Radial point interpolation finite element method for free vibration of linear elastic solid and application

Abstract

The invention discloses a linear elastic solid free vibration radial point interpolation finite element method and application, and belongs to the technical field of solid mechanics and engineering numerical simulation. Discretizing a problem domain into a standard finite element grid, and deriving a nodal function based on a radial basis function; constructing a local strengthening interpolation primary function to finish local numerical value approximation; establishing a displacement vector field global interpolation format on the premise of satisfying unit decomposition characteristics; obtaining a free vibration equation in combination with a Galerkin weak form and a displacement boundary condition; the inherent frequency and the vibration mode are obtained through MATLAB solving, and analysis is completed by comparing results of different methods; according to the method, high-order interpolation can be realized without adding unit nodes, the multi-order inherent frequency and vibration mode can still be accurately solved under low grid density, the calculation precision and efficiency are remarkably improved, the resource consumption is reduced, the method is adaptive to vibration safety design and optimization of a complex structure, and the method has important engineering application value.

Description

Technical Field

[0001] This invention relates to the field of solid mechanics and engineering numerical simulation technology, and in particular to a radial point interpolation finite element method for free vibration of linear elastic solids and its application. Background Technology

[0002] Dynamics has wide applications in the development of national economy and science and technology. Free vibration analysis, as one of the core research contents of structural dynamics, aims to obtain the natural frequencies and mode shapes of a structure. These two parameters directly determine the response characteristics and safety stability of the structure under complex dynamic loads. In engineering practice, whether it is a high-speed rotating mechanical component, reciprocating industrial equipment, or large structures such as high-rise buildings, offshore platforms, bridges, and aircraft, they may encounter resonance risks during operation or service. When the external excitation frequency is close to the structure's natural frequency, the structure will generate severe vibrations, leading to performance degradation and fatigue damage at best, and catastrophic accidents such as rupture and overturning at worst. Therefore, accurately obtaining the free vibration characteristics of a structure is a prerequisite for structural dynamics design, resonance avoidance, vibration reduction and noise reduction optimization, and has irreplaceable significance for ensuring the reliability and durability of engineering structures.

[0003] However, linear elastic solid structures in engineering practice often have complex topological forms and variable boundary conditions, making it difficult to derive exact solutions to most free vibration problems analytically. Traditionally, experimental methods for obtaining the free vibration characteristics of structures, such as excitation testing and modal analysis, while providing direct data, have significant limitations: on the one hand, the tests require specialized excitation equipment, high-precision sensing systems, and harsh testing environments, resulting in extremely high upfront equipment investment and ongoing operational costs; on the other hand, for large and complex structures, such as cross-sea bridges, nuclear power plant containment structures, or structural prototypes in the design phase, experimental schemes are often difficult to implement, and the test results are easily affected by environmental interference, resulting in poor repeatability and versatility.

[0004] The Finite Element Method (FEM) is currently the most widely used technique for solving free vibration problems of structures. However, it faces irreconcilable contradictions in practical applications: free vibration analysis is highly sensitive to the accuracy of displacement field interpolation, while the classical finite element method, when using low-order elements, has limited approximation capabilities of the interpolation function, leading to large errors in the calculation of natural frequencies. Higher-order vibration modes are also prone to distortion, and numerical accuracy is heavily dependent on mesh quality. To compensate for these accuracy deficiencies, mesh refinement is often used in engineering to improve computational reliability. However, this results in a geometrical increase in the number of degrees of freedom, requiring significant computational resources and greatly extending computation time. This approach is often insufficient to meet the efficiency requirements of engineering design, especially for multi-order free vibration analysis of complex three-dimensional structures. Furthermore, the classical finite element method is sensitive to mesh distortion. In the free vibration analysis of irregular structures, a decrease in mesh quality further amplifies numerical errors, making it difficult to guarantee the stability of the results. Summary of the Invention

[0005] The purpose of this invention is to provide a radial point interpolation finite element method for the free vibration of linear elastic solids and its application. This method addresses the challenges of complex topologies and variable boundary conditions in practical engineering applications of linear elastic solids. It also addresses the fact that most free vibration problems cannot be solved analytically with accurate solutions, and that low-order element interpolation has insufficient approximation capabilities, leading to large errors in natural frequency calculations, distortion of high-order mode shapes, and a high dependence on mesh quality for numerical accuracy. While mesh refinement is necessary to improve accuracy, it results in a geometric increase in degrees of freedom, consuming significant computational resources and time, and is sensitive to mesh distortion, making it difficult to balance accuracy and efficiency.

