Smoothed finite element-newmark method for transient acoustic scattering from underwater targets

Abstract

The application discloses a smooth finite element-Newmark calculation method for underwater target transient acoustic scattering, and belongs to the technical field of underwater transient acoustic scattering numerical simulation. First, a calculation domain of an underwater transient acoustic scattering field is discretized into a standard quadrilateral element grid, and an acoustic pressure field interpolation format is constructed. The quadrilateral element is divided into multiple smooth domains, a smooth acoustic pressure gradient field is obtained through acoustic generalized gradient smoothing technology, and a smooth acoustic pressure gradient matrix is obtained through Gaussian integration. A system matrix equation is constructed in combination with a transient acoustic scattering control equation and a Galerkin weighted residual value method. A parameter-optimized Newmark time integration method is used to complete time domain discretization and recursive solving, and numerical results of the transient acoustic scattering field are obtained. The application softens the system stiffness matrix through gradient smoothing technology, reduces spatial discretization error, eliminates time domain false numerical damping, and has higher precision than the standard finite element method under the same grid, thereby providing an efficient and reliable scheme for underwater transient acoustic scattering engineering calculation.

Description

Technical Field

[0001] This invention relates to the field of underwater transient acoustic scattering numerical simulation technology, and in particular to a smooth finite element-Newmark calculation method for transient acoustic scattering of underwater targets. Background Technology

[0002] The problem of transient acoustic scattering from underwater targets is one of the core scientific issues in the field of underwater acoustics. Its research results directly support the technological development in multiple fields, including underwater defense and offense, marine resource exploration, underwater engineering inspection, underwater acoustic communication, and marine environmental perception. In core engineering scenarios such as active sonar detection, underwater target identification and classification, and submarine acoustic stealth design, transient scattering signals contain key characteristics such as the target's geometry and material properties. High-precision numerical simulation of these signals is the core foundation for the research and development of related technologies.

[0003] The finite element method (FEM) is currently the most commonly used numerical method for solving transient acoustic scattering problems of underwater targets. However, it has inherent limitations in simulating acoustic wave propagation: the stiffness matrix of the numerical model system constructed by the standard FEM exhibits "overstiffness," deviating from the stiffness of the real system. This leads to significant dispersion errors during spatial discretization, manifesting as spurious oscillations in the numerical solution and a decrease in computational accuracy. To meet engineering accuracy requirements, a meshing criterion of "at least 6 elements per wavelength" must be followed. As the computational frequency and computational domain scale increase, the required number of meshes rises sharply, and the computational scale and memory consumption grow exponentially. Under conventional computing resources, it is difficult to achieve efficient solutions to large-scale, broadband transient acoustic scattering problems.

[0004] Meanwhile, solving transient acoustic scattering problems requires time-domain discretization using time integration methods. However, the traditional Newmark integration method, if parameters are not properly selected, can introduce spurious numerical damping, generating additional time-discretion errors and further reducing the reliability of the calculation results. The spatial dispersion error and the time-discretion integration error are coupled, becoming the core bottleneck restricting the accuracy and efficiency of underwater transient acoustic scattering numerical simulations. Summary of the Invention

[0005] The purpose of this invention is to propose a smooth finite element-Newmark calculation method for transient acoustic scattering of underwater targets. This method overcomes the shortcomings of the standard finite element method in solving the transient acoustic scattering problem of underwater targets, such as large spatial discretization dispersion errors, low accuracy due to spurious numerical damping in temporal discretization, and excessive computational cost for large-scale problems. By reducing spatial discretization errors through acoustic generalized gradient smoothing techniques and eliminating spurious numerical damping in temporal discretization by optimizing the Newmark integral parameters, this invention achieves higher accuracy in transient acoustic scattering field calculation under the same mesh conditions, providing an efficient and reliable numerical solution for engineering applications.

