Post-processing method of displacement eigenvalues

Abstract

The present application discloses a post-processing method for displacement eigenvalues. It includes: constructing a matrix eigenvalue problem using unit data, material parameters and boundary conditions of multiple units of a structure, solving the problem to obtain eigenvalues ​​and eigenvectors corresponding to the finite element type, and determining the corresponding discrete eigenfunction using the finite element basis function corresponding to the finite element type; determining the reconstructed discrete k-order derivative function based on the value of the k-order derivative function of the discrete eigenfunction at each integral point in each unit; determining the posterior error estimator corresponding to the finite element type using the eigenvalue, eigenfunction and reconstructed discrete k-order derivative function corresponding to the finite element type; post-processing the eigenvalue according to the post-processing method and the posterior error estimator corresponding to the finite element type to obtain the final eigenvalue. The present application solves the technical problem that it is difficult to obtain high-precision eigenvalues ​​when the related art uses non-coordinated finite elements to process the vibration problem of a structure.

Description

Technical Field

[0001] The present application relates to the technical field of structural mechanics, and in particular to a post-processing method for displacement eigenvalues. Background Art

[0002] The problem of film vibration and thin plate vibration has a wide range of application backgrounds in engineering, physics, and materials science. Generally, when the resonant frequency of a structure such as a film or a thin plate is close to its natural frequency, resonance will occur. This phenomenon is also widely used in engineering, such as the musical resonance of musical instruments, orbital resonance between satellites, and resonance of the basilar membrane in the ear of animals. In engineering, it is usually necessary to utilize this phenomenon or try to avoid the loss caused by the resonance phenomenon. Therefore, studying the natural frequency of a structure is an important basis for seismic analysis and structural design.

[0003] At present, the study of the natural frequency of structures such as films or thin plates can be converted into the eigenvalue problem of different differential operators, and the study of the eigenvalue of the corresponding differential operator is an important tool for studying the natural vibration mode of the structure, stability analysis, etc. Among them, the vibration problem of the film can be converted into the eigenvalue problem of the Laplace operator, and the vibration problem of the thin plate can be converted into the eigenvalue problem of the reharmonic operator.

[0004] However, the most commonly used method for solving vibration problems in engineering is the finite element method, and its calculation accuracy is mainly determined by the approximation and coordination of the finite element discrete space. To obtain high-precision calculation results, it is necessary to solve a large discrete system, so the calculation time and cost are relatively large. However, by designing a high-precision eigenvalue post-processing method, only a small discrete system can be solved and post-processed to obtain numerical results that meet the accuracy requirements, reducing the calculation time and cost. At the same time, the existing eigenvalue post-processing methods are mainly designed for discrete eigenvalues ​​corresponding to coordinated elements, so there is a lack of effective eigenvalue post-processing methods for non-coordinated finite elements. In addition, due to the compatibility error of non-coordinated finite elements, it brings great difficulties to the design of efficient eigenvalue post-processing methods. Therefore, how to overcome this difficulty to design a high-precision algorithm for discrete eigenvalues ​​of non-coordinated finite elements is a major problem that needs to be solved urgently.

[0005] To address the above-mentioned problems, no effective solution has been proposed yet. Summary of the invention

[0006] The embodiment of the present application provides a post-processing method for displacement eigenvalues ​​to at least solve the technical problem in the related art that it is difficult to obtain high-precision eigenvalues ​​when using non-coordinated finite elements to process the vibration problem of a structure.

[0007] According to one aspect of an embodiment of the present application, a post-processing method for displacement eigenvalues ​​is provided, comprising: step S1, obtaining unit data, material parameters, boundary conditions and post-processing methods of multiple units obtained by dividing a structure, wherein the structure comprises: a film or a thin plate, the unit data comprises: unit shape, node coordinates, number of units, unit dimension, the boundary conditions comprise: force constraints, displacement constraints, the material parameters comprise: a first material parameter associated with the film, a second material parameter associated with the thin plate, and the first material parameter comprises: surface tension coefficient, unit mass density, the second material parameter comprises: bending stiffness, unit mass density, the post-processing method comprises: reconstructing eigenvalues ​​or combining eigenvalues; step S2, determining the finite element type and derivative order k used to solve the eigenvalue problem corresponding to the structure according to the material parameters, wherein the types of finite element types comprise: Crouzeix-Raviart element or augmented Crouzeix-Raviart element corresponding to the film, coordinated linear element, Morley element corresponding to the thin plate , the derivative order k includes: the derivative order k1 corresponding to the film or the derivative order k2 corresponding to the thin plate; step S3, looping each unit of the structure, obtaining the unit area, each integration point and the integral weight of each integration point of the current unit, and combining the unit data obtained in step S1 with the finite element type and the derivative order k obtained in step S2 to determine the values ​​of the finite element basis function corresponding to the finite element type and the k-order derivative of the finite element basis function at each integration point in the current unit; step S4, using the finite element basis function corresponding to the finite element type and the k-order derivative of the finite element basis function at each integration point in the current unit obtained in step S3 to determine the unit stiffness matrix and unit mass matrix of the current unit; step S5, determining the degree of freedom of the finite element, and combining the unit stiffness matrix and unit mass matrix of each unit obtained from steps S3 to S4 to obtain the overall stiffness matrix A and the overall mass matrix M; step S6, processing the overall stiffness matrix A and the overall mass matrix M obtained in step S5 according to the type of boundary conditions to obtain the corresponding eigenvalue problem ; Step S7, solving the matrix eigenvalue problem , and obtain the discrete eigenvalues ​​corresponding to the finite element type and the discrete eigenvector x, and determine the discrete eigenfunction corresponding to the finite element type in combination with the finite element basis function corresponding to the finite element type obtained in step S3; step S8, determine the k-order derivative function of the discrete eigenfunction corresponding to the finite element type, and use the value of the k-order derivative function of the discrete eigenfunction corresponding to the finite element type at each integral point in each unit to determine the reconstructed discrete k-order derivative function belonging to the Crouzeix-Raviart element discrete space; step S9, use the discrete eigenvalues ​​corresponding to the finite element type , discrete characteristic function and reconstructed discrete k-order derivative function to determine the posterior error estimator corresponding to the finite element; step S10, according to the post-processing method determined in step S1, and using the posterior error estimator corresponding to the finite element type obtained in step S9 to calculate the discrete characteristic value corresponding to the corresponding finite element type After post-processing, the final eigenvalues ​​are obtained.

[0008] Optionally, the above step S4 includes the following steps: traverse each unit and execute the following steps: step S41, using the values ​​of the finite element basis function corresponding to the finite element type obtained in step S3 at each integration point in the current unit and the material coefficient obtained in step S1 to determine the unit mass matrix; step S42, using the values ​​of the k-th order derivative of the finite element basis function corresponding to the finite element type obtained in step S3 at each integration point in the current unit and the material coefficient obtained in step S1 to determine the unit stiffness matrix.

[0009] Optionally, the above step S5 includes the following steps: step S51, determining the finite element type; step S52, when the finite element type is Crouzeix-Raviart element, taking the integral average of the function on each unit edge in the unit as the degree of freedom of the Crouzeix-Raviart element; step S53, when the finite element type is augmented Crouzeix-Raviart element, taking the integral average of the function on each unit edge in the unit and the integral average of the function on the unit as the degree of freedom of the augmented Crouzeix-Raviart element; step S54, when the finite element type is a coordinated linear element, the function value at each unit vertex in the current unit is used as the degree of freedom of the coordinated linear element; step S55, when the finite element type is a Morley element, the function value at each unit vertex in the unit and the integral average of the normal derivative of the function on each unit edge in the unit are used as the degree of freedom of the Morley element; step S56, according to the degrees of freedom of various finite elements obtained in the above steps S52 to S55, the unit stiffness matrix and unit mass matrix of each unit are assembled to obtain the overall stiffness matrix A and the overall mass matrix M.

[0010] Optionally, the above step S6 includes the following steps: traverse each boundary condition and execute the following steps: step S61, determine the type of the current boundary condition; step S62, when the type of the boundary condition is a displacement constraint, process the overall stiffness matrix A and the overall mass matrix M according to the following rules: step S621, determine the finite element type corresponding to the unit to which the current boundary condition belongs; step S622, when the finite element type is Crouzeix-Raviart element or augmented Crouzeix-Raviart element, take all degrees of freedom corresponding to the boundary edge as boundary degrees of freedom, and delete the rows and columns corresponding to the boundary degrees of freedom in the overall stiffness matrix A and the overall mass matrix M obtained in step S5; step S623, when the finite element type is Morley element, take all degrees of freedom corresponding to the boundary edge and the vertices on the boundary edge as boundary degrees of freedom, and delete the rows and columns corresponding to the boundary degrees of freedom in the overall stiffness matrix A and the overall mass matrix M; step S63, when the type of the boundary condition is a force constraint, keep the overall stiffness matrix A and the overall mass matrix M unchanged.

[0011] Optionally, the above step S7 includes the following steps: Step S71, solving the matrix eigenvalue problem , and obtain the discrete eigenvalues ​​corresponding to the finite element type and discrete characteristic vector x; step S72, loop each degree of freedom corresponding to the finite element type and determine the type of degree of freedom; step S73, when the type of degree of freedom is a boundary degree of freedom, determine the value of the discrete characteristic function on the degree of freedom using the corresponding boundary conditions; step S74, when the type of degree of freedom is a non-boundary degree of freedom, determine the value of the discrete characteristic function on the degree of freedom using the discrete characteristic vector x; step S75, determine the discrete characteristic function corresponding to the finite element type using the value of the discrete characteristic function on the degree of freedom obtained in steps S73 and S74 and the finite element basis function corresponding to the finite element type.

[0012] Optionally, the above step S8 includes the following steps: step S81, determining the k-th derivative function of the discrete characteristic function corresponding to the finite element type; step S82, traversing each unit edge of the unit to determine the type of the unit edge, wherein the type of the unit edge includes: common edge, boundary edge; step S83, when the type of the unit edge is a common edge, solving the k-th derivative of the discrete characteristic function at the midpoint of the unit edge in two adjacent units, and taking the average of the two k-th derivatives as the value of the Crouzeix-Raviart element degree of freedom corresponding to the reconstructed discrete k-th derivative function at the unit edge. ; Step S84, when the type of the unit edge is a boundary edge, determine two midpoints that are collinear with the midpoint of the unit edge from multiple midpoints of other unit edges that share a vertex with the unit edge, and use the average value of the k-order derivatives of the two midpoints for interpolation to obtain the value of the Crouzeix-Raviart element degree of freedom of the reconstructed discrete k-order derivative function corresponding to the unit edge; Step S85, combine the values ​​of the Crouzeix-Raviart element degree of freedom of the reconstructed discrete k-order derivative function corresponding to each unit edge obtained in step S83 and step S84 to obtain the reconstructed discrete k-order derivative function.

