Time-varying finite element modeling method for extremely low frequency sound field under internal wave dynamic process

Abstract

This invention specifically relates to a time-varying finite element modeling method for extremely low frequency (UHF) sound fields under dynamic internal wave processes, belonging to the field of underwater acoustic physics. The method first simulates the internal wave propagation process using the modified Korteweg-de Vries equation based on background field environmental information, and then obtains the vertical modes of the internal waves using the Taylor-Goldstein equation to construct a dynamic sound velocity field. Secondly, in a fluid-structure interaction model containing multiple layers of seabed media, the wave equation is solved using the finite element method to achieve time-varying calculation of the UHF sound field. This invention can precisely characterize the spatiotemporal evolution of the sound velocity field under complex terrain and dynamic internal wave disturbances, accurately simulate the distribution characteristics of the UHF sound field throughout the entire internal wave propagation process, and is suitable for sound field prediction and analysis in sea areas with high internal wave incidence, such as the South China Sea.

Description

Technical Field

[0001] This invention relates to the field of underwater acoustic physics, specifically to a time-varying finite element modeling method for extremely low frequency sound fields under internal wave dynamic processes. Background Technology

[0002] As a relatively common sub-mesoscale ocean phenomenon, internal ocean waves can cause dramatic fluctuations in local seawater temperature, salinity, and sound speed during their propagation and evolution, thereby causing underwater acoustic field anomalies and having a significant impact on underwater acoustic detection (such as decreased acoustic field correlation and acoustic field phase perturbation).

[0003] The South my country Sea is the sea area with the most frequent and strongest internal waves in the world. Internal waves are initially generated in the Luzon Strait through tidal-land interactions. During their westward propagation, they continuously deepen nonlinearly, evolving into highly nonlinear internal solitary waves when they reach the continental shelf area near Dongsha Island. These internal solitary waves subsequently become shallower, split, or intensify during their ascent. These dynamic processes of internal waves affect the degree of sound field fluctuations and the detection performance of sonar equipment at different spatiotemporal scales.

[0004] Currently, domestic and international scholars have mainly conducted research on the influence of sound fields in internal wave environments. In terms of internal wave sound velocity field modeling, most studies divide the water body into two layers with unequal densities and sound velocities (i.e., a two-layer ocean model). This simplifies the model calculation process to some extent, but the constructed sound velocity field differs significantly from reality. Secondly, some studies use the classic Garrett-Munk (GM) internal wave spectrum to model the sound velocity disturbances caused by random internal waves. However, as a statistically averaged model, the GM spectrum is difficult to accurately simulate the dynamic processes of internal waves (especially in environments with drastic topographic changes and highly nonlinear internal solitary waves). Furthermore, most of the aforementioned works focus on first-mode internal solitary waves or linear internal waves, while the internal solitary waves observed in the northern South China Sea typically appear as first-mode internal solitary wave trains. Additionally, some second-mode internal solitary waves exist in winter, and there is very little research specifically targeting the sound velocity field modeling of these two types of internal waves.

[0005] On the other hand, with the increasing concealment of modern underwater targets, their radiated noise spectrum is mainly concentrated in the low-frequency and extremely low-frequency bands, thus driving sonar operating frequencies to shift to these bands to enhance their detection capabilities. However, the propagation characteristics of extremely low-frequency sound are more sensitive to changes in the marine environment (sound speed field, seabed topography, and seabed sediment). In continental shelf areas where internal waves are frequent, the terrain slope is large, the seabed sediment layering is complex, and the internal wave dynamic processes change rapidly. The simultaneous existence of these factors makes it difficult for mainstream sound field models (ray acoustics, normal mode, and parabolic equation models) to accurately characterize the propagation of extremely low-frequency sound under complex terrain and dynamic sound speed fields.

[0006] Based on the above background, this invention establishes a time-varying finite element modeling method for extremely low frequency sound fields under internal wave dynamic processes, and uses this method to simulate and calculate the sound fields under three typical internal wave environments. Summary of the Invention

[0007] To address the problem of calculating extremely low frequency sound fields under dynamic internal wave environments, this invention provides a refined modeling method for the internal wave sound velocity field and a finite element modeling method for the time-varying extremely low frequency sound field under complex multi-layered seabed topography. This method can accurately simulate the changes in the extremely low frequency sound field during the entire process of solitary wave propagation and evolution in typical continental slope areas of the South China Sea.

[0008] Other features and advantages of the invention will become apparent from the following detailed description, or may be learned in part by practice of the invention.

[0009] According to a first aspect of the present invention, a time-varying finite element modeling method for extremely low frequency sound fields under internal wave dynamic processes is provided, the method comprising: Step 1: Simulation of the internal wave dynamic propagation process, specifically including: Step 1.1: Using a properly corrected Korteweg-de Vries equation and in conjunction with background field information, solve the evolution of the vertical displacement of the internal wave at the depth of the stratum as a function of time and horizontal distance; Step 1.2: Solve the Taylor-Goldstein equation to obtain the vertical mode function and linear wave velocity of the internal wave. Then, extend the vertical displacement at the jump layer to the entire water layer to obtain the linear wave velocity and vertical mode function of the internal wave. Step 2: Substitute the internal wave linear velocity and vertical mode function obtained in Step 1.1 into the wave solution in Step 1.2 to obtain the vertical displacement field of the internal wave in the entire water layer. Combine the vertical displacement field of the internal wave in the entire water layer with the background sound velocity field and interpolate along the depth dimension to construct the sound velocity field under the dynamic process of the internal wave. Step 3: Finite element solution of the extremely low frequency sound field. In a multi-layer fluid-structure interaction model containing a seawater layer and at least one layer of seabed medium, the seawater layer is geometrically divided according to the isovelocity layer of the internal wave sound velocity field. Then, the wave equation is discretized and solved using the finite element method to obtain the sound pressure distribution of the extremely low frequency sound field in the study area under the internal wave dynamic process.

