Method for solving three-dimensional water wave band gap attenuation coefficient based on high-order boundary element

Abstract

The application discloses a kind of based on high-order boundary element's three-dimensional water wave band gap attenuation coefficient solving method, belong to water wave regulation field, including the following steps: S1, based on linear potential flow theory establishes three-dimensional fixed structure infinite periodic array model, fluid is divided into several identical units, and the control equation and boundary condition that fluid domain velocity potential satisfies are obtained;S2, high-order boundary element method is selected to solve three-dimensional Laplace equation, boundary condition is substituted into boundary integral equation and simple Green function is selected, by numerical dispersion into high-order grid unit, and linear homogeneous equation set is established;S3, by organizing equation set, equation set has solution problem is converted into local minimum value problem to solve complex Bloch wave number, and the attenuation coefficient of water wave band in band gap is obtained.The application adopts the above solving method, realizes the fast and accurate solution of three-dimensional fixed structure band gap structure and attenuation coefficient in band gap.

Description

Technical Field

[0001] This invention relates to the field of water wave manipulation, and in particular to a method for solving the three-dimensional water wave bandgap attenuation coefficient based on higher-order boundary element method. Background Technology

[0002] The core of water wave control lies in the design of water wave crystals, based on their band structure. By designing the bandgap frequency range of a gradient array, wave blocking, attenuation, or focusing can be achieved. Water wave crystals have broad application prospects. For example, they can utilize bandgap properties to achieve high reflection within specific frequency ranges, thereby protecting coastlines from wave erosion; they can control waves and harvest energy through periodic structure reflections; current research mainly focuses on narrowband frequency ranges, but extending to broadband wave control will enhance their application potential in energy harvesting and coastal protection; and they can regulate wave propagation to improve the safety and efficiency of offshore operations (such as drilling and shipping).

[0003] Commonly used research methods include: plane wave expansion method: suitable for... xy Periodic structures within a plane simplify complex three-dimensional problems to two-dimensional problems through Fourier expansion; multiple scattering method: used to analyze the band structure of vertical cylindrical arrays, representing the wave field as the sum of scattered waves from the cylinder; transfer matrix method: suitable for... xz For planar problems, the wave fields on both sides of the structure are connected by reflection and transmission coefficients; the finite element method can be used to solve complex three-dimensional structures, but the computational load is huge.

[0004] These methods perform well in simple structures, but practical marine engineering often involves complex structures, necessitating the development of new methods to analyze the interaction between water waves and three-dimensional structures. Handling complex marine engineering structures suffers from high computational complexity and low efficiency. Furthermore, predicting the vibration isolation effect of existing arrays, or constructing arrays to target attenuation effects, requires the attenuation coefficient of the corresponding structure within the bandgap. Therefore, new computational methods are needed to solve for the band structure and bandgap attenuation coefficient of three-dimensional structures. Summary of the Invention

[0005] The purpose of this invention is to provide a method for solving the attenuation coefficient of three-dimensional water wave bandgap based on high-order boundary element method. By adding periodic Bloch boundary conditions, the method can quickly and accurately solve the band structure and attenuation coefficient within the bandgap of three-dimensional fixed structures, which facilitates the construction of water wave gradient arrays and realizes water wave control.

[0006] To achieve the above objectives, this invention provides a method for solving the three-dimensional water wave bandgap attenuation coefficient based on higher-order boundary element method, comprising the following steps:

[0007] S1. Based on the linear potential flow theory, a three-dimensional fixed structure infinite periodic array model is established. The fluid is divided into several identical units, and the control equations and boundary conditions satisfied by the velocity potential of the fluid domain are obtained.

[0008] S2. The three-dimensional Laplace equation is solved by using the high-order boundary element method. The boundary conditions are substituted into the boundary integral equation and a simple Green's function is selected. The equations are then numerically discretized into high-order grid elements to establish a linear homogeneous system of equations.