[0013] Optionally, the above step S9 includes the following steps: step S91, calculating the difference between the value of the reconstructed discrete k-order derivative function corresponding to the non-conforming finite element at each integration point in the current unit and the value of the k-order derivative of the discrete characteristic function corresponding to the non-conforming finite element at each integration point in the current unit, and using the square of the difference and the integral weight corresponding to each integration point to determine the first integral value in the area where the structure is located, wherein the non-conforming finite element includes: Crouzeix-Raviart element, augmented Crouzeix-Raviart element, Morley element; step S92, when using Crouzeix-Raviart element or augmented Crouzeix-Raviart element When the Crouzeix-Raviart element is used to solve the eigenvalue problem corresponding to the thin film, each unit is traversed, and the first-kind a posteriori error estimator corresponding to the Crouzeix-Raviart element or the augmented Crouzeix-Raviart element is determined according to the following steps: Step S921, according to the finite element type, a first-kind polynomial that depends only on the k+1-order derivative of the discrete eigenfunction is determined; Step S922, the reconstructed discrete k+1-order derivative value on the current unit is calculated using the reconstructed discrete k-order derivative function corresponding to the Crouzeix-Raviart element or the augmented Crouzeix-Raviart element, and the reconstructed discrete k+1-order derivative value is calculated using the The obtained reconstructed discrete k+1 order derivative value is used to calculate the value of the first type polynomial in the current unit; step S923, using the value of the first type polynomial in the current unit, the value of the discrete characteristic function corresponding to the Crouzeix-Raviart element or the augmented Crouzeix-Raviart element at each integral point in the current unit, and the integral weight corresponding to each integral point to determine the second integral value; step S924, calculating the product of the discrete characteristic value corresponding to the Crouzeix-Raviart element or the augmented Crouzeix-Raviart element and the sum of the second integral values ​​in all units, and using the first integral value minus twice the product , obtain the first kind of a posteriori error estimator corresponding to the Crouzeix-Raviart element or the augmented Crouzeix-Raviart element; step S93, when using the Crouzeix-Raviart element or the augmented Crouzeix-Raviart element to solve the eigenvalue problem corresponding to the film, traverse each unit and determine the second kind of a posteriori error estimator corresponding to the Crouzeix-Raviart element or the augmented Crouzeix-Raviart element according to the following method: step S931, determine the second kind of polynomial that only depends on the k+1 order derivative of the discrete eigenfunction according to the coordinated linear element;Step S932, using the reconstructed discrete k-order derivative function corresponding to the Crouzeix-Raviart element or the augmented Crouzeix-Raviart element to calculate the reconstructed discrete k+1-order derivative value on the current unit, and using the obtained reconstructed discrete k+1-order derivative value to calculate the value of the second type polynomial in the current unit and the value of the second type polynomial at the integral point on each unit edge of the current unit; Step S933, traverse each unit edge on the current unit, determine the average value of the second type polynomial in the two adjacent units to which the current unit edge belongs, and calculate the product of the average value and the jump of the normal derivative of the discrete characteristic function corresponding to the Crouzeix-Raviart element or the augmented Crouzeix-Raviart element on the current unit edge, and use the obtained product result and the integral weights corresponding to each integral point to calculate the third integral value of the current unit edge; Step S934, calculate Cr The product of the discrete characteristic function and the discrete eigenvalue corresponding to the Crouzeix-Raviart element or the augmented Crouzeix-Raviart element is added to the Laplace of the corresponding discrete characteristic function, and the value of the product of the addition result and the second-type polynomial in the current unit is calculated, and the fourth integral value is obtained by combining the integral weights corresponding to each integral point in the current unit; step S935, the first integral value is added to the sum of the third integral values ​​of all unit edges, and then subtracted from the sum of the fourth integral values ​​of all units. The second-type posterior error estimator corresponding to the Crouzeix-Raviart element or the augmented Crouzeix-Raviart element is obtained; step S94, in the case of using Morley element to solve the eigenvalue problem corresponding to the thin plate, the first integral value in step S91 is determined as the third-type posterior error estimator corresponding to the Morley element. ;

[0014] Optionally, the above-mentioned step S10 includes the following steps: step S101, when the post-processing method determined in step S1 is the reconstruction of eigenvalues, the discrete eigenvalues ​​and the posterior error estimator corresponding to the non-conforming finite element type are added to obtain the final eigenvalues; step S102, when the post-processing method determined in step S1 is the combined eigenvalues, the discrete eigenvalues ​​corresponding to any two finite element types are linearly combined according to a preset combination coefficient to obtain the final eigenvalue, wherein the combination coefficient is determined by the posterior error estimator corresponding to the finite element.

[0015] According to another aspect of an embodiment of the present application, a non-volatile storage medium is also provided, which includes a stored computer program, wherein the device where the non-volatile storage medium is located executes the above-mentioned post-processing method of the displacement characteristic value by running the computer program.

[0016] According to another aspect of an embodiment of the present application, a computer program product is further provided. The computer program product includes a stored computer program, wherein the computer program implements the above-mentioned post-processing method of displacement characteristic values ​​when executed by a processor.

[0017] In an embodiment of the present application, corresponding a posteriori error estimators are designed for different types of non-conforming elements, and the discrete eigenvalues ​​calculated by each type of non-conforming element are post-processed with error approximation using the a posteriori error estimators of the non-conforming elements. This makes it possible to achieve higher accuracy of discrete eigenvalues ​​obtained by calculating the matrix eigenvalue problem using the non-conforming elements, thereby solving the technical problem in the related art that it is difficult to obtain high-precision eigenvalues ​​when using non-conforming finite elements to process the vibration problem of a structure. BRIEF DESCRIPTION OF THE DRAWINGS

[0018] The drawings described herein are used to provide a further understanding of the present application and constitute a part of the present application. The illustrative embodiments of the present application and their descriptions are used to explain the present application and do not constitute an improper limitation on the present application. In the drawings:

[0019] Figure 1 It is a hardware structure block diagram of a computer terminal (or mobile device) for implementing an optional post-processing method of displacement characteristic values ​​according to an embodiment of the present application;

[0020] Figure 2 is a flow chart of an optional post-processing method of displacement characteristic values ​​according to an embodiment of the present application;

[0021] Figure 3 is a flow chart of an optional step S5 according to an embodiment of the present application;

[0022] Figure 4 is a flow chart of an optional step S6 according to an embodiment of the present application;

[0023] Figure 5 is a flow chart of an optional step S7 according to an embodiment of the present application;

[0024] Figure 6 is a flowchart of an optional step S8 according to an embodiment of the present application;

[0025] Figure 7 is a schematic diagram of an optional adjacent unit according to an embodiment of the present application;

[0026] Figure 8 is a flowchart of an optional step S9 according to an embodiment of the present application;

[0027] Fig. 9 is a schematic diagram of an optional H-type film according to an embodiment of the present application;

[0028] Fig.10 is a schematic diagram of an optional grid model 1 according to an embodiment of the present application;

[0029] Fig.11 is a schematic diagram of an optional grid model 2 according to an embodiment of the present application;

[0030] Fig.12 is a schematic diagram of an optional grid model 3 according to an embodiment of the present application;

[0031] Fig.13 is a schematic diagram of an optional grid model 4 according to an embodiment of the present application;

[0032] Fig.14 is a schematic diagram of an optional grid model 5 according to an embodiment of the present application;

[0033] Fig.15 is a schematic diagram of an optional grid model 6 according to an embodiment of the present application;

[0034] Fig.16 It is a schematic diagram of error comparison between eigenvalues ​​obtained by analyzing grid models 1-6 and eigenvalues ​​obtained by analyzing a reference grid model according to an optional embodiment of the present application. DETAILED DESCRIPTION

[0035] In order to enable those skilled in the art to better understand the present application, the technical solutions in the embodiments of the present application will be clearly and completely described below in conjunction with the drawings in the embodiments of the present application. Obviously, the described embodiments are only part of the embodiments of the present application, not all of the embodiments. Based on the embodiments in the present application, all other embodiments obtained by ordinary technicians in this field without creative work should fall within the scope of protection of this application.

[0036] It should be noted that the terms "first", "second", etc. in the specification and claims of the present application and the above-mentioned drawings are used to distinguish similar objects, and are not necessarily used to describe a specific order or sequence. It should be understood that the data used in this way can be interchangeable where appropriate, so that the embodiments of the present application described herein can be implemented in an order other than those illustrated or described herein. In addition, the terms "including" and "having" and any of their variations are intended to cover non-exclusive inclusions, for example, a process, method, system, product or device comprising a series of steps or units is not necessarily limited to those steps or units clearly listed, but may include other steps or units that are not clearly listed or inherent to these processes, methods, products or devices.

[0037] In addition, the relevant information (including but not limited to user device information, user personal information, etc.) and data (including but not limited to data for display, data for analysis, etc.) involved in this application are all information and data authorized by the user or fully authorized by all parties. For example, an interface is set up between this system and the relevant users or organizations. Before obtaining relevant information, it is necessary to send an acquisition request to the aforementioned user or organization through the interface, and obtain relevant information after receiving the consent information fed back by the aforementioned user or organization.

[0038] In order to better understand the embodiments of the present application, the technical terms involved in the embodiments of the present application are explained as follows:

[0039] Crouzeix-Raviart (CR) is a non-conforming finite element defined on triangular and tetrahedral meshes. It is often used to deal with problems with low regularity solutions, especially when dealing with complex geometric shapes and irregular mesh structures. The characteristic of this element is that its corresponding finite element solution is a weakly continuous function, and the asymptotic lower bound of the eigenvalue can be obtained.

[0040] The augmented Crouzeix-Raviart element is an extension of the Crouzeix-Raviart element, which can improve the approximation properties of the finite element function within the element. The finite element solution corresponding to this finite element is also a weakly continuous function, and the asymptotic lower bound of the eigenvalue can also be obtained.

