Underwater target low and medium frequency sound scattering field forecasting method based on overlapping finite element method

Abstract

The invention discloses an overlapped finite element method-based underwater target low and medium frequency sound scattering field forecasting method, which comprises the following steps of: generating a virtual node in an overlapped finite element, distributing a group of quadratic complete shift polynomial basis functions in a local analysis domain of each field node to construct local field approximation, and multiplying the local field approximation by a unit decomposition function of the overlapped finite element, constructing a high-order overlapping finite element approximation function; the method comprises the following steps: cutting off an infinite fluid domain by using an artificial boundary, and introducing a Dirichlet-to-Neumann mapping technology to realize a non-reflection boundary condition. For the problem of underwater target sound scattering in middle and low frequency bands, an overlapped finite element numerical calculation model of elastic target sound scattering is established. According to the method, high-order field variable approximation is realized by using the standard low-order finite element grid, the numerical dispersion error and the numerical anisotropy are effectively inhibited, and the low and medium frequency sound scattering field of the elastic target in the infinite water area can be accurately forecasted.

Description

Technical Field

[0001] The invention relates to the field of underwater target acoustic scattering prediction, and in particular to a method for predicting medium and low frequency acoustic scattering fields of underwater targets by using an advanced overlapping finite element method. Background Art

[0002] The scattered echoes of underwater acoustic targets are the source of information for active sonar, and the scattered acoustic field contains a lot of characteristic information of the target, such as material properties and geometric dimensions. Therefore, the accurate prediction of the acoustic scattering characteristics of underwater targets is of great value in many fields such as marine environmental monitoring, fisheries, and military applications.

[0003] The traditional finite element method is currently the most commonly used method for dealing with practical engineering applications because it has a relatively complete mathematical theoretical foundation and benefits from the good operability of commercial software. However, when dealing with acoustic problems, the traditional low-order finite element method has significant numerical dispersion errors, which causes the numerical accuracy of the finite element model to drop sharply with the increase of the calculation frequency. In addition, the traditional low-order finite element numerical solution also has significant numerical anisotropy, that is, the calculation accuracy of the finite element model has significant differences at different sound wave propagation angles.

[0004] In order to solve the above problems, the present invention proposes a method for predicting the medium and low frequency acoustic scattering fields of underwater targets based on the overlapping finite element method, so as to use low-order traditional finite element grids to achieve high-order field variable approximation and accurately predict the medium and low frequency acoustic scattering fields of elastic targets in infinite waters. Summary of the invention

[0005] In view of the above defects or improvement needs of the prior art, the present invention provides a method for predicting the low-frequency acoustic scattering field of underwater targets based on the overlapping finite element method, which generates virtual nodes in the unit and allocates a set of quadratic complete shift polynomial basis functions in the support domain of each field node to construct a local field approximation, which is multiplied by the unit decomposition function of the overlapping finite element to construct a high-order overlapping finite element approximation function, so as to solve the problems of large numerical dispersion error and numerical anisotropy of calculation accuracy in the traditional low-order finite element mesh model in the underwater acoustic scattering problem.

[0006] To achieve the above object, the present invention provides a method for predicting low-frequency acoustic scattering fields of underwater targets based on overlapping finite element method, comprising the following steps:

[0007] S1: geometric modeling of the underwater target, establishing a spherical artificial boundary surrounding the target outside the underwater target, and truncating the infinite fluid domain into a finite calculation domain;

[0008] S2: In the calculation domain of step S1, the discrete size of the overlapping finite elements is determined according to the frequency band to be analyzed for the acoustic scattering problem;

[0009] S3: using the overlapping finite elements of discrete size obtained in step S2 to discretize the underwater target and the fluid domain within the artificial boundary, respectively, to establish a structural grid and a fluid grid;

[0010] S4: setting corresponding physical parameters of materials and media for the structure grid and fluid grid obtained in step S3, and setting boundary condition parameters for the acoustic scattering problem;

[0011] S5: Based on the physical parameters and boundary condition parameters given in step S4, virtual nodes are generated on the fluid unit and the structural unit respectively, and a set of quadratic complete shift polynomial basis functions are allocated in the support domain of each field node to construct a high-order overlapping finite element local field approximation, and then a unit shape function based on overlapping finite elements is constructed;

[0012] S6: Based on the unit shape function of the overlapping finite element in step S5, combining the interaction between the structure and the acoustic field in the grouping in the low and medium frequency bands, establishing an overlapping finite element numerical discrete model for the acoustic scattering problem of underwater elastic targets and obtaining a discrete system equation, solving the discrete system equation to obtain unknown coefficients;

[0013] S7: Perform visualization post-processing based on the unknown coefficients solved in step S6, extract the field variable value (sound pressure) and its gradient (sound speed) at any position, and express them through a graphical interface.

[0014] Further, the frequency band in step S2 is the highest frequency of the incident wave;

[0015] When determining the discretization size of overlapping finite elements, the number of elements required per wavelength should be no less than 2 elements.

