Method for obtaining autonomous propulsion performance of flexible deformation body based on resistance correction panel method

Abstract

This invention provides a method for obtaining the autonomous propulsion performance of a flexible deformable body based on the drag-corrected surface element method, comprising the following steps: (1) determining the physical properties and initial parameters of the flexible deformable body, as well as the initial conditions and flow parameters of the fluid; (2) performing direct numerical simulation of the flexible deformable body using the classical surface element method to obtain the fluid pressure on the surface; (3) calculating the drag compensation applied to the center of mass of the flexible deformable body through an equivalent drag model; (4) calculating the autonomous forward velocity of the flexible deformable body during stable movement by balancing the pressure and equivalent drag experienced by the flexible deformable body during its movement; (5) obtaining the hydrodynamic power and efficiency of the flexible deformable body based on the forces and movement speed of each part of the flexible deformable body. This invention retains the speed and convenience of the surface element method in calculating fluid-structure interaction problems, while significantly improving the error of the classical surface element method in practical applications.

Description

Technical Field

[0001] This invention relates to the field of biomimetic machine technology, specifically to a method for obtaining the autonomous propulsion performance of flexible deformable bodies based on the resistance correction surface element method. Background Technology

[0002] Designing high-speed, high-efficiency flexible deformable bodies (e.g., biomimetic robotic fish) can significantly improve propulsion efficiency and energy utilization, offering greater stealth and adaptability compared to traditional propeller-driven systems. This not only reveals the propulsion mechanisms of fish but also has wide applications in underwater robotics, environmental monitoring, and other fields, providing new technological support for underwater engineering. To save on construction costs, the performance of flexible deformable bodies needs to be roughly assessed before production. Traditional numerical methods typically employ the submerged boundary method or the moving mesh method for calculation; however, the former has a limited applicable Reynolds number range, while the latter requires excessive computation time.

[0003] In conclusion, it is necessary to further innovate existing technologies. Summary of the Invention

[0004] To address the technical problems existing in the background art, this invention proposes a method for obtaining the autonomous propulsion performance of flexible deformable bodies based on the drag-corrected surface element method. By using coefficients and formulas determined according to the drag scaling relationship, and combining the pressure calculated by the classical surface element method, the swimming performance of the flexible deformable body is finally calculated, providing a reliable basis for the preliminary design and iteration of flexible deformable bodies (e.g., biomimetic robotic fish).

[0005] To address the aforementioned technical problems, this invention provides a method for obtaining the autonomous propulsion performance of a flexible deformable body based on the drag-corrected surface element method, comprising the following steps:

[0006] (1) Determine the physical properties and initial parameters of the flexible deformable body, as well as the initial conditions and flow parameters of the fluid;

[0007] (2) The classical surface element method is used to perform direct numerical simulation of flexible deformable bodies in order to obtain the fluid pressure on the surface of the object;

[0008] (3) Calculate the resistance compensation applied to the center of mass of the flexible deformable body using the equivalent resistance model;

[0009] (4) Calculate the autonomous forward speed of the flexible deformable body during stable swimming by balancing the pressure and equivalent resistance it experiences during swimming.

[0010] (5) Based on the forces and swimming speed of each part of the flexible deformable body, the hydrodynamic power and efficiency of the flexible deformable body are obtained.

[0011] The method for obtaining the autonomous propulsion performance of a flexible deformable body based on the drag correction surface element method, wherein: the physical characteristics of the flexible deformable body in step (1) include the density, shape, size, and width and height distributed along the body length of the flexible deformable body.

[0012] The method for obtaining the autonomous propulsion performance of a flexible deformable body based on the drag-corrected surface element method, wherein the specific process of using the classical surface element method to perform direct numerical simulation of the flexible deformable body in step (2) is as follows:

[0013] (2.1) Construct the initial geometric model;

[0014] (2.2) Determine the centerline deformation mode of the flexible deformable body;

[0015] (2.3) Establish a coupled model of the fluid domain and the solid domain;

[0016] (2.4) Solve for the flow field in the fluid domain and the force conditions of the flexible deformable body.