[0041] Morley element is a non-conforming finite element proposed by Leonard SD Morley. It was originally used to solve fourth-order partial differential equations such as plate bending problems. It was designed for biharmonic equations in two-dimensional space and was later extended to higher dimensions to solve higher-order partial differential equations. The characteristic of Morley element is that it uses piecewise quadratic polynomials as the shape function space, and its degrees of freedom are defined as the integral average of the function values ​​at the unit vertices and the normal derivatives on the unit edges. Morley element is the finite element with the least degrees of freedom for solving fourth-order partial differential equations and is widely used in engineering calculations.

[0042] Shape functions are local functions defined on finite elements, which are used to construct approximate solutions inside the element. Shape functions usually have local support, that is, they are non-zero in some regions inside the element and zero in other regions. Linear combinations of shape functions can be used to approximate field variables (such as displacement, pressure, etc.) inside the element.

[0043] Basis functions usually refer to a set of linearly independent functions that constitute a function space. They are used to construct a global approximate solution space. The basis function corresponding to a specific degree of freedom of a finite element refers to a function in the finite element discrete function space whose value in that degree of freedom is non-zero and whose values ​​in all other degrees of freedom are all zero.

[0044] Unit mass density: refers to the ratio of the mass of a sample of a certain size to its surface area, and its expression is: , where MPA represents unit mass density, m represents total mass, and A represents total area.

[0045] Surface Tension Coefficient: It is the partial derivative of Gibbs free energy with respect to area when temperature and pressure remain constant, where the unit of Gibbs free energy is energy unit.

[0046] Bending Rigidity: refers to the ability of an object to resist bending deformation. Usually, the ability of a material to resist bending deformation can be expressed by the product of the elastic modulus of the material and the moment of inertia of the cross section of the bent member around its neutral axis.

[0047] Dirichlet boundary conditions: also known as the "first-kind boundary conditions" of ordinary differential equations or partial differential equations, specify the values ​​of the solutions of differential equations at the boundaries. Depending on whether the value of the solution of the differential equation at the boundary is zero, Dirichlet boundary conditions can be divided into homogeneous and non-homogeneous. The homogeneous Dirichlet boundary condition means that the value of the solution of the differential equation at the boundary is zero, that is, if u is the function to be solved, then for all boundary points x, u(x)=0; while the non-homogeneous boundary condition means that the value of the solution of the differential equation at the boundary is not zero, which means that there is a given non-zero function g, and the solution on the boundary must satisfy u=g.

[0048] Neumann boundary conditions are used to describe the behavior of partial differential equations on the boundary of the solution domain. Its general form can be expressed as:

[0049]

[0050] Where u is the function to be solved, Indicates that the function u is along the boundary The derivative of the unit external normal direction n is called the normal derivative or boundary derivative, and g is a known function that represents a given value on the boundary, which can be a constant, a variable, or a more complex expression. In the finite element method, the treatment of Neumann boundary conditions usually involves modifying the contribution to the boundary integral during the assembly of the right-hand side of the discretized linear system to reflect the derivative conditions on the boundary.

[0051] Example 1

[0052] According to an embodiment of the present application, a method embodiment of a post-processing method for displacement eigenvalues ​​is provided. It should be noted that the steps shown in the flowchart of the accompanying drawings can be executed in a computer system such as a set of computer executable instructions, and although a logical order is shown in the flowchart, in some cases, the steps shown or described can be executed in an order different from that shown here.

[0053] The method embodiments provided in the embodiments of the present application can be executed in a mobile terminal, a computer terminal or a similar computing device. Figure 1 FIG. 1 shows a hardware structure block diagram of a computer terminal (or mobile device) for implementing a post-processing method for displacement characteristic values. Figure 1 As shown, the computer terminal 10 (or mobile device 10) may include one or more (102a, 102b, ..., 102n are used to illustrate) processors 102 (the processor 102 may include but is not limited to a processing device such as a microprocessor MCU or a programmable logic device FPGA), a memory 104 for storing data, and a transmission device 106 for communication functions. In addition, it may also include: a display, an input / output interface (I / O interface), a universal serial bus (USB) port (which may be included as one of the ports of the BUS bus), a network interface, a power supply and / or a camera. It can be understood by those skilled in the art that Figure 1 The structure shown is only for illustration and does not limit the structure of the above electronic device. Figure 1 More or fewer components as shown, or with Figure 1 Different configurations are shown.

[0054] It should be noted that the one or more processors 102 and / or other data processing circuits described above may generally be referred to herein as "data processing circuits". The data processing circuits may be embodied in whole or in part as software, hardware, firmware, or any other combination thereof. In addition, the data processing circuit may be a single independent processing module, or may be incorporated in whole or in part into any of the other components in the computer terminal 10 (or mobile device). As involved in the embodiments of the present application, the data processing circuit acts as a processor control (e.g., selection of a variable resistor terminal path connected to an interface).

[0055] The memory 104 can be used to store software programs and modules of application software, such as program instructions / data storage devices corresponding to the post-processing method of displacement characteristic values ​​in the embodiment of the present application. The processor 102 executes various functional applications and data processing by running the software programs and modules stored in the memory 104, that is, the post-processing method of displacement characteristic values ​​of the above-mentioned application program is realized. The memory 104 may include a high-speed random access memory, and may also include a non-volatile memory, such as one or more magnetic storage devices, flash memory, or other non-volatile solid-state memory. In some examples, the memory 104 may further include a memory remotely arranged relative to the processor 102, and these remote memories may be connected to the computer terminal 10 via a network. Examples of the above-mentioned network include, but are not limited to, the Internet, an intranet, a local area network, a mobile communication network, and combinations thereof.

[0056] The transmission device 106 is used to receive or send data via a network. The specific example of the above network may include a wireless network provided by a communication provider of the computer terminal 10. In one example, the transmission device 106 includes a network adapter (Network Interface Controller, NIC), which can be connected to other network devices through a base station so as to communicate with the Internet. In one example, the transmission device 106 can be a radio frequency (RF) module, which is used to communicate with the Internet wirelessly.

[0057] The display may be, for example, a touch screen liquid crystal display (LCD) that enables a user to interact with a user interface of the computer terminal 10 (or mobile device).

[0058] In the above operating environment, Figure 2 is a flow chart of an optional post-processing method of displacement characteristic values ​​according to an embodiment of the present application, such as Figure 2 As shown, the method at least includes steps S1-S10, wherein:

[0059] Step S1, obtaining unit data, material parameters, boundary conditions and post-processing methods of multiple units obtained by dividing the structure.

[0060] Specifically, when simulating and analyzing structural mechanics, the unit shape, number of units, and unit dimensions of the divided structure play a vital role in the accuracy of the simulation results. In order to simplify the structure to meet the accuracy requirements and quickly solve the simulation results, the structure can be divided according to the actual situation by selecting a grid of a suitable shape, and the number of units and unit dimensions obtained by the specific division can be determined according to the size and dimension of the divided grid shape and the actual area. In addition, the boundary conditions, material coefficients, and body forces of the unit can be determined according to the constraints, material coefficients, and body forces of the actual structure. Among them, the structures analyzed in this application include: thin films, thin plates; the above-mentioned boundary conditions include: force constraints, displacement constraints; the above-mentioned material parameters include: first material parameters related to the film, and second material parameters related to the thin plate, and the first material parameters include: surface tension coefficient, unit mass density, and the second material parameters include: bending stiffness, unit mass density.

[0061] In addition, since the purpose of this application is to obtain high-precision eigenvalues, after using the finite element method to solve the eigenvalue problem to obtain the corresponding eigenvalues, a post-processing method is used to process the eigenvalues, wherein the post-processing method includes: reconstructing eigenvalues ​​and combining eigenvalues.

[0062] Step S2, determining the finite element type and the derivative order k used to solve the eigenvalue problem corresponding to the structure according to the material parameters.

[0063] Specifically, the type of eigenvalue problem corresponding to the structure can be determined based on the material parameters, and then the finite element type and derivative order k used to solve the eigenvalue problem can be determined. When the material parameter is the first material parameter, that is, the structure is a thin film, the corresponding eigenvalue problem is of the form , and the finite element type used to solve the eigenvalue problem, that is, the finite element type of the unit is Crouzeix-Raviart element (abbreviated as CR element) or augmented Crouzeix-Raviart element (abbreviated as augmented CR element), that is, the finite element type of multiple units that divide each structure is CR element or augmented CR element, and the corresponding derivative order is k1, where the value of k1 is 1.

[0064] When the material parameter is the second material parameter, that is, the structure is a thin plate, the corresponding eigenvalue problem is as follows: The eigenvalue problem is solved, and the finite element type used to solve the eigenvalue problem, that is, the finite element type of the unit is Morley element, that is, the finite element type of multiple units that divide each structure is Morley unit, and the corresponding derivative order is k2, where the value of k2 is 2.

[0065] Step S3, looping over each unit of the structure, obtaining the unit area, each integration point and the integration weight of each integration point of the current unit, and combining the unit data obtained in step S1 with the finite element type and the derivative order k obtained in step S2 to determine the finite element basis function corresponding to the finite element type and the k-order derivative of the finite element basis function at each integration point in the current unit.

[0066] That is to say, for each unit obtained by dividing the structure, the following steps are executed in a loop: the unit area of ​​the current unit, each integration point and the integration weight of each integration point are obtained, and the finite element basis function corresponding to the finite element type is determined in combination with the unit data unit of the current unit obtained in step S1 and the finite element type of the current unit determined in step S2 (i.e., the finite element type), and then the k-order derivative of the finite element basis function corresponding to the finite element type is calculated at each integration point in the current unit in combination with the derivative order k determined in step S2.

[0067] Step S4, determining the unit stiffness matrix and unit mass matrix of the current unit using the finite element basis function corresponding to the finite element type obtained in step S3 and the values ​​of the k-order derivatives of the finite element basis function at each integration point in the current unit.

[0068] As an optional implementation, in the technical solution provided in the above step S4, each unit can be traversed, and the unit stiffness matrix and unit mass matrix of the unit can be obtained through the following steps, including:

[0069] Step S41, determining the unit mass matrix using the values ​​of the finite element basis functions corresponding to the finite element obtained in step S3 at each integration point in the current unit and the material coefficients obtained in step S1.

[0070] Step S42, determining the unit stiffness matrix using the values ​​of the k-th order derivative of the finite element basis function corresponding to the finite element obtained in step S3 at each integration point in the current unit and the material coefficient obtained in step S1.