[0016] The specific method for determining the discrete size of the overlapping finite elements in step S2 is as follows:

[0017] In practical engineering applications, the mesh size of the traditional finite element method is usually determined according to the so-called rule of thumb, that is, at least 6 to 10 elements per wavelength. Therefore, according to the rule of thumb:

[0018] kh=constant (1)

[0019]

[0020] Where k is the wave number, h is the unit size, and n is res represents the grid resolution, λ is the wavelength;

[0021] A rule of thumb is that the grid resolution n res , that is, the number of elements required for each wavelength λ, is the same for any wave number k. resThe larger the grid resolution n, the smaller the dimensionless wave numbers k and h, and the smaller the relative error. Therefore, the number of cells required for each wavelength is not less than 2 cells, that is, the grid resolution n res ≥2, unit size

[0022] Furthermore, the step S5 of constructing the unit shape function based on overlapping finite elements comprises the following steps:

[0023] S51: Generate virtual nodes at the midpoints of each edge of each overlapping finite element, and use the quadratic Lagrangian interpolation function and predefined weight coefficients to construct the interpolation function in the local field approximation

[0024] S52: Allocate a set of quadratic complete shift polynomial basis functions in the local analysis domain of each field node to construct a local field approximation, and perform a normalization operation on the basis functions using the unit size to improve the stability of the system matrix;

[0025] S53: Take the local field approximation ψ I The weighted average of (x) constructs a global approximation of the field variable p, and obtains the unit decomposition function ρ of the overlapping finite element k , the unit decomposition function ρ k The global parameter β is included, and the value of the global parameter β is selected as β = 0.01;

[0026] S54: Based on the unit decomposition function ρ k The element shape function H in the overlapping finite element is constructed with the local basis function f.

[0027] Furthermore, in step S51, in the overlapping finite element method, the local field approximation ψ of node I I (x):

[0028]

[0029] In the formula, is an interpolation function that satisfies the unit decomposition property, u J is the node function, N is the number of unit vertices, and x = (x, y, z) is the spatial coordinate;

[0030] Generate virtual nodes in each overlapping finite element to construct the interpolation function in the local field approximation

[0031]

[0032] In the formula, is the shape function of the traditional quadratic finite element, M n is the number of nodes in the traditional quadratic finite element, is the predefined weight coefficient;

[0033] For the node function u J It is expressed as:

[0034]

[0035] In the formula, f J is the local basis function, a J are the unknown coefficients to be solved.

[0036] In step S52, a set of second-order complete shift polynomial basis functions is assigned to each field node to construct a local field approximation, and the unit size h is used to perform a normalization operation on it to improve the stability of the algorithm. That is:

[0037]

[0038] In the formula, is a shift polynomial, (x J ,y J ,z J ) are the coordinates of the cell vertices.

[0039] In step S53, the local field approximation ψ I The weighted average of (x) constructs a global approximation of the field variable p, and the weight function is the interpolation function h of the standard low-order finite element. I ,get

[0040]

[0041] Substituting equations (3) and (4) into equation (7), the global approximation of the sound pressure p in the overlapping finite element discrete model is:

[0042]

[0043] Using the weight coefficient expression and the low-order element shape function h I and the corresponding high-order unit shape function Substituting the mathematical relationship between into the above formula (8), we can obtain the unit decomposition function ρ of the overlapping finite element k :

[0044]

[0045] Where J represents the node connected to node K through the unit edge, is the shape function corresponding to the midpoint of edge JK in the standard quadratic finite element;

[0046] Unit decomposition function ρ of overlapping finite elements k The global parameter β is included, and the value of the global parameter β is selected as β = 0.01;

[0047] In step S54, based on the unit decomposition function ρ k The element shape function H in the overlapping finite element is constructed with the local basis function f:

[0048] H={ρ1,ρ2,ρ3,ρ4}×f (10).

[0049] Furthermore, in step S6, the method for establishing the overlapping finite element numerical discrete model of the underwater elastic target acoustic scattering problem is to establish a control equation of the underwater target acoustic scattering problem, introduce the Dirichlet-to-Neumann mapping technology on the artificial boundary to realize the non-reflecting boundary condition, consider the interaction between the structure and the acoustic field, establish the discrete Galerkin weak form of the control equation of the underwater elastic target acoustic scattering problem, obtain the overlapping finite element numerical discrete model of the underwater elastic target acoustic scattering problem, use but not limited to the Gaussian elimination method and other linear equation solvers to solve the overlapping finite element numerical discrete model of the underwater target acoustic scattering problem, and obtain the unknown solution coefficients.

[0050] Furthermore, the step S6 comprises the following steps:

[0051] S6.1, establish the governing equations for underwater rigid target acoustic scattering:

[0052] The steady-state acoustic field distribution in a closed problem domain Ω follows the Helmholtz equation, and we get:

[0053]

[0054] In the formula, is the gradient operator;

[0055] There exists a Dirichlet boundary condition Γ on the boundary Γ of the problem domain D , Neumann boundary condition Γ N and Robin boundary condition Γ R ,Right now:

[0056]

[0057] In the formula, and A n They represent the sound pressure, sound particle velocity and acoustic admittance applied on the corresponding boundary respectively; n is the unit external normal direction vector, ρ f is the fluid density, ω is the circular frequency, represents the imaginary unit;

[0058] Based on Robin mixed boundary conditions:

[0059]

[0060] In the formula, f is a given value;

[0061] At the same time, the sound particle velocity v and the sound pressure p satisfy the following relationship:

[0062]

[0063] S6.2: Introduce Dirichlet-to-Neumann mapping technology to achieve non-reflecting boundary conditions:

[0064] Since the Sommerfeld far-field radiation condition needs to be satisfied in the external acoustic field scattering problem, we get:

[0065]

[0066] The Dirichlet-to-Neumann mapping technique is introduced on the spherical artificial boundary B, and the sound pressure on the DtN boundary is:

[0067]

[0068] In the formula, are the polar coordinates of any point on the boundary of DtN, θ' and is the integration variable, represents the spherical Hankel function of the first kind of order n+1 / 2, represents the first kind of associated Legendre function of order j and order n, R is the radius of the boundary of the sphere DtN, and the coefficient c n The expression is:

[0069]

[0070] Derivative (16) with respect to variable r

[0071]

