculation efficiency. The present invention proposes a new matrix fast processing method, which can transform the generalized eigenvalue problem into an ordinary eigenvalue problem. This method is significantly superior to the traditional method in terms of calculation accuracy and efficiency, especially when dealing with complex structures and boundary conditions, showing strong adaptability and stability. The feasibility and superiority of this method are verified through two example tests of LMR filters and piezoelectric rate gyroscopes, providing a novel and effective solution for the characteristic frequency analysis of piezoelectric materials. In summary, the research in this paper provides a new and efficient calculation tool for the characteristic frequency analysis of piezoelectric materials, with broad application prospects and potential value.

[0058] The present invention provides a method for solving the intrinsic frequency of piezoelectric materials based on the Spectral Element Method (SEM). This method significantly improves the calculation efficiency and accuracy through the transformation of the generalized eigenvalue problem and matrix fast processing. In a specific embodiment, the specific implementation manner is further described in detail according to the following steps:

[0059] Step 1: Construction of an efficient solution model for the spectral element method. The piezoelectric model is based on the solid-state multi-physical field effect and is used for the mutual conversion of electrical energy and pressure. In the absence of external load excitation, the intrinsic relationship between mechanical strain and electric field is given by the following equation:

[0060]

[0061]

[0062] where τ is the stress tensor, and ρ and D are the mass density and electric displacement vector of the piezoelectric material respectively, and u is the displacement vector at the location where deformation occurs. The intrinsic relationship coupling between the mechanical strain tensor S and the electric field is given by the following equation:

[0063] D = ∈E + e:S

[0064] T = c:S - e T E

[0065] where e, S, ε, E, and c are the piezoelectric coupling coefficient, mechanical strain tensor, dielectric constant, electric field strength, and elastic coefficient matrix respectively, T represents the stress tensor, and e T is the transpose matrix of the piezoelectric coupling coefficient matrix e. The equation D = ∈E + e:S represents the piezoelectric effect, which indicates that when a stress field is applied to an object, an electrical signal will be generated. The formula T = c:S - e TE represents the inverse piezoelectric effect, which means that when an electric field is applied to a piezoelectric material, a mechanical force will be generated.

[0066] The spectral element method is different from the traditional finite element method in that the spectral element method uses high-order polynomial interpolation functions. After integrating the above formula using the spectral element method, the mass matrix is obtained:

[0067] M ij = 0 for i ≠ j.

[0068] Step 2: Transformation and optimization of the generalized eigenvalue problem. In this embodiment, the embodiments of the present invention transform the generalized eigenvalue problem into an ordinary eigenvalue problem through a new matrix fast processing method. The specific steps are as follows:

[0069] In the traditional solution process, the standard form of the generalized eigenvalue problem:

[0070] ω 2 MU + ωCU + KU = 0

[0071] where U is the displacement vector, describing the displacements of the nodes in the system, M is the diagonal mass matrix, C is the symmetric damping matrix, and K is the symmetric stiffness matrix. These types of equations are usually used to describe the vibration behavior of the system under the action of external forces. To solve the characteristic frequencies and modes of the system, we need to transform the above second-order differential equation into a generalized eigenvalue problem.

[0072]

[0073] K g U t = wM g U t

[0074] Although the linearization technique is somewhat effective in solving the generalized eigenvalue problem, there are also obvious challenges in its application process.

[0075] First of all, the linearization technique will lead to a significant increase in the system scale, especially in the case of the presence of multi-scale features. To solve the generalized eigenvalue problem, traditional methods often need to construct a sufficiently large initial finite element space, which not only increases the computational complexity but also significantly improves the memory requirements, bringing greater resource consumption and performance bottlenecks to practical applications. To address such challenges, a series of numerical algorithm optimization schemes have been proposed in the prior art, including but not limited to the following methods:

[0076] (1) By iteratively solving the eigenvalues in a smaller subspace, the overall computational amount is reduced;

[0077] (2) By constructing a Krylov subspace to efficiently approximate the eigenvalues and eigenvectors, the computational efficiency is improved;

[0078] (3) By using the model degradation technique, a large-scale system is simplified into an equivalent small-scale system, thereby reducing the computational complexity.

