A method for solving three-dimensional water wave energy band structure based on high-order boundary elements

Abstract

The present invention discloses a method for solving three-dimensional water wave energy band structure based on high-order boundary elements, which belongs to the field of water wave control and includes the following steps: S1. Based on the linear potential flow theory, a model of an infinite periodic array of three-dimensional fixed structures is established, the fluid is divided into a number of identical units, and the control equations and boundary conditions satisfied by the velocity potential of the fluid domain are obtained; S2. The high-order boundary element method is used to solve the three-dimensional Laplace equation, the boundary conditions are substituted into the boundary integral equation and a simple Green's function is selected. By numerically discretizing into high-order grid units, a linear homogeneous system of equations is established; S3. By arranging the system of equations, the solvable problem of the system of equations is converted into an eigenvalue problem to solve the real Bloch wave number and obtain a water wave energy band structure diagram. The present invention adopts the above-mentioned method for solving three-dimensional water wave energy band structure based on high-order boundary elements to achieve a rapid and accurate solution of the water wave energy band structure of three-dimensional fixed structures of arbitrary shape and underwater terrain.

Description

Technical Field

[0001] The present invention relates to the field of water wave regulation, and in particular to a method for solving three-dimensional water wave energy band structure based on high-order boundary elements. Background Art

[0002] The core of water wave control lies in the design of water wave crystals, whose theoretical foundation is their band structure. By designing the bandgap frequency range of a gradient array, water waves can be blocked, attenuated, or focused. Water wave crystals have a wide range of applications. For example, they can exploit the bandgap properties to achieve high reflection within a specific frequency range, thereby protecting coastal areas from wave erosion; they can also achieve wave control and energy harvesting by reflecting waves through periodic structures; while current research focuses on narrowband frequency ranges, expanding to broadband wave control will enhance its potential for energy harvesting and coastal protection; and they can also 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: applicable to Periodic structures in a plane can be simplified into two-dimensional problems through Fourier expansion. Multiple scattering method: used to analyze the band structure of vertical cylindrical arrays, and the wave field is expressed as the sum of cylindrical scattered waves. Transfer matrix method: applicable to Plane problems, connecting the wave fields on both sides of the structure through reflection and transmission coefficients; Finite element method: can be used to solve complex three-dimensional structures, but the calculation amount is huge.

[0004] While these methods work well for simple structures, practical ocean engineering often involves complex structures, requiring the development of new methods to analyze the interaction between water waves and three-dimensional structures. This complexity is computationally expensive and inefficient, leading to the need for new computational methods to address the band structure of three-dimensional structures. Summary of the Invention

[0005] The purpose of the present invention is to provide a method for solving the three-dimensional water wave band structure based on high-order boundary elements, add periodic Bloch boundary conditions, realize the rapid and accurate solution of the band structure of three-dimensional fixed structures, effectively improve the efficiency of band structure solution, facilitate the construction of water wave gradient arrays, and realize water wave control.

[0006] To achieve the above object, the present invention provides a method for solving the three-dimensional water wave energy band structure based on high-order boundary elements, comprising the following steps:

[0007] S1. Based on the linear potential flow theory, a three-dimensional fixed structure infinite period 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. Use the high-order boundary element method to solve the three-dimensional Laplace equation, substitute the boundary conditions into the boundary integral equation and select the simple Green's function, and establish the linear homogeneous equation system by numerical discretization into high-order grid elements;

[0009] S3. By arranging the equation group, the solvable problem of the equation group is transformed into an eigenvalue problem to solve the real Bloch wave number and obtain the water wave band structure diagram.

[0010] Preferably, in step S1, the control equation is:

[0011] (1);

[0012] in, Represents the rectangular coordinates in three-dimensional space, is the Hamiltonian operator, represents the velocity potential, represents the fluid domain of the cell;

[0013] The boundary conditions are:

[0014] (2);

[0015] in, represents the normal derivative at the boundary;

[0016] (3);

[0017] in, Represents water depth;

[0018] (4);

[0019] in, represents the angular frequency, which is the wave number function, represents the acceleration due to gravity;

[0020] (5);

[0021] in, Representatives at the border Directional derivative, Representatives at the border Directional derivative, represent The Bloch wave number in the direction, represent The Bloch wave number in the direction, L is the length of the unit, W is the width of the cell, .

