Numerical hybrid method for forward scattering acoustic field of underwater regular shaped objects

Abstract

The present application belongs to the technical field of underwater acoustic channel and physical field, and relates to a numerical hybrid calculation method of forward scattering acoustic field of a regular shape object underwater. The method comprises: obtaining an initial projection profile of a three-dimensional geometric model of the regular shape object under a specified projection plane and an incident direction; dividing discrete points of the initial projection profile into upper boundary points and lower boundary points; interpolating and completing the upper boundary points and the lower boundary points, so that the upper boundary points and the lower boundary points have the same number and share the same horizontal coordinates; discretizing the completed projection profile by using trapezoidal plane elements; calculating a forward scattering far field pattern function based on the discretization result; and calculating a forward scattering target strength of the object. The method provided by the present application can cover the shape range of the scattering object applicable to the analytical method, and does not need to calculate complex series expressions composed of special functions such as Bessel functions and spherical harmonic waves, thereby obviously improving the calculation efficiency.

Description

Technical Field

[0001] This invention belongs to the field of underwater acoustic channel and physical field technology, and relates to a numerical hybrid calculation method for the forward scattered sound field of underwater regular-shaped objects. Background Technology

[0002] Active sonar detects and identifies underwater objects by actively emitting sound wave signals and then receiving and detecting the scattered sound wave signals from all potential sound scattering sources, including underwater objects. The sound scattering characteristics of underwater objects are the physical basis of this detection mode. Establishing a scattered sound field model of underwater objects and exploring their scattering characteristics can provide important references for obtaining the sound scattering characteristics of underwater objects. Since the external shape, internal structure, and material properties vary greatly depending on the application scenario, the sound scattering characteristics of underwater objects also vary greatly. Based on the detection principle, active sonar can be divided into four basic operating modes: monostatic (transmitter and receiver combined or echo), bistatic (transmitter and receiver separated at small angles), multistatic (multi-transmitter and multi-receiver, single-transmitter and multi-receiver), and forward scattering.

[0003] Forward acoustic scattering from underwater objects is a special type of separate transmit and receive scattering. Its distinguishing feature from combined transmit and receive scattering and small-angle separate transmit and receive scattering is the transmit and receive separation angle (…). β Mostly obtuse angles (usually) (The scattered wave is difficult to separate due to interference and superposition with the directly transmitted wave), and there may even be a limit to this. (That is, the case of strict forward scattering, in which the forward-scattered wave and the directly transmitted wave completely interfere and superimpose and are masked by the directly transmitted wave). However, since forward scattering is one of the basic working modes of active sonar, the modeling and analysis of the forward scattering characteristics of underwater objects can be carried out within the same acoustic scattering modeling research framework, just like other detection methods.

[0004] As is well known, scattering problems involving 11 types of curved surfaces, including spheres, cylinders, elliptical cylinders, and paraboloids of revolution, have closed-form solutions. Scattering problems involving other curved surfaces can only be solved using numerical or approximate methods. Existing series methods for modeling acoustic scattering from underwater objects can be categorized into three main types: analytical, numerical, and approximate. Among these, the PSW (wave series) method and BIEM (boundary integral equation method) are analytical methods, while the T-matrix method, FDTD (finite-difference time-domain) method, FEM (finite element method), BEM (boundary element method), FEM-BEM (finite element-boundary element method), and WSM (wave superposition method) are numerical methods. The KA (Kirchhoff approximation) method is a typical approximate method. The applicability of analytical methods is limited by the surface shape of the scattering object; they are only applicable to 11 types of curved surfaces, such as spheres, infinitely long cylinders, and elliptical cylinders, and are not suitable for other regularly shaped or complex scattering objects. Furthermore, the convergence speed of analytical methods for sound field series solutions slows down under high-frequency conditions, and series divergence issues arise. Numerical methods offer high computational efficiency and accuracy at lower incident sound wave frequencies, but as the incident sound wave frequency increases, the computational efficiency and accuracy requirements for computational resources also significantly increase, hindering rapid engineering implementation. The KA method is more suitable for calculating the forward-scattered sound field of underwater objects with a normal transverse incident attitude, and is not applicable to other incident attitudes, thus significantly limiting its application scope. Summary of the Invention

[0005] To address the shortcomings of existing underwater object acoustic scattering modeling techniques in calculating forward-scattered sound fields, this invention proposes a Hybrid Kirchhoff approximation (HKA) method. This method integrates geometric projection contour acquisition with the forward scattering KA method, making it suitable for the rapid calculation of forward-scattered sound fields of regularly shaped objects.