[0072] In the formula, is a DtN core, which makes the Sommerfeld far-field radiation condition become a boundary condition on the artificial boundary B;

[0073] S6.3: Establish the discrete Galerkin weak form of the governing equations for underwater rigid target acoustic scattering:

[0074] Multiplying equation (11) with the test function w in the entire domain yields the equivalent integral form:

[0075]

[0076] Using Green's theorem we get:

[0077]

[0078] Substituting boundary conditions (12), (13) and (18) into the above equation, we obtain the Galerkin weak form of the governing equation for underwater rigid target acoustic scattering:

[0079]

[0080] Where M is the DtN operator obtained by equation (18);

[0081] After the shape function matrix H of the overlapping finite element is introduced, the coupled overlapping finite element-DtN numerical calculation model for the external acoustic field scattering problem of the rigid target is obtained:

[0082]

[0083] The above formula can be written in matrix form:

[0084] [K f -k 2 M f +ikC f +K DtN ]P=F f (twenty three)

[0085] In the formula, K DtN =∫ B H T MHdΓ, M f =∫ Ω H T HdΩ;

[0086] S6.4: Discrete Galerkin weak form of the governing equations for acoustic scattering from underwater elastic targets

[0087] When analyzing in the mid- and low-frequency bands, the elastic effect of the structure is introduced based on the interaction between the structure and the acoustic field in step S6.3;

[0088] The discrete Galerkin weak form of the governing equations describing the motion of an elastic body is:

[0089]

[0090] Where B is the strain matrix, D is the material parameter matrix, ε is the strain vector within the unit, is the structural density, u is the displacement vector, b is the body force vector, t is the traction force vector, Γ t is the boundary of traction force;

[0091] The control equations in the above discrete form are written in matrix form:

[0092]

[0093] Among them, K e =∫ Ω B T DBdΩ; U={u0,u1,...};

[0094] At the coupling interface between fluid and structure The effect of the sound pressure in the fluid medium on the elastic structure can be considered as an additional load f e ,Right now:

[0095]

[0096] Where σ is stress, H e is the overlapping finite element shape function of the elastic structural part;

[0097] The additional load of the vibration of the structure surface on the sound field is expressed as:

[0098]

[0099] In the formula, H f is the overlapping finite element shape function of the fluid part;

[0100] Define the coupling matrix H ef for:

[0101]

[0102] The sound pressure p can be expressed as the sum of the incident sound pressure pi and the scattered sound pressure ps:

[0103] p=p i +p s (29)

[0104] Combining equations (23), (25) to (29), the governing equation for underwater elastic body acoustic scattering can be written in discrete Galerkin form as:

[0105]

[0106] According to formula (30), the global coefficient matrices are assembled to obtain an overlapping finite element numerical discrete model of the underwater elastic target acoustic scattering problem. The overlapping finite element numerical discrete model of the underwater target acoustic scattering problem is solved using a linear equation solver such as but not limited to the Gaussian elimination method to obtain the unknown solution coefficients.

[0107] Furthermore, in step S7, the visualization post-processing method includes but is not limited to a scattered sound pressure distribution cloud map, a target sound scattering morphological function, and a scattered sound pressure gradient distribution cloud map.

[0108] This method constructs local field approximation by generating virtual nodes in the unit and assigning a set of quadratic complete shift polynomial basis functions in the support domain of each field node. The function is multiplied with the unit decomposition function of overlapping finite elements to construct a high-order overlapping finite element approximation function. It is used to solve the problems of large numerical dispersion error and numerical anisotropy of calculation accuracy in the traditional low-order finite element mesh model in underwater sound scattering problems.

[0109] In addition, an artificial boundary is used to truncate the infinite fluid domain, and the Dirichlet-to-Neumann mapping technology is introduced to realize the non-reflection boundary condition. In addition, for the acoustic scattering problem of underwater targets, especially when analyzing in the medium and low frequency bands, the interaction between the structure and the acoustic field cannot be ignored. The present invention takes into account the acoustic-elastic coupling effect, derives the control equations of the structure-acoustic coupling problem, and constructs a set of overlapping finite element method numerical calculation models for solving the acoustic scattering problem of underwater elastic bodies.

[0110] In general, compared with the prior art, the above technical solutions conceived by the present invention have the following advantages:

[0111] The present invention proposes a method for predicting the low-frequency acoustic scattering field of underwater targets based on the overlapping finite element method. By allocating a set of quadratic complete shifted polynomial basis functions in the support domain of each field node to construct a high-order overlapping finite element local field approximation, a high-order field variable approximation can be achieved using a low-order traditional finite element grid, thereby significantly suppressing the numerical dispersion effect. In addition, the prediction method proposed by the present invention is insensitive to grid deformity, and its calculation accuracy is insensitive to the direction of sound wave propagation, that is, there is almost no numerical anisotropy.

[0112] In general, the prediction method proposed in the present invention can effectively suppress numerical dispersion errors and numerical anisotropy, and significantly improves the calculation accuracy compared with the traditional finite element method. It can significantly expand the applicable calculation frequency range of standard low-order finite element grids, greatly reduce the workload of grid division, and thus improve the analysis efficiency. It can accurately predict the medium and low frequency scattered sound fields of elastic targets in infinite waters. BRIEF DESCRIPTION OF THE DRAWINGS

[0113] Figure 1 The figure is a flow chart of the method in the implementation of the present invention.

[0114] Figure 2 A schematic diagram of modeling the underwater elastic solid sphere sound scattering problem provided in an embodiment of the present invention.

[0115] Figure 3 The embodiment of the present invention provides a model for discretizing the underwater elastic solid sphere sound scattering problem using a standard low-order finite element grid.

