Reinforced meshless Galerkin algorithm for reducing frequency dispersion error of wave propagation analysis
By enhancing the meshless Galerkin algorithm to reduce dispersion error in wave propagation analysis, the problems of large dispersion error and low computational efficiency in existing technologies are solved, achieving high-precision and high-efficiency wave propagation analysis.
Patent Information
- Application Number
- CN202511696475.3
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-11-19
- Publication Date
- 2026-02-17
AI Technical Summary
Existing wave propagation analysis methods suffer from large dispersion errors and low computational efficiency under non-uniform media and complex boundary conditions. In particular, high-frequency simulations have high hardware requirements and are time-consuming, making it difficult to meet the timeliness requirements of engineering design.
We employ an enhanced meshless Galerkin algorithm, which reduces dispersion error by constructing local enhanced interpolation basis functions and a global interpolation scheme, combined with the Galerkin weighted residual method. We then establish the system matrix equation for dispersion analysis of the wave propagation problem and solve it numerically.
It effectively reduces dispersion error in numerical calculations, especially significantly reducing errors in high-frequency wave propagation scenarios, improving calculation accuracy and efficiency, and is suitable for multi-directional wave propagation analysis.
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Abstract
Description
Technical Field
[0001] This invention relates to the field of wave propagation numerical analysis and dispersion error prediction technology, and in particular to an enhanced meshless Galerkin algorithm for reducing dispersion error in wave propagation analysis. Background Technology
[0002] Wave propagation is a core research subject in wave dynamics. Its applications not only cover many important fields such as mechanical structure vibration analysis, underwater acoustic detection, and medical ultrasound diagnosis, but also extend to emerging cutting-edge directions such as geophysical monitoring and quantum wave function analysis.
[0003] While traditional analytical methods can provide accurate mathematical solutions, they strictly rely on the assumptions of medium homogeneity (requiring parameters such as density and elastic modulus to be spatially constant) and boundary regularity (applicable only to ideal geometric boundaries such as circles and rectangles). However, they have significant limitations in practical engineering applications. For example, scenarios with complex boundaries or non-homogeneous medium characteristics, such as fault structures in oil exploration or gradual changes in acoustic impedance in human tissue, often exceed the processing capabilities of analytical methods.
[0004] To overcome the limitations of analytical solutions, numerical simulation techniques such as the finite difference method, finite element method, and spectral element method, through spatial discretization and combined with high-performance computing, can not only simulate wave field evolution in three-dimensional non-uniform media (including anisotropic attenuation and multiple scattering effects), but also achieve real-time inversion and parameter optimization. This provides a solution that combines accuracy and efficiency for wave propagation analysis under complex working conditions, becoming an irreplaceable technical support for modern engineering research, from basic theoretical verification to industrial application development.
[0005] However, among various numerical methods, the finite element method, the most widely used, still suffers from inherent numerical error problems. In engineering practice, to ensure computational accuracy, wave dynamics finite element models must follow the "rule of thumb" (at least six elements are required per wavelength range), which causes the model size to increase geometrically with increasing frequency. High-frequency simulations not only place stringent demands on hardware, with single solutions potentially taking tens of hours, but also often exceed the memory capacity of ordinary workstations; especially in scenarios requiring repeated calculations, such as multiphysics coupling or parameter inversion, it is difficult to meet the timeliness requirements of engineering designs.
[0006] A deeper analysis of the error mechanism reveals that the core cause of numerical errors in wave propagation numerical simulations stems from the dispersion effect triggered by the discretization process. Specifically, this manifests as a significant deviation between the calculated wave number and the theoretical solution, with the error amplitude increasing exponentially with frequency. Essentially, when the continuous wave equation is discretized into a numerical model, the discrete system exhibits "excessive stiffness"—that is, the stiffness matrix of the discrete model is more stiff than that of the actual physical system. This increased stiffness directly leads to a deviation between the numerical wave velocity and the actual wave velocity, forming the so-called "numerical dispersion."
[0007] In view of the problems of grid dependence and prominent dispersion error in existing numerical methods for wave propagation analysis, this invention proposes an enhanced meshless Galerkin algorithm to reduce dispersion error in wave propagation analysis. Summary of the Invention
[0008] The purpose of this invention is to provide an enhanced meshless Galerkin algorithm to reduce dispersion errors in wave propagation analysis, thereby obtaining more accurate numerical calculation results.
[0009] To achieve the above objectives, this invention provides an enhanced meshless Galerkin algorithm for reducing dispersion errors in wave propagation analysis, comprising the following steps: S1. Discretize the wave propagation problem domain into nodes; select interpolation basis functions and weight functions; construct an approximate interpolation function based on the standard meshless Galerkin algorithm, determine the unknown coefficients to be solved in the approximate interpolation function, and establish the initial global interpolation scheme of the standard meshless Galerkin algorithm for the physical field of wave propagation to be solved. S2. Based on the initial global interpolation scheme of the standard meshless Galerkin algorithm obtained in S1, by selecting local reinforcement interpolation basis functions, nodal degrees of freedom containing local reinforcement interpolation basis functions and coefficients to be determined are constructed at the nodes, thus completing the local numerical approximation of the reinforced meshless Galerkin algorithm: S3. Based on the initial global interpolation scheme of the standard meshless Galerkin algorithm obtained in S1 and the local numerical approximation of the enhanced meshless Galerkin algorithm obtained in S2, under the premise of satisfying the unit decomposition characteristic, the global interpolation scheme of the enhanced meshless Galerkin algorithm for the wave propagation physical field is constructed. S4. Based on the enhanced meshless Galerkin algorithm obtained in S3, the global interpolation scheme is combined with the partial differential control equations of the wave propagation problem and the Galerkin weighted residual method to obtain the system matrix equations for the dispersion analysis of the wave propagation problem. S5. Solve the system matrix equation for the dispersion analysis of the wave propagation problem obtained in S4 to obtain the dispersion error in the wave propagation direction; S6. Based on the dispersion error obtained in S5 in the wave propagation direction, and compared with the commonly used calculation methods for wave propagation problems, the dispersion analysis of the wave propagation problem is completed.
