An enhanced meshless computational method for reducing dispersion errors in wave propagation problems

Abstract

The present invention discloses an enhanced gridless computing method for reducing the dispersion error of wave propagation problems, which relates to the field of numerical calculation and dispersion error prediction of wave propagation problems. First, the wave propagation problem domain is discretized into a series of scattered points, and an interpolation format of a general scalar field is constructed based on radial basis functions, and the interpolation basis function of each scattered point is derived; in order to improve the interpolation accuracy of the interpolation format obtained, a new node degree of freedom is constructed at each scattered point, and the node degree of freedom is a local numerical approximation composed of a local enhanced interpolation basis function and a coefficient to be determined. Based on the obtained local numerical approximation, a global interpolation format of a general scalar field is constructed under the premise of satisfying the unit decomposition characteristics. Utilizing the constructed global interpolation format, combined with the partial differential governing equation of the wave propagation problem, the Galerkin weighted residual method is used to obtain a numerical calculation model for the dispersion analysis of the wave propagation problem, and the dispersion error of the wave propagation problem is calculated. The present invention can significantly reduce the dispersion error.

Description

[0001] Methodology

[0002] The present invention relates to the field of numerical calculation of wave propagation problems and dispersion error prediction, and in particular to an enhanced gridless calculation method for reducing dispersion errors in wave propagation problems.

[0003] Background Methods

[0004] Wave propagation is an important research topic in the field of wave dynamics. It has extremely important application backgrounds in many engineering fields such as mechanical engineering, underwater acoustic engineering, earthquake detection, medical ultrasound imaging, and even national defense and military industry. The most common numerical method for solving wave propagation problems is the finite element method, but the finite element solution of wave propagation problems often has certain numerical errors. In actual engineering calculations, in order to ensure a certain level of calculation accuracy, the established wave dynamics finite element calculation model must meet the so-called "rule of thumb", that is, a certain number of units must be guaranteed within a wavelength range. This causes the scale of the finite element calculation model of the wave problem to increase rapidly with the increase of the calculation frequency; for large-scale three-dimensional wave propagation problems, if the results are to be guaranteed to have sufficient solution accuracy, the mesh of the constructed finite element model of the wave problem will be very large, which is not applicable in actual engineering.

[0005] In addition to the standard finite element method, the meshless method is also an important numerical method for solving wave dynamics problems. Compared with the finite element method, the main advantage of the meshless method is that it does not require meshing the wave propagation problem domain to be solved. Only a series of distributed scattered points are needed to complete the discretization of the computational problem domain. Since the dispersion error of the finite element method in solving wave propagation problems is closely related to the quality of the mesh used, a mesh of poor quality will naturally lead to a larger dispersion error. When the meshless method is used for numerical calculations of wave propagation problems, since no mesh is required, its ability to control dispersion error is significantly stronger than that of the standard finite element method. The meshless method based on radial basis functions is a classic meshless numerical method that has been widely used in the numerical calculation of wave propagation problems. Although it can obtain smaller dispersion errors than the standard finite element method when performing numerical calculations of wave propagation problems, it still cannot completely eliminate the dispersion errors of wave dynamics calculations.

[0006] Therefore, an enhanced meshless computing method for reducing the dispersion error of wave propagation problems is provided to solve the above problems. Summary of the Invention

[0007] The purpose of the present invention is to provide an enhanced meshless computing method for reducing dispersion errors in wave propagation problems. The original interpolation space is enhanced by using appropriate basis functions, so that a higher-order interpolation format than the standard radial basis function meshless method can be obtained without adding nodes. The dispersion error can be significantly reduced when performing dispersion analysis, and more accurate numerical results can be obtained when analyzing wave propagation problems.

