Solution method for oscillating hydrofoil motion response and fluid-structure coupling deformation

Abstract

The application relates to the technical field of tidal current energy utilization, and discloses a solution method for oscillating hydrofoil motion response and fluid-structure coupling deformation, which comprises the following steps: defining hydrofoil motion modes and key parameters of structures and fluids, and clearly defining the core features of given pitching motion, free heaving motion and bending-torsion coupling deformation; establishing a heaving motion dynamics equation based on Newton's second law, and constructing a bending-torsion coupling deformation control equation in combination with the principle of structural mechanics; building a three-dimensional numerical simulation model containing a block grid and a boundary layer encryption design, and meeting the calculation requirements through spanwise and time discretization processing; solving fluid distribution load and total lift based on the Navier-Stokes equation; solving the heaving motion response by using the fourth-order Runge-Kutta method, and solving the bending-torsion coupling deformation by using the Newmark- β time integration method and the finite difference method. In this way, the strong coupling characteristics of hydrofoil motion and fluid-structure coupling deformation can be accurately captured, thereby providing technical support for the design optimization and performance improvement of an oscillating hydrofoil type tidal current energy device.

Description

Technical Field

[0001] This application relates to the field of tidal energy utilization technology, for example to a method for solving the motion response and fluid-structure interaction deformation of an oscillating hydrofoil. Background Technology

[0002] Tidal energy, as a renewable and clean marine energy source, boasts significant advantages such as high energy density, strong stability, and good predictability. Oscillating hydrofoil tidal energy generation devices are one of the core pieces of equipment for capturing tidal energy. They convert fluid kinetic energy into mechanical energy through the periodic oscillating motion of the hydrofoil within the tidal current, thereby generating electricity. The energy capture efficiency and structural reliability of these devices directly determine their engineering application value.

[0003] As a key component of oscillating hydrofoil tidal energy devices, the oscillating hydrofoil employs a composite motion mode of "given pitch motion and free heave motion." The pitch axis is typically positioned near the hydrofoil's pressure center to reduce pitch moment and optimize energy capture efficiency. During actual operation, the hydrofoil not only undergoes free heave motion under fluid loads but also experiences bending and torsional deformation (i.e., fluid-structure interaction deformation) due to the strong coupling between its flexible characteristics and the fluid loads. This coupling effect of motion and deformation directly alters the hydrofoil's actual angle of attack distribution, fluid load characteristics, and motion attitude, thus significantly impacting the device's energy capture efficiency and structural fatigue life. Therefore, accurately describing the motion response and fluid-structure interaction deformation characteristics of the oscillating hydrofoil is a crucial prerequisite for the optimized design and performance improvement of tidal energy devices.

[0004] However, existing technologies have many shortcomings in the motion and deformation analysis of oscillating hydrofoils, making it difficult to meet the simulation accuracy and reliability requirements of engineering applications:

[0005] Firstly, traditional studies often adopt the assumption of rigid hydrofoils, ignoring the flexible deformation of hydrofoils, or simplifying the deformation to bending deformation in a single direction, without considering the coupling effect of bending and torsion. This leads to a large deviation between the numerical simulation results and the actual working conditions, and cannot accurately reflect the influence of fluid-structure interaction on the motion and load distribution of hydrofoils.

[0006] Secondly, there is a strong coupling relationship between fluid load and hydrofoil motion and deformation. Existing solution methods mostly adopt a one-way coupling strategy (i.e., first calculate the fluid load, and then apply it as an external load to the structure to solve the deformation), without considering the reaction of the hydrofoil attitude change on the fluid flow field after deformation, resulting in insufficient accuracy of the coupled solution.

[0007] Third, the numerical simulation model for the oscillating hydrofoil has defects. For example, unreasonable setting of the computational domain boundary leads to insufficient development of the incoming flow or insufficient diffusion of the wake. The mesh generation is not adapted to the wall requirements of the turbulence model, resulting in low accuracy of fluid load calculation. The time and space discretization scheme does not meet the stability conditions, which makes the solution process prone to divergence. These problems all affect the reliability of the simulation results.

[0008] Fourth, there is a lack of a systematic and complete solution framework. Existing technologies mostly focus on single aspects of motion response or fluid-structure interaction deformation, and have not formed a complete solution method from parameter definition, model establishment, load solving to motion and deformation iterative calculation and mesh updating, which is difficult to apply directly to engineering design and optimization.

[0009] Furthermore, as tidal energy devices develop towards larger scale and greater scale, the performance requirements for oscillating hydrofoils are constantly increasing. Traditional simplified models and solution methods can no longer meet the needs of device optimization design for accuracy and comprehensiveness.

[0010] It should be noted that the information disclosed in the background section above is only used to enhance the understanding of the background of this application, and therefore may include information that does not constitute prior art known to those skilled in the art. Summary of the Invention

[0011] To provide a basic understanding of some aspects of the disclosed embodiments, a brief summary is given below. This summary is not intended as a general commentary, nor is it intended to identify key / important components or describe the scope of protection of these embodiments, but rather as a prelude to the detailed description that follows.

[0012] This disclosure provides a method for solving the motion response and fluid-structure interaction deformation of an oscillating hydrofoil, so as to achieve an accurate description of the full-dimensional characteristics of the hydrofoil, including "given pitch motion + free heave motion + bending-torsional coupling deformation", taking into account both calculation accuracy and stability, and providing reliable technical support for the optimized design and performance improvement of oscillating hydrofoil tidal energy devices.

[0013] In some embodiments, the method for solving the motion response and fluid-structure interaction deformation of the oscillating hydrofoil includes: S1: defining the motion mode and core parameters of the hydrofoil, and constructing the heave motion dynamic equation and bending-torsional coupling deformation control equation of the hydrofoil; S2: building a three-dimensional numerical simulation model of the hydrofoil, including setting the computational domain, mesh generation, and discretization; S3: based on the Navier-Stokes equations for incompressible fluids, using the finite volume method to solve the distributed fluid load on the hydrofoil surface, and integrating the distributed fluid load to obtain the total fluid lift at the current moment; S4: based on the total fluid lift, total hydrofoil mass, and heave parameters at the current moment, using the fourth-order Runge-Kutta method to solve the heave motion response at the next moment; S5: based on the bending parameters, torsional parameters, force distribution parameters, and structural parameters of the hydrofoil at the current moment, using the Newmark- β The time integration method and the finite difference method, combined with the fixed-free boundary conditions, are assembled into a system of linear equations; the Gaussian elimination method is used to solve for the bending and torsional deformations of each spanwise node at the next moment.

