Enhanced moving least square algorithm for reducing frequency dispersion error of wave propagation analysis
By enhancing the moving least squares algorithm and utilizing local enhanced interpolation basis functions and global interpolation schemes, the problem of dispersion error in wave propagation analysis is solved, achieving higher accuracy calculation results. This method is applicable to fields such as mechanical structure vibration analysis, underwater acoustic detection, seismic wave inversion, and medical ultrasound diagnosis.
Patent Information
- Application Number
- CN202511696877.3
- 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
In existing technologies, dispersion errors in wave propagation analysis are difficult to reduce effectively, resulting in insufficient computational accuracy. This is especially true for large-scale three-dimensional wave propagation problems, where the computational load is large, the time consumption is long, and the timeliness requirements of engineering design cannot be met.
An enhanced moving least squares algorithm is adopted, which combines local enhanced interpolation basis functions and global interpolation schemes with the Galerkin weighted residual method to construct the system matrix equation for dispersion analysis of wave propagation problems, thereby reducing dispersion error.
Without adding nodes, it significantly reduces dispersion error, improves calculation accuracy, and enhances wave propagation analysis capabilities, making it suitable for practical engineering applications.
Smart Images
Figure CN121542559A_ABST
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 moving least squares algorithm for reducing dispersion error in wave propagation analysis. Background Technology
[0002] Wave propagation is a core topic in wave dynamics research, with significant applications in key areas such as mechanical structure vibration analysis, underwater acoustic detection, seismic wave inversion, medical ultrasound diagnosis, and sonar technology.
[0003] While analytical methods can provide accurate mathematical solutions, they are limited by their inherent mathematical constraints (such as requirements for homogeneous media and regular boundaries), making them suitable only for one-dimensional models or two-dimensional scenarios with simple geometric boundaries, such as wave propagation analysis in an infinitely large flat plate. However, in engineering practice, complex boundary conditions (such as fault geological structures in oil exploration) and non-homogeneous media (such as the continuous variation of acoustic impedance in human tissue) are common challenges. Therefore, numerical simulation techniques have become an indispensable research tool. These methods, through spatial domain discretization approximation combined with high-performance computing technology, provide a reliable technical approach for wave propagation analysis.
[0004] In numerical simulations of wave propagation problems, the finite element method (FEM) is one of the most commonly used methods. However, this method suffers from numerical errors during the solution process. 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). This results in a geometrical increase in model size with increasing frequency: when dealing with large-scale three-dimensional wave propagation problems, the required finite element mesh becomes exceptionally large to guarantee the accuracy of the results. This massive computational load not only requires supercomputing hardware support, with a single solution potentially taking tens of hours, but also often exceeds the memory capacity of ordinary workstations. More importantly, in scenarios involving multiphysics coupling or parameter inversion that require repeated calculations, the traditional finite element method is almost unable to meet the timeliness requirements of engineering designs.
[0005] In numerical simulations of wave propagation, the primary source of numerical error lies in the dispersion effect caused by the discretization process. Specifically, this manifests as a significant deviation between the numerically calculated wavenumber and the theoretically accurate solution, with this dispersion error amplifying dramatically as the frequency increases. 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 rigid 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, resulting in what is known as "numerical dispersion."
[0006] In order to overcome the mesh dependence of the finite element method and solve the problem that existing technologies cannot completely eliminate the dispersion error in wave propagation analysis, this invention discloses an enhanced moving least squares algorithm to reduce the dispersion error in wave propagation analysis. Summary of the Invention
[0007] The purpose of this invention is to provide an enhanced moving least squares algorithm that reduces dispersion error in wave propagation analysis. This algorithm can significantly reduce dispersion error during dispersion analysis, thus providing more accurate numerical calculation results when analyzing wave propagation problems.
[0008] To achieve the above objectives, this invention provides an enhanced moving least squares 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; fit local approximate interpolation functions based on moving least squares method, and derive the interpolation basis function format of the nodes based on standard moving least squares. S2. Based on S1, the interpolation basis function format of the standard moving least squares algorithm is obtained. By selecting a locally enhanced interpolation basis function, nodal degrees of freedom containing the locally enhanced interpolation basis function and the coefficients to be determined are constructed at the nodes, thus completing the local numerical approximation of the enhanced moving least squares algorithm: S3. Based on the interpolation basis function of the standard moving least squares algorithm obtained in S1 and the local numerical approximation of the enhanced moving least squares algorithm obtained in S2, construct the global interpolation scheme of the general scalar field to obtain the global interpolation scheme of the enhanced moving least squares algorithm. S4. Based on the global interpolation scheme of the enhanced moving least squares algorithm obtained in S3, combined with the partial differential control equations of the wave propagation problem and the Galerkin weighted residual method, the system matrix equations for the dispersion analysis of the wave propagation problem are obtained. 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.
