Three-dimensional structure free vibration solving method under equal water depth condition

Abstract

The invention discloses a method for solving free vibration of a three-dimensional structure under an equal water depth condition, and relates to the technical field of ocean engineering, the method comprises the following steps: discretizing a geometric model of the structure to obtain a finite element mesh model and a boundary element wet surface mesh model; constructing a boundary integral equation based on the boundary element wet surface grid model and a potential flow theory, introducing a finite water depth mirror image Green function without considering a free surface effect, and solving a flow field velocity potential of a structure in the equation under an equal water depth condition; calculating the additional mass generated by the fluid on the structure when the structure freely vibrates in the flow field based on the flow field velocity potential; and obtaining a free vibration equation of the structure according to the finite element mesh model, adding additional mass, and solving to obtain the vibration intrinsic mode and the vibration frequency of each order of the structure in the flow field. According to the method, the simulation requirement for the fluid-solid coupling effect of the structure in actual engineering can be met, simulation calculation can be carried out only by carrying out finite element modeling on the structure, and certain calculation precision is achieved.

Description

Technical Field

[0001] The present invention relates to the technical field of ocean engineering, and particularly to a method for solving the free vibration of a three-dimensional structure under equal water depth conditions. Background Art

[0002] Since the air density is small and its influence on the structure vibration can be ignored, the natural vibration mode generated by the free vibration of the structure in air is approximately regarded as the natural vibration mode in vacuum, also known as the dry mode. When the structure is in water, the density of water is much greater than that of air, and the fluid-structure interaction effect between water and the structure must be considered. Analyzing the free vibration problem of the structure in this case is called wet mode analysis.

[0003] Determining the wet frequency and wet mode of the structure in water is an important issue in fluid-structure interaction research. There are corresponding analytical solutions for the wet frequency and wet mode of some simple-shaped structures such as cylinders and spheres. For complex-shaped structures, numerical methods are generally used for solution, such as the structure-acoustic finite element coupling method, the boundary element-finite element hybrid method, etc. Although the theoretical accuracy of the boundary element-finite element hybrid method is not as good as that of the structure-acoustic finite element coupling method in solving the wet mode problem, the boundary element-finite element hybrid method can significantly reduce the amount of calculation and increase the calculation efficiency by introducing the boundary element method to solve the flow field problem in a reduced dimension, so it is widely used.

[0004] Taking the commercial software MSC Nastran as an example, the virtual mass method of Nastran for solving the wet mode problem essentially applies the boundary element-finite element hybrid method. However, the virtual mass method of Nastran can only solve the wet mode of a three-dimensional structure under infinite water depth conditions, which is a simplification compared to the finite water depth problem in actual engineering. Therefore, there is an urgent need for a method that can solve the wet mode of the structure under finite water depth. Summary of the Invention

[0005] In view of the above problems and technical requirements, the inventor of the present invention proposes a method for solving the free vibration of a three-dimensional structure under equal water depth conditions. The technical solution of the present invention is as follows:

[0006] A method for solving the free vibration of a three-dimensional structure under equal water depth conditions, comprising the following steps:

[0007] Discretize the structural geometric model to obtain a finite element mesh model and a boundary element wet surface mesh model;

[0008] Based on the boundary element wet surface mesh model and potential flow theory, construct a boundary integral equation, and introduce a finite water depth mirror Green's function without considering the free surface effect to solve the flow field velocity potential of the structure under equal water depth conditions in the equation;

[0009] Calculate the added mass generated by the fluid on the structure during its free vibration in the flow field based on the flow field velocity potential;

[0010] The free vibration equation of the structure is obtained based on the finite element mesh model, and after adding the additional mass, the natural vibration modes and vibration frequencies of each order of the structure in the flow field are solved.

[0011] Its further technical solution is that the method for obtaining the boundary element wet surface mesh model includes:

[0012] According to the draft depth of the structure, the surface mesh below the waterline is extracted from the finite element mesh model to form the boundary element wet surface mesh model, and the wet surface mesh information includes at least element numbers, element connectivity, node numbers, and node coordinates.

[0013] Its further technical solution is that the boundary integral equation is constructed based on the boundary element wet surface mesh model and potential flow theory, including:

[0014] The fluid is regarded as an ideal fluid without rotation and viscosity, and the velocity potential of the structure in the flow field is expressed as It satisfies the Laplace equation and corresponding boundary conditions within the entire flow domain. Generally speaking, based on Green's third formula, with the Green's function G(p,q) as the fundamental solution of the Laplace equation, a source-dipole mixed distribution boundary integral equation about the potential function is constructed and discretely expressed; among them, the source-dipole mixed distribution boundary integral equation about the potential function is expressed as:

[0015]

[0016] Among them, respectively represent the potential functions of the field point p(x,y,z) and the source point , s q represents the small surface element where the source point q is located, n q represents the normal vector, and it is stipulated that the positive direction is from the inside of the structure to the outside of the flow field.