[0014] In some embodiments, the solution system for the motion response and fluid-structure interaction deformation of the oscillating hydrofoil includes: a definition module configured to define the motion mode and core parameters of the hydrofoil, and construct the heave motion dynamics equations and bending-torsional coupling deformation control equations of the hydrofoil; a model building module configured to build a three-dimensional numerical simulation model of the hydrofoil, including computational domain setting, mesh generation, and discretization processing; a first calculation module configured to solve the distributed fluid load on the hydrofoil surface using the finite volume method based on the Navier-Stokes equations for incompressible fluids, and integrate the distributed fluid load to obtain the total fluid lift at the current moment; a second calculation module configured to solve the heave motion response at the next moment using the fourth-order Runge-Kutta method based on the total fluid lift, total mass of the hydrofoil, and heave parameters at the current moment; and a third calculation module configured to solve the heave motion response at the next moment using the Newmark- β The time integration method and the finite difference method, combined with the fixed-free boundary conditions, are assembled into a system of linear equations; the Gaussian elimination method is used to solve for the bending and torsional deformations of each spanwise node at the next moment.

[0015] The solution method for the motion response and fluid-structure interaction deformation of the oscillating hydrofoil provided in this disclosure can achieve the following technical effects:

[0016] The coupling relationship description is more comprehensive. For the first time, it systematically considers the full-dimensional coupling relationship of "pitch motion-heave motion-bending-torsional coupling deformation" of a semi-active oscillating hydrofoil, breaking through the limitations of traditional rigid hydrofoil assumptions or simplifications based on single-direction deformation. The simulation results are more consistent with actual working conditions. Using block mesh generation and boundary layer refinement techniques, it adapts to the wall requirements of the SST k-ω turbulence model, significantly improving the accuracy of fluid load calculation. Combined with the fourth-order Runge-Kutta method and Newmark- β The time integration method (second-order accuracy) solves for motion and deformation, balancing computational efficiency and stability, resulting in higher accuracy and stability while avoiding divergence in the solution process. A more complete solution system is constructed, encompassing parameter definition, model establishment, load solving, and motion and deformation calculation, forming a standardized and reusable solution scheme that can be directly applied to engineering design without requiring additional intermediate steps. It supports flexible adjustment of hydrofoil structural parameters, motion parameters, and fluid parameters, quickly adapting to the simulation needs of semi-active oscillating hydrofoils of different sizes and operating conditions. This provides accurate data support for structural optimization, energy capture efficiency improvement, and structural fatigue life assessment of tidal energy devices, promoting the large-scale engineering application of oscillating hydrofoil tidal energy devices.

[0017] The above general description and the description below are exemplary and illustrative only and are not intended to limit this application. Attached Figure Description

[0018] One or more embodiments are illustrated by way of example with reference to the accompanying drawings. These illustrations and drawings do not constitute a limitation on the embodiments. Elements having the same reference numerals in the drawings are shown as similar elements. The drawings are not to be scaled. And wherein:

[0019] Figure 1 This is a schematic diagram of the first method for solving the motion response and fluid-structure interaction deformation of an oscillating hydrofoil provided in this embodiment of the present disclosure;

[0020] Figure 2 This is a schematic diagram of the structure of the electric motor-driven oscillating hydrofoil tidal energy generation device provided in the embodiments of this disclosure;

[0021] Figure 3 This is a schematic diagram of the second method for solving the motion response and fluid-structure interaction deformation of an oscillating hydrofoil provided in this embodiment of the present disclosure;

[0022] Figure 4 This is an accuracy verification diagram comparing the calculation results of the solution method provided in this embodiment with experimental data containing error bands;

[0023] Figure 5 This is a comparison chart of the cumulative average computation time of the solution method provided in this embodiment and the traditional method as the simulation progresses;

[0024] Figure 6 This is a comparison chart of the time history response stability of the solution method provided in this embodiment and the traditional explicit method under extreme conditions;

[0025] Figure 7 This is a comparison diagram of the total system energy evolution during the simulation process between the solution method provided in this embodiment and the traditional method;

[0026] Figure 8 This is a schematic diagram of a solution system for the motion response and fluid-structure interaction deformation of an oscillating hydrofoil provided in an embodiment of this disclosure.

[0027] Figure label:

[0028] 80. Solution system for the motion response and fluid-structure interaction deformation of oscillating hydrofoils; 81. Definition and construction module; 82. Model building module; 83. First calculation module; 84. Second calculation module; 85. Third calculation module. Detailed Implementation

[0029] To provide a more detailed understanding of the features and technical content of the embodiments of this disclosure, the implementation of the embodiments of this disclosure will be described in detail below with reference to the accompanying drawings. The accompanying drawings are for illustrative purposes only and are not intended to limit the embodiments of this disclosure. In the following technical description, for ease of explanation, several details are used to provide a full understanding of the disclosed embodiments. However, one or more embodiments may still be implemented without these details. In other cases, well-known structures and devices may be simplified in their depiction to simplify the drawings.

[0030] The terms "first," "second," etc., used in the specification and accompanying drawings of this disclosure are used to distinguish similar objects and are not necessarily used to describe a specific order or sequence. It should be understood that such data can be interchanged where appropriate for the embodiments of this disclosure described herein. Furthermore, the terms "comprising" and "having," and any variations thereof, are intended to cover non-exclusive inclusion.

[0031] Unless otherwise stated, the term "multiple" means two or more.

[0032] In this embodiment of the disclosure, the character " / " indicates that the objects before and after it are in an "or" relationship. For example, A / B means: A or B.

[0033] The term "and / or" describes an association between objects, indicating that three relationships can exist. For example, A and / or B means: A or B, or A and B.

[0034] The term "correspondence" can refer to an association or binding relationship. The correspondence between A and B means that there is an association or binding relationship between A and B.

[0035] Combination Figure 1 As shown, this disclosure provides a first method for solving the motion response and fluid-structure interaction deformation of an oscillating hydrofoil, including:

[0036] S1: Define the motion mode and core parameters of the hydrofoil, and construct the heave motion dynamics equation and bending-torsional coupling deformation control equation of the hydrofoil;

[0037] S2: Build a three-dimensional numerical simulation model of the hydrofoil, including setting the computational domain, mesh generation, and discretization.

[0038] S3: Based on the Navier-Stokes equations for incompressible fluids, the distributed fluid load on the hydrofoil surface is solved using the finite volume method, and the distributed fluid load is integrated to obtain the total fluid lift at the current moment.

[0039] S4: Based on the total fluid lift, total hydrofoil mass, and heave parameters at the current moment, the fourth-order Runge-Kutta method is used to solve the heave motion response at the next moment;

[0040] S5: Based on the bending parameters, torsional parameters, force distribution parameters, and hydrofoil structural parameters at the current moment, using Newmark- β The time integration method and the finite difference method, combined with the fixed-free boundary conditions, are assembled into a system of linear equations; the Gaussian elimination method is used to solve for the bending and torsional deformations of each spanwise node at the next moment.