[0006] To achieve the above objectives, this invention provides a radial point interpolation finite element method for the free vibration of a linear elastic solid, comprising the following steps: S1. Discretize the problem domain of the linear elastic solid into a standard finite element mesh, and derive the nodal shape functions constructed using radial basis functions within the element; S2. Select appropriate local reinforcement interpolation basis functions, construct nodal degrees of freedom at the nodes consisting of local reinforcement interpolation basis functions and coefficients to be determined, and complete the local numerical approximation of the enhanced radial point interpolation finite element calculation method; S3. Based on the nodal shape function constructed using radial basis functions obtained in S1 and the local numerical approximation obtained in S2, a global interpolation scheme for the displacement vector field within the element in the elastic dynamics problem is constructed under the premise of satisfying the unit decomposition property.

[0007] S4. Based on the global interpolation scheme of the displacement vector field obtained in S3, combined with the Galerkin weak form of the elastic dynamics problem, and with the application of displacement boundary conditions, the free vibration equation of the system is obtained: ;in, Here is the stiffness matrix of the system; It is the angular frequency; The system's quality matrix; This is the displacement magnitude vector; S5. Use MATLAB software to solve the free vibration equation of the system obtained in S4, and calculate the natural frequency and natural mode shape of the system. S6. Based on the system natural frequency calculation results obtained in S5, compare the system natural frequency calculation results obtained by different calculation methods to complete the free vibration analysis of the linear elastic solid.

[0008] Preferably, the specific content of S1 is as follows: Within a unit, any scalar field function defined within the computational domain The global interpolation format is expressed as follows: ; In the formula, Scalar field function The interpolation format; The position vector at the interpolation point; The radial basis functions used; The basis functions of the Lagrange polynomials used; and These are the number of radial basis functions and polynomial basis functions used, respectively. and These represent the vectors of undetermined coefficients associated with the radial basis functions and the polynomial basis functions, respectively. make This holds true at each node within the unit, constructing a nodal shape function with the following expression: ; In the formula, The shape function matrix obtained using radial basis functions, ,in For the first The shape function corresponding to each node; Let be the nodal displacement matrix, where For the first The displacement values ​​of each node.

[0009] Preferably, the specific content of S2 is as follows: An enhanced interpolation scheme is constructed using the enhanced radial point interpolation finite element algorithm. The expression is as follows: ; In the formula, To enhance the local numerical approximation of the radial point interpolation finite element algorithm, and These are the local reinforcement interpolation basis functions constructed at the nodes and the coefficients to be determined, respectively. Lagrange polynomial basis functions are used as local reinforcement interpolation basis functions, as shown in the following expression: ; In the formula, Let x be the spatial coordinates, x for one dimension, (x, y) for two dimensions, and (x, y, z) for three dimensions. The final local enhancement interpolation basis function is constructed using dimensionless relative coordinate values, as shown in the following expression: ; In the formula, and These are dimensionless relative coordinate values. This represents the average spacing between field points.

[0010] Preferably, the specific content of S3 is as follows: The enhanced interpolation scheme in the enhanced radial point interpolation finite element algorithm is expressed as: ; In the formula, Let be the local enhancement interpolation function matrix, where ; For the corresponding unknown coefficient vector; The element node displacement vector; Based on the expression of the enhanced interpolation scheme in the enhanced radial point interpolation finite element algorithm, the displacement vector field within the element in the final elastic dynamics problem is obtained. The expression for the global interpolation format is as follows: ; In the formula, This is the global interpolation format for the displacement vector field; To enhance the element interpolation function matrix in the radial point interpolation finite element algorithm; The element node displacement vector; These are the unknown coefficients in the local numerical approximation; This represents the total degrees of freedom vector of the element; , , These represent displacements in the x, y, and z directions, respectively.