[0008] To achieve the above object, the present invention provides an enhanced gridless computing method for reducing dispersion error in wave propagation problems, comprising the following steps:

[0009] Step S1: Discretize the wave propagation problem domain into a series of scattered points and derive the node interpolation basis functions of the standard radial point interpolation meshless algorithm;

[0010] Step S2: Select appropriate local enhanced interpolation basis functions, construct node degrees of freedom at the nodes consisting of the local enhanced interpolation basis functions and the coefficients to be determined, and complete the local numerical approximation of the enhanced radial point interpolation meshless algorithm;

[0011] Step S3: Based on the node interpolation basis functions of the standard radial point interpolation meshless algorithm obtained in step S1 and the local numerical approximation obtained in step S2, a global interpolation format for the general scalar field is constructed under the premise of satisfying the unit decomposition property;

[0012] S4: Based on the global interpolation format of the general scalar field obtained in step S3, combined with the partial differential governing equations of the wave propagation problem and the Galerkin weighted residual method, the system matrix equation for the dispersion analysis of the wave propagation problem is obtained;

[0013] S5: Based on the given field point distribution, the system matrix equation obtained in step S4 is solved to calculate the dispersion errors of different calculation methods along different wave propagation directions;

[0014] S6: Based on the dispersion errors in different wave propagation directions obtained in step S5, a curve of the dispersion error changing with the dimensionless wave number is plotted, and the dispersion error results obtained by different calculation methods are compared to complete the dispersion analysis of the wave propagation problem.

[0015] Preferably, step S1 specifically includes the following steps:

[0016] For any scalar field, numerical interpolation p h (x) is expressed as:

[0017]

[0018] Where x represents the position vector of the interpolation point, m is the number of field nodes involved in the interpolation, R(x) is the radial basis function, S(x) is the Lagrange polynomial basis function, and linear basis functions are often used for interpolation; n is the number of terms in the Lagrange polynomial basis function, and a and b represent the undetermined coefficient vectors related to the radial basis function and the polynomial basis function, respectively.

[0019] The global interpolation of any scalar field is expressed as:

[0020]

[0021] Where N represents the node interpolation basis function matrix of the standard radial point interpolation meshless algorithm.

[0022] Preferably, step S2 specifically includes the following steps:

[0023] p(x) represents the interpolation format of the standard radial point interpolation meshless algorithm. In the enhanced radial point interpolation meshless algorithm, the enhanced interpolation format is expressed as:

[0024]

[0025] Where N i represents the number of field points involved in the interpolation of the enhanced radial point interpolation meshless algorithm, represents the local numerical approximation of the enhanced radial point interpolation meshless algorithm, ψ(x) and a i Indicates that the local enhanced interpolation basis function and the coefficients to be determined are constructed at the node;

[0026] The Lagrange polynomial basis function is used as the local enhanced interpolation basis function:

[0027]

[0028] Construct the local enhanced interpolation basis function through dimensionless relative coordinate values:

[0029]

[0030] Where, is the dimensionless relative coordinate value, and h represents the average distance between field points.

[0031] Preferably, in step S3, the global interpolation format in the final enhanced radial point interpolation gridless algorithm is expressed as:

[0032]

[0033] Preferably, step S4 specifically includes the following steps:

[0034] The governing equation for small amplitude waves in an ideal static uniform fluid medium is expressed as:

[0035]

[0036] in, represents the Laplace operator, P represents the variable in the wave propagation problem domain, c represents the propagation speed of the sound wave, and t represents time;

[0037] The sound pressure distribution in the wave propagation problem domain is expressed as:

[0038] P(x)=p(x)e jωt ;

[0039] Where p(x) is the spatial distribution of the sound pressure amplitude, j is the imaginary unit, and ω is the angular frequency. According to the above two formulas, the small amplitude wave equation under the frequency condition of the ideal fluid medium is:

[0040]

[0041] Where k represents the wave number and:

[0042]

[0043] The propagation problem involves the Dirichlet boundary condition DBC, the Newman boundary condition NBC, and the Robin boundary condition RBC. The specific expressions of the three boundary conditions are:

[0044] p=p D onΓ D ;

[0045]

[0046] Among them, Γ D , Γ N and Γ R Denote Dirichlet boundary condition, Newman boundary condition and Robin boundary condition respectively, n denotes the normal unit vector on the corresponding boundary; ρ denotes the density of the ideal fluid medium, v n and A n They represent the vibration velocity of the particle under the Newman boundary condition and the admittance coefficient under the Robin boundary condition respectively;