[0041] I. Defining the core parameters of the hydrofoil

[0042] A linear motor-type oscillating hydrofoil tidal current power generation device, such as... Figure 2 As shown. Before conducting numerical simulations, it is necessary to clarify the core parameters such as the motion mode parameters, structural parameters, and fluid medium parameters of the hydrofoil, so as to provide the basic input for subsequent construction of the heave motion dynamics model and simulation calculations.

[0043] Bending parameters include bending deflection, deflection velocity, and deflection acceleration, among other bending-related parameters. Torsional parameters include torsion angle, torsion angular velocity, and torsion angular acceleration, among other torsion-related parameters. Force distribution parameters include distributed lift and distributed torque, among other force distribution-related parameters.

[0044] 1.1 Motion Mode Parameters

[0045] The hydrofoil employs a composite mode of "given pitch motion + free heave motion," while also considering bending and torsional deformation caused by fluid-structure interaction.

[0046] (1) Pitch motion: This is a pre-defined periodic motion, and the equation of motion is: ,in, The amplitude of the pitch angle;a Angular frequency, based on Calculations show that The pitch period;

[0047] (2) Heave motion: Free translational motion (along the vertical direction) driven by fluid load, initial displacement initial velocity ;

[0048] (3) Fluid-structure interaction deformation: including spanwise deformation z Vertical bending deflection at the point and elastic torsion angle The total twist angle of the hydrofoil at a certain spanwise position is .in, This refers to the elastic torsional angle (deformation) caused by fluid-structure interaction.

[0049] 1.2 Structural Parameters

[0050] (1) Geometric parameters: span String length Maximum thickness of the airfoil;

[0051] (2) Mass and inertia parameters: Total mass Unit length quality Unit span polar moment of inertia ;

[0052] (3) Stiffness and damping parameters: Bending stiffness of the cross section Torsional stiffness heave stiffness heave damping coefficient Bending damping coefficient Torsional damping coefficient .

[0053] 1.3 Fluid Parameters

[0054] The working medium is seawater at normal temperature and pressure. Specific parameters:

[0055] (1) Density Dynamic viscosity ;

[0056] (2) Ignore the effects of volume forces such as gravity and Coriolis force, i.e., volume force vector .

[0057] II. Constructing the Heave and Sinking Dynamics Equations of the Hydrofoil

[0058] Considering only the free translation of the hydrofoil along the vertical direction (Y-axis), neglecting rotation about the X and Z axes and translation along the X and Z axes, the dynamic equilibrium relationship established based on Newton's second law is specifically defined as the product of the hydrofoil's total mass and heave acceleration, equivalent to the total fluid lift acting on the hydrofoil surface at the current moment. This describes the single-degree-of-freedom translational law of the hydrofoil under fluid load. The specific equation is:

[0059] ;

[0060] in, For heave acceleration ( ), Heave speed ( ), h (t) represents the heave displacement ( ), for t The total fluid lift acting on the hydrofoil at all times ( ).

[0061] 2.2 Control Equations for Bending-Torsion Coupling Deformation

[0062] 2.2.1 Bending Deformation Control Equation

[0063] The force equilibrium relationship of the infinitesimal element distributed along the span of the hydrofoil is given by the following equation: an elastic restoring force term consisting of the spatial fourth derivative of the hydrofoil's cross-sectional bending stiffness and bending deflection; a first inertial force term consisting of the time second derivative of the unit span mass and deflection; and a first damping force term consisting of the time first derivative of the bending damping coefficient and deflection. The algebraic sum of the elastic restoring force term, the first inertial force term, and the first damping force term is constrained to be equal to the lift per unit span fluid distribution at that location. Specifically:

[0064] Vertical bending deflection of hydrofoil sections along its span ω ( z,t It satisfies a fourth-order partial differential equation:

[0065] ;

[0066] in, for t Time to the z Lift per unit span of fluid distribution at location ( ), z ∈[0,L].

[0067] 2.2.2 Control Equation for Torsional Deformation

[0068] The moment balance relationship along the hydrofoil span is represented by the following equation: an elastic torsional moment term consisting of the spatial second derivative of the hydrofoil section torsional stiffness and the elastic torsion angle; a second inertial moment term consisting of the time second derivative of the polar moment of inertia per unit span and the torsion angle; and a second damping moment term consisting of the time first derivative of the torsional damping coefficient and the torsion angle. The linear combination of the elastic torsional moment term, the second inertial moment term, and the second damping moment term is constrained to be equal to the fluid distribution torque per unit span at that location. Specifically:

[0069] Elastic torsion angles of hydrofoil cross sections along the span Satisfies the second-order partial differential equation:

[0070] ;

[0071] in, for t Time to the z Torque per unit length of fluid distribution at location ( ), z ∈[0,L].

[0072] III. Numerical Simulation Model Construction

[0073] 3.1 Computation Domain Settings

[0074] With the root of the hydrofoil ( z Establish a global rectangular coordinate system with the origin at (=0).

[0075] X-axis: Incoming flow direction (horizontal to the right) and its range (1m ≥ 2c from the inlet to the leading edge of the hydrofoil, 2.5m ≥ 5c from the trailing edge of the hydrofoil to the outlet).

[0076] Y-axis: Vertically upward, and its range (≥50c, to avoid flow field boundary interference);

[0077] Z-axis: span, and its range (covering the entire span of the hydrofoil).

[0078] 3.2 Mesh Generation

[0079] Use ANSYS ICEM to perform block mesh generation. Specific requirements:

[0080] Mesh partitioning: The computational domain is divided into the hydrofoil near-field region, transition region, and far-field region, and the extent of each region can be set as needed;

[0081] Mesh type: The near field uses a dense tetrahedral unstructured mesh, and the far field uses a sparse tetrahedral unstructured mesh;

[0082] Boundary layer settings: An M-layer boundary layer mesh is set on the hydrofoil wall, with the first layer mesh height and mesh growth rate set to ensure... (Compatible with SST k-ω turbulence model);

[0083] Mesh quality: orthogonality, distortion, total number of meshes, and the number of nodes in the spanning mesh should be consistent with the number of nodes in the subsequent discretization.

[0084] 3.3 Discretization Processing

[0085] 3.3.1 Spread Discretization

[0086] Divide the hydrofoil span into N equal segments, with spanwise step length... The position of the extension node is ;

[0087] 3.3.2 Time Discreteness

[0088] Select time step (≤0.01T), the time series is Set the total simulation duration and total number of time steps.

[0089] IV. Fluid Load Solution: Based on the Navier-Stokes equations for incompressible fluids, ANSYS Fluent is used to solve the flow field and extract the fluid load on the hydrofoil surface.