[0006] To achieve the above objectives, this invention provides a smooth finite element-Newmark calculation method for transient acoustic scattering of underwater targets, comprising the following steps: Step S1: Discretize the spatial computational domain of the transient acoustic scattering field of the underwater target into a standard quadrilateral cell grid, and construct an acoustic pressure field interpolation scheme based on the quadrilateral cell; Step S2: Divide each quadrilateral unit into several smooth regions, and use acoustic generalized gradient smoothing technology to smooth each smooth region to obtain the smoothed sound pressure gradient field. Step S3: Numerically integrate the smoothed sound pressure gradient field using the Gaussian integral formula to calculate the smoothed sound pressure gradient matrix; Step S4: Based on the smooth acoustic pressure gradient matrix and the partial differential control equations of underwater transient acoustic scattering, and combining the virtual displacement principle and the Galerkin weighted residual method, construct the system matrix equations for the underwater transient acoustic scattering problem. Step S5: Select the integration parameters of the Newmark time integration method, discretize the system matrix equation in the time domain, and obtain the discretized time domain recursive equation. Step S6: Based on the initial conditions, the discretized time-domain recursive equation is solved recursively to obtain the full-time-domain numerical calculation results of the transient acoustic scattering field of the underwater target; Step S7: Analyze the numerical calculation results, compare the differences in results from different calculation methods, and complete the solution and characteristic analysis of the transient acoustic scattering problem of underwater targets.

[0007] Preferably, in step S1, the sound pressure field interpolation format based on the quadrilateral cell grid is as follows: within each quadrilateral cell, the sound pressure interpolation format at any point is expressed as: ; in, The sound pressure interpolation result at any point within the quadrilateral element. , is the quadrilateral unit Interpolation shape function for each node, For the first Unknown node sound pressure level of each node; The sound pressure interpolation format at any point within the entire spatial computational domain is expressed as follows: ; in, This represents the total number of nodes in the quadrilateral element within the spatial computational domain. This is the sound pressure interpolation result at any point within the computation domain.

[0008] Preferably, in step S2, each quadrilateral unit is divided into 4 smooth regions, and acoustic generalized gradient smoothing is performed on each smooth region. The specific process is as follows: The constitutive relationship between sound pressure and sound particle vibration velocity in the sound field is as follows: ; in, The imaginary unit, The density of the underwater acoustic fluid medium. The angular frequency of the sound wave The velocity of the sound particle vibration; The acoustic particle vibration velocity field within each smoothed domain is smoothed, and the expression for the smoothed acoustic particle vibration velocity field is as follows: ; in, The velocity field of the smoothed acoustic particles. The original sound particle vibration velocity field, For the spatial region of a smooth domain, It is a smooth function and satisfies ; The smoothing function used is the constant smoothing function, expressed as follows: ; in, For smooth regions The area; Using Green's formula, the smoothed sound particle vibration velocity field is converted into a sound pressure gradient-related form, as shown in the following expression: ; in, For the boundary of a smooth region, Let be the outward normal unit vector of the boundary of the smooth domain; For a two-dimensional acoustic problem, the outward normal unit vector The expression is as follows: ; in, The outward normal unit vector along directional components, The outward normal unit vector along The directional component.

[0009] Preferably, in step S3, the calculation process of the smooth sound pressure gradient matrix is ​​as follows: Based on the spatial computational domain-based sound pressure field interpolation scheme, the smoothed sound particle vibration velocity field is expressed as: ; in, For the first The smooth acoustic pressure gradient matrix corresponding to each node; The smooth sound pressure gradient matrix is ​​calculated using the Gaussian integral formula, as shown below: ; in, The total number of line segments that define the boundary of the smooth domain. The number of Gaussian integration points set on each boundary segment, when using a one-point Gaussian integral. , Let the spatial location of the Gaussian integration point be... These are the weighting coefficients corresponding to the Gaussian integration points.