[0010] In some exemplary embodiments, the repaired positive Korteweg-de Vries equation specifically includes:

[0011] In the formula, This represents the vertical displacement of the internal wave at the mezzanine. and Representing the horizontal coordinate and time respectively; The linear wave velocity of the internal wave; and These are the nonlinear coefficients and dispersion coefficients of the internal wave, respectively. This indicates an externally forced modification.

[0012] In some exemplary embodiments, the Taylor-Goldstein equation is specifically as follows:

[0013]

[0014]

[0015] in, Indicates the buoyancy frequency. Represents gravitational acceleration. and Let represent the linear wave velocity and background horizontal velocity at different times or distances during the propagation of the internal wave, respectively. This represents the vertical mode function of the internal wave during propagation. The coordinates represent the depth direction. Equations 2 and 3 represent the rigid boundary condition and the free boundary condition of the sea surface, respectively.

[0016] In some exemplary embodiments, the internal wave includes a single isolated wave in the first mode, a single isolated wave in the second mode, or a wave train in the first mode; for a single isolated wave, the solution of the modified KdV equation is in the form of a hyperbolic secant function; for a wave train, the solution of the modified KdV equation is in the form of a Dnoidal wave solution.

[0017] In some exemplary embodiments, the construction of the sound velocity field under the internal wave dynamic process is specifically as follows: For isolated waves in the first and second modes, the vertical displacement field of the waves within the entire water layer is as follows:

[0018] In the formula, Corresponding to solitary waves within the first and second modes, respectively. Indicates the amplitude of the internal solitary wave; For the isolated wave train in the first mode, the vertical displacement field of the wave within the entire water layer is as follows:

[0019] Using background sound velocity field Displacement field of the entire water layer with internal waves Constructing the sound velocity field under the internal wave dynamic process:

[0020] In the formula, This indicates interpolation along the depth dimension.

[0021] In some exemplary embodiments, the discretization solution of the wave equation using the finite element method specifically includes:

[0022] In the formula, , and These represent the sound pressure, sound velocity, and density of seawater, respectively. and These represent the external force on the volume and the mass acceleration injected into the finite space, respectively. For Hamiltonian operators; Accordingly, the boundary conditions are as follows:

[0023] In the formula, and These represent the sound pressure level and normal acceleration at the boundary, respectively. This represents the gradient of sound pressure along the normal direction; According to the weighted residual method, the weight function Multiplying each of the two equations above and summing them after integral transformation, we obtain the weak form of the wave equation as follows:

[0024] Discretize the above equation and assume the space Divided into a finite number of discrete units and having If there are nodes, then the space The sound pressure at any node within the range is expressed as The form of the weight function is as follows: , The finite element form of the wave equation is obtained as follows:

[0025]

[0026]

[0027]

[0028] In the formula, and These are the mass matrix and the stiffness matrix, respectively; The total contribution of the sound source; Under the finite element theory, space The system is divided into a finite number of triangular elements, and the sound pressure level within each element depends only on the nodes of that element; let the nodes... , and The three vertices of one of the triangular units have the following coordinates: , and The sound pressure value at the node , and The sound pressure level within the unit can then be calculated using the following formula:

[0029] In the formula, For nodes sound pressure value right The sound pressure contribution at the point; The sound pressure value at any location within the study area can be calculated using the finite element method described above. After interpolation, the sound pressure field data under a standard grid can be obtained.

[0030] According to a second aspect of the present invention, a storage medium is provided having a computer program stored thereon, which, when executed by a processor, implements the time-varying finite element modeling method for extremely low frequency sound fields under the internal wave dynamic process described in the first aspect above.

[0031] According to a third aspect of the present invention, a computer program product is provided, on which a computer program is stored, wherein when the computer program is executed by a processor, the time-varying finite element modeling method for extremely low frequency sound fields under the internal wave dynamic process described in the first aspect is implemented.

[0032] According to a fourth aspect of the present invention, an electronic device is provided, comprising: Processor; and Memory for storing the executable instructions of the processor; The processor is configured to implement the time-varying finite element modeling method for the extremely low frequency sound field under the internal wave dynamic process described in the first aspect above by executing the executable instructions.

[0033] The time-varying finite element modeling method for extremely low frequency sound fields under internal wave dynamic processes provided by the embodiments of the present invention has the following advantages compared with the prior art: 1. Sound velocity field modeling is more in line with actual ocean scenarios: In view of the shortcomings of existing technologies, such as large deviation between the two-layer ocean model and the actual sound velocity field, and difficulty in achieving dynamic and refined simulation of the GM internal wave spectrum, this invention is based on a first-order positron KdV equation and Taylor-Goldstein modal equation, and simultaneously covers three typical internal wave types in the South China Sea: single wave of the first mode, wave train of the first mode, and internal solitary wave of the second mode. It accurately simulates the vertical displacement field in the dynamic propagation process of internal waves. The dynamic sound velocity field constructed by combining the background sound velocity field can truly reflect the spatiotemporal disturbance characteristics of internal waves (especially strongly nonlinear internal solitary waves) on the sound velocity of seawater, and solves the problem of missing modeling of specific internal wave types.

[0034] 2. Significantly improved accuracy in solving extremely low frequency sound fields: Addressing the shortcomings of mainstream sound field models (ray acoustics, normal wave, parabolic equation models) in accurately depicting the propagation of extremely low frequency sound in complex terrain, dynamic sound velocity fields, and multi-layered seabed environments, this invention employs the finite element method. By constructing a perfectly matched PML layer, dividing into fine meshes, and establishing a seawater-seabed current-structure coupling model, it fully considers the long wavelength (hundreds to thousands of meters) and strong medium penetration characteristics of extremely low frequency sound waves. It effectively captures the comprehensive influence of seabed layering, topographic slope changes, and internal wave dynamic processes on the sound field, achieving accurate time-varying solutions for extremely low frequency sound fields.