[0022] Preferably, in step S2, a simple Green's function that satisfies the three-dimensional Laplace equation is selected:

[0023] (6);

[0024] in It's the venue. is the source point. Using Green’s theorem, we get the boundary integral equation:

[0025] (7);

[0026] in is the fixed angle coefficient, It is Green's formula G On-site The normal partial derivative of is the velocity potential On-site Substituting the boundary conditions (2)-(5) into equation (7) yields the following:

[0027] (8);

[0028] in, , , and Representatives are on the left side of the unit , front side , right side and rear side The venue, , ;

[0029] After discretizing the boundary into high-order grid cells, formula (8) is expanded to obtain:

[0030] (13);

[0031] in, Represents the index variable, N Represents the number of units, K Represents the number of discrete element nodes, is a node The interpolation function, are the normalized local coordinates within the cell, is the Jacobian ratio matrix, Representative The velocity potential of each node, and Represents the left and front units respectively The velocity potential of each node, and Represents the left and front units respectively The velocity potential normal derivative of each node.

[0032] Preferably, in step S3, by adjusting the column order, the following form is obtained:

[0033] (14);

[0034] in, represents the unknown velocity potential at each node of the free surface, Indicates something similar to The remaining unknown quantities, Indicates that the source points are distributed on the water surface The coefficient matrix of N × N , here N Indicates the number of water surface nodes, Indicates that the source points are distributed on the water surface The coefficient matrix of N × M , here M represents the number of nodes excluding the water surface, Indicates that the source points are distributed on surfaces other than the water surface The coefficient matrix of M × N , Indicates that the source points are distributed on surfaces other than the water surface The coefficient matrix of M × M , Indicates that the source points are distributed on the water surface and contains The coefficient matrix of N × N , Indicates that the source points are distributed outside the water surface. The coefficient matrix of M × M ;

[0035] By eliminating the formula (14) Get the shape The eigenvalue problem of is as follows:

[0036] (15);

[0037] By substituting real numbers To Matrix , repeat the above operation to obtain different eigenvalues ,Will Arrange them in ascending order and number them accordingly. The same number By connecting them, we finally get the band structure diagram, and the dispersion equations for frequency and wave number are as follows:

[0038] (16);

[0039] in, Represents the acceleration due to gravity.

[0040] Therefore, the present invention adopts the above-mentioned three-dimensional water wave band structure solution method based on high-order boundary elements, adds periodic Bloch boundary conditions, and realizes the rapid and accurate solution of the band structure of three-dimensional fixed structures, effectively improves the efficiency of band structure solution, facilitates the construction of water wave gradient arrays, and realizes water wave control.

[0041] The technical solution of the present invention is further described in detail below through the accompanying drawings and embodiments. BRIEF DESCRIPTION OF THE DRAWINGS

[0042] Figure 1 Schematic diagram of a periodic unit of a method for solving the three-dimensional water wave energy band structure based on high-order boundary elements of the present invention;

[0043] Figure 2 A schematic diagram of a quadrilateral grid unit of a method for solving the three-dimensional water wave energy band structure based on high-order boundary elements according to the present invention;

[0044] Figure 3 A schematic diagram of a triangular grid unit of a method for solving the three-dimensional water wave energy band structure based on high-order boundary elements according to the present invention;

[0045] Figure 4 This is a comparison chart of the energy band results of a three-dimensional water wave energy band structure solution method based on high-order boundary elements in a space unit with a length, width and height of 1 meter;

[0046] Figure 5 This is a comparison chart of the energy band results of a bottom-mounted cylindrical unit with a radius of 0.25 meters in a spatial unit with a length, width, and height of 1 meter using a high-order boundary element method for solving the three-dimensional water wave energy band structure of the present invention;

[0047] Figure 6 This is a comparison chart of the energy band results of a three-dimensional water wave energy band structure solution method based on high-order boundary elements in a space unit with a length of two meters and a width and height of one meter;