[0116] Figure 4A relationship diagram between field nodes and virtual nodes in a hexahedral overlapping finite element in a Cartesian coordinate system provided by an embodiment of the present invention.

[0117] Figure 5 A relationship diagram between field nodes and virtual nodes in a hexahedral overlapping finite element in a natural coordinate system provided by an embodiment of the present invention.

[0118] Figure 6 A scattered sound pressure directivity diagram when f=2000 Hz is provided for an embodiment of the present invention.

[0119] Figure 7 A scattered sound pressure directivity diagram when f=3000 Hz is provided for an embodiment of the present invention.

[0120] Figure 8 A schematic diagram of post-processing visualization of sound pressure and frequency is provided for an embodiment of the present invention. DETAILED DESCRIPTION

[0121] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below can be combined with each other as long as they do not conflict with each other.

[0122] The specific embodiment of the present invention is for an underwater three-dimensional elastic solid sphere irradiated by a plane wave. The radius of the elastic solid sphere is 0.6m and the material is aluminum. Although the exemplary embodiments of the present disclosure are shown in the accompanying drawings, it should be understood that the present invention can be used for various underwater targets and should not be limited by the embodiments described here. The present invention can effectively predict the acoustic scattering field of various underwater complex targets.

[0123] Reference Figure 1 As shown, the present invention provides a method for predicting low-frequency acoustic scattering fields of underwater targets based on overlapping finite element method, comprising the following steps:

[0124] S1: geometric modeling of the underwater target, establishing a spherical artificial boundary surrounding the target outside the underwater target, and truncating the infinite fluid domain into a finite calculation domain;

[0125] S2: In the calculation domain of step S1, the discrete size of the overlapping finite elements is determined according to the frequency band to be analyzed for the acoustic scattering problem (i.e., the highest frequency of the incident wave);

[0126] S3: using the overlapping finite elements of discrete size obtained in step S2 to discretize the underwater target and the fluid domain within the artificial boundary, respectively, to establish a structural grid and a fluid grid;

[0127] S4: setting corresponding physical parameters of materials and media for the structure grid and fluid grid obtained in step S3, and setting boundary condition parameters for the acoustic scattering problem;

[0128] S5: Based on the physical parameters and boundary condition parameters given in step S4, virtual nodes are generated on the fluid unit and the structural unit respectively, and a set of quadratic complete shift polynomial basis functions are allocated in the support domain of each field node to construct a high-order overlapping finite element local field approximation, and then a unit shape function based on overlapping finite elements is constructed;

[0129] S6: Based on the element shape function of the overlapping finite element in step S5, an overlapping finite element numerical discrete model for underwater elastic target acoustic scattering problem is established and a discrete system equation is obtained, and the discrete system equation is solved to obtain unknown coefficients;

[0130] S7: Perform visualization post-processing based on the unknown coefficients solved in step S6, extract the field variable value (sound pressure) and its gradient (sound speed) at any position, and express them through a graphical interface.

[0131] The following is a detailed description of each of the above steps:

[0132] In step S1, a solid sphere object in this problem is established, and a spherical surface with a radius of R = 1.2 m is used as the DtN artificial boundary to cut the infinite fluid domain into a finite computational domain. Figure 2 The structure region is shown in green and the fluid domain is shown in blue.

[0133] In step S2, the highest calculation frequency analyzed in this specific embodiment is f=3500 Hz, and the overlapping finite element grid size is about h=0.14 m.

[0134] The specific method for determining the discrete size of the overlapping finite elements in step S2 is as follows:

[0135] In practical engineering applications, the mesh size of the traditional finite element method is usually determined according to the so-called rule of thumb, that is, at least 6 to 10 elements per wavelength. Therefore, according to the rule of thumb:

[0136] kh=constant (1)

[0137]

[0138] Where k is the wave number, h is the unit size, and n is res represents the grid resolution, λ is the wavelength;

[0139] A rule of thumb is that the grid resolution n res, that is, the number of elements required for each wavelength λ, is the same for any wave number k. res The larger the grid resolution n, the smaller the dimensionless wave numbers k and h, and the smaller the relative error. Therefore, the number of cells required for each wavelength is not less than 2 cells, that is, the grid resolution n res ≥2, unit size

[0140] In step S3, this specific embodiment uses standard eight-node hexahedral units to discretize the fluid domain and the structural domain, such as Figure 3 Although the attached Figure 3 The disclosed embodiment shown in the figure only uses eight-node hexahedral units, however, it should be understood that the disclosed invention can be implemented by flexibly using unit types including but not limited to four-node tetrahedral units, six-node triangular prism units, eight-node hexahedral units, etc. according to the structural mechanical properties and geometric shapes of the underwater target, and should not be limited by the unit types used in the embodiments described here.

[0141] In step S4, the material of the solid ball in this specific embodiment is aluminum, and the Young's modulus is set to E=70 GPa, the Poisson's ratio is μ=0.3, and the density is The fluid medium is water, with a density of ρ f =1000kg / m 3 .

[0142] In step S5, constructing the element shape function based on overlapping finite elements comprises the following steps:

[0143] S51: Generate virtual nodes at the midpoints of each edge of each overlapping finite element, and use the quadratic Lagrangian interpolation function and predefined weight coefficients to construct the interpolation function in the local field approximation

[0144] S52: Allocate a set of quadratic complete shift polynomial basis functions in the local analysis domain of each field node to construct a local field approximation, and perform a normalization operation on the basis functions using the unit size to improve the stability of the system matrix;

[0145] S53: Take the local field approximation ψ I The weighted average of (x) constructs a global approximation of the field variable p, and obtains the unit decomposition function ρ of the overlapping finite element k , the unit decomposition function ρ k The global parameter β is included, and the value of the global parameter β is selected as β = 0.01;

[0146] S54: Based on the unit decomposition function ρ k The element shape function H in the overlapping finite element is constructed with the local basis function f.

