Three-dimensional water wave energy band structure solving method based on high-order boundary elements

Abstract

The invention discloses a three-dimensional water wave energy band structure solving method based on high-order boundary elements, and belongs to the field of water wave regulation and control, and the method comprises the following steps: S1, building a three-dimensional fixed structure infinite period array model based on a linear potential flow theory, dividing a fluid into a plurality of same units, obtaining a control equation and boundary conditions satisfied by the fluid domain velocity potential; s2, a high-order boundary element method is selected to solve a three-dimensional Laplacian equation, boundary conditions are substituted into a boundary integral equation, a simple Green function is selected, high-order grid units are obtained through numerical discretization, and a linear homogeneous equation set is established; and S3, by arranging the equation set, converting a solution problem of the equation set into a characteristic value problem to solve a real Bloch wave number, and obtaining a water wave energy band structure diagram. According to the three-dimensional water wave energy band structure solving method based on the high-order boundary elements, rapid and accurate solving of a three-dimensional fixed structure in any shape and a water bottom topography water wave energy band structure is achieved.

Description

Technical Field

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

[0002] The core of water wave control lies in the design of water wave crystals, whose theoretical basis is the energy band structure of water wave crystals. Furthermore, by designing the forbidden band frequency range of the gradient array, the blocking, attenuation or focusing of water waves can be achieved. Water wave crystals have broad application prospects. For example, the high reflection in a specific frequency range can be realized by using the forbidden band characteristics to protect the coast from wave erosion; the control and energy collection of waves can be realized by reflecting waves through a periodic structure; current research mainly focuses on the narrowband frequency range, and the extension to broadband wave control will enhance its application potential in energy collection and coastal protection; regulating wave propagation to improve the safety and efficiency of offshore operations (such as drilling and shipping), etc.

[0003] Common research methods include: the plane wave expansion method: applicable to periodic structures in a plane, which simplifies complex three-dimensional problems into two-dimensional problems through Fourier expansion; the multiple scattering method: used to analyze the energy band structure of a vertical cylinder array, representing the wave field as the sum of cylinder scattering waves; the transfer matrix method: applicable to plane problems, connecting the wave fields on both sides of the structure through reflection and transmission coefficients; the finite element method: can be used to solve complex three-dimensional structures, but the calculation amount is huge.

[0004] These methods perform well in simple structures, but in actual ocean engineering, complex structures are often involved, and new methods need to be developed to analyze the interaction between water waves and three-dimensional structures. There are problems of large calculation amount and low efficiency when dealing with complex ocean engineering structures, so new calculation methods need to be developed to solve the energy band structure of three-dimensional structures. Summary of the Invention

[0005] The object of the present invention is to provide a method for solving the three-dimensional water wave energy band structure based on the high-order boundary element method, adding periodic Bloch boundary conditions to achieve the fast and accurate solution of the energy band structure of three-dimensional fixed structures, effectively improving the efficiency of solving the energy band structure, facilitating the construction of a water wave gradient array, and realizing 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 the high-order boundary element method, including the following steps: S1. Establish an infinite periodic array model of a three-dimensional fixed structure based on the linear potential flow theory, divide the fluid into several identical units, and obtain the control equation and boundary conditions satisfied by the velocity potential of the fluid domain; S2. Select the high-order boundary element method to solve the three-dimensional Laplace equation. Substitute the boundary conditions into the boundary integral equation and select a simple Green's function. Numerically discretize it into high-order grid cells to establish a linear homogeneous equation system. S3. By reorganizing the equation system, transform the problem of having a solution to the equation system into an eigenvalue problem to solve for the real Bloch wave number and obtain the water wave energy band structure diagram.

[0007] Preferably, in step S1, the governing equation is: (1); where represents the rectangular coordinates in three-dimensional space, is the Hamiltonian operator, represents the velocity potential, represents the fluid domain of the element; The boundary conditions are: (2); where represents the normal derivative on the boundary; (3); where represents the water depth; (4); where represents the angular frequency, which is a function of the wave number , represents the gravitational acceleration; (5); where represents the partial derivative in the direction on the boundary, represents the partial derivative in the direction on the boundary, represents the Bloch wave number in the direction, represents L the Bloch wave number in the W direction, .

