Method and system for evaluating hydrodynamic performance of a floating elastic disc coupled with a bottom-mounted cylinder

Abstract

The application discloses a floating elastic disc and bottom-coupling cylinder coupling water dynamic force evaluation method and system, comprising: establishing a multi-structure wave action model; defining a global coordinate system, a disc local polar coordinate system and a cylinder local polar coordinate system; dividing a fluid region into a disc lower internal domain, an external domain and a cylinder non-inlet flow region; constructing a velocity potential expression satisfying seabed impermeability, a free liquid surface, water elasticity coupling, disc free boundary and cylinder wall impermeability conditions; through vertical characteristic function expansion and coordinate translation addition theorem, converting a diffraction interference relationship into an algebraic equation group; solving coefficients and calculating a wave diffraction field, disc response, free surface elevation, disc vertical exciting force and cylinder horizontal exciting force. The method can simultaneously consider wave interference between a flexible floating structure and a rigid fixed structure, water elasticity deformation and multiple scattering feedback, and provides a semi-analytical evaluation means for water dynamic response prediction of a floating photovoltaic unit and a single-pile foundation adjacent arrangement.

Description

Technical Field

[0001] This invention relates to the fields of marine engineering hydrodynamics, wave diffraction analysis, and response evaluation of floating elastic structures, and particularly to a method, system, electronic device, and computer-readable storage medium for hydrodynamic evaluation of a floating elastic disk coupled with a bottom-sitting cylinder. Background Technology

[0002] In marine engineering, offshore energy development, floating platform layout, breakwater design, and safety assessment of offshore floating facilities, floating elastic structures and fixed rigid structures are often situated in adjacent wave environments. Upon wave incidence, the floating elastic structure undergoes a hydroelastic response, while diffraction and reflection occur around the fixed rigid structure. The superposition of scattered waves, diffracted waves, and radiation responses induced by structural deformation between the two structures results in a distinct multibody interference characteristic in the local wave field, structural load, and deformation response.

[0003] With the development of marine renewable energy facilities such as offshore wind power and floating photovoltaic systems, the arrangement of circular or near-circular flexible floating units adjacent to monopile foundations is becoming increasingly common. A monopile foundation can typically be approximated as a bottom-supporting cylinder extending along the water depth direction and fixed to the seabed; thin-film or flexible-supported floating photovoltaic units have large planar dimensions and small thicknesses, and under wave action, they may behave as floating elastic plates or elastic disks. Incident waves diffracting through the monopile foundation alter the local wave environment near the flexible floating unit, while the deformation, scattering, and radiation of the flexible floating unit also feed back into the pressure distribution and horizontal excitation force around the monopile. Therefore, under adjacent arrangement conditions, floating photovoltaic units and monopile foundations should not be simply regarded as independent hydrodynamic objects.

[0004] Existing theories for single floating elastic disks or plates can typically describe the hydroelastic response of isolated flexible floating bodies under regular wave action. However, when a rigid, bottom-supported cylinder is present nearby, the single-unit model struggles to reflect the influence of the cylinder's diffracted waves on the disk's edge matching conditions, the pressure distribution beneath the disk, the disk's deflection field, and the overall vertical load on the disk. Existing diffracted analysis of bottom-supported cylinders usually treats the structure as a rigid fixed body, which can describe the impermeable boundary of the cylinder wall and horizontal wave forces, but it is difficult to simultaneously reflect the flexible response generated by the adjacent floating elastic disk and its reaction force on the cylinder.

[0005] Furthermore, for adjacent systems containing circular floating elastic structures and fixed cylindrical structures, directly using general numerical discretization methods typically requires discretizing the free surface, elastic plate region, cylindrical boundary, and the interface conditions of multiple structures. This results in high computational cost and modeling complexity, and is also not conducive to quickly scanning parameters such as structural spacing, incident direction, cylinder dimensions, and disk stiffness. For regular basic configurations composed of circular boundaries, it is necessary to establish a compact semi-analytical coupled evaluation method using local polar coordinates, eigenfunction expansion, and coordinate translation addition theorem.

[0006] Therefore, a coupled hydrodynamic evaluation method is needed that can simultaneously handle floating elastic disks and bottomed cylinders within a unified analytical framework. This method should satisfy both the hydroelastic boundary conditions of the elastic thin plate and the rigid impermeable boundary conditions of the bottomed cylinder. Furthermore, it should be able to describe the wave interference and multiple scattering feedback relationships between the two types of structures through multi-coordinate system transformation, thereby outputting multiple response quantities such as the vertical excitation force of the disk, the horizontal excitation force of the cylinder, the deflection field of the disk, and the wave field of the surrounding free surface. Summary of the Invention

[0007] To address the aforementioned problems in the existing technology, this invention provides a method and system for evaluating the hydrodynamic coupling of a floating elastic disk and a bottom-sitting cylinder.