[0048] Figure 7 This is a comparison chart of the energy band results of a vertical square bottom structure with a length and width of 0.25 meters in a watershed unit with a length, width and height of 1 meter using a high-order boundary element method for solving the three-dimensional water wave energy band structure of the present invention;

[0049] Figure 8 This is a comparison chart of the energy band results of a vertical C-shaped cylindrical bottom structure in a watershed unit with a length, width, and height of 1 meter using a high-order boundary element method for solving the three-dimensional water wave energy band structure of the present invention;

[0050] Figure 9 A C-shaped cylinder with a wall thickness of 0.05 meters and a radius of 0.25 meters according to a method for solving the three-dimensional water wave energy band structure based on high-order boundary elements of the present invention;

[0051] Figure 10 This is a diagram of the energy band structure of a truncated cylinder with a radius of 0.25 meters and a draft of 0.5 meters in a watershed unit with a length, width, and height of 1 meter, according to a method for solving the energy band structure of three-dimensional water waves based on high-order boundary elements of the present invention;

[0052] Figure 11 This is a diagram showing the force results in the longitudinal surge direction on each column of truncated cylinders in a five-row periodic array with a row-column spacing of 1 meter, a radius of 0.25 meters, and a draft of 0.5 meters in a three-dimensional water wave energy band structure solution method based on high-order boundary elements of the present invention;

[0053] Figure 12 This is a diagram of the force results in the heave direction on each column of truncated cylinders in a periodic array with 1 meter row and column spacing, 0.25 meter radius, and 0.5 meter draft in a three-dimensional water wave energy band structure solution method based on high-order boundary elements in the present invention. DETAILED DESCRIPTION

[0054] To make the objectives, technical solutions, and advantages of the embodiments of the present invention more clear, the technical solutions of the embodiments of the present invention will be clearly and completely described below in conjunction with the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Generally, the components of the embodiments of the present invention described and shown in the drawings herein can be arranged and designed in various different configurations.

[0055] The embodiments of the present invention will be described in detail below with reference to the accompanying drawings.

[0056] Example

[0057] A method for solving the three-dimensional water wave energy band structure based on high-order boundary elements includes the following steps:

[0058] S1. Based on the linear potential flow theory, a three-dimensional fixed structure infinite period array model is established, such as Figure 1 As shown, at a finite depth of A periodically arranged array of truncated cylinders is fixed in water, using a three-dimensional rectangular coordinate system, where The plane is fixed at the mean position of the free surface, zThe axis is vertically upward, and a periodic array of several identical truncated cylinders is formed in Direction and The fluid is divided into the same units, each unit is infinite in the direction of The length in the direction is L ,exist The width in the direction is W ,exist z The depth in the direction is , and each unit contains the same truncated cylinder, and the waves in adjacent units have a Bloch phase relationship;

[0059] The control equations and boundary conditions satisfied by the velocity potential of the fluid domain are obtained. The control equations are:

[0060] (1);

[0061] in, Represents the rectangular coordinates in three-dimensional space, is the Hamiltonian operator, represents the velocity potential, represents the fluid domain of the cell;

[0062] The boundary conditions are:

[0063] (2);

[0064] in, represents the normal derivative at the boundary;

[0065] (3);

[0066] in, Represents water depth;

[0067] (4);

[0068] in, ω represents the angular frequency, which is the wave number function, represents the acceleration due to gravity;

[0069] (5);

[0070] in, Representatives at the border Directional derivative, Representatives at the border Directional derivative, represent The Bloch wave number in the direction, represent The Bloch wave number in the direction,L is the length of the unit, W is the width of the cell, .