[0071] Step S5, determining the overall degrees of freedom of the finite element, and combining the unit stiffness matrix and unit mass matrix of each unit obtained from steps S3 to S4 to obtain the overall stiffness matrix A and the overall mass matrix M.

[0072] As an optional implementation, Figure 3 is a flow chart of an optional step S5 according to an embodiment of the present application, such as Figure 3 As shown. By traversing each unit and performing the following steps in sequence, the overall stiffness matrix A and the overall mass matrix M are obtained, including:

[0073] Step S51, determining the finite element type.

[0074] Specifically, the finite element type defines that all units obtained by dividing the structure are the same finite element. Therefore, when the overall stiffness matrix A and the overall mass matrix M of the structure are obtained, the finite element types of all units can be determined. Wherein, when the above finite element type is Crouzeix-Raviart element, step S52 is executed; when the above finite element type is augmented Crouzeix-Raviart element, step S53 is executed; when the finite element type is coordinated linear element, step S54 is executed; when the finite element type is Morley element, step S55 is executed.

[0075] Step S52, taking the integrated average value of the function on each unit edge in the unit as the degree of freedom of the Crouzeix-Raviart element.

[0076] Specifically, the degrees of freedom of the Crouzeix-Raviart element are defined on the edges of the element, so the expression for the degrees of freedom of the Crouzeix-Raviart element can be written as: , where e represents the unit edge and u is a given function.

[0077] Therefore, the degrees of freedom of the Crouzeix-Raviart element can be understood as three degrees of freedom for each triangular element, one for each edge of the element, which are defined by integrating the function on the edge, ensuring the incompatibility of the discrete function. For example, for a triangular element, the basis functions of the Crouzeix-Raviart element are related to the barycentric coordinates of the triangle and satisfy specific local properties, so that the average value of the integral on the element edge is 1, while the average value of the integral on other edges is zero.

[0078] Step S53, taking the integral average of the function on each unit edge in the unit and the integral average of the function on the unit as the degrees of freedom of the augmented Crouzeix-Raviart element.

[0079] Specifically, the degrees of freedom of the augmented Crouzeix-Raviart element are defined on the edge of the element and inside the element, which can be obtained by integrating the local support of the basis function. Therefore, the expression of the degrees of freedom of the augmented Crouzeix-Raviart element can be written as: and , where K represents the unit, e represents the unit edge, and u is a given function.

[0080] Step S54, taking the function value at each unit vertex in the unit as the degree of freedom of the coordinated linear element.

[0081] Specifically, the degrees of freedom of a conforming linear element are defined at the element nodes. In the two-dimensional case, the finite element space of a conforming linear element can be spanned by the hat functions (also called shape functions) at the nodes.

[0082] Step S55, taking the function value of the function at each unit vertex and the integral average of the normal derivative of the function at each unit edge as the degree of freedom of the Morley element.

[0083] Specifically, the degrees of freedom of the Morley element are defined on the edges and vertices of the element, that is, for each element, its degrees of freedom are the integrated average of the normal derivative on the edge and the function value at the vertex of the element.

[0084] Step S56, assembling the unit stiffness matrix and unit mass matrix of each unit according to the degrees of freedom of various finite elements obtained in the above steps S52 to S55, to obtain the overall stiffness matrix A and the overall mass matrix M.

[0085] It should be noted that the function mentioned in the above steps S52-S55 is any function that can define a corresponding point value or integral value, which can be a discrete function or a continuous function. The embodiment of the present application does not impose any specific restrictions on the form of the function.

[0086] Step S6, processing the overall stiffness matrix A and the overall mass matrix M obtained in step S5 according to the type of boundary conditions to obtain the corresponding matrix eigenvalue problem .

[0087] As an optional implementation, Figure 4 is a flow chart of an optional step S6 according to an embodiment of the present application, such as Figure 4 As shown, for each boundary condition of the structure, the overall stiffness matrix A and the overall mass matrix M are processed according to the following rules, including:

[0088] Step S61, determining the type of current boundary condition.

[0089] Specifically, the types of the above-mentioned boundary conditions include: displacement constraint and force constraint. Among them, the displacement constraint specifies the function value on the boundary, so it belongs to the Dirichlet boundary condition. Therefore, when the type of the boundary condition is the displacement constraint, execute step S62; and the force constraint is a force condition on the boundary, which belongs to the Neumann boundary condition. Therefore, when the type of the boundary condition is the force constraint, execute step S63.

[0090] Step S62, processing the overall stiffness matrix A and the overall mass matrix M according to the following rules:

[0091] Step S621, determine the finite element type corresponding to the unit to which the current boundary condition belongs, wherein when the finite element type is Crouzeix-Raviart element or augmented Crouzeix-Raviart element, execute step S622; when the finite element type is Morley element, execute step S623.

[0092] Step S622: All degrees of freedom corresponding to the boundary edges are taken as boundary degrees of freedom, and the rows and columns corresponding to the boundary degrees of freedom in the overall stiffness matrix A and the overall mass matrix M are deleted.

[0093] That is to say, all degrees of freedom of the unit boundary edges corresponding to the current boundary conditions are taken as boundary degrees of freedom, and the rows and columns corresponding to the boundary degrees of freedom in the overall stiffness matrix A and the overall mass matrix M obtained in step S5 are deleted to modify the overall stiffness matrix A and the overall mass matrix M.

[0094] Step S623: All degrees of freedom corresponding to the boundary edges and vertices on the boundary edges are taken as boundary degrees of freedom, and the rows and columns corresponding to the boundary degrees of freedom in the overall stiffness matrix A and the overall mass matrix M are deleted.

[0095] That is to say, the degrees of freedom of the integrated average value on the boundary edge corresponding to the current boundary condition and all the degrees of freedom of the vertices on the boundary edge are taken as boundary degrees of freedom, and the rows and columns corresponding to the boundary degrees of freedom in the overall stiffness matrix A and the overall mass matrix M obtained in step S5 are deleted to modify the overall stiffness matrix A and the overall mass matrix M.

[0096] Step S63, keeping the overall stiffness matrix A and the overall mass matrix M unchanged.

[0097] Therefore, after traversing each boundary condition in sequence through the above steps S61-S63, the adjusted overall stiffness matrix A and the overall mass matrix M can be obtained, and the matrix eigenvalue problem is constructed by the adjusted overall stiffness matrix A and the overall mass matrix M.

[0098] Step S7, solving the matrix eigenvalue problem , and obtain the discrete eigenvalues ​​corresponding to the finite element type and the discrete eigenvector x, and determine the discrete eigenfunction corresponding to the finite element type in combination with the finite element basis function corresponding to the finite element type obtained in step S3.

[0099] As an optional implementation, Figure 5 is a flow chart of an optional step S7 according to an embodiment of the present application, such as Figure 5 As shown. The discrete characteristic function corresponding to the finite element type can be obtained by following the following steps, including:

[0100] Step S71, solving the matrix eigenvalue problem , and obtain the discrete eigenvalues ​​corresponding to the finite element type and the discrete eigenvector x.

[0101] Step S72, looping through each degree of freedom corresponding to the finite element type to determine the type of the degree of freedom;

[0102] Step S73, when the type of the degree of freedom is a boundary degree of freedom, determining the value of the discrete characteristic function on the degree of freedom using the corresponding boundary condition;

[0103] Step S74, when the type of the degree of freedom is a non-boundary degree of freedom, using the discrete characteristic vector x to determine the value of the discrete characteristic function on the degree of freedom;

[0104] Step S75, determining the discrete characteristic function corresponding to the finite element type using the values ​​of the discrete characteristic function on the degrees of freedom obtained in steps S73 and S74 and the finite element basis functions corresponding to the finite element type.

[0105] Step S8, determine the k-th order derivative function of the discrete characteristic function corresponding to the finite element type, and use the value of the k-th order derivative function of the discrete characteristic function corresponding to the finite element type at each integration point in each unit to determine the reconstructed discrete k-th order derivative function belonging to the Crouzeix-Raviart element discrete space.

[0106] As an optional implementation, Figure 6 is a flow chart of an optional step S8 according to an embodiment of the present application, such as Figure 6 For each unit obtained by dividing the structure, the reconstructed discrete k-order derivative function is obtained through the following steps, including:

[0107] Step S81, determining the k-th order derivative function of the discrete characteristic function corresponding to the finite element type;

[0108] Step S82, traverse each cell edge of the cell and determine the type of the cell edge.

[0109] The types of the unit edges include: common edges and boundary edges. When the unit edge type is a common edge, step S83 is executed; when the unit edge type is a boundary edge, step S84 is executed.

[0110] Step S83, solving the k-order derivative of the discrete characteristic function at the midpoint of the unit edge in two adjacent units, and taking the average of the two k-order derivatives as the value of the Crouzeix-Raviart element degree of freedom corresponding to the reconstructed discrete k-order derivative function at the unit edge.

[0111] That is to say, if For unit and The midpoint of the common side of The value at the midpoint of the triangle side can be written as:

[0112]

[0113] Where q is the k-order derivative function of the discrete characteristic function, so, Represents the discrete characteristic function in the unit The unit edge The k-order derivative function on , Represents the discrete characteristic function in the unit The unit edge The k-th derivative function on .

[0114] Step S84, determine two midpoints that are collinear with the midpoint of the unit edge from multiple midpoints of other unit edges that share a vertex with the unit edge, and use the average value of the k-order derivatives of these two midpoints for interpolation to obtain the value of the Crouzeix-Raviart element degree of freedom of the reconstructed discrete k-order derivative function corresponding to the current unit edge.

[0115] Specifically, Figure 7 Take the two adjacent cells shown as an example, m is the boundary cell The midpoint of an edge e on the An adjacent unit , record unit and unit The common edge of ,unit The other side of , therefore, the expression for reconstructing the value of the Crouzeix-Raviart element degree of freedom corresponding to the discrete k-th order derivative function at the current unit edge can be written as:

[0116]

[0117] Step S85, obtaining the reconstructed discrete k-order derivative function according to the values ​​of the Crouzeix-Raviart element degrees of freedom corresponding to each unit edge of the reconstructed discrete k-order derivative function determined in step S83 and step S84.