[0079] However, when dealing with large-scale and multi-scale problems, these optimization schemes still face the problems of low computational efficiency and poor numerical stability, and it is difficult to meet the requirements of actual engineering applications.

[0080] In view of the above deficiencies of the prior art, the present invention innovatively proposes a new algorithm based on the spectral element method. This algorithm makes full use of the diagonalization property of the mass matrix in the spectral element method. By transforming the generalized eigenvalue problem into an ordinary eigenvalue problem, the calculation process is significantly simplified, and the computational and storage requirements are effectively reduced. This method not only overcomes the limitations of traditional linearization techniques in large-scale multi-scale problems, but also can greatly improve the computational efficiency and stability while maintaining high accuracy, and is applicable to the eigenfrequency analysis of complex piezoelectric structures.

[0081]

[0082] K n U n = ωU n

[0083] It should be noted that since the mass matrix of this method is a diagonal matrix, the new stiffness matrix obtained through the above transformation has the same sparsity as the original stiffness matrix. This means that the sparse structure will not be changed during matrix multiplication, thus avoiding the increase in computational overhead due to the reduction of sparsity.

[0084] Step 3: Application verification in piezoelectric devices. In this embodiment, the method of the present invention is applied to a variety of actual piezoelectric devices, including low-frequency filters (LMR filters), surface acoustic wave filters (SAW filters), and piezoelectric gyroscopes, etc. The specific application verification steps are as follows:

[0085] Low-frequency filter (LMR filter): For the LMR filter with a simple structure and low degrees of freedom, the eigenfrequency analysis is performed using the method of the present invention. As Figure 3 and Figure 4 , by comparing with the results of the finite element method (FEM), the advantages of this method in terms of computational time and accuracy are verified.

[0086] First, the accuracy of the low-degree-of-freedom results is tested through this model. The LMR filter is usually used to filter low-frequency signals, and its simple structure and low degree of freedom make it suitable for initial verification of the algorithm accuracy. Specifically, the parameters of the model are set and the eigenfrequencies are calculated by transforming the generalized eigenvalue problem into an ordinary eigenvalue problem. The calculation results show that the obtained eigenfrequencies are highly consistent with the theoretical values, and the errors are within an acceptable range, as shown in Table 1 below.

[0087] Table 1 Comparison of SEM and COMSOL Results of LMR Filter

[0088] <![CDATA[SEM(GH z )]]> <![CDATA[FEM(GH z )]]> ERROR 1 2.247731368 2.247915438 0.00005 2 2.269404285 2.260586562 0.00390 3 2.276421232 2.279329154 0.00048 <![CDATA 4 > 2.278218571 2.279329154 0.00048 5 2.295903526 2.291132701 0.00208 6 2.298649343 2.292690338 0.00259 7 2.318677139 2.318338347 0.00014 8 2.326711629 2.320396921 0.00272 9 2.329980855 2.321810801 0.00351 10 2.325144129 2.323225734 0.00082

[0089] Surface Acoustic Wave Filter (SAW Filter): For the SAW filter with complex structure and high degree of freedom, the spectral element method of different orders is used to solve the eigenfrequencies. Experiments show that as the order increases, the calculation error gradually decreases, verifying the high accuracy and adaptability of the spectral element method, as Figure 5 and Figure 6 shown.

[0090] Secondly, this paper selects the SAW filter model to test the change of the eigenfrequency solution error with the increase of the SEM order. The SAW filter is widely used in high-frequency signal processing, and its complex structure and high degree of freedom affect the accuracy of the eigenfrequencies. By setting different SEM orders, this embodiment observes that the error of the eigenfrequency solution gradually decreases with the increase of the order. This shows that the increase of the SEM order can significantly improve the calculation accuracy, as Figure 7 shown.