[0017] The method for obtaining the autonomous propulsion performance of a flexible deformable body based on the drag-corrected surface element method, wherein the centerline deformation mode of the flexible deformable body in step (2.2) includes a function of the centerline position of the flexible deformable body changing with time:

[0018]

[0019] The method for obtaining the autonomous propulsion performance of flexible deformable bodies based on the drag-corrected surface element method, wherein the specific process of establishing the coupled model of the fluid domain and the solid domain in step (2.3) is as follows:

[0020] (2.3.1) Constructing the equations of motion for the fluid domain

[0021] The equations of motion in the fluid domain include the inviscid momentum equation and the continuity equation. In this case, the inviscid momentum equation can degenerate into the Laplace equation:

[0022] The inviscid momentum equation is as follows:

[0023]

[0024] In the above formula, u is the fluid velocity vector, t is time, and ρ is... f Where ρ is the fluid density and p is the pressure.

[0025] The continuity equation is as follows:

[0026]

[0027] (2.3.2) Constructing the equations of motion for the solid domain

[0028] The equations of motion for a solid domain are determined by both the momentum equation and the angular momentum equation:

[0029] The momentum equation is:

[0030] The equation for angular momentum is:

[0031] In the above formula, V is the volume of the flexible deformable body; ρ is the density of the flexible deformable body; ∫ρdV is the integral of the density of the flexible deformable body with respect to the volume of the flexible deformable body, which is the mass m of the flexible deformable body; F is the net external force of the fluid acting on the solid; I is the moment of inertia of the flexible deformable body relative to the center of mass; Ω is the angular velocity vector of the flexible deformable body rotating about the center of mass; and M is the net external torque of the fluid acting on the solid.

[0032] The method for obtaining the autonomous propulsion performance of a flexible deformable body based on the drag-corrected surface element method, wherein the specific process of solving the flow field in the fluid domain and the force condition of the flexible deformable body in step (2.4) is as follows:

[0033] (2.4.1) Initialize the physical parameters of the fluid domain, and initialize the physical parameters and deformation parameters of the solid domain;

[0034] (2.4.2) Predicting the position of the flexible deformable body

[0035] Predicting the acceleration 'a' of a flexible deformable body using Newton's iterative method G , and the angular acceleration α of the flexible deformable body G By combining the velocity and angular velocity of the previous moment with the deformation of the centerline of the flexible deformable body at the current moment, the predicted position of the flexible deformable body can be obtained.

[0036] (2.4.3) Calculate the velocity and pressure fields in the fluid domain

[0037] Based on the predicted position of the flexible deformable body as the boundary condition, the velocity field and pressure field of the fluid domain are solved using the motion equations of the fluid domain.

[0038] (2.4.4) Calculate the forces and moments acting on the solid boundary.

[0039] Calculate the resultant pressure T on the solid boundary. p Furthermore, by combining the resultant resistance force D applied to the center of mass using the equivalent resistance model, the net external force F and net external torque M experienced by the flexible deformable body are calculated.

[0040]

[0041] In the above formula, S is the area of ​​the flexible deformable body, and n is the direction vector of the surface of the flexible deformable body;

[0042] (2.4.5) Calculation of overall motion

[0043] Based on the calculated net external force F and net external torque M, the acceleration a of the corresponding flexible deformable body can be determined. G , and the angular acceleration α of the flexible deformable body G Angular acceleration:

[0044] ;

[0045] (2.4.6) Update the fluid domain and solid domain

[0046] The velocity field of the fluid domain and the position and velocity of the flexible deformable body are updated by using the motion equations of the fluid domain and the motion equations of the solid domain.

[0047] (2.4.7) Iterative calculation

[0048] Repeat steps (2.4.2)-(2.4.6) above to perform iterative calculations of the time step until the predetermined simulation time or convergence condition is reached.

[0049] The method for obtaining the autonomous propulsion performance of a flexible deformable body based on the drag correction surface element method, wherein the equivalent drag model in step (3) is determined by the CFD method in terms of its coefficients and form, and the coefficients of the equivalent drag model are related to the shape of the flexible deformable body and the deformation mode of the centerline, and the form of the equivalent drag model is shown in the following formula:

[0050]

[0051] In the above formula, D x Let D be the equivalent drag force applied in the x-direction at the center of mass, and let the resultant drag force applied at the center of mass by the equivalent drag model be D = D x ·e x e x It refers to the unit vector in the x-direction; f is the wander frequency; A tail The amplitude of the tail swing; u c λ is the autonomous forward velocity of the flexible deformable body; L is the wavelength of the flexible deformable body; F is the length of the flexible deformable body. B ξ is a shape correction factor based on an empirical formula. λ d1 and d2 are parameters determined by CFD calculations related to the shape, centerline deformation mode, and wavelength of the flexible deformable body; d1 and d2 are parameters determined by CFD calculations related to the shape and centerline deformation mode of the flexible deformable body; Re is the dimensionless Reynolds number, expressed as: From fluid density ρ f The body length L of the flexible deformable body, the floating frequency f, and the fluid dynamic viscosity μ f Sure.

