Prediction method of low-frequency acoustic scattering field of underwater targets based on overlapping finite element method

Abstract

The present invention discloses a method for predicting the low- and medium-frequency acoustic scattering fields of underwater targets based on the overlapping finite element method. The method generates virtual nodes in overlapping finite elements, allocates 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 multiplies the function with the unit decomposition function of the overlapping finite element to construct a high-order overlapping finite element approximation function. The infinite fluid domain is truncated using an artificial boundary, and the Dirichlet-to-Neumann mapping technology is introduced to achieve non-reflecting boundary conditions. For the problem of underwater target acoustic scattering in the low- and medium-frequency bands, an overlapping finite element numerical calculation model for acoustic scattering of elastic targets is established. The present invention uses a standard low-order finite element grid to achieve high-order field variable approximation, effectively suppresses numerical dispersion errors and numerical anisotropy, and can accurately predict the low- and medium-frequency acoustic scattering fields of elastic targets in infinite waters.

Description

Technical Field

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

[0002] The scattered echoes from underwater acoustic targets are the source of information for active sonar. Furthermore, the scattered acoustic field contains a wealth of target characteristics, such as material properties and geometric dimensions. Therefore, accurately predicting the acoustic scattering characteristics of underwater targets is of great value in a variety of fields, including marine environmental monitoring, fisheries, and military applications.

[0003] The traditional finite element method (FEM), with its relatively comprehensive mathematical foundation and the ease of use of commercial software, is currently the most commonly used approach for practical engineering applications. However, when dealing with acoustic problems, the traditional low-order FEM suffers from significant numerical dispersion errors, causing the numerical accuracy of the finite element model to drop sharply with increasing computational frequency. Furthermore, the traditional low-order FEM numerical solution exhibits significant numerical anisotropy, meaning that the computational accuracy of the finite element model varies significantly at different sound wave propagation angles.

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

[0005] In response to the above-mentioned 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. The method 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. The local field approximation is multiplied by the unit decomposition function of the overlapping finite element to construct a high-order overlapping finite element approximation function. The method is used to solve the problems of large numerical dispersion error and numerical anisotropy of calculation accuracy in the traditional low-order finite element grid model in underwater acoustic scattering problems.

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

[0007] S1: Geometric modeling of the underwater target is performed, and a spherical artificial boundary is established outside the underwater target to enclose the target, thereby truncating the infinite fluid domain into a finite computational domain.

[0008] S2: Within the computational 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 sizes 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 the corresponding physical parameters of the material and medium for the structural grid and fluid grid obtained in step S3, and setting the boundary condition parameters of the sound 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 assigned 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 the overlapping finite element is constructed;

[0012] S6: Based on the element shape function of the overlapping finite element in step S5, the interaction between the structure and the acoustic field is combined in the grouping in the medium and low frequency bands, and an overlapping finite element numerical discrete model of the underwater elastic target sound scattering problem is established and a discrete system equation is obtained. The discrete system equation is solved to obtain the 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] Furthermore, 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 the wave number. 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 value, the smaller the dimensionless wave number k and h, and the smaller the relative error; therefore, the number of units required for each wavelength is not less than 2 units, that is, the grid resolution n res ≥2, unit size

[0022] Furthermore, the step S5 of constructing the element 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 within 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 is I (x):

[0028]

[0029] Where, 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] Where, is the shape function of the traditional quadratic finite element, M n is the number of nodes in the traditional quadratic finite element, is a predefined weight coefficient;

[0033] For the node function u J Expressed as:

[0034]

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

[0036] In step S52, a set of quadratic complete shifted polynomial basis functions is assigned to each field node to construct a local field approximation, and the local field approximation is normalized using the unit size h to improve the stability of the algorithm.

[0037]

[0038] Where, 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 takes 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 element shape function Substituting the mathematical relationship between them into the above formula (8), we can get 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 the 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 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.