[0010] Preferably, S1 is as follows: S11. Discretize the wave propagation problem domain into nodes, and define the influence domain for each node. The influence domain is a circular or rectangular region, and the specific rules are as follows: (1) Define the support region of the integration point: a circular or rectangular region centered on the integration point; (2) Define the node influence domain: the area that each node can influence when performing numerical integration. The size of the node influence domain is closely related to the support domain of the integration point. (3) The weight function value of nodes within the influence domain is non-zero, and the weight function value of nodes outside the influence domain is zero. The weight function expression is as follows: ; in, Represents the weight function; Indicates the radius of the influence region; S12. Based on the standard meshless Galerkin number algorithm, a numerical interpolation scheme is constructed, the details of which are as follows: First, based on the influence domain rules defined in S11, the influence domain for interpolation is determined; the number of nodes participating in interpolation is determined by the support domain of the integration points during numerical integration. Then, based on the nodes within the selected influence domain, the numerical interpolation of the standard meshless Galerkin algorithm is constructed as follows: ; in, Represents the physical variables that can describe the wave propagation problem; This represents the position vector at the point to be interpolated; Represents the interpolation basis function; Indicates the first The unknown coefficients to be determined for each interpolation node; This represents a vector consisting of the unknown coefficients to be determined. Represents the interpolation basis function vector of the standard meshless Galerkin algorithm; Indicates the number of nodes participating in the interpolation; S13. For one-dimensional, two-dimensional, and three-dimensional problems, the Lagrange linear basis functions are used as the interpolation basis functions of the standard meshless Galerkin algorithm, as shown below: ; S14. The unknown coefficients of the numerical interpolation scheme of the standard meshless Galerkin algorithm are determined by the least squares fitting method, as shown below: ; in, Indicates the corresponding node in the influence domain A vector or matrix consisting of the function values at each node. Specifically, it is expressed as follows: ; Specifically, it is expressed as follows: ; S15. Combining the numerical interpolation of the standard meshless Galerkin algorithm in S13 with the unknown coefficients determined in S14, the initial global interpolation scheme of the standard meshless Galerkin algorithm for the physical field to be determined is established as follows: ; in, It is a nodal interpolation function constructed by the standard meshless Galerkin; It is the first one obtained by construction Interpolation function for each node.
[0011] Preferably, S2 is as follows: S21. Select composite basis functions consisting of first-order Lagrange linear basis functions and trigonometric functions as basis functions, for each node. Construct the enhanced interpolation basis function as follows: ; in, Indicates at node Strengthening interpolation basis functions constructed at the location; and Indicates the first The coordinates of each node; and Indicates the fundamental wavelength; S22. Using dimensionless relative coordinate values, optimize the local enhancement interpolation basis function in S21, as shown below: ; in, and These are dimensionless relative coordinate values. This represents the average spacing between nodes; S23. Based on S1, the initial global interpolation scheme of the standard meshless Galerkin algorithm is obtained. Combined with the local reinforcement interpolation basis function optimized in S22, the local reinforcement interpolation scheme of the reinforced meshless Galerkin algorithm is constructed as follows: ; in, This represents the local reinforcement interpolation scheme for the enhanced meshless Galerkin algorithm; The term represents the number of nodes involved in interpolation in the reinforced meshless Galerkin algorithm; the second term in the formula represents the local numerical approximation of the reinforced meshless Galerkin algorithm. The local numerical approximation of the enhanced meshless Galerkin algorithm is achieved by using the aforementioned local enhancement interpolation scheme.
[0012] Preferably, in S3, the global interpolation scheme of the meshless Galerkin algorithm is enhanced as follows: ; in, This represents the global interpolation scheme for enhancing the meshless Galerkin algorithm; This represents the enhanced interpolation function obtained through construction.
[0013] Preferably, S4 is as follows: S41. Establish the governing equations for the wave propagation problem and transform them into frequency domain small-amplitude wave equations based on time-harmonic characteristics. S42. Introduce boundary conditions for the wave propagation problem; process the frequency domain small amplitude wave equation obtained in S41 using the Galerkin weighted residual method to obtain an integral form equation containing boundary terms; S43. Combining the global interpolation scheme of the enhanced meshless Galerkin algorithm obtained in S3 and the integral form equation with boundary terms obtained in S42, the system matrix equation for the dispersion analysis of the wave propagation problem is established.
[0014] Preferably, S41 is as follows: First, for the wave propagation problem in an ideal, stationary, homogeneous fluid medium, the governing equations for small-amplitude waves are established as follows: ; in, Represents the Laplace operator; Represent the variables in the problem domain of wave propagation; Indicates the speed at which sound waves propagate; Indicates time; Secondly, based on the time-harmonic characteristics of the small-amplitude wave propagation problem, the spatial distribution of variables in the wave propagation problem domain is determined as follows: ; in, The spatial distribution of variable amplitude; The imaginary unit; Angular frequency; Finally, substituting the above variable distributions into the governing equations, we obtain the small-amplitude wave equation in the frequency domain, as shown below: ; in, Let represent the wave number, and satisfy: .