[0008] This invention is implemented by proposing a hydrodynamic evaluation method for a floating elastic disk coupled with a bottom-sitting cylinder, characterized by the following steps: S1. Establish a hydrodynamic calculation model consisting of a floating elastic disk and a bottom-supporting cylinder, wherein the floating elastic disk floats freely on the water surface and the bottom-supporting cylinder is fixed to the seabed and extends along the water depth direction. S2. Define a global coordinate system, a first local polar coordinate system with the center of the floating elastic disk as the origin, and a second local polar coordinate system with the center of the base cylinder as the origin, and determine the center distance and azimuth angle between the floating elastic disk and the base cylinder; S3. Divide the fluid region into an inner region located below the floating elastic disk, an outer region located outside the floating elastic disk and outside the base cylinder, and a non-flow-into-the-flow region occupied by the base cylinder; S4. Based on the linear potential flow theory, construct the incident wave velocity potential, the outer domain diffraction velocity potential and the inner domain velocity potential, and make the velocity potential satisfy the governing equation, the seabed impermeability condition, the free liquid surface condition, the floating elastic disk hydroelastic coupling condition, the floating elastic disk free boundary condition and the bottom cylinder wall impermeability condition. S5. The velocity potential is expanded into a series using the vertical characteristic function, angular harmonic function, Bessel function or modified Bessel function, and the wave field transformation relationship between the first local polar coordinate system and the second local polar coordinate system is established using the coordinate translation addition theorem.

[0009] S6. Apply an impermeable condition to the wall of the bottom cylinder, and apply a velocity potential continuity condition and a normal velocity continuity condition to the virtual interface corresponding to the edge of the floating elastic disk, to obtain a system of algebraic equations containing undetermined expansion coefficients. S7. Truncate and solve the system of algebraic equations to obtain the undetermined expansion coefficients; S8. Calculate the hydrodynamic response of the floating elastic disk and the bottom cylinder under the coupling action based on the undetermined expansion coefficient.

[0010] In the above technical solution, preferably, in step S4, the free boundary conditions of the floating elastic disk include zero bending moment and zero equivalent shear force at the edge of the disk.

[0011] In the above technical solution, preferably, in step S5, the coordinate translation addition theorem is used to transform the diffraction wave field represented by the center of the floating elastic disk to the local coordinate system of the bottom cylinder, or to transform the diffraction wave field represented by the center of the bottom cylinder to the local coordinate system of the floating elastic disk.

[0012] In the above technical solution, preferably, in step S6, after applying an impermeable condition to the wall of the base cylinder, the undetermined diffraction coefficient corresponding to the base cylinder is expressed as a function of the incident wave coefficient and the diffraction coefficient of the floating elastic disk.

[0013] In the above technical solution, preferably, in step S6, at the virtual interface, the velocity potential continuity condition and the normal velocity continuity condition are converted into an algebraic relationship about the outer domain coefficients and the inner domain coefficients by angular mode decoupling and vertical orthogonal projection.

[0014] In the above technical solution, preferably, in step S7, the order of the diagonal mode and the order of the vertical characteristic function are finitely truncated to construct a matrix equation and solve for the undetermined expansion coefficients.

[0015] In the above technical solution, preferably, in step S8, the hydrodynamic response includes at least one of the following: wave diffraction velocity potential, free surface elevation, hydrodynamic pressure, vertical displacement of the floating elastic disk, vertical excitation force of the floating elastic disk, horizontal excitation force of the bottom-sitting cylinder, and response amplification or attenuation characteristics caused by wave interference.

[0016] Compared with the prior art, the present invention has at least the following beneficial effects: 1. It can simultaneously handle the hydroelastic response of a floating elastic disk and the rigid diffraction boundary of a bottomed cylinder within the same theoretical framework.

[0017] 2. By setting up local coordinate systems for a disk and a cylinder, and introducing the coordinate translation addition theorem, wave interference and multibody diffraction coupling between two structures can be described.

[0018] 3. By using analytical elimination based on the impermeability condition of the cylindrical wall, the scale of unknowns in the coupled equation system is reduced, making it easier to construct a stable matrix solution.

[0019] 4. By utilizing the continuity of velocity potential and normal velocity at the virtual interface at the edge of the disk, the outer diffraction field can be effectively coupled with the inner hydroelastic field below the disk.

[0020] 5. It can be used to evaluate the impact of near-field fixed obstacles on the response of floating elastic structures, providing a calculation basis for the layout of floating structures, design of breakwater structures, safety assessment of marine engineering, and prediction of hydrodynamic loads.

[0021] 6. It can simultaneously output the vertical excitation force of the disk, the horizontal excitation force of the base cylinder, the disk deflection field, and the surrounding free surface wave field, which facilitates the identification of response amplification, attenuation, peak shift, and incident direction sensitivity between adjacent structures.

[0022] 7. For regular adjacent configurations formed by circular boundaries, this invention uses semi-analytical series expansion and matrix solving to replace the discretization of the complete spatial grid, which facilitates rapid scanning of parameters such as structural spacing, cylinder radius, incident direction and disk bending stiffness.

[0023] This invention also proposes a hydrodynamic evaluation system coupled with a floating elastic disk and a bottom-sitting cylinder, characterized in that it includes: The model building module is used to establish the geometric model and coordinate relationship between the floating elastic disk and the base cylinder; The watershed division module is used to divide the inner and outer regions below the disk and the non-flow-into-the-base cylinder. The potential function construction module is used to construct the velocity potential expansion that satisfies the linear potential flow control equation and boundary conditions; The coordinate translation module is used to establish wavefield transformation relationships between different local coordinate systems based on the coordinate translation addition theorem. The equation assembly module is used to assemble a system of algebraic equations based on the impermeability condition of the cylinder wall and the matching condition of the virtual interface at the edge of the disk. The solver module is used to solve for the undetermined expansion coefficients; A hydrodynamic output module is used to output the coupled hydrodynamic response of a floating elastic disk and a bottom-sitting cylinder. The coupled hydrodynamic response includes at least one of the following: free surface elevation, vertical displacement of the disk, vertical excitation force of the disk, and horizontal excitation force of the bottom-sitting cylinder. The system is used to execute the above-described method.