[0009] 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 values of nodes within the influence domain are non-zero, while the weight function values of nodes outside the influence domain are zero. The weight functions are as follows: ; in, Represents the weight function; Indicates the radius of the influence region; S12. Based on the standard moving least squares algorithm, construct a numerical interpolation method, 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, numerical interpolation 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 the vector of unknown coefficients to be determined. The interpolation basis function vector represents the standard moving least squares 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 for the standard moving least squares algorithm, as shown below: ; S14. Construct the weighted squared error function and determine the coefficients of the interpolation basis function for the standard moving least squares algorithm; The weighted squared error function is shown below: ; 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; The coefficients of the interpolation basis function for the standard moving least squares algorithm are 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: ; matrix Specifically, it is expressed as follows: ; S15, Matrix , Substituting the coefficient formulas of the interpolation basis functions of the standard moving least squares algorithm into the numerical interpolation scheme of S12, we obtain the interpolation basis function schemes of the standard moving least squares algorithm for the nodes, as shown below: ; in, It is the interpolation basis function vector of the standard moving least squares algorithm; It is the first Interpolation function for each node.
[0010] Preferably, S2 is as follows: S21. Choosing the Lagrange polynomial basis functions as the local reinforcement interpolation basis functions, the expressions for one-dimensional, two-dimensional, and three-dimensional problems are as follows: ; in, This represents the local reinforcement interpolation basis function corresponding to the local numerical approximation constructed at the node; , , These represent the coordinate values of a node in a three-dimensional Cartesian coordinate system; S22. Using dimensionless relative coordinate values, construct the local enhancement interpolation basis functions 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 interpolation basis function format of the standard moving least squares algorithm is obtained. Combined with the locally enhanced interpolation basis function optimized in S22, the local interpolation format of the enhanced moving least squares algorithm is constructed as follows: ; in, The local interpolation scheme for the enhanced moving least squares algorithm is as follows: The term represents the number of nodes involved in the interpolation in the enhanced moving least squares algorithm; the second term in the formula represents the local numerical approximation of the enhanced moving least squares algorithm.
[0011] Preferably, in S3, the global interpolation scheme of the enhanced moving least squares algorithm is obtained as follows: ; in, The global interpolation format representing the enhanced moving least squares algorithm; This represents the enhanced interpolation function obtained through construction.
[0012] 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 moving least squares algorithm obtained in S3 and the integral form equation with boundary terms obtained in S42, the system matrix equation for dispersion analysis of the wave propagation problem is established.
[0013] 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. Describe the variables in the problem domain of wave propagation. Indicates the speed of sound wave propagation. Indicates time; Secondly, based on the time-harmonic characteristics of the small-amplitude wave propagation problem, the distribution of variables in the wave propagation problem domain is as follows: ; in, For the spatial distribution of variables; 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: .
[0014] 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; Indicates the spatial distribution of variables; 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: ; Then, both sides of the small-amplitude wave equation in the frequency domain are multiplied by the test function. and in the wave propagation problem domain Integrating the above, we obtain the integral form of the governing equation, as shown below: ; Finally, by integrating the integral of the control equations obtained above by parts, we obtain the integral equations containing boundary terms, as shown below: ; in, Representing the problem domain The boundary, The outward normal unit vector on the boundary is expressed as follows: ; in, They represent the outward normal unit vectors respectively. Along Projections from three different directions.
[0015] Preferably, in S43, the system matrix equation for the dispersion analysis of the wave propagation problem is as follows: ; in, 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: .
[0016] 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 moving least squares algorithm, at node , there is 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 vectors corresponding to the unknown node coefficients are shown below: ; 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: ; Where, vector ; Representation of 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.
[0017] 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.
[0018] Therefore, the present invention employs the above-mentioned enhanced moving least squares algorithm for reducing dispersion error in wave propagation analysis, which has the following beneficial effects: (1) The present invention uses appropriate basis functions to strengthen the original interpolation space, so as to obtain a higher order interpolation format than the standard moving least squares method without adding nodes.
[0019] (2) Based on the design of high-order interpolation, the enhanced moving least squares calculation method proposed in this invention has a more powerful ability to perform wave propagation analysis compared with the standard moving least squares method, which can effectively reduce the dispersion error generated by numerical calculation and finally obtain more accurate calculation results when performing wave propagation analysis.