[0008] Preferably, in step S2, select a simple Green's function that satisfies the three-dimensional Laplace equation: (6); where is the field point, is the source point. Using Green's theorem, the boundary integral equation is obtained as: (7); where is the solid angle coefficient, is Green's formula G at the field point normal derivative, is the velocity potential at the field point normal derivative. Substituting the boundary conditions (2)-(5) into equation (7) gives: (8); where, , , and represent the field points on the left side , front side , right side and back side of the element, , ; After discretizing the boundary into high-order grid elements, expanding formula (8) gives: (13); where, represents the index variable, N represents the number of elements, K represents the number of discrete element nodes, is the interpolation function of node , is the normalized local coordinate within the element, is the Jacobian matrix, represents the velocity potential of the th node, and represent the velocity potentials of the th nodes on the left and front sides of the element respectively, and represent the normal derivatives of the velocity potentials of the th nodes on the left and front sides of the element respectively.

[0009] Preferably, in step S3, by adjusting the column order, the following form is obtained: (14); where, represents the unknown velocity potentials of the nodes on the free surface, represents the remaining unknowns similar to , represents the coefficient matrix when the source points are distributed on the water surface , the matrix dimension is N × N , hereN represents the number of water surface nodes, represents when the source points are distributed on the water surface of the coefficient matrix, the matrix dimension N × M , where M represents the number of nodes other than the water surface, represents when the source points are distributed on other surfaces outside the water surface of the coefficient matrix, the matrix dimension M × N , represents when the source points are distributed on other surfaces outside the water surface of the coefficient matrix, the matrix dimension M × M , represents when the source points are distributed on the water surface and includes of the coefficient matrix, the matrix dimension N × N , represents when the source points are distributed outside the water surface and include of the coefficient matrix, the matrix dimension M × M ; By eliminating in formula (14), an eigenvalue problem in the form of is obtained, and the equation is as follows: (15); By substituting the real number into the matrix , repeating the above operations, different eigenvalues are obtained. Arrange in ascending order and number them. Connect the with the same number corresponding to different . Finally, the energy band structure diagram is obtained, and the dispersion equation between the frequency and the wave number is as follows: (16); where represents the acceleration due to gravity.

[0010] Therefore, the present invention adopts the above-mentioned method for solving the three-dimensional water wave energy band structure based on the high-order boundary element, adds the periodic Bloch boundary condition, realizes the fast and accurate solution of the energy band structure of the three-dimensional fixed structure, effectively improves the efficiency of solving the energy band structure, facilitates the construction of the water wave gradual array, and realizes the water wave control.

[0011] Next, through the drawings and embodiments, the technical solution of the present invention will be further described in detail. Description of the Drawings