[0024] The present invention also proposes an electronic device, including a processor and a memory, wherein the memory stores a computer program, characterized in that the computer program, when executed by the processor, implements the above-described method.

[0025] The present invention also proposes a computer-readable storage medium having a computer program stored thereon, characterized in that the computer program implements the above-described method when executed by a processor. Attached Figure Description

[0026] Figure 1 This is a schematic diagram of the planar arrangement and coordinate system of the floating elastic disk and the base cylinder in an embodiment of the present invention; Figure 2 This is a schematic diagram of the side cross-section of the floating elastic disk and the bottom cylinder and the watershed division in an embodiment of the present invention; Figure 3 This is a flowchart of the coupled hydrodynamic evaluation method in an embodiment of the present invention; Figure 4 A comparison of the vertical excitation force of a floating elastic disk as a function of dimensionless wavenumber when the incident direction is 0°. Figure 5 A comparison of the vertical excitation force of a floating elastic disk as a function of dimensionless wavenumber when the incident direction is 180°. Figure 6 A comparison of the horizontal excitation force of the bottom cylinder as a dimensionless wavenumber when the incident direction is 0°. Figure 7 A comparison of the horizontal excitation force of the bottom cylinder as a dimensionless wavenumber when the incident direction is 180°. Figure 8 A comparison chart showing the change of the center deflection of a floating elastic disk with dimensionless wavenumber when the incident direction is 0°. Figure 9 This is a comparison chart showing the change in the center deflection of a floating elastic disk with dimensionless wavenumber when the incident direction is 180°. Detailed Implementation

[0027] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0028] This invention provides a method for evaluating the hydrodynamic coupling of a floating elastic disk and a bottom-sitting cylinder, comprising the following steps.

[0029] First, establish the geometric model. Assume a floating, elastic disk is freely floating on the water surface, with a radius of... It can be considered as an isotropic linear elastic thin plate; assuming the base cylinder is fixed to the seabed, its radius is... This creates a rigid boundary in the direction of water depth, preventing flow. Let the water depth be... The center distance between the floating elastic disk and the base cylinder is and satisfy equation (1): (1) This ensures that the two structures do not overlap in the horizontal plane.

[0030] Secondly, establish a coordinate system. Define a global coordinate system. The still water surface is The seabed is Defined at the center of a floating elastic disk. The first local polar coordinate system with the origin and the center of the base cylinder Second local polar coordinate system with origin Definition by point to The azimuth is ,Depend on point to The azimuth is .

[0031] Next, the watershed is divided. The entire fluid region is divided into outer regions. and inner domain The outer domain is the water area not covered by the floating elastic disk and located outside the bottom cylinder, while the inner domain is the water area below the floating elastic disk. The interior of the bottom cylinder is a non-flowable region and is not involved in the solution of the fluid velocity potential.

[0032] Under the assumptions of incompressible, inviscid, and irrotational linear potential flow, the velocity potential... Satisfying the Laplace equation shown in equation (2): (2) And it satisfies the impenetrable seabed condition shown in equation (3): (3) The free surface in the outer domain satisfies the linear free surface condition shown in equation (4): (4) The disk-covered area satisfies the hydroelastic coupling condition between the elastic thin plate and the fluid pressure shown in equation (5): (5) Parameters in equations (4) and (5) , and Defined according to formula (6): (6) in, Wave angular frequency , acceleration due to gravity , Free surface wavenumber parameter , For bending stiffness parameters , The equivalent draft of the plate , Bending stiffness per unit width of a floating elastic disk , fluid density , Elastic modulus of floating elastic disk material , For the thickness of the floating elastic disk , Poisson's ratio, Mass per unit area of ​​a floating elastic disk , Density of the material for the floating elastic disk .

[0033] The bottom cylindrical wall surface satisfies the impermeability condition shown in equation (7): (7) Furthermore, the incident wave velocity potential, the disk diffraction velocity potential, the cylinder diffraction velocity potential, and the velocity potential of the inner region below the disk are expanded into series. Let the incident wave amplitude be... The incident direction angle is The propagation wavenumber in open water is Then, the incident wave velocity potential in the global coordinate system can be written as equation (8): (8) In the local polar coordinate system of the disk and the local polar coordinate system of the cylinder, the incident wave is expanded into equation (9) and equation (10), respectively: (9) (10) in, For the first type of modified Bessel function, and These are the equivalent incident coefficients from the perspectives of the disk and the cylinder, respectively. Let the center of the disk be... Located at the origin of the global coordinate system Then the center of the cylinder The coordinates are .make The incident coefficient can be uniformly written as: (11) The external velocity potential is obtained by superimposing the incident wave, the disk diffracted wave, and the cylinder diffracted wave. The disk diffracted wave and the cylinder diffracted wave are represented by equations (12) and (13), respectively: (12) (13) in, This is a modified Bessel function of the second kind. The wavenumber corresponding to the external domain vertical mode. Let be the external domain vertical characteristic function (where Corresponding propagation mode ), and These are the diffraction coefficients for the disk and the cylinder, respectively. The velocity potential in the inner region below the disk is expressed by equation (14): (14) Wherein, represents the wavenumber of the hydroelastic mode in the inner domain. For the vertical characteristic function of the interior domain, These are the undetermined coefficients for the inner domain. All velocity potentials employ a boundary-normalized radial basis, resulting in expansion coefficients... At the interface, it is directly equal to the velocity potential amplitude of that mode (because... hour , , hour External domain vertical characteristic function and outer domain wavenumber Determine according to formula (15): (15) Inner domain vertical characteristic function and the internal domain water elastic modal wavenumber Determine according to formula (16): (16) In specific calculations, the outer domain wavenumber It is obtained by solving the characteristic equation in equation (15), where Take the propagation root that satisfies the radiation condition; in this modified Bessel function representation, the propagation mode can be written as ,in This is the actual propagation wavenumber in the usual sense. Take the root of the corresponding evanescent wave mode; the wavenumber of the hydroelastic mode in the inner domain. It is obtained by solving the characteristic equation in equation (16), where and For complex root modes in hydroelasticity problems, The remaining internal modal roots are represented by the following equations. The above characteristic equations can be solved using Newton's iteration method, secant method, or complex plane root search method, and then sorted according to modal order for the series expansion of equations (12) to (14).