[0020] (3) The enhanced moving least squares calculation method proposed in this invention has great potential application value in engineering practice.
[0021] 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
[0022] Figure 1 This is a flowchart of an enhanced moving least squares algorithm for reducing dispersion error in wave propagation analysis according to the present invention; Figure 2This is a schematic diagram of the node influence domain in the enhanced moving least squares algorithm in this embodiment of the invention; Figure 3 This is a schematic diagram of the support domain of the integral point in the enhanced moving least squares algorithm in this embodiment of the 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 moving least squares algorithm and the enhanced moving least squares algorithm used for wave propagation dispersion analysis in this embodiment of the invention; Figure 7 The 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 moving least squares algorithm in this embodiment of the invention; Figure 10 The curves showing the variation of dispersion error with dimensionless dispersion in different wave propagation directions, calculated by the enhanced moving least squares algorithm in this embodiment of the invention. Detailed Implementation
[0023] The technical solution of the present invention will be further described below with reference to the accompanying drawings and embodiments.
[0024] Example This embodiment calculates and analyzes dispersion error using a wave propagation computational domain containing uniformly distributed nodes. While exemplary embodiments of this disclosure are presented in the accompanying drawings, it should be understood that the invention can also be applied to scenarios with non-uniformly distributed nodes, and the invention should not be limited to the embodiments described herein. Practical applications show that the present invention can control dispersion error within a very small range when performing dispersion analysis on wave propagation problems with different node distributions.
[0025] like Figure 1 As shown, the present invention provides an enhanced moving least squares 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; fit local approximate interpolation functions based on moving least squares method, and derive the interpolation basis function format of the nodes based on standard moving least squares.
[0026] 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.
[0027] (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.
[0028] (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.
[0029] S12. For any scalar field, construct a numerical interpolation method based on the standard moving least squares algorithm, as detailed below: (1) Determine the influence domain and number of nodes involved in the interpolation.
[0030] 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.
[0031] (2) Based on the nodes within the selected influence domain, construct numerical interpolation as follows: (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 a vector consisting of the unknown coefficients to be determined. This represents the interpolation basis function vector of the standard moving least squares algorithm.
[0032] 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 . .
[0033] S13. For one-dimensional, two-dimensional, and three-dimensional problems, the Lagrange linear basis functions are used as the interpolation basis functions for the standard moving least squares algorithm, as shown below: (3); S14. Construct the weighted squared error function and determine the coefficients of the interpolation basis function for the standard moving least squares algorithm.
[0034] First, the weighted squared error function is 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.
[0035] Secondly, the extreme values of the weighted squared difference function are found to determine the coefficients of the interpolation basis function based on standard moving least squares.
[0036] 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).
[0037] Therefore, we can conclude that: (6).
[0038] 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).
[0039] matrix Specifically, it is expressed as follows: (9).
[0040] 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.
[0041] The results obtained The matrix is a 3× n The matrix is shown below: (11).
[0042] S15, Matrix , Substituting equation (7) into equation (2), we obtain the interpolation basis function format for the standard moving least squares algorithm of the node, as shown below: (12); in, It is the interpolation basis function vector of the standard moving least squares algorithm; It is the first one obtained by construction Interpolation function for each node.
[0043] S2. Based on S1, the interpolation basis function format of the standard moving least squares algorithm is obtained. By selecting the local reinforcement interpolation basis function, the node degrees of freedom containing the local reinforcement interpolation basis function and the coefficients to be determined are constructed at the node, thus completing the local numerical approximation of the enhanced moving least squares algorithm.
[0044] S21. Choosing the Lagrange polynomial basis functions as the local reinforcement interpolation basis functions, the expressions for one-dimensional, two-dimensional, and three-dimensional problems are as follows: (13); in, This represents the local reinforcement interpolation basis function corresponding to the local numerical approximation constructed at the node; , , These represent the coordinates of a node in a three-dimensional Cartesian coordinate system.
[0045] S22. To ensure sufficient stability of the system matrix equations, dimensionless relative coordinate values are used to construct local reinforcement interpolation basis functions, as shown below: (14); in, , and These are dimensionless relative coordinate values. This represents the average spacing between nodes.