[0047] When a small amplitude wave propagates in an ideal static fluid, the particle vibration velocity v and the sound pressure gradient The relationship is expressed as:

[0048]

[0049] In the ideal fluid medium, the small amplitude wave equation under the frequency condition is multiplied by a weight function w on both sides and integrated over the entire problem domain Ω. The corresponding control equation is expressed as:

[0050]

[0051] By integrating the corresponding control equation by parts, the control equation can be re-expressed as:

[0052]

[0053] Where Γ represents the boundary of the problem domain Ω, and n represents the external normal unit vector on the boundary Γ;

[0054]

[0055] Among them, n i (i=x,y,z) represents the projection of the external normal unit vector n along the three different directions of x, y, and z respectively;

[0056] Applying the Dirichlet boundary condition, Newman boundary condition and Robin boundary condition, it can be expressed as:

[0057]

[0058] Substituting the global interpolation format of the enhanced radial point interpolation meshless algorithm into the above formula, we get:

[0059]

[0060] The matrix form is expressed as:

[0061] [Kk 2 M+jρωC]a=-jρωF;

[0062]

[0063] M=∫ Ω N T NdΩ;

[0064]

[0065] Where a is the node coefficient to be determined, K, M, C, and F represent the stiffness matrix, mass matrix, damping matrix, and node force vector in the finite element calculation model of the general wave propagation problem, respectively, and B is the gradient matrix.

[0066] Preferably, in step S5, when performing dispersion analysis of wave propagation problems, boundary conditions are not considered, and the matrix is ​​expressed as:

[0067] [Kk 2 M[a=0;

[0068] If there are n nodes in the enhanced radial point interpolation meshless algorithm punknown node coefficients, then the vector a is expressed as:

[0069]

[0070] Where k h represents the wave number calculated by numerical method, n represents the unit vector in the wave propagation direction, represents the position vector of the field point in the wave propagation direction, Indicates that n p The magnitude vector corresponding to the unknown node coefficients;

[0071]

[0072] D mass and D stiff represents the coefficient matrix related to the mass matrix M and the stiffness matrix K, where k represents the exact wave number;

[0073] [D stiff -k 2 D mass ]a i =0;

[0074] The condition for the existence of a non-zero solution of the above equation is that the determinant of its coefficient matrix is ​​zero:

[0075] det[D stiff -k 2 D mass ]=0;

[0076] According to the above formula, we can get: k 2 is the matrix D stiff / D mass The characteristic value of

[0077]

[0078] Matrix D mass and D stiff are all numerical wave numbers k h The dispersion error of wave propagation analysis is defined as:

[0079]

[0080] Therefore, the present invention adopts an enhanced gridless computing method for reducing dispersion errors in wave propagation problems using the above structure, which has the following beneficial effects:

[0081] (1) The present invention uses appropriate basis functions to enhance the original interpolation space and realizes high-order interpolation without increasing unit nodes.

[0082] (2) The enhanced radial point interpolation gridless algorithm for calculating wave propagation problems disclosed in the present invention can significantly reduce the dispersion error when performing dispersion analysis, so it can obtain more accurate numerical results when analyzing wave propagation problems.

[0083] The method scheme of the present invention is further described in detail below through the drawings and examples. BRIEF DESCRIPTION OF THE DRAWINGS

[0084] Figure 1 It is a flow chart of an enhanced gridless computing method for reducing dispersion errors in wave propagation problems according to the present invention;

[0085] Figure 2 Schematic diagram of the integration point support domain in the enhanced radial point interpolation meshless algorithm in an embodiment of the present invention;

[0086] Figure 3 Schematic diagram of the field point influence domain in the enhanced radial point interpolation meshless algorithm in an embodiment of the present invention;

[0087] Figure 4 A schematic diagram of field point distribution in a standard finite element algorithm using triangular elements for wave propagation dispersion analysis provided by an embodiment of the present invention;

[0088] Figure 5 Schematic diagram of field point distribution in the standard radial basis function meshless method and the enhanced radial basis function meshless algorithm for wave propagation dispersion analysis provided as comparative examples of the present invention;

[0089] Figure 6 A graph showing the variation of dispersion errors in different wave propagation directions along with dimensionless dispersion obtained by using a standard finite element algorithm of triangular elements provided for comparative examples of the present invention;

[0090] Figure 7 A graph showing the variation of dispersion errors in different wave propagation directions along with dimensionless dispersion obtained by using the radial basis function meshless algorithm provided in the comparative example of the present invention;

[0091] Figure 8 This is a curve diagram showing the variation of dispersion errors in different wave propagation directions along with dimensionless dispersion obtained by using the enhanced radial basis function meshless algorithm provided in the comparative example of the present invention. DETAILED DESCRIPTION

[0092] The method scheme of the present invention is further described below through the drawings and examples.