[0009] Choose a simple Green's function that satisfies the three-dimensional Laplace equation:

[0010] (6)

[0011] in It is a venue. Since it is the source point, using Green's theorem, the boundary integral equation is:

[0012] (7)

[0013] in, Represents the left side of the unit. Represents the front side of the unit. Represents the right side of the unit. Represents the rear side of the unit. Represents the bottom of the water. Represents the mean free water surface. Represents the wet surface. It is the fixed angle factor. It is Green's theorem G At the scene The normal partial derivative, It is velocity potential At the scene The normal partial derivative, when substituted into equation (7) using the boundary conditions, yields:

[0014] (8)

[0015] in, ， ， and The representatives are located on the left side of the unit. Front side right side and rear side The venue, , ;

[0016] After discretizing into higher-order mesh elements, expanding formula (8) yields:

[0017] (13)

[0018] in, Represents an index variable. N Represents the number of units. K Represents the number of discrete unit nodes. Index variables representing discrete unit nodes. Representing the Interpolation function for each node, These are standardized local coordinates within the cell. It's the Accord matrix. Representing the The velocity potential of each node and Representing the left and front units respectively The velocity potential of each node and Representing the left and front units respectively The normal derivative of the velocity potential of each node;

[0019] The dispersion equations for frequency and wavenumber are as follows:

[0020] (14)

[0021] in, Represents gravitational acceleration. Represents wave number;

[0022] S3. By rearranging the system of equations, the solvable problem of the system of equations is transformed into a local minimum problem, and the complex Bloch wave number is solved to obtain the attenuation coefficient of the water wave energy band within the band gap.

[0023] Formula (13) is a system of linear homogeneous equations, which can be simplified as follows:

[0024] (15)

[0025] Among them, the Bloch wavenumber , and frequency These are the coefficients to be determined, given first. =0, then [0, π / L Discretize the real numbers into several equal parts. Assign values ​​by substitution. For the matrix determinant value |det( A Numerical search is performed on the minimum value, and then on the real numbers. Perform the assignment, repeat the operation to obtain all the results. ,Will Number them in ascending order, and assign them to different... Same number By connecting the band structures, the final band structure diagram is obtained; when a band gap exists in the band structure, the wavenumber of the band gap is determined. The range is discretized into several parts, and values ​​are assigned to them to determine the range. The value, when determined back, As variables, for Perform the assignment and numerical search for the matrix determinant value |det( A Find the minimum value of |) to obtain and Results; Definition of Bloch transmission coefficient , which is the water wave attenuation coefficient of the water wave band structure within the band gap.

[0026] Preferably, in step S1, the governing equation is:

[0027] (1)

[0028] in, Represents rectangular coordinates in three-dimensional space. It is the Hamiltonian operator. Represents velocity potential, The fluid domain of the unit;

[0029] The boundary conditions are:

[0030] (2)

[0031] in, Represents the normal derivative at the boundary;

[0032] (3)

[0033] in, Represents water depth;

[0034] (4)

[0035] in, Represents angular frequency, which is the wavenumber. The function, Represents gravitational acceleration;

[0036] (5)

[0037] in, Represents the boundary Partial derivative of direction, Represents the boundary Partial derivative of direction, represent Bloch wavenumber in the direction, represent Bloch wavenumber in the direction, L It is the length of the unit. W It is the width of the unit. .

[0038] Therefore, this invention adopts the above-mentioned method for solving the attenuation coefficient of three-dimensional water wave bandgap based on high-order boundary elements, and adds periodic Bloch boundary conditions to achieve fast and accurate solution of the band structure and attenuation coefficient within the bandgap of three-dimensional fixed structures, which facilitates the construction of water wave gradient arrays and realizes water wave control.