[0046] S23. Based on S1, the interpolation basis function format of the standard moving least squares algorithm is obtained. Combined with the locally enhanced interpolation basis function optimized in S22, the local interpolation format of the enhanced moving least squares algorithm is constructed as follows: (15); in, The local interpolation scheme for the enhanced moving least squares algorithm is as follows: This represents the number of nodes involved in interpolation in the enhanced moving least squares algorithm; the second term in the formula represents the local numerical approximation of the enhanced moving least squares algorithm. This represents the local reinforcement interpolation basis function corresponding to the local numerical approximation constructed at each node.
[0047] S3. Based on the interpolation basis functions of the standard moving least squares algorithm obtained in S1 and the local numerical approximation of the enhanced moving least squares algorithm obtained in S2, a global interpolation scheme for a general scalar field is constructed under the premise of satisfying the unity decomposition property. Finally, the global interpolation scheme of the enhanced moving least squares algorithm is obtained, as shown below: (16); in, The global interpolation format representing the enhanced moving least squares algorithm; This represents the enhanced interpolation function obtained through construction.
[0048] S4. Based on the global interpolation scheme of the enhanced moving least squares algorithm obtained in S3, combined with the partial differential control equations of the wave propagation problem and the Galerkin weighted residual method, the system matrix equations for the dispersion analysis of the wave propagation problem are obtained.
[0049] 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.
[0050] First, for the wave propagation problem in an ideal, stationary, homogeneous fluid medium, the governing equations for small-amplitude waves are as follows: (17); in, Represents the Laplace operator. Represents variables (such as sound pressure) in the problem domain of wave propagation. Indicates the speed of sound wave propagation. Indicates time.
[0051] Secondly, considering the time-harmonic characteristics of small-amplitude wave propagation, the distribution of variables (such as sound pressure) in the wave propagation problem domain can be expressed as: (18); in, Spatial distribution of variables (such as sound pressure variables); The imaginary unit; ω is the angular frequency.
[0052] Finally, substituting equation (18) into equation (17), we obtain the small-amplitude wave equation in the frequency domain, as shown below: (19); in, Let represent the wave number, and satisfy: (20); 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.
[0053] 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 one); in, Represent the Dirichlet boundary conditions; This represents the spatial distribution of variables (such as sound pressure variables) in the wave propagation problem domain. This represents the sound pressure value defined at the Dirichlet boundary; Newman boundary conditions: (twenty two); 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: (twenty three); in, Represent Robin's boundary conditions; This represents the admittance coefficient on the Robin boundary condition.
[0054] 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: (twenty four); Then, multiply both sides of equation (19) by a test function. And throughout the entire discussion of the wave propagation problem Integrating the above, we obtain the integral form of the corresponding control equation, as shown below: (25); Finally, by integrating the integral of the control equations obtained above by parts, we obtain the integral equations containing boundary terms, as shown below: (26); in, Representing the problem domain The boundary, The outward normal unit vector on the boundary is shown below: (27); in, They represent the outward normal unit vectors respectively. Along Projections from three different directions.
[0055] S43. Combining the global interpolation scheme of the enhanced moving least squares algorithm obtained in S3 and the integral form equation with boundary terms obtained in S42, the system matrix equation for dispersion analysis of the wave propagation problem is established.
[0056] First, applying the boundary conditions of equations (21)–(23), equation (26) can be expressed as: (28); Secondly, substituting the global interpolation scheme from the enhanced moving least squares algorithm obtained in S3 into equation (27), we get: (29); The system matrix equation for dispersion analysis of wave propagation problems can be expressed as: (30); 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: (31); in, The gradient matrix; The mass matrix in the meshless computational model of the wave propagation problem is specifically represented as: (32); The damping matrix in a meshless computational model of wave propagation is specifically represented as: (33); The nodal force vector in a meshless computational model of the wave propagation problem is represented as follows: (34); S5, based on such Figure 6 Given the node distribution shown, the system matrix equation for the dispersion analysis of the wave propagation problem obtained in step S4 is solved, and the dispersion error along different wave propagation directions is calculated using different calculation methods.
[0057] 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: (35); S52. If in the enhanced moving least squares algorithm, at each node there is... If there are unknown node coefficients, then the vector of unknown coefficients to be solved is... It can be represented as (36); 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: (37); The magnitude vector corresponding to each node is exactly the same without considering the boundary conditions; therefore, the vector... It will appear repeatedly in equation (37).
[0058] S53. For a given node distribution, substituting equation (36) from S52 into the simplified system matrix equation (35) for the dispersion analysis of the wave propagation problem, we get: (38); in, Representation of stiffness matrix The relevant coefficient matrix is specifically represented as follows: (39); 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: (40); in, and Representing the mass matrix respectively The Middle row and number The elements of the column.