[0093] Unless otherwise defined, technical terms or scientific terms used in the present invention shall have the same meaning as commonly understood by one of ordinary skill in the art to which the present invention belongs.

[0094] The words “include” or “comprising” and similar words used in the present invention mean that the elements before the word include the elements listed after the word, and do not exclude the possibility of also including other elements. The orientation or position relationship indicated by the terms “inside”, “outside”, “upper”, “lower”, etc. is based on the orientation or position relationship shown in the accompanying drawings. It is only for the convenience of describing the present invention and simplifying the description, and does not indicate or imply that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation. Therefore, it cannot be understood as a limitation of the present invention. When the absolute position of the described object changes, the relative position relationship may also change accordingly. In the present invention, unless otherwise clearly stipulated and limited, the terms such as “attachment” should be understood in a broad sense. For example, it can be a fixed connection, a detachable connection, or an integral whole; it can be directly connected or indirectly connected through an intermediate medium, and it can be the internal connection of two elements or the interaction relationship between two elements. For ordinary technicians in this field, the specific meanings of the above terms in the present invention can be understood according to the specific circumstances.

[0095] Example

[0096] like Figure 1 As shown, the present invention provides an enhanced gridless computing method for reducing dispersion errors in wave propagation problems, comprising the following steps:

[0097] Step S1: Figure 2 and Figure 3 As shown in the figure, the wave propagation problem domain is discretized into a series of scattered points, and the node interpolation basis functions of the standard radial point interpolation meshless algorithm are derived;

[0098] Step S1 specifically includes the following steps:

[0099] For any scalar field, numerical interpolation p h (x) is expressed as:

[0100]

[0101] Where x represents the position vector of the interpolation point, m is the number of field nodes involved in the interpolation, and the number of field points involved in the interpolation is determined by the support domain of the integration point during numerical integration. The support domain of the integration point is defined as a circular area or a rectangular area centered on the integration point; Figure 2 As shown, the rectangular area used in this embodiment has a size of 1.5h (h is the average spacing of the field point distribution) centered at the integration point. When performing numerical integration, the area that each field point can affect is called the field point influence domain. The size of the field point influence domain is closely related to the integration point support domain. Figure 3As shown, the influence domain of each field point in this embodiment is also a rectangular area with a size of 1.5h centered at the field point. R(x) is the radial basis function, S(x) is the Lagrange polynomial basis function, and linear basis functions are often used for interpolation. n is the number of terms in the Lagrange polynomial basis function, and a and b represent the undetermined coefficient vectors related to the radial basis function and the polynomial basis function, respectively.

[0102] For one-dimensional, two-dimensional and three-dimensional problems, the Lagrangian linear basis functions used can be expressed as:

[0103]

[0104] In formula (1), the radial basis function R(x) used can be expressed as:

[0105]

[0106] Where r i Indicates the distance between the point to be interpolated and the field point involved in the interpolation, α c and q are two unknown coefficients, here we take α c =2 and q=1.03, h represents the average spacing between the field points involved in the interpolation.

[0107] In order to determine the unknown coefficient vectors a and b, we must first ensure that the interpolation in equation (1) is valid at the field nodes involved in the interpolation. Then

[0108]

[0109] Where R M is the matrix related to the radial basis function, S M is the matrix associated with the polynomial basis functions.