[0118] Therefore, the above expression for reconstructing the discrete k-th order derivative function can be written as:

[0119]

[0120] In the formula represents the basis function corresponding to the Crouzeix-Raviart element and the unit edge e, Represents the value of the Crouzeix-Raviart element degrees of freedom corresponding to the reconstructed discrete k-th order derivative function at the current unit edge.

[0121] Step S9, using the discrete eigenvalues ​​corresponding to the finite element type , discrete characteristic functions and reconstructed discrete k-th order derivative functions to determine the a posteriori error estimator corresponding to the finite element type.

[0122] As an optional implementation, Figure 8 is a flow chart of an optional step S9 according to an embodiment of the present application, such as Figure 8 As shown, for each unit obtained by dividing the structure, the a posteriori error estimator corresponding to the finite element type is obtained according to the following process, including:

[0123] Step S91, calculate the difference between the value of the reconstructed discrete k-th order derivative function corresponding to the non-conforming finite element at each integration point in the current unit and the value of the k-th order derivative of the discrete characteristic function corresponding to the non-conforming finite element at each integration point in the current unit, and use the square of the difference and the total weight of the integral corresponding to each integration point to determine the first integral value in the area where the structure is located.

[0124] The non-conforming finite element includes: Crouzeix-Raviart element, augmented Crouzeix-Raviart element or Morley element. Therefore, the expression of the first integral value can be written as:

[0125]

[0126] In the formula represents the reconstructed discrete k-order derivative function corresponding to the non-conforming finite element, represents the k-order derivative of the discrete characteristic function corresponding to the non-conforming finite element, Indicates the force-bearing area of ​​the structure.

[0127] Step S92, when using Crouzeix-Raviart element or augmented Crouzeix-Raviart element to solve the eigenvalue problem corresponding to the film, determine the first kind of posterior error estimator corresponding to the Crouzeix-Raviart element or augmented Crouzeix-Raviart element according to the following steps, including:

[0128] Step S921, determining a first kind polynomial that depends only on the k+1 order derivative of the discrete characteristic function according to the finite element type;

[0129] Step S922, using the reconstructed discrete k-order derivative function corresponding to the Crouzeix-Raviart element or the augmented Crouzeix-Raviart element to calculate the reconstructed discrete k+1-order derivative value on the current element, and using the obtained reconstructed discrete k+1-order derivative value to calculate the value of the first kind polynomial on the current element;

[0130] Step S923, determining a second integral value by using the value of the first kind polynomial on the current unit, the value of the discrete characteristic function corresponding to the Crouzeix-Raviart element or the augmented Crouzeix-Raviart element at each integral point on the current unit, and the integral weight corresponding to each integral point;

[0131] Step S924, calculate the product of the discrete eigenvalue corresponding to the Crouzeix-Raviart element or the augmented Crouzeix-Raviart element and the sum of the second integral values ​​of all units, and use the first integral value minus twice the product to obtain the first type posterior error estimator corresponding to the Crouzeix-Raviart element or the augmented Crouzeix-Raviart element.

[0132] Therefore, the expression of the first-kind a posteriori error estimator corresponding to the Crouzeix-Raviart element can be written as:

[0133] Wherein, the above K represents a unit, represents the stress-bearing area of ​​the structure, represents the polynomial of the first kind corresponding to the Crouzeix-Raviart element, represents the discrete characteristic function corresponding to the Crouzeix-Raviart element, represents the discrete eigenvalue corresponding to the Crouzeix-Raviart element, represents the reconstructed discrete k-order derivative value corresponding to the Crouzeix-Raviart element, represents the value of the first kind polynomial corresponding to the Crouzeix-Raviart element on the current element, Represents the second integral value corresponding to the Crouzeix-Raviart element.

[0134] The expression of the first-kind a posteriori error estimator corresponding to the augmented Crouzeix-Raviart element can be written as:

[0135]

[0136] Wherein, the above K represents a unit, represents the stress-bearing area of ​​the structure, represents the polynomial of the first kind corresponding to the augmented Crouzeix-Raviart element, represents the discrete characteristic function corresponding to the augmented Crouzeix-Raviart element, represents the discrete eigenvalue corresponding to the augmented Crouzeix-Raviart element, represents the reconstructed discrete k-order derivative value corresponding to the augmented Crouzeix-Raviart element, represents the value of the first kind polynomial corresponding to the augmented Crouzeix-Raviart element on the current element, represents the second integral value corresponding to the augmented Crouzeix-Raviart element.

[0137] It should be noted that the above The expression can be written as:

[0138]

[0139]

[0140] And in the above formula,

[0141]

[0142]

[0143]

[0144]

[0145]

[0146]

[0147] in, represents the centroid of the triangular element, represents the coordinates of the i-th vertex, I represents the identity transformation, Representation vector , Representation vector , represents the interpolation corresponding to the augmented Crouzeix-Raviart element.

[0148] Step S93, when using Crouzeix-Raviart element or augmented Crouzeix-Raviart element to solve the eigenvalue problem corresponding to the film, determine the second kind of posterior error estimator corresponding to the Crouzeix-Raviart element or augmented Crouzeix-Raviart element according to the following method:

[0149] Step S931: determine a second type polynomial that depends only on the k+1 order derivative of the discrete characteristic function based on the coordinated linear element.

[0150] Step S932, using the reconstructed discrete k-order derivative function corresponding to the Crouzeix-Raviart element or the augmented Crouzeix-Raviart element to calculate the reconstructed discrete k+1-order derivative value on the current unit, and using the obtained reconstructed discrete k+1-order derivative value to calculate the value of the second kind polynomial in the current unit and the value of the second kind polynomial at the integral point on each unit edge of the current unit;

[0151] Step S933, traverse each unit edge on the current unit, determine the average value of the second kind polynomial in the two adjacent units to which the current unit edge belongs, and calculate the product of the average value and the jump of the normal derivative of the discrete characteristic function corresponding to the Crouzeix-Raviart element or the augmented Crouzeix-Raviart element on the current unit edge, and use the obtained product result and the integral weights corresponding to each integral point to calculate the third integral value of the current unit edge;

[0152] Step S934, calculating the product of the discrete characteristic function corresponding to the Crouzeix-Raviart element or the augmented Crouzeix-Raviart element and the discrete characteristic value, and adding the product to the Laplace of the corresponding discrete characteristic function, calculating the value of the product of the addition result and the second kind polynomial in the current unit, and combining the integral weights corresponding to each integral point in the current unit to obtain a fourth integral value;

[0153] Step S935, add twice the sum of the third integral values ​​of all unit edges to the first integral value, and then subtract twice the sum of the fourth integral values ​​of all units to obtain a second type posterior error estimator corresponding to the Crouzeix-Raviart element or augmented Crouzeix-Raviart element.

[0154] Therefore, the expression of the second-type a posteriori error estimator corresponding to the Crouzeix-Raviart element can be written as:

[0155]

[0156] In the formula represents a polynomial of the second kind, Denotes the third integral value corresponding to the Crouzeix-Raviart element.

[0157] The expression of the second type of posterior error estimator corresponding to the augmented Crouzeix-Raviart element can be written as:

[0158]

[0159] In the formula represents a polynomial of the second kind, represents the third integral value corresponding to the augmented Crouzeix-Raviart element, represents a polynomial of the second kind.

[0160] Among them, the above The expression can be written as:

[0161]

[0162] In the formula represents the barycentric coordinates of the triangular element, Represents the counterclockwise unit tangent vector of the i-th edge of the triangle element.

[0163] Step S94, when the Morley element is used to solve the eigenvalue problem corresponding to the thin plate, a third-kind a posteriori error estimator corresponding to the Morley element is determined based on the first integral value in step S91.

[0164] Therefore, when k in the first integral value in step S91 is set to 2, the third type of posterior error estimator corresponding to the Morley element can be obtained based on the first integral value in step S91, and its expression can be written as:

[0165]

[0166] Among them, the above represents the stress-bearing area of ​​the structure, Represents the discrete characteristic function corresponding to the Morley element.

[0167] Step S10, according to the post-processing method determined in step S1, and using the a posteriori error estimator corresponding to the finite element type obtained in step S9 to calculate the discrete eigenvalues ​​of the corresponding finite element type After post-processing, the final eigenvalues ​​are obtained.

[0168] As an optional implementation, in the technical solution provided in the above step S10, the discrete eigenvalues ​​of the corresponding finite element type can be calculated according to the following rules: Perform post-processing, including:

[0169] Step S101, when the post-processing method determined in step S1 is to reconstruct eigenvalues, the discrete eigenvalues ​​and the posterior error estimator corresponding to the non-conforming finite element are added to obtain the final eigenvalues.

[0170] For the Crouzeix-Raviart element, if the reconstructed eigenvalue method is used to calculate the discrete eigenvalue After post-processing, the final eigenvalue expression can be written as:

[0171]

[0172] For the augmented Crouzeix-Raviart element, if the reconstructed eigenvalue method is used to calculate the discrete eigenvalue After post-processing, the final eigenvalue expression can be written as:

[0173]

[0174] For Morley element, if the reconstructed eigenvalue method is used to calculate the discrete eigenvalue After post-processing, the final eigenvalue expression can be written as:

[0175]

[0176] Step S102, when the post-processing method determined in step S1 is the combined eigenvalue, the discrete eigenvalues ​​corresponding to any two finite element types are linearly combined according to a preset combination coefficient to obtain a final eigenvalue, wherein the combination coefficient is determined by the a posteriori error estimator corresponding to the finite element.

[0177] For example, for the Crouzeix-Raviart element, if the combined eigenvalue method is used to calculate the discrete eigenvalues

[0178] After post-processing, the final eigenvalue expression can be written as:

[0179]

[0180] In the above formula ,and is the discrete characteristic function corresponding to the Crouzeix-Raviart element The average projection in the coordinated linear element space, High-precision gradient reconstruction technique for coordination elements It should be noted that the calculation process of the discrete eigenvalues ​​and discrete eigenfunctions corresponding to the coordination element is a prior art, and this application will not explain the calculation process in detail.

[0181] Based on the scheme defined by the above steps S1 to S10, it can be known that, in an embodiment, the present application designs corresponding a posteriori error estimators for different types of non-conforming elements, and uses the a posteriori error estimators of each type of non-conforming element to post-process the eigenvalues ​​calculated by the non-conforming element of that type, so that the discrete eigenvalues ​​obtained by using the non-conforming elements to calculate the eigenvalue problem of the structure are more accurate, thereby solving the technical problem that it is difficult to obtain high-precision eigenvalues ​​when the related art uses non-conforming finite elements to process the vibration problem of the structure.