[0010] Preferably, in step S4, the process of constructing the system matrix equation is as follows: The partial differential governing equations for underwater transient sound propagation are expressed as follows: ; in, The speed of sound wave propagation underwater. sound pressure Regarding time The second derivative; Based on the principle of virtual displacement, the weak integral form of the governing equations is established, as follows: ; in, This represents the imaginary displacement of the sound pressure. By using integration by parts and Gauss's divergence theorem, the weak integral form is transformed into: ; in, The global boundary of the spatial computational domain. The unit vector of the outward normal to the boundary of the computation domain; By substituting the sound pressure field interpolation scheme into the above formula using the Galerkin weighted residual method, the system matrix equation is obtained, as shown below: ; in, for The sound pressure vector of each unknown node. Let be the second derivative vector of the nodal sound pressure vector with respect to time; For the system quality matrix, A vector composed of nodal interpolation shape functions; Here is the system's smooth stiffness matrix. The smooth sound pressure gradient matrix; For the system load vector, This is used to compute the normal acoustic particle velocity vector at the boundary node of the domain.

[0011] Preferably, in step S5, the time-domain discretization process of the Newmark time integration method is as follows: Set Newmark time integration parameters , The time integration step size is Construct the recursive relationship between sound pressure vectors at different times: ; Among them, superscript and They represent Time and Physical quantity at time. The vector is the first derivative of the nodal sound pressure vector with respect to time; based on Time and The dynamic equilibrium equations at time t: ; Substituting the sound pressure recursive relation into the dynamic equilibrium equation, we obtain the discretized time-domain recursive equation: .

[0012] Preferably, in step S6, the specific process of recursive solution is as follows: Set initial time sound pressure vector at time First derivative vector Second derivative vector ; Solving through time-domain recurrence equations Nodal sound pressure vector at time 1 Then, the first derivative of the sound pressure at the corresponding moment is calculated using the sound pressure recursion relationship. and second derivative ; Using the calculation result at the current moment as the initial condition for the next moment, the calculation is iterated until all time steps are completed, and the full-time domain numerical calculation result of the transient acoustic scattering field of the underwater target is obtained.

[0013] Preferably, in step S7, the result analysis process specifically involves: comparing the calculation results of the transient acoustic scattering field with the standard finite element method at the same time and under the same grid, analyzing numerical oscillations and calculation errors, extracting the spatial distribution characteristics and temporal evolution law of the transient acoustic scattering field of the underwater target, and completing the quantitative analysis of the transient acoustic scattering characteristics of the underwater target.

[0014] Preferably, the acoustic stiffness matrix of the standard finite element method is softened using acoustic generalized gradient smoothing technology, making the system stiffness matrix closer to the real system stiffness and reducing the dispersion error caused by spatial discretization; by setting , The Newmark integral parameters are used to eliminate spurious numerical damping in the time-domain discretization process, avoid time discretization errors, and achieve higher accuracy in transient acoustic scattering field calculation than the standard finite element method under the same grid conditions.

[0015] Therefore, the present invention employs the above-mentioned smooth finite element-Newmark calculation method for transient acoustic scattering of underwater targets, which has the following advantages: (1) The present invention uses acoustic generalized gradient smoothing technology to appropriately soften the "overly stiff" acoustic stiffness matrix in the standard finite element method, so that the constructed system stiffness matrix is ​​closer to the real system stiffness, which greatly reduces the dispersion error generated in the spatial discretization process, effectively suppresses the false oscillation of the numerical solution, and obtains higher spatial calculation accuracy under the same grid density. (2) This invention optimizes the parameters of the Newmark time integration method and selects... , It completely eliminates spurious numerical damping in the time-domain discretization process, avoids additional calculation errors caused by time discretization, and ensures the unconditional stability of time integrals, thus achieving high-precision solution of transient acoustic scattering fields in the entire time domain. (3) This invention effectively controls both spatial and temporal discretization errors. Without significantly increasing computational costs, it greatly improves the computational accuracy and reliability of transient acoustic scattering problems of underwater targets, reduces the stringent requirements of grid density for large-scale, broadband transient acoustic scattering problems, and effectively reduces the computational scale. It provides an efficient and reliable numerical computation scheme for transient acoustic scattering simulation in engineering fields such as underwater attack and defense and marine exploration.