[0017] Its further technical solution is that the source-dipole mixed distribution boundary integral equation about the potential function is discretely expressed, including:

[0018] Following the discrete idea of the boundary element, the flow field boundary S can be discretized into N elements, and the discrete expression form of equation (1) is as follows:

[0019]

[0020] j represents the discrete small surface element.

[0021] When the on-site point p approaches the boundary of the flow field along the negative direction of the normal vector, considering the influence of the dipole principal value integral -2π, the discrete format of the source-dipole mixed distribution boundary integral equation is further obtained as follows:

[0022]

[0023] Its further technical solution is that when the structure makes free vibrations in still water and the free surface effect caused by waves is not considered, the corresponding boundary conditions are expressed as:

[0024]

[0025] Among them, U represents the normal velocity of the wet surface, n represents the normal direction, and S H represents the wet surface of the boundary element, z = 0 represents the still water surface, and z = -h represents the seabed.

[0026] Its further technical solution is that the construction method of the mirror Green's function with finite water depth without considering the free surface effect includes:

[0027] Since both the still water surface and the seabed are impenetrable rigid walls, to construct the boundary integral equation for the potential function the idea of the mirror Green's function needs to be introduced. By constructing countless mirror layers m in the water depth direction, the boundary conditions of the still water surface and the seabed can be satisfied. For the problem of solving the added mass of a three-dimensional structure under the condition of equal water depth, the mirror Green's function can be constructed from the simple Green's function:

[0028]

[0029] Among them:

[0030] m = 0 represents the original still water surface, and h is the depth from the still water surface to the seabed.

[0031] So far, substituting the mirror Green's function into the source-dipole mixed distribution boundary integral equation (3), the velocity potential of the flow field can be solved Also known After separating the time term, the velocity potential φ in the frequency domain can be obtained.

[0032] Its further technical solution is that based on the velocity potential of the flow field, the added mass generated by the fluid on the structure during its free vibration in the flow field is calculated, and the calculation formula is:

[0033]

[0034] Among them, ρ is the fluid density, and φ j is the jth velocity potential in the frequency domain, and Re(φ j ) represents the real part of the velocity potential, and n iDenote the normal vector as $\vec{n}$, and $s$ is the wetted surface $S$ of the boundary element. H The small curved surface element.

[0035] Its further technical solution is as follows. According to structural mechanics, the equation of undamped free vibration of a structure in a vacuum is:

[0036]

[0037] where $[M]$ and $[K]$ are the mass matrix and stiffness matrix of the structure respectively, and $\{\ddot{\delta}\}$ and $\{\delta\}$ are the nodal acceleration and displacement of the structure respectively. The solution of the above equation can be written as: $\{\delta\}=\{\delta_0\}\sin\omega t$. Where $\{\delta_0\}$ is independent of time, and $\omega$ is the circular frequency. Therefore, we can obtain: 0} 0}

[0038] ([K] - \omega^2[M])\{\delta_0\} = \{0\}\ (8) 2} 0} = \{0\}\ (8)

[0039] Adding the additional mass matrix $[A]=\{a_{ij}\}$ to the above equation, we can obtain the equation of free vibration of the structure in the flow field: ij}

[0040] ([K] - \omega^2([M]+[A]))\{\delta_0\} = \{0\}\ (9) 2 ([M]+[A]))\{\delta_0\} 0} = \{0\}\ (9)

[0041] Solving the above equation can determine the wet modes and wet frequencies of the structure in the flow field.

[0042] The beneficial technical effects of the present invention are:

[0043] The present invention discloses a method for solving the free vibration of a three-dimensional structure under the condition of equal water depth. The method extracts the boundary element wet surface grid from the finite element mesh model, constructs a boundary integral equation without considering the free surface effect based on the potential flow theory by using the mirror Green's function, and thus calculates the additional mass generated by the free vibration of the structure in the flow field. Then, by coupling the finite element free vibration equation of the structure, the vibration frequencies and vibration modes of each order of the three-dimensional structure affected by the flow field can be solved. The solution method proposed by the present invention can meet the simulation requirements of the fluid-structure interaction effect in practical engineering, only need to perform finite element modeling on the structure to carry out simulation calculations, and has a certain calculation accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] Figure 1 is a flowchart of a method for solving the free vibration of a three-dimensional structure under the condition of equal water depth provided by the present application.