[0182] In practical applications, thin films are an important form of materials, and the study and control of their vibration behavior has a wide range of applications in engineering, such as structural design, precision acousto-optical instrument manufacturing, biomedical materials, and many other aspects. Among them: In acoustic and optical devices, the vibration mode of the film will directly affect the propagation of sound waves and light waves, thereby affecting the performance of the equipment; and in the field of materials engineering, the vibrational dynamics of the film is of great significance for the development of new materials, the evaluation of its reliability and durability, and the optimization of manufacturing processes. One of the core contents of studying the vibration problem of thin films is the solution of eigenvalues, that is, the solution of the natural frequency and vibration mode of the structure. Therefore, the post-processing method of the displacement eigenvalue will be used to solve Fig. 9 The natural frequency (i.e., the eigenvalue obtained by solving the matrix eigenvalue problem of the H-type membrane) shown (whose boundary condition is the zero boundary value condition) is compared with the results of the commercial software Abaqus to prove the effectiveness of the embodiment of the present application in reconstructing high-precision eigenvalues.

[0183] Specifically, in the verification process, different numbers of grids are used to divide the above Fig. 9 The H-type film shown in the figure is Figure 10-15 The grid models 1-6 shown, wherein the above Fig.10 (i.e. mesh model 1) is the initial mesh model, Figure 11-15 (i.e. other grid models 2-6) are all obtained by encrypting the previous grid, so the number of grids and model degrees of freedom of grid models 1-6 are shown in Table 1 below.

[0184] Table 1

[0185]

[0186] Based on Table 1 above, we can first use the non-conforming Crouzeix-Raviart element to solve the eigenvalue problem corresponding to the H-type membrane on the benchmark grid model, and then use the solution results is regarded as the reference value; then, on the grid models 1-6, the eigenvalue problem corresponding to the H-type membrane is solved by using the non-conforming Crouzeix-Raviart element, and the post-processed eigenvalues ​​obtained by using the first-kind a posteriori error estimate to solve the corresponding eigenvalue problem using the non-conforming Crouzeix-Raviart element are post-processed to obtain the post-processed eigenvalues ​​(i.e. ), using the second-kind a posteriori error estimate to post-process the eigenvalues ​​obtained by solving the corresponding eigenvalue problem using the non-coordinated Crouzeix-Raviart element, and obtain the post-processed eigenvalues ​​(i.e. ), and compare the error between the solution and the above reference value, and get Fig.16 The relative error diagram is shown in the figure and the relative error results are shown in the following Table 2.

[0187] Table 2

[0188]

[0189] Combining the above Table 2 and Fig.16 It is not difficult to see that for mesh models with different degrees of freedom, the final eigenvalue obtained by post-processing the eigenvalue directly solved by the displacement eigenvalue post-processing method provided in the embodiment of the present application has a lower error than the reference value. Therefore, it is proved that the eigenvalue calculated by the post-processing method provided in the embodiment of the present application is better than the eigenvalue solved by the direct non-coordinated Crouzeix-Raviart element when the mesh scale is the same. The accuracy is higher.

[0190] Example 2

[0191] According to an embodiment of the present application, a non-volatile storage medium is further provided, in which a program is stored, wherein when the program is running, the device where the non-volatile storage medium is located is controlled to execute the post-processing method of the displacement characteristic value in Example 1.

[0192] Optionally, the device where the non-volatile storage medium is located implements the following steps by running the program:

[0193] Step S1, obtaining unit data, material parameters, boundary conditions and post-processing methods of multiple units obtained by dividing the structure, wherein the structure includes: a film or a thin plate, the unit data includes: unit shape, node coordinates, unit quantity, unit dimension, boundary conditions include: force constraints, displacement constraints, material parameters include: a first material parameter related to the film, a second material parameter related to the thin plate, and the first material parameter includes: surface tension coefficient, unit mass density, the second material parameter includes: bending stiffness, unit mass density, and the post-processing method includes: reconstructing eigenvalues ​​or combining eigenvalues;

[0194] Step S2, determining the finite element type and the derivative order k used to solve the eigenvalue problem corresponding to the structure according to the material parameters, wherein the finite element type includes: Crouzeix-Raviart element or augmented Crouzeix-Raviart element corresponding to the film, a compatible linear element, and a Morley element corresponding to the thin plate, and the derivative order k includes: derivative order k1 corresponding to the film or derivative order k2 corresponding to the thin plate;

[0195] Step S3, looping through each unit of the structure, obtaining the unit area, each integration point and the integration weight of each integration point of the current unit, and combining the unit data obtained in step S1 with the finite element type and the derivative order k obtained in step S2 to determine the value of the finite element basis function corresponding to the finite element type and the k-order derivative of the finite element basis function at each integration point in the current unit;

[0196] Step S4, determining the unit stiffness matrix and unit mass matrix of the current unit using the finite element basis function corresponding to the finite element type obtained in step S3 and the value of the k-order derivative of the finite element basis function at each integration point in the current unit;

[0197] Step S5, determining the degrees of freedom of the finite element, and combining the unit stiffness matrix and unit mass matrix of each unit obtained from steps S3 to S4 to obtain the overall stiffness matrix A and the overall mass matrix M;

[0198] Step S6, processing the overall stiffness matrix A and the overall mass matrix M obtained in step S5 according to the type of boundary conditions to obtain the corresponding matrix eigenvalue problem ;

[0199] Step S7, solving the matrix eigenvalue problem , and obtain the discrete eigenvalues ​​corresponding to the finite element type and the discrete characteristic vector x, and determine the discrete characteristic function corresponding to the finite element type in combination with the finite element basis function corresponding to the finite element type obtained in step S3;

[0200] Step S8, determining the k-th order derivative function of the discrete characteristic function corresponding to the finite element type, and using the value of the k-th order derivative function of the discrete characteristic function corresponding to the finite element type at each integration point in each unit, determining the reconstructed discrete k-th order derivative function belonging to the Crouzeix-Raviart element discrete space;

[0201] Step S9, using the discrete eigenvalues ​​corresponding to the finite element type , discrete characteristic functions and reconstructed discrete k-th order derivative functions to determine a posteriori error estimator corresponding to the finite element type;

[0202] Step S10, according to the post-processing method determined in step S1, and using the a posteriori error estimator corresponding to the finite element type obtained in step S9 to calculate the discrete eigenvalue corresponding to the corresponding finite element type After post-processing, the final eigenvalues ​​are obtained.

[0203] According to an embodiment of the present application, a computer program product is also provided, which includes a stored computer program, wherein when the computer program is executed by a processor, the post-processing method of the displacement characteristic value in Example 1 is implemented.

[0204] Optionally, the computer program executes the following steps:

[0205] Step S1, obtaining unit data, material parameters, boundary conditions and post-processing methods of multiple units obtained by dividing the structure, wherein the structure includes: a film or a thin plate, the unit data includes: unit shape, node coordinates, unit quantity, unit dimension, boundary conditions include: force constraints, displacement constraints, material parameters include: a first material parameter related to the film, a second material parameter related to the thin plate, and the first material parameter includes: surface tension coefficient, unit mass density, the second material parameter includes: bending stiffness, unit mass density, and the post-processing method includes: reconstructing eigenvalues ​​or combining eigenvalues;

[0206] Step S2, determining the finite element type and the derivative order k used to solve the eigenvalue problem corresponding to the structure according to the material parameters, wherein the finite element type includes: Crouzeix-Raviart element or augmented Crouzeix-Raviart element corresponding to the film, a compatible linear element, and a Morley element corresponding to the thin plate, and the derivative order k includes: derivative order k1 corresponding to the film or derivative order k2 corresponding to the thin plate;

[0207] Step S3, looping through each unit of the structure, obtaining the unit area, each integration point and the integration weight of each integration point of the current unit, and combining the unit data obtained in step S1 with the finite element type and the derivative order k obtained in step S2 to determine the value of the finite element basis function corresponding to the finite element type and the k-order derivative of the finite element basis function at each integration point in the current unit;

[0208] Step S4, determining the unit stiffness matrix and unit mass matrix of the current unit using the finite element basis function corresponding to the finite element type obtained in step S3 and the value of the k-order derivative of the finite element basis function at each integration point in the current unit;

[0209] Step S5, determining the degrees of freedom of the finite element, and combining the unit stiffness matrix and unit mass matrix of each unit obtained from steps S3 to S4 to obtain the overall stiffness matrix A and the overall mass matrix M;

[0210] Step S6, processing the overall stiffness matrix A and the overall mass matrix M obtained in step S5 according to the type of boundary conditions to obtain the corresponding matrix eigenvalue problem ;

[0211] Step S7, solving the matrix eigenvalue problem , and obtain the discrete eigenvalues ​​corresponding to the finite element type and the discrete characteristic vector x, and determine the discrete characteristic function corresponding to the finite element type in combination with the finite element basis function corresponding to the finite element type obtained in step S3;

[0212] Step S8, determining the k-th order derivative function of the discrete characteristic function corresponding to the finite element type, and using the value of the k-th order derivative function of the discrete characteristic function corresponding to the finite element type at each integration point in each unit, determining the reconstructed discrete k-th order derivative function belonging to the Crouzeix-Raviart element discrete space;

[0213] Step S9, using the discrete eigenvalues ​​corresponding to the finite element type , discrete characteristic functions and reconstructed discrete k-th order derivative functions to determine a posteriori error estimator corresponding to the finite element type;

[0214] Step S10, according to the post-processing method determined in step S1, and using the a posteriori error estimator corresponding to the finite element type obtained in step S9 to calculate the discrete eigenvalue corresponding to the corresponding finite element type After post-processing, the final eigenvalues ​​are obtained.

[0215] According to an embodiment of the present application, a processor is further provided, which is used to run a program, wherein the post-processing method of the displacement characteristic value in Example 1 is executed when the program is running.

