Linear elastic solid harmonic response radial point interpolation finite element method and application

Abstract

The invention discloses a radial point interpolation finite element method for linear elastic solid harmonic response and application, and belongs to the technical field of solid mechanics and engineering numerical simulation. The method comprises the following steps: discretizing a linear elastic solid problem domain into a standard finite element grid, deriving a nodal function through a radial basis function, selecting a local enhanced interpolation basis function to construct a newly added degree of freedom and complete local numerical approximation, and constructing a global interpolation format of a displacement vector field in an element in combination with unit decomposition characteristics; based on the interpolation format and a Galerkin weak form of an elastic dynamic problem, applying a displacement boundary condition to obtain a system forced vibration equation; acquiring steady-state dynamic response and frequency domain response characteristics under different frequency harmonic excitation; according to the method, high-order interpolation can be realized without increasing nodes, the numerical precision is improved, the computing resource and time consumption is reduced and a more reliable numerical solution is provided under the condition that the grid density is relatively small, and the method has important application value in engineering practice.

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 the harmonic response of linear elastic solids and its application. Background Technology

[0002] Dynamics problems have wide applications in the development of national economy and science and technology. Harmonic response analysis, as one of the core technologies in structural dynamics and engineering numerical simulation, aims to obtain the steady-state frequency domain response characteristics of structures under periodic harmonic excitation, thereby providing crucial basis for vibration-resistant design, fatigue life assessment, and stability optimization of engineering structures. In modern industrial production and infrastructure construction, harmonic response analysis is applied across multiple core areas: the harmonic response of rotors and shells in high-speed rotating motors, steam turbines, and other power machinery directly determines the smoothness and service life of the equipment; large structures such as high-rise buildings, bridges, and offshore platforms require harmonic response analysis to avoid resonance damage caused by periodic loads such as strong winds, waves, and earthquakes; the harmonic response characteristics of fuselages and engine components of aerospace vehicles under harmonic excitations such as airflow pulsation and mechanical vibration directly affect flight safety and reliability. Furthermore, harmonic response analysis is also a core means of solving problems such as vibration noise and structural fatigue in the design of precision instruments, automobile chassis, and chemical pipelines.

[0003] However, harmonic response analysis in practical engineering faces two major challenges: First, obtaining structural harmonic response characteristics through experimental methods requires building complex excitation systems and sensing test platforms, and conducting multiple loading tests covering a wide frequency range. This not only consumes a lot of manpower, material resources, and time, but also makes it difficult to accurately measure the harmonic response of complex internal structures, such as closed pipes and core component connections. Especially for harmonic response testing of large three-dimensional structures or under extreme conditions, the experimental feasibility is extremely low. Second, numerical methods have become the mainstream choice for harmonic response analysis, among which the finite element method (FEM) is the most widely used. However, the classical finite element method has significant limitations in harmonic response analysis: the interpolation characteristics of low-order elements lead to insufficient accuracy in describing the frequency domain response, especially near the resonance frequency, where the calculation errors of displacement and stress peak values ​​are large, making it difficult to accurately identify the weak points of the structure. If the finite element mesh is densified to improve accuracy, the number of degrees of freedom will increase exponentially, which will not only significantly increase the consumption of computing resources, but also significantly prolong the calculation cycle. For engineering designs that require rapid iterative optimization or broadband harmonic response analysis of large and complex structures, this contradiction between "accuracy and efficiency" is particularly prominent.

[0004] More importantly, the core requirement of harmonic response analysis is to accurately capture the steady-state response of a structure under different frequency excitations, especially the accurate identification of the resonant frequency and the quantitative calculation of the peak response. Numerical errors in the classical finite element method can directly lead to resonant frequency shifts and peak amplitude distortions, thereby misleading engineering designs and causing safety hazards such as excessive structural vibration and fatigue failure. Therefore, developing a numerical method for harmonic response analysis that can achieve high-order accuracy interpolation without excessive mesh refinement, while balancing computational efficiency and numerical reliability, has become an urgent need to solve practical engineering problems and a key breakthrough for promoting the in-depth application of harmonic response analysis technology in complex structural design. Summary of the Invention

