Gravity long wave simulation method based on space-time joint collocation type meshless method

Abstract

The invention belongs to the technical field of gravity long wave simulation, and discloses a gravity long wave simulation method based on a space-time joint collocation type meshless method, which comprises the following steps of: establishing a gravity wavelength control equation; converting the gravity wavelength control equation into a polar coordinate form; obtaining a complete basic function for the polar coordinate expression of the gravity wavelength control equation by using a variable separation method; linearly superposing the primary functions to form a gravity long wave approximate solution; introducing a time axis into the two-dimensional space coordinate system, and establishing a space-time coordinate system; and obtaining a corresponding solution according to any space-time coordinate in the research area through space-time joint solution. According to the gravity long wave simulation method based on the space-time joint collocation type meshless method, a boundary meshless method and a semi-analytical method are combined, the simulation precision in a whole computational domain is improved, and discrete errors introduced by a time difference method are avoided; by constructing a semi-analytical solution, the calculation efficiency and precision are improved, and the propagation and reflection characteristics of waves are better captured.

Description

Technical Field

[0001] The invention relates to the technical field of gravity long wave simulation, and in particular to a gravity long wave simulation method based on a time-space joint point-matching type gridless method. Background Art

[0002] When simulating long gravity waves, traditional grid-based methods, such as the finite difference method and the finite element method, face some obstacles and difficulties, especially in irregular calculation areas or when high accuracy is required. The first is the influence of the grid. The characteristic of the grid method is that it must be solved after the grid is formed, and the accuracy of the solution depends on the quality of the grid division. If the grid division is too coarse, it will lead to the inability to correctly simulate the shape of the wave; if the grid division is too fine, it will consume a lot of time and computing resources. The second is error accumulation. In today's mainstream numerical methods, in order to deal with transient problems, the time difference method is often used to simulate the change of time. The time difference method inevitably introduces truncation errors when simulating long gravity waves. This error comes from the approximation of the time integral formula, because high-order terms are usually ignored in this method. As the calculation time progresses, the error will gradually accumulate, which may affect the accuracy of the entire simulation result. There is also boundary condition processing. The grid method usually relies on grid nodes that are uniformly or non-uniformly distributed throughout the calculation domain. When the mesh node positions do not match the complex shape of the actual boundary conditions, additional numerical errors will be introduced, affecting the accuracy and stability of the numerical solution, and additional processing will be required to approximate or accurately describe the boundary conditions.

[0003] Therefore, based on the above technical defects, the present invention proposes a gravity long wave simulation method based on a space-time joint point-matching gridless method, aiming to accurately and efficiently simulate the gravity long wave problem. Summary of the invention

[0004] The purpose of the present invention is to provide a gravity long wave simulation method based on a time-space joint point-matching type gridless method, which combines a boundary gridless method and a semi-analytical method to improve the simulation accuracy in the entire calculation domain; establish a time-space coordinate system to avoid the discrete error introduced by the time difference method by simultaneously processing time and space variables; combine the advantages of analytical solutions and numerical solutions to improve the calculation efficiency and accuracy by constructing a semi-analytical solution; and better capture the propagation and reflection characteristics of waves in the process of solving wave equations.

[0005] To achieve the above object, the present invention provides a gravity long wave simulation method based on a time-space joint point-matching meshless method, comprising the following steps:

[0006] S1. Establish the gravity wavelength control equation;

[0007] S2. Convert the gravity wavelength control equation into polar coordinate form;

[0008] S3. Use the separation of variables method to obtain the complete basic function for the polar coordinate expression of the gravity wavelength control equation;

[0009] S4, linearly superimpose the above basis functions to form an approximate solution of gravity long waves;

[0010] S5. Introduce the time axis into the two-dimensional space coordinate system to establish a space-time coordinate system;

[0011] S6. Through the joint solution of time and space, the corresponding solution is obtained according to any space-time coordinates in the study area.

[0012] Preferably, in step S1, the gravity long wave control equation is as follows:

[0013]

[0014] Among them, ζ(x, y, t) represents the vertical displacement of the free surface of the liquid during vibration; represents the speed of wave propagation in water, g represents gravity, and h0 is the depth of water when the liquid surface is in equilibrium.

[0015] Preferably, in step S2, the gravity wavelength control equation of step S1 is converted into polar coordinate form as shown below:

[0016]

[0017] Among them, r is the polar diameter and θ is the polar angle.