[0216] Optionally, the following steps are performed when the program is running:

[0217] Step S1, obtaining unit data, material parameters, boundary conditions and post-processing methods of multiple units obtained by dividing the structure, wherein the structure includes: a film or a thin plate, the unit data includes: unit shape, node coordinates, unit quantity, unit dimension, boundary conditions include: force constraints, displacement constraints, material parameters include: a first material parameter related to the film, a second material parameter related to the thin plate, and the first material parameter includes: surface tension coefficient, unit mass density, the second material parameter includes: bending stiffness, unit mass density, and the post-processing method includes: reconstructing eigenvalues ​​or combining eigenvalues;

[0218] Step S2, determining the finite element type and the derivative order k used to solve the eigenvalue problem corresponding to the structure according to the material parameters, wherein the finite element type includes: Crouzeix-Raviart element or augmented Crouzeix-Raviart element corresponding to the film, a compatible linear element, and a Morley element corresponding to the thin plate, and the derivative order k includes: derivative order k1 corresponding to the film or derivative order k2 corresponding to the thin plate;

[0219] Step S3, looping through each unit of the structure, obtaining the unit area, each integration point and the integration weight of each integration point of the current unit, and combining the unit data obtained in step S1 with the finite element type and the derivative order k obtained in step S2 to determine the value of the finite element basis function corresponding to the finite element type and the k-order derivative of the finite element basis function at each integration point in the current unit;

[0220] Step S4, determining the unit stiffness matrix and unit mass matrix of the current unit using the finite element basis function corresponding to the finite element type obtained in step S3 and the value of the k-order derivative of the finite element basis function at each integration point in the current unit;

[0221] Step S5, determining the degrees of freedom of the finite element, and combining the unit stiffness matrix and unit mass matrix of each unit obtained from steps S3 to S4 to obtain the overall stiffness matrix A and the overall mass matrix M;

[0222] Step S6, processing the overall stiffness matrix A and the overall mass matrix M obtained in step S5 according to the type of boundary conditions to obtain the corresponding matrix eigenvalue problem ;

[0223] Step S7, solving the matrix eigenvalue problem , and obtain the discrete eigenvalues ​​corresponding to the finite element type and the discrete characteristic vector x, and determine the discrete characteristic function corresponding to the finite element type in combination with the finite element basis function corresponding to the finite element type obtained in step S3;

[0224] Step S8, determining the k-th order derivative function of the discrete characteristic function corresponding to the finite element type, and using the value of the k-th order derivative function of the discrete characteristic function corresponding to the finite element type at each integration point in each unit, determining the reconstructed discrete k-th order derivative function belonging to the Crouzeix-Raviart element discrete space;

[0225] Step S9, using the discrete eigenvalues ​​corresponding to the finite element type , discrete characteristic functions and reconstructed discrete k-th order derivative functions to determine a posteriori error estimator corresponding to the finite element type;

[0226] Step S10, according to the post-processing method determined in step S1, and using the a posteriori error estimator corresponding to the finite element type obtained in step S9 to calculate the discrete eigenvalue corresponding to the corresponding finite element type After post-processing, the final eigenvalues ​​are obtained.

[0227] According to an embodiment of the present application, an electronic device is also provided, wherein the electronic device includes one or more processors; a memory for storing one or more programs, and when the one or more programs are executed by one or more processors, the one or more processors are implemented to run the programs, wherein the program is configured to execute the post-processing method of the displacement characteristic value in the above-mentioned embodiment 1 when running.

[0228] Optionally, the processor is configured to implement the following steps by executing a computer program:

[0229] Step S1, obtaining unit data, material parameters, boundary conditions and post-processing methods of multiple units obtained by dividing the structure, wherein the structure includes: a film or a thin plate, the unit data includes: unit shape, node coordinates, unit quantity, unit dimension, boundary conditions include: force constraints, displacement constraints, material parameters include: a first material parameter related to the film, a second material parameter related to the thin plate, and the first material parameter includes: surface tension coefficient, unit mass density, the second material parameter includes: bending stiffness, unit mass density, and the post-processing method includes: reconstructing eigenvalues ​​or combining eigenvalues;

[0230] Step S2, determining the finite element type and the derivative order k used to solve the eigenvalue problem corresponding to the structure according to the material parameters, wherein the finite element type includes: Crouzeix-Raviart element or augmented Crouzeix-Raviart element corresponding to the film, a compatible linear element, and a Morley element corresponding to the thin plate, and the derivative order k includes: derivative order k1 corresponding to the film or derivative order k2 corresponding to the thin plate;

[0231] Step S3, looping through each unit of the structure, obtaining the unit area, each integration point and the integration weight of each integration point of the current unit, and combining the unit data obtained in step S1 with the finite element type and the derivative order k obtained in step S2 to determine the value of the finite element basis function corresponding to the finite element type and the k-order derivative of the finite element basis function at each integration point in the current unit;

[0232] Step S4, determining the unit stiffness matrix and unit mass matrix of the current unit using the finite element basis function corresponding to the finite element type obtained in step S3 and the value of the k-order derivative of the finite element basis function at each integration point in the current unit;

[0233] Step S5, determining the degrees of freedom of the finite element, and combining the unit stiffness matrix and unit mass matrix of each unit obtained from steps S3 to S4 to obtain the overall stiffness matrix A and the overall mass matrix M;

[0234] Step S6, processing the overall stiffness matrix A and the overall mass matrix M obtained in step S5 according to the type of boundary conditions to obtain the corresponding matrix eigenvalue problem ;

[0235] Step S7, solving the matrix eigenvalue problem , and obtain the discrete eigenvalues ​​corresponding to the finite element type and the discrete eigenvector x, and determine the discrete eigenfunction corresponding to the finite element type in combination with the finite element basis function corresponding to the finite element type obtained in step S3;

[0236] Step S8, determining the k-th order derivative function of the discrete characteristic function corresponding to the finite element type, and using the value of the k-th order derivative function of the discrete characteristic function corresponding to the finite element type at each integration point in each unit, determining the reconstructed discrete k-th order derivative function belonging to the Crouzeix-Raviart element discrete space;

[0237] Step S9, using the discrete eigenvalues ​​corresponding to the finite element type , discrete characteristic functions and reconstructed discrete k-th order derivative functions to determine a posteriori error estimator corresponding to the finite element type;

[0238] Step S10, according to the post-processing method determined in step S1, and using the a posteriori error estimator corresponding to the finite element type obtained in step S9 to calculate the discrete eigenvalue corresponding to the corresponding finite element type After post-processing, the final eigenvalues ​​are obtained.

[0239] The serial numbers of the above-mentioned embodiments of the present application are for description only and do not represent the advantages or disadvantages of the embodiments.

[0240] In the above embodiments of the present application, the description of each embodiment has its own emphasis. For parts that are not described in detail in a certain embodiment, please refer to the relevant description of other embodiments.

[0241] In the several embodiments provided in this application, it should be understood that the disclosed technical content can be implemented in other ways. Among them, the device embodiments described above are only schematic. For example, the division of the units can be a logical function division. There may be other division methods in actual implementation. For example, multiple units or components can be combined or integrated into another system, or some features can be ignored or not executed. Another point is that the mutual coupling or direct coupling or communication connection shown or discussed can be through some interfaces, indirect coupling or communication connection of units or modules, which can be electrical or other forms.

[0242] The units described as separate components may or may not be physically separated, and the components shown as units may or may not be physical units, that is, they may be located in one place or distributed on multiple units. Some or all of the units may be selected according to actual needs to achieve the purpose of the present embodiment.

[0243] In addition, each functional unit in each embodiment of the present application may be integrated into one processing unit, or each unit may exist physically separately, or two or more units may be integrated into one unit. The above-mentioned integrated unit may be implemented in the form of hardware or in the form of software functional units.

[0244] If the integrated unit is implemented in the form of a software functional unit and sold or used as an independent product, it can be stored in a computer-readable storage medium. Based on this understanding, the technical solution of the present application, or the part that contributes to the relevant technology or all or part of the technical solution, can be embodied in the form of a software product. The computer software product is stored in a storage medium, including a number of instructions to enable a computer device (which can be a personal computer, server or network device, etc.) to execute all or part of the steps of the method described in each embodiment of the present application. The aforementioned storage medium includes: U disk, read-only memory (ROM, Read-Only Memory), random access memory (RAM, Random Access Memory), mobile hard disk, disk or optical disk, etc. Various media that can store program codes.

[0245] The above is only a preferred implementation of the present application. It should be pointed out that for ordinary technicians in this technical field, several improvements and modifications can be made without departing from the principles of the present application. These improvements and modifications should also be regarded as the scope of protection of the present application.

Claims

1. A post-processing method for displacement eigenvalues, characterized in that: The method comprises the following steps: Step S1, obtaining unit data, material parameters, boundary conditions and post-processing methods of multiple units obtained by dividing the structure, wherein the structure includes: a film or a thin plate, the unit data includes: unit shape, node coordinates, unit quantity, unit dimension, boundary conditions include: force constraints, displacement constraints, material parameters include: a first material parameter related to the film, a second material parameter related to the thin plate, and the first material parameter includes: surface tension coefficient, unit mass density, the second material parameter includes: bending stiffness, unit mass density, and the post-processing method includes: reconstructing eigenvalues ​​or combining eigenvalues; Step S2, determining the finite element type and the derivative order k used to solve the eigenvalue problem corresponding to the structure according to the material parameters, wherein the finite element type includes: Crouzeix-Raviart element or augmented Crouzeix-Raviart element corresponding to the film, a compatible linear element, and a Morley element corresponding to the thin plate, and the derivative order k includes: derivative order k1 corresponding to the film or derivative order k2 corresponding to the thin plate; Step S3, looping through each unit of the structure, obtaining the unit area, each integration point and the integration weight of each integration point of the current unit, and combining the unit data obtained in step S1 with the finite element type and the derivative order k obtained in step S2 to determine the value of the finite element basis function corresponding to the finite element type and the k-order derivative of the finite element basis function at each integration point in the current unit; Step S4, determining the unit stiffness matrix and unit mass matrix of the current unit using the finite element basis function corresponding to the finite element type obtained in step S3 and the value of the k-order derivative of the finite element basis function at each integration point in the current unit; Step S5, determining the degrees of freedom of the finite element, and combining the unit stiffness matrix and the unit mass matrix of each unit obtained from steps S3 to S4 to obtain the overall stiffness matrix A and the overall mass matrix M; Step S6, processing the overall stiffness matrix A and the overall mass matrix M obtained in step S5 according to the type of boundary conditions to obtain the corresponding matrix eigenvalue problem ; Step S7, solving the matrix eigenvalue problem , and obtain the discrete eigenvalues ​​corresponding to the finite element type and the discrete characteristic vector x, and determine the discrete characteristic function corresponding to the finite element type in combination with the finite element basis function corresponding to the finite element type obtained in step S3; Step S8, determining the k-th order derivative function of the discrete characteristic function corresponding to the finite element type, and using the value of the k-th order derivative function of the discrete characteristic function corresponding to the finite element type at each integration point in each unit, determining the reconstructed discrete k-th order derivative function belonging to the Crouzeix-Raviart element discrete space; Step S9, using the discrete eigenvalues ​​corresponding to the finite element type , discrete characteristic functions and reconstructed discrete k-th order derivative functions to determine a posteriori error estimator corresponding to the finite element type; Step S10, according to the post-processing method determined in step S1, and using the a posteriori error estimator corresponding to the finite element type obtained in step S9 to calculate the discrete eigenvalue corresponding to the corresponding finite element type After post-processing, the final eigenvalues ​​are obtained.