[0052] The method for obtaining the autonomous propulsion performance of a flexible deformable body based on the drag-corrected surface element method, wherein the flexible deformable body in step (4) is subjected to the combined action of pressure and equivalent drag during the swimming process. The pressure is calculated by the classical surface element method, and the equivalent drag is calculated by the equivalent drag model. When the average values ​​of pressure and equivalent drag are equal over one cycle, it is considered that stable swimming has been achieved.

[0053]

[0054] In the above formula, It represents the average pressure over a period of time, which acts as a thrust. This represents the average resistance over a period of time.

[0055] The autonomous forward speed of a flexible deformable fish during stable swimming is obtained through the thrust-resistance balance equation. If the thrust is greater than the resistance, the fish accelerates; if the thrust is less than the resistance, the fish decelerates.

[0056] The method for obtaining the autonomous propulsion performance of a flexible deformable body based on the drag-corrected surface element method, wherein the swimming velocity of the flexible deformable body in step (5) is defined as the average velocity over one period in a steady state. The periodic average hydrodynamic power that needs to be calculated It consists of two parts: first, the power required for the pressure exerted on the body surface; and second, the negative work done by the equivalent resistance. The efficiency of the flexible deformable body is defined by referencing the energy consumption per unit distance in biomechanics, namely, the ratio of power to the product of the flexible deformable body's mass and swimming speed.

[0057]

[0058] In the above formula, V represents the surface velocity vector of the flexible deformable body; D is the resultant drag force applied to the center of mass; u c It is the autonomous forward velocity vector.

[0059] By adopting the above technical solution, the present invention has the following beneficial effects:

[0060] This invention presents a method for obtaining the autonomous propulsion performance of flexible deformable bodies based on the drag-corrected surface element method. Through numerical simulation, it rapidly calculates the pressure and drag experienced by the flexible deformable body (e.g., a biomimetic robotic fish), thereby determining its swimming speed and other dynamic properties. This provides important reference data for the preliminary design and rapid iteration of flexible deformable bodies, guiding the creation of high-efficiency flexible deformable body models and the setting of motion parameters. This modified surface element method boasts high computational efficiency and is applicable to high Reynolds numbers (~10^4). Compared to traditional computational fluid dynamics (CFD) methods, such as finite element or finite volume-based solutions, it offers faster computation speed and significantly reduces computational costs. Compared to methods such as the immersed boundary method, it has a wider applicable Reynolds number range.

[0061] This invention optimizes autonomously propelled flexible deformable bodies, accurately capturing their dynamic behavior in flow fields and enabling in-depth analysis of the propulsion mechanism. Furthermore, it efficiently calculates the pressure distribution and drag characteristics of the flexible deformable body under different movement modes, thus accurately predicting propulsion efficiency, energy consumption, and flow field effects, providing theoretical support for the design of flexible deformable body propulsion systems. Additionally, this invention is applicable to a special case where the flexible wavelength approaches infinity: rigid structures.

[0062] Compared with existing technologies, this invention has significant advantages in terms of computational efficiency, applicability, and engineering application value. On the one hand, its high computational efficiency makes it suitable for rapid iterative optimization, significantly shortening the development cycle of flexible deformable bodies. On the other hand, its applicability to high Reynolds number flows expands the method's applicable scenarios, enabling the analysis of a wider range of biomimetic swimming problems. Based on the method of this invention, the design of biomimetic underwater robots can be further optimized, propulsion efficiency improved, energy consumption reduced, and adaptability enhanced in complex hydrodynamic environments, providing important theoretical and technical support for the development of high-efficiency underwater robot technology. Attached Figure Description

[0063] To more clearly illustrate the specific embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the specific embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained from these drawings without creative effort.

[0064] Figure 1 This is a flowchart of the method for obtaining the autonomous propulsion performance of a flexible deformable body based on the drag-corrected surface element method according to the present invention.