[0147] Specifically, in step S51, in the overlapping finite element method, the local field approximation ψ of node I I (x):

[0148]

[0149] In the formula, is an interpolation function that satisfies the unit decomposition property, u J is the node function, N is the number of unit vertices, and x = (x, y, z) is the spatial coordinate;

[0150] Generate virtual nodes in each overlapping finite element to construct the interpolation function in the local field approximation

[0151]

[0152] In the formula, is the shape function of the traditional quadratic finite element, M n is the number of nodes in the traditional quadratic finite element, is the predefined weight coefficient;

[0153] For the node function u J It is expressed as:

[0154]

[0155] In the formula, f J is the local basis function, a J are the unknown coefficients to be solved.

[0156] In step S52, a set of second-order complete shift polynomial basis functions is assigned to each field node to construct a local field approximation, and the unit size h is used to perform a normalization operation on it to improve the stability of the algorithm. That is:

[0157]

[0158] In the formula, is a shift polynomial, (x J ,y J ,z J ) are the coordinates of the cell vertices.

[0159] In step S53, the local field approximation ψ I The weighted average of (x) constructs a global approximation of the field variable p, and the weight function is the interpolation function h of the standard low-order finite element. I ,get

[0160]

[0161] Substituting equations (3) and (4) into equation (7), the global approximation of the sound pressure p in the overlapping finite element discrete model is:

[0162]

[0163] Using the weight coefficient expression and the low-order element shape function h I and the corresponding high-order element shape function Substituting the mathematical relationship between into the above formula (8), we can obtain the unit decomposition function ρ of the overlapping finite element k :

[0164]

[0165] Where J represents the node connected to node K through the unit edge, is the shape function corresponding to the midpoint of edge JK in the standard quadratic finite element;

[0166] Unit decomposition function ρ of overlapping finite elements k The global parameter β is included, and the value of the global parameter β is selected as β = 0.01;

[0167] In step S54, based on the unit decomposition function ρ k The element shape function H in the overlapping finite element is constructed with the local basis function f:

[0168] H={ρ1,ρ2,ρ3,ρ4}×f (10).

[0169] That is, an overlapping finite element calculation model is established based on the eight-node hexahedral finite element mesh used. Figure 4 As shown in the figure, for any low-order hexahedral element in the Cartesian coordinate system, it is mapped to the natural coordinate system, and the virtual nodes shown in red are generated at the midpoints of each side of the element to construct the unit decomposition function, and a set of quadratic complete shift polynomial basis functions are allocated on the local analysis domain corresponding to the field nodes shown in black to construct the local field approximation, and finally the field variable approximation in the overlapping finite element is obtained. Figure 4 The disclosed embodiment shown in FIG. 1 only shows the construction of virtual nodes in an eight-node hexahedral element. However, it should be understood that the same virtual node generation method can be used to implement the overlapping finite element method unit decomposition function ρ in the above-mentioned various traditional low-order finite elements. k The structure of the unit decomposition function in each type of overlapping finite element is consistent with formula (9).

[0170] In step S6, the method for establishing the overlapping finite element numerical discrete model of the underwater elastic target acoustic scattering problem is to establish the control equation of the underwater target acoustic scattering problem, introduce the Dirichlet-to-Neumann mapping technology on the artificial boundary to realize the non-reflecting boundary condition, consider the interaction between the structure and the acoustic field, establish the discrete Galerkin weak form of the control equation of the underwater elastic target acoustic scattering problem, obtain the overlapping finite element numerical discrete model of the underwater elastic target acoustic scattering problem, use but not limited to the Gaussian elimination method and other linear equation solvers to solve the overlapping finite element numerical discrete model of the underwater target acoustic scattering problem, and obtain the unknown solution coefficients.

[0171] Step S6 comprises the following steps:

[0172] S6.1, establish the governing equations for underwater rigid target acoustic scattering:

[0173] The steady-state acoustic field distribution in a closed problem domain Ω follows the Helmholtz equation, and we get:

[0174]

[0175] In the formula, is the gradient operator;

[0176] There exists a Dirichlet boundary condition Γ on the boundary Γ of the problem domain D , Neumann boundary condition Γ N and Robin boundary condition Γ R ,Right now:

[0177]

[0178] In the formula, and A n They represent the sound pressure, sound particle velocity and acoustic admittance applied on the corresponding boundary respectively; n is the unit external normal direction vector, ρ f is the fluid density, ω is the circular frequency, represents the imaginary unit;

[0179] Based on Robin mixed boundary conditions:

[0180]

[0181] In the formula, f is a given value;

[0182] At the same time, the sound particle velocity v and the sound pressure p satisfy the following relationship:

[0183]

[0184] S6.2: Introduce Dirichlet-to-Neumann mapping technology to achieve non-reflecting boundary conditions:

[0185] Since the Sommerfeld far-field radiation condition needs to be satisfied in the external acoustic field scattering problem, we get:

[0186]

[0187] The Dirichlet-to-Neumann mapping technique is introduced on the spherical artificial boundary B, and the sound pressure on the DtN boundary is:

[0188]

[0189] In the formula, are the polar coordinates of any point on the boundary of DtN, θ' and is the integration variable, represents the spherical Hankel function of the first kind of order n+1 / 2, represents the first kind of associated Legendre function of order j and order n, R is the radius of the boundary of the sphere DtN, and the coefficient c n The expression is:

[0190]

[0191] Derivative (16) with respect to variable r

[0192]

[0193] In the formula, is a DtN core, which makes the Sommerfeld far-field radiation condition become a boundary condition on the artificial boundary B;

[0194] S6.3: Establish the discrete Galerkin weak form of the governing equations for underwater rigid target acoustic scattering:

[0195] Multiplying equation (11) with the test function w in the entire domain yields the equivalent integral form:

[0196]

[0197] Using Green's theorem we get:

[0198]

[0199] Substituting boundary conditions (12), (13) and (18) into the above equation, we obtain the Galerkin weak form of the governing equation for underwater rigid target acoustic scattering:

[0200]

[0201] Where M is the DtN operator obtained by equation (18);

[0202] After the shape function matrix H of the overlapping finite element is introduced, the coupled overlapping finite element-DtN numerical calculation model for the external acoustic field scattering problem of the rigid target is obtained:

[0203]

[0204] The above formula can be written in matrix form:

[0205] [K f -k 2 M f +ikC f +K DtN ]P=F f (twenty three)

[0206] In the formula, K DtN =∫ B H T MHdΓ, M f =∫ Ω H T HdΩ;

[0207] S6.4: Discrete Galerkin weak form of the governing equations for acoustic scattering from underwater elastic targets

[0208] When analyzing in the mid- and low-frequency bands, the elastic effect of the structure is introduced based on the interaction between the structure and the acoustic field in step S6.3;

[0209] The discrete Galerkin weak form of the governing equations describing the motion of an elastic body is:

[0210]

[0211] Where B is the strain matrix, D is the material parameter matrix, ε is the strain vector within the unit, is the structural density, u is the displacement vector, b is the body force vector, t is the traction force vector, Γ t is the boundary of traction force;

[0212] The control equations in the above discrete form are written in matrix form:

[0213]

[0214] Among them, K e =∫ Ω B T DBdΩ; U={u0,u1,...};

[0215] At the coupling interface between fluid and structure The effect of the sound pressure in the fluid medium on the elastic structure can be considered as an additional load f e ,Right now:

[0216]

[0217] Where σ is stress, H e is the overlapping finite element shape function of the elastic structural part;

[0218] The additional load of the vibration of the structure surface on the sound field is expressed as:

[0219]

[0220] In the formula, H f is the overlapping finite element shape function of the fluid part;

[0221] Define the coupling matrix H ef for:

[0222]

[0223] The sound pressure p can be expressed as the sum of the incident sound pressure pi and the scattered sound pressure ps:

[0224] p=p i +p s (29)

[0225] Combining equations (23), (25) to (29), the governing equation for underwater elastic body acoustic scattering can be written in discrete Galerkin form as:

[0226]

[0227] According to formula (30), the global coefficient matrices are assembled to obtain an overlapping finite element numerical discrete model of the underwater elastic target acoustic scattering problem. The overlapping finite element numerical discrete model of the underwater target acoustic scattering problem is solved using a linear equation solver such as but not limited to the Gaussian elimination method to obtain the unknown solution coefficients.

[0228] In step S7, the sound field is post-processed for visualization according to the solution coefficients obtained in step S6. Figure 5 As shown, this specific embodiment provides a scattered sound pressure directivity diagram on the xy plane, Figure 6 The target morphology function calculation results are given. However, it should be understood that the data obtained by the present invention can be further processed by visualization including but not limited to scattered sound pressure distribution cloud map, target sound scattering morphology function, scattered sound pressure gradient distribution cloud map, etc., as well as extracting corresponding data to generate table and other types of files.

[0229] The following is an explanation with charts:

[0230] Depend on Figure 5 It can be seen that at f = 2000Hz, the maximum forward scattering sound pressure predicted by the standard finite element model is less than the exact solution, while the error of the backscattering petal shape is large. When the calculation frequency increases to f = 3000Hz, the error of the result predicted by the standard finite element model is significant. In contrast, the overlapping finite element method can accurately predict the directivity of the scattered sound field of the elastic solid sphere.

[0231] Figure 6 The morphological function of the scattered sound field of the elastic solid sphere is given, that is, the relationship between the forward scattered sound pressure amplitude and the calculation frequency. It can be seen that when the frequency is greater than f=1000, it can be observed that the numerical solution of the traditional finite element method produces a phase lag, and this lag effect will gradually accumulate with the increase of the calculated wave number. In contrast, the overlapping finite element method proposed in the present invention can still predict the sound scattering morphological function with higher accuracy when the calculation frequency is relatively high, indicating that the overlapping finite element method proposed in the present invention expands the applicable calculation frequency range of the standard low-order finite element grid.

[0232] The data of the traditional finite element method and the overlapping finite element method of the present invention when the same calculation accuracy is obtained are shown in Table 1.

[0233] Table 1 Comparison of computational costs of different methods for achieving the same computational accuracy

[0234]

[0235] As can be seen from Table 1, the number of units required for the overlapping finite element method proposed in the present invention is only 1.4% of that of the traditional finite element method, and the number of global degrees of freedom required is only 8.8% of that of the traditional finite element method, indicating that the overlapping finite element method proposed in the present invention can greatly reduce the workload of meshing. In addition, the number of integral points required for the overlapping finite element method is only 2.2% of that of the traditional finite element method, indicating that the time cost of constructing the global system matrix is ​​also smaller. Therefore, the overlapping finite element method proposed in the present invention has higher analysis efficiency.

[0236] It will be easily understood by those skilled in the art that the above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included in the protection scope of the present invention.