[0015] Preferably, S42 is as follows: First, the boundary conditions for the wave propagation problem are determined as follows: Dirichlet boundary conditions: ; in, Represent the Dirichlet boundary conditions; This represents the amplitude value of the variable specified on the Dirichlet boundary; Newman boundary conditions: ; in, Represent the Newman boundary conditions; Represents the outward normal unit vector on the boundary; This represents the density of an ideal fluid medium; This represents the vibrational velocity of a particle under the Newman boundary condition; Robin's boundary conditions: ; in, Represent Robin's boundary conditions; Represents the admittance coefficient on the Robin boundary conditions; Secondly, during the propagation of a small-amplitude wave in an ideal stationary fluid, the particle vibration velocity... and sound pressure gradient The relationship is as follows: ; Finally, the frequency domain wave equation is processed using the Galerkin weighted residual method to obtain the integral form equation containing boundary terms, as shown below: ; in, Representing the problem domain The boundary.
[0016] Preferably, in S43, the system matrix equation for the dispersion analysis of the wave propagation problem is as follows: ; in, Let the vector be the unknown coefficients to be determined. The stiffness matrix in a meshless computational model of the wave propagation problem is specifically represented as: ;in, The gradient matrix; The mass matrix in the meshless computational model of the wave propagation problem is specifically represented as: ; The damping matrix in a meshless computational model of a general wave propagation problem is specifically represented as: ; This represents the nodal force vector in a meshless computational model of a general wave propagation problem, specifically expressed as: .
[0017] Preferably, S5 is as follows: S51. When performing dispersion analysis on wave propagation problems, the boundary conditions and damping effects are ignored, and the system matrix equation for dispersion analysis of wave propagation problems is simplified as follows: ; S52. Suppose that in the enhanced meshless Galerkin algorithm, there are nodes with... If there are unknown node coefficients, then the vector of unknown coefficients to be solved is... As shown below: ; in, Represents the numerical wavenumber; Represents the unit vector along the direction of wave propagation; Indicates and The magnitude vector corresponding to the coefficients of each unknown node, and From basic unit subvectors Constituting a periodic repetition; S53. Substituting the unknown node coefficient vector from S52 into the simplified system matrix equation for the dispersion analysis of the wave propagation problem, we obtain: ; in, Represents the stiffness matrix The relevant coefficient matrix is specifically represented as follows: ;in, and Representing the stiffness matrix The Middle row and number Column elements; and These represent the index offsets; Representation and mass matrix The relevant coefficient matrix is specifically represented as follows: ;in, and Representing the mass matrix respectively The Middle row and number Column elements; S54. Combining mathematical conditions and the definition of error, complete the calculation of dispersion error.
[0018] Preferably, S54 is as follows: First, for the equation in S53, the condition for the existence of a non-zero solution is that the determinant of the coefficient matrix is zero, that is: ; at this time, It is a matrix The eigenvalues, namely: ; Secondly, based on wavenumber sum of numerical wavenumbers The dispersion error is defined as: ; Therefore, the dispersion error in wave propagation analysis is derived.
[0019] Therefore, the present invention employs the above-mentioned enhanced meshless Galerkin algorithm for reducing dispersion error in wave propagation analysis, which has the following beneficial effects: (1) Without increasing the number of unit nodes, the present invention achieves a high-order interpolation effect by using a combination of Lagrange linear basis functions and trigonometric functions as local reinforcement basis functions.
[0020] (2) The enhanced meshless Galerkin algorithm proposed in this invention has a more powerful ability to perform wave propagation analysis, which can effectively reduce the dispersion error generated by numerical calculation. Especially in the high-frequency wave propagation scenario with a large dimensionless wave number, it avoids the problem of the dispersion error rising sharply with the wave number in the traditional method. Finally, when performing wave propagation analysis, more accurate calculation results can be obtained.
[0021] (3) The enhanced meshless Galerkin algorithm proposed in this invention shows excellent error control capability in multi-directional wave propagation scenarios and has great potential application value in engineering practice.
[0022] 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
[0023] Figure 1 This is a flowchart of an enhanced meshless Galerkin algorithm for reducing dispersion error in wave propagation analysis according to the present invention; Figure 2 This is a schematic diagram of the node influence domain in the enhanced meshless Galerkin algorithm of this invention embodiment; Figure 3 This is a schematic diagram of the integration point support domain in the enhanced meshless Galerkin algorithm of this invention. Figure 4 This is a schematic diagram of the node distribution in the standard finite element algorithm using triangular elements for wave propagation dispersion analysis in an embodiment of the present invention; Figure 5 This is a schematic diagram of the node distribution in the standard finite element algorithm using quadrilateral elements for wave propagation dispersion analysis in an embodiment of the present invention; Figure 6 This is a schematic diagram of the node distribution in the standard meshless Galerkin algorithm and the enhanced meshless Galerkin algorithm used for wave propagation dispersion analysis in this embodiment of the invention; Figure 7The curves showing the variation of dispersion error with dimensionless dispersion in different wave propagation directions, calculated using the standard finite element algorithm with triangular elements in this embodiment of the invention. Figure 8 The curves showing the variation of dispersion error with dimensionless dispersion in different wave propagation directions, calculated using the standard finite element algorithm with quadrilateral elements in this embodiment of the invention. Figure 9 The curves showing the variation of dispersion error with dimensionless dispersion in different wave propagation directions, calculated by the standard meshless Galerkin algorithm in this embodiment of the invention. Figure 10 The curves showing the variation of dispersion error with dimensionless frequency in different wave propagation directions are obtained by the enhanced meshless Galerkin numerical algorithm using a first-order Lagrange linear polynomial as the local enhanced interpolation basis function in this embodiment of the invention. Figure 11 The graph shows the variation of dispersion error with dimensionless dispersion in different wave propagation directions, calculated by the enhanced meshless Galerkin algorithm using a combination of first-order Lagrange linear polynomials and trigonometric functions as local enhanced interpolation basis functions in this embodiment of the invention. Detailed Implementation
[0024] The technical solution of the present invention will be further described below with reference to the accompanying drawings and embodiments.