[0065] Figure 2 This is a flowchart of the numerical simulation of the modified surface element method involved in the method for obtaining the autonomous propulsion performance of a flexible deformable body based on the drag-modified surface element method of the present invention.

[0066] Figure 3 This is a schematic diagram of the initial geometric model of the robotic fish involved in the method for obtaining the autonomous propulsion performance of a flexible deformable body based on the drag correction surface element method of the present invention.

[0067] Figure 4 This is a schematic diagram of the centerline deformation of the robotic fish involved in the method for obtaining the autonomous propulsion performance of a flexible deformable body based on the drag correction surface element method of the present invention. Detailed Implementation

[0068] The technical solution of the present invention will now be clearly and completely described with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of the present invention. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0069] The present invention will be further explained below with reference to specific embodiments.

[0070] like Figure 1 As shown in the figure, this embodiment provides a method for obtaining the autonomous propulsion performance of a flexible deformable body based on the drag correction element method, which includes the following steps:

[0071] S100. Determine the physical properties and initial parameters of the flexible deformable body, as well as the initial conditions and flow parameters of the fluid.

[0072] First, determine the physical properties of the flexible deformable body, including its density, shape, size, and the width and height distributed along its length; these parameters will be used to construct the initial geometric model of the flexible deformable body.

[0073] ρ: Density of flexible deformable body (kg / m³) 3 );

[0074] L: Length of the flexible deformable body (m);

[0075] w(x): Width (m) distributed along the body length;

[0076] h(x): Height (m) distributed along the body length;

[0077] S200. Direct numerical simulation of flexible deformable bodies is performed using the modified surface element method to obtain the fluid pressure on the surface of the object. The steps are as follows:

[0078] S201. Construct the initial geometric model;

[0079] S202. Determine the centerline deformation mode of the flexible deformable body.

[0080] The model should determine the time-varying deformation pattern of the flexible deformable body along its pre-defined centerline, as its propulsion mode; the centerline deformation pattern of the flexible deformable body includes a function of the centerline position of the flexible deformable body changing with time.

[0081]

[0082] S203. Establish a coupled model of the fluid domain and the solid domain.

[0083] In the calculation process based on the modified surface element method, the coupled calculation of the fluid domain and the solid domain is a key part; this process involves transferring the motion of the solid boundary to the fluid domain and influencing the motion of the flexible deformable body through the reaction force of the fluid domain, while applying an equivalent drag model on the solid centroid to correct the lack of viscosity in the surface element method.

[0084] S2031, Constructing the equations of motion for the fluid domain

[0085] The equations of motion in the fluid domain include the inviscid momentum equation and the continuity equation. In this case, the inviscid momentum equation can degenerate into the Laplace equation:

[0086] The inviscid momentum equation is as follows:

[0087]

[0088] In the above formula, u is the fluid velocity vector, t is time, and ρ is... f Where ρ is the fluid density and p is the pressure.

[0089] The continuity equation is as follows:

[0090]

[0091] S2032, Constructing the motion equations of the solid domain

[0092] The equations of motion for a solid domain are determined by both the momentum equation and the angular momentum equation:

[0093] The momentum equation is:

[0094] The equation for angular momentum is:

[0095] In the above formula, V is the volume of the flexible deformable body; ρ is the density of the flexible deformable body; ∫ρdV is the integral of the density of the flexible deformable body with respect to the volume of the flexible deformable body, which is the mass m of the flexible deformable body; F is the net external force of the fluid acting on the solid; I is the moment of inertia of the flexible deformable body relative to the center of mass; Ω is the angular velocity vector of the flexible deformable body rotating about the center of mass; and M is the net external torque of the fluid acting on the solid.

[0096] S204. Solve for the flow field and the force conditions of the flexible deformable body in the fluid domain, such as... Figure 2 As shown, it includes the following steps:

[0097] S2041. Initialize the physical parameters of the fluid domain, and initialize the physical and deformation parameters of the solid domain;

[0098] S2042, Predicting the position of flexible deformable bodies

[0099] Using Newton's iterative method to predict the acceleration 'a' of a flexible deformable bodyG , and the angular acceleration α of the flexible deformable body G By combining the velocity and angular velocity of the previous moment with the deformation of the centerline of the flexible deformable body at the current moment, the predicted position of the flexible deformable body can be obtained.