[0071] S2. Use the high-order boundary element method to solve the three-dimensional Laplace equation, substitute the boundary conditions into the boundary integral equation and select the simple Green's function, and establish the linear homogeneous equation system by numerical discretization into high-order grid elements;

[0072] Choose a simple Green's function that satisfies the three-dimensional Laplace equation:

[0073] (6);

[0074] in It's the venue. is the source point. Using Green’s theorem, we get the boundary integral equation:

[0075] (7);

[0076] in is the fixed angle coefficient, It is Green's formula G On-site The normal partial derivative of is the velocity potential On-site Substituting the boundary conditions (2)-(5) into equation (7) yields the following:

[0077] (8);

[0078] in, , , and Representatives are on the left side of the unit , front side , right side and rear side The venue, , ;

[0079] like Figure 2 and Figure 3 As shown, the boundary is discretized into quadrilateral and triangular high-order boundary element units, and the element area d s The relationship between isoparametric unit area and isoparametric unit area is as follows:

[0080] (9);

[0081] in, are the normalized local coordinates within the cell, is the Jacobian ratio matrix, which is expressed as follows:

[0082] (10);

[0083] The normal velocity, velocity potential, and coordinates of the liquid at any point on the wetted surface can be expressed as the series and form of the product of the corresponding physical value of the unit and the quadratic interpolation function:

[0084] (11);

[0085] in, K represents the number of nodes of the element, Indicates the The velocity potential normal derivative of each node, Representative The velocity potential of each node, Indicates the The coordinates of the nodes, is a node The interpolation function is as follows:

[0086] (12);

[0087] Expanding formula (8) yields:

[0088] (13);

[0089] in, Represents the index variable, N Represents the number of units, and Represents the left and front units respectively The velocity potential of each node, and Represents the left and front units respectively The velocity potential normal derivative of each node.

[0090] S3. By rearranging the system of equations, the solvable problem of the system of equations is transformed into an eigenvalue problem to solve the real Bloch wave number. By adjusting the column order, the following form is obtained:

[0091] (14);

[0092] in, represents the unknown velocity potential at each node of the free surface, Indicates something similar to The remaining unknown quantities, Indicates that the source points are distributed on the water surface The coefficient matrix of N × N , here NIndicates the number of water surface nodes, Indicates that the source points are distributed on the water surface The coefficient matrix of N × M , here M represents the number of nodes excluding the water surface, Indicates that the source points are distributed on surfaces other than the water surface The coefficient matrix of M × N , Indicates that the source points are distributed on surfaces other than the water surface The coefficient matrix of M × M , Indicates that the source points are distributed on the water surface and contains The coefficient matrix of N × N , Indicates that the source points are distributed outside the water surface. The coefficient matrix of M × M ;

[0093] By eliminating the formula (14) Get the shape The eigenvalue problem of is as follows:

[0094] (15);

[0095] By substituting real numbers To Matrix , repeat the above operation to obtain different eigenvalues ,Will Arrange them in ascending order and number them accordingly. The same number By connecting them, we finally get the band structure diagram, and the dispersion equations for frequency and wave number are as follows:

[0096] (16);

[0097] in, Represents the acceleration due to gravity.

[0098] By comparing the band structure results of different three-dimensional vertical bottom-standing structures and two-dimensional structures, the accuracy of the three-dimensional band structure solution method was preliminarily verified. Figure 4-Figure 8 as well as Figure 10-12 The dimensionless wave number is , dimensionless Bloch wave number for ,like Figure 4 and Figure 5 As shown in the figure, the results of the present invention are compared with the results of the airspace obtained by the method in the paper (McIver, 2000) (hereinafter referred to as the existing method 1) and the results of the bottom cylinder with a radius of 0.25 meters. The results of the present invention are in good agreement with the results of the existing method 1. Then, the finite element method is used to solve the Helmholtz equation and establish an infinite periodic array model of a two-dimensional structure, as shown in the figure. Figure 6-Figure 9 As shown in the figure, the two-dimensional finite element results and the results of this method for the three-dimensional base structure are compared respectively (the three-dimensional base structure is an airspace with a length of 2 meters and a width and height of 1 meter, a base cuboid, and a C-shaped cylinder with a wall thickness of 0.05 meters and a radius of 0.25 meters). The results are in good agreement, which preliminarily verifies the accuracy of this method.

[0099] like Figure 11 and Figure 12 As shown in the figure, the wave excitation forces in the longitudinal and heave directions on the periodic truncated cylinder array are 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 periodic truncated cylinders and their band structure diagram ( Figure 10 ), it was found that within the passband, the forces varied alternately. However, within the bandgap, the forces varied in steps according to the cylinder number, with the front cylinder being significantly higher than the rear cylinder.