[0025] Example This embodiment focuses on a wave propagation computational domain containing uniformly distributed nodes to calculate and analyze dispersion errors. Although exemplary embodiments of the invention are shown in the accompanying drawings, it should be noted that the application scenarios of this invention should not be limited to the specific scenarios described in this embodiment; the invention is equally applicable to other non-uniformly distributed nodes. This invention can control dispersion errors within a very small range when performing dispersion analysis on wave propagation problems with different node distributions.
[0026] like Figure 1 As shown, the present invention provides an enhanced meshless Galerkin algorithm for reducing dispersion errors in wave propagation analysis, comprising the following steps: S1. Discretize the wave propagation problem domain into nodes and select interpolation basis functions and weight functions; construct an approximate interpolation function based on the standard meshless Galerkin algorithm; determine the unknown coefficients to be solved in the approximate interpolation function, and then establish the initial global interpolation scheme of the standard meshless Galerkin algorithm for the physical field of wave propagation to be solved.
[0027] S11. Discretize the wave propagation problem domain into nodes, and define the influence domain for each node. The influence domain is a circular or rectangular region, such as... Figure 2 As shown, the specific rules are as follows: (1) Define the support region of the integration point: a circular or rectangular region centered on the integration point, such as Figure 3 As shown.
[0028] (2) Define the influence domain of a node: the area that each node can influence when performing numerical integration. The size of the influence domain of a node is closely related to the support domain of the integration point.
[0029] (3) The weight function value of the node within the influence domain is non-zero, while the weight function value of the node outside the influence domain is zero. The weight function expression is as follows: (1); in, Represents the weight function; This indicates the radius of the influence region.
[0030] S12. For any scalar field, construct a numerical interpolation scheme based on the standard meshless Galerkin algorithm, as detailed below: (1) Determine the influence domain and number of nodes involved in the interpolation.
[0031] Based on the influence domain rules defined in S11, the influence domain for interpolation is determined; the number of nodes participating in interpolation is determined by the support domain of the integration points during numerical integration.
[0032] (2) Based on the nodes within the selected influence domain, construct the numerical interpolation scheme of the standard meshless Galerkin algorithm, as shown below: (2); in, This refers to physical variables (such as sound pressure or displacement) that can describe wave propagation problems. This represents the position vector at the point to be interpolated; Represents the interpolation basis function; Indicates the first The unknown coefficients to be determined for each interpolation node; Indicates the number of nodes participating in the interpolation; This represents the vector of unknown coefficients to be determined. This represents the interpolation basis function vector of the standard meshless Galerkin algorithm.
[0033] This embodiment uses a rectangular region, with the center of the rectangular support region as the integration point, and the distance from the center of the rectangular region to the side length of the rectangular region is... ,in, Let be the average spacing between nodes; correspondingly, the influence domain of each node is also a rectangular area, with the node at the center of the rectangular node influence domain, and the distance from the center of the rectangular node influence domain to the side length is . .
[0034] S13. For one-dimensional, two-dimensional, and three-dimensional problems, the Lagrange linear basis functions are used as the interpolation basis functions of the standard meshless Galerkin algorithm, as shown below: (3).
[0035] S14. By using the least squares fitting method, determine the unknown coefficients to be solved for the numerical interpolation scheme of the standard meshless Galerkin algorithm, so that the shape function can be expressed as a linear combination of node values.
[0036] First, construct the weighted squared error function as follows: (4); in, This represents the constructed squared error function; Indicates coordinates as The total number of nodes contained within the influence domain of a node; Indicates coordinates as The function value at the node; This represents the position vector at the point to be interpolated.
[0037] Secondly, the extreme values of the weighted squared difference function are found to determine the unknown coefficients of the numerical interpolation scheme of the standard meshless Galerkin algorithm.
[0038] In order for the weighted squared error function defined in formula (4) to reach its minimum value, the weighted squared error must be about The extreme point, i.e. (5).
[0039] Therefore, we can conclude that: (6).
[0040] Therefore, we get: (7); in, Indicates the corresponding node in the influence domain A vector or matrix consisting of the function values at each node. Specifically, it is expressed as follows: (8).
[0041] matrix Specifically, it is expressed as follows: (9).
[0042] This embodiment uses a two-dimensional problem as an example, and gives a matrix. and For the specific expression, the Lagrange linear basis function used for interpolation in the two-dimensional problem is: The result obtained in this way The matrix is a 3x3 square matrix, as shown below: (10); in, and Indicates the first The coordinates of each node.
[0043] The results obtained The matrix is a 3× n The matrix is shown below: (11).
[0044] S15. Establish the global interpolation format of the standard meshless Galerkin algorithm for the physical field to be determined.