[0045] Figure 2 is a schematic diagram of the wet surface grid provided by the present application.

[0046] Figure 3 It is a schematic diagram of a ship in finite water depth provided by this application.

[0047] Figure 4 It is a schematic diagram of the vertical mirror image of the ship provided by this application with respect to the still water surface and the seabed.

[0048] Figure 5 It is a schematic diagram of the coordinate transformation of the field source points between the ship and the mirror image provided by this application.

[0049] Figure 6 (a) - (d) are schematic diagrams of the calculation results of the wet modes of the ship provided by this application. Specific implementation manners

[0050] The following further describes the specific implementation manners of the present invention with reference to the accompanying drawings.

[0051] This embodiment provides a method for solving the free vibration of a three - dimensional structure under the condition of equal water depth. Taking a ship as an example of the three - dimensional structure, as Figure 1 described in the process, to solve the free vibration problem of the ship under the condition of equal water depth, first, the geometric model of the ship needs to be discretized. The purpose of the discretization process is to obtain a finite - element mesh model and a boundary - element wet - surface mesh model. Among them, the boundary - element wet - surface mesh model can be directly generated from the three - dimensional geometric model, or as Figure 2 shown, by picking up the meshes below the waterline surface on the surface of the finite - element mesh model and creating a surface set to form the boundary - element wet - surface mesh model. The wet - surface mesh information needs to include element numbers, element connectivity, node numbers, and node coordinates, etc. Secondly, as Figure 3 shown, set the draft d and the water depth h as the initial conditions to prepare for applying the boundary - element method to solve the added mass of the ship in still water. Since the finite - element analysis type is modal analysis, no boundary conditions and loads need to be applied to the finite - element mesh model.

[0052] Next, for each surface element, based on the wet - surface mesh information, draft d, water depth h, and other information, a boundary - integral equation corresponding to the boundary conditions based on Equation (4) is generated. The form of the boundary - integral equation is as shown in Equation (3). In order to eliminate the influence of the still water surface on the calculation of the velocity potential in the boundary - integral equation, during the numerical calculation process, the part of the ship below the water surface is mirrored successively with respect to the still water surface and the seabed. The schematic diagram of the vertical mirror image of the hull with respect to the still water surface and the seabed is as Figure 4As shown, first, the hull part ① is mirrored with respect to the still water surface m = 0 to obtain the mirrored hull ②, and then the hull part ① is mirrored with respect to the seabed to obtain the mirrored hull ③. Similarly, the mirrored hull ③ can be mirrored with respect to the constructed still water surface m = -1 and the corresponding mirrored seabed, which will not be described in detail here. The distribution of the mirrored field points and mirrored source points generated by the mirroring principle between the field point p and the source point q on the wet surface is shown in Figure 5 As shown in the figure, the positional relationship between the field (source) point and the mirror field (source) point can be expressed by the mirror Green's function in equation (5), where m in equation (5) can be taken as 10 in order to take into account both the efficiency and accuracy of numerical calculation. All other quantities except are known. Traversing each face element on the wet surface, a boundary integral equation system of the form AX = B can be formed, where A and B are known. According to the standard matrix solution method, the equation can be solved to obtain the velocity potential Known It includes incident potential, diffraction potential and radiation potential. Since the free surface effect caused by waves is not considered, the incident potential and diffraction potential are both zero. The velocity potential obtained is Finally, the radiation potential φ is substituted into equation (6) to obtain the additional mass a generated by the fluid when the ship vibrates freely in water. ij .

[0053] According to the finite element mesh model, the free vibration equation (8) of the ship structure can be obtained, and the additional mass matrix [A] = {a ij}After that, the wet modal vibration equation of the ship structure considering the fluid effect is obtained (9). By solving the equation, the wet frequency and wet mode of the ship can be finally obtained. Figure 6 (a) to (d) are respectively the 7th to 10th order wet modes of the ship under constant water depth conditions obtained by applying the present invention, and their corresponding wet frequencies are 0.4377 Hz (first-order vertical bending), 0.8089 Hz (first-order lateral bending), 0.9368 Hz (first-order torsion), and 0.9689 Hz (second-order vertical bending).

[0054] The above is only a preferred embodiment of the present application, and the present invention is not limited to the above embodiments. It is understood that other improvements and changes directly derived or associated by those skilled in the art without departing from the spirit and concept of the present invention should be considered to be included in the protection scope of the present invention.