[0018] Preferably, in step S4, the basis functions are linearly superimposed to form a gravity long wave approximate solution as shown below:

[0019]

[0020] in, D 1 ,D 2 ,D 3 ,D 4 is an arbitrary constant; m, s are orders; J0 is λ q ,0th order Bessel function of the first kind, I0 is λ q ,0th order modified Bessel function of the first kind.

[0021] Preferably, in step S5, a time axis is introduced into a two-dimensional space coordinate system to establish a space-time coordinate system; points are randomly distributed in the space-time coordinate system, and the initial value and boundary value are discretized into n1 points and n2 points; the space-time coordinates and boundary values ​​of the discrete points are brought into the gravity long wave approximate solution, and the solution is obtained by the basis function that satisfies the control equation, and a group of linear algebraic equations is established and expressed in the matrix operation form of Cλ=D without dividing the grid.

[0022] Preferably, in step S6, the linear algebraic equation group Cλ=D is left-divided to obtain undetermined coefficients that satisfy initial conditions and boundary conditions, and the corresponding solution is obtained by joint space-time solution according to any space-time coordinates in the study area.

[0023] Therefore, the present invention adopts the above-mentioned gravity long wave simulation method based on the time-space joint point-matching type gridless method, and the beneficial effects are as follows:

[0024] (1) Using nodes instead of grids: Using a set of nodes instead of grids to discretize any problem domain and boundary, there is no need to generate and adjust internal grids, which saves a lot of time required to generate complex grids;

[0025] (2) High accuracy: The approximate solution is formed by combining basis functions that satisfy the governing equation, which naturally satisfies the strong form of the equation in the entire computational domain, making the numerical solution have high accuracy in the entire domain. At the same time, it reduces the error accumulation in the numerical calculation and improves the computational efficiency.

[0026] (3) Boundary condition processing: By arranging points on the boundary and directly enforcing the boundary conditions to be satisfied at these points, complex boundary conditions can be processed more accurately.

[0027] The technical solution of the present invention is further described in detail below through the accompanying drawings and embodiments. BRIEF DESCRIPTION OF THE DRAWINGS

[0028] Figure 1 It is a schematic diagram of the research area of ​​the present invention;

[0029] Figure 2 It is a schematic diagram of time-space point distribution of the present invention;

[0030] Figure 3 This is a schematic diagram of the boundary points in Example 1;

[0031] Figure 4 is the distribution diagram of MAE under different m,s orders in Example 1;

[0032] Figure 5 is the distribution of MAE at different m orders when s=5 in Example 1;

[0033] Figure 6is the MAE and E under different numbers of boundary points in the embodiment L2 Distribution of

[0034] Figure 7 This is a schematic diagram of the boundary points in Example 2;

[0035] Figure 8 is a graph showing the relationship between error and simulation time in Example 2;

[0036] Fig. 9 It is a flow chart of a gravity long wave simulation method based on a space-time joint point-matching type gridless method of the present invention. DETAILED DESCRIPTION

[0037] The technical solution of the present invention is further described below through the accompanying drawings and embodiments.

[0038] like Fig. 9 As shown, the present invention provides a gravity long wave simulation method based on a time-space joint point-matching type meshless method, comprising the following steps:

[0039] S1. Establish the gravity wavelength control equation.

[0040] Gravity long waves are gravity waves with wavelengths much greater than the depth of the liquid, such as the long waves in a large pool. When the bottom of the pool is horizontal, the governing equation for gravity long waves is as follows:

[0041]

[0042] Among them, ζ(x, y, t) represents the vertical displacement of the free surface of the liquid during vibration; represents the speed of wave propagation in water, g represents gravity, and h0 is the depth of water when the liquid surface is in equilibrium.

[0043] S2. Convert the gravity wavelength control equation of step S1 into polar coordinate form as follows:

[0044]

[0045] Among them, r is the polar diameter and θ is the polar angle.

[0046] S3. The polar coordinate expression of the gravity wavelength control equation is used to obtain the complete basic function using the separation of variables method, as shown in Table 1.

[0047] Table 1 Complete basic functions

[0048]

[0049]

[0050] S4. Linearly superimpose the above basis functions to form an approximate solution of gravity long waves, as shown below:

[0051]

[0052] in, D 1 ,D 2 ,D 3 ,D 4 is an arbitrary constant; m, s are orders; J0 is λ q ,0th order Bessel function of the first kind, I0 is λ q ,0th order modified Bessel function of the first kind.