[0050] Furthermore, 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] Where, is the gradient operator;

[0055] There is 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] Where, and A n Represent the sound pressure, sound particle velocity and sound 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] Where, 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: Introducing 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 sound field scattering problem, we can obtain:

[0065]

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

[0067]

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

[0069]

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

[0071]

[0072] Where, 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 the acoustic scattering problem of underwater rigid targets:

[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 equations, we obtain the Galerkin weak form of the governing equation for the underwater rigid target acoustic scattering problem:

[0079]

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

[0081] After introducing the shape function matrix H of the overlapping finite element, the coupled overlapping finite element-DtN numerical calculation model for the external sound 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] Where, K DtN =∫ B H T MHdΓ, M f =∫ Ω H T HdΩ;

[0086] S6.4: Establishing the Discrete Galerkin Weak Formulation 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 equation 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 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 structure part;

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

[0098]

[0099] Where 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 a local field approximation by generating virtual nodes in the element and assigning a set of quadratic complete shift polynomial basis functions within the support domain of each field node. The function is multiplied with the unit decomposition function of the overlapping finite element 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] Furthermore, an artificial boundary is used to truncate the infinite fluid domain, and the Dirichlet-to-Neumann mapping technique is introduced to achieve non-reflecting boundary conditions. Furthermore, for the acoustic scattering problem of underwater targets, especially when analyzing in the mid- and low-frequency bands, the interaction between the structure and the acoustic field cannot be ignored. This paper considers the acoustic-elastic coupling effect, derives the governing equations for the structure-acoustic coupling problem, and constructs a numerical calculation model using the overlapping finite element method to solve 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] This paper proposes a method for predicting the low- and medium-frequency acoustic scattering fields from underwater targets based on the overlapping finite element method. By assigning a set of quadratic complete shifted polynomial basis functions within the support domain of each field node to construct a high-order overlapping finite element local field approximation, this method can approximate high-order field variables using a low-order traditional finite element mesh, significantly suppressing numerical dispersion effects. Furthermore, the proposed prediction method is insensitive to mesh distortion, and its computational accuracy is insensitive to the direction of acoustic wave propagation, meaning that numerical anisotropy is virtually absent.

[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 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 Flow chart of the method in the implementation of the present invention.

[0114] Figure 2 Schematic diagram of modeling the underwater elastic solid sphere sound scattering problem provided by 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 diagram illustrating the relationship 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 diagram illustrating the relationship 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 The embodiment of the present invention provides a scattered sound pressure directivity diagram when f=2000 Hz.

[0119] Figure 7 The embodiment of the present invention provides a scattered sound pressure directivity diagram when f=3000 Hz.

[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 objectives, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely for the purpose of explaining 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 may be combined with each other as long as they do not conflict with each other.

[0122] The specific embodiment of the present invention is based on an underwater three-dimensional elastic solid sphere illuminated by a plane wave. The sphere has a radius of 0.6 m and is made of aluminum. While the figures show exemplary embodiments of the present disclosure, it should be understood that the present invention can be applied to various underwater targets and is not limited to the embodiments described herein. The present invention can effectively predict the acoustic scattering field of various complex underwater targets.

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

[0124] S1: Geometric modeling of the underwater target is performed, and a spherical artificial boundary is established outside the underwater target to enclose the target, thereby truncating the infinite fluid domain into a finite computational domain.

[0125] S2: Within the computational 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 sizes 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 the corresponding physical parameters of the material and medium for the structural grid and fluid grid obtained in step S3, and setting the boundary condition parameters of the sound 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 assigned 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 the overlapping finite element 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 of the underwater elastic target sound 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, the solid sphere object in this problem is established, and the sphere with a radius of R = 1.2m 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 the wave number. 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 value, the smaller the dimensionless wave number k and h, and the smaller the relative error; therefore, the number of units required for each wavelength is not less than 2 units, 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 elements. However, it should be understood that the disclosed invention can be implemented flexibly using element types including but not limited to four-node tetrahedral elements, six-node triangular prism elements, eight-node hexahedral elements, etc. based on the structural mechanical properties and geometric shape of the underwater target, and should not be limited by the element 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 within 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 of node I is ψ I (x):

[0148]

[0149] Where, 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] Where, is the shape function of the traditional quadratic finite element, M n is the number of nodes in the traditional quadratic finite element, is a predefined weight coefficient;

[0153] For the node function u J Expressed as:

[0154]

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

[0156] In step S52, a set of quadratic complete shifted polynomial basis functions is assigned to each field node to construct a local field approximation, and the local field approximation is normalized using the unit size h to improve the stability of the algorithm.