[0039] Compared with existing technologies, this technology has the following advantages and beneficial effects:

[0040] 1. Based on the existing high-order boundary element method, this invention adds periodic Bloch boundary conditions, which solves the technical problem that the existing method is not applicable to the solution of the band structure of three-dimensional water wave crystals, and realizes the fast and accurate solution of the band structure of three-dimensional fixed structures;

[0041] 2. This invention solves the technical problem of solving the attenuation coefficient within the bandgap of a three-dimensional water wave crystal, enabling rapid and accurate solving of the attenuation coefficient within the bandgap of a three-dimensional fixed structure, thus achieving vibration isolation effect prediction, or array construction for target attenuation effect;

[0042] 3. In engineering applications, it provides a fast and accurate solution method for calculating the wave attenuation coefficient of arbitrarily shaped complex three-dimensional fixed structures and underwater topographic energy band structures, which facilitates the construction of water wave gradient arrays and enables water wave control, i.e., spatial separation and enhancement of waves. It has broad prospects for engineering applications.

[0043] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. Attached Figure Description

[0044] Figure 1 This is a schematic diagram of the periodic element of the method for solving the three-dimensional water wave bandgap attenuation coefficient based on the high-order boundary element method of the present invention.

[0045] Figure 2 This is a schematic diagram of the quadrilateral mesh element used in the present invention to solve the three-dimensional water wave bandgap attenuation coefficient based on the high-order boundary element method.

[0046] Figure 3 This is a schematic diagram of a triangular mesh element for the method of solving the three-dimensional water wave bandgap attenuation coefficient based on high-order boundary elements in this invention.

[0047] Figure 4 for =0, When = 0, the coefficient matrix of the spatial structure is 1 meter in length, width, and height. A Determinant value and its real part follow the wave number k Change diagram;

[0048] Figure 5 for Figure 4 A magnified view of graph M corresponding to the second minimum value;

[0049] Figure 6 for = , When = 0, the coefficient matrix of the spatial structure is 1 meter in length, width, and height. A Determinant value and its real part follow the wave number k Change diagram;

[0050] Figure 7 for Figure 6 A magnified view of graph C corresponding to the first local minimum;

[0051] Figure 8 This is a comparison of the bandgap attenuation coefficient calculation method for three-dimensional water wave energy bandgap based on high-order boundary element method in this invention, using a spatial unit with a length, width, and height of 1 meter.

[0052] Figure 9 The diagram shows a comparison of the band structure results for a vertical cylindrical base structure with a radius of 0.25 meters within a watershed unit with dimensions of 1 meter in length, width, and height, based on the method for solving the three-dimensional water wave band gap attenuation coefficient using the higher-order boundary element method of this invention.

[0053] Figure 10 The present invention provides a method for solving the attenuation coefficient of three-dimensional water wave band gap based on higher-order boundary elements. The method is used in a watershed unit with a length, width, and height of 1 meter and a radius of 0.25 meters. The comparison diagram shows the band structure results and the schematic diagram of the real part of the first band gap.

[0054] Figure 11 The present invention provides a method for solving the three-dimensional water wave bandgap attenuation coefficient based on higher-order boundary element method. This method is applied to a vertical cylindrical base structure with a radius of 0.25 meters within a watershed unit with dimensions of 1 meter in length, width, and height. Comparison diagram of real and imaginary parts;

[0055] Figure 12 The present invention provides a method for solving the three-dimensional water wave bandgap attenuation coefficient based on the higher-order boundary element method. This method is applied to a vertical cylindrical base structure with a radius of 0.25 meters within a watershed unit with dimensions of 1 meter in length, width, and height. wave number k Change diagram;

[0056] Figure 13 The present invention relates to a method for solving the three-dimensional water wave bandgap attenuation coefficient based on the higher-order boundary element method. This method is applied to a truncated cylindrical structure with a radius of 0.25 meters and a draft of 0.5 meters within a watershed unit with dimensions of 1 meter in length, width, and height. wave number k Change diagram;

[0057] Figure 14 This is a diagram showing the force results in the longitudinal direction of each column of a truncated cylinder with a row and column spacing of 1 meter, a radius of 0.25 meters, and a draft of 0.5 meters, in the three-dimensional water wave band gap attenuation coefficient solution method based on higher-order boundary elements of this invention.