[0045] matrix , Substituting equation (7) into equation (2), we obtain the global interpolation scheme of the standard meshless Galerkin algorithm, as shown below: (12); in, It is a nodal interpolation function constructed by the standard meshless Galerkin; It is the first one obtained by construction Interpolation function for each node.
[0046] S2. Based on the initial global interpolation scheme of the standard meshless Galerkin algorithm obtained in S1, by selecting local reinforcement interpolation basis functions, nodal degrees of freedom containing local reinforcement interpolation basis functions and coefficients to be determined are constructed at the nodes, thus completing the local numerical approximation of the reinforced meshless Galerkin algorithm.
[0047] For two-dimensional wave propagation analysis, first-order Lagrange polynomial basis functions are often used, and for nodes... i At this point, the forms of the enhanced interpolation basis functions are as follows: (13).
[0048] To improve computational accuracy, this invention uses a composite basis function composed of first-order Lagrange polynomial basis functions and trigonometric functions as the basis functions.
[0049] S21. For two-dimensional wave propagation analysis, a composite basis function consisting of a first-order Lagrange linear basis function and trigonometric functions is selected as the basis function. For the nodes... Construct the enhanced interpolation basis function as follows: (14); in, Indicates at node Strengthening interpolation basis functions constructed at the location; and Indicates the first The coordinates of each node; and Indicates the fundamental wavelength. ;in, This represents the average spacing between nodes.
[0050] S22. Using dimensionless relative coordinate values, optimize the local reinforcement interpolation basis function in S21 to ensure sufficient stability of the system matrix equation. The optimized local reinforcement interpolation basis function is shown below: (15); in, and These are dimensionless relative coordinate values.
[0051] S23. Based on S1, the initial global interpolation scheme of the standard meshless Galerkin algorithm is obtained. Combined with the local reinforcement interpolation basis function optimized in S22, the local reinforcement interpolation scheme of the reinforced meshless Galerkin algorithm is constructed as follows: (16); in, This represents the local reinforcement interpolation scheme for the enhanced meshless Galerkin algorithm; The term represents the number of nodes involved in interpolation in the reinforced meshless Galerkin algorithm; the second term in the formula represents the local numerical approximation of the reinforced meshless Galerkin algorithm; the local numerical approximation of the reinforced meshless Galerkin algorithm is completed through the above local reinforced interpolation scheme.
[0052] S3. Based on the initial global interpolation scheme of the standard meshless Galerkin algorithm obtained in S1 and the local numerical approximation of the enhanced meshless Galerkin algorithm obtained in S2, and under the premise of satisfying the unit decomposition property, the global interpolation scheme of the enhanced meshless Galerkin algorithm for the wave propagation physical field is constructed as follows: (17); in, This represents the global interpolation scheme for enhancing the meshless Galerkin algorithm; This represents the enhanced interpolation function obtained through construction.
[0053] S4. Based on the enhanced meshless Galerkin algorithm obtained in S3, the global interpolation scheme is combined with the partial differential control equations of the wave propagation problem and the Galerkin weighted residual method to obtain the system matrix equations for the dispersion analysis of the wave propagation problem.
[0054] S41. Establish the governing equations for the wave propagation problem and transform them into frequency domain small-amplitude wave equations based on time-harmonic characteristics.
[0055] First, for the wave propagation problem in an ideal, stationary, homogeneous fluid medium, the governing equations for small-amplitude waves are established as follows: (18); in, Represents the Laplace operator. Represents the sound pressure in the problem domain of wave propagation. Indicates the speed of sound wave propagation. Indicates time.
[0056] Secondly, based on the time-harmonic characteristics of the small-amplitude wave propagation problem, the sound pressure distribution in the wave propagation problem domain is determined as follows: (19); in, The spatial distribution of sound pressure amplitude; The imaginary unit; ω is the angular frequency.
[0057] Finally, substituting equation (19) into equation (18), we obtain the small-amplitude wave equation in the frequency domain, as shown below: (20); in, Let represent the wave number, and satisfy: (twenty one).
[0058] S42. Introduce boundary conditions for the wave propagation problem; process the frequency domain small amplitude wave equation obtained in S41 using the Galerkin weighted residual method to obtain an integral form equation containing boundary terms.
[0059] First, determine the boundary conditions for the wave propagation problem, including the Dirichlet Boundary Condition (DBC), the Neumann Boundary Condition (NBC), and the Robin Boundary Condition (RBC). These three boundary conditions are as follows: Dirichlet boundary conditions: (twenty two); in, Represent the Dirichlet boundary conditions; This represents the sound pressure value defined at the Dirichlet boundary.
[0060] Newman boundary conditions: (twenty three); in, Represent the Newman boundary conditions; Represents the outward normal unit vector on the boundary; This represents the density of an ideal fluid medium; This represents the vibrational velocity of a particle under the Newman boundary condition.
[0061] Robin's boundary conditions: (twenty four); in, Represent Robin's boundary conditions; This represents the admittance coefficient on the Robin boundary condition.
[0062] Secondly, during the propagation of a small-amplitude wave in an ideal stationary fluid, the particle vibration velocity... and sound pressure gradient The relationship is as follows: (25).
[0063] Then, multiply both sides of equation (20) by a test function. And throughout the entire problem domain of wave propagation. Integrating the above, we obtain the integral form of the corresponding control equation, as shown below: (26).