[0011] Preferably, the specific content of S4 is as follows: The Galerkin weak form of the elastic dynamics problem is constructed as follows: ; In the formula, The matrix represents the differential operators; The elastic matrix of the material; It is a displacement vector field; Acceleration vector field; It is an acceleration vector field; For the variation of the displacement vector field; It is a volume force vector; The mass density of the material; The damping coefficient; To act on the boundary Area force on; For linear elastic solids, this is the physical solution domain. Differential operator matrix The expression is as follows: ; The elastic matrix of the material The expression is as follows: ; In the formula, E and These are Young's modulus and Poisson's ratio of the material, respectively. The elastic modulus of the material; The Poisson's ratio of the material; Two-dimensional problems are divided into plane stress problems and plane strain problems, with the following expressions: ; Substituting the global interpolation scheme from the enhanced radial point interpolation finite element algorithm into the Galerkin weak form of the elastic dynamics problem, we obtain the equations of motion in elastic dynamics, as follows: ; ; In the formula, , , and These are the mass matrix, damping matrix, stiffness matrix, and load vector of the element, respectively. , , and These are the system's mass matrix, damping matrix, stiffness matrix, and load vector, respectively. The strain matrix; Let be the system's displacement vector; The velocity vector of the system; Let this be the acceleration vector of the system; In free vibration analysis, it is assumed that the displacement field has simple harmonic characteristics. The expression is as follows: ; In the formula, For displacement magnitude vectors, It is the angular frequency. The imaginary unit; Will Substituting into the equations of motion and neglecting the effect of damping, we obtain the equations of free vibration of the system in the problem of elastic dynamics, as follows: ; In the formula, Here is the stiffness matrix of the system; It is the angular frequency; The system's quality matrix; This is the displacement magnitude vector.

[0012] A radial point interpolation finite element method is applied to the free vibration of a linear elastic solid. The analysis results of the free vibration of the linear elastic solid are obtained by using the radial point interpolation finite element method as described above.

[0013] Therefore, the radial point interpolation finite element application and method for the time-domain transient response of linear elastic solids described above has the following beneficial effects: (1) This invention constructs nodal shape functions by radial basis functions and combines them with local reinforcement interpolation basis functions, thereby achieving high-order interpolation without increasing the number of element nodes. It can accurately describe the displacement vector field during the free vibration of linear elastic solids, especially in the calculation of natural frequencies and mode shape reconstruction. It can obtain multiple natural frequencies that are highly consistent with the reference solution under low mesh density. The fidelity of high-order mode shapes is significantly better than that of the classical finite element method, effectively avoiding the problems of high-order mode shape distortion and excessive frequency calculation error in traditional methods, and providing accurate data support for subsequent resonance risk assessment. (2) The method proposed in this invention does not require improving accuracy by densifying the mesh. It can quickly converge to the accurate solution using only a low-density mesh, which significantly reduces the computational degree of freedom and resource consumption. For solving the multi-order natural frequencies and mode shapes of large and complex structures, it can greatly shorten the calculation cycle and meet the needs of rapid iterative optimization in engineering design. (3) The enhanced radial point interpolation finite element calculation method proposed in this invention has great potential application value in engineering practice. The accurate natural frequency and mode shape data obtained by this method can be directly applied to the core scenario of structural dynamics design. Its reliable numerical results can help engineers accurately predict the vibration response of the structure under dynamic load, optimize the structural stiffness distribution and mass configuration, reduce the risk of fatigue damage and structural failure caused by resonance from the source, and provide key technical support for the vibration safety and performance optimization of various engineering structures.