[0034] To handle the interaction between two local polar coordinate systems, the coordinate translation addition theorem is introduced to transform the wave field represented by the center of the disk to the cylindrical local coordinate system, or vice versa. For the source coordinate system... The target coordinate system is The conversion can be performed using equation (17): (17) For the source coordinate system is The target coordinate system is The conversion can be performed using equation (18): (18) Equations (17) and (18) can unify the impermeability condition of the cylindrical wall and the matching condition of the virtual interface at the edge of the disk into the same algebraic solution framework. In the first local polar coordinate system, the total velocity potential of the external domain is composed of the incident wave, the diffracted wave of the floating elastic disk, and the diffracted wave of the bottomed cylinder after coordinate translation, and its final form is Equation (19): (19) On the bottom cylindrical wall, the condition that the radial derivative is zero is applied to the total velocity potential. By utilizing the orthogonality of the vertical characteristic function and the independence of the angular mode, the cylinder diffraction coefficient is expressed as a function of the incident wave coefficient and the disk diffraction coefficient, thereby realizing the analytical elimination of the unknown coefficients of the cylinder.

[0035] Specifically, the total velocity potential in the external domain within the cylindrical coordinate system is: (20) Take the radial partial derivative of equation (20) and substitute it into... Using the impermeability condition in equation (7), an explicit expression for the cylinder diffraction coefficient can be obtained: (twenty one) in, For Kroneck symbol (when) (Takes 1 if the condition is met, otherwise takes 0), used to convert the incident wave (only... The modalities are uniformly incorporated into the full modal summation framework. The apostrophe indicates that the function is differentiated with respect to its independent variable. Equation (21) shows that the cylindrical diffraction response is jointly determined by the direct incident wave excitation and the secondary scattering excitation from the disk diffraction wave.

[0036] The virtual interface corresponding to the edge of the floating elastic disk place ( This refers to the outer side of the disk, i.e., the outer region. (This refers to the inner side of the disk, i.e., the inner domain side). The continuity condition for the potential function is applied to both the outer domain velocity potential and the inner domain velocity potential. (twenty two) And the continuity condition for normal velocity: (twenty three) Combining this with the free boundary conditions of the disk, we obtain a system of closed algebraic equations concerning the coefficients of the disk's outer and inner domains. In actual calculations, the vertical mode cutoff for the outer domain is... The vertical mode of the interior domain is truncated as Diagonal mode truncation is After angular mode decoupling, vertical orthogonal projection, and elimination of external coefficients, the internal coefficient equation shown in equation (24) can be formed: (twenty four) in, These are the elements of the fluid dynamics matrix determined by coordinate translation relationships, vertical projection relationships, and virtual interface matching relationships. Let be the known right-hand side term formed by the incident wave and the reflection from the cylinder. The above matrix elements and right-hand side terms are calculated according to equation (25): (25) Among them, the angular interference matrix and Substituting equation (21) into equations (22) and (23) and performing angular decoupling and phase collapse, we obtain the following elements: ; ):

[0037] Pressure driving force and speed driving force It consists of the incident wave and its component reflected by the cylinder (only) (Non-zero):

[0038] To complete the equations corresponding to the hydroelastic modes in the inner domain, the edge of the disk must also satisfy the conditions of zero bending moment and zero equivalent shear force as shown in equations (26) and (27): (26) (27) Combining equations (24), (26), and (27), we can assemble them into the super matrix equation shown in equation (28): (28) in, Includes all unknown coefficients in the inner domain Seeking Then, for each vertical mode in the outer domain outer domain diffraction coefficient of the disk Restore using the following formula:

[0039] cylinder diffraction coefficient Then, according to equation (21), Direct calculation is performed. This allows for the calculation of the coupled wave field, hydrodynamic pressure, vertical response of the disk, vertical excitation force of the disk, and horizontal excitation force of the base cylinder between the floating elastic disk and the base cylinder. Since the governing equations and physical quantities in this embodiment are directly expressed in dimensional form using the International System of Units (SI), the vertical and horizontal excitation forces directly calculated from the expansion coefficients below are dimensional forces. .

[0040] For the vertical excitation force on the disk, assume that the frequency domain pressure satisfies a linear Bernoulli relationship and use a time factor. Then the vertical force at the bottom of the disk can be written as equation (29): (29) Substituting equation (14) into equation (29) and utilizing angular orthogonality, only axisymmetric modes are obtained. It contributes to the total vertical force. Using the equation of elastic motion of the plate water to eliminate the component of the hydrostatic restoring force, we obtain equation (30): (30) Among the factors The correction factor for net dynamic excitation force after excluding hydrostatic restoring force.