[0035] 3. The modeling method boasts strong adaptability and practicality: Focusing on key sea areas with high internal wave incidence and complex topography, such as the northern continental shelf of the South China Sea, it is compatible with dynamic internal wave scenarios of different modes and propagation stages. Through a complete process of "internal wave simulation - sound velocity field construction - sound field solution," it can fully reproduce the sound field changes during the propagation and evolution of internal waves. This method provides reliable technical support for the detection of extremely low frequency radiated noise from underwater targets, the performance optimization of sonar equipment, and the assessment of the marine acoustic environment, meeting the high-precision requirements of modern underwater detection for sound field modeling in complex environments.

[0036] 4. Possessing both theoretical and engineering application value: At the theoretical level, it improves the theoretical system of coupled modeling of internal wave dynamic processes and extremely low frequency sound fields, filling the gap in the modeling of sound velocity fields of multimodal internal isolated waves (especially wave trains and second modes) and the application of finite element method in this scenario; at the engineering application level, its simulation results can directly provide data support for sonar operating frequency adaptation, detection range planning and anti-interference strategy formulation, which is of great practical significance for improving underwater detection capabilities and ensuring marine security.

[0037] It should be understood that the above general description and the following detailed description are exemplary and explanatory only, and are not intended to limit the invention. Attached Figure Description

[0038] The accompanying drawings, which are incorporated in and constitute a part of this specification, illustrate embodiments consistent with the invention and, together with the description, serve to explain the principles of the invention. It is obvious that the drawings described below are merely some embodiments of the invention, and those skilled in the art can obtain other drawings based on these drawings without any inventive effort.

[0039] Figure 1 Flowchart for implementing the time-varying finite element modeling method for extremely low frequency sound fields under internal wave dynamic processes; Figure 2 (a) Initial field sound velocity profile, topographic profile and (b) seabed topographic map of the study area; Figure 3 The initial waveforms are the three types of internal isolated waves; Figure 4 Time history diagrams of isovelocity changes at a depth of 100m during the propagation of three types of internal solitary waves: (a) a single internal solitary wave in the first mode, (b) a single internal solitary wave in the second mode, and (c) a time history diagram of isovelocity changes at a depth of 100m during the propagation of the internal solitary wave train in the first mode. Figure 5 This is a schematic diagram of the seawater-seabed stratification. Figure 6 The geometric division of the study domain is shown for the propagation of three types of internal solitary waves to a distance of 10 km from the sound source; (a) a single internal solitary wave of the first mode, (b) a single internal solitary wave of the second mode, and (c) a geometric division of the study domain for the propagation of the internal solitary wave train of the first mode to a distance of 10 km from the sound source. Figure 7 The sound velocity profiles and pseudo-color images of the sound field when the isolated wave propagates to a distance of 15km, 10km and 5km from the sound source in the first mode are shown. Figure 8 The sound velocity profiles and pseudo-color images of the sound field when the isolated wave in the second mode propagates to a distance of 15km, 10km and 5km from the sound source. Figure 9 The sound velocity profiles and pseudo-color images of the sound field when the isolated wave train in the first mode propagates to a distance of 15km, 10km and 5km from the sound source are shown. Detailed Implementation

[0040] Exemplary embodiments will now be described more fully with reference to the accompanying drawings. However, these exemplary embodiments can be implemented in many forms and should not be construed as limited to the examples set forth herein; rather, they are provided so that the invention will be more comprehensive and complete, and will fully convey the concept of the exemplary embodiments to those skilled in the art. The described features, structures, or characteristics may be combined in any suitable manner in one or more embodiments.

[0041] Furthermore, the accompanying drawings are merely illustrative of the invention and are not necessarily drawn to scale. The same reference numerals in the drawings denote the same or similar parts, and therefore repeated descriptions of them will be omitted. Some block diagrams shown in the drawings are functional entities and do not necessarily correspond to physically or logically independent entities. These functional entities can be implemented in software, in one or more hardware modules or integrated circuits, or in different network and / or processor devices and / or microcontroller devices.

[0042] To address the shortcomings and deficiencies of existing technologies, this example implementation provides a time-varying finite element modeling method for extremely low frequency sound fields under internal wave dynamic processes. Based on the initial background field information of the research sea area and the internal wave propagation and modal control equations, the dynamic sound velocity field is simulated. On this basis, the finite element method is combined to model the extremely low frequency sound field under complex seabed topography, thereby realizing the solution of the time-varying extremely low frequency sound field under internal wave dynamic processes.

[0043] refer to Figure 1 As shown, the specific steps may include: Step 1: Simulation of the dynamic propagation process of internal waves Internal waves, during their propagation, disturb the seawater stratification, causing significant vertical displacements (amplitudes up to hundreds of meters) of isodense surfaces. This results in drastic changes in the sound velocity of the seawater in the vertical direction, altering the sound propagation path. Particularly for internal solitary waves with large amplitudes and strong nonlinearity, the resulting changes in the vertical sound velocity field are even more pronounced. Because internal waves exhibit strong structural characteristics in the vertical direction but weak variability in the horizontal direction, the sound velocity field and sound field of internal waves are modeled on a two-dimensional vertical profile (i.e., the XZ coordinate system).