[0014] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. Attached Figure Description

[0015] Figure 1 This invention provides a radial point interpolation finite element method for free vibration of a linear elastic solid and its application flowchart. Figure 2 This is a schematic diagram of a two-dimensional cantilever beam model provided in an embodiment of the present invention; Figure 3 The low-density mesh used in the classical finite element method and the enhanced radial point interpolation finite element calculation method for the analysis of free vibration of linear elastic solids provided in the embodiments of the present invention; Figure 4 The medium-density mesh used in the classical finite element method and the enhanced radial point interpolation finite element calculation method for analyzing the free vibration of linear elastic solids provided in the embodiments of the present invention; Figure 5 The high-density mesh used in the classical finite element method and the enhanced radial point interpolation finite element calculation method for the analysis of free vibration of linear elastic solids provided in the embodiments of the present invention. Detailed Implementation

[0016] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the embodiments of the present invention will be further described in detail below with reference to the accompanying drawings and examples. It should be understood that the specific embodiments described herein are merely illustrative of the embodiments of the present invention and are not intended to limit the embodiments of the present invention. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without creative effort are within the scope of protection of this application. Examples of the embodiments are shown in the accompanying drawings, wherein the same or similar reference numerals denote the same or similar elements or elements having the same or similar functions throughout.

[0017] It should be noted that the terms "comprising" and "having," and any variations thereof, are intended to cover non-exclusive inclusion, such as a process, method, system, product, or server that includes a series of steps or units, not necessarily limited to those steps or units explicitly listed, but may include other steps or units not explicitly listed or inherent to such process, method, product, or device.

[0018] The following is combined with Figures 1-5 The embodiments of the present invention will be described in detail below.

[0019] Example like Figure 1 As shown, this invention provides a radial point interpolation finite element method for free vibration of a linear elastic solid, comprising the following steps: S1. Discretize the problem domain of the linear elastic solid into a standard finite element mesh, and derive the nodal shape functions constructed using radial basis functions within the element; Within a unit, any scalar field function defined within the computational domain The global interpolation format is expressed as follows: ; In the formula, Scalar field function The interpolation format; The position vector at the interpolation point; The radial basis functions used; The basis functions of the Lagrange polynomials used; and These are the number of radial basis functions and polynomial basis functions used, respectively. and These represent the vectors of undetermined coefficients associated with the radial basis functions and the polynomial basis functions, respectively. make This holds true at each node within the unit, constructing a nodal shape function with the following expression: ; In the formula, The shape function matrix obtained using radial basis functions, ,in For the first The shape function corresponding to each node; Let be the nodal displacement matrix, where For the first The displacement values ​​of each node.

[0020] S2. Select appropriate local reinforcement interpolation basis functions, construct nodal degrees of freedom at the nodes consisting of local reinforcement interpolation basis functions and coefficients to be determined, and complete the local numerical approximation of the enhanced radial point interpolation finite element calculation method; An enhanced interpolation scheme is constructed using the enhanced radial point interpolation finite element algorithm. The expression is as follows: ; In the formula, To enhance the local numerical approximation of the radial point interpolation finite element algorithm, and These are the local reinforcement interpolation basis functions constructed at the nodes and the coefficients to be determined, respectively. Lagrange polynomial basis functions are used as local reinforcement interpolation basis functions, as shown in the following expression: ; In the formula, Let x be the spatial coordinates, x for one dimension, (x, y) for two dimensions, and (x, y, z) for three dimensions. The final local enhancement interpolation basis function is constructed using dimensionless relative coordinate values, as shown in the following expression: ; In the formula, and These are dimensionless relative coordinate values. This represents the average spacing between field points.

[0021] S3. Based on the nodal shape function constructed using radial basis functions obtained in S1 and the local numerical approximation obtained in S2, a global interpolation scheme for the displacement vector field within the element in the elastic dynamics problem is constructed under the premise of satisfying the unit decomposition property.