[0006] The present invention provides a numerical mixing calculation method for the forward-scattered sound field of an underwater regularly shaped object, comprising the following steps:

[0007] (1) Obtain the initial projection profile of the three-dimensional geometric model of the regular-shaped object under a specified projection plane and incident direction, wherein the incident direction includes the incident zenith angle. and incident azimuth The initial projected contour is represented by a series of discrete points;

[0008] (2) Divide the discrete points of the initial projected contour into upper boundary points and lower boundary points;

[0009] (3) Perform interpolation to fill in the upper and lower boundary points so that the number of upper and lower boundary points is the same and they share the same horizontal coordinate;

[0010] (4) Use trapezoidal plane elements to discretize the completed projected contour, where each trapezoidal plane element is bounded by its adjacent upper boundary points and its corresponding lower boundary points;

[0011] (5) Based on the discretization result, according to the scattering zenith angle and scattering azimuth angle The value of ,

[0012] Calculate the far-field shape function of forward scattering ;

[0013] (6) Calculate the forward scattering target intensity of the object:

[0014] .

[0015] Furthermore, the three-dimensional geometric model is established in 3D CAD software based on a standard three-dimensional Cartesian coordinate system; the projection plane is... flat.

[0016] Furthermore, in step (2), the method for dividing the upper and lower boundary points is as follows:

[0017] Find the minimum value of the x-coordinate of the discrete point. and maximum value and its corresponding endpoint coordinates and ;

[0018] Establish the equations of the lines containing the two endpoints:

[0019] ;

[0020] For each discrete point Set its x-coordinate Substituting into the equation of the line, if its ordinate... If the value is greater than the calculated value of the line equation, it is classified as an upper boundary point; if it is less than the value, it is classified as a lower boundary point.

[0021] Furthermore, in step (3), interpolation includes:

[0022] For each point in the set of points on the upper boundary Find the x-coordinates that satisfy the following conditions in the set of lower boundary points. Two points of the relation and ,in Insert a new lower boundary point between these two lower boundary points, with its x-coordinate being... The vertical axis is:

[0023] ;

[0024] For each point in the lower boundary point set Find the x-coordinates in the set of upper boundary points that satisfy Two points of the relation and ,in Insert a new upper boundary point between these two upper boundary points, with its x-coordinate being... The vertical axis is:

[0025] ;

[0026] After interpolation, the sets of upper and lower boundary points are represented in matrix form. and ,in, This is a shared x-coordinate vector; and These represent the upper and lower boundaries, respectively. and These represent the number of upper boundary points and the number of lower boundary points, respectively.

[0027] Further, in step (4), the discretization of the trapezoidal plane element includes: for each adjacent x-coordinate interval Establish the equations of the upper and lower boundary lines:

[0028] ;

[0029] in,

[0030] .

[0031] Furthermore, in step (5), the calculation of the far-field morphology function of the forward-scattered sound field is divided into the following four cases:

[0032] when and When the morphological function is calculated, the formula is:

[0033] ;

[0034] in The imaginary unit, The cross-sectional area of ​​the projected profile. The wavelength of the incident wave;

[0035] when and When the morphological function is calculated, the formula is: ;

[0036] in, , ;

[0037] when and When the morphological function is calculated, the formula is:

[0038] ;

[0039] in:

[0040] ;

[0041] ;

[0042] when When the morphological function is calculated, the formula is:

[0043] ;

[0044] in:

[0045] ;

[0046] ;

[0047] Let be the wave number of the incident wave. For the sound source frequency, The velocity of sound in the water medium of the outer region.