[0059] S54. Combining mathematical conditions and the definition of error, complete the calculation of dispersion error.
[0060] First, for equation (38) in S35, the condition for the existence of a non-zero solution is that the determinant of the coefficient matrix is zero, that is: (41); From equation (41), it can be seen that It is a matrix The eigenvalues, namely: (42); Secondly, it can be seen from equations (40) and (41) in S54 that the matrix and Both are numerical wavenumbers The function of, therefore, for any given exact wavenumber The corresponding numerical wavenumber can be calculated using equation (42). .
[0061] In wave propagation analysis, dispersion error is defined as: (43); Equation (43) can be used to determine the magnitude of dispersion error when performing wave propagation analysis using different numerical algorithms.
[0062] 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.
[0063] To compare and verify the ability of the enhanced moving least squares algorithm proposed in this invention to control dispersion error, this embodiment calculates 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 moving least squares algorithm.
[0064] like Figure 7 As shown, the dispersion errors in each wave propagation direction calculated using the finite element method with standard triangular elements are relatively large, and the corresponding dispersion errors also increase rapidly with the increase of the dimensionless wave number.
[0065] like Figure 8 As shown, the dispersion error calculated using the finite element method with standard quadrilateral elements is also relatively large, and it shows a significant upward trend with the increase of dimensionless wavenumber.
[0066] like Figure 9 As shown, compared with the finite element method using standard triangular elements, the standard moving least squares algorithm calculates smaller dispersion errors in each wave propagation direction. Therefore, it demonstrates stronger control over dispersion errors compared to the finite element method using standard triangular elements. However, its ability to control dispersion errors is not significantly better than that of the finite element method based on standard quadrilateral elements.
[0067] like Figure 10 As shown, compared with other methods, the enhanced moving least squares calculation method proposed in this invention has the strongest control capability for dispersion error in wave propagation analysis, because the dispersion error values calculated in each different wave propagation direction are significantly smaller than those of other calculation methods.
[0068] In summary, the enhanced moving least squares calculation method presented in this invention has a more powerful ability to perform wave propagation analysis, which can effectively reduce the dispersion error generated by numerical calculation and ultimately obtain more accurate calculation results when performing wave propagation analysis. Therefore, the enhanced moving least squares calculation method proposed in this invention has great potential application value in engineering practice.
[0069] Therefore, the present invention adopts the above-mentioned enhanced moving least squares algorithm to reduce dispersion error in wave propagation analysis, which has the following advantages: it achieves high-order interpolation through local enhanced numerical approximation without increasing the number of unit nodes; the enhanced moving least squares algorithm for calculating wave propagation problems can significantly reduce dispersion error when performing dispersion analysis, thus it can obtain numerical calculation results with better calculation accuracy when performing wave propagation problem analysis.
[0070] 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. An enhanced moving least squares algorithm for reducing dispersion error in wave propagation analysis, characterized in that, Includes the following steps: S1. Discretize the wave propagation problem domain into nodes; select interpolation basis functions and weight functions; fit local approximate interpolation functions based on the moving least squares algorithm, and derive the interpolation basis function format of the nodes based on the standard moving least squares algorithm; S2. Based on S1, the interpolation basis function format of the standard moving least squares algorithm is obtained. By selecting a locally enhanced interpolation basis function, nodal degrees of freedom containing the locally enhanced interpolation basis function and the coefficients to be determined are constructed at the nodes, thus completing the local numerical approximation of the enhanced moving least squares algorithm: S3. Based on the interpolation basis function of the standard moving least squares algorithm obtained in S1 and the local numerical approximation of the enhanced moving least squares algorithm obtained in S2, construct the global interpolation scheme of the general scalar field to obtain the global interpolation scheme of the enhanced moving least squares algorithm. S4. Based on the global interpolation scheme of the enhanced moving least squares algorithm obtained in S3, combined with the partial differential control equations of the wave propagation problem and the Galerkin weighted residual method, the system matrix equations for the dispersion analysis of the wave propagation problem are obtained. 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.