[0022] The enhanced interpolation scheme in the enhanced radial point interpolation finite element algorithm is expressed as: ; In the formula, Let be the local enhancement interpolation function matrix, where ; For the corresponding unknown coefficient vector; The element node displacement vector; Based on the expression of the enhanced interpolation scheme in the enhanced radial point interpolation finite element algorithm, the displacement vector field within the element in the final elastic dynamics problem is obtained. The expression for the global interpolation format is as follows: ; In the formula, This is the global interpolation format for the displacement vector field; To enhance the element interpolation function matrix in the radial point interpolation finite element algorithm; The element node displacement vector; These are the unknown coefficients in the local numerical approximation; This represents the total degrees of freedom vector of the element; , , These represent displacements in the x, y, and z directions, respectively.

[0023] S4. Based on the global interpolation scheme of the displacement vector field obtained in S3, combined with the Galerkin weak form of the elastic dynamics problem, and with the application of displacement boundary conditions, the free vibration equation of the system is obtained. The Galerkin weak form of the elastic dynamics problem is constructed as follows: ; In the formula, The matrix represents the differential operators; The elastic matrix of the material; It is a displacement vector field; Acceleration vector field; It is an acceleration vector field; For the variation of the displacement vector field; It is a volume force vector; The mass density of the material; The damping coefficient; To act on the boundary Area force on; For linear elastic solids, this is the physical solution domain. Differential operator matrix The expression is as follows: ; The elastic matrix of the material The expression is as follows: ; In the formula, E and These are Young's modulus and Poisson's ratio of the material, respectively. The elastic modulus of the material; The Poisson's ratio of the material; Two-dimensional problems are divided into plane stress problems and plane strain problems, with the following expressions: ; Substituting the global interpolation scheme from the enhanced radial point interpolation finite element algorithm into the Galerkin weak form of the elastic dynamics problem, we obtain the equations of motion in elastic dynamics, as follows: ; ; In the formula, , , and These are the mass matrix, damping matrix, stiffness matrix, and load vector of the element, respectively. , , and These are the system's mass matrix, damping matrix, stiffness matrix, and load vector, respectively. The strain matrix; Let be the system's displacement vector; The velocity vector of the system; Let this be the acceleration vector of the system; In free vibration analysis, it is assumed that the displacement field has simple harmonic characteristics. The expression is as follows: ; In the formula, For displacement magnitude vectors, It is the angular frequency. The imaginary unit; Will Substituting into the equations of motion and neglecting the effect of damping, we obtain the equations of free vibration of the system in the problem of elastic dynamics, as follows: ; In the formula, Here is the stiffness matrix of the system; It is the angular frequency; The system's quality matrix; This is the displacement magnitude vector.

[0024] S5. Use MATLAB software to solve the free vibration equation of the system obtained in S4, and calculate the natural frequency and natural mode shape of the system. S6. Based on the system natural frequency calculation results obtained in S5, compare the system natural frequency calculation results obtained by different calculation methods to complete the free vibration analysis of the linear elastic solid.

[0025] The free vibration analysis results of linear elastic solids were obtained by using the radial point interpolation finite method described above.

[0026] Comparative Example To compare and verify the effectiveness and correctness of the enhanced radial basis function interpolation finite element method presented in this scheme for the analysis of free vibration of online elastic solids, as well as its ability to control numerical calculation errors, a two-dimensional cantilever beam is analyzed for free vibration using both the classical finite element method (FEM-Q4) and the enhanced radial basis function interpolation finite element method (RBF-N3) disclosed in this scheme. The geometric dimensions of the cantilever beam are as follows: Figure 2 As shown, the Young's modulus of the material used is Poisson's ratio is mass density is The cantilever beam is under plane stress. The leftmost boundary of the cantilever beam is fixed. The two different calculation methods use finite element meshes of different sizes, such as... Figures 3-5 As shown, where Figure 3 The grid density is 10×1. Figure 4 The grid density is 20×2. Figure 5 The grid density is 40×4.