[0058] Figure 15 This diagram illustrates the heave forces acting on each column of a truncated cylinder in a 5-column periodic array with a row and column spacing of 1 meter, a radius of 0.25 meters, and a draft of 0.5 meters, as described in the present invention's method for solving the attenuation coefficient of three-dimensional water wave band gap based on higher-order boundary elements. Detailed Implementation

[0059] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. The components of the embodiments of the present invention described and shown in the accompanying drawings can generally be arranged and designed in various different configurations.

[0060] The embodiments of the present invention will now be described in detail with reference to the accompanying drawings.

[0061] Example

[0062] A method for solving the attenuation coefficient of three-dimensional water wave bandgap based on high-order boundary element method includes the following steps:

[0063] S1. Based on linear potential flow theory, establish a three-dimensional fixed structure infinite periodic array model, such as... Figure 1 As shown, at a finite depth of A periodically arranged array of truncated cylinders is fixed in the water, using a three-dimensional Cartesian coordinate system. The plane is fixed at its average position on the free surface. z A periodic array of several identical truncated cylinders, with the axis pointing vertically upwards. direction and Extending infinitely in all directions, the fluid is divided into identical units, each unit in The length in the direction is L ,exist Width in the direction is W ,exist zDepth in the direction is Furthermore, each unit contains the same truncated cylinder, and the waves in adjacent units exhibit a Bloch phase relationship.

[0064] The governing equations and boundary conditions satisfied by the velocity potential in the fluid domain are obtained. The governing equations are:

[0065] (1)

[0066] in, Represents rectangular coordinates in three-dimensional space. It is the Hamiltonian operator. Represents velocity potential, The fluid domain of the unit;

[0067] The boundary conditions are:

[0068] (2)

[0069] in, Represents the normal derivative at the boundary;

[0070] (3)

[0071] in, Represents water depth;

[0072] (4)

[0073] in, Represents angular frequency, which is the wavenumber. The function, Represents gravitational acceleration;

[0074] (5)

[0075] in, Represents the boundary Partial derivative of direction, Represents the boundary Partial derivative of direction, represent Bloch wavenumber in the direction, represent y Bloch wavenumber in the direction, L It is the length of the unit. W It is the width of the unit. .

[0076] S2. The three-dimensional Laplace equation is solved by using the high-order boundary element method. The boundary conditions are substituted into the boundary integral equation and a simple Green's function is selected. The equations are then numerically discretized into high-order grid elements to establish a linear homogeneous system of equations.

[0077] Choose a simple Green's function that satisfies the three-dimensional Laplace equation:

[0078] (6)

[0079] in It is a venue. Since it is the source point, using Green's theorem, the boundary integral equation is:

[0080] (7)

[0081] in It is the fixed angle factor. It is Green's theorem G At the scene The normal partial derivative, It is velocity potential At the scene Substituting the normal partial derivatives of boundary conditions (2)-(5) into equation (7), we obtain:

[0082] (8)

[0083] in, ， ， and The representatives are located on the left side of the unit. Front side right side and rear side The venue, , ;

[0084] like Figure 2 and Figure 3 As shown, the boundary is discretized into higher-order boundary element units of quadrilaterals and triangles, and the area of ​​the infinitesimal element is d. s The area of ​​isoparametric elements has the following relationship:

[0085] (9)

[0086] in, These are standardized local coordinates within the cell. The Jacobian matrix is ​​expressed as follows:

[0087] (10)

[0088] The normal velocity, velocity potential, and coordinates of a liquid at any point on the wetted surface can all be expressed as a series sum of the product of the corresponding physical values ​​of its element and a quadratic interpolation function:

[0089] (11)

[0090] in, K Represents the number of discrete unit nodes. Index variables representing discrete unit nodes. Indicates the first The normal derivative of the velocity potential of each node Representing the The velocity potential of each node Indicates the first The coordinates of each node, Representing the The interpolation function for each node is shown below:

[0091] (12)

[0092] Expanding formula (8) gives:

[0093] (13)

[0094] in, Represents an index variable. N Represents the number of units. and Representing the left and front units respectively The velocity potential of each node and Representing the left and front units respectively The velocity potential normal derivative of each node.