[0064] Finally, by integrating the integral of the control equations obtained above by parts, we obtain the integral equations containing boundary terms, as shown below: (27); in, Representing the problem domain The boundary, The outward normal unit vector on the boundary is shown below: (28); in, They represent the outward normal unit vectors respectively. Along Projections from three different directions.
[0065] S43. Combining the global interpolation scheme of the enhanced meshless Galerkin algorithm obtained in S3 and the integral equation with boundary terms obtained in S42, the system matrix equation for the dispersion analysis of the wave propagation problem is established.
[0066] First, applying the boundary conditions of equations (22)–(24), equation (27) can be expressed as: (29).
[0067] Secondly, substituting the global interpolation scheme of the enhanced meshless Galerkin algorithm obtained in S3 into equation (29), we can obtain: (30).
[0068] The system matrix equation for dispersion analysis of wave propagation problems can be expressed as: (31); in, Let the vector be the unknown coefficients to be determined. The stiffness matrix in a meshless computational model of the wave propagation problem is specifically represented as: (32); in, The gradient matrix; The mass matrix in the meshless computational model of the wave propagation problem is specifically represented as: (33); The damping matrix in a meshless computational model of wave propagation is specifically represented as: (34); The nodal force vector in a meshless computational model of the wave propagation problem is represented as follows: (35).
[0069] S5. Based on the given node distribution, solve the system matrix equation for the dispersion analysis of the wave propagation problem obtained in step S4, and calculate the dispersion error along different wave propagation directions using different calculation methods.
[0070] S51. When performing dispersion analysis on wave propagation problems, if boundary conditions and damping effects are not considered, the system matrix equation for dispersion analysis of wave propagation problems can be simplified to: (36).
[0071] S52. In the enhanced meshless Galerkin algorithm, each node has If there are unknown node coefficients, then the vector of unknown coefficients to be solved is... It can be represented as: (37); in, Represents the numerical wavenumber; Represents the unit vector along the direction of wave propagation; Indicates and The magnitude vectors corresponding to the unknown node coefficients are shown below: (38); Further define vector Since, without considering boundary conditions, the magnitude vector corresponding to each node is exactly the same, therefore, the vector It appears repeatedly in equation (38).
[0072] S53. For a given node distribution, such as Figure 6 As shown, substituting equation (37) from S52 into the simplified system matrix equation (36) for the dispersion analysis of the wave propagation problem, we can obtain: (39); in, Represents the stiffness matrix The relevant coefficient matrix is specifically represented as follows: (40); in, and Representing the stiffness matrix The Middle row and number Column elements; and These represent the index offsets; Representation and mass matrix The relevant coefficient matrix is specifically represented as follows: (41); in, and Representing the mass matrix respectively The Middle row and number The elements of the column.
[0073] S54. Combining mathematical conditions and the definition of error, complete the calculation of dispersion error.
[0074] First, for equation (39) in S35, the condition for the existence of a non-zero solution is that the determinant of the coefficient matrix is zero, that is: (42).
[0075] From equation (42), it can be seen that It is actually a matrix eigenvalues, i.e. (43).
[0076] From equations (42) and (43) in S54, it can be seen that the matrix and All are numerical wavenumbers The function of , therefore, for any given wavenumber The corresponding numerical wavenumber can be calculated using equation (43). .
[0077] Secondly, in wave propagation analysis, dispersion error is defined as: (44).
[0078] Equation (44) can be used to determine the magnitude of dispersion error when performing wave propagation analysis using different numerical algorithms.
[0079] S6. Based on the dispersion error obtained in step S5 in different wave propagation directions, plot the dispersion error as a function of the dimensionless wave number, compare the dispersion error results obtained by different calculation methods, and complete the dispersion analysis of the wave propagation problem.
[0080] To compare and verify the ability of the enhanced meshless Galerkin algorithm proposed in this invention to control dispersion error, this embodiment not only presents the calculation results obtained when using different local enhanced interpolation basis functions for local enhanced interpolation, but also calculates the results using the standard triangular finite element algorithm with the same node distribution (node distribution as shown in the figure). Figure 4 ), standard quadrilateral finite element algorithm (node distribution as follows) Figure 5 (As shown) and the dispersion error along different wave propagation directions calculated by the standard meshless Galerkin algorithm.
[0081] like Figure 7 , Figure 8 As shown, the dispersion errors in each wave propagation direction calculated using the finite element method with standard triangular and standard quadrilateral elements are relatively large, and the corresponding dispersion errors also increase rapidly with the increase of the dimensionless wave number.
[0082] Compared to finite element algorithms using standard triangular elements, the standard meshless Galerkin algorithm (such as...) Figure 9 As shown in the figure, the dispersion error is relatively small in all wave propagation directions. Therefore, compared with the finite element algorithm using standard triangular elements, it exhibits stronger control over dispersion error. However, the algorithm's ability to control dispersion error is not significantly better than that of the finite element method based on standard quadrilateral elements.
[0083] from Figure 11The dispersion error results show that the enhanced meshless Galerkin algorithm proposed in this invention has the strongest control over dispersion error in wave propagation analysis when using a local enhanced interpolation basis function composed of a first-order Lagrange polynomial and trigonometric functions. This is significantly better than other methods because the dispersion error values calculated by this algorithm are significantly smaller than those of other calculation methods in different wave propagation directions.