[0005] The purpose of this invention is to provide a radial point interpolation finite element method for the harmonic response of linear elastic solids and its application. This method addresses the problems of experimental methods, which not only require a large investment of manpower, resources, and time to build complex testing systems, but also make it difficult to accurately measure the harmonic response of complex internal structures. Furthermore, the method is not feasible for testing the broadband harmonic response of large three-dimensional structures or under extreme conditions, and the low-order element interpolation leads to large numerical errors near the resonance frequency, which can easily cause resonance frequency shifts and peak distortion. In addition, improving accuracy by refining the mesh will exponentially increase the number of degrees of freedom, which will significantly increase the consumption of computational resources and the cycle time.

[0006] To achieve the above objectives, this invention provides a radial point interpolation finite element method for the harmonic response of linear elastic solids, 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 within the element using radial basis functions; select local reinforcement interpolation basis functions, construct new degrees of freedom at the nodes, and complete the local numerical approximation; based on the obtained nodal shape functions and local numerical approximation, construct a global interpolation scheme for the displacement vector field within the element while satisfying the unit decomposition property. S2. Based on the global interpolation scheme of the displacement vector field obtained in S1, combined with the Galerkin weak form of the elastic dynamics problem, and with the application of displacement boundary conditions, the forced 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; Let the amplitude vector be the simple harmonic excitation force. S3. Use MATLAB software to solve the forced vibration equation of the system obtained in S2, calculate the steady-state dynamic response of the system under simple harmonic excitation at different frequencies, and obtain the harmonic response characteristics of the system. S4. Based on the harmonic response analysis results obtained in S3, compare the steady-state dynamic response calculation results obtained by different calculation methods to complete the harmonic response analysis of linear elastic solids.

[0007] Preferably, S1 specifically includes: S101. 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. S102. 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. S103. Based on the nodal shape function constructed using radial basis functions obtained in S101 and the local numerical approximation obtained in S102, 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.

[0008] Preferably, in S101, the problem domain of the linear elastic solid is discretized into a standard finite element mesh, and the specific content of deriving the nodal shape functions constructed using radial basis functions within the element 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.

[0009] Preferably, in S102, a local reinforcement interpolation basis function is selected, and nodal degrees of freedom consisting of the local reinforcement interpolation basis function and the coefficients to be determined are constructed at the nodes. The specific content of the local numerical approximation of the enhanced radial point interpolation finite element calculation method 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, in S103, based on the nodal shape functions constructed using radial basis functions obtained in S101 and the local numerical approximation obtained in S102, and under the premise of satisfying the unit decomposition property, the specific content of the global interpolation scheme for the displacement vector field within the element in the elastic dynamics problem 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, S2 specifically includes: 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 forced vibration analysis, under harmonic excitation at different frequencies, the system generates a harmonic response in steady state. and The expression is as follows: ; ; In the formula, This is the displacement magnitude vector; Let the amplitude vector be the simple harmonic excitation force. It is the angular frequency; The imaginary unit; Will and Substituting into the equations of motion and neglecting the damping effect, we obtain the equations of forced 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; Let be the amplitude vector of the harmonic excitation force.

[0012] A radial point interpolation finite element method is applied to the harmonic response of a linear elastic solid. The harmonic response analysis results of the linear elastic solid are obtained by using the radial point interpolation finite element method 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 combines radial basis functions with local reinforcement interpolation basis functions to achieve high-order interpolation without additional nodes, which can accurately capture the key characteristics of harmonic response. In resonant frequency identification, it can accurately match the reference solution under low grid density, effectively avoiding the problems of resonant frequency shift and peak amplitude distortion in classical methods. It has higher accuracy in calculating steady-state displacement and stress response over a wide frequency range, especially in the resonant interval, where it can accurately restore the peak value and curve trend of the response, providing reliable frequency domain data support for structural vibration resistance design. (2) In response to the need for wide frequency range harmonic response analysis in engineering, this method can converge quickly without the need for mesh refinement, which significantly reduces the computational degree of freedom and resource consumption; for solving the multi-frequency excitation response of large and complex structures, it can greatly shorten the calculation cycle, meet the need for rapid iterative optimization in engineering design, and solve the core contradiction of "accuracy-efficiency" in classical finite element method. (3) The enhanced radial point interpolation finite element calculation method proposed in this invention has great potential application value in engineering practice. For engineering structures with complex topology and varied boundary conditions, this method has low dependence on mesh quality and can maintain stable calculation accuracy even under irregular mesh or complex load boundaries. Its fast convergence characteristics and numerical reliability can be adapted to the harmonic response analysis of various linear elastic solids, and are especially suitable for the accurate solution of harmonic response with complex internal structure, thus expanding the scope of application of harmonic response numerical simulation.