[0044] The simulation of the internal wave dynamic propagation process consists of two steps: First, the evolution of the internal wave vertical displacement (waveform) at the mezzanine depth with time and horizontal distance is solved using a one-dimensional corrected Korteweg de Vries (KdV) equation. Second, the vertical mode control equations of the internal wave are solved to obtain the vertical mode functions of the internal wave, and the internal wave vertical displacement at the mezzanine depth is extended to all depth layers, thus obtaining the internal wave vertical displacement field under continuously stratified seawater. The specific solution process is as follows: Step 1: Solving the Internal Wave One-Well Corrected KdV Equation The Kortweg-de Vries (KdV) equation is a governing equation derived under the assumption of shallow water and small amplitude to describe the propagation of one-dimensional waves. This equation considers the background field density and the continuous stratification of seawater and is widely used to explain internal solitary wave disturbances occurring near oceanic strata. However, due to the dramatic topographic variations in the continental shelf areas where internal waves are concentrated, internal waves exhibit strong spatial variability. Therefore, it is necessary to consider factors such as the oceanic medium and horizontal topographic variations, and the original KdV equation is modified as follows: (1) In the formula, This represents the vertical displacement (waveform) of the internal wave at the mezzanine. and Representing the horizontal coordinate and time respectively; The linear wave velocity of the internal wave; and These are the nonlinear coefficients and dispersion coefficients of the internal wave, respectively. This indicates a correction term for external forcing (such as changes in topography or seawater medium).

[0045] Correspondingly, the coefficients of the nonlinear terms Dispersion term coefficient and corrections The expressions are as follows: (2) (3) (4) In the formula, and These represent the local seawater depth and the coordinates in the depth direction (upward is the positive direction); and These represent the linear wave velocity of the internal wave and the background horizontal velocity, respectively, in the initial state. and These represent the linear wave velocity and background horizontal velocity at different times (or at different distances) during the propagation of the internal wave, respectively. and These represent the internal wave vertical mode functions in the initial state and the subsequent propagation process (obtained in the second step).

[0046] The propagation process of an internal wave can be viewed as multiple discrete "time snapshots". The solution to equation (1) at a certain moment (snapshot) is called the steady-state solution of the internal solitary wave. For a single internal solitary wave, its solution takes the following form: (5) (6) In the formula, This represents the amplitude of the internal solitary wave at the initial moment (negative value for concave internal solitary waves, positive value for convex internal solitary waves). It is a hyperbolic secant function; The nonlinear wave velocity (i.e., propagation speed) of the internal wave. The characteristic half-wave width of the internal wave.

[0047] For an internally isolated wave train, its solution (the Dnoidal wave solution of the KdV equation) takes the following form: (7) In the formula, This represents the amplitude of the initial wave / leader soliton internal wave in the wave train at the initial moment; Wave number; The modulus is Jacobian elliptic function; The nonlinear parameter can be determined by the following formula: (8) in, The spatiotemporal ratio coefficient; and These are the first and second kind of complete elliptic integral functions, respectively. (where is the elliptic modulus), and its expressions are as follows: (9) (10) Correspondingly, the nonlinear wave velocity (propagation speed) of the internal isolated wave train in equation (7) for: (11) Step 2: Solving the governing equations for the internal wave vertical mode In the first step, equations (5) and (7) respectively give the wave solution forms of the internal solitary wave single wave and the internal solitary wave train during their propagation and evolution. It describes the waveform (vertical displacement) at the jump layer during the propagation and evolution of an internal isolated wave single wave / wave train. With horizontal distance and time The changes in the internal solitary wave are significant. As a marine phenomenon with strong vertical structure, the influence range covers the entire water layer, therefore it is necessary to extend the wave solution at the interlayer depth to the entire water layer.

[0048] Under continuous stratified seawater conditions, the linear wave velocity of internal waves and vertical mode functions This can be obtained by solving the following Taylor-Goldstein (TG) equation: (12) (13) (14) in, Indicates the buoyancy frequency (Brunt-Väisälä frequency); This represents the acceleration due to gravity. Equations (13) and (14) represent the rigid boundary condition and the free boundary condition of the sea surface, respectively.

[0049] The aforementioned TG equation is essentially a homogeneous eigenvalue equation, which can be solved using the central difference method. It is assumed that the ocean is vertically divided into... The portion includes the sea surface and the seabed. Layers, with the thickness between each layer being ( Then, the TG equation under the central difference scheme can be transformed into a generalized eigenvalue problem of the following form: (15) in: (16) (17) (18) (19) (20) In the formula, , , , , , ( ), , , , , ( ).

[0050] By solving the homogeneous eigenvalue equations above, a set of eigenvalues ​​and their corresponding eigenvectors can be obtained. Arranging these eigenvalues ​​in descending order of magnitude, the largest eigenvalue and its corresponding eigenvector represent the linear wave velocity of the internal wave in the first mode. With vertical mode function The second largest eigenvalue and its corresponding eigenvector are the linear wave velocity of the internal wave in the second mode. With vertical mode function .

[0051] Step 2: Construction of the sound velocity field under the internal wave dynamic process For isolated waves in the first mode and the second mode, the linear wave velocity obtained in the second step of step 1 will be... Substitute into equation (5) and compare with the obtained vertical mode function Multiplying these together, we can obtain the vertical displacement field of the waves within the entire water layer as follows: (twenty one) In the formula, These correspond to isolated waves in the first and second modes, respectively.

[0052] Similarly, the linear wave velocity of the inner wave in the first mode... Substitute into equation (7) and compare with the vertical mode function Then, the vertical displacement field of the entire water layer for the isolated wave train in the first mode can be obtained: (twenty two) Furthermore, the background sound velocity field (the sound velocity field at the initial moment) is utilized. Displacement field of the entire water layer with internal waves The sound velocity field under the internal wave dynamic process can be constructed: (twenty three) In the formula, This indicates interpolation along the depth dimension.