[0041] For the horizontal excitation force of the bottom cylinder, let... And the effective potential amplitude coefficient of the cylindrical surface is defined according to equation (31): (31) This utilizes the even symmetry of the modified Bessel function. Therefore Time is written as .

[0042] Then the cylinder is in the global direction and The horizontal excitation forces in the directions are respectively Equations (32) and (33): (32) (33) Equations (30), (32), and (33) enable the method to directly output the vertical force on the disk and the horizontal force on the cylinder (both dimensional forces) after obtaining the series expansion coefficients. Therefore, it is not necessary to perform numerical integration on the complete spatial pressure field again.

[0043] vertical deflection of the disk The velocity potential of the inner domain can be given by kinematic boundary conditions:

[0044] To facilitate understanding of the above formulas, the meanings of the main symbols in this invention are explained below.

[0045] Geometric and coordinate parameters: Represents the radius of the floating elastic disk , Indicates the radius of the base cylinder , Indicates water depth , Indicates the center of the floating elastic disk Center of the base cylinder Horizontal center distance between . Represents the global coordinate system. This represents the first local polar coordinate system with the center of the floating elastic disk as the origin. This represents a second local polar coordinate system with the center of the base cylinder as the origin. Indicates by point to azimuth angle, Indicates by point to The azimuth angle.

[0046] Wave, fluid, and elastic parameters: Indicates the amplitude of the incident wave , Indicates the incident wave direction angle; to avoid confusion, It only indicates the incident direction, and is different from the elastic stiffness parameter. . Indicates the angular frequency of the wave , Represents fluid density , Represents gravitational acceleration . , and Defined according to equation (6), where Free surface wavenumber parameter , For bending stiffness parameters , The equivalent draft of the plate . Indicates the bending stiffness per unit width of a floating elastic disk , Represents the elastic modulus of the floating elastic disk material , Indicates the thickness of the floating elastic disk , This represents the mass per unit area of ​​a floating elastic disk. , Indicates the density of the floating elastic disk material , This represents the Poisson's ratio of an elastic disk.

[0047] The units for all the quantities mentioned above have been indicated in the definitions above. It has been determined by equation (16). and Dimensional wavenumber In equations (30) to (33) , and The excitation force is dimensional, and the unit is . .

[0048] Operators, velocity potentials and mode functions: In equation (2) Describing the three-dimensional Laplace operator, in equation (5) This represents the biharmonic operator for bending of a thin plate acting on the horizontal coordinate. Represents the complex amplitude of the velocity potential. This represents the incident wave velocity potential in the global coordinate system. This represents the incident wave velocity potential in the first local polar coordinate system. This represents the incident wave velocity potential in the second local polar coordinate system. This represents the external diffraction velocity potential generated by the floating elastic disk. This represents the external diffraction velocity potential generated by the base cylinder. This represents the velocity potential in the inner region below the disk. This represents the total velocity potential of the external domain in the local coordinate system of the base cylinder. Represents the external domain vertical characteristic function. This represents the vertical characteristic function of the inner domain.

[0049] Wavenumber and Function: Indicates the outer domain Dimensional wavenumber corresponding to the vertical mode And determined by the characteristic equation in equation (15); where The complex wavenumber corresponding to the propagation mode can be written in this modified Bessel function representation as follows: , This refers to the actual propagation wavenumber in the usual sense. This represents the wave number corresponding to the evanescent wave mode. Indicates the inner domain number Dimensional wavenumbers corresponding to the water elastic modes And determined by the characteristic equation in equation (16); in the summation of the inner domain Used to represent complex root modes that occur in hydroelasticity problems. This represents the modified Bessel function of the first kind. This indicates a modified Bessel function of the second kind, with the apostrophe indicating that the function is differentiated with respect to its independent variable.

[0050] Expansion coefficients and summation index: Denotes the incident wave in the local coordinate system of the floating elastic disk. Angular expansion coefficient, Denotes the incident wave in the local coordinate system of the base cylinder. Angular expansion coefficient. Indicates the first The phase compensation factor of the incident wave relative to the origin of a local coordinate system and the global coordinate system. The first term in the diffraction velocity potential of the outer domain of a floating elastic disk is... The first vertical mode and the second The expansion coefficients corresponding to the angular mode. The first part represents the diffraction velocity potential in the outer domain of the base cylinder. The first vertical mode and the second The expansion coefficients corresponding to the angular mode. The first velocity potential in the inner region below the disk represents the... The first vertical mode and the second The expansion coefficients corresponding to the angular mode. , Indicates the vertical modality index. , , Indicates angular modal index, Indicates the vertical mode cutoff order. This indicates the angular mode cutoff order.

[0051] Matrix, sign function, and load parameters: and Both represent the Kronecker symbol, which takes the value 1 when the two subscripts are equal, and 0 otherwise. and This represents the angular interference matrix (elements are shown in the above formula). and This represents the driving force generated by the incident wave and the reflection from the cylinder. This represents the matrix elements determined by coordinate translation, boundary matching, and vertical projection. This represents the known right-hand side term in the corresponding test mode. This represents the assembled supermatrix. Indicates the expansion coefficient of the inner domain The unknown vector formed by the arrangement This represents the equivalent excitation vector. This represents the integral region on the lower surface of the floating elastic disk. This represents the dimensionless vertical excitation force acting on the floating elastic disk. (Given by equations (29) and (30), where the dynamic factor in equation (30) excludes the contribution of hydrostatic restoring force). and These represent the base cylinder in the global context. direction and Dimensional horizontal excitation force in the direction , Indicates the outer domain The integral of the vertical characteristic function along the water depth direction. Indicates the first surface of the bottom cylindrical surface The first vertical mode and the second The effective potential amplitude coefficient corresponding to the angular mode.