[0053] S5. Introduce the time axis into the two-dimensional space coordinate system and establish a space-time coordinate system.

[0054] like Figure 1 The computational domain shown in the figure introduces the time axis into the two-dimensional space coordinate system to establish the space-time coordinate system. In the space-time coordinate system, points are randomly distributed, and the initial value and boundary value are discretized into n1 points and n2 points, as shown in Figure 2 As shown. The space-time coordinates and boundary values ​​of discrete points are brought into the approximate solution of gravity long waves, and the solution is obtained by satisfying the basis functions of the control equations. A set of linear algebraic equations is established and expressed as a matrix operation form of Cλ=D, without the need to divide the grid, and the complex calculation area can be effectively processed.

[0055] Among them, the matrix D is a (n1+n2)×1 matrix, which is composed of boundary values ​​and initial values. The matrix C is a (n1+n2)×4(m+s+2ms) matrix, as shown below:

[0056]

[0057] The coefficient λ to be determined is a 4(m+s+2ms)×1 matrix, as shown below:

[0058]

[0059] S6. Divide the linear algebraic equation system Cλ=D by left division to obtain the unknown coefficients that satisfy the initial and boundary conditions, and obtain the corresponding solutions according to any space-time coordinates in the study area. By solving the problem in space and time, it is more efficient to deal with dynamic change problems, and at the same time reduces the error accumulated by the traditional time difference method over time.

[0060] According to a gravity long wave simulation method based on a spatiotemporal joint point-matching meshless method proposed in the present invention, the accuracy of the method is analyzed through the following examples.

[0061] Example 1

[0062] Assuming the wave speed c = 1, the gravity long wave control equation is as follows:

[0063]

[0064] The study areas are as follows:

[0065]

[0066] The boundary conditions are as follows:

[0067]

[0068] The initial conditions are as follows:

[0069] ζ(x,y,t=0)=0 (10)

[0070] The points of the calculation area are as follows: Figure 3 As shown, the analytical solution expression of this embodiment is as follows:

[0071]

[0072] The accuracy is measured by the distribution of the maximum absolute error (MAE) and L 2 Norm Error As shown below:

[0073] MAE=maxζ A (r i ,θ i ,t i )-ζ N (r i ,θ i ,t i ), 1≤i≤N (12)

[0074]

[0075] Among them, A (r i ,θ i ,t i ) and ζ N (r i ,θ i ,t i ) represents the analytical solution and numerical solution of the i-th point, and N represents the number of internal points. The distribution of MAE under different m and s orders is as follows: Figure 4 As shown. Combined Figure 4 and Figure 5 , when the order of s is 5 and the order of m is greater than 12, MAE stabilizes at 10 -9 When s = 5 and m = 12, the MAE is the smallest, which is 1.071×10-9 .

[0076] like Figure 6 As shown in the figure, the distribution of errors under different numbers of boundary points is shown. It can be seen that when the number of boundary points is greater than 800, both errors are relatively stable, and MAE is stable at 10 -9 Magnitude, Stable at 10 -11 Therefore, the method proposed in the present invention has good stability.

[0077] Example 2

[0078] Assuming the wave speed c = 1, the governing equation of gravity long waves is as follows:

[0079]

[0080] The study areas are as follows:

[0081]

[0082] The boundary conditions are as follows:

[0083]

[0084] The initial conditions are as follows:

[0085] ζ(x,y,t=0)=0 (18)

[0086] The analytical solution expression of this embodiment is as follows:

[0087]

[0088] In this embodiment, the number of boundary points is 3925, the orders of m and s are 10 and 12 respectively, and the points of the calculation area are as follows: Figure 7 shown.

[0089] like Figure 8 As shown in the figure, the relationship between error and simulation time is shown. It can be seen that as time increases, MAE and are relatively stable, with MAE being stable at 10 -5 Magnitude, Stable at 10 -6 As shown in Table 2, the accuracy of the proposed method is compared with other methods. It can be seen that at different time scales, the accuracy of the proposed method is better than that of the finite element method and the radial basis function method (multiple quadratic functions).