[0027] Considering three different meshes, as shown in Tables 1-3, the differences between the first ten natural frequencies of the cantilever beam obtained by the traditional method FEM-Q4 and the method RBF-N3 used in this invention, and the given reference solution, are presented. The results in the tables show that when using a lower density finite element mesh, the natural frequency results calculated by the classical finite element method differ significantly from the reference solution, exhibiting a large numerical error. In contrast, the calculation results of the enhanced radial basis function point interpolation finite element method disclosed in this scheme are much more accurate. As the mesh density increases, the calculation errors of both methods decrease, but the results of the classical finite element method still show a certain gap from the reference solution, exhibiting a large numerical error. The calculation results of the enhanced radial basis function point interpolation finite element method disclosed in this scheme converge to the reference solution more quickly, providing a fairly accurate and reliable numerical solution.

[0028] Table 1. Comparison of natural frequencies at a mesh density of 10×1

[0029] Table 2. Comparison of natural frequencies at a mesh density of 20×2

[0030] Table 3. Comparison of natural frequencies at a mesh density of 40×4

[0031] This indicates that the enhanced radial point interpolation finite element method proposed in this scheme has a more powerful ability to perform free vibration analysis of linear elastic solids, which can significantly reduce the numerical error in the calculation results and ultimately obtain a more accurate numerical solution. Therefore, the enhanced radial point interpolation finite element method proposed in this scheme has great potential application value in engineering practice.

[0032] Therefore, the present invention adopts the above-mentioned radial point interpolation finite element method and application for the free vibration of linear elastic solids, which realizes high-order interpolation without increasing the number of element nodes. Compared with the classical finite element method, it can significantly reduce numerical errors and obtain more accurate numerical results when performing free vibration analysis of linear elastic solids.

[0033] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit them. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the technical solutions of the present invention, and these modifications or equivalent substitutions cannot cause the modified technical solutions to deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. A radial point interpolation finite element method for free vibration of a linear elastic solid, characterized in that, Specifically, the following steps are included: S1. Discretize the problem domain of the linear elastic solid into a standard finite element mesh, and derive the nodal shape functions constructed using radial basis functions within the element; S2. Select the local reinforcement interpolation basis function, construct the nodal degrees of freedom at the nodes consisting of the local reinforcement interpolation basis function and the coefficients to be determined, and complete the local numerical approximation of the enhanced radial point interpolation finite element calculation method; S3. Based on the nodal shape function constructed using radial basis functions obtained in S1 and the local numerical approximation obtained in S2, a global interpolation scheme for the displacement vector field within the element in the elastic dynamics problem is constructed under the premise of satisfying the unit decomposition property. S4. Based on the global interpolation scheme of the displacement vector field obtained in S3, combined with the Galerkin weak form of the elastic dynamics problem, and with the application of displacement boundary conditions, the free vibration equation of the system is obtained: ;in, Here is the stiffness matrix of the system; It is the angular frequency; The system's quality matrix; This is the displacement magnitude vector; S5. Use MATLAB software to solve the free vibration equation of the system obtained in S4, and calculate the natural frequency and natural mode shape of the system. S6. Based on the system natural frequency calculation results obtained in S5, compare the system natural frequency calculation results obtained by different calculation methods to complete the free vibration analysis of the linear elastic solid.

2. The radial point interpolation finite element method for free vibration of a linear elastic solid according to claim 1, characterized in that, The specific content of S1 is as follows: Within a unit, any scalar field function defined within the computational domain The global interpolation format is expressed as follows: ； In the formula, Scalar field function The interpolation format; The position vector at the interpolation point; The radial basis functions used; The basis functions of the Lagrange polynomials used; and These are the number of radial basis functions and polynomial basis functions used, respectively. and These represent the vectors of undetermined coefficients associated with radial basis functions and polynomial basis functions, respectively. make This holds true at each node within the unit, constructing a nodal shape function with the following expression: ； In the formula, The shape function matrix obtained using radial basis functions, ,in For the first The shape function corresponding to each node; Let be the nodal displacement matrix, where For the first The displacement values ​​of each node.