[0053] Step 3: Finite element solution of the ultra-low frequency sound field Based on the sound velocity field under the internal wave dynamic process obtained in step 2, the seawater sound velocity profile at each time point is geometrically divided according to the isovelocity layer (the medium parameters in the isovelocity layer are completely the same), and this is used as the medium for sound propagation.

[0054] Because extremely low frequency (ULF) sound waves have long wavelengths (typically hundreds to thousands of meters) and strong penetrating power, the influence of the seabed medium on ULF sound propagation cannot be ignored. Studies have shown that sedimentary layers exist between water bodies and bedrock, especially in continental shelf areas where internal waves are frequent. The seabed consists of multiple layers of sediment with varying degrees of consolidation, and the excitation of shear waves within these layers leads to a rapid increase in propagation loss. Considering both signal and environmental factors, the propagation characteristics of ULF sound differ significantly from those of conventional frequency bands. Mainstream sound field calculation models cannot adequately address these issues; therefore, the finite element method is used to solve for the ULF sound field, as detailed below: The core idea of ​​the finite element method is to transform the wave equation (a higher-order partial differential equation) with boundary conditions into a weakly expressed integral equation, and then solve the equation using numerical discretization. It is assumed that the sound field is located in space. The wave equation then takes the following form: (twenty four) In the formula, , and These represent the sound pressure, sound velocity, and density of seawater, respectively. and These represent the external force on the volume and the mass acceleration injected into the finite space, respectively. For Hamiltonian operators.

[0055] Accordingly, the boundary conditions are as follows: (25) In the formula, and These represent the sound pressure level and normal acceleration at the boundary, respectively. This represents the gradient of sound pressure along the normal direction.

[0056] According to the weighted residual method, the weight function Multiplying by equations (24) and (25) respectively, and summing after integral transformation, we obtain the weak form of the wave equation as follows: (26) Furthermore, the above equation needs to be discretized. Hypothesis space Divided into a finite number of discrete units and having If there are nodes, then the space The sound pressure at any node within the range can be expressed as The form of the weight function is as follows: ( Substituting this into equation (26), we can obtain the finite element form of the wave equation as follows: (27) (28) (29) (30) In the formula, and These are the mass matrix and the stiffness matrix, respectively; The total contribution of the sound source.

[0057] Under the finite element theory, space The system is divided into a finite number of triangular elements, and the sound pressure level within each element depends only on the nodes of that element. Let the nodes... , and The three vertices of one of the triangular units have the following coordinates: , and The sound pressure value at the node , and The sound pressure level within the unit can then be calculated using the following formula: (31) In the formula, For nodes sound pressure value right The sound pressure contribution at the point.

[0058] The sound pressure value at any location within the study area can be calculated using the finite element method described above. After interpolation, the sound pressure field data under a standard grid can be obtained.

[0059] The following will describe in more detail each step of the phased array radar design method in this exemplary embodiment, with reference to the accompanying drawings and embodiments.

[0060] Example 1 Background environmental information is as follows: Figure 2 As shown in b, the study area is the sea area northeast of Dongsha Island in the northern South China Sea (this sea area is prone to internal waves). Figure 2 The location marked by the red line in b is the survey line (the direction of the survey line is the average propagation direction of isolated waves in this sea area), with a survey length of approximately 20 km. The depth decreases from ~620 m to ~270 m from east to west, which is a typical continental slope topography. Figure 2 a). The sound speed profile in the initial state is as follows: Figure 2 As shown by the green curve in section a, the distribution is basically negative gradient, with a mixed layer present at depths shallower than 15 m. Internal waves include three common types found in actual oceans: single isolated waves of the first mode, single isolated waves of the second mode, and wave trains of the first mode. The initial amplitude is set to 100 m. Furthermore, the seafloor topography data in this example comes from the SRTM15_PLUS global high-resolution topography and depth dataset, the background hydrological data (temperature, salinity, and current profiles) comes from the HYCOM mixed coordinate ocean model dataset, and the sound velocity is calculated using the Chen-Millero-L formula.

[0061] The specific implementation process of the time-varying finite element modeling method for the ultra-low frequency sound field under the internal wave dynamic process is as follows: Step 1: Simulation of the dynamic propagation process of internal waves Internal waves, during their propagation, disturb the seawater stratification, causing significant vertical displacements (amplitudes up to hundreds of meters) of isodense surfaces. This results in drastic changes in the sound velocity of the seawater in the vertical direction, altering the sound propagation path. Particularly for internal solitary waves with large amplitudes and strong nonlinearity, the resulting changes in the vertical sound velocity field are even more pronounced. Because internal waves exhibit strong structural characteristics in the vertical direction but weak variability in the horizontal direction, the sound velocity field and sound field of internal waves are modeled on a two-dimensional vertical profile (i.e., the XZ coordinate system).

[0062] The simulation of the internal wave dynamic propagation process consists of two steps: First, the evolution of the internal wave vertical displacement (waveform) at the mezzanine depth with time and horizontal distance is solved using a one-dimensional corrected Korteweg de Vries (KdV) equation. Second, the vertical mode control equations of the internal wave are solved to obtain the vertical mode functions of the internal wave, and the internal wave vertical displacement at the mezzanine depth is extended to all depth layers, thus obtaining the internal wave vertical displacement field under continuously stratified seawater. The specific solution process is as follows: Step 1: Solving the Internal Wave One-Well Corrected KdV Equation The Kortweg-de Vries (KdV) equation is a governing equation derived under the assumption of shallow water and small amplitude to describe the propagation of one-dimensional waves. This equation considers the background field density and the continuous stratification of seawater and is widely used to explain internal solitary wave disturbances occurring near oceanic strata. However, due to the dramatic topographic variations in the continental shelf areas where internal waves are concentrated, internal waves exhibit strong spatial variability. Therefore, it is necessary to consider factors such as the oceanic medium and horizontal topographic variations, and the original KdV equation is modified as follows: (32) In the formula, This represents the vertical displacement (waveform) of the internal wave at the mezzanine. and Representing the horizontal coordinate and time respectively; The linear wave velocity of the internal wave; and These are the nonlinear coefficients and dispersion coefficients of the internal wave, respectively. This indicates a correction term for external forcing (such as changes in topography or seawater medium).