[0095] frequency and wave number The dispersion formula is as follows:

[0096] (14)

[0097] in Represents gravitational acceleration. Represents wave number.

[0098] S3. By rearranging the system of equations, the problem of finding a solvable system of equations is transformed into a local minimum problem, and the complex Bloch wavenumber is solved.

[0099] Formula (13) is a system of linear homogeneous equations, which can be simplified as follows:

[0100] (15)

[0101] Among them, the Bloch wavenumber , and frequency ωThese are the coefficients to be determined. Typically, the problem of finding a solution to a system of linear homogeneous equations is solving for the determinant |det( A ( , , ω The root of ))| = 0. However, with the matrix A As the dimension increases, the matrix A Determinant | det( A The equation becomes a high-order equation with the degree matching the matrix dimension, making direct root finding impossible. This invention employs a numerical search for local minima to solve for the coefficients.

[0102] because and The solution process is similar, so only the following will be introduced here. The solution process. First, give... =0, will Discretize into several parts at equal intervals, for real numbers Assign values ​​by substitution. For the matrix determinant value |det( A Numerical search is performed using the minimum value. Figures 4-15 In the equation, the dimensionless wavenumber is: Dimensionless Bloch wavenumber for L / ,like Figures 4-7 As shown, the numerical search process for local minima of spatial structure is illustrated, and then the real numbers are... By assigning values ​​and repeating the above steps, you can obtain all the results. k First, k Number them in ascending order, and assign them to different... Same number k By connecting the links, the final band structure diagram is obtained. The spatial band structure diagram is shown below. Figure 8 As shown, a band structure with a base cylinder is arranged in this space as follows: Figure 9 As shown.

[0103] Clearly, there is no bandgap in the airspace. Taking a bottom-mounted cylinder as an example, the solution within the bandgap first involves determining the wavenumber of the bandgap. The range is discretized into several parts, and values ​​are assigned to them, then the determination is made. Whether it is 0 or 1 depends on Figure 10 As shown, the first prohibited item At this time only A variable is assigned a value, and the matrix determinant value |det( is searched numerically). A The minimum value of )| is finally obtained. and like Figure 11 As shown. Define the Bloch transmission coefficient. This refers to the water wave attenuation coefficient within the bandgap of the water wave bandgap structure. The Bloch transmission coefficient of the bottom cylindrical bandgap is as follows: Figure 12 As shown.

[0104] By comparing the band structure results of different three-dimensional vertical base structures and two-dimensional structures, the accuracy of the three-dimensional band structure solution method is preliminarily verified. Figure 8 and Figure 9 As shown, the results of this invention are compared with those of the method in the paper (McIver, 2000) (hereinafter referred to as Existing Method 1) for the spatial domain and the base cylinder with a radius of 0.25 meters. The results of this invention are in good agreement with those of Existing Method 1, which verifies the accuracy of this method.

[0105] like Figure 14 and Figure 15 As shown, the wave excitation forces in the longitudinal and helical directions of the periodically truncated cylindrical array were calculated using the method in the paper (Yue, 2019) (hereinafter referred to as Existing Method 2). By comparing the changes in the wave excitation forces on the periodically truncated cylinder with its band structure diagram ( Figure 13 Within the passband, the forces alternated. However, within the foreband, the forces varied in a stepped manner with the cylinder numbers, with the front cylinders showing significantly higher forces than the rear cylinders. Based on... Figure 14 and Figure 15 Attenuation ratio within the bandgap and Figure 13 The attenuation coefficients are close to those in the theoretical results, indicating that although using a finite number of structures for water wave control will deviate from the theoretical results, the attenuation effect can be predicted in general, and the number of structures required can be estimated in advance.

[0106] Therefore, this invention adopts the above-mentioned method for solving the attenuation coefficient of three-dimensional water wave bandgap based on high-order boundary elements, and adds periodic Bloch boundary conditions to achieve fast and accurate solution of the band structure and attenuation coefficient within the bandgap of three-dimensional fixed structures, which facilitates the construction of water wave gradient arrays and realizes water wave control.