2. The method according to claim 1, characterized in that The step S4 comprises the following steps: Traverse each cell and perform the following steps: Step S41, determining the unit mass matrix using the values ​​of the finite element basis functions corresponding to the finite element type obtained in step S3 at each integration point in the current unit and the material coefficients obtained in step S1; Step S42, determining the element stiffness matrix using the values ​​of the k-th order derivative of the finite element basis function corresponding to the finite element type obtained in step S3 at each integration point in the current element and the material coefficient obtained in step S1.

3. The method according to claim 1, characterized in that The step S5 comprises the following steps: Step S51, determining the finite element type; Step S52, when the finite element type is Crouzeix-Raviart element, the integral average value of the function on each element edge in the element is used as the degree of freedom of the Crouzeix-Raviart element; Step S53, when the finite element type is an augmented Crouzeix-Raviart element, the integral average of the function on each element edge in the element and the integral average of the function on the element are used as the degrees of freedom of the augmented Crouzeix-Raviart element; Step S54, when the finite element type is a coordinated linear element, the function value at each vertex of the function in the element is used as the degree of freedom of the coordinated linear element; Step S55, when the finite element type is Morley element, the integral average of the function value at each unit vertex in the unit and the normal derivative of the function at each unit edge in the unit is used as the degree of freedom of the Morley element; Step S56, assembling the unit stiffness matrix and unit mass matrix of each unit according to the degrees of freedom of various finite elements obtained in the above steps S52 to S55, to obtain the overall stiffness matrix A and the overall mass matrix M.

4. The method according to claim 1, characterized in that The step S6 comprises the following steps: Traverse each boundary condition and perform the following steps: Step S61, determining the type of current boundary condition; Step S62: When the type of the boundary condition is displacement constraint, the overall stiffness matrix A and the overall mass matrix M are processed according to the following rules: Step S621, determining the finite element type corresponding to the unit to which the current boundary condition belongs; Step S622, when the finite element type is Crouzeix-Raviart element or augmented Crouzeix-Raviart element, all degrees of freedom corresponding to the boundary edges are taken as boundary degrees of freedom, and the rows and columns corresponding to the boundary degrees of freedom in the global stiffness matrix A and the global mass matrix M obtained in step S5 are deleted; Step S623, when the finite element type is Morley element, all degrees of freedom corresponding to the boundary edges and vertices on the boundary edges are taken as boundary degrees of freedom, and the rows and columns corresponding to the boundary degrees of freedom in the global stiffness matrix A and the global mass matrix M are deleted; Step S63, when the type of the boundary condition is force constraint, keep the overall stiffness matrix A and the overall mass matrix M unchanged.

5. The method according to claim 1, characterized in that The step S7 comprises the following steps: Step S71, solving the matrix eigenvalue problem , and obtain the discrete eigenvalues ​​corresponding to the finite element type and the discrete eigenvector x; Step S72, looping through each degree of freedom corresponding to the finite element type to determine the type of the degree of freedom; Step S73, when the type of the degree of freedom is a boundary degree of freedom, determining the value of the discrete characteristic function on the degree of freedom using the corresponding boundary condition; Step S74, when the type of the degree of freedom is a non-boundary degree of freedom, using the discrete characteristic vector x to determine the value of the discrete characteristic function on the degree of freedom; Step S75, determining the discrete characteristic function corresponding to the finite element type using the values ​​of the discrete characteristic function on the degrees of freedom obtained in steps S73 and S74 and the finite element basis functions corresponding to the finite element type.

6. The method according to claim 1, characterized in that The step S8 comprises the following steps: Step S81, determining the k-th order derivative function of the discrete characteristic function corresponding to the finite element type; Step S82, traversing each cell edge of the cell to determine the type of the cell edge, wherein the type of the cell edge includes: a common edge and a boundary edge; Step S83, when the type of the unit edge is a common edge, solving the k-order derivative of the discrete characteristic function at the midpoint of the unit edge in two adjacent units, and taking the average of the two k-order derivatives as the value of the Crouzeix-Raviart element degree of freedom corresponding to the reconstructed discrete k-order derivative function at the unit edge; Step S84, when the type of the unit edge is a boundary edge, two midpoints that are collinear with the midpoint of the unit edge are determined from multiple midpoints of other unit edges that share a vertex with the unit edge, and the average value of the k-order derivatives of the two midpoints is used for extrapolation to obtain the value of the Crouzeix-Raviart element degree of freedom corresponding to the reconstructed discrete k-order derivative function at the unit edge; Step S85, combining the values ​​of the Crouzeix-Raviart element degrees of freedom corresponding to each unit edge of the reconstructed discrete k-order derivative function obtained in step S83 and step S84, to obtain the reconstructed discrete k-order derivative function.

7. The method according to claim 1, characterized in that The step S9 comprises the following steps: Step S91, calculating the difference between the value of the reconstructed discrete k-order derivative function corresponding to the non-conforming finite element at each integral point in the current unit and the value of the k-order derivative of the discrete characteristic function corresponding to the non-conforming finite element at each integral point in the current unit, and using the square of the difference and the integral weight corresponding to each integral point to determine the first integral value in the area where the structure is located, wherein the non-conforming finite element includes: Crouzeix-Raviart element, augmented Crouzeix-Raviart element, Morley element; Step S92, when using Crouzeix-Raviart element or augmented Crouzeix-Raviart element to solve the eigenvalue problem corresponding to the film, traverse each unit and determine the first-kind a posteriori error estimator corresponding to the Crouzeix-Raviart element or augmented Crouzeix-Raviart element according to the following steps: Step S921, determining a first kind polynomial that depends only on the k+1 order derivative of the discrete characteristic function according to the finite element type; Step S922, using the reconstructed discrete k-order derivative function corresponding to the Crouzeix-Raviart element or the augmented Crouzeix-Raviart element to calculate the reconstructed discrete k+1-order derivative value on the current element, and using the obtained reconstructed discrete k+1-order derivative value to calculate the value of the first kind polynomial on the current element; Step S923, determining a second integral value by using the value of the first kind polynomial on the current unit, the value of the discrete characteristic function corresponding to the Crouzeix-Raviart element or the augmented Crouzeix-Raviart element at each integral point on the current unit, and the integral weight corresponding to each integral point; Step S924, calculating the product of the discrete eigenvalue corresponding to the Crouzeix-Raviart element or the augmented Crouzeix-Raviart element and the sum of the second integral values ​​of all the elements, and subtracting twice the product from the first integral value to obtain a first-kind a posteriori error estimator corresponding to the Crouzeix-Raviart element or the augmented Crouzeix-Raviart element; Step S93, when using Crouzeix-Raviart element or augmented Crouzeix-Raviart element to solve the eigenvalue problem corresponding to the film, traverse each unit and determine the second kind of posterior error estimator corresponding to the Crouzeix-Raviart element or augmented Crouzeix-Raviart element according to the following method: Step S931, determining a second type polynomial that depends only on the k+1 order derivative of the discrete characteristic function according to the coordinated linear element; Step S932, using the reconstructed discrete k-order derivative function corresponding to the Crouzeix-Raviart element or the augmented Crouzeix-Raviart element to calculate the reconstructed discrete k+1-order derivative value on the current unit, and using the obtained reconstructed discrete k+1-order derivative value to calculate the value of the second kind polynomial in the current unit and the value of the second kind polynomial at the integral point on each unit edge of the current unit; Step S933, traverse each unit edge on the current unit, determine the average value of the second kind polynomial in the two adjacent units to which the current unit edge belongs, and calculate the product of the average value and the jump of the normal derivative of the discrete characteristic function corresponding to the Crouzeix-Raviart element or the augmented Crouzeix-Raviart element on the current unit edge, and use the obtained product result and the integral weights corresponding to each integral point to calculate the third integral value of the current unit edge; Step S934, calculating the product of the discrete characteristic function corresponding to the Crouzeix-Raviart element or the augmented Crouzeix-Raviart element and the discrete characteristic value, and adding the product to the Laplace of the corresponding discrete characteristic function, calculating the value of the product of the addition result and the second kind polynomial in the current unit, and combining the integral weights corresponding to each integral point in the current unit to obtain a fourth integral value; Step S935, add twice the sum of the third integral values ​​of all unit edges to the first integral value, and then subtract twice the sum of the fourth integral values ​​of all units, to obtain a second-type posterior error estimator corresponding to the Crouzeix-Raviart element or the augmented Crouzeix-Raviart element; Step S94, when the Morley element is used to solve the eigenvalue problem corresponding to the thin plate, a third-kind a posteriori error estimator corresponding to the Morley element is determined based on the first integral value in step S91.

8. The method according to claim 1, characterized in that The step S10 comprises the following steps: Step S101, when the post-processing method determined in step S1 is to reconstruct eigenvalues, add the discrete eigenvalues ​​and the posterior error estimator corresponding to the non-conforming finite element type to obtain the final eigenvalues; Step S102, when the post-processing method determined in step S1 is the combined eigenvalue, the discrete eigenvalues ​​corresponding to any two finite element types are linearly combined according to a preset combination coefficient to obtain a final eigenvalue, wherein the combination coefficient is determined by the a posteriori error estimator corresponding to the finite element.

9. A non-volatile storage medium, characterized in that: The non-volatile storage medium stores a computer program, wherein the device where the non-volatile storage medium is located executes the displacement characteristic value post-processing method according to any one of claims 1 to 8 by running the computer program.

10. A computer program product, characterized in that include: A computer program, wherein when the computer program is executed by a processor, the post-processing method of the displacement characteristic value described in any one of claims 1 to 8 is implemented.

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