[0048] Furthermore, after each change of the incident attitude angle, steps (1) to (6) are repeated to update the target intensity.

[0049] Furthermore, the regular-shaped object refers to an object whose shape encloses a regular simply connected region.

[0050] Compared with existing technologies, the advantages of this invention are as follows: The method provided by this invention can cover the range of scattering object shapes applicable to analytical methods, and does not require the calculation of special functions such as Bessel functions and spherical harmonics, or the complex series expressions formed by their combinations, thus significantly improving computational efficiency. Compared with existing numerical methods, the method of this invention does not require surface element subdivision of the scattering object or volume element subdivision of the scattering object, nor does it require meshing of the external solution domain. Instead, it only requires discretization of the projected profile based on trapezoidal plane elements, greatly saving modeling degrees of freedom, making it suitable for rapid calculation under high-frequency conditions. Compared with existing forward scattering KA methods, the significant advantage of this invention is that it obtains the object's projected profile at any incident angle through geometric projection profile-assisted acquisition, requiring only the calculation of the forward scattered sound field under normal transverse incidence conditions for different projected profiles. Especially when calculating the forward scattered sound field under oblique incidence (or illumination) conditions, it does not require secondary projection of the normal transverse projection profile as in existing KA methods, simplifying the problem-solving process and improving computational efficiency. Attached Figure Description

[0051] Figure 1 This is a flowchart of the numerical mixing calculation method for the forward-scattered sound field of an underwater regularly shaped object in an embodiment of the present invention;

[0052] Figure 2 A schematic diagram of the trapezoidal plane element discretization of the scatterer's projected profile;

[0053] Figure 3 The model is a three-dimensional geometric model of a flattened rotating ellipsoid (semi-major axis 38.0 m, semi-minor axis 4.0 m).

[0054] Figure 4 The projection section of the flattened ellipsoid after boundary point interpolation and completion; where (a) is the incident attitude angle. ψ =90°; (b) Incident attitude angle ψ = 45°;

[0055] Figure 5 Target intensity ‒ scattering azimuth angle ( ) curve; where, (a) incident attitude angle ψ = 90°, (b) incident attitude angle ψ = 45°;

[0056] Figure 6 Target intensity ‒ Scattering zenith angle ( ) curve; where, (a) incident attitude angle ψ = 90°, (b) incident attitude angle ψ = 45°;

[0057] Figure 7A three-dimensional geometric model of a finite-length stainless steel cylinder (length 0.74 m, diameter 0.22 m).

[0058] Figure 8 Example of the projected profile of a finite-length cylinder under measurement condition 1; where, (a) incident attitude angle ψ =0°; (b) Incident attitude angle ψ = 60°; (c) Incident attitude angle ψ = 90°; (d) Incident attitude angle ψ = 150°;

[0059] Figure 9 For measuring working condition 1 TS - ψ Curves; where (a) comparison of numerical calculation results; (b) comparison of numerical calculation results with experimental data;

[0060] Figure 10 Example of the projected profile of a finite-length cylinder under measurement condition 2; where, (a) incident attitude angle ψ =0°; (b) Incident attitude angle ψ = 60°; (c) Incident attitude angle ψ = 90°; (d) Incident attitude angle ψ = 150°;

[0061] Figure 11 For measuring working condition 2 TS - ψ The curves; where (a) is the strictly forward scattering direction and (b) is the large angular scattering direction. Detailed Implementation

[0062] To more clearly understand the technical content of this invention, its solution will now be described in detail with reference to the accompanying drawings and specific embodiments. The accompanying drawings show a preferred embodiment of this invention, but its scope of protection is not limited thereto, and can be flexibly adjusted according to specific usage requirements in practical applications. This embodiment aims to help those skilled in the art to fully and deeply understand the inventive concept of this solution.