[0012] Figure 1 Schematic diagram of the periodic unit of a three-dimensional water wave energy band structure solving method based on high-order boundary elements according to the present invention; Figure 2 Schematic diagram of the quadrilateral mesh unit of a three-dimensional water wave energy band structure solving method based on high-order boundary elements according to the present invention; Figure 3 Schematic diagram of the triangular mesh unit of a three-dimensional water wave energy band structure solving method based on high-order boundary elements according to the present invention; Figure 4 Comparison diagram of the energy band results of the airspace unit with a length, width, and height of 1 meter for a three-dimensional water wave energy band structure solving method based on high-order boundary elements according to the present invention; Figure 5 Comparison diagram of the energy band results of the bottom-mounted cylindrical unit with a radius of 0.25 meters in the airspace unit with a length, width, and height of 1 meter for a three-dimensional water wave energy band structure solving method based on high-order boundary elements according to the present invention; Figure 6 Comparison diagram of the energy band results of the airspace unit with a length of 2 meters and a width and height of 1 meter for a three-dimensional water wave energy band structure solving method based on high-order boundary elements according to the present invention; Figure 7 Comparison diagram of the energy band results of the vertical square bottom-mounted structure with a length and width of 0.25 meters in the watershed unit with a length, width, and height of 1 meter for a three-dimensional water wave energy band structure solving method based on high-order boundary elements according to the present invention; Figure 8 Comparison diagram of the energy band results of the vertical C-shaped cylindrical bottom-mounted structure in the watershed unit with a length, width, and height of 1 meter for a three-dimensional water wave energy band structure solving method based on high-order boundary elements according to the present invention; Figure 9 C-shaped cylinder with a wall thickness of 0.05 meters and a radius of 0.25 meters for a three-dimensional water wave energy band structure solving method based on high-order boundary elements according to the present invention; Figure 10 Energy band structure diagram of the truncated cylinder with a radius of 0.25 meters and a draft of 0.5 meters in the watershed unit with a length, width, and height of 1 meter for a three-dimensional water wave energy band structure solving method based on high-order boundary elements according to the present invention; Figure 11 Result diagram of the longitudinal oscillation direction force on each column of the 5-column periodic array of truncated cylinders with a row and column spacing of 1 meter, a radius of 0.25 meters, and a draft of 0.5 meters for a three-dimensional water wave energy band structure solving method based on high-order boundary elements according to the present invention; Figure 12 Result diagram of the heave direction force on each column of the 5-column periodic array of truncated cylinders with a row and column spacing of 1 meter, a radius of 0.25 meters, and a draft of 0.5 meters for a three-dimensional water wave energy band structure solving method based on high-order boundary elements according to the present invention. Detailed implementation manners

[0013] To make the objectives, technical solutions and advantages of the embodiments of the present invention clearer, the following will clearly and completely describe the technical solutions in the embodiments of the present invention with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are some, but not all, of the embodiments of the present invention. Usually, the components of the embodiments of the present invention described and illustrated in the accompanying drawings here can be arranged and designed in various different configurations.

[0014] The following will describe in detail the implementation manners of the present invention with reference to the accompanying drawings.

[0015] Embodiment A method for solving the three-dimensional water wave energy band structure based on the high-order boundary element includes the following steps: S1. Establish an infinite periodic array model of a three-dimensional fixed structure based on the linear potential flow theory. As shown, in water with a finite depth of, a periodically arranged truncated cylinder array is fixed. A three-dimensional rectangular coordinate system is adopted, where the plane is fixed at the average position of the free surface, the axis is vertically upward, and the periodic array of several identical truncated cylinders extends infinitely in the direction and the direction. The fluid is divided into identical units. The length of each unit in the direction is, the width in the direction is, the depth in the direction is, and each unit contains the same truncated cylinder. There is a Bloch phase relationship between the waves in adjacent units; Figure 1 obtain the control equation and boundary conditions satisfied by the velocity potential of the fluid domain. The control equation is: z (1); wherein, represents the rectangular coordinates in three-dimensional space, is the Hamiltonian operator, represents the velocity potential, and represents the fluid domain of the unit; The boundary conditions are: L (2); wherein, represents the normal derivative at the boundary; W z (3); wherein, represents the water depth; ​​​​​​​​​​​​​​​​​​ (4); Among them, ω represents the angular frequency, which is a function of the wave number ; represents the acceleration due to gravity; (5); Among them, represents the partial derivative in the direction, represents the partial derivative in the direction, represents the Bloch wave number in the direction, represents L the Bloch wave number in the W direction, is the length of the unit,