[0110]

[0111] Taking the two-dimensional problem as an example, S M It can be expressed as

[0112]

[0113] In order to ensure the uniqueness of the solution, the polynomial basis function used needs to meet the following additional constraints:

[0114]

[0115] Combining equations (4) and (7), we can get the following matrix equation:

[0116]

[0117] Substitute the coefficient vectors a and b obtained by equation (8) into the global interpolation represented by equation (1):

[0118]

[0119] According to formula (9), the interpolation function at the field point involved in the interpolation can be expressed as

[0120]

[0121] Since there are actually only m field points involved in the interpolation, we only need to use the first m terms in equation (10) as the field point interpolation basis functions.

[0122] According to equations (8) and (9), the unknown coefficient vectors a and b can be obtained by the following formulas:

[0123]

[0124]

[0125] Finally, the global interpolation of any scalar field is expressed as:

[0126]

[0127] Where N represents the node interpolation basis function matrix of the standard radial point interpolation meshless algorithm.

[0128] Step S2: Select appropriate local enhanced interpolation basis functions, construct node degrees of freedom at the nodes consisting of the local enhanced interpolation basis functions and the coefficients to be determined, and complete the local numerical approximation of the enhanced radial point interpolation meshless algorithm;

[0129] Step S2 specifically includes the following steps:

[0130] In formula (13), p(x) represents the interpolation format of the standard radial point interpolation meshless algorithm. In the enhanced radial point interpolation meshless algorithm, the enhanced interpolation format is expressed as:

[0131]

[0132] Where N i represents the number of field points involved in the interpolation of the enhanced radial point interpolation meshless algorithm, represents the local numerical approximation of the enhanced radial point interpolation meshless algorithm, ψ(x) and a i Indicates that the local enhanced interpolation basis function and the coefficients to be determined are constructed at the node;

[0133] The Lagrange polynomial basis function is used as the local enhanced interpolation basis function:

[0134]

[0135] In order to ensure that the obtained system matrix equation has sufficient stability, the local enhanced interpolation basis function is constructed through the dimensionless relative coordinate values:

[0136]

[0137] Where, is the dimensionless relative coordinate value, and h represents the average distance between field points.

[0138] Step S3: Based on the node interpolation basis functions of the standard radial point interpolation meshless algorithm obtained in step S1 and the local numerical approximation obtained in step S2, a global interpolation format for the general scalar field is constructed under the premise of satisfying the unit decomposition property;

[0139] In step S3, the global interpolation format in the final enhanced radial point interpolation meshless algorithm is expressed as:

[0140]

[0141] S4: Based on the global interpolation format of the general scalar field obtained in step S3, combined with the partial differential governing equations of the wave propagation problem and the Galerkin weighted residual method, the system matrix equation for the dispersion analysis of the wave propagation problem is obtained;

[0142] Step S4 specifically includes the following steps:

[0143] The governing equation for small amplitude waves in an ideal static uniform fluid medium is expressed as:

[0144]

[0145] in, represents the Laplace operator, P represents the variable in the wave propagation problem domain, c represents the propagation speed of the sound wave, and t represents time;

[0146] The sound pressure distribution in the wave propagation problem domain is expressed as:

[0147] P(x)=p(x)e jωt ; (19)

[0148] Where p(x) is the spatial distribution of the sound pressure amplitude, j is the imaginary unit, and ω is the angular frequency. According to the above two formulas, the small amplitude wave equation under the frequency condition of the ideal fluid medium is:

[0149]

[0150] Where k represents the wave number and:

[0151]

[0152] The propagation problem involves the Dirichlet boundary condition DBC, the Newman boundary condition NBC, and the Robin boundary condition RBC. The specific expressions of the three boundary conditions are:

[0153] p=p D onΓ D ; (twenty two)

[0154]

[0155] Among them, Γ D , Γ N and Γ R Denote Dirichlet boundary condition, Newman boundary condition and Robin boundary condition respectively, n denotes the normal unit vector on the corresponding boundary; ρ denotes the density of the ideal fluid medium, v n and A n They represent the vibration velocity of the particle under the Newman boundary condition and the admittance coefficient under the Robin boundary condition respectively;

[0156] When a small amplitude wave propagates in an ideal static fluid, the relationship between the particle vibration velocity v and the sound pressure gradient ▽p is expressed as:

[0157]