[0063] The present invention provides a numerical mixing and calculation method (HKA) for the forward-scattered sound field of underwater regularly shaped objects, such as... Figure 1 As shown, the specific steps include:

[0064] (1) Obtaining the initial projected profile

[0065] Using the geometric projection function in the modeling software, set the projection plane and the incident direction (including the incident zenith angle). and incident azimuth ), obtained in The initial projected profile represented in the plane. This projected profile is actually a closed curve formed by connecting a series of discrete points in sequence. Assume there are a total of discrete points representing this closed curve. If there are a number of discrete points, then the coordinates of these discrete points can be conveniently represented by a matrix. ,in Let x be the x-coordinate of the discrete point. y is the ordinate of the discrete point.

[0066] (2) Division of the upper and lower boundaries of the outline

[0067] Obviously, the directly exported projection profile cannot be used directly to calculate the forward-scattered sound field; it needs to be preprocessed before it can be used for subsequent discretization. The preprocessing here consists of two steps: the first step is to divide the upper and lower boundaries of the projection profile, that is, to divide the points on the profile into upper boundary points and lower boundary nodes; the second step is to fill in the mutual reference between the upper and lower boundary points.

[0068] Iterate through the x-coordinates of the search points and find the minimum value. and maximum value And its position number in the horizontal axis sequence is denoted as and Obviously there is and Find the corresponding position by referring to the position number. and The corresponding ordinate value, and denoted as and After this operation, the coordinates of the two endpoints of the projected contour are obtained. and .

[0069] Establish the equations of the lines containing the two endpoints of the projected profile: (1).

[0070] For any other point on the projected contour Set its x-coordinate Substitute into equation (1) and its ordinate For comparison, if:

[0071] (2);

[0072] This point is then classified as the upper boundary point;

[0073] like:

[0074] (3);

[0075] Then this point is classified as the lower boundary point.

[0076] After traversing all boundary points, we obtain the sets of upper and lower boundary points, which are represented by matrices respectively. and ,in , , , , and These represent the upper and lower boundaries, respectively. and These represent the number of upper boundary points and the number of lower boundary points, respectively.

[0077] (3) Interpolation and completion of boundary points

[0078] In general, the number of upper boundary points Number of lower boundary points The boundary points are not identical, or even if the number is the same, the x-coordinates of the upper and lower boundary points are mostly different, thus failing to directly support the discretization of the trapezoidal plane elements of the projected contour. Therefore, it is necessary to fill in the boundary points through interpolation. The specific implementation method is as follows:

[0079] ① For a point on the upper boundary By traversing and searching the set of lower boundary points, we find the one that satisfies... The two lower boundary points of the relation and ,in Insert a new lower boundary point between these two lower boundary points, with its x-coordinate being... The vertical axis is:

[0080] (4).

[0081] ②For a point on the lower boundary By traversing and searching the set of upper boundary points, we find the one that satisfies... The two upper boundary points of the relation and ,in Insert a new upper boundary point between these two upper boundary points, with its x-coordinate being... The vertical axis is:

[0082] (5);

[0083] After interpolation, the sets of upper and lower boundary points are represented in matrix form. and ,in, For a shared x-coordinate vector, , , .

[0084] (4) Discretization of the projected profile

[0085] Based on the complete set of upper boundary points and lower boundary point set , refer to Figure 2 The method shown uses trapezoidal planar elements as basic units to discretize the projected contour. For any two adjacent upper boundary points... and and the two coordinate points of its corresponding lower boundary and Calculate the equations of the upper and lower boundary lines:

[0086] (6);

[0087] in,

[0088] .

[0089] In this way, the trapezoidal plane element enclosed by these four coordinate points can be completely described by its four boundary lines.