[0090] Table 2 The performance of each method under different time scales contrast

[0091] method Finite Element Method Radial Basis Function Method (Multiple Quadratic Functions) The present invention Points 15981 1209 3925 t＝12 <![CDATA[1.18×10 -4 ]]> <![CDATA[2.81×10 -4 ]]> <![CDATA[2.07×10 -6 ]]> t＝24 <![CDATA[3.70×10 -5 ]]> <![CDATA[2.48×10 -5 ]]> <![CDATA[2.70×10 -6 ]]> t＝36 <![CDATA[2.67×10 -4 ]]> <![CDATA[3.97×10 -4 ]]> <![CDATA[1.62×10 -6 ]]> t＝48 <![CDATA[5.27×10 -4 ]]> <![CDATA[6.98×10 -4 ]]> <![CDATA[7.98×10 -7 ]]> t＝60 <![CDATA[4.07×10 -4 ]]> <![CDATA[5.90×10 -4 ]]> <![CDATA[2.85×10 -6 ]]>

[0092] Therefore, the present invention adopts the above-mentioned gravity long wave simulation method based on the time-space joint point-matching type gridless method, combines the boundary gridless method and the semi-analytical method, and improves the simulation accuracy in the entire calculation domain; establishes a time-space coordinate system, and simultaneously processes time and space variables to avoid the discrete error introduced by the time difference method; combines the advantages of analytical solutions and numerical solutions, and improves the calculation efficiency and accuracy by constructing a semi-analytical solution; in the process of solving the wave equation, better captures the propagation and reflection characteristics of the wave.

[0093] Finally, it should be noted that the above embodiments are only used to illustrate the technical solution of the present invention rather than to limit it. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that they can still modify or replace the technical solution of the present invention with equivalents, and these modifications or equivalent replacements cannot cause the modified technical solution to deviate from the spirit and scope of the technical solution of the present invention.

Claims

1. A gravity long wave simulation method based on a spatiotemporal joint point-matching meshless method, characterized in that: The following steps are involved: S1. Establish the gravity wavelength control equation; S2. Convert the gravity wavelength control equation into polar coordinate form; S3. Use the separation of variables method to obtain the complete basic function for the polar coordinate expression of the gravity wavelength control equation; S4, linearly superimpose the above basis functions to form an approximate solution of gravity long waves; S5. Introduce the time axis into the two-dimensional space coordinate system to establish a space-time coordinate system; S6. Through the joint solution of time and space, the corresponding solution is obtained according to any space-time coordinates in the study area.

2. The method for simulating long-wave gravity based on the spatiotemporal joint point-matching meshless method according to claim 1, characterized in that: In step S1, the gravity long wave governing equation is as follows: Among them, ζ(x, y, t) represents the vertical displacement of the free surface of the liquid during vibration; represents the speed of wave propagation in water, g represents gravity, and h0 is the depth of water when the liquid surface is in equilibrium.

3. The method for simulating gravity long waves based on the space-time joint collocation type meshless method according to claim 1, characterized in that: In step S2, the gravity wavelength control equation of step S1 is converted into polar coordinate form as follows: Among them, r is the polar diameter and θ is the polar angle.

4. The method for simulating long-wave gravity based on the spatiotemporal joint point-matching meshless method according to claim 1, characterized in that: In step S4, the basis functions are linearly superimposed to form an approximate solution of gravity long waves, as shown below: in, D 1 ,D 2 ,D 3 ,D 4 is an arbitrary constant; m, s are orders; J0 is λ q ,0th order Bessel function of the first kind, I0 is λ q ,0th order modified Bessel function of the first kind.

5. The method for simulating long waves of gravity based on a spatiotemporal joint point-matching meshless method according to claim 1, characterized in that: In step S5, the time axis is introduced into the two-dimensional space coordinate system to establish a space-time coordinate system; points are randomly distributed in the space-time coordinate system, and the initial value and boundary value are discretized into n1 points and n2 points; the space-time coordinates and boundary values ​​of the discrete points are brought into the gravity long wave approximate solution, and the solution is obtained by the basis function that satisfies the control equation, and a group of linear algebraic equations is established and expressed in the matrix operation form of Cλ=D without dividing the grid.

6. The method for simulating long waves of gravity based on the space-time joint collocation type meshless method according to claim 5, characterized in that: In step S6, the linear algebraic equation group Cλ=D is divided by left division to obtain the unknown coefficients that meet the initial conditions and boundary conditions, and the corresponding solution is obtained by joint time and space solving according to any time and space coordinates in the study area.

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