[0016] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. Attached Figure Description

[0017] Figure 1 This is a flowchart of a smooth finite element-Newmark calculation method for transient acoustic scattering of an underwater target in an embodiment of the present invention; Figure 2 This is a schematic diagram of the quadrilateral unit and smooth domain division in an embodiment of the present invention; Figure 3 This is a schematic diagram of the shape function values ​​of the nodes and integration points of the standard quadrilateral element in an embodiment of the present invention; Figure 4 This is a schematic diagram of the shape function values ​​of the quadrilateral unit nodes and integration points of the two smooth domains in an embodiment of the present invention; Figure 5 This is a schematic diagram of the shape function values ​​of the quadrilateral unit nodes and integration points divided into three smooth regions in an embodiment of the present invention; Figure 6This is a schematic diagram of the shape function values ​​of the quadrilateral unit nodes and integration points in the four smooth domain divisions of this invention. Figure 7 This is a schematic diagram of the computational domain for the transient acoustic scattering problem of an underwater submarine model in an embodiment of the present invention; Figure 8 This is a schematic diagram of the quadrilateral cell mesh division of the computational domain in an embodiment of the present invention; Figure 9 The transient sound field distribution at different times is obtained from the standard finite element-Newmark method. Figure 10 The transient sound field distribution results at different times obtained by the method of the present invention are shown. Detailed Implementation

[0018] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of the present invention. The components of the embodiments of the present invention described and shown in the accompanying drawings can generally be arranged and designed in various different configurations. Therefore, the following detailed description of the embodiments of the present invention provided in the accompanying drawings is not intended to limit the scope of the claimed invention, but merely to illustrate selected embodiments of the invention. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without inventive effort are within the scope of protection of the present invention.

[0019] It should be noted that similar labels and letters in the following figures indicate similar items. Therefore, once an item is defined in one figure, it does not need to be further defined and explained in subsequent figures.

[0020] Example like Figure 1-10 As shown, this embodiment proposes a smooth finite element-Newmark calculation method for transient acoustic scattering of underwater targets. This embodiment uses the transient acoustic scattering problem of an underwater submarine model with point sound source incidence as the solution object, and provides a detailed description of the method. The calculation scenario of this embodiment is as follows: Figure 7 As shown, the spatial computational domain is a square two-dimensional domain with a side length of 1500m. The submarine model is located at the geometric center of the computational domain, and the point sound source is located at the upper left vertex of the computational domain, capable of emitting short-time pulsating sound waves. The entire computational domain is discretized into a standard quadrilateral element mesh, and the mesh division is as follows: Figure 8 As shown.

[0021] like Figure 1 As shown, the smooth finite element-Newmark calculation method for transient acoustic scattering of underwater targets in this embodiment specifically includes the following steps: Step S1: Discretize the square computational domain of the transient acoustic scattering field of the underwater submarine model into a standard quadrilateral element mesh, and construct the acoustic pressure field interpolation scheme using the bilinear interpolation shape function of the quadrilateral elements. Within a single quadrilateral element, the acoustic pressure interpolation scheme at any point is as follows: ; in, This represents the sound pressure interpolation result at any point within the quadrilateral element. For the quadrilateral unit number Interpolation shape function for each node, For the first Unknown node sound pressure level of each node Spatial location coordinates; The sound pressure interpolation format at any point within the entire spatial computational domain is expressed as follows: ; in, This represents the total number of nodes in the quadrilateral element within the spatial computational domain. This is the sound pressure interpolation result at any point within the computation domain.

[0022] Step S2: As Figure 2 As shown in (d), each quadrilateral unit is divided into four smooth regions, and acoustic generalized gradient smoothing is performed on each smooth region. The specific process is as follows: Based on the constitutive relationship between sound pressure and sound particle vibration velocity in the sound field: ; in, The imaginary unit, The density of the underwater acoustic fluid medium. The angular frequency of the sound wave The velocity of the sound particle vibration; The sound particle vibration velocity field within each smooth region is smoothed, and the smoothed sound particle vibration velocity field is as follows: ; in, The velocity field of the smoothed acoustic particles. The original sound particle vibration velocity field, For the spatial region of a smooth domain, It is a smooth function and satisfies ; The smooth function is specifically: ; in, For smooth regions The area; Combining Green's formula, the volume integral is transformed into a boundary integral, and the smoothed acoustic particle vibration velocity field is transformed into: ; In the formula, For the boundary of a smooth region, Let be the outward normal unit vector of the smooth domain boundary. For the two-dimensional acoustic problem in this embodiment, , , They are the outward normal unit vectors along , The directional component.