[0100] S2043, Computational fluid domain velocity and pressure fields

[0101] Based on the predicted position of the flexible deformable body as the boundary condition, the velocity field and pressure field of the fluid domain are solved using the equation of motion of the fluid domain.

[0102] S2044. Calculate the forces and moments acting on the solid boundary.

[0103] Calculate the resultant pressure T on the solid boundary. p Furthermore, by combining the resultant resistance force D applied to the center of mass using the equivalent resistance model, the net external force F and net external torque M acting on the flexible deformable body are calculated:

[0104]

[0105] In the above formula, S is the area of ​​the flexible deformable body, and n is the direction vector of the surface of the flexible deformable body;

[0106] Furthermore, the net external force F and net external torque M acting on the flexible deformable body are calculated using the equivalent resistance model.

[0107] S2045, Calculation of Overall Motion

[0108] The acceleration a of the flexible deformable body is calculated based on the net external force F and net external torque M. G , and the angular acceleration α of the flexible deformable body G Angular acceleration:

[0109] ;

[0110] S2046, Updated Fluid and Solid Domains

[0111] The velocity field of the fluid domain and the position and velocity of the flexible deformable body are updated by using the motion equations of the fluid domain and the motion equations of the solid domain.

[0112] S2047, Iterative Calculation

[0113] Repeat steps S2042-S2046 above to iteratively calculate the time step until the predetermined simulation time or convergence condition is reached.

[0114] The following example demonstrates how to numerically solve for the flow field and the force conditions of a flexible deformable body in a fluid domain:

[0115] Assume the initial conditions of the flexible deformable body are as follows:

[0116] The density of the flexible deformable body is ρ = 1000 kg / m³ 3 ;

[0117] The length of the flexible deformable body is L = 1m;

[0118] The deformation form of the centerline is as follows

[0119] Tail swing amplitude A tail =A(1)=0.1m;

[0120] Deformed wavelength λ = 1m;

[0121] Swimming frequency f = 1 Hz;

[0122] Fluid density ρ f =1000kg / m 3 ;

[0123] Reynolds number Re = 50000;

[0124] Initialize the parameters of the fluid domain and the flexible deformable body:

[0125] ① Calculate the initial fluid velocity field:

[0126] u(t=0=0;

[0127] ② Using Newton's iterative method to calculate the angular acceleration 'a' of a flexible deformable body. G ,α G The process involves making predictions and calculating new positions. Newton's Iteration is a numerical method for solving nonlinear equations f(x) = 0, based on the idea of ​​local linearization of the function. The core step is to approximate the function's zero using the tangent line at the current point and iteratively approach the exact solution.

[0128] ;

[0129] u c (t)=u c (t-Δt)+a G Δt,Ω=Ω(t-Δt)+α G Δt;

[0130]

[0131] ③ Based on the predicted new location, the potential flow function φ is calculated using the classical surface element method, and the Laplace equation is solved:

[0132]

[0133] ④ Calculate the pressure on the solid boundary and provide a correction for drag:

[0134]

[0135] ⑤ Calculate the resultant force T of the fluid pressure component acting on the flexible deformable body based on the pressure distribution. p Calculate the fluid torque M and equivalent drag D, and then calculate the corresponding overall motion:

[0136]

[0137] ⑥ Update the fluid domain and solid domain:

[0138] By iterating the above steps repeatedly, coupled calculations of the fluid domain and the solid domain are achieved until the convergence condition or the predetermined simulation time is reached.

[0139]

[0140] The velocity and force of the flexible deformable body at different times are calculated by numerical simulation in order to compare the swimming performance of different designs.

[0141] S300, calculate the resistance compensation applied to the center of mass of the flexible deformable body through the equivalent resistance model;

[0142] The equivalent drag model's coefficients and form are determined using CFD methods, involving the dimensionless Reynolds number Re. Its form is as follows: The coefficients are related to the shape of the flexible deformable body and the deformation mode of the centerline. The equivalent drag model is shown in the following equation:

[0143]

