Method for calculating deformation of component with large length-width ratio based on local radial basis function collocation method

Abstract

The invention discloses a method for calculating deformation of a component with a large length-width ratio based on a local radial basis function point collocation method. The method comprises the following steps: 1) obtaining mechanical parameters of the inspected component; 2) configuring a dot matrix of a plurality of observation points in the interior and the boundary of the inspected component; 3) respectively determining local influence points for each observation point in the dot matrix; 4) obtaining an interpolation matrix of the calculation field quantity at each observation point according to a local radial basis function collocation method; 5) establishing a control equation of component deformation, and determining boundary conditions according to working conditions of the components; 6) constructing an equation for calculating a displacement field, and calculating the displacement value of each observation point; and 7) according to engineering requirements, calculating a stress field and a strain field by using the displacement value of the observation point. According to the method, the precision of structural deformation calculation is improved, the deformation calculation efficiency of the component with the large length-width ratio is greatly improved, the influence of shape parameters on the calculation precision is greatly reduced, and therefore the stability of deformation calculation is improved.

Description

Technical Field

[0001] The invention belongs to the technical field of mechanical properties of geotechnical components, and relates to a calculation method for calculating the stress deformation of components with large length-width ratios such as rods and plates based on an optimized local radial basis function collocation method. Background Art

[0002] Components such as rods and plates are common load-bearing components in the field of civil engineering. Accurately calculating the stress and deformation of components and predicting the structural force feedback are the primary conditions and determining factors for guiding the optimization of structural design and construction. Traditional component force calculations can usually be done using the finite element method. However, the calculation accuracy of this method depends to a large extent on a reasonable calculation grid, and the calculation efficiency is low, especially for large deformation problems.

[0003] The local radial basis function collocation method is a new type of meshless algorithm. It applies the radial basis function to the collocation method and localizes the algorithm, making the algorithm more efficient and the calculation results more stable. However, this method usually requires a uniform collocation lattice to be set in the entire calculation domain. For components with a large length-to-width ratio, such as rods and plates, the collocation point spacing can only be set very small due to the length of the short side. Such a collocation point spacing is too small for the long side, and the entire lattice is too dense, resulting in a decrease in calculation efficiency. Especially in three-dimensional space, this low efficiency is intolerable. At the same time, the calculation accuracy of radial basis function algorithms is often affected by the shape parameters in the function. Unreasonable shape parameters will lead to large calculation errors. So far, scholars in the industry have not been able to propose a good method to obtain the optimal shape parameters.

[0004] The local radial basis function collocation method itself has good characteristics such as simple principle, fast calculation speed and high calculation efficiency. If the above-mentioned problem of too dense collocation matrix and shape parameter selection for the calculation of components with large aspect ratio can be overcome, and the calculation efficiency and calculation stability of the algorithm can be further improved, then this method can be effectively applied to the rapid calculation and engineering design of the forces of geotechnical components such as rods and plates. Summary of the invention

[0005] The object of the present invention is to provide a simple, efficient, gridless method for calculating the stress deformation of components with large aspect ratios such as rods and plates based on a local radial basis function collocation method.

[0006] The present invention is achieved through the following technical solutions.

[0007] The present invention provides a method for calculating deformation of a component with a large aspect ratio based on a local radial basis function collocation method, comprising the following steps:

[0008] (1) Obtain the mechanical parameters of the component under investigation, including elastic modulus E, Poisson's ratio ν, density ρ, etc.

[0009] (2) Configure a matrix of observation points inside and on the boundary of the component under investigation. The configured matrix is ​​arranged along the coordinate axis. Use N x ,N y ,N z Represents the number of point arrays along the three coordinate axes, so the total number of observation points is n = N x ×N y ×N z .

[0010] (3) Determine the local influence point for each observation point in the lattice. The local influence point is used to interpolate and calculate the first-order partial derivative value of the field quantity u at the corresponding observation point.

[0011] The local influence point selects the three closest points on the same line as the observation point according to the direction of the partial derivative. i ,y i ,z i ), calculate the partial derivative of the field quantity u When , select the local influence point along the x direction. Similarly, calculate and When , select points along the y and z directions respectively. At this time, the same observation point should have three groups of local influence points in different directions.

[0012] (4) According to the local radial basis function collocation method, the interpolation matrix of the calculated field quantity u at each observation point is obtained.

[0013] (5) Establish the control equation for component deformation and determine the boundary conditions based on the working conditions of the component.

[0014] (6) Substitute the control equations and boundary conditions into the corresponding interpolation matrix to construct a set of equations for calculating the displacement field and calculate the displacement value of each observation point.