[0052] The following combination Figures 1 to 9 The following embodiments further illustrate the present invention. It should be understood that the following embodiments are used to illustrate the technical solutions of the present invention, and are not intended to limit the scope of protection of the present invention.

[0053] Example 1: This embodiment provides a method for evaluating the coupled hydrodynamics of a floating elastic disk and a fixed base cylinder. This method is applicable to predicting the hydrodynamic response when a fixed base cylinder exists near a floating elastic disk under regular wave action.

[0054] like Figure 1As shown, a floating elastic disk floats freely on the water surface, while a bottom-supported cylinder is fixed to the seabed and extends along the water depth direction. To describe the wave interference relationship between the two, this embodiment establishes a global coordinate system, a local polar coordinate system for the floating elastic disk, and a local polar coordinate system for the bottom-supported cylinder, and divides the fluid region into an inner region below the disk, an outer region outside the disk and outside the cylinder, and a region where the cylinder cannot enter the flow.

[0055] like Figure 2 As shown, the coupled hydrodynamic evaluation method in this embodiment may include steps such as parameter input, coordinate system and watershed establishment, outer domain velocity potential construction, inner domain velocity potential construction, bottom cylindrical wall condition processing, disk edge matching, disk free boundary condition processing, matrix equation assembly and solution, and hydrodynamic response output.

[0056] Step 1: Input the geometric and wave parameters. The geometric parameters include the radius of the floating elastic disk. Radius of the base cylinder , water depth Center distance Azimuth or Wave parameters include the incident wave angular frequency. Incident wave direction angle and incident wave amplitude Structural parameters include the disk's bending stiffness. Mass per unit area Compared to Poisson .

[0057] Step 2: Establish coordinate systems and watersheds. Establish a global coordinate system using the still water surface and seabed as vertical boundaries. Establish local polar coordinate systems using the center of the disk and the center of the cylinder, respectively. Define the water area below the disk as the inner region, the water area outside the disk and outside the cylinder as the outer region, and the area inside the cylinder as the non-flow region.

[0058] Step 3: Construct the external velocity potential. The total external velocity potential includes the incident wave velocity potential, the diffraction velocity potential caused by the floating elastic disk, and the diffraction velocity potential caused by the bottom-mounted cylinder. The incident wave velocity potential is expanded according to equations (8) to (11), and the disk diffraction velocity potential and the cylinder diffraction velocity potential are expanded according to equations (12) and (13), respectively. The external velocity potential satisfies the governing equations shown in equations (2) to (4), the impenetrable seabed condition, and the free surface condition of the external region.

[0059] Step four, construct the inner domain velocity potential. The inner domain velocity potential is located below the disk and satisfies the elastic thin-plate hydroelastic coupling condition shown in equation (5). The inner domain velocity potential is expanded according to equation (14), where... These are the main unknown coefficients for directly solving the subsequent matrix equations.

[0060] Step 5: Apply boundary conditions to the bottom cylindrical wall. At this point, the radial derivative of the total velocity potential along the cylinder is zero. The disk diffracted wave field has been translated using equation (17) and incorporated into the external total velocity potential of equation (20). The radial partial derivative of equation (20) is then substituted into... Using the impermeability condition in equation (7), the cylinder diffraction coefficient shown in equation (21) can be solved. The explicit expression of this is achieved. This step ensures that the cylinder coefficients are no longer independent unknowns in the final matrix equations, but are determined by the incident wave coefficients and the diffraction coefficients of the outer domain of the disk.

[0061] Step six: Apply virtual interface matching conditions to the disk edge. At this point, the velocity potential of the outer domain is continuous with that of the inner domain, and the radial velocity of the outer domain is continuous with that of the inner domain, thus satisfying equations (22) and (23). The cylindrical diffracted wave field has been translated by equation (18) and incorporated into the total velocity potential of the outer domain in equation (19). Substituting equations (19) and (14) into equations (22) and (23), and using angular orthogonality and vertical orthogonality projection, we obtain the inner domain coefficient equations shown in equations (24) and (25).

[0062] Step 7: Apply free boundary conditions to the disk. The edge of the disk satisfies zero bending moment and zero equivalent shear force, corresponding to equations (26) and (27) respectively. These two sets of conditions supplement two pure mechanical equations for each angular mode, so that the fluid matching equation shown in equation (24) can form a closed system together with the disk boundary mechanical equation.

[0063] Step 8: Assemble and solve the matrix equations. Select the angular truncation order. and vertical cutoff order Equations (24), (26), and (27) are then processed according to the unknown coefficients. Assembled in index order, we obtain the matrix equation shown in equation (28):

[0064] in, The coefficient matrix consists of fluid matching relationships, coordinate translation relationships, and disk free boundary conditions. Let be the vector of coefficients of the inner domain to be determined. Let be the equivalent excitation vector formed by the reflection of the incident wave and the base cylinder. After solving, the complete truncation expansion coefficients of the inner domain velocity potential are obtained, and the diffraction coefficients of the outer domain disk and the base cylinder are recovered accordingly.