[0107] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit them. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the technical solutions of the present invention, and these modifications or equivalent substitutions cannot cause the modified technical solutions to deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. A method for solving the three-dimensional water wave bandgap attenuation coefficient based on high-order boundary element method, characterized in that: Includes the following steps: S1. Based on the linear potential flow theory, a three-dimensional fixed structure infinite periodic array model is established. The fluid is divided into several identical units, and the control equations and boundary conditions satisfied by the velocity potential of the fluid domain are obtained. S2. The three-dimensional Laplace equation is solved by using the high-order boundary element method. The boundary conditions are substituted into the boundary integral equation and a simple Green's function is selected. The equations are then numerically discretized into high-order grid elements to establish a linear homogeneous system of equations. Choose a simple Green's function that satisfies the three-dimensional Laplace equation: (6) in It is a venue. Since it is the source point, using Green's theorem, the boundary integral equation is: (7) in, Represents the left side of the unit. Represents the front side of the unit. Represents the right side of the unit. Represents the rear side of the unit. Represents the bottom of the water. Represents the mean free water surface. Represents the wet surface. It is the fixed angle factor. It is Green's theorem G At the scene The normal partial derivative, It is velocity potential At the scene The normal partial derivative, when substituted into equation (7) using the boundary conditions, yields: (8) in, ， ， and The representatives are located on the left side of the unit. Front side right side and rear side The venue, , ; After discretizing into higher-order mesh elements, expanding formula (8) yields: (13) in, Represents an index variable. N Represents the number of units. K Represents the number of discrete unit nodes. Index variables representing discrete unit nodes. Representing the Interpolation function for each node, These are standardized local coordinates within the cell. It's the Accord matrix. Representing the The velocity potential of each node and Representing the left and front units respectively The velocity potential of each node and Representing the left and front units respectively The normal derivative of the velocity potential of each node; The dispersion equations for frequency and wavenumber are as follows: (14) in, Represents gravitational acceleration. Represents wave number; S3. By rearranging the system of equations, the solvable problem of the system of equations is transformed into a local minimum problem, and the complex Bloch wave number is solved to obtain the attenuation coefficient of the water wave energy band within the band gap. Formula (13) is a system of linear homogeneous equations, which can be simplified as follows: (15) Among them, the Bloch wavenumber , and frequency These are the coefficients to be determined, given first. =0, will Discretize into several parts at equal intervals, for real numbers Assign values ​​by substitution. For the matrix determinant value |det( A Numerical search is performed on the minimum value, and then on the real numbers. Perform the assignment, repeat the operation to obtain all the results. ,Will Number them in ascending order, and assign them to different... Same number By connecting the band structures, the final band structure diagram is obtained; when a band gap exists in the band structure, the wavenumber of the band gap is determined. The range is discretized into several parts, and values ​​are assigned to them to determine the range. The value, when determined back, As variables, for Perform the assignment and numerical search for the matrix determinant value |det( A Find the minimum value of |) to obtain and Results; Definition of Bloch transmission coefficient , which is the water wave attenuation coefficient of the water wave band structure within the band gap.

2. The method for solving the three-dimensional water wave bandgap attenuation coefficient based on high-order boundary element method according to claim 1, characterized in that: In step S1, the governing equation is: (1) in, Represents rectangular coordinates in three-dimensional space. It is the Hamiltonian operator. Represents velocity potential, The fluid domain of the unit; The boundary conditions are: (2) in, Represents the normal derivative at the boundary; (3) in, Represents water depth; (4) in, Represents angular frequency, which is the wavenumber. The function, Represents gravitational acceleration; (5) in, Represents the boundary Partial derivative of direction, Represents the boundary Partial derivative of direction, represent Bloch wavenumber in the direction, represent Bloch wavenumber in the direction, L It is the length of the unit. W It is the width of the unit. .

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