[0091] Piezoelectric Piston: For the complex geometric structure and multi-scale characteristics of the piezoelectric piston, the method of the present invention is used to solve and analyze the high-degree-of-freedom model. The results show that even under high-degree-of-freedom conditions, the method of the present invention can still quickly and stably solve the eigenfrequencies, and the error is controlled within 1e-5, showing excellent calculation performance and robustness, as Figure 8 and 9 shown.

[0092] Finally, the embodiment of the present invention uses the piston mode suppression model commonly used in the acoustic field to test the accuracy of the large-degree-of-freedom results. This model is used to suppress unwanted spurious modes and improve the frequency selectivity of the system. Although the traditional eigenvalue solving method has a large computational amount under high degrees of freedom, we can quickly and accurately solve the eigenfrequencies through eigenvalue transformation. The results show that even under large degrees of freedom, the calculation results are consistent with the theoretical values, verifying the robustness and effectiveness of this method. As shown in Table 2 below.

[0093] Table 2 Comparison of SEM and COMSOL Results

[0094] <![CDATA[SEM(GH z )]]> <![CDATA[FEM(GH z )]]> <![CDATA[Erro r > 1 1.894866431 1.894872872 1e-5 2 1.895242757 1.895248516 1e-6 3 1.895699908 1.895786391 1e-4 <![CDATA 4 > 1.896107384 1.896158624 1e-5 5 1.897293693 1.897262819 1e-5 6 1.901749813 1.901718625 1e-5 7 1.902479983 1.902479268 1e-6 8 1.903121491 1.903198164 1e-5 <![CDATA 9 > 1，904205134 1，904248290 1e-5 10 1.905239847 1.905198276 1e-4

[0095] Through the application verification of the above actual devices, the wide applicability and superior performance of the present invention in the analysis of various piezoelectric materials are demonstrated.

[0096] In summary, the present invention proposes a method for analyzing the natural frequencies of piezoelectric materials based on the Spectral Element Method (SEM), which is particularly suitable for solving the eigenvalues of piezoelectric devices such as Lamb wave resonators (LMR filters) and surface acoustic wave (SAW filters). Compared with the traditional Finite e Element Method (FEM), the method of the present invention not only significantly improves the calculation efficiency but also maintains excellent calculation accuracy when dealing with complex geometric structures and boundary conditions. By transforming the generalized eigenvalue problem into an ordinary eigenvalue problem, the present invention effectively simplifies the calculation process and reduces the overall calculation burden. Experimental verification shows that in the applications of LMR filters and SAW filters, the spectral element method significantly shortens the calculation time and reduces the memory requirements; in the case of a large number of degrees of freedom, such as in the piezoelectric piston model, the spectral element method can still efficiently and accurately solve the natural frequencies, demonstrating its superior stability and adaptability.

[0097] By utilizing the orthogonality of high-order interpolation polynomials and the characteristics of Gauss-Lobatto-Legendre (GLL) integration points, the present invention constructs a diagonalized mass matrix, simplifies the frequency-domain analysis, and improves the calculation stability and accuracy. The present invention provides a powerful and efficient calculation tool for the analysis of the natural frequencies of piezoelectric materials, with broad application prospects, and can be used for the design and optimization of piezoelectric devices such as sensors, actuators, and filters. In future work, the spectral element method can be further extended to other types of piezoelectric devices, and further algorithm optimization can be explored to improve the calculation performance and application scope.

[0098] Further referring to Figure 10 As an implementation of the methods shown in the above figures, the present application provides an embodiment of a device, and this device embodiment corresponds to Figure 10 the method embodiment shown, and this device can be specifically applied to various electronic devices.

[0099] In a second aspect, an embodiment of the present invention also discloses a device for solving the eigenvalues of piezoelectric materials based on the spectral element method, as Figure 10 shown, and this device includes: a solution model construction module 101, an eigenvalue conversion module 102, and an application verification module 103.