[0144] In the above formula, D x Let D be the equivalent drag force applied in the x-direction at the center of mass, and let the resultant drag force applied at the center of mass by the equivalent drag model be D = D x ·e x ,e x It refers to the unit vector in the x-direction; f is the wander frequency; A tail The amplitude of the tail swing; u c λ is the autonomous forward velocity of the flexible deformable body; L is the wavelength of the flexible deformable body; F is the length of the flexible deformable body. B ξ is a shape correction factor based on an empirical formula. λ d1 and d2 are parameters determined by CFD calculations related to the shape, centerline deformation mode, and wavelength of the flexible deformable body; d1 and d2 are parameters determined by CFD calculations related to the shape and centerline deformation mode of the flexible deformable body; Re is the dimensionless Reynolds number, expressed as: From fluid density ρ fThe body length L of the flexible deformable body, the floating frequency f, and the fluid dynamic viscosity μ f Sure.

[0145] S400. Calculate the autonomous forward speed of the flexible deformable body during stable swimming by balancing the pressure and equivalent resistance it experiences during swimming.

[0146] During its movement, the flexible deformable body is subjected to both pressure and equivalent drag. The pressure is calculated using the classical surface element method, and the equivalent drag is calculated using the equivalent drag model. Stable movement is considered achieved when the average values ​​of the pressure and equivalent drag over one cycle are equal.

[0147]

[0148] In the above formula, It represents the average pressure over a period of time, which acts as a thrust. This represents the average resistance over a period of time.

[0149] The autonomous forward speed of a flexible deformable fish during stable swimming is obtained through the thrust-resistance balance equation. If the thrust is greater than the resistance, the fish accelerates; if the thrust is less than the resistance, the fish decelerates.

[0150] S500. Based on the forces and swimming speed of each part of the flexible deformable body, the hydrodynamic power and efficiency of the flexible deformable body are obtained.

[0151] The velocity of a flexible deformable body in a steady state is defined as the average velocity over one period. The periodic average hydrodynamic power that needs to be calculated It consists of two parts: first, the power required for the pressure exerted on the body surface; and second, the negative work done by the equivalent resistance. The efficiency of a flexible deformable body can be defined by referring to the cost of transport (COT), a metric used in biomechanics to measure energy consumption per unit distance. COT is the ratio of power to the product of the flexible deformable body's mass and its swimming speed.

[0152]

[0153] Among them, F p V is the distributed pressure force acting on the flexible deformable body; V represents the velocity vector relative to the surface of the flexible deformable body; D is the equivalent drag force applied to the center of mass; u c It is the autonomous forward velocity vector.

[0154] The following analysis, using a robotic fish as a flexible deformable body, explains the basis for the equivalent drag model and the determination of its coefficients:

[0155] The classical surface element method is a simplified model for idealized conditions of inviscid fluids. However, in practical applications, although robotic fish typically move at high speeds, resulting in large Reynolds numbers and dominant inertial forces, viscous effects cannot be ignored. These effects are mainly manifested in boundary layer effects and overall viscous drag. Therefore, it is necessary to correct this by establishing an equivalent drag model. The formula of this model is based on a comprehensive consideration of multiple factors, including the main dimensionless numbers controlling the flow field characteristics, the swimming form and shape characteristics of the robotic fish, and classical empirical drag formulas, and its effectiveness has been verified through CFD simulations.

[0156] The main dimensionless numbers controlling the flow field characteristics are:

[0157] Reynolds number, The key dimensionless parameter governing the main characteristics of the Navigator-Stokes equations is the ratio of inertial force to viscous force. On the other hand, the Stochar number... Commonly used to describe wake structure, representing the ratio of unsteady force to inertial force. Since the swimming speed of the robotic fish fluctuates during motion, U, representing the average swimming speed, is replaced with the instantaneous swimming speed u. c The resistance encountered during swimming is mainly related to the two dimensionless numbers mentioned above.

[0158] The swimming pattern and physical characteristics of robotic fish have an impact on:

[0159] During actual swimming, the different shapes and deformations of the robotic fish's midline affect the magnitude of the drag it experiences. This effect is reflected in parameters d1, d2, and ξ. λ In practice, the specific values ​​need to be determined in advance by combining CFD simulations, such as the dynamic mesh method.

[0160] Classical Empirical Formula for Resistance

[0161] In engineering practice, predecessors summarized empirical formulas for the resistance experienced by streamlined objects in fluids through experimental research. Here, this type of resistance is considered to be closely related to the shape and is denoted as F. B The specific influencing factors are mainly determined by the geometric characteristics of the flexible deformable body.