[0016] S2. Select the high-order boundary element method to solve the three-dimensional Laplace equation. Substitute the boundary conditions into the boundary integral equation and select a simple Green's function. Discretize it numerically into high-order grid elements to establish a linear homogeneous equation system; Select a simple Green's function that satisfies the three-dimensional Laplace equation: (6); Among them is the field point, is the source point. Using Green's theorem, the boundary integral equation is obtained as: (7); Among them is the solid angle coefficient, is Green's formula G at the field point normal partial derivative, is the velocity potential at the field point normal partial derivative. Substitute the boundary conditions (2)-(5) into equation (7) to obtain: (8); Among them, , , and represent the field points on the left side , front side , right side and back side of the unit respectively, , ; Such as Figure 2 andFigure 3 As shown, the boundary is discretized into quadrilateral and triangular high-order boundary element cells, and the differential element area d s and the isoparametric element area have the following relationship: (9); where is the normalized local coordinate within the element, is the Jacobian matrix, and the expression is as follows: (10); The normal velocity, velocity potential, and coordinates of the liquid at any point can all be expanded as the series sum of the product of the corresponding physical values of the element where it is located and the quadratic interpolation function: (11); where K represents the number of nodes of the element, denotes the th derivative of the velocity potential in the normal direction at the th node, represents the velocity potential at the th node, denotes the coordinates of the th node, is the interpolation function of node (12); Expanding formula (8) gives: (13); where represents the index variable, N represents the number of elements, and respectively represent the velocity potentials at the th nodes of the left and front elements, and respectively represent the derivatives of the velocity potentials in the normal direction at the th nodes of the left and front elements.

[0017] S3. By organizing the equations, the problem of having solutions to the equations is transformed into an eigenvalue problem to solve for the real Bloch wave number. By adjusting the column order, the following form is obtained: (14); where represents the unknown velocity potentials of each node on the free surface, denotes the remaining unknowns similar to , represents the coefficient matrix of when the source points are distributed on the water surface, and the matrix dimensionN × N , here N represents 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 other surfaces other than the water surface The coefficient matrix of M × N , Indicates that the source points are distributed on other 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 formula (14) Get the shape The eigenvalue problem of is as follows: (15); By substituting real numbers To Matrix , repeat the above operation to get different eigenvalues ,Will Arrange the numbers from small to large and number the corresponding The same number After connecting, 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.

[0018] By comparing the band structure results of different three-dimensional vertical bottom-mounted structures and two-dimensional structures, the accuracy of the three-dimensional band structure solution method was preliminarily verified. Figures 4 - 8 as well as Figures 10 - 12 The dimensionless wave number is , dimensionless Bloch wave number for ,like Figure 4 and Figure 5As shown, the air domain results obtained by the method in the paper (McIver, 2000) (hereinafter referred to as the existing method 1) and the results of the bottom-mounted cylinder with a radius of 0.25 m are respectively compared with the results obtained by the present invention, and the results of the present invention are in good agreement with those of the existing method 1. Then, the finite element method is used to solve the Helmholtz equation, and a two-dimensional infinite periodic array model of the structure is established. As Figures 6 - 9 shown, the two-dimensional finite element results and the results of the present method for the three-dimensional bottom-mounted structure are respectively compared (the three-dimensional bottom-mounted structures are the air domain and the bottom-mounted cuboid with a length of 2 m, a width of 1 m and a height of 1 m, and the C-shaped cylinder with a wall thickness of 0.05 m and a radius of 0.25 m), and the results are in good agreement, preliminarily verifying the accuracy of the present method.

[0019] As Figure 11 and Figure 12 shown, the wave exciting forces in the surge direction and the heave direction acting on the periodic truncated cylinder array are calculated by the method in the paper (Yue, 2019) (hereinafter referred to as the existing method 2). By comparing the variation of the wave exciting forces acting on the periodic truncated cylinder and its energy band structure diagram ( Figure 10 ), it is found that within the passband range, the forces change alternately. However, within the stopband range, the forces change stepwise with the cylinder number, and the front side is significantly higher than the rear side cylinders.

[0020] Therefore, the present invention adopts the above-mentioned three-dimensional water wave energy band structure solving method based on the high-order boundary element, adds the periodic Bloch boundary condition, realizes the rapid and accurate solution of the energy band structure of the three-dimensional fixed structure, effectively improves the efficiency of the energy band structure solution, facilitates the construction of the water wave gradual change array, and realizes the water wave control.