[0090] 4.1 Governing Equations

[0091] Mass conservation equation (continuity equation): ;

[0092] Momentum equation: ;

[0093] in, For the fluid velocity vector ( ), p For fluid pressure ( ).

[0094] 4.2 Solver Setup

[0095] Solver type: Pressure-based implicit solver;

[0096] Turbulence model: SST k-ω model;

[0097] Pressure-velocity coupling algorithm: SIMPLEC;

[0098] Spatial discretization scheme: Pressure adopts the PRESTO! scheme, and momentum adopts the second-order upwind scheme;

[0099] Residual convergence criterion: All variable residuals ≤ 10^{-5}.

[0100] 4.3 Load Extraction

[0101] The core steps of writing a Fluent UDF (user-defined function) to extract distributed loads across nodes in the spanwise direction are as follows:

[0102] Traverse all computational cells on the hydrofoil surface and obtain the spanwise coordinates of the cell centroids using the C_FACE_CENTROID macro. z ;

[0103] Matching extension nodes z i , will the same z i The unit load is averaged within the range to obtain the lift distributed per unit span. and distributed torque ;

[0104] The total lift was calculated using the trapezoidal integral method: .

[0105] Example: t At 0 = 0s, each branching node (uniform in development), then .

[0106] V. Solution of Motion Response and Fluid-Structure Coupling Deformation

[0107] 5.1 Solving for Heave Motion (Fourth-Order Runge-Kutta Method)

[0108] Known t n heave displacement at any moment h n ,speed Total lift Iterative solution t n+1 Moment h n+1 and Specific steps:

[0109] Calculate intermediate state variables: Based on the current heave velocity, total fluid lift, and total hydrofoil mass, calculate the estimated values ​​of the derivative of the heave velocity and the estimated value of the heave acceleration at the starting point, midpoint, and ending point within the current time step. The four estimated values ​​of the derivative of the heave velocity correspond to the current state, two intermediate state based on half-step prediction, and one state for the next time step based on full-step prediction.

[0110] Weighted iterative update: The four estimated values ​​of heave velocity derivative and heave acceleration are respectively subjected to weighted summation. The estimated value of heave velocity derivative corresponding to the intermediate time state is given a higher weight than the initial state and the final state, so as to obtain the average heave velocity slope and the average heave acceleration slope.

[0111] The specific formula is as follows:

[0112] ,

[0113] ,

[0114] ,

[0115] ,

[0116] in, , , Obtained through linear interpolation or intermediate-step flow field simulation;

[0117] calculate t n+1 Time-in-time state variables: Multiply the average heave velocity slope and the average heave acceleration slope by the time step to obtain the heave displacement increment and heave velocity increment, respectively. Then, superimpose these increments onto the current heave displacement and heave velocity to obtain the heave motion response at the next moment. The specific formula is as follows:

[0118] ;

[0119] ;

[0120] Example: t When 0 = 0s, h 0=0、 =0、 Substituting into h 1 ≈ 0.00016 m, ≈0.0519 m / s.

[0121] 5.2 Solving for Bending-Torsion Coupled Deformation (Newmark- β (and finite difference method)

[0122] Bending and torsion are decoupled into two independent linear systems and solved in parallel, with the two systems only indirectly coupled through fluid loads.

[0123] 5.2.1 Newmark- β Time integration

[0124] Based on Newmark- βThe time integration principle is used to establish a linear predictive relationship between the displacement at the next moment and the displacement, velocity, and acceleration at the current moment; and a correction logic for velocity and acceleration is constructed, which is configured to update the acceleration at the next moment by back-calculating the integral constant and displacement increment after solving for the displacement at the next moment, and then use the updated acceleration to correct the velocity at the next moment.

[0125] Specifically, it includes:

[0126] First, let's review the dynamic equations and rearrange them into their standard form:

[0127] Bending deformation (fourth-order PDE): ;

[0128] Torsional deformation (second-order PDE): ;

[0129] in, , , For the first time derivative, , It is the second derivative in time.

[0130] For Newmark- β The law has a basic formula:

[0131] ;

[0132] In order to eliminate the equation and Transform the above formula and use Indicate them:

[0133] ;

[0134] The integral constant is defined as:

[0135] ;

[0136] ;

[0137] For the bending-torsional coupling deformation control equations given above, select... (Constant average acceleration method, unconditionally stable), by substituting the acceleration and velocity expressions, the governing equations for bending-torsional coupling deformation are rewritten as:

[0138] ;

[0139] Among them, payload Effective torque They are respectively:

[0140] ;

[0141] Solve for bending deflection Then, update the deflection velocity. and deflection acceleration formula:

[0142] ;

[0143] Solve for the angle of twist Next, update the torsional angular velocity. and angular acceleration formula:

[0144] .

[0145] 5.2.2 Spatial Discretization using the Finite Difference Method

[0146] Spatial derivatives are processed using the central difference scheme, with nodes... i The differential is transformed into an algebraic expression.

[0147] For the i Each node (step size) ),have:

[0148] The fourth-order derivative discretization scheme for bending is as follows: Using the central difference method, the fourth-order partial derivative terms in the bending deformation control equation with respect to spanwise position are transformed into a linear combination of the bending deflection values ​​of the target node and its four spanwise adjacent nodes. The fourth power of the spanwise step size of the mesh is used as the normalized denominator. The specific formula is as follows: ;

[0149] Torsional Second Derivative Discretization Scheme: The construction logic of the torsional second derivative discretization scheme is as follows: using the central difference method, the second-order partial derivative terms in the torsional deformation control equation with respect to spanwise position are transformed into a linear combination of the elastic torsional angle values ​​of the target node and its two spanwise adjacent nodes, and the square of the mesh spanwise step size is used as the normalized denominator. The specific formula is: .

[0150] After substituting into the equivalent static equation, the first i The equation for each node is:

[0151] bending: ;

[0152] Twist: .

[0153] 5.2.3 Boundary Condition Handling

[0154] For the fixed boundary conditions at the hydrofoil root and the free boundary conditions at the finite element tip, virtual computational nodes are introduced outside the physical boundaries. Based on the displacement and derivative constraints at the boundaries, a linear mapping equation between the virtual nodes and the internal physical nodes is established to eliminate unknown terms at the boundaries of the difference equation system. Specifically, this includes:

[0155] The virtual node method is used to handle boundary conditions. i =0 and i =Equation at N.

[0156] For bending boundary conditions and torsional boundary conditions, we have:

[0157] Wing root ( i =0, fixed support): ;

[0158] Wing tip ( i =N, free): That is, ;

[0159] The torsion angle satisfies That is, , .