[0063] Correspondingly, the coefficients of the nonlinear terms Dispersion term coefficient and corrections The expressions are as follows: (33) (34) (35) In the formula, and These represent the local seawater depth and the coordinates in the depth direction (upward is the positive direction); and These represent the linear wave velocity of the internal wave and the background horizontal velocity, respectively, in the initial state. and These represent the linear wave velocity and background horizontal velocity at different times (or at different distances) during the propagation of the internal wave, respectively. and These represent the internal wave vertical mode functions in the initial state and the subsequent propagation process (obtained in the second step).

[0064] The propagation process of an internal wave can be viewed as multiple discrete "time snapshots". The solution of equation (32) at a certain moment (snapshot) is called the steady-state solution of the internal solitary wave. For a single internal solitary wave, its solution takes the following form: (36) (37) In the formula, This represents the amplitude of the internal solitary wave at the initial moment (negative value for concave internal solitary waves, positive value for convex internal solitary waves). It is a hyperbolic secant function; The nonlinear wave velocity (i.e., propagation speed) of the internal wave. The characteristic half-wave width of the internal wave.

[0065] For an internally isolated wave train, its solution (the Dnoidal wave solution of the KdV equation) takes the following form: (38) In the formula This represents the amplitude of the initial wave / leader soliton internal wave in the wave train at the initial moment; Wave number; The modulus is Jacobian elliptic function; The nonlinear parameter can be determined by the following formula: (39) in The spatiotemporal ratio coefficient; and These are the first and second kind of complete elliptic integral functions, respectively. (where is the elliptic modulus), and its expressions are as follows: (40) (41) Correspondingly, the nonlinear wave velocity (propagation speed) of the internal isolated wave train in equation (38) for: (42) Step 2: Solving the governing equations for the internal wave vertical mode In the first step, equations (36) and (38) respectively give the wave solution forms of the internal solitary wave single wave and the internal solitary wave train in their propagation and evolution process. It describes the waveform (vertical displacement) at the jump layer during the propagation and evolution of an internal isolated wave single wave / wave train. With horizontal distance and time The changes in the internal solitary wave are significant. As a marine phenomenon with strong vertical structure, the influence range covers the entire water layer, therefore it is necessary to extend the wave solution at the interlayer depth to the entire water layer.

[0066] Under continuous stratified seawater conditions, the linear wave velocity of internal waves and vertical mode functions This can be obtained by solving the following Taylor-Goldstein (TG) equation: (43) (44) (45) in Indicates the buoyancy frequency (Brunt-Väisälä frequency); This represents the acceleration due to gravity. Equations (44) and (45) represent the rigid boundary condition and the free boundary condition of the sea surface, respectively.

[0067] The aforementioned TG equation is essentially a homogeneous eigenvalue equation, which can be solved using the central difference method. It is assumed that the ocean is vertically divided into... The portion includes the sea surface and the seabed. Layers, with the thickness between each layer being ( Then, the TG equation under the central difference scheme can be transformed into a generalized eigenvalue problem of the following form: (46) in: (47) (48) (49) (50) (51) In the formula, , , , , , ( ), , , , , ( ).

[0068] By solving the homogeneous eigenvalue equations above, a set of eigenvalues ​​and their corresponding eigenvectors can be obtained. Arranging these eigenvalues ​​in descending order of magnitude, the largest eigenvalue and its corresponding eigenvector represent the linear wave velocity of the internal wave in the first mode. With vertical mode function The second largest eigenvalue and its corresponding eigenvector are the linear wave velocity of the internal wave in the second mode. With vertical mode function .

[0069] like Figure 3 The figure shows the waveforms of three types of internal solitary waves at the initial moment (all amplitudes are...). The first mode of internal solitary waves exhibits a symmetrical "bell-shaped" waveform. The second mode's internal solitary waves show "convex" and "concave" waveforms at shallow (~60 m) and deep (~400 m) depths, respectively. The first mode's internal solitary wave train contains three solitons. The leading soliton's wave (head wave) has the largest amplitude and shortest wavelength (corresponding to a larger nonlinear coefficient), while subsequent wavelets show gradually decreasing amplitudes and increasing wavelengths (corresponding to decreasing nonlinear coefficients). It can be seen that the three types of internal solitary waves differ significantly in both the horizontal and vertical directions, thus affecting the sound field to varying degrees.

[0070] Figure 4 Three types of internal isolated wave edges are further given. Figure 2 The complete time-varying process of the internal wave waveform (vertical displacement) at a depth of 100 m during the "climbing" of the continental slope in Figure a is shown. It can be seen that due to the shallowing effect, the amplitude and wavelength of the isolated wave and wave train in the first mode gradually decrease during their propagation. Figure 4 (a and 4c); while in the second mode, the waveform of the isolated wave at a depth of 100 m gradually changes from "convex upward" to "concave downward" during propagation.

[0071] Step 2: Construction of the sound velocity field under the internal wave dynamic process For the isolated wave in the first mode and the second mode, substituting the linear wave velocity and vertical mode obtained in the second step of step 1 into equation (36), the vertical displacement field of the wave in the entire water layer can be obtained as follows: (52) In the formula These correspond to solitary waves within the first and second modes, respectively.