[0065] Step 9: Output hydrodynamic response. Based on the obtained velocity potential coefficient, calculate the wave field distribution, hydrodynamic pressure below the floating elastic disk, vertical displacement response of the disk, free surface elevation, vertical excitation force of the disk, and horizontal excitation force of the bottom-sitting cylinder in the target water area. Among them, the vertical excitation force of the disk can be calculated according to equation (30), and the bottom-sitting cylinder is calculated globally. direction and The horizontal excitation force in the direction can be calculated according to equations (32) and (33). Based on the calculation results under different center distances, cylinder radii, wave frequencies and incident directions, the influence of the bottom-sitting cylinder on the hydrodynamic response of the floating elastic disk, as well as the feedback influence of the floating elastic disk on the horizontal excitation force of the bottom-sitting cylinder, can be evaluated.

[0066] Example 2 This embodiment illustrates the feasibility and application of the method described in Embodiment 1. A collinear arrangement is selected as a typical arrangement, with the center of the floating elastic disk being... The center of the base cylinder is When changing the structural spacing or the cylinder radius, by changing... Indicates the base of the cylindrical edge The positional change along the axial direction. The incident direction can be taken as... and Two opposite directions were used to evaluate the difference in coupling response when the wave first acts on the floating elastic disk and when it first acts on the bottom cylinder.

[0067] In one optional calculation setting, the radius of the floating elastic disk is taken. Radius of the base cylinder , water depth The net distance between the outer edge of the disk and the outer edge of the cylinder center distance The disk structure parameters can be set according to... and Settings, where and Defined by equation (6). The angular truncation order is used. and vertical cutoff order At the same time, it can output the vertical excitation force of the floating elastic disk, the horizontal excitation force of the bottom cylinder, the deflection of the disk center, and the local free surface elevation between the disk and the cylinder.

[0068] Figure 3 The figure illustrates one form of calculation output for the typical arrangement described above. It simultaneously displays the vertical excitation force of the floating elastic disk, the horizontal excitation force of the base cylinder, and the disk's center deflection, and allows comparison of the coupled system results with those of the corresponding isolated structures. This output enables the identification of response amplification, attenuation, peak position changes, and incident direction differences caused by the coupling effects of adjacent structures. Figure 3This invention is only intended to illustrate that the method of the present invention can output and compare multiple types of hydrodynamic responses, and does not limit the specific structural dimensions, wave parameters or cutoff order to which the present invention is applicable.

[0069] To verify the numerical stability of the method, the following can be fixed: and change and fixed and change The changes in the vertical excitation force of the disk, the horizontal excitation force of the cylinder, the disk deflection, and the free surface elevation were compared respectively. When the response quantity tends to stabilize with increasing truncation order, the truncation order used for subsequent evaluation can be determined. This convergence test proves that the matrix assembly and series truncation method of the present invention is suitable for simultaneously calculating the overall load, structural deformation, and local wave field.

[0070] To test the adaptability of the method to extreme cases, the radius of the base cylinder can be set to be much smaller than the radius of the floating elastic disk, and the center distance between the two can be sufficiently large, so that the disturbance of the base cylinder on the wave field near the disk tends to disappear. Under this condition, the method described in Example 1 degenerates into an isolated hydroelastic problem of a floating elastic disk, and the disk deflection field and vertical excitation force can be recovered. Alternatively, the radius of the floating elastic disk can be set to be much smaller than the radius of the base cylinder, and the center distance between the two can be sufficiently large, so that the disturbance of the floating elastic disk on the diffracted wave field of the cylinder tends to disappear. Under this condition, the method described in Example 1 degenerates into an isolated diffracted problem of a base cylinder, and the horizontal excitation force of the base cylinder can be recovered.

[0071] In the typical arrangement described above, changing the incident direction, structural spacing, radius of the base cylinder, or bending stiffness of the floating elastic disk can yield response peaks and peak locations at different frequencies. The calculation results can be used to identify the following situations: when the floating elastic disk is located on the wave-facing side, its scattered waves may enhance the horizontal excitation force of the downstream base cylinder; when the base cylinder is located on the wave-facing side, its diffracted waves may enhance the local wave field and central deflection of the downstream floating elastic disk; increasing the structural spacing does not necessarily lead to a monotonically reduced coupling response; changing the cylinder radius and disk bending stiffness will alter the scattering intensity, structural deformation level, and phase superposition relationship. Therefore, this invention can provide a calculation basis for spacing selection, stiffness selection, and load assessment in adjacent arrangements of circular flexible floating photovoltaic units and monopile foundations.

[0072] Example 3 This embodiment provides a hydrodynamic assessment system coupled with a floating elastic disk and a bottom-mounted cylinder. The system includes a model building module, a watershed division module, a potential function construction module, a coordinate translation module, an equation assembly module, a solution module, and a hydrodynamic output module.

[0073] The model building module is used to input and store parameters for the floating elastic disk, the bottom-sitting cylinder, water depth, and incident waves. The watershed delineation module is used to determine the inner, outer, and non-flow-prone regions based on the positions of the disk and cylinder. The potential function construction module is used to generate series expressions for the incident wave, the disk-diffracted wave, the cylinder-diffracted wave, and the wave field in the inner region below the disk. The coordinate translation module is used to perform wave field transformations between the first and second local polar coordinate systems. The equation assembly module is used to form matrix equations based on the boundary conditions. The solution module is used to solve for the undetermined expansion coefficients. The hydrodynamic output module outputs results such as free surface elevation, pressure, displacement, and excitation force.