[0160] 5.2.4 Solving the system of equations

[0161] Substituting the time integral and spatial discretization schemes into the bending-torsional coupling deformation control equations, and combining them with the fixed-free boundary conditions, a linear system of equations is assembled: Ax = b The Gaussian elimination method is used to solve for the output nodes. and .

[0162] Example: t At 1 = 0.02 s, the wingtip ( z =2 m) ≈0.0012 m, ≈0.005 rad.

[0163] The solution method for the motion response and fluid-structure interaction deformation of the oscillating hydrofoil provided in this disclosure provides a more comprehensive description of the coupling relationship. For the first time, it systematically considers the full-dimensional coupling relationship of "pitch motion-heave motion-bending-torsional coupling deformation" in a semi-active oscillating hydrofoil, overcoming the limitations of traditional rigid hydrofoil assumptions or simplifications based on single-direction deformation. The simulation results are more closely aligned with actual working conditions. The use of block mesh generation and boundary layer refinement techniques adapts to the wall requirements of the SST k-ω turbulence model, significantly improving the accuracy of fluid load calculation. Combined with the fourth-order Runge-Kutta method and Newmark- βThe time integration method (second-order accuracy) solves for motion and deformation, balancing computational efficiency and stability, resulting in higher accuracy and stability while avoiding divergence in the solution process. A more complete solution system is constructed, encompassing parameter definition, model establishment, load solving, and motion and deformation calculation, forming a standardized and reusable solution scheme that can be directly applied to engineering design without requiring additional intermediate steps. It supports flexible adjustment of hydrofoil structural parameters, motion parameters, and fluid parameters, quickly adapting to the simulation needs of semi-active oscillating hydrofoils of different sizes and operating conditions. This provides accurate data support for structural optimization, energy capture efficiency improvement, and structural fatigue life assessment of tidal energy devices, promoting the large-scale engineering application of oscillating hydrofoil tidal energy devices.

[0164] Combination Figure 3 As shown, this disclosure provides a second method for solving the motion response and fluid-structure interaction deformation of an oscillating hydrofoil, including:

[0165] S1: Define the motion mode and core parameters of the hydrofoil, and construct the heave motion dynamics equation and bending-torsional coupling deformation control equation of the hydrofoil;

[0166] S2: Build a three-dimensional numerical simulation model of the hydrofoil, including setting the computational domain, mesh generation, and discretization.

[0167] S3: Based on the Navier-Stokes equations for incompressible fluids, the distributed fluid load on the hydrofoil surface is solved using the finite volume method, and the distributed fluid load is integrated to obtain the total fluid lift at the current moment.

[0168] S4: Based on the total fluid lift, total hydrofoil mass, and heave parameters at the current moment, the fourth-order Runge-Kutta method is used to solve the heave motion response at the next moment;

[0169] S5: Based on the bending parameters, torsional parameters, force distribution parameters, and hydrofoil structural parameters at the current moment, using Newmark- β The time integration method and the finite difference method, combined with the fixed-free boundary conditions, are assembled into a system of linear equations; the Gaussian elimination method is used to solve for the bending and torsional deformations of each spanwise node at the next moment.

[0170] S6: Based on the heave response, bending deformation, and torsional deformation obtained at the next moment, the initial positions of the mesh nodes are superimposed, and the positions of the mesh nodes are updated by the heave translation, bending deformation, and torsional deformation at the current moment; linear interpolation is used to complete the non-node mesh positions, and continuous simulation is completed through time step iteration.

[0171] 5.3 Simulation Verification and Comprehensive Performance Analysis

[0172] To fully verify the performance of the solution method proposed in this application, this embodiment uses the NACA0012 standard hydrofoil model. R e =2×10 5 Simulation tests were conducted under working conditions, and the results were compared with traditional strongly coupled simulation methods (such as conventional bidirectional fluid-structure interaction (FSI) based on dynamic mesh) in terms of solution accuracy, computational efficiency, and numerical stability.

[0173] (1) Accuracy verification: In order to verify the accuracy more rigorously, Figure 4 The lift coefficient calculated by the method of this application ( C L The time-history curves were compared with classic flume experimental data. The results show that the phase of the calculated results in this application is synchronized with the experimental data, and the peak relative error is only 4.2% (less than the 5% threshold allowed by engineering). This indicates that by constructing the "heave motion dynamics equation" and the "bending-torsion coupling deformation control equation" and combining the "virtual node method" to handle boundary conditions, this application can accurately reproduce the nonlinear load characteristics under fluid-structure interaction, solving the problem of insufficient accuracy of simplified models in existing technologies.

[0174] (2) Computational Efficiency Verification: On the same computing hardware platform (configured with an Intel i7 processor and 32GB of memory), the computational time of the two methods was compared. Traditional strongly coupled methods require multiple iterative convergences of the fluid and structural domains within each time step, resulting in an average computation time of approximately 12.5 hours for a single oscillation cycle. The method in this application benefits from the fourth-order Runge-Kutta method and the Newmark- β The efficient decoupling solution strategy significantly reduces the average time to calculate a single oscillation cycle to 7.8 hours. Figure 5 The graph compares the cumulative computation time consumed by the two methods as the physical simulation cycle progresses. The slope of the curves in the graph intuitively reflects the computation rate. It can be seen that the traditional strongly coupled method (red square line) has a large slope due to its cumbersome internal iterations and rapid increase in computation time; while the efficient decoupling method of this application (blue circle line) has a much slower increase in computation time and a significantly smaller slope. After completing 5 simulation cycles, the method of this application has saved a significant amount of computational resources, improved the overall computational efficiency by approximately 38%, and shortened the optimization design cycle.

[0175] (3) Numerical stability verification: In order to fully verify the robustness of the method of this application under complex sea conditions, this embodiment focuses on large angle of attack and high reduction frequency ( k Numerical stability stress tests were conducted under extreme conditions of 0.2, and an in-depth analysis was performed in conjunction with the evolution law of the total energy of the system.

[0176] First, regarding the time-domain response (see...) Figure 6 In this case, the traditional explicit finite difference method suffers from mesh distortion due to excessive structural deformation, which hinders the simulation. t =1.5 T The previous method exhibited severe numerical oscillations and eventually diverged, causing computational interruption; however, this application employs the unconditionally stable Newmark- β Implicit integral scheme (parameters set to) ), and combined with the virtual node method, eliminated boundary singularities, thus achieving full-time domain ( t =0~20 T All of them maintained good convergence and stability, and no numerical divergence was observed.

[0177] Furthermore, analyzing from the physical essence of energy conservation (see...) Figure 7 The dimensionless total energy of the system calculated by the method in this application always remains within a reasonable physical range (10). 2 ~ 10 3 The proposed method exhibits stable fluctuations without energy drift; in stark contrast, traditional methods show a non-physical exponential explosive increase in system energy during the divergence phase, rapidly exceeding the failure threshold. In summary, the dual verification in both the time and energy domains fully demonstrates the excellent robustness of the proposed method, effectively suppressing numerical error accumulation and adapting to the simulation requirements of large deformations in strongly nonlinear fluid-structure interaction.