[0100] In a specific embodiment, a solution model construction module 101 is configured to construct an efficient solution model for the spectral element method, establish a piezoelectric model based on solid-state multi-physical effects, and determine an internal relationship equation between mechanical strain and electric field; use a high-order polynomial interpolation function of the spectral element method to integrate to obtain a mass matrix, and make the mass matrix satisfy a preset condition; an eigenvalue conversion module 102 is configured to perform conversion and optimization of a generalized eigenvalue problem, convert a standard form of the generalized eigenvalue problem into a new matrix form, and then through a new matrix fast processing algorithm and maintain the symmetry of the matrix, convert the generalized eigenvalue problem into an ordinary eigenvalue problem.

[0101] It further includes: an application verification module 103, configured to apply the above modules to the complex structure analysis of piezoelectric materials for application verification in piezoelectric devices, where the piezoelectric devices include low-frequency filters, surface acoustic wave filters, and piezoelectric gyroscopes. When the application verification module is used for the complex structure analysis of piezoelectric materials, it can maintain high precision at different degrees of freedom, and the calculation error in a high-degree-of-freedom model is controlled within 1e-5.

[0102] In the solution model construction module 101, the spectral element method uses high-order interpolation polynomials and Gauss-Lobatto–Legendre integration points, so that the mass matrix exhibits a diagonalized characteristic in frequency-domain analysis. Compared with the traditional finite element method, the calculation time is reduced by 40%, and the memory occupancy is reduced by 30%. The mass matrix satisfies M ij = 0 for i ≠ j.

[0103] The functions of the above modules correspond to the methods and will not be elaborated here.

[0104] The above description is only a preferred embodiment of the present invention and an explanation of the applied technical principles. Those skilled in the art should understand that the scope of the invention involved in the present invention is not limited to the technical solutions formed by the specific combination of the above technical features, but also covers other technical solutions formed by any combination of the above technical features or their equivalent features without departing from the above inventive concept. For example, the technical solutions formed by mutually replacing the above features with the (but not limited to) technical features with similar functions disclosed in the present invention.

Claims

1. A method for solving the characteristic value of piezoelectric material based on spectral element method, characterized in that: The method comprises the following steps: Constructing an efficient solution model of the spectral element method: establishing a piezoelectric model based on solid-state multi-physics effects, and determining the intrinsic relationship equation between mechanical strain and electric field; using the high-order polynomial interpolation function of the spectral element method, integrating to obtain a mass matrix, so that the mass matrix satisfies preset conditions; The generalized eigenvalue problem is converted and optimized: the standard form of the generalized eigenvalue problem is converted into a new matrix form, and then the generalized eigenvalue problem is converted into a common eigenvalue problem through a new matrix fast processing algorithm while maintaining the symmetry of the matrix.

2. The method for solving the characteristic value of piezoelectric material based on the spectral element method according to claim 1 is characterized in that: In the step of constructing an efficient solution model of the spectral element method, the spectral element method uses high-order interpolation polynomials and Gauss-Lobatto-Legendre integration points to make the mass matrix present a diagonal characteristic in the frequency domain analysis, and the mass matrix satisfies M ij =0for i≠j.

3. The method for solving the characteristic value of piezoelectric material based on the spectral element method according to claim 2, characterized in that: In the step of constructing the efficient solution model of the spectral element method, the intrinsic relationship equation between mechanical strain and electric field is: Where τ is the stress tensor, ρ and D elements are the density and potential displacement vector of the piezoelectric material, respectively, and u is the displacement vector where the deformation occurs; The intrinsic relationship between the mechanical strain tensor S and the electric field is coupled by the equation representing the piezoelectric effect D = ∈E + e:S and the equation representing the inverse piezoelectric effect T = c:Se T E is given by, where e, S, ε, E and c are the piezoelectric coupling coefficient, mechanical strain tensor, dielectric constant, electric field intensity and elastic coefficient matrix, respectively, T represents the stress tensor, e T is the transposed matrix of the piezoelectric coupling coefficient matrix e.