[0084] Further comparison shows that, compared to Figure 10 The example shown uses only a first-order Lagrange polynomial as the local reinforcement interpolation basis function. When using a local reinforcement interpolation basis function composed of a first-order Lagrange polynomial and trigonometric functions, the reinforcement meshless Galerkin algorithm of this invention has a more powerful ability to perform wave propagation analysis, which can effectively reduce the dispersion error generated by numerical calculation. Finally, when performing wave propagation analysis, more accurate calculation results can be obtained. Therefore, the reinforcement meshless Galerkin algorithm proposed in this invention has great potential application value in engineering practice.
[0085] Therefore, the present invention adopts the above-mentioned enhanced meshless Galerkin algorithm for reducing dispersion error in wave propagation analysis. The beneficial effect is that the present invention significantly reduces dispersion error without significantly increasing computational complexity, achieving a good balance between high accuracy and high efficiency in wave propagation analysis.
[0086] 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 strengthened meshless Galerkin algorithm for reducing dispersion error in wave propagation analysis, characterized in that, The method comprises the following steps: S1, discretizing a wave propagation problem domain into nodes; selecting an interpolation basis function and a weight function; constructing an approximate interpolation function based on a standard meshless Galerkin algorithm; determining unknown coefficients of the approximate interpolation function, and establishing an initial global interpolation format of the standard meshless Galerkin algorithm for a to-be-solved wave propagation physical field; S2, based on the initial global interpolation format of the standard meshless Galerkin algorithm obtained in S1, constructing a node degree of freedom at the node by selecting a locally reinforced interpolation basis function and to-be-solved coefficients, and completing local numerical approximation of the reinforced meshless Galerkin algorithm; S3, based on the initial global interpolation format of the standard meshless Galerkin algorithm obtained in S1 and the local numerical approximation of the reinforced meshless Galerkin algorithm obtained in S2, constructing a global interpolation format of the reinforced meshless Galerkin algorithm for the wave propagation physical field under the premise of meeting unit decomposition characteristics; S4, based on the global interpolation format of the reinforced meshless Galerkin algorithm obtained in S3, combining a partial differential control equation of the wave propagation problem and a Galerkin weighted residual method, and obtaining a system matrix equation of the wave propagation problem frequency dispersion analysis; S5, solving the system matrix equation of the wave propagation problem frequency dispersion analysis obtained in S4, and obtaining a frequency dispersion error in a wave propagation direction; S6, based on the frequency dispersion error in the wave propagation direction obtained in S5, and compared with a commonly used calculation method of the wave propagation problem, completing frequency dispersion analysis of the wave propagation problem.
2. The enhanced meshless Galerkin algorithm for reducing dispersion error in wave propagation analysis according to claim 1, wherein S1 is specifically: S11, discretizing a wave propagation problem domain into nodes, and defining an influence domain for the nodes, wherein the influence domain is a circular or rectangular region, and the specific rules are as follows: (1) defining a support domain of an integral point, which is a circular or rectangular region centered on the integral point; (2) defining a node influence domain, which is a region that can be influenced by each node during numerical integration, and the size of the node influence domain is closely related to the integral point support domain; (3) the weight function value of the node in the influence domain is non-zero, and the weight function value of the node outside the influence domain is zero, and the weight function expression is as follows: ； wherein, represents a weight function; represents a radius of the influence domain; S12, based on the standard meshless Galerkin algorithm, constructing a numerical interpolation format, and the specific content is as follows: First, according to the influence domain rules defined in S11, determine the influence domain participating in interpolation; the number of nodes participating in interpolation is determined by the integral point support domain during numerical integration; Then, based on the selected nodes in the influence domain, construct the numerical interpolation of the standard meshless Galerkin algorithm, as follows: ； wherein, represents a physical variable capable of describing the wave propagation problem; represents a position vector at the point to be interpolated; represents an interpolation basis function; represents the unknown coefficient of the interpolation node; represents the unknown coefficient vector; represents the interpolation basis function vector of the standard meshless Galerkin algorithm; represents the number of nodes participating in the interpolation; S13, for one-dimensional problems, two-dimensional problems and three-dimensional problems, the Lagrange linear basis function is used as the interpolation basis function of the standard meshless Galerkin algorithm, as follows: ； S14, the least square fitting method is used to determine the to-be-solved unknown coefficients of the numerical interpolation format of the standard meshless Galerkin algorithm, as follows: ； wherein, represents a vector consisting of function values at the nodes in the node influence domain corresponding to ; is specifically represented as: ; is specifically represented as: ; S15, combining the numerical interpolation of the standard meshless Galerkin algorithm in S13 and the to-be-solved unknown coefficients determined in S14, establishing an initial global interpolation format of the standard meshless Galerkin algorithm for a to-be-solved physical field, as follows: ； wherein, is the nodal interpolation function constructed by the standard meshless Galerkin method; is the interpolation function of the first node constructed by the present application.
3. The enhanced meshless Galerkin algorithm for reducing dispersion error in wave propagation analysis according to claim 1, wherein S2 is specifically: S21, select the composite basis function composed of the first order Lagrange linear basis function and the trigonometric function as the basis function, and construct the enhanced interpolation basis function for the nodes as follows: ； wherein, denotes a strengthened interpolation basis function constructed at a node ; and denotes a coordinate value at the th node; and denotes a fundamental wavelength; S22, using dimensionless relative coordinate values to optimize the local reinforced interpolation basis function in S21, as follows: ； wherein and are dimensionless relative coordinate values, denotes the average distance between nodes; S23, based on S1 to get the initial global interpolation format of standard meshless Galerkin algorithm, combined with S22 optimized local reinforcement interpolation basis function, to build a local reinforcement interpolation format of enhanced meshless Galerkin algorithm, as follows: ； wherein, represents a local enrichment interpolation scheme of the enriched meshless Galerkin algorithm; represents the number of nodes participating in interpolation in the enriched meshless Galerkin algorithm; the second term in the equation represents a local numerical approximation of the enriched meshless Galerkin algorithm; Through the above local reinforcement interpolation format, the local numerical approximation of the enhanced meshless Galerkin algorithm is completed.