[0090] (5) Calculation of far-field morphology function of forward scattering

[0091] The scattered sound field formed by an arbitrary object in an infinite water space after being irradiated by an incident sound wave can be represented by its far-field morphology function. It means that among them For scattering zenith angle, The azimuth angle is the scattering angle. The method of this invention, considering the values ​​of the scattering zenith angle and the scattering azimuth angle, calculates the far-field morphology function of the forward-scattered sound field in the following four cases:

[0092] a. and :

[0093] (7);

[0094] in, The imaginary unit, The cross-sectional area of ​​the projected profile. The wavelength of the incident wave is 1. Let be the wave number of the incident wave. For the sound source frequency, The velocity of sound in the water medium of the outer region.

[0095] b. and : (8);

[0096] in, , .

[0097] c. and :

[0098] (9);

[0099] in:

[0100] ;

[0101] .

[0102] d. :

[0103] (10);

[0104] in:

[0105] ;

[0106] .

[0107] (6) Calculation of the intensity of forward-scattering targets

[0108] Target strength is commonly used in engineering practice. TS To measure the ability of an underwater object to scatter incident sound waves, the forward scattering far-field shape function obtained in step (5) can be used to directly calculate the forward scattering target intensity of the object, i.e.:

[0109] (11).

[0110] Each time the incident attitude angle is changed, repeat steps (1)-(6) to calculate the corresponding forward scattering far-field morphology function or target intensity.

[0111] To demonstrate the application effects of the HKA method of this invention in more detail, two typical implementation examples are given here. In the first embodiment, the scatterer is an oblate spheroid of rotation. This scatterer model is often used as a regularized approximation model for complex underwater objects, and its rapid calculation and prediction of the forward-scattered sound field has certain application value. The forward scattering of the oblate spheroid of rotation is calculated under a fixed incident attitude angle. and As a result, the feasibility of the HKA method is demonstrated by comparing the results with the BIEM reference solution. The second embodiment relates to a scattering measurement experiment in an anechoic pool. The main task of the experiment was to measure the forward scattering target intensity of a finite-length stainless steel cylinder. For two measurement conditions in the experiment, the forward scattering of the finite-length cylindrical object was calculated. TS At the incident attitude angle ψ The results of the changes were compared with the experimental data analysis results to confirm the practicality of the HKA method of the present invention.

[0112] Example 1: This example uses an oblate spheroid of rotation as an example to further illustrate the method of the present invention. The specific steps are as follows:

[0113] (1) Three-dimensional geometric model of the flattened rotating ellipsoid as follows Figure 3 As shown, its rotational symmetry axis is located at x The axis has a semi-major axis length of 38.0 m and a semi-minor axis length of 4.0 m. Under transverse irradiation ( ψ = 90°) and 45° from the direct cross ( ψ = 45°) Under two working conditions, the initial projection contour is obtained, the upper and lower boundaries are divided, and the boundary points are interpolated and filled in sequentially according to steps (1)-(3) to obtain Figure 4 The projected profile section shown can be used for subsequent discretization. Due to the symmetry of the ellipsoid itself, the projected profile section is elliptical in both working conditions.

[0114] (2) Based on Figure 4 The projected outline shown is referenced. Figure 2 Discretize the elliptical projection section into a series of trapezoidal plane elements and substitute the coordinates of the four vertices of each trapezoidal plane element into formula (4) to solve the equations of the lines corresponding to the upper and lower boundaries, obtaining the slope and intercept. Then substitute the coordinates of these boundary points and their corresponding slope and intercept values ​​into the formula in step (5), letting... , Obtain the far-field morphology function of the scattering The result, in which the sound source frequency f 0 = 1.0 kHz, velocity of sound in the outer water medium c 0 = 1500 m / s. Finally, substituting the numerical value of the scattering morphology function into formula (11), we obtain... Figure 5 The target intensity ‒ scattering azimuth angle shown by the solid red line in the middle ( The curve shows that, under both incident attitudes, the HKA calculation results differ in the forward scattering direction ( Within an azimuth angle range of ±20° centered on the BIEM numerical results (blue discontinuous line), the approximation effect of the KA method is better than that of the existing KA method (black dotted line).