4. The method for solving the characteristic value of piezoelectric material based on the spectral element method according to claim 1, characterized in that: The steps of converting and optimizing the generalized eigenvalue problem specifically include: Transform the standard form of the generalized eigenvalue problem into a new matrix form ω 2 MU+ωCU+KU=0 is converted to make Get K g U t =wM g U t , where U is the displacement vector describing the displacement of the nodes in the system, M is the diagonal mass matrix, C is the symmetric damping matrix, and K is the symmetric stiffness matrix; Then, the new matrix is ​​quickly processed and transformed into a common eigenvalue problem while ensuring the symmetry of the matrix, that is, K n U n =wU n , and the new stiffness matrix has the same sparsity as the original stiffness matrix.

5. The method for solving the characteristic value of piezoelectric material based on the spectral element method according to claim 1, characterized in that: The specific steps of solving the eigenvalue include: Data preparation, export the COMOSL mesh file ".mphbin", record the material parameters and solution parameters through the CSV file; Matrix assembly, processing mesh data and material parameters, and assembling the mass matrix and stiffness matrix; Matrix format processing, using Intel mkl library and eigen library to process the matrix format; First try to use the conventional finite element method to solve. If it fails, adjust the solution parameters and use the arpack library to calculate the generalized eigenvalues. If successful, get the result directly. Then use the optimized spectral element method to transform the generalized eigenvalues ​​into ordinary eigenvalues, and then adjust the parameters to use the arpack library to calculate the ordinary eigenvalues ​​to finally get the results.

6. The method for solving the characteristic value of piezoelectric material based on the spectral element method according to claim 1, characterized in that: Also includes: The above steps are applied to the complex structural analysis of piezoelectric materials for application verification in piezoelectric devices, wherein the piezoelectric devices include low-frequency filters, surface acoustic wave filters, and piezoelectric gyroscopes. In the complex structural analysis of piezoelectric materials, high precision can be maintained at different degrees of freedom, and the calculation error in the high-degree-of-freedom model is controlled within 1e-5.

7. The method for solving the characteristic value of piezoelectric material based on the spectral element method according to claim 2, characterized in that: By utilizing the orthogonality of high-order interpolation functions and the selection of GLL integration points in the spectral element method, a sparse and diagonalized mass matrix is ​​constructed. Compared with the traditional finite element method, the calculation time is reduced by 40% and the memory usage is reduced by 30%.

8. A device for solving characteristic values ​​of piezoelectric materials based on spectral element method, characterized in that: The device includes: A solution model building module is used to build an efficient solution model of the spectral element method, establish a piezoelectric model based on solid-state multi-physics effects, and determine the intrinsic relationship equation between mechanical strain and electric field; use the high-order polynomial interpolation function of the spectral element method to integrate and obtain a mass matrix so that the mass matrix meets preset conditions; The eigenvalue conversion module is used to convert and optimize the generalized eigenvalue problem, convert the standard form of the generalized eigenvalue problem into a new matrix form, and then use a new matrix fast processing algorithm to maintain the symmetry of the matrix to convert the generalized eigenvalue problem into an ordinary eigenvalue problem.

9. The device for solving the characteristic value of piezoelectric material based on the spectral element method according to claim 8, characterized in that: In the solution model building module, the spectral element method uses high-order interpolation polynomials and Gauss-Lobatto–Legendre integration points to make the mass matrix present a diagonal characteristic in the frequency domain analysis. Compared with the traditional finite element method, the calculation time is reduced by 40%, the memory usage is reduced by 30%, and the mass matrix satisfies M ij =0 for i≠j.

10. The device for solving the characteristic value of piezoelectric material based on the spectral element method according to claim 8, characterized in that: Also includes: An application verification module is used to apply the above module to the complex structure analysis of piezoelectric materials to perform application verification in piezoelectric devices. The piezoelectric devices include low-frequency filters, surface acoustic wave filters, and piezoelectric gyroscopes. When the application verification module is used for the complex structure analysis of piezoelectric materials, it can maintain high precision in different degrees of freedom, and the calculation error in the high-degree-of-freedom model is controlled within 1e-5.

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