[0178] VI. Grid Node Position Update

[0179] based on t n+1 Based on the heave and torsional deformation results at any given time, update the hydrofoil mesh node coordinates using the following steps:

[0180] Initial position: ,in, Set the initial Cartesian coordinates for the nodes;

[0181] Heave and translation: superimposed heave and translation displacements. ;

[0182] Bending deformation: superimposed vertical bending deflection, ;

[0183] Torsional deformation: Total torsion angle of the node rotating about the X-axis Coordinate correction:

[0184] ;

[0185] ;

[0186] ;

[0187] in, For nodes relative to the span z i The local y-coordinate of the midpoint of the chord;

[0188] Non-node mesh: Linear interpolation is used to complete the coordinates of non-node meshes to ensure mesh continuity and quality.

[0189] It should be noted that the specific implementation methods for S1 to S5 can be found in the previous text and will not be repeated here.

[0190] In this way, the mesh node positions are updated based on the motion and deformation results, and continuous simulation is achieved through time-step iteration. This method can accurately capture the strong coupling characteristics of hydrofoil motion and fluid-structure interaction deformation, providing technical support for the design optimization and performance improvement of oscillating hydrofoil tidal energy devices.

[0191] Combination Figure 8 As shown, this embodiment of the present disclosure provides a solution system 80 for the motion response and fluid-structure interaction deformation of an oscillating hydrofoil, including: a definition construction module 81, a model building module 82, a first calculation module 83, a second calculation module 84, and a third calculation module 85. The system comprises three modules: a definition module 81, configured to define the hydrofoil's motion mode and core parameters, and to construct the hydrofoil's heave dynamics equations and bending-torsional coupling deformation control equations; a model building module 82, configured to build a three-dimensional numerical simulation model of the hydrofoil, including computational domain setting, mesh generation, and discretization; a first calculation module 83, configured to solve for the distributed fluid load on the hydrofoil surface using the finite volume method based on the Navier-Stokes equations for incompressible fluids, and to integrate the distributed fluid load to obtain the total fluid lift at the current moment; a second calculation module 84, configured to solve for the heave response at the next moment using the fourth-order Runge-Kutta method based on the current total fluid lift, the hydrofoil's total mass, and the current heave parameters; and a third calculation module 85, configured to calculate the heave response at the next moment using the Newmark- β The time integration method and the finite difference method, combined with the fixed-free boundary conditions, are assembled into a system of linear equations; the Gaussian elimination method is used to solve for the bending and torsional deformations of each spanwise node at the next moment.

[0192] The solution system for the motion response and fluid-structure interaction deformation of the oscillating hydrofoil provided in this disclosure provides a more comprehensive description of the coupling relationship. For the first time, it systematically considers the full-dimensional coupling relationship of "pitch motion-heave motion-bending-torsional coupling deformation" in a semi-active oscillating hydrofoil, overcoming the limitations of traditional rigid hydrofoil assumptions or simplifications based on single-direction deformation. The simulation results are more closely aligned with actual working conditions. The use of block mesh generation and boundary layer refinement techniques adapts to the wall requirements of the SST k-ω turbulence model, significantly improving the accuracy of fluid load calculation. Combined with the fourth-order Runge-Kutta method and Newmark- β The time integration method (second-order accuracy) solves for motion and deformation, balancing computational efficiency and stability, resulting in higher accuracy and stability while avoiding divergence in the solution process. A more complete solution system is constructed, encompassing parameter definition, model establishment, load solving, and motion and deformation calculation, forming a standardized and reusable solution scheme that can be directly applied to engineering design without requiring additional intermediate steps. It supports flexible adjustment of hydrofoil structural parameters, motion parameters, and fluid parameters, quickly adapting to the simulation needs of semi-active oscillating hydrofoils of different sizes and operating conditions. This provides accurate data support for structural optimization, energy capture efficiency improvement, and structural fatigue life assessment of tidal energy devices, promoting the large-scale engineering application of oscillating hydrofoil tidal energy devices.

[0193] The technical solutions of this disclosure can be embodied in the form of a software product. This computer software product is stored in a storage medium and includes one or more instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute all or part of the steps of the method described in this disclosure. The aforementioned storage medium can be a non-transitory storage medium, such as a USB flash drive, external hard drive, read-only memory (ROM), random access memory (RAM), magnetic disk, or optical disk, etc., and other media capable of storing program code.

[0194] The foregoing description and accompanying drawings fully illustrate embodiments of this disclosure to enable those skilled in the art to practice them. Other embodiments may include structural, logical, electrical, procedural, and other changes. The embodiments represent only possible variations. Individual components and functions are optional unless explicitly required, and the order of operation may vary. Parts and features of some embodiments may be included or substituted for parts and features of other embodiments. Moreover, the terminology used in this application is for descriptive purposes only and is not intended to limit the application. Without further limitations, an element defined by the phrase "comprising a…" does not exclude the presence of additional identical elements in the process, method, or apparatus that includes said element. Throughout this document, each embodiment may emphasize differences from other embodiments, and similar or identical parts between embodiments may be referred to mutually. For methods, products, etc., disclosed in the embodiments, if they correspond to the method section disclosed in the embodiments, the relevant details may be referred to the description of the method section.

[0195] Those skilled in the art will recognize that the units and algorithm steps of the various examples described in conjunction with the embodiments disclosed herein can be implemented in electronic hardware, or a combination of computer software and electronic hardware. Whether these functions are implemented in hardware or software depends on the specific application and design constraints of the technical solution. Those skilled in the art can use different methods to implement the described functions for each specific application, but such implementation should not be considered beyond the scope of the embodiments of this disclosure. Those skilled in the art will clearly understand that, for the sake of convenience and brevity, the specific working processes of the systems, devices, and units described above can be referred to the corresponding processes in the foregoing method embodiments, and will not be repeated here.

[0196] The methods and products disclosed in the embodiments herein (including but not limited to devices and equipment) can be implemented in other ways. For example, the device embodiments described above are merely illustrative. For instance, the division of units may be merely a logical functional division, and in actual implementation, there may be other division methods. For example, multiple units or components may be combined or integrated into another system, or some features may be ignored or not executed. In addition, the mutual coupling or direct coupling or communication connection shown or discussed may be through some interfaces, and the indirect coupling or communication connection of devices or units may be electrical, mechanical, or other forms. The units described as separate components may or may not be physically separate. The components shown as units may or may not be physical units, that is, they may be located in one place or distributed across multiple network units. Some or all of the units can be selected to implement this embodiment according to actual needs. In addition, the functional units in the embodiments of this disclosure may be integrated into one processing unit, or each unit may exist physically separately, or two or more units may be integrated into one unit.