[0157]

[0158] Where, 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 takes 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 them into the above formula (8), we can get 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 the 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. A set of quadratic complete shift polynomial basis functions are assigned to the local analysis domain corresponding to the field nodes shown in black to construct the local field approximation, and finally the field variable approximation within the overlapping finite element is obtained. Figure 4 The disclosed embodiment shown in FIG 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 various traditional low-order finite elements mentioned above. k The structure of , and the expression 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 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 acoustic 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 to 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] Where, is the gradient operator;

[0176] There is 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] Where, and A n Represent the sound pressure, sound particle velocity and sound 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] Where, 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: Introducing 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 sound field scattering problem, we can obtain:

[0186]

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

[0188]

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

[0190]

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

[0192]

[0193] Where, 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 the acoustic scattering problem of underwater rigid targets:

[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 equations, we obtain the Galerkin weak form of the governing equation for the underwater rigid target acoustic scattering problem:

[0200]

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

[0202] After introducing the shape function matrix H of the overlapping finite element, the coupled overlapping finite element-DtN numerical calculation model for the external sound 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] Where, K DtN =∫ B H T MHdΓ, M f =∫ Ω H T HdΩ;

[0207] S6.4: Establishing the Discrete Galerkin Weak Formulation 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 equation 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 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 structure part;

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

[0219]

[0220] Where 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 based on 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 to include but not limited to visualization of scattered sound pressure distribution cloud maps, target sound scattering morphology functions, scattered sound pressure gradient distribution cloud maps, and other types of files such as extracting corresponding data to generate tables.

[0229] The following is an explanation with the help of charts:

[0230] Depend on Figure 5 It can be seen that at f = 2000Hz, the maximum forward scattered sound pressure predicted by the standard finite element model is lower than the exact solution, while the backscattered petal shape has a large error. When the calculation frequency increases to f = 3000Hz, the standard finite element model's prediction error becomes 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 acoustic field of an elastic solid sphere—the relationship between the forward scattered sound pressure amplitude and the computational frequency—is presented. It can be seen that when the frequency is greater than f = 1000, a phase lag is observed in the numerical solution of the traditional finite element method, and this lag effect gradually accumulates with increasing computational wavenumbers. In contrast, the overlapping finite element method proposed in this invention is still able to predict the acoustic scattering morphological function with high accuracy even at relatively high computational frequencies, demonstrating that the overlapping finite element method proposed in this invention expands the applicable computational frequency range of standard low-order finite element meshes.

[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 achieved are shown in Table 1.

[0233] Table 1 Comparison of computational costs of different methods to achieve the same computational accuracy

[0234]

[0235] As can be seen from Table 1, the number of elements 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 integration 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 merely 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 scope of protection of the present invention.