[0158] In the ideal fluid medium, the small amplitude wave equation under the frequency condition is multiplied by a weight function w on both sides and integrated over the entire problem domain Ω. The corresponding control equation is expressed as:

[0159]

[0160] By integrating the corresponding control equation by parts, the control equation can be re-expressed as:

[0161]

[0162] Where Γ represents the boundary of the problem domain Ω, and n represents the external normal unit vector on the boundary Γ;

[0163]

[0164] Among them, n i (i=x,y,z) represents the projection of the external normal unit vector n along the three different directions of x, y, and z respectively;

[0165] Applying the Dirichlet boundary condition, Newman boundary condition and Robin boundary condition, it can be expressed as:

[0166]

[0167] Substituting the global interpolation format of the enhanced radial point interpolation meshless algorithm into the above formula, we get:

[0168]

[0169] The matrix form of formula (30) is expressed as:

[0170] [Kk 2 M+jρωC]a=-jρωF; (31)

[0171]

[0172] M=∫ Ω N T NdΩ; (33)

[0173]

[0174] Where a is the node coefficient to be determined, K, M, C, and F represent the stiffness matrix, mass matrix, damping matrix, and node force vector in the finite element calculation model of the general wave propagation problem, respectively, and B is the gradient matrix.

[0175] S5: Based on the given field point distribution, the system matrix equation obtained in step S4 is solved to calculate the dispersion errors of different calculation methods along different wave propagation directions;

[0176] In step S5, when performing dispersion analysis of wave propagation problems, the boundary conditions are not considered and the matrix of formula (31) is expressed as:

[0177] [Kk 2 M]a=0; (36)

[0178] If there are n nodes in the enhanced radial point interpolation meshless algorithm p unknown node coefficients, then the vector a is expressed as:

[0179]

[0180] Where k h represents the wave number calculated by numerical method, n represents the unit vector in the wave propagation direction, represents the position vector of the field point in the wave propagation direction, Indicates that n p The magnitude vector corresponding to the unknown node coefficients;

[0181]

[0182] In formula (20), the amplitude vector corresponding to each node is exactly the same without considering the boundary, that is, the vector It will appear repeatedly in formula (20).

[0183] For the attached Figure 4 and Figure 5 Given the field point distribution in , substitute equation (37) into equation (36), and we can get

[0184] [D stiff -k 2 D mass ]a i =0; (39)

[0185] Where D mass and D stiff represents the coefficient matrix related to the mass matrix M and the stiffness matrix K, where k represents the exact wave number;

[0186]

[0187] The condition for the existence of a non-zero solution for formula (39) is that the determinant of its coefficient matrix is ​​zero:

[0188] det[D stiff -k 2 D mass ]=0;(42)

[0189] According to the above formula, we can get: k 2 is the matrix D stiff / D mass The characteristic value of

[0190]

[0191] Matrix D mass and D stiff are all numerical wave numbers k h Therefore, for any given accurate wave number k, the corresponding numerical wave number k can be calculated by formula (43): h The dispersion error for wave propagation analysis is defined as:

[0192]

[0193] S6: Based on the dispersion errors in different wave propagation directions obtained in step S5, a curve of the dispersion error changing with the dimensionless wave number is plotted, and the dispersion error results obtained by different calculation methods are compared to complete the dispersion analysis of the wave propagation problem.

[0194] Comparative Example

[0195] In order to compare and verify the ability of the enhanced radial point interpolation meshless calculation method proposed in this scheme to control the dispersion error, the dispersion errors along different wave propagation directions are calculated using the standard triangle finite element algorithm and the standard radial basis function meshless algorithm with the same node distribution. The node distribution is as follows: Figure 5 As shown;