[0074] Example 4 This embodiment provides an electronic device and a computer-readable storage medium. The electronic device includes a processor and a memory, in which a computer program is stored. When the processor executes the computer program, it can implement the hydrodynamic evaluation method for the coupling of a floating elastic disk and a bottom-sitting cylinder as described in Embodiment 1.

[0075] A computer program is stored on a computer-readable storage medium. When executed by a processor, the computer program can implement the method described in Embodiment 1.

Claims

1. A method for evaluating the hydrodynamic coupling of a floating elastic disk and a bottom-sitting cylinder, characterized in that, The steps include the following: S1. Establish a hydrodynamic calculation model consisting of a floating elastic disk and a bottom-supporting cylinder, wherein the floating elastic disk floats freely on the water surface and the bottom-supporting cylinder is fixed to the seabed and extends along the water depth direction. S2. Define a global coordinate system, a first local polar coordinate system with the center of the floating elastic disk as the origin, and a second local polar coordinate system with the center of the base cylinder as the origin, and determine the center distance and azimuth angle between the floating elastic disk and the base cylinder; S3. Divide the fluid region into an inner region located below the floating elastic disk, an outer region located outside the floating elastic disk and outside the base cylinder, and a non-flow-into-the-flow region occupied by the base cylinder; S4. Based on the linear potential flow theory, construct the incident wave velocity potential, the outer domain diffraction velocity potential and the inner domain velocity potential, and make the velocity potential satisfy the governing equation, the seabed impermeability condition, the free liquid surface condition, the floating elastic disk hydroelastic coupling condition, the floating elastic disk free boundary condition and the bottom cylinder wall impermeability condition. S5. The velocity potential is expanded into a series using the vertical characteristic function, angular harmonic function, Bessel function or modified Bessel function, and the wave field transformation relationship between the first local polar coordinate system and the second local polar coordinate system is established using the coordinate translation addition theorem. S6. Apply an impermeable condition to the wall of the bottom cylinder, and apply a velocity potential continuity condition and a normal velocity continuity condition to the virtual interface corresponding to the edge of the floating elastic disk, to obtain a system of algebraic equations containing undetermined expansion coefficients. S7. Truncate and solve the system of algebraic equations to obtain the undetermined expansion coefficients; S8. Calculate the hydrodynamic response of the floating elastic disk and the bottom cylinder under the coupling action based on the undetermined expansion coefficient.

2. The method according to claim 1, characterized in that, In step S4, the free boundary conditions of the floating elastic disk include zero bending moment and zero equivalent shear force at the edge of the disk.

3. The method according to claim 1, characterized in that, In step S5, the coordinate translation addition theorem is used to transform the diffracted wave field represented by the center of the floating elastic disk to the local coordinate system of the base cylinder, or to transform the diffracted wave field represented by the center of the base cylinder to the local coordinate system of the floating elastic disk.

4. The method according to claim 1, characterized in that, In step S6, after applying an impermeable condition to the wall of the base cylinder, the undetermined diffraction coefficient corresponding to the base cylinder is expressed as a function of the incident wave coefficient and the diffraction coefficient of the floating elastic disk.

5. The method according to claim 1, characterized in that, In step S6, at the virtual interface, the velocity potential continuity condition and the normal velocity continuity condition are transformed into an algebraic relationship with respect to the outer domain coefficients and the inner domain coefficients through angular mode decoupling and vertical orthogonal projection.

6. The method according to claim 1, characterized in that, In step S7, the diagonal modal order and the vertical eigenfunction order are finitely truncated to construct a matrix equation and solve for the undetermined expansion coefficients.

7. The method according to claim 1, characterized in that, In step S8, the hydrodynamic response includes at least one of the following: wave diffraction velocity potential, free surface elevation, hydrodynamic pressure, vertical displacement of the floating elastic disk, vertical excitation force of the floating elastic disk, horizontal excitation force of the bottom-sitting cylinder, and response amplification or attenuation characteristics caused by wave interference.

8. A hydrodynamic assessment system coupled with a floating elastic disk and a bottom-sitting cylinder, characterized in that, include: The model building module is used to establish the geometric model and coordinate relationship between the floating elastic disk and the base cylinder; The watershed division module is used to divide the inner and outer regions below the disk and the non-flow-into-the-base cylinder. The potential function construction module is used to construct the velocity potential expansion that satisfies the linear potential flow control equation and boundary conditions; The coordinate translation module is used to establish wavefield transformation relationships between different local coordinate systems based on the coordinate translation addition theorem. The equation assembly module is used to assemble a system of algebraic equations based on the impermeability condition of the cylinder wall and the matching condition of the virtual interface at the edge of the disk. The solver module is used to solve for the undetermined expansion coefficients; A hydrodynamic output module is used to output the coupled hydrodynamic response of a floating elastic disk and a bottom-sitting cylinder. The coupled hydrodynamic response includes at least one of the following: free surface elevation, vertical displacement of the disk, vertical excitation force of the disk, and horizontal excitation force of the bottom-sitting cylinder. The system is used to perform the method according to any one of claims 1 to 7.

9. An electronic device comprising a processor and a memory, wherein the memory stores a computer program, characterized in that, When the computer program is executed by the processor, it can implement the method described in any one of claims 1 to 7.

10. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the computer program is executed by a processor, it implements the method described in any one of claims 1 to 7.

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