Claims

1. A method for predicting low-frequency acoustic scattering fields of underwater targets based on overlapping finite element method, characterized in that: The following steps are involved: S1: geometric modeling of the underwater target, establishing a spherical artificial boundary surrounding the target outside the underwater target, and truncating the infinite fluid domain into a finite calculation domain; S2: In the calculation domain of step S1, the discrete size of the overlapping finite elements is determined according to the frequency band to be analyzed for the acoustic scattering problem; S3: using the overlapping finite elements of discrete size obtained in step S2 to discretize the underwater target and the fluid domain within the artificial boundary, respectively, to establish a structural grid and a fluid grid; S4: setting corresponding physical parameters of materials and media for the structure grid and fluid grid obtained in step S3, and setting boundary condition parameters for the acoustic scattering problem; S5: Based on the physical parameters and boundary condition parameters given in step S4, virtual nodes are generated on the fluid unit and the structural unit respectively, and a set of quadratic complete shift polynomial basis functions are allocated in the support domain of each field node to construct a high-order overlapping finite element local field approximation, and then a unit shape function based on overlapping finite elements is constructed; S6: Based on the unit shape function of the overlapping finite element in step S5, combining the interaction between the structure and the acoustic field in the analysis in the medium and low frequency bands, establishing an overlapping finite element numerical discrete model for the acoustic scattering problem of underwater elastic targets and obtaining a discrete system equation, solving the discrete system equation to obtain unknown coefficients; S7: Perform visualization post-processing based on the unknown coefficients solved in step S6, extract the field variable value and its gradient at any position, and express them through a graphical interface.

2. The method for predicting low-frequency acoustic scattering field of underwater targets based on overlapping finite element method according to claim 1, characterized in that: The frequency band in step S2 is the highest frequency of the incident wave; When determining the discretization size of overlapping finite elements, the number of elements required per wavelength should be no less than 2 elements.

3. The method for predicting low-frequency acoustic scattering field of underwater targets based on overlapping finite element method as claimed in claim 2, characterized in that: The specific method for determining the discrete size of the overlapping finite elements in step S2 is as follows: As a rule of thumb: kh=constant (1) Where k is the wave number, h is the unit size, and n is res represents the grid resolution, λ is the wavelength; Since the number of elements required for each wavelength λ is the same for any wave number k, when the grid resolution n res The larger the grid resolution n, the smaller the dimensionless wave numbers k and h, and the smaller the relative error. Therefore, the number of cells required for each wavelength is not less than 2 cells, that is, the grid resolution n res ≥2, unit size 4. The method for predicting low-frequency acoustic scattering field of underwater targets based on overlapping finite element method according to claim 1, characterized in that: The step S5 of constructing the unit shape function based on overlapping finite elements comprises the following steps: S51: Generate virtual nodes at the midpoints of each edge of each overlapping finite element, and use the quadratic Lagrangian interpolation function and predefined weight coefficients to construct the interpolation function in the local field approximation S52: Allocate a set of quadratic complete shift polynomial basis functions in the local analysis domain of each field node to construct a local field approximation, and perform a normalization operation on the basis functions using the unit size to improve the stability of the system matrix; S53: Take the local field approximation ψ I The weighted average of (x) constructs a global approximation of the field variable p, and obtains the unit decomposition function ρ of the overlapping finite element k , the unit decomposition function ρ k The global parameter β is included, and the value of the global parameter β is selected as β = 0.01; S54: Based on the unit decomposition function ρ k The element shape function H in the overlapping finite element is constructed with the local basis function f.

5. The method for predicting low-frequency acoustic scattering field of underwater targets based on overlapping finite element method as claimed in claim 4, characterized in that: In step S51, in the overlapping finite element method, the local field approximation ψ of node I is I (x): In the formula, is an interpolation function that satisfies the unit decomposition property, u J is the node function, N is the number of unit vertices, and x = (x, y, z) is the spatial coordinate; Generate virtual nodes in each overlapping finite element to construct the interpolation function in the local field approximation In the formula, is the shape function of the traditional quadratic finite element, M n is the number of nodes in the traditional quadratic finite element, is the predefined weight coefficient; For the node function u J It is expressed as: In the formula, f J is the local basis function, a J are the unknown coefficients to be solved.

6. The method for predicting low-frequency acoustic scattering field of underwater targets based on overlapping finite element method as claimed in claim 5, characterized in that: In step S52, a set of second-order complete shift polynomial basis functions is allocated to each field node to construct a local field approximation, and the unit size is used to perform a normalization operation on the basis functions, that is, In the formula, is a shift polynomial, (x J ,y J ,z J ) are the coordinates of the cell vertices.

7. The method for predicting low-frequency acoustic scattering field of underwater targets based on overlapping finite element method as claimed in claim 6, characterized in that: In step S53, the local field approximation ψ I The weighted average of (x) constructs a global approximation of the field variable p, and the weight function is the interpolation function h of the standard low-order finite element. I ,get Substituting equations (3) and (4) into equation (7), the global approximation of the sound pressure p in the overlapping finite element discrete model is: Using the weight coefficient expression and the low-order element shape function h I and the corresponding high-order element shape function Substituting the mathematical relationship between into the above formula (8), we can obtain the unit decomposition function ρ of the overlapping finite element k : Where J represents the node connected to node K through the unit edge, is the shape function corresponding to the midpoint of edge JK in the standard quadratic finite element; Unit factorization function ρ of overlapping finite elements k The global parameter β is included, and the value of the global parameter β is selected as β = 0.01; In step S54, based on the unit decomposition function ρ k The element shape function H in the overlapping finite element is constructed with the local basis function f: H={ρ1,ρ2,ρ3,ρ4}×f (10).