[0196] Figure 6-Figure 8 Given are curves of the dispersion errors along different wave propagation directions obtained by different calculation methods as the dimensionless wave number changes. It can be seen from the figure that the dispersion errors in each wave propagation direction calculated by the finite element method using standard triangular elements are relatively large, and as the dimensionless wave number increases, the corresponding dispersion errors will also increase rapidly; compared with the finite element algorithm using standard triangular elements, the dispersion errors in each wave propagation direction calculated by the standard radial basis function meshless method are smaller, so it has a stronger ability to control the dispersion error than the finite element algorithm using standard triangular elements. However, it can be seen from the dispersion error results in the figure that, compared with the finite element method algorithm using standard triangular elements and the standard radial basis function meshless algorithm, the enhanced radial point interpolation meshless calculation method given in this scheme has the strongest control ability over the dispersion error in wave propagation analysis, because the dispersion errors calculated in different wave propagation directions are significantly smaller than those of the other two methods. This shows that the enhanced radial point interpolation meshless calculation method given in this scheme has a more powerful ability to perform wave propagation analysis, enabling it to effectively reduce the dispersion error generated by numerical calculations, and ultimately obtain more accurate calculation results when performing wave propagation analysis. Therefore, the enhanced radial point interpolation meshless calculation method proposed in this scheme has great potential application value in engineering practice.

[0197] Therefore, the present invention adopts the above-mentioned enhanced gridless calculation method for reducing the dispersion error of wave propagation problems. The enhanced radial point interpolation gridless calculation method provided has a more powerful ability to perform wave propagation analysis, so that it can effectively reduce the dispersion error generated by numerical calculations, and finally obtain more accurate calculation results when performing wave propagation analysis.

[0198] Finally, it should be noted that the above embodiments are only used to illustrate the method scheme of the present invention and not to limit it. Although the present invention has been described in detail with reference to the preferred embodiments, ordinary method personnel in this field should understand that they can still modify or replace the method scheme of the present invention with equivalents, and these modifications or equivalent replacements cannot cause the modified method scheme to deviate from the spirit and scope of the method scheme of the present invention.

Claims

1. An enhanced meshless computing method for reducing dispersion errors in wave propagation problems, characterized by: The following steps are involved: Step S1: Discretize the wave propagation problem domain into a series of scattered points and derive the node interpolation basis functions of the standard radial point interpolation meshless algorithm; Step S2: Select appropriate local enhanced interpolation basis functions, construct node degrees of freedom at the nodes consisting of the local enhanced interpolation basis functions and the coefficients to be determined, and complete the local numerical approximation of the enhanced radial point interpolation meshless algorithm; Step S3: Based on the node interpolation basis functions of the standard radial point interpolation meshless algorithm obtained in step S1 and the local numerical approximation obtained in step S2, a global interpolation format for the general scalar field is constructed under the premise of satisfying the unit decomposition property; S4: Based on the global interpolation format of the general scalar field obtained in step S3, combined with the partial differential governing equations of the wave propagation problem and the Galerkin weighted residual method, the system matrix equation for the dispersion analysis of the wave propagation problem is obtained; S5: Based on the given field point distribution, the system matrix equation obtained in step S4 is solved to calculate the dispersion errors of different calculation methods along different wave propagation directions; S6: Based on the dispersion errors in different wave propagation directions obtained in step S5, a curve of the dispersion error changing with the dimensionless wave number is plotted, and the dispersion error results obtained by different calculation methods are compared to complete the dispersion analysis of the wave propagation problem.

2. The enhanced meshless computing method for reducing dispersion error in wave propagation problems according to claim 1, characterized in that: Step S1 specifically includes the following steps: For any scalar field, numerical interpolation p h (x) is expressed as: Where x represents the position vector of the interpolation point, m is the number of field nodes involved in the interpolation, R(x) is the radial basis function, S(x) is the Lagrange polynomial basis function, and linear basis functions are often used for interpolation; n is the number of terms in the Lagrange polynomial basis function, and a and b represent the undetermined coefficient vectors related to the radial basis function and the polynomial basis function, respectively. The global interpolation of any scalar field is expressed as: Where N represents the node interpolation basis function matrix of the standard radial point interpolation meshless algorithm.

3. The enhanced meshless computing method for reducing dispersion error in wave propagation problems according to claim 2, characterized in that: Step S2 specifically includes the following steps: p(x) represents the interpolation format of the standard radial point interpolation meshless algorithm. In the enhanced radial point interpolation meshless algorithm, the enhanced interpolation format is expressed as: Where N i represents the number of field points involved in the interpolation of the enhanced radial point interpolation meshless algorithm, represents the local numerical approximation of the enhanced radial point interpolation meshless algorithm, ψ(x) and a i Indicates that the local enhanced interpolation basis function and the coefficients to be determined are constructed at the node; The Lagrange polynomial basis function is used as the local enhanced interpolation basis function: Construct the local enhanced interpolation basis function through dimensionless relative coordinate values: Where, and is the dimensionless relative coordinate value, and h represents the average distance between field points.