[0115] (3) Maintain the incident angle and Unchanged and let Substituting the far-field morphology function of the scattering from step (5) into formula (11) yields the following result. Figure 6 Target strength shown TS The result is that when the ellipsoid is illuminated by an incident wave in a normal transverse direction, the HKA yields... The curve almost completely overlaps with the curve of KA. When the ellipsoid is irradiated obliquely by an incident wave, the HKA curve... The curve differs significantly from the curve of KA. However, in the forward scattering direction ( Within an azimuth angle range of ±20° centered on the BIEM numerical results (blue discontinuous line), the approximation effect is still better than the existing KA method (black dotted line).

[0116] This embodiment demonstrates that the HKA method retains and appropriately improves the performance of the existing forward scattering KA method.

[0117] Example 2: This example uses a finite-length stainless steel cylinder as an example to illustrate the method of the present invention in detail. The specific steps are as follows:

[0118] (1) Figure 7 A three-dimensional geometric model of a finite-length stainless steel cylinder, 0.74 m in length and 0.22 m in diameter, used in the scattering measurement experiment of the anechoic pool is presented, consistent with the actual physical dimensions. The center frequency of the signal emitted by the sound source is... f 0 = 30kHz, measured speed of sound under water pool conditions c 0 ≈ 1486 m / s. Condition 1 in the experiment corresponds to a measurement state where the sound source, cylinder, and receiving hydrophone are all at a depth of 2.0 m underwater. Condition 2 corresponds to a measurement state where the sound source and receiving hydrophone are still at a depth of 2.0 m underwater, but the depth of the cylinder is adjusted to 1.5 m underwater. In both measurement conditions, the cylinder rotates counterclockwise around its center position (equivalent to a changing incident attitude angle). The horizontal distances from the sound source to the target and from the target to the receiving hydrophone remain constant.

[0119] (2) For measurement condition 1, from Figure 7 Starting with the geometric model in the image, steps (1) to (3) were completed sequentially to obtain the desired result. Figure 8 The projected profile cross-section of a finite-length cylinder is shown. Since the sound source and the cylinder are at the same depth, the incident angle... The changes do not affect the symmetry of the projected outline, but only alter its size and shape.

[0120] For each incident attitude angle, completing steps (1)-(6) sequentially will yield the forward scattering in the direction of the hydrophone at the receiving end during the experiment. TS Once the calculations for all attitude angle values ​​are completed, the result is obtained. TS - ψ curve. Figure 9 Figure (a) shows the strictly forward scattering direction obtained by the HKA, KA, and BIEM methods. TS - ψ The calculation results of the HKA method and the BIEM method are very close, while the error of the KA method increases significantly as the attitude angle gradually deviates from the positive lateral direction. Figure 9 (b) shows a comparison curve between the calculation results of the HKA method and the experimental data measurement results, revealing that in the case of a normal transverse incident attitude ( The calculated results of HKA within an angle range of ±60° centered at 90° show good consistency with the experimental measurement results.

[0121] (3) For measurement condition 2, since the sound source and the cylinder are no longer on the same horizontal plane, the rotation of the finite-length cylinder simultaneously affects the symmetry, shape, and size of the projected profile section, such as... Figure 10 As shown. The measurement scenario at this time is different from that of condition 1, and the corresponding numerical calculation scenario is also more complex than that of condition 1.

[0122] For each incident attitude angle value, steps (1)-(6) are performed sequentially to obtain... Figure 11 The results, among which Figure 11 In the middle (a), the strictly forward scattering direction is... TS - ψ curve, Figure 11 (b) shows the direction of large-angle scattering (near forward scattering). TS - ψ The curve represents the strictly forward scattering direction. It can be seen that the numerical calculation results of the HKA method and the corresponding experimental measurement results generally show good consistency. However, due to certain measurement errors in distance, depth, and cylinder attitude during the experiment, the experimental data did not achieve complete consistency with the HKA numerical results across all incident attitude angle ranges as expected. Nevertheless, this does not affect the effectiveness of this embodiment in demonstrating the practicality of the HKA method.