4. The enhanced meshless Galerkin algorithm for reducing dispersion error in wave propagation analysis according to claim 1, wherein, In S3, the global interpolation format of the enhanced meshless Galerkin algorithm, as follows: ； wherein, represents a global interpolation scheme of the enhanced meshless Galerkin algorithm; represents the constructed enhanced interpolation function.
5. The enhanced meshless Galerkin algorithm for reducing dispersion error in wave propagation analysis according to claim 1, wherein, S4 is specifically: S41, the control equation of wave propagation problem is established, and the small amplitude wave equation in frequency domain is transformed based on the time harmonic characteristic; S42, the boundary condition of wave propagation problem is introduced; the small amplitude wave equation in frequency domain obtained by S41 is processed by Galerkin weighted residual method to obtain the integral form equation containing boundary term; S43, combined with the global interpolation format of the enhanced meshless Galerkin algorithm obtained by S3, the integral form equation containing boundary term obtained by S42, the system matrix equation of wave propagation problem dispersion analysis is established.
6. The enhanced meshless Galerkin algorithm for reducing dispersion error in wave propagation analysis according to claim 5, wherein S41 is specifically: First, for the wave propagation problem in ideal static uniform fluid medium, the control equation of small amplitude wave is established, as follows: ； wherein, represents the Laplacian operator; represents a variable in the wave propagation problem domain; represents the propagation velocity of the sound wave; represents time; Secondly, based on the time harmonic characteristic of small amplitude wave propagation problem, the spatial distribution of variables in wave propagation problem domain is determined, as follows: ； wherein is the spatial distribution of the variable amplitude; is the imaginary unit; is the angular frequency; Finally, the above variable distribution is substituted into the control equation to obtain the small amplitude wave equation in frequency domain, as follows: ； wherein represents the wave number and satisfies: .
7. The enhanced meshfree Galerkin algorithm for reducing dispersion error in wave propagation analysis according to claim 1, wherein S42 is specifically: First, the boundary condition of wave propagation problem is determined, as follows: Dirichlet boundary condition: ; wherein denotes a Dirichlet boundary condition; denotes a prescribed amplitude value of the variable on the Dirichlet boundary; Neumann boundary condition: ; wherein, represents the Neumann boundary condition; represents the outward normal unit vector on the boundary; represents the density of the ideal fluid medium; represents the vibration velocity of the particle on the Neumann boundary condition; Robin boundary condition: ; wherein denotes a Robin boundary condition; denotes an admittance coefficient on the Robin boundary condition; Secondly, the relationship between the particle vibration velocity and the sound pressure gradient is shown as follows: ； Finally, the frequency domain wave equation is processed by Galerkin weighted residual method to obtain the integral form equation containing boundary term, as follows: ； wherein, represents the boundary of the problem domain .
8. The enhanced meshfree Galerkin algorithm for reducing dispersion error in wave propagation analysis according to claim 1, wherein, In S43, the system matrix equation of wave propagation problem dispersion analysis, as follows: ； wherein, is the unknown coefficient vector to be solved; denotes the stiffness matrix in the meshless computational model for wave propagation problems, and is specifically expressed as: ; wherein, is the gradient matrix; denotes the mass matrix in the meshless computational model for wave propagation problems, and is specifically expressed as: ; denotes the damping matrix in the meshless computational model for general wave propagation problems, and is specifically expressed as: ; denotes the node force vector in the meshless computational model for general wave propagation problems, and is specifically expressed as: .
9. The enhanced meshfree Galerkin algorithm for reducing dispersion error in wave propagation analysis according to claim 1, wherein, S5 is specifically: S51, when performing wave propagation problem dispersion analysis, ignore the boundary condition and damping effect, simplify the system matrix equation of wave propagation problem dispersion analysis, as follows: ； S52, set the unknown node coefficient in the enhanced meshless Galerkin algorithm at the node unknown coefficient vector to be solved as follows: ； wherein, denotes the numerical wavenumber; denotes the unit vector in the direction of wave propagation; denotes the amplitude vector corresponding to the unknown node coefficients, and is formed by the basis unit vectors periodically repeated. S53, the unknown node coefficient vector in S52 is substituted into the simplified system matrix equation of wave propagation problem dispersion analysis to obtain: ； wherein denotes the stiffness matrix the associated coefficient matrix, in particular given by ;in, and Representing the stiffness matrix The Middle row and number Column elements; and These represent the index offsets; Representation and mass matrix The relevant coefficient matrix is specifically represented as follows: ;in, and Representing the mass matrix respectively The Middle row and number Column elements; S54, combined with mathematical conditions and error definition, the dispersion error calculation is completed.
10. The enhanced meshless Galerkin algorithm for reducing dispersion error in wave propagation analysis according to claim 9, wherein, S54 is specifically: First, for the equation in S53, the condition for the existence of a non-zero solution is that the determinant of the coefficient matrix is zero, i.e. ; At this time, is the eigenvalue of the matrix , i.e., ; Second, based on the wave number and the numerical wave number , the dispersion error is defined as: ; Thus, the dispersion error of wave propagation analysis is obtained.