[0023] The quadrilateral element is divided into four smooth regions, and the interpolation shape function values ​​at its nodes and integration points are as follows: Figure 6 As shown.

[0024] Step S3: Based on the sound pressure field interpolation scheme, the smoothed sound particle vibration velocity field is represented as: ; in, For the first The smooth acoustic pressure gradient matrix corresponding to each node; The smooth sound pressure gradient matrix is ​​calculated using the Gaussian integral formula: ; in, The number of line segments that divide the boundary of the smooth region. The number of Gaussian integration points set on each boundary segment is taken in this embodiment. , Let the spatial location of the Gaussian integration point be... These are the weighting coefficients corresponding to the Gaussian integration points.

[0025] Step S4: Based on the smooth acoustic pressure gradient matrix and the partial differential governing equations for underwater transient acoustic scattering, and combining the virtual displacement principle and the Galerkin weighted residual method, construct the system matrix equations for the underwater transient acoustic scattering problem. The expression for the partial differential governing equations for underwater transient sound propagation is as follows: ; in, For the Laplace operator, The speed of sound in water is taken as 1500 m / s. sound pressure Regarding time The second derivative; Based on the principle of virtual displacement, the weak integral form of the governing equations is established, and its expression is: ; in, This represents the imaginary displacement of the sound pressure. By using integration by parts and Gauss's divergence theorem, the weak integral form is transformed into: ; in, The global boundary of the spatial computational domain. The unit vector of the outward normal to the boundary of the computation domain; Using the Galerkin weighted residual method, substituting the sound pressure field interpolation scheme into the above formula, we obtain the system matrix equation, which is expressed as: ; in, for The sound pressure vector of each unknown node. Let be the second derivative vector of the nodal sound pressure vector with respect to time; For the system quality matrix, A vector composed of nodal interpolation shape functions; Here is the system's smooth stiffness matrix. The smooth sound pressure gradient matrix; The system load vector is determined by the boundary conditions of the sound source at the incident point. This is used to compute the normal acoustic particle velocity vector at the boundary node of the domain.

[0026] Step S5: Set Newmark time integration parameters , This parameter combination enables the Newmark integration method to be free of spurious numerical damping and remain unconditionally stable; setting the time integration step size s, construct the recursive relationship of sound pressure vectors at different times: ; Among them, superscript and They represent Time and Physical quantity at time. The vector is the first derivative of the nodal sound pressure vector with respect to time; based on Time and The dynamic equilibrium equations at time t: ; Substituting the sound pressure recursive relation into the dynamic equilibrium equation, we obtain the discretized time-domain recursive equation: .

[0027] Step S6: Set the initial time sound pressure vector at time First derivative vector Second derivative vector ; Solving through time-domain recurrence equations Nodal sound pressure vector at time 1 Then, the first derivative of the sound pressure at the corresponding moment is calculated using the sound pressure recursion relationship. and second derivative ; Using the calculation result at the current moment as the initial condition for the next moment, the calculation is iterated until all time steps within t=1.0s are completed, and the full-time domain numerical calculation result of the transient acoustic scattering field of the underwater submarine model is obtained.

[0028] Step S7: Analyze the numerical calculation results, compare the differences between the results of different calculation methods, and complete the solution and characteristic analysis of the transient acoustic scattering problem of underwater targets. Based on the full-time domain numerical calculation results, extract the spatial distribution data of the transient acoustic scattering field at four times t=0.7s, 0.8s, 0.9s, and 1.0s, and compare the calculation results of the standard finite element-Newmark method and the method of this invention under the same grid and time step.