[0162] CFD simulation verification

[0163] The effectiveness of the correction was verified by CFD calculations using typical shapes and wave patterns to ensure the applicability of the resistance correction model.

[0164] Appendix Figure 3The initial geometric shape of the robotic fish is shown, and the centerline of the robotic fish is marked. The length of the centerline is called the length of the robotic fish. A three-dimensional coordinate system is established to represent it, with the X, Y, and Z axes representing the horizontal and vertical directions, respectively. The Y direction, which is distributed along the body length, is called the width, and the Z direction, which is distributed along the body length, is called the height.

[0165] Appendix Figure 4 Detailed explanation of the overall form The maximum positive distance that the controlled A(x) can reach in one period at the tail is called the tail amplitude; where λ is the deformed wavelength and f is the swimming frequency.

[0166] This embodiment can quickly simulate the dynamic characteristics of the robotic fish during its swimming process, thereby calculating the swimming performance under different designs and providing a reliable basis for its swimming performance and further optimization under different conditions.

[0167] This invention uses the classical surface element method to numerically simulate and calculate the pressure on the flexible deformable body, and combines it with a modified model to obtain the equivalent drag applied to the center of mass. The forward swimming speed of the flexible deformable body after force equilibrium is calculated, thereby quickly obtaining the swimming performance of the biomimetic robotic fish at medium to high Reynolds numbers.

[0168] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for obtaining the autonomous propulsion performance of a flexible deformable body based on the drag-corrected surface element method, characterized in that, Includes the following steps: (1) Determine the physical properties and initial parameters of the flexible deformable body, as well as the initial conditions and flow parameters of the fluid; (2) The classical surface element method is used to perform direct numerical simulation of flexible deformable bodies in order to obtain the fluid pressure on the surface of the object; The specific process is as follows: (2.1) Construct the initial geometric model; (2.2) Determine the centerline deformation mode of the flexible deformable body; (2.3) Establish a coupling model of the fluid domain and the solid domain; (2.4) Solve for the flow field and the force conditions of the flexible deformable body in the fluid domain; the specific process is as follows: (2.4.1) Initialize the physical parameters of the fluid domain, and initialize the physical parameters and deformation parameters of the solid domain; (2.4.2) Predicting the location of the flexible deformable body Predicting the acceleration of flexible deformable bodies using Newton's iterative method Angular acceleration of flexible deformable bodies By combining the velocity and angular velocity of the previous moment with the deformation of the centerline of the flexible deformable body at the current moment, the predicted position of the flexible deformable body can be obtained. (2.4.3) Calculate the velocity and pressure fields in the fluid domain Using the predicted position of the flexible deformable body as the boundary condition, the velocity and pressure fields of the fluid domain are solved using the motion equations of the fluid domain. (2.4.4) Calculate the forces and moments acting on the solid boundary. Calculate the resultant pressure force on the solid boundary. Furthermore, the resultant resistance force applied to the center of mass is combined with the equivalent resistance model. Calculate the net external force on a flexible deformable body Resultant external torque ; ; In the above formula, The area of ​​the flexible deformable body. It is the surface direction vector of the flexible deformable body. It is pressure; (2.4.5) Calculation of overall motion Based on the calculated net external force Resultant external torque Find the acceleration of the corresponding flexible deformable body. Angular acceleration of flexible deformable bodies : ； In the above formula, The autonomous forward velocity of the flexible deformable body. ω is the angular velocity vector of the flexible deformable body rotating about its center of mass; (2.4.6) Update the fluid domain and solid domain The velocity field of the fluid domain and the position and velocity of the flexible deformable body are updated by using the motion equations of the fluid domain and the motion equations of the solid domain. (2.4.7) Iterative calculation Repeat steps (2.4.2)-(2.4.6) above to iteratively calculate the time step until the predetermined simulation time or convergence condition is reached; (3) Calculate the resistance compensation applied to the center of mass of the flexible deformable body using the equivalent resistance model; (4) Calculate the autonomous forward speed of the flexible deformable body during stable swimming by balancing the pressure and equivalent resistance it experiences during swimming. (5) Based on the forces and swimming speed of each part of the flexible deformable body, the hydrodynamic power and efficiency of the flexible deformable body are obtained.