Claims

1. A method for predicting the low-frequency acoustic scattering field of underwater targets based on the overlapping finite element method, characterized in that: The following steps are involved: S1: Geometric modeling of the underwater target is performed, and a spherical artificial boundary is established outside the underwater target to enclose the target, thereby truncating the infinite fluid domain into a finite computational domain. S2: Within the computational 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 sizes 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 the corresponding physical parameters of the material and medium for the structural grid and fluid grid obtained in step S3, and setting the boundary condition parameters of the sound 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 assigned 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 the overlapping finite element is constructed; S6: Based on the element shape function of the overlapping finite element in step S5, the interaction between the structure and the acoustic field is combined in the analysis in the medium and low frequency bands to establish an overlapping finite element numerical discrete model of the underwater elastic target sound scattering problem and obtain a discrete system equation, and solve the discrete system equation to obtain the 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 the low-frequency acoustic scattering field of underwater targets based on the 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 the low-frequency acoustic scattering field of underwater targets based on the overlapping finite element method according to 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 the wave number. 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 value, the smaller the dimensionless wave number k and h, and the smaller the relative error; therefore, the number of units required for each wavelength is not less than 2 units, that is, the grid resolution n res ≥2, unit size 4. The method for predicting the low-frequency acoustic scattering field of underwater targets based on the overlapping finite element method according to claim 1, characterized in that: The step S5 of constructing the element 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 within 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 the low-frequency acoustic scattering field of underwater targets based on the overlapping finite element method according to claim 4, characterized in that: In step S51, in the overlapping finite element method, the local field approximation of node I is ψ I (x): Where, 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 Where, is the shape function of the traditional quadratic finite element, M n is the number of nodes in the traditional quadratic finite element, is a predefined weight coefficient; For the node function u J Expressed as: Where, f J is the local basis function, a J are the unknown coefficients to be solved.

6. The method for predicting the low-frequency acoustic scattering field of underwater targets based on the overlapping finite element method according to claim 5, characterized in that: In step S52, a set of quadratic complete shifted polynomial basis functions is allocated to each field node to construct a local field approximation, and the local field approximation is normalized using the unit size, that is: Where, is a shift polynomial, (x J ,y J ,z J ) are the coordinates of the cell vertices.

7. The method for predicting the low-frequency acoustic scattering field of underwater targets based on the overlapping finite element method according to 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 takes 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 them into the above formula (8), we can get 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 the edge JK in the standard quadratic finite element; 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; 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 the low-frequency acoustic scattering field of underwater targets based on the overlapping finite element method according to claim 7, characterized in that: In 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 acoustic 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 the Gaussian elimination method linear equation solver 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 the low-frequency acoustic scattering field of underwater targets based on the overlapping finite element method according to 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: Where, is the gradient operator, k represents the wave number; There is 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: Where, and A n Represent the sound pressure, sound particle velocity and sound 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: Where, f is a given value; At the same time, the sound particle velocity v and the sound pressure p satisfy the following relationship: S6.2: Introducing 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 sound field scattering problem, we can obtain: The Dirichlet-to-Neumann mapping technique is introduced on the spherical artificial boundary B. The sound pressure on the DtN boundary is: Where, are the polar coordinates of any point on the DtN boundary, θ' and is the integration variable, represents the n+1 / 2 order spherical Hankel function of the first kind, represents the first kind of associated Legendre function of order j and order n, R is the radius of the spherical DtN boundary, and the coefficient c n The expression is: Derivative of formula (16) with respect to variable r Where, 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 the acoustic scattering problem of underwater rigid targets: Multiplying Equation (11) with the test function w in the entire domain yields the equivalent integral form: Using Green's theorem we get: Substituting boundary conditions (12), (13) and (18) into the above equations, we obtain the Galerkin weak form of the governing equation for the underwater rigid target acoustic scattering problem: Where M is the DtN operator obtained by equation (18), is the speed of sound particles; After introducing the shape function matrix H of the overlapping finite element, the coupled overlapping finite element-DtN numerical calculation model for the external sound 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) Where, K DtN =∫ B H T MHdΓ, M f =∫ Ω H T HdΩ; S6.4: Establish the discrete Galerkin weak form of the governing equations for acoustic scattering from underwater elastic targets. When analyzing the low and medium frequency bands, consider the interaction between the structure and the acoustic field and, based on step S6.3, introduce the elastic effect of the structure. The discrete Galerkin weak form of the governing equation 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 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 structure part; The additional load of the structural surface vibration on the acoustic field is expressed as: Where 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 the overlapping finite element numerical discrete model of the underwater elastic target acoustic scattering problem. The Gaussian elimination method linear equation solver is used to solve the overlapping finite element numerical discrete model of the underwater target acoustic scattering problem to obtain the unknown solution coefficients.

10. The method for predicting the 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 a scattered sound pressure distribution cloud map, a target sound scattering morphological function, and a scattered sound pressure gradient distribution cloud map.

Images