[0029] from Figure 9 It can be seen that the transient sound field distribution calculated by the standard finite element method-Newmark method exhibits obvious spurious numerical oscillations, the sound field distribution is not smooth, and the calculation error is significant. The root cause lies in the "overly stiff" system stiffness matrix of the standard finite element method, leading to severe spatial discretization dispersion errors. From Figure 10 As can be seen, the transient sound field distribution calculated by the method of this invention has no obvious spurious oscillations, the sound field distribution is smooth and continuous, and the calculation accuracy and reliability are significantly improved. The core reason is that the acoustic generalized gradient smoothing technique effectively softens the system stiffness matrix, making it closer to the real system stiffness, and greatly reducing spatial discretization errors; at the same time, the optimized Newmark integral parameters eliminate spurious numerical damping, avoid time discretization errors, and achieve high-precision solution of transient sound scattering field.

[0030] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit them. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the technical solutions of the present invention, and these modifications or equivalent substitutions cannot cause the modified technical solutions to deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. A smooth finite element-Newmark calculation method for transient acoustic scattering of underwater targets, characterized in that, Includes the following steps: Step S1: Discretize the spatial computational domain of the transient acoustic scattering field of the underwater target into a standard quadrilateral cell grid, and construct an acoustic pressure field interpolation scheme based on the quadrilateral cell; Step S2: Divide each quadrilateral unit into several smooth regions, and use acoustic generalized gradient smoothing technology to smooth each smooth region to obtain the smoothed sound pressure gradient field. Step S3; The smoothed sound pressure gradient field was numerically integrated using the Gaussian integral formula to obtain the smoothed sound pressure gradient matrix. Step S4: Based on the smooth acoustic pressure gradient matrix and the partial differential control equations of underwater transient acoustic scattering, and combining the virtual displacement principle and the Galerkin weighted residual method, construct the system matrix equations for the underwater transient acoustic scattering problem. Step S5: Select the integration parameters of the Newmark time integration method, discretize the system matrix equation in the time domain, and obtain the discretized time domain recursive equation. Step S6: Based on the initial conditions, the discretized time-domain recursive equation is solved recursively to obtain the full-time-domain numerical calculation results of the transient acoustic scattering field of the underwater target; Step S7: Analyze the numerical calculation results, compare the differences in results from different calculation methods, and complete the solution and characteristic analysis of the transient acoustic scattering problem of underwater targets.

2. The smooth finite element-Newmark calculation method for transient acoustic scattering of underwater targets according to claim 1, characterized in that: In step S1, the sound pressure field interpolation format based on the quadrilateral element mesh is specifically as follows: Within each quadrilateral element, the sound pressure interpolation format at any point is expressed as: ； in, This represents the sound pressure interpolation result at any point within the quadrilateral element. For the quadrilateral unit number Interpolation shape function for each node, For the first Unknown node sound pressure level of each node; The sound pressure interpolation format at any point within the entire spatial computational domain is expressed as follows: ； in, This represents the total number of nodes in the quadrilateral element within the spatial computational domain. This is the sound pressure interpolation result at any point within the computation domain.

3. The smooth finite element-Newmark calculation method for transient acoustic scattering of underwater targets according to claim 1, characterized in that: In step S2, each quadrilateral unit is divided into 4 smooth regions, and acoustic generalized gradient smoothing is performed on each smooth region. The specific process is as follows: The constitutive relationship between sound pressure and sound particle vibration velocity in the sound field is as follows: ； in, The imaginary unit, The density of the underwater acoustic fluid medium. The angular frequency of the sound wave The velocity of the sound particle vibration; The acoustic particle vibration velocity field within each smoothed domain is smoothed, and the expression for the smoothed acoustic particle vibration velocity field is as follows: ； in, The velocity field of the smoothed acoustic particles. The original sound particle vibration velocity field, For the spatial region of a smooth domain, It is a smooth function and satisfies ; The smoothing function used is the constant smoothing function, expressed as follows: ； in, For smooth regions The area; Using Green's formula, the smoothed sound particle vibration velocity field is converted into a sound pressure gradient-related form, as shown in the following expression: ； in, For the boundary of a smooth region, Let be the outward normal unit vector of the boundary of the smooth domain; For a two-dimensional acoustic problem, the outward normal unit vector The expression is as follows: ； in, The outward normal unit vector along directional components, The outward normal unit vector along The directional component.