[0100] Therefore, the present invention adopts the above-mentioned three-dimensional water wave band structure solution method based on high-order boundary elements, adds periodic Bloch boundary conditions, and realizes the rapid and accurate solution of the band structure of three-dimensional fixed structures, effectively improves the efficiency of band structure solution, facilitates the construction of water wave gradient arrays, and realizes water wave control.

[0101] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention rather than to limit the same. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that they can still modify or replace the technical solutions of the present invention with equivalents, and these modifications or equivalent replacements 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 three-dimensional water wave energy band structure based on high-order boundary elements, characterized by: The following steps are involved: S1. Based on the linear potential flow theory, a three-dimensional fixed structure infinite period array model is established, and the fluid is divided into several identical units. h A periodically arranged array of truncated cylinders is fixed in the water. A three-dimensional rectangular coordinate system is used, and each unit contains the same truncated cylinder. The waves in adjacent units have a Bloch phase relationship. The control equations and boundary conditions satisfied by the velocity potential of the fluid domain are obtained. S2. Use the high-order boundary element method to solve the three-dimensional Laplace equation, substitute the boundary conditions into the boundary integral equation and select the simple Green's function, and establish the linear homogeneous equation system by numerical discretization into high-order grid elements; S3. By arranging the equations, the solvable problem of the equations is transformed into an eigenvalue problem to solve the real Bloch wave number and obtain the water wave band structure diagram; In step S2, a simple Green's function that satisfies the three-dimensional Laplace equation is selected: (6) in It's the venue. is the source point. Using Green’s theorem, we get the boundary integral equation: (7) in is the fixed angle coefficient, It is Green's formula G On-site The normal partial derivative of is the velocity potential On-site Substituting the boundary conditions into equation (7) yields the following: (8) in, , , and Representatives are on the left side of the unit , front side , right side and rear side The venue, , ; After discretizing the boundary into high-order grid cells, formula (8) is expanded to obtain: (13) in, Represents the index variable, Represents the number of units, Represents the number of discrete element nodes, is a node The interpolation function, are the normalized local coordinates within the cell, is the Jacobian ratio matrix, Representative The velocity potential of each node, and Represents the left and front units respectively The velocity potential of each node, and Represents the left and front units respectively The velocity potential normal derivative of each node; In step S3, by adjusting the column order, the following form is obtained: (14) in, represents the unknown velocity potential at each node of the free surface, Indicates something similar to The remaining unknown quantities, Indicates that the source points are distributed on the water surface The coefficient matrix of N × N , here N Indicates the number of water surface nodes, Indicates that the source points are distributed on the water surface The coefficient matrix of N × M , here M represents the number of nodes excluding the water surface, Indicates that the source points are distributed on surfaces other than the water surface The coefficient matrix of M × N , Indicates that the source points are distributed on surfaces other than the water surface The coefficient matrix of M × M , Indicates that the source points are distributed on the water surface and contains The coefficient matrix of N × N , Indicates that the source points are distributed outside the water surface. The coefficient matrix of M × M ; By eliminating the formula (14) Get the shape The eigenvalue problem of is as follows: (15) By substituting real numbers To Matrix , repeat the above operation to obtain different eigenvalues ,Will Arrange them in ascending order and number them accordingly. The same number By connecting them, we finally get the band structure diagram, and the dispersion equations for frequency and wave number are as follows: (16) in, Represents the acceleration due to gravity.

2. The method for solving the three-dimensional water wave energy band structure based on high-order boundary elements according to claim 1, characterized in that: In step S1, the control equation is: (1) in, Represents the rectangular coordinates in three-dimensional space, is the Hamiltonian operator, represents the velocity potential, represents the fluid domain of the cell; The boundary conditions are: (2) in, represents the normal derivative at the boundary; (3) in, Represents water depth; (4) in, represents the angular frequency, which is the wave number function, represents the acceleration due to gravity; (5) in, Representatives at the border Directional derivative, Representatives at the border Directional derivative, represent The Bloch wave number in the direction, represent The Bloch wave number in the direction, is the length of the unit, is the width of the cell, .

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