[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 is a flowchart illustrating the radial point interpolation finite element method for the harmonic response of a linear elastic solid and its application. 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 triangular finite element mesh diagram used for calculation by the classical finite element method and the enhanced radial point interpolation finite element calculation method for harmonic response analysis of linear elastic solids provided in the embodiments of the present invention; Figure 4 The quadrilateral finite element mesh diagram used for calculation by the classical finite element method and the enhanced radial point interpolation finite element calculation method for harmonic response analysis of linear elastic solids provided in the embodiments of the present invention; Figure 5 The displacement frequency response curve at point A on a cantilever beam under harmonic excitation, calculated by the classical finite element method and the enhanced radial point interpolation finite element calculation method 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 1: like Figure 1 As shown, this invention provides a radial point interpolation finite element method for the harmonic response of linear elastic solids, 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 within the element using radial basis functions; select appropriate local reinforcement interpolation basis functions, construct new degrees of freedom at the nodes, and complete the local numerical approximation; based on the obtained nodal shape functions and local numerical approximation, construct a global interpolation scheme for the displacement vector field within the element while satisfying the unit decomposition property. S101. 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 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.

[0020] S102. 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] S103. Based on the nodal shape function constructed using radial basis functions obtained in S101 and the local numerical approximation obtained in S102, 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] S2. Based on the global interpolation scheme of the displacement vector field obtained in S1, combined with the Galerkin weak form of the elastic dynamics problem, and with the application of displacement boundary conditions, the forced vibration equation of the system is obtained, as detailed below: 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 forced vibration analysis, under harmonic excitation at different frequencies, the system generates a harmonic response in steady state. and The expression is as follows: ; ; In the formula, This is the displacement magnitude vector; Let the amplitude vector be the simple harmonic excitation force. It is the angular frequency; The imaginary unit; Will and Substituting into the equations of motion and neglecting the damping effect, we obtain the equations of forced 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; Let be the amplitude vector of the harmonic excitation force.

[0024] S3. Use MATLAB software to solve the forced vibration equation of the system obtained in S2, calculate the steady-state dynamic response of the system under simple harmonic excitation at different frequencies, and obtain the frequency domain response characteristics of the system. S4. Based on the harmonic response analysis results obtained in S3, compare the steady-state dynamic response calculation results obtained by different calculation methods to complete the harmonic response analysis of linear elastic solids.

[0025] The harmonic response analysis results of linear elastic solids were obtained 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 online elastic solid harmonic response analysis, as well as its ability to control numerical calculation errors, harmonic response analysis of a two-dimensional cantilever beam is performed using both the classical finite element method (FEM-T3, 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, and the rightmost end is subjected to harmonic excitation forces of different frequencies in the y-direction. The finite element mesh used in the calculation is as follows: Figure 3 and Figure 4 As shown, in this embodiment, the FEM-T3 method uses a triangular mesh, while FEM-Q4 and the radial point interpolation finite element method (RBF-N3) for the harmonic response of linear elastic solids disclosed in this scheme both use quadrilateral meshes. Both meshes have the same node distribution. Harmonic response analysis is performed on the cantilever beam, calculating the steady-state dynamic response of the cantilever beam under simple harmonic excitation. Based on the calculation results, the displacement frequency domain response curve at point A on the cantilever beam is plotted (e.g., ...). Figure 5 (As shown).