[0197] The flowcharts and block diagrams in the accompanying drawings illustrate the architecture, functionality, and operation of possible implementations of systems, methods, and computer program products according to embodiments of this disclosure. In this regard, each block in a flowchart or block diagram may represent a module, segment, or portion of code containing one or more executable instructions for implementing a specified logical function. In some alternative implementations, the functions marked in the blocks may occur in a different order than that shown in the drawings. For example, two consecutive blocks may actually be executed substantially in parallel, and they may sometimes be executed in reverse order, depending on the functions involved. In the descriptions corresponding to the flowcharts and block diagrams in the accompanying drawings, the operations or steps corresponding to different blocks may also occur in a different order than disclosed in the description, and sometimes there is no specific order between different operations or steps. For example, two consecutive operations or steps may actually be executed substantially in parallel, and they may sometimes be executed in reverse order, depending on the functions involved. Each block in a block diagram and / or flowchart, and combinations of blocks in a block diagram and / or flowchart, can be implemented using a dedicated hardware-based system that performs the specified function or action, or using a combination of dedicated hardware and computer instructions.

Claims

1. A method for solving the motion response and fluid-structure interaction deformation of an oscillating hydrofoil, characterized in that, include: S1: Define the motion mode and core parameters of the hydrofoil, and construct the heave motion dynamics equation and bending-torsional coupling deformation control equation of the hydrofoil; S2: Build a three-dimensional numerical simulation model of the hydrofoil, including setting the computational domain, mesh generation, and discretization. S3: Based on the Navier-Stokes equations for incompressible fluids, the distributed fluid load on the hydrofoil surface is solved using the finite volume method, and the distributed fluid load is integrated to obtain the total fluid lift at the current moment. S4: Based on the current total fluid lift, hydrofoil mass, and heave parameters, the fourth-order Runge-Kutta method is used to solve the heave motion response at the next moment; wherein, the step of using the fourth-order Runge-Kutta method to solve the heave motion response at the next moment based on the current total fluid lift, hydrofoil mass, and heave parameters includes: constructing intermediate state variables: based on the current heave velocity, total fluid lift, and hydrofoil mass, calculate the four lift parameters at the starting point, midpoint, and ending point within the current time step. The heave velocity derivative estimation value and heave acceleration estimation value are calculated. The four heave velocity derivative estimation values ​​correspond to the current state, two intermediate state based on half-step prediction, and one next state based on full-step prediction, respectively. A weighted iterative update step is performed: the four heave velocity derivative estimation values ​​and heave acceleration estimation values ​​are weighted and summed, where the heave velocity derivative estimation value corresponding to the intermediate state is assigned a higher weight than the starting and ending states, resulting in the average heave velocity slope and average heave acceleration slope. The next time-moment response calculation step involves multiplying the average heave velocity slope and average heave acceleration slope by the time step to obtain the heave displacement increment and heave velocity increment, respectively. These increments are then superimposed on the current heave displacement and heave velocity to obtain the heave motion response at the next time moment. S5: Based on the bending parameters, torsional parameters, force distribution parameters, and hydrofoil structural parameters at the current moment, using Newmark- β The time integration method and the finite difference method, combined with the fixed-free boundary conditions, are assembled into a system of linear equations. Gaussian elimination is then used to solve for the bending and torsional deformations of each spanwise node at the next time step. The bending parameters, torsional parameters, force distribution parameters, and hydrofoil structural parameters at the current time step are used to solve for the deformations using Newmark- β The time integration method and the finite difference method, combined with the fixed-free boundary conditions, are assembled into a system of linear equations, including: based on the bending parameters, torsional parameters, force distribution parameters, and hydrofoil structural parameters at the current moment, using Newmark- β Iterative formulas for displacement, acceleration, torsional angle, and torsional angular acceleration are established using the time integration method; the second and fourth spatial derivatives in the torsional deformation equation are discretized using the finite difference method, transforming the differentials at the spanwise nodes into algebraic expressions; and the virtual node method is used to handle boundary conditions. i =0 and i The equation at point N; where... i For the spanwise nodes, the discretized derivative scheme is substituted into the bending-torsional coupling deformation equations to assemble the linear equation system.

2. The method for solving the motion response and fluid-structure interaction deformation of an oscillating hydrofoil according to claim 1, characterized in that, The construction logic of the iterative formula is as follows: Based on Newmark- β The time integration principle is used to establish a linear predictive relationship between the displacement at the next moment and the displacement, velocity, and acceleration at the current moment; and a correction logic for velocity and acceleration is constructed, which is configured to update the acceleration at the next moment by back-calculating the integral constant and displacement increment after solving for the displacement at the next moment, and then use the updated acceleration to correct the velocity at the next moment. The construction logic of the fourth derivative discrete scheme of bending is as follows: using the central difference method, the fourth partial derivative terms of the bending deformation control equation with respect to the spanwise position are transformed into a linear combination of the bending deflection values ​​of the target node and its four spanwise adjacent nodes, and the fourth power of the spanwise step size of the mesh is used as the normalized denominator. The construction logic of the torsional second derivative discrete scheme is as follows: using the central difference method, the second-order partial derivative terms in the torsional deformation control equation with respect to the spanwise position are transformed into a linear combination of the elastic torsional angle values ​​of the target node and the two spanwise adjacent nodes, and the square of the spanwise step of the grid is used as the normalized denominator. The virtual node method processing logic is as follows: for the fixed boundary conditions of the hydrofoil root and the free boundary conditions of the wingtip, virtual computing nodes are introduced outside the physical boundary, and a linear mapping equation between the virtual nodes and the internal physical nodes is established based on the displacement and derivative constraint relationship at the boundary to eliminate the unknown terms at the boundary of the difference equation system.

3. The method for solving the motion response and fluid-structure interaction deformation of an oscillating hydrofoil according to claim 2, characterized in that, The heave dynamics equations and bending-torsional coupling deformation control equations for constructing the hydrofoil include: The heave motion dynamics equation is characterized as a dynamic equilibrium relationship based on Newton's second law, specifically defined as the product of the total mass of the hydrofoil and the heave acceleration, which is equivalent to the total fluid lift acting on the surface of the hydrofoil at the current moment, thereby describing the single-degree-of-freedom translational law of the hydrofoil under fluid load. The bending deformation control equation is characterized as the force balance relationship of the infinitesimal elements distributed along the span of the hydrofoil. This equation includes: an elastic restoring force term composed of the spatial fourth derivative of the hydrofoil section bending stiffness and bending deflection; a first inertial force term composed of the time second derivative of the unit span mass and deflection; and a first damping force term composed of the bending damping coefficient and the time first derivative of the deflection. The algebraic sum of the elastic restoring force term, the first inertial force term, and the first damping force term is constrained to be equal to the lift force per unit span fluid distribution at that location. The torsional deformation control equation is characterized as a moment balance relationship distributed along the hydrofoil span. This equation includes: an elastic anti-torsional moment term composed of the spatial second derivative of the hydrofoil section torsional stiffness and the elastic torsion angle; a second inertial moment term composed of the time second derivative of the polar moment of inertia per unit span and the torsion angle; and a second damping moment term composed of the torsional damping coefficient and the time first derivative of the torsion angle. The linear combination of the elastic anti-torsional moment term, the second inertial moment term, and the second damping moment term is constrained to be equal to the fluid distribution torque per unit span at that location.