Claims

1. A numerical method for calculating the forward-scattered sound field of an underwater regularly shaped object, characterized in that, Includes the following steps: (1) Obtain the initial projection profile of the three-dimensional geometric model of the regular-shaped object under a specified projection plane and incident direction, wherein the incident direction includes the incident zenith angle. and incident azimuth The initial projected contour is represented by a series of discrete points; (2) Divide the discrete points of the initial projected contour into upper boundary points and lower boundary points; (3) Perform interpolation to fill in the upper and lower boundary points so that the number of upper and lower boundary points is the same and they share the same horizontal coordinate; (4) Use trapezoidal plane elements to discretize the completed projected contour, where each trapezoidal plane element is bounded by its adjacent upper boundary points and its corresponding lower boundary points; (5) Based on the discretization result, according to the scattering zenith angle and scattering azimuth angle The value of is used to calculate the far-field morphology function of forward scattering. ; (6) Calculate the forward scattering target intensity of the object: 。 2. The numerical mixing calculation method for the forward-scattered sound field of an underwater regularly shaped object according to claim 1, characterized in that, The three-dimensional geometric model is established based on the standard three-dimensional Cartesian coordinate system in 3D CAD software; the projection plane is... flat.

3. The numerical mixing calculation method for the forward-scattered sound field of an underwater regularly shaped object according to claim 1, characterized in that, In step (2), the method for dividing the upper and lower boundary points is as follows: Find the minimum value of the x-coordinate of the discrete point. and maximum value and its corresponding endpoint coordinates and ; Establish the equations of the lines containing the two endpoints: ； For each discrete point Set its x-coordinate Substituting into the equation of the line, if its ordinate... If the value is greater than the calculated value of the line equation, it is classified as an upper boundary point; if it is less than the value, it is classified as a lower boundary point.

4. The numerical mixing calculation method for the forward-scattered sound field of an underwater regularly shaped object according to claim 1, characterized in that, In step (3), interpolation completion includes: For each point in the set of points on the upper boundary Find the x-coordinates that satisfy the following conditions in the set of lower boundary points. Two points of the relation and ,in Insert a new lower boundary point between these two lower boundary points, with its x-coordinate being... The vertical axis is: ； For each point in the lower boundary point set Find the x-coordinates in the set of upper boundary points that satisfy Two points of the relation and ,in Insert a new upper boundary point between these two upper boundary points, with its x-coordinate being... The vertical axis is: ； After interpolation, the sets of upper and lower boundary points are represented in matrix form. and ,in, This is a shared x-coordinate vector; and These represent the upper and lower boundaries, respectively. and These represent the number of upper boundary points and the number of lower boundary points, respectively.

5. The numerical mixing calculation method for the forward-scattered sound field of an underwater regularly shaped object according to claim 1, characterized in that, In step (4), the discretization of the trapezoidal plane element includes: for each adjacent x-coordinate interval Establish the equations of the upper and lower boundary lines: ； in, 。 6. The numerical mixing calculation method for the forward-scattered sound field of an underwater regularly shaped object according to claim 5, characterized in that, In step (5), the calculation of the far-field morphology function of the forward-scattered sound field is divided into the following four cases: when and When the morphological function is calculated, the formula is: ； in The imaginary unit, The cross-sectional area of ​​the projected profile. The wavelength of the incident wave; when and When the morphological function is calculated, the formula is: ； in, , ; when and When the morphological function is calculated, the formula is: ； in: ； ； when When the morphological function is calculated, the formula is: ； in: ； ； Let be the wave number of the incident wave. For the sound source frequency, The velocity of sound in the water medium of the outer region.

7. The numerical mixing calculation method for the forward-scattered sound field of an underwater regularly shaped object according to any one of claims 1-6, characterized in that, Each time the incident attitude angle is changed, steps (1) to (6) are repeated to update the target intensity.

8. The numerical mixing calculation method for the forward-scattered sound field of an underwater regularly shaped object according to any one of claims 1-6, characterized in that, The term "regularly shaped object" refers to an object whose shape encloses a simply connected region that is regular in shape.

Images