[0072] Similarly, the linear wave velocity of the inner wave in the first mode... With vertical mode function Substituting into equation (38), we can obtain the vertical displacement field of the entire water layer for the isolated wave train in the first mode: (53) Furthermore, the background sound velocity field (the sound velocity field at the initial moment) is utilized. Displacement field of the entire water layer with internal waves The sound velocity field under the internal wave dynamic process can be constructed: (54) In the formula This indicates interpolation along the depth dimension.

[0073] Figure 7 ac、 Figure 8 ac and Figure 9 The ac data presents the sound velocity field distribution at different times during the propagation of the three types of internal solitary waves. It can be seen that the perturbation range of the sound velocity field by various types of internal waves during their propagation exhibits spatiotemporal variability. The above method can provide a relatively accurate description of the complete dynamic process of internal waves.

[0074] Step 3: Finite element solution of the ultra-low frequency sound field Because extremely low frequency (ULF) sound waves have long wavelengths (typically hundreds to thousands of meters) and strong penetrating power, the influence of the seabed medium on ULF sound propagation cannot be ignored. Studies have shown that sedimentary layers exist between water bodies and bedrock, especially in continental shelf areas where internal waves are frequent. The seabed consists of multiple layers of sediment with varying degrees of consolidation, and the excitation of shear waves within these layers leads to a rapid increase in propagation loss. Considering both signal and environmental factors, the propagation characteristics of ULF sound differ significantly from those of conventional frequency bands. Mainstream sound field calculation models cannot adequately address these issues; therefore, the finite element method is used to solve for the ULF sound field, as detailed below: The core idea of ​​the finite element method is to transform the wave equation (a higher-order partial differential equation) with boundary conditions into a weakly expressed integral equation, and then solve the equation using numerical discretization. It is assumed that the sound field is located in space. The wave equation then takes the following form: (55) In the formula , and These represent the sound pressure, sound velocity, and density of seawater, respectively. and These represent the external force on the volume and the mass acceleration injected into the finite space, respectively. For Hamiltonian operators.

[0075] Accordingly, the boundary conditions are as follows: (56) In the formula and These represent the sound pressure level and normal acceleration at the boundary, respectively. This represents the gradient of sound pressure along the normal direction.

[0076] According to the weighted residual method, the weight function Multiplying these equations by (55) and (56) respectively, and then summing them after integral transformation, we obtain the weak form of the wave equation as follows: (57) Furthermore, the above equation needs to be discretized. Hypothesis space Divided into a finite number of discrete units and having If there are nodes, then the space The sound pressure at any node within the range can be expressed as The form of the weight function is as follows: ( Substituting this into equation (26), we can obtain the finite element form of the wave equation as follows: (58) (59) (60) (61) In the formula and These are the mass matrix and the stiffness matrix, respectively; The total contribution of the sound source.

[0077] Under the finite element theory, space The system is divided into a finite number of triangular elements, and the sound pressure level within each element depends only on the nodes of that element. Let the nodes... , and The three vertices of one of the triangular units have the following coordinates: , and The sound pressure value at the node , and The sound pressure level within the unit can then be calculated using the following formula: (62) In the formula For nodes sound pressure value right The sound pressure contribution at the point.

[0078] The finite element method described above can be used to calculate the sound pressure value at any location within the study area. After interpolation, sound pressure field data under a standard grid can be obtained. Based on experimental results in existing literature or materials, the seabed will be modeled as Figure 5 The five-layer seabed model shown, from shallowest to deepest, consists of clay layers (thickness...). ,density Longitudinal wave speed transverse wave speed of sound ), silty clay layer (thickness) ,density Longitudinal wave speed transverse wave speed of sound ), sandy silt layer (thickness) ,density Longitudinal wave speed transverse wave speed of sound ), semi-hard substrate (thickness) ,density Longitudinal wave speed transverse wave speed of sound ) and bedrock (thickness) ,density Longitudinal wave speed transverse wave speed of sound Furthermore, a fluid-solid coupling boundary is defined between the seawater layer and the clay layer. For example... Figure 6 The diagram shows the geometric division of the seawater layer and the seabed layer when three types of internal solitary waves propagate to the intermediate distance. The seawater layer is further divided into 50 layers based on the isovelocity lines of the internal wave sound velocity field.

[0079] Under the aforementioned layered model, the extremely low-frequency sound field during the propagation of three types of internal solitary waves was simulated using the finite element method (sound source depth 50 m, frequency 10 Hz). The results are as follows: Figure 7 df、 Figure 8 df and Figure 9As shown in df, it can be seen that due to the influence of the continental slope topography, sound propagation exhibits multiple reflections between the sea surface and the seabed. At the same time, the sound field energy distribution behind the inner wave crest exhibits spatiotemporal variability. However, since the wavelength of the sound wave at the 10 Hz frequency is relatively long (~150 m), which is comparable to the vertical scale (100 m) of the inner wave's disturbance to the sound velocity field, the scattering, refraction, and modal coupling effects of the sound wave are relatively weak.

[0080] This embodiment demonstrates that the method of the present invention can be used to simulate time-varying extremely low-frequency sound fields under complex marine environments and internal wave dynamic processes. The present invention provides corresponding dynamic sound velocity field modeling methods for three common types of internal solitary wave dynamic processes in the South China Sea, and establishes a time-varying extremely low-frequency sound field calculation model under multi-layered seabed and internal wave dynamic environments based on finite element theory. This model can accurately simulate and describe the distribution of extremely low-frequency sound fields under typical internal wave environments throughout the entire process.

[0081] Furthermore, the above figures are merely illustrative of the processes included in the method according to exemplary embodiments of the present invention, and are not intended to be limiting. It is readily understood that the processes shown in the above figures do not indicate or limit the temporal order of these processes. Additionally, it is readily understood that these processes may be executed synchronously or asynchronously, for example, in multiple modules.