4. The method for solving the motion response and fluid-structure interaction deformation of an oscillating hydrofoil according to claim 1, characterized in that, The three-dimensional numerical simulation model for constructing the hydrofoil includes: The computational domain is set to satisfy the following conditions: distance from inlet to hydrofoil leading edge ≥ 2c, distance from hydrofoil trailing edge to outlet ≥ 5c, lateral range ≥ 50c, and hydrofoil chordal midpoint corresponding to... X =0, extending to cover the entire region Z∈[0,L]; The mesh generation adopts tetrahedral block mesh, and the hydrofoil wall is set with multiple boundary layer meshes. The height of the first layer mesh is a preset height, the mesh growth rate is a preset growth rate, the mesh orthogonality is greater than or equal to the first preset value, and the twist is less than or equal to the second preset value. Spanwise discretization divides the hydrofoil span L into N segments, resulting in N+1 nodes, with node positions... z i = i Δ z ; Where, Δ z = L / N ; Time discretization t n = n Δ t Δ t Satisfy Δ t ≤0.01 T ,in, T The pitching motion period.

5. The method for solving the motion response and fluid-structure interaction deformation of an oscillating hydrofoil according to claim 1, characterized in that, The Navier-Stokes equations based on incompressible fluids are used to solve for the distributed fluid load on the hydrofoil surface using the finite volume method. The distributed fluid load is then integrated to obtain the total fluid lift at the current moment, including: A load extraction program was written using Fluent UDF, which uses begin_f_loop and end_f_loop macro loops to traverse the hydrofoil surface elements and obtain the unit span distribution lift at each spanwise node. Q z ( z i , t n and distributed torque T ( z i , t n ); The Navier-Stokes equations include: the mass conservation equations. and momentum equation ,in For the fluid velocity vector, p For fluid pressure, This is a volume force vector.

6. The method for solving the motion response and fluid-structure interaction deformation of an oscillating hydrofoil according to any one of claims 1 to 5, characterized in that, The method further includes: Based on the heave response, bending deformation and torsional deformation obtained at the next moment, the initial position of the mesh node is superimposed, and the position of the mesh node is updated by the heave translation, bending deformation and torsional deformation at the current moment. Linear interpolation is used to complete the non-node mesh positions, and continuous simulation is completed through time step iteration.

7. The method for solving the motion response and fluid-structure interaction deformation of an oscillating hydrofoil according to claim 6, characterized in that, The update of grid node positions includes: The initial position of the grid node is Superimposed heave and translation Bending deformation ; When torsional deformation is superimposed, the node around x Total torsion angle of the shaft rotation The coordinates are corrected to: ； ； ； in, For nodes relative to the span z i The local y-coordinate of the midpoint of the chord; For bending deflection; The total angle of twist; It is the elastic torsional angle caused by fluid-structure interaction; This represents the heave or displacement at the next moment.

8. A solution system for the motion response and fluid-structure interaction deformation of an oscillating hydrofoil, characterized in that, include: Define the building block, which is configured to define the motion mode and core parameters of the hydrofoil, and construct the heave motion dynamics equation and bending-torsional coupling deformation control equation of the hydrofoil; The model building module is configured to build a three-dimensional numerical simulation model of the hydrofoil, including setting the computational domain, mesh generation, and discretization. The first calculation module is configured to solve the distributed fluid load on the hydrofoil surface using the finite volume method based on the Navier-Stokes equations for incompressible fluids, and to integrate the distributed fluid load to obtain the total fluid lift at the current moment. The second calculation module is configured to solve for the heave motion response at the next moment using the fourth-order Runge-Kutta method based on the total fluid lift, total hydrofoil mass, and heave parameters at the current moment. The step of solving for the heave motion response at the next moment using the fourth-order Runge-Kutta method based on the total fluid lift, total hydrofoil mass, and heave parameters at the current moment includes: a step of constructing intermediate state variables: based on the heave velocity, total fluid lift, and total hydrofoil mass at the current moment, calculating the starting point, midpoint, and ending point within the current time step. The system generates four estimated values ​​of heave velocity derivatives and heave acceleration; these four estimated values ​​correspond to the current state, two intermediate states based on half-step prediction, and one next state based on full-step prediction. A weighted iterative update step is performed: the four estimated values ​​of heave velocity derivatives and heave acceleration are weighted and summed, with the estimated value of the heave velocity derivative corresponding to the intermediate state given a higher weight than the starting and ending states, resulting in the average heave velocity slope and average heave acceleration slope. The next-moment response calculation step involves multiplying the average heave velocity slope and average heave acceleration slope by the time step to obtain the heave displacement increment and heave velocity increment, respectively. These increments are then superimposed onto the current heave displacement and heave velocity to obtain the heave motion response at the next moment. The third calculation module is configured to use Newmark- based on the bending parameters, torsional parameters, force distribution parameters, and hydrofoil structural parameters at the current moment. β The time integration method and the finite difference method, combined with the fixed-free boundary conditions, are assembled into a system of linear equations. Gaussian elimination is then used to solve for the bending and torsional deformations of each spanwise node at the next time step. The bending parameters, torsional parameters, force distribution parameters, and hydrofoil structural parameters at the current time step are used to solve for the deformations using Newmark- β The time integration method and the finite difference method, combined with the fixed-free boundary conditions, are assembled into a system of linear equations, including: based on the bending parameters, torsional parameters, force distribution parameters, and hydrofoil structural parameters at the current moment, using Newmark- β Iterative formulas for displacement, acceleration, torsional angle, and torsional angular acceleration are established using the time integration method; the second and fourth spatial derivatives in the torsional deformation equation are discretized using the finite difference method, transforming the differentials at the spanwise nodes into algebraic expressions; and the virtual node method is used to handle boundary conditions. i =0 and i The equation at point N; where... i For the spanwise nodes, the discretized derivative scheme is substituted into the bending-torsional coupling deformation equations to assemble the linear equation system.

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