[0021] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit them. Although the present invention has been described in detail with reference to the preferred embodiments, those of ordinary skill in the art should understand that they can still modify or equivalently replace the technical solutions of the present invention, and these modifications or equivalent replacements cannot make the modified technical solutions 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 energy band structure based on the high-order boundary element, characterized in that: It includes the following steps: S1. Establish an infinite periodic array model of three-dimensional fixed structures based on the linear potential flow theory, divide the fluid into several identical units, and obtain the control equations and boundary conditions satisfied by the velocity potential of the fluid domain; S2. Select the high-order boundary element method to solve the three-dimensional Laplace equation, substitute the boundary conditions into the boundary integral equation and select a simple Green's function, discretize it numerically into high-order grid elements, and establish a linear homogeneous equation system; S3. By reorganizing the equation system, transform the problem of the equation system having a solution into an eigenvalue problem to solve for the real Bloch wave number, and obtain the water wave energy band structure diagram.

2. A three-dimensional water wave energy band structure solving method based on high-order boundary elements according to claim 1, characterized in that: In step S1, the control equation is: (1)； Among them, represents the rectangular coordinates in three-dimensional space, is the Hamiltonian operator, represents the velocity potential, represents the fluid domain of the element; The boundary conditions are: (2)； Among them, represents the normal derivative at the boundary; (3)； Among them, represents water depth; (4)； Among them, ω represents the angular frequency, which is a function of the wave number ; represents the acceleration due to gravity; (5)； Among them, represents the partial derivative in the direction, represents the partial derivative in the direction, represents the Bloch wave number in the direction, represents L the Bloch wave number in the W direction, .

3. A three-dimensional water wave energy band structure solving method based on high-order boundary elements according to claim 2, characterized in that: In step S2, select a simple Green's function that satisfies the three-dimensional Laplace equation: (6)； where is the field point, is the source point. Using Green's theorem, the boundary integral equation is obtained as follows: (7)； wherein is the solid angle coefficient, is Green's formula G at the field point normal derivative, is the velocity potential at the field point normal derivative. Substituting the boundary conditions (2)-(5) into equation (7) gives: (8)； Among them, , , and represent the field points on the left side , front side , right side and back side of the unit, , ; After discretizing the boundary into high-order grid elements, expand formula (8) to obtain: (13)； Among them, represents the index variable, N represents the number of elements, K represents the number of discrete element nodes, is the node interpolation function, is the normalized local coordinate within the element, is the Jacobian matrix, represents the th node's velocity potential, and respectively represent the velocity potential of the th node of the left and front elements, and respectively represent the normal derivative of the velocity potential of the th node of the left and front elements.

4. A three-dimensional water wave energy band structure solving method based on high-order boundary elements according to claim 3, characterized in that: In step S3, by adjusting the column order, the following form is obtained: (14)； Among them, represents the unknown velocity potential of each node on the free surface, denotes the remaining unknown quantities similar to ; represents the coefficient matrix when the source points are distributed on the water surface, with the matrix dimension × N × N where N represents the number of nodes on the water surface, represents the coefficient matrix when the source points are distributed on the water surface, with the matrix dimension × N × M where M represents the number of nodes other than those on the water surface, represents the coefficient matrix when the source points are distributed on a surface other than the water surface, with the matrix dimension × M × N , represents the coefficient matrix when the source points are distributed on a surface other than the water surface, with the matrix dimension × M × M , represents the coefficient matrix when the source points are distributed on the water surface and contains and with the matrix dimension N × N , represents the coefficient matrix when the source points are distributed outside the water surface and contains with the matrix dimension M × M ; By eliminating the in formula (14), we obtain an eigenvalue problem in the following form: (15)； By substituting real numbers into the matrix , repeating the above operations, different eigenvalues are obtained. Arrange in ascending order and number them. Connect the corresponding with the same number . Finally, the band structure diagram is obtained. The dispersion equation for frequency and wave number is as follows: (16)； Among them, represents the acceleration due to gravity.

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