2. The enhanced moving least squares algorithm for reducing dispersion error in wave propagation analysis according to claim 1, characterized in that, S1 specifically refers to: 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 values of nodes within the influence domain are non-zero, while the weight function values of nodes outside the influence domain are zero. The weight functions are as follows: ； in, Represents the weight function; Indicates the radius of the influence region; S12. Based on the standard moving least squares algorithm, construct a numerical interpolation method, 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, numerical interpolation 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. The interpolation basis function vector represents the standard moving least squares 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 for the standard moving least squares algorithm, as shown below: ； S14. Construct the weighted squared error function and determine the coefficients of the interpolation basis function for the standard moving least squares algorithm; The weighted squared error function is shown below: ； 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; The coefficients of the interpolation basis function for the standard moving least squares algorithm are 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: ; matrix Specifically, it is expressed as follows: ; S15, Matrix , Substituting the coefficient formulas of the interpolation basis functions of the standard moving least squares algorithm into the numerical interpolation scheme of S12, we obtain the interpolation basis function schemes of the standard moving least squares algorithm for the nodes, as shown below: ； in, It is the interpolation basis function vector of the standard moving least squares algorithm; It is the first Interpolation function for each node.
3. The enhanced moving least squares algorithm for reducing dispersion error in wave propagation analysis according to claim 1, characterized in that, S2 specifically refers to: S21. Choosing the Lagrange polynomial basis functions as the local reinforcement interpolation basis functions, the expressions for one-dimensional, two-dimensional, and three-dimensional problems are as follows: ； in, This represents the local reinforcement interpolation basis function corresponding to the local numerical approximation constructed at the node; , , These represent the coordinate values of a node in a three-dimensional Cartesian coordinate system; S22. Using dimensionless relative coordinate values, construct the local enhancement interpolation basis functions 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 interpolation basis function format of the standard moving least squares algorithm is obtained. Combined with the locally enhanced interpolation basis function optimized in S22, the local interpolation format of the enhanced moving least squares algorithm is constructed as follows: ； in, The local interpolation scheme for the enhanced moving least squares algorithm is as follows: The term represents the number of nodes involved in the interpolation in the enhanced moving least squares algorithm; the second term in the formula represents the local numerical approximation of the enhanced moving least squares algorithm.
4. The enhanced moving least squares algorithm for reducing dispersion error in wave propagation analysis according to claim 1, characterized in that, In S3, the global interpolation scheme for the enhanced moving least squares algorithm is obtained as follows: ； in, The global interpolation format representing the enhanced moving least squares algorithm; This represents the enhanced interpolation function obtained through construction.
5. The enhanced moving least squares algorithm for reducing dispersion error in wave propagation analysis according to claim 1, characterized in that, S4 specifically refers to: 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 moving least squares algorithm obtained in S3 and the integral form equation with boundary terms obtained in S42, the system matrix equation for dispersion analysis of the wave propagation problem is established.
6. The enhanced moving least squares algorithm for reducing dispersion error in wave propagation analysis according to claim 5, characterized in that, S41 specifically refers to: 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. Describe the variables in the problem domain of wave propagation. Indicates the speed of sound wave propagation. Indicates time; Secondly, based on the time-harmonic characteristics of the small-amplitude wave propagation problem, the distribution of variables in the wave propagation problem domain is as follows: ； in, For the spatial distribution of variables; 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: 。 7. The enhanced moving least squares algorithm for reducing dispersion error in wave propagation analysis according to claim 1, characterized in that, S42 specifically refers to: First, the boundary conditions for the wave propagation problem are determined as follows: Dirichlet boundary conditions: ； in, Represent the Dirichlet boundary conditions; Indicates the spatial distribution of variables; 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: ； Then, both sides of the small-amplitude wave equation in the frequency domain are multiplied by the test function. and in the wave propagation problem domain Integrating the above, we obtain the integral form of the governing equation, as shown below: ； Finally, by integrating the integral of the control equations obtained above by parts, we obtain the integral equations containing boundary terms, as shown below: ； in, Representing the problem domain The boundary, The outward normal unit vector on the boundary is expressed as follows: ； in, They represent the outward normal unit vectors respectively. Along Projections from three different directions.
8. The enhanced moving least squares algorithm for reducing dispersion error in wave propagation analysis according to claim 1, characterized in that, In S43, the system matrix equation for the dispersion analysis of the wave propagation problem is as follows: ； in, 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: .
9. The enhanced moving least squares algorithm for reducing dispersion error in wave propagation analysis according to claim 1, characterized in that, S5 specifically refers to: 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 moving least squares algorithm, at node , there is 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 vectors corresponding to the unknown node coefficients are shown below: ； 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: ； Where, vector ; Representation of 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.
10. The enhanced moving least squares algorithm for reducing dispersion error in wave propagation analysis according to claim 9, characterized in that, S54 specifically refers to: 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.