[0082] Other embodiments of the invention will readily occur to those skilled in the art upon consideration of the specification and practice of the invention herein. This application is intended to cover any variations, uses, or adaptations of the invention that follow the general principles of the invention and include common knowledge or customary techniques in the art not disclosed herein. The specification and embodiments are to be considered exemplary only, and the true scope and spirit of the invention are indicated by the claims.

[0083] It should be understood that the present invention is not limited to the precise structure described above and shown in the accompanying drawings, and various modifications and changes can be made without departing from its scope. The scope of the invention is defined only by the appended claims.

Claims

1. A time-varying finite element modeling method for very low frequency acoustic field under internal wave dynamic process, characterized in that, The method includes: Step 1: Simulation of the internal wave dynamic propagation process, specifically including: Step 1.1: Using a properly corrected Korteweg-de Vries equation and in conjunction with background field information, solve the evolution of the vertical displacement of the internal wave at the depth of the stratum as a function of time and horizontal distance; Step 1.2: Solve the Taylor-Goldstein equation to obtain the vertical mode function and linear wave velocity of the internal wave. Then, extend the vertical displacement at the jump layer to the entire water layer to obtain the linear wave velocity and vertical mode function of the internal wave. Step 2: Substitute the internal wave linear velocity and vertical mode function obtained in Step 1.1 into the wave solution in Step 1.2 to obtain the vertical displacement field of the internal wave in the entire water layer. Combine the vertical displacement field of the internal wave in the entire water layer with the background sound velocity field and interpolate along the depth dimension to construct the sound velocity field under the dynamic process of the internal wave. Step 3: Finite element solution of the extremely low frequency sound field. In a multi-layer fluid-structure interaction model containing a seawater layer and at least one layer of seabed medium, the seawater layer is geometrically divided according to the isovelocity layer of the internal wave sound velocity field. Then, the wave equation is discretized and solved using the finite element method to obtain the sound pressure distribution of the extremely low frequency sound field in the study area under the internal wave dynamic process.

2. The method according to claim 1, characterized in that, The repaired positive Korteweg-de Vries equation is specifically as follows: In the formula, This represents the vertical displacement of the internal wave at the mezzanine. and Representing the horizontal coordinate and time respectively; The linear wave velocity of the internal wave; and These are the nonlinear coefficients and dispersion coefficients of the internal wave, respectively; This indicates an externally forced modification term.

3. The method according to claim 2, characterized in that, The Taylor-Goldstein equation is as follows: in, Indicates the buoyancy frequency. Represents gravitational acceleration. and Let these represent the linear wave velocity and the background horizontal velocity at different times or distances during the propagation of the internal wave, respectively. This represents the vertical mode function of the internal wave during propagation. The coordinates represent the depth direction. Equations 2 and 3 represent the rigid boundary condition and the free boundary condition of the sea surface, respectively.

4. The method according to claim 3, characterized in that, The internal waves include a single isolated wave in the first mode, a single isolated wave in the second mode, or a wave train of isolated waves in the first mode; for a single isolated wave, the solution of the modified KdV equation is in the form of a hyperbolic secant function; for a wave train of isolated waves, the solution of the modified KdV equation is in the form of a Dnoidal wave solution.

5. The method according to claim 4, characterized in that, The construction of the sound velocity field under the internal wave dynamic process is as follows: For isolated waves in the first and second modes, the vertical displacement field of the waves within the entire water layer is as follows: In the formula, Corresponding to solitary waves within the first and second modes, respectively. Indicates the amplitude of the internal solitary wave; For the isolated wave train in the first mode, the vertical displacement field of the wave within the entire water layer is as follows: Using background sound velocity field Displacement field of the entire water layer with internal waves Constructing the sound velocity field under the internal wave dynamic process: In the formula, This indicates interpolation along the depth dimension.

6. The method according to claim 5, characterized in that, The method of using the finite element method to discretize and solve the wave equation is as follows: In the formula, , and These represent the sound pressure, sound velocity, and density of seawater, respectively. and These represent the external force on the volume and the mass acceleration injected into the finite space, respectively. For Hamiltonian operators; Accordingly, the boundary conditions are as follows: In the formula, and These represent the sound pressure level and normal acceleration at the boundary, respectively. This represents the gradient of sound pressure along the normal direction; According to the weighted residual method, the weight function Multiplying each of the two equations above and summing them after integral transformation, we obtain the weak form of the wave equation as follows: Discretize the above equation and assume the space Divided into a finite number of discrete units and having If there are nodes, then the space The sound pressure at any node within the range is expressed as The form of the weight function is as follows: , The finite element form of the wave equation is obtained as follows: In the formula, and These are the mass matrix and the stiffness matrix, respectively; The total contribution of the sound source; Under the finite element theory, space The system is divided into a finite number of triangular elements, and the sound pressure level within each element depends only on the nodes of that element; let the nodes... , and The three vertices of one of the triangular units have the following coordinates: , and The sound pressure value at the node , and The sound pressure level within the unit can then be calculated using the following formula: In the formula, For nodes sound pressure value right The sound pressure contribution at the point; The sound pressure value at any location within the study area can be calculated using the finite element method described above. After interpolation, the sound pressure field data under a standard grid can be obtained.

7. A storage medium storing a computer program thereon, wherein the computer program, when executed by a processor, implements the time-varying finite element modeling method for extremely low frequency sound fields under internal wave dynamic processes as described in claims 1-6.

8. A computer program product having a computer program stored thereon, wherein the computer program, when executed by a processor, implements the time-varying finite element modeling method for extremely low frequency sound fields under internal wave dynamic processes as described in claims 1-6.

9. An electronic device, comprising: processor; as well as Memory for storing the executable instructions of the processor; The processor is configured to implement the time-varying finite element modeling method for the extremely low frequency sound field under the internal wave dynamic process as described in claims 1-6 by executing the executable instructions.

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