2. The method for obtaining the autonomous propulsion performance of a flexible deformable body based on the drag correction element method according to claim 1, characterized in that: The physical properties of the flexible deformable body in step (1) include its density, shape, and size.

3. The method for obtaining the autonomous propulsion performance of a flexible deformable body based on the drag-corrected surface element method according to claim 1, characterized in that, The centerline deformation mode of the flexible deformable body in step (2.2) includes a function of the centerline position of the flexible deformable body changing with time: 。 4. The method for obtaining the autonomous propulsion performance of a flexible deformable body based on the drag-corrected surface element method as described in claim 1, characterized in that, The specific process of establishing the coupled model of the fluid domain and the solid domain in step (2.3) is as follows: (2.3.1) Constructing the equations of motion for the fluid domain The equations of motion in the fluid domain include the inviscid momentum equation and the continuity equation. In this case, the inviscid momentum equation can degenerate into the Laplace equation: The inviscid momentum equation is as follows: ； In the above formula, It is the fluid velocity vector. It is time. It is the fluid density. It is pressure; The continuity equation is as follows: ； (2.3.2) Constructing the equations of motion for the solid domain The equations of motion for a solid domain are determined by both the momentum equation and the angular momentum equation: The momentum equation is: ; The equation for angular momentum is: ; In the above formula, For the volume of a flexible deformable body; The density of a flexible deformable body; The integral of the density of a flexible deformable body with respect to its volume is the mass of the flexible deformable body. ; It is the net external force exerted by the fluid on the solid; Let be the moment of inertia of the flexible deformable body relative to its center of mass; The angular velocity vector of the flexible deformable body rotating about its center of mass; It is the net external torque exerted by the fluid on the solid; The autonomous forward velocity of the flexible deformable body.

5. The method for obtaining the autonomous propulsion performance of a flexible deformable body based on the drag-correction surface element method as described in claim 1, characterized in that, The equivalent drag model in step (3) has its coefficients and form determined by the CFD method. The coefficients of the equivalent drag model are related to the shape of the flexible deformable body and the deformation mode of the centerline. The form of the equivalent drag model is shown in the following formula: ； In the above formula, Let x be the equivalent drag force applied to the center of mass in the x-direction, and let the resultant drag force of the equivalent drag model be applied to the center of mass. , It refers to the unit vector in the x-direction; The swimming frequency; This refers to the tail swing amplitude; The autonomous forward velocity of the flexible deformable body; The wavelength of the flexible deformable body; The length of the flexible deformable body; This is a shape correction factor based on an empirical formula; These are parameters determined by CFD calculations that relate to the shape, centerline deformation mode, and wavelength of the flexible deformable body. The parameters are determined by CFD calculations to match the shape and centerline deformation mode of the flexible deformable body. The Reynolds number, which is dimensionless, is expressed as: , due to fluid density Length of flexible deformable body swimming frequency and fluid dynamic viscosity Sure.

6. The method for obtaining the autonomous propulsion performance of a flexible deformable body based on the drag-corrected surface element method as described in claim 5, characterized in that, In step (4), the flexible deformable body is subjected to both pressure and equivalent resistance during its movement. The pressure is calculated using the classical surface element method, and the equivalent resistance is calculated using the equivalent resistance model. When the average values ​​of the pressure and equivalent resistance are equal over one cycle, stable movement is considered to have been achieved. ； In the above formula, It represents the average pressure over a period of time, which acts as a thrust. This represents the average resistance over a period of time. The autonomous forward velocity of a flexible deformable body during stable swimming is obtained through the thrust-resistance balance equation. If the thrust is greater than the resistance, it accelerates; if the thrust is less than the resistance, it decelerates.

7. The method for obtaining the autonomous propulsion performance of a flexible deformable body based on the drag-corrected surface element method according to claim 1, characterized in that, In step (5), the velocity of the flexible deformable body is defined as the average velocity over one period in a steady state. The periodic average hydrodynamic power that needs to be calculated It consists of two parts: first, the power required for the pressure exerted on the body surface; and second, the negative work done by the equivalent resistance. The efficiency of the flexible deformable body is defined by referencing the energy consumption per unit distance in biomechanics, namely, the ratio of power to the product of the flexible deformable body's mass and swimming speed. ； ； In the above formula, It represents the surface velocity vector of a flexible deformable body; It is the resultant force of resistance applied to the center of mass; It is the autonomous forward velocity vector.

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