[0027] Figure 5The figure presents the displacement-frequency response curves at point A on a cantilever beam calculated using both the classical finite element method and the enhanced radial basis function interpolation finite element method. A reference solution curve is also provided for comparative analysis of the results from the two algorithms. The figure shows a significant discrepancy between the displacement-frequency response curve calculated by the classical finite element method and the reference solution curve, indicating a large numerical error. In contrast, the enhanced radial basis function interpolation finite element method presented in this paper demonstrates very high numerical accuracy, maintaining consistency with the reference solution. This indicates that the enhanced radial basis function interpolation finite element method proposed in this paper can significantly reduce numerical errors in the calculation results when performing harmonic response analysis of linear elastic solids, thereby more accurately calculating the steady-state dynamic response of the system and obtaining accurate resonance frequencies and peak values. This provides more reliable numerical results for the evaluation and design of structural systems. Therefore, the enhanced radial basis function interpolation finite element method proposed in this paper has significant potential application value in engineering practice.

[0028] Therefore, the present invention employs the radial point interpolation finite element method and its application for the harmonic response of linear elastic solids described above, which achieves 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 harmonic response analysis of linear elastic solids.

[0029] 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 the harmonic response 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 within the element using radial basis functions; select local reinforcement interpolation basis functions, construct new degrees of freedom at the nodes, and complete the local numerical approximation; based on the obtained nodal shape functions and local numerical approximation, construct a global interpolation scheme for the displacement vector field within the element while satisfying the unit decomposition property. S2. Based on the global interpolation scheme of the displacement vector field obtained in S1, combined with the Galerkin weak form of the elastic dynamics problem, and with the application of displacement boundary conditions, the forced 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; Let the amplitude vector be the simple harmonic excitation force. S3. Use MATLAB software to solve the forced vibration equation of the system obtained in S2, calculate the steady-state dynamic response of the system under simple harmonic excitation at different frequencies, and obtain the harmonic response characteristics of the system. S4. Based on the harmonic response analysis results obtained in S3, compare the steady-state dynamic response calculation results obtained by different calculation methods to complete the harmonic response analysis of linear elastic solids.

2. The radial point interpolation finite element method for the harmonic response of a linear elastic solid according to claim 1, characterized in that, S1 specifically includes: S101. 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. S102. 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. S103. Based on the nodal shape function constructed using radial basis functions obtained in S101 and the local numerical approximation obtained in S102, 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.

3. The radial point interpolation finite element method for the harmonic response of a linear elastic solid according to claim 2, characterized in that, In S101, the problem domain of the linear elastic solid is discretized into a standard finite element mesh. The specific details of deriving the nodal shape functions constructed using radial basis functions within the element are 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.

4. The radial point interpolation finite element method for the harmonic response of a linear elastic solid according to claim 2, characterized in that, In S102, a local reinforcement interpolation basis function is selected. At the nodes, nodal degrees of freedom are constructed, consisting of the local reinforcement interpolation basis function and the coefficients to be determined. The specific details of the local numerical approximation of the enhanced radial point interpolation finite element method 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.

5. The radial point interpolation finite element method for the harmonic response of a linear elastic solid according to claim 2, characterized in that, Based on the nodal shape functions constructed using radial basis functions obtained in S101 and the local numerical approximation obtained in S102, and under the premise of satisfying the unit decomposition property, the specific content of the global interpolation scheme for the displacement vector field within the element in the elastic dynamics problem 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.

6. The radial point interpolation finite element method for the harmonic response of a linear elastic solid according to claim 5, characterized in that, S2 specifically includes: 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 forced vibration analysis, under harmonic excitation at different frequencies, the system generates a harmonic response in steady state. and The expression is as follows: ； ； In the formula, This is the displacement magnitude vector; Let the amplitude vector be the simple harmonic excitation force. It is the angular frequency; The imaginary unit; Will and Substituting into the equations of motion and neglecting the damping effect, we obtain the equations of forced 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; Let be the amplitude vector of the harmonic excitation force.

7. A radial point interpolation finite element method for the harmonic response of a linear elastic solid, comprising applying the radial point interpolation finite element method for the harmonic response of a linear elastic solid as described in any one of claims 1-6, characterized in that: The harmonic response analysis results of linear elastic solids were obtained using the radial point interpolation finite element method.

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