4. The enhanced meshless computing method for reducing dispersion error in wave propagation problems according to claim 3, characterized in that: In step S3, the global interpolation format in the final enhanced radial point interpolation meshless algorithm is expressed as:

5. The enhanced meshless computing method for reducing dispersion error in wave propagation problems according to claim 4, characterized in that: Step S4 specifically includes the following steps: The governing equation for small amplitude waves in an ideal static uniform fluid medium is expressed as: in, represents the Laplace operator, P represents the variable in the wave propagation problem domain, c represents the propagation speed of the sound wave, and t represents time; The sound pressure distribution in the wave propagation problem domain is expressed as: P(x)=p(x)e jωt ; Where p(x) is the spatial distribution of the sound pressure amplitude, j is the imaginary unit, and ω is the angular frequency. According to the above two formulas, the small amplitude wave equation under the frequency condition of the ideal fluid medium is: Where k represents the wave number and: The propagation problem involves the Dirichlet boundary condition DBC, the Newman boundary condition NBC, and the Robin boundary condition RBC. The specific expressions of the three boundary conditions are: p=p D onΓ D ; Among them, Γ D , Γ N and Γ R Denote Dirichlet boundary condition, Newman boundary condition and Robin boundary condition respectively, n denotes the normal unit vector on the corresponding boundary; ρ denotes the density of the ideal fluid medium, v n and A n They represent the vibration velocity of the particle under the Newman boundary condition and the admittance coefficient under the Robin boundary condition respectively; When a small amplitude wave propagates in an ideal static fluid, the relationship between the particle vibration velocity v and the sound pressure gradient ▽p is expressed as: In the ideal fluid medium, the small amplitude wave equation under the frequency condition is multiplied by a weight function w on both sides and integrated over the entire problem domain Ω. The corresponding control equation is expressed as: By integrating the corresponding control equation by parts, the control equation can be re-expressed as: Where Γ represents the boundary of the problem domain Ω, and n represents the external normal unit vector on the boundary Γ; Among them, n i (i=x,y,z) represents the projection of the external normal unit vector n along the three different directions of x, y, and z respectively; Applying the Dirichlet boundary condition, Newman boundary condition and Robin boundary condition, it can be expressed as: Substituting the global interpolation format of the enhanced radial point interpolation meshless algorithm into the above formula, we get: The matrix form is expressed as: [Kk 2 M+jρωC]a=-jρωF; M=∫ Ω N T NdΩ; Where a is the node coefficient to be determined, K, M, C, and F represent the stiffness matrix, mass matrix, damping matrix, and node force vector in the finite element calculation model of the general wave propagation problem, respectively, and B is the gradient matrix.

6. The enhanced gridless computing method for reducing dispersion error in wave propagation problems according to claim 5, characterized in that: In step S5, when performing dispersion analysis of wave propagation problems, the boundary conditions are not considered and the matrix is ​​expressed as: If there are n nodes in the enhanced radial point interpolation meshless algorithm p unknown node coefficients, then the vector a is expressed as: Where k h represents the wave number calculated by numerical method, n represents the unit vector in the wave propagation direction, represents the position vector of the field point in the wave propagation direction, Indicates that n p The magnitude vector corresponding to the unknown node coefficients; D mass and D stiff represents the coefficient matrix related to the mass matrix M and the stiffness matrix K, where k represents the exact wave number; [D stiff -k 2 D mass ]a i =0; The condition for the existence of a non-zero solution of the above equation is that the determinant of its coefficient matrix is ​​zero: it[D stiff -k 2 D mass ]=0; According to the above formula, we can get: k 2 is the matrix D stiff / D mass The characteristic value of Matrix D mass and D stiff are all numerical wave numbers k h The dispersion error of wave propagation analysis is defined as:

Images