3. The radial point interpolation finite element method for free vibration of a linear elastic solid according to claim 2, characterized in that, The specific details of S2 are as follows: An enhanced interpolation scheme is constructed using the enhanced radial point interpolation finite element algorithm. The expression is as follows: ； In the formula, To enhance the local numerical approximation of the radial point interpolation finite element algorithm, and These are the local reinforcement interpolation basis functions constructed at the nodes and the coefficients to be determined, respectively. Lagrange polynomial basis functions are used as local reinforcement interpolation basis functions, as shown in the following expression: ； In the formula, Let x be the spatial coordinates, x for one dimension, (x, y) for two dimensions, and (x, y, z) for three dimensions. The final local enhancement interpolation basis function is constructed using dimensionless relative coordinate values, as shown in the following expression: ； In the formula, and These are dimensionless relative coordinate values. This represents the average spacing between field points.

4. The radial point interpolation finite element method for free vibration of a linear elastic solid according to claim 3, characterized in that, The specific details of S3 are as follows: The enhanced interpolation scheme in the enhanced radial point interpolation finite element algorithm is expressed as: ； In the formula, Let be the local enhancement interpolation function matrix, where ; For the corresponding unknown coefficient vector; The element node displacement vector; Based on the expression of the enhanced interpolation scheme in the enhanced radial point interpolation finite element algorithm, the displacement vector field within the element in the final elastic dynamics problem is obtained. The expression for the global interpolation format is as follows: ； In the formula, This is the global interpolation format for the displacement vector field; To enhance the element interpolation function matrix in the radial point interpolation finite element algorithm; The element node displacement vector; These are the unknown coefficients in the local numerical approximation; This represents the total degrees of freedom vector of the element; , , These represent displacements in the x, y, and z directions, respectively.

5. The radial point interpolation finite element method for free vibration of a linear elastic solid according to claim 4, characterized in that, The specific details of S4 are as follows: The Galerkin weak form of the elastic dynamics problem is constructed as follows: ； In the formula, The matrix represents the differential operators; The elastic matrix of the material; It is a displacement vector field; Acceleration vector field; It is an acceleration vector field; For the variation of the displacement vector field; It is a volume force vector; The mass density of the material; The damping coefficient; To act on the boundary Area force on; For linear elastic solids, this is the physical solution domain. Differential operator matrix The expression is as follows: ； The elastic matrix of the material The expression is as follows: ； In the formula, E and These are Young's modulus and Poisson's ratio of the material, respectively. The elastic modulus of the material; The Poisson's ratio of the material; Two-dimensional problems are divided into plane stress problems and plane strain problems, with the following expressions: ； Substituting the global interpolation scheme from the enhanced radial point interpolation finite element algorithm into the Galerkin weak form of the elastic dynamics problem, we obtain the equations of motion in elastic dynamics, as follows: ； ； In the formula, , , and These are the mass matrix, damping matrix, stiffness matrix, and load vector of the element, respectively. , , and These are the system's mass matrix, damping matrix, stiffness matrix, and load vector, respectively. The strain matrix; Let be the system's displacement vector; The velocity vector of the system; Let this be the acceleration vector of the system; In free vibration analysis, it is assumed that the displacement field has simple harmonic characteristics. The expression is as follows: ； In the formula, For displacement magnitude vectors, It is the angular frequency. The imaginary unit; Will Substituting into the equations of motion and neglecting the effect of damping, we obtain the equations of free vibration of the system in the problem of elastic dynamics, as follows: ； In the formula, Here is the stiffness matrix of the system; It is the angular frequency; The system's quality matrix; This is the displacement magnitude vector.

6. A radial point interpolation finite element method for free vibration of a linear elastic solid, employing the radial point interpolation finite element method for free vibration of a linear elastic solid as described in any one of claims 1-5, characterized in that: The free vibration analysis results of linear elastic solids were obtained using the radial point interpolation finite element method.

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