Claims

1. A method for solving the free vibration of a three-dimensional structure under constant water depth conditions, characterized in that: The method comprises: Discretize the structural geometric model to obtain a finite element mesh model and a boundary element wet surface mesh model; Based on the boundary element wet surface mesh model and potential flow theory, a boundary integral equation is constructed, and a finite water depth mirror Green's function that does not consider the free surface effect is introduced to solve the flow field velocity potential of the structure under the condition of equal water depth in the equation; Calculating the additional mass of the structure generated by the fluid when the structure freely vibrates in the flow field based on the velocity potential of the flow field; The free vibration equation of the structure is obtained according to the finite element mesh model, and after adding the additional mass, the natural modes and vibration frequencies of each order of vibration of the structure in the flow field are obtained by solving the equation.

2. The method for solving the free vibration of a three-dimensional structure under constant water depth conditions according to claim 1 is characterized in that: The method for obtaining the boundary element wet surface mesh model comprises: According to the draft of the structure, the surface mesh below the waterline is extracted from the finite element mesh model to form a boundary element wet surface mesh model, and the wet surface mesh information at least includes unit number, unit connectivity, node number and node coordinates.

3. The method for solving the free vibration of a three-dimensional structure under constant water depth conditions according to claim 1 is characterized in that: Based on the boundary element wet surface mesh model and potential flow theory, a boundary integral equation is constructed, including: The fluid is regarded as an ideal irrotational and inviscid fluid, and the velocity potential of the structure in the flow field is expressed as The Laplace equations and corresponding boundary conditions are satisfied in the entire basin; According to Green's third formula, taking Green's function G(p,q) as the basic solution of Laplace equation, we construct the potential function The source pair mixed distribution boundary integral equation is obtained and expressed in a discretized form.

4. The method for solving the free vibration of a three-dimensional structure under constant water depth conditions according to claim 3 is characterized in that: Regarding the potential function The discretization expression of the source-pair mixed distribution boundary integral equation includes: Following the discrete idea of ​​boundary element, the flow field boundary S is discretized into N units, and the discrete expression of the source-pair mixed distribution boundary integral equation is obtained; When the field point p approaches the flow field boundary along the negative direction of the normal vector in the flow field, considering the influence of the dipole principal value integral -2π, the discrete format of the source-dipole mixed distribution boundary integral equation is further obtained as follows: in, Represent the field point p(x,y,z) and the source point respectively Potential function; ΔS j represents a discrete small surface unit; s q represents the small surface unit where the source point q is located; n q Represents the normal vector, and the positive direction is defined as the direction from the inside of the structure to the outside of the flow field.

5. The method for solving the free vibration of a three-dimensional structure under constant water depth conditions according to claim 3 is characterized in that: When the structure is in free vibration in still water, the free surface effect caused by waves is not considered, and the corresponding boundary conditions are expressed as: Where U is the normal velocity of the wet surface, n is the normal direction, and S H represents the boundary element wet surface, z=0 represents the still water surface, and z=-h represents the seabed.

6. The method for solving the free vibration of a three-dimensional structure under constant water depth conditions according to claim 1 is characterized in that: The method for constructing the finite depth mirror Green's function without considering the free surface effect includes: Since both the still water surface and the seabed are impenetrable rigid walls, in order to construct the potential function The boundary integral equation of the mirror image Green's function is introduced. In the water depth direction, countless mirror images are constructed to satisfy the static water surface and seabed boundary conditions. The mirror image Green's function is expressed as: in: m = 0 represents the original still water surface, and h is the depth from the still water surface to the sea bottom; (x,y,z) represents the location of the field point on the wet surface, Indicates the location of the source point on the wet surface.

7. The method for solving the free vibration of a three-dimensional structure under constant water depth conditions according to claim 1 is characterized in that: The additional mass generated by the fluid on the structure when it vibrates freely in the flow field is calculated based on the velocity potential of the flow field. The calculation formula is: Where ρ is the fluid density, φ j is the jth frequency domain velocity potential obtained by separating the time term from the velocity potential of the flow field, Re(φ j ) represents the real part of the velocity potential, n i represents the normal vector, s is the boundary element wet surface S H Small surface unit.

8. The method for solving the free vibration of a three-dimensional structure under constant water depth conditions according to claim 1 is characterized in that: The free vibration equation of the structure including the additional mass is expressed as: ([K]-ω 2 ([M]+[A])){δ0}={0} Where [M], [K], and [A] are the mass matrix, stiffness matrix, and additional mass matrix of the structure, respectively; {δ0} is independent of time and represents the structural displacement; ω represents the circular frequency.

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