8. The method for predicting low-frequency acoustic scattering field of underwater targets based on overlapping finite element method as claimed in claim 7, characterized in that: In the step S6, the method for establishing the overlapping finite element numerical discrete model of the underwater elastic target sound scattering problem is to establish the control equation of the underwater target sound scattering problem, introduce the Dirichlet-to-Neumann mapping technology on the artificial boundary to realize the non-reflecting boundary condition, consider the interaction between the structure and the sound field, establish the discrete Galerkin weak form of the control equation of the underwater elastic target sound scattering problem, obtain the overlapping finite element numerical discrete model of the underwater elastic target sound scattering problem, use but not limited to the Gaussian elimination method and other linear equation solvers to solve the overlapping finite element numerical discrete model of the underwater target sound scattering problem, and obtain the unknown solution coefficients.

9. The method for predicting low-frequency acoustic scattering field of underwater targets based on overlapping finite element method as claimed in claim 8, characterized in that: The step S6 comprises the following steps: S6.1, establish the governing equations for underwater rigid target acoustic scattering: The steady-state acoustic field distribution in a closed problem domain Ω follows the Helmholtz equation, and we get: ▽ 2 p+k 2 p=0 (11) Where ▽ is the gradient operator; There exists a Dirichlet boundary condition Γ on the boundary Γ of the problem domain D , Neumann boundary condition Γ N and Robin boundary condition Γ R ,Right now: In the formula, and A n They represent the sound pressure, sound particle velocity and acoustic admittance applied on the corresponding boundary respectively; n is the unit external normal direction vector, ρ f is the fluid density, ω is the circular frequency, represents the imaginary unit; Based on Robin mixed boundary conditions: ▽p·n+ikp=f (13) In the formula, f is a given value; At the same time, the sound particle velocity v and the sound pressure p satisfy the following relationship: ▽p=-iρ f ωv (14) S6.2: Introduce Dirichlet-to-Neumann mapping technology to achieve non-reflecting boundary conditions: Since the Sommerfeld far-field radiation condition needs to be satisfied in the external acoustic field scattering problem, we get: The Dirichlet-to-Neumann mapping technique is introduced on the spherical artificial boundary B, and the sound pressure on the DtN boundary is: In the formula, are the polar coordinates of any point on the boundary of DtN, θ' and is the integration variable, represents the spherical Hankel function of the first kind of order n+1 / 2, represents the first kind of associated Legendre function of order j and order n, R is the radius of the boundary of the sphere DtN, and the coefficient c n The expression is: Derivative (16) with respect to variable r In the formula, is a DtN core, which makes the Sommerfeld far-field radiation condition become a boundary condition on the artificial boundary B; S6.3: Establish the discrete Galerkin weak form of the governing equations for underwater rigid target acoustic scattering: Multiplying equation (11) with the test function w in the entire domain yields the equivalent integral form: ∫ Ω -w(▽ 2 p+k 2 p) dΩ=0 (19) Using Green's theorem we get: ∫ Ω ▽w·▽p dΩ+k 2 ∫ Ω wp dΩ-∫ Γ w(▽p·n) dΓ=0 (20) Substituting boundary conditions (12), (13) and (18) into the above equation, we obtain the Galerkin weak form of the governing equation for underwater rigid target acoustic scattering: Where M is the DtN operator obtained by equation (18); After the shape function matrix H of the overlapping finite element is introduced, the coupled overlapping finite element-DtN numerical calculation model for the external acoustic field scattering problem of the rigid target is obtained: The above formula can be written in matrix form: [K f -k 2 M f +ikC f +K DtN ]P=F f (23) In the formula, K DtN =∫ B H T MHdΓ,K f =∫ Ω ▽H▽HdΩ;M f =∫ Ω H T HdΩ; S6.4: Establish the discrete Galerkin weak form of the governing equations for underwater elastic target acoustic scattering problems. When analyzing in the low and medium frequency bands, combine the interaction between the structure and the acoustic field and introduce the elastic effect of the structure based on step S6.3; The discrete Galerkin weak form of the governing equations describing the motion of an elastic body is: Where B is the strain matrix, D is the material parameter matrix, ε is the strain vector within the unit, is the structural density, u is the displacement vector, b is the body force vector, t is the traction force vector, Γ t is the boundary of traction force; The control equations in the above discrete form are written in matrix form: Among them, K e =∫ Ω B T DBdΩ; U={u0,u1,...}; At the coupling interface between fluid and structure The effect of the sound pressure in the fluid medium on the elastic structure can be considered as an additional load f e ,Right now: Where σ is stress, H e is the overlapping finite element shape function of the elastic structural part; The additional load of the vibration of the structure surface on the sound field is expressed as: In the formula, H f is the overlapping finite element shape function of the fluid part; Define the coupling matrix H ef for: The sound pressure p can be expressed as the sum of the incident sound pressure pi and the scattered sound pressure ps: p=p i +p s (29) Combining equations (23), (25) to (29), the governing equation for underwater elastic body acoustic scattering can be written in discrete Galerkin form as: According to formula (30), the global coefficient matrices are assembled to obtain an overlapping finite element numerical discrete model of the underwater elastic target acoustic scattering problem. The overlapping finite element numerical discrete model of the underwater target acoustic scattering problem is solved using a linear equation solver such as but not limited to the Gaussian elimination method to obtain the unknown solution coefficients.

10. The method for predicting low-frequency acoustic scattering field of underwater targets based on overlapping finite element method according to claim 1, characterized in that: In step S7, the visualization post-processing method includes but is not limited to a scattered sound pressure distribution cloud map, a target sound scattering morphological function, and a scattered sound pressure gradient distribution cloud map.

Images