4. The smooth finite element-Newmark calculation method for transient acoustic scattering of underwater targets according to claim 3, characterized in that: In step S3, the calculation process of the smooth sound pressure gradient matrix is ​​as follows: Based on the spatial computational domain-based sound pressure field interpolation scheme, the smoothed sound particle vibration velocity field is expressed as: ； in, For the first The smooth acoustic pressure gradient matrix corresponding to each node; The smooth sound pressure gradient matrix is ​​calculated using the Gaussian integral formula, as shown below: ； in, The total number of line segments that define the boundary of the smooth domain. The number of Gaussian integration points set on each boundary segment, when using a one-point Gaussian integral. , Let the spatial location of the Gaussian integration point be... These are the weighting coefficients corresponding to the Gaussian integration points.

5. The smooth finite element-Newmark calculation method for transient acoustic scattering of underwater targets according to claim 4, characterized in that: In step S4, the specific process of constructing the system matrix equation is as follows: The partial differential governing equations for underwater transient sound propagation are expressed as follows: ； in, The speed of sound wave propagation underwater. sound pressure Regarding time The second derivative; Based on the principle of virtual displacement, the weak integral form of the governing equations is established, as follows: ； in, This represents the imaginary displacement of the sound pressure. By using integration by parts and Gauss's divergence theorem, the weak integral form is transformed into: ； in, The global boundary of the spatial computational domain. The unit vector of the outward normal to the boundary of the computation domain; By substituting the sound pressure field interpolation scheme into the above formula using the Galerkin weighted residual method, the system matrix equation is obtained, as shown below: ； in, for The sound pressure vector of each unknown node. Let be the second derivative vector of the nodal sound pressure vector with respect to time; For the system quality matrix, A vector composed of nodal interpolation shape functions; Here is the system's smooth stiffness matrix. The smooth sound pressure gradient matrix; For the system load vector, This is used to compute the normal acoustic particle velocity vector at the boundary node of the domain.

6. The smooth finite element-Newmark calculation method for transient acoustic scattering of underwater targets according to claim 5, characterized in that: In step S5, the time-domain discretization process of the Newmark time integration method is as follows: Set Newmark time integration parameters , The time integration step size is Construct the recursive relationship between sound pressure vectors at different times: ； Among them, superscript and They represent Time and Physical quantity at time. The vector is the first derivative of the nodal sound pressure vector with respect to time; based on Time and The dynamic equilibrium equations at time t: ； Substituting the sound pressure recursive relation into the dynamic equilibrium equation, we obtain the discretized time-domain recursive equation: 。 7. The smooth finite element-Newmark calculation method for transient acoustic scattering of underwater targets according to claim 6, characterized in that: In step S6, the specific process of recursively solving is as follows: Set initial time sound pressure vector at time First derivative vector Second derivative vector ; Solving through time-domain recurrence equations Nodal sound pressure vector at time 1 Then, the first derivative of the sound pressure at the corresponding moment is calculated using the sound pressure recursion relationship. and second derivative ; Using the calculation result at the current moment as the initial condition for the next moment, the calculation is iterated until all time steps are completed, and the full-time domain numerical calculation result of the transient acoustic scattering field of the underwater target is obtained.

8. The smooth finite element-Newmark calculation method for transient acoustic scattering of underwater targets according to claim 1, characterized in that: In step S7, the result analysis process is as follows: compare the calculation results of the transient acoustic scattering field with the standard finite element method at the same time and under the same grid, analyze the numerical oscillation and calculation error, extract the spatial distribution characteristics and temporal evolution law of the transient acoustic scattering field of the underwater target, and complete the quantitative analysis of the transient acoustic scattering characteristics of the underwater target.

9. The smooth finite element-Newmark calculation method for transient acoustic scattering of underwater targets according to claim 1, characterized in that: The acoustic stiffness matrix of the standard finite element method is softened using acoustic generalized gradient smoothing technology, making the system stiffness matrix closer to the real system stiffness and reducing the dispersion error caused by spatial discretization; by setting... , The Newmark integral parameters are used to eliminate spurious numerical damping in the time-domain discretization process, avoid time discretization errors, and achieve higher accuracy in transient acoustic scattering field calculation than the standard finite element method under the same grid conditions.

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