[0015] (7) According to the needs of the project, the displacement values ​​of the observation points are used to calculate the stress field and strain field. Substituting the displacement field into the geometric equation can obtain the strain component field. Substituting the strain component into the constitutive equation can obtain the stress component field.

[0016] Preferably, in step (4), the field quantity u is interpolated and expressed as:

[0017]

[0018] Among them, x = (x, y, z) is the coordinate of the observation point, x k is the coordinate of the local influence point, α k is the weight function, and φ is the radial basis function. The radial basis function used in deformation calculation is the multi-quadratic surface MQ radial basis function:

[0019]

[0020] Where r = ‖xx k ‖ is the Euclidean distance between the observation point and the local influence point, and c is the shape parameter. If the distance between the two points is d, the c value can be any value between 15d and 10000d.

[0021] make φ′=[φ(‖x-x1‖)φ(‖x-x2‖)φ(‖x-x3‖)], then the interpolation matrix of the field quantity u at any point in the field is:

[0022]

[0023] Let the partial differential operator for calculating the field quantity be L, Lφ′=[Lφ(‖x-x1‖)Lφ(‖x-x2‖)Lφ(‖x-x3‖)], and the partial differential form interpolation matrix of the field quantity u is:

[0024]

[0025] Preferably, in step (5), the control equation for component deformation is selected as the elastic control equation:

[0026]

[0027] in, is the Laplace operator, and the equation contains three displacement field quantities: {u x ,u y ,u z}, are the displacement components in the three coordinate axis directions. In addition, the displacement boundary conditions of the component are expressed as:

[0028]

[0029] in is the known displacement component of the boundary. The stress boundary condition of the component is expressed as:

[0030]

[0031] Among them, {T x , T y , T z} are the known stress components on the boundary, {l x , l y , l z} is the cosine value of the angle between the boundary normal vector and each coordinate axis. x , σ y , σ z} and τ xy =τ yx , τ yz =τ zy , τxz =τ zx are the normal stress and shear stress at the boundary point.

[0032] The present invention proposes to select local influence points on a line according to the deflection direction, and then use the local radial basis function collocation method to calculate the stress deformation of components with large aspect ratios such as rods and plates. The beneficial effects of the present invention are:

[0033] (1) In the present invention, the local influence points are distributed on a line, which converts the high-dimensional problem into a one-dimensional problem. Therefore, the present invention improves the accuracy of structural deformation calculation.

[0034] (2) In the present invention, since the selection of local influence points in different directions is completely independent, the lattice density in different directions can be different. For components with a large length-to-width ratio such as rods and plates, different lattice densities are selected in the length and width directions, which avoids the problem of over-dense lattices in traditional methods and greatly improves the calculation efficiency.

[0035] (3) In the present invention, only three local influence points are used for deformation calculation. The distance between the observation point and the local influence point is constant, which greatly reduces the influence of the shape parameters on the calculation accuracy. This will make the value range of the shape parameters very wide and improve the stability of the deformation calculation. BRIEF DESCRIPTION OF THE DRAWINGS

[0036] Figure 1 It is a flow chart of the present invention.

[0037] Figure 2 Schematic diagram of the method for selecting local influence points of the present invention.

[0038] Figure 3 Schematic diagram of the cantilever beam dimensions in an embodiment of the present invention.

[0039] Figure 4 is the relative error of the calculation results under different shape parameters in the embodiment of the present invention. DETAILED DESCRIPTION

[0040] The present invention will be further described below in conjunction with the accompanying drawings and embodiments.

[0041] Figure 1 This is a flowchart for calculating the stress and deformation of components with large length-width ratios such as rods and plates based on the optimized local radial basis function collocation method. The specific steps are as follows:

[0042] (1) Obtain the mechanical parameters of the component under investigation. The main mechanical parameters include: elastic modulus E, Poisson's ratio v, density ρ, etc.

[0043] (2) Configure a number of observation points in a matrix inside and on the boundary of the component under investigation. For a regular rectangular component, the configured matrix can be arranged regularly along the coordinate axis. x ,N y ,N z Represents the number of point arrays along the three coordinate axes, so the total number of observation points is n = N x ×N y ×N z For irregular components, the mirror method is needed to mirror the irregular components to the regular computational domain. (Related mirror method reference: M. Li, PH Wen. Finite block method for transient heat conduction analysis in functionally graded media. Int. J. Numer. Meth. Eng., 99: 372–390, 2014.)

[0044] (3) Determine the local influence point for each observation point in the dot matrix. The local influence point is used to interpolate the first-order partial derivative value of the corresponding observation point. The local influence point selects the three closest points on the same line as the observation point according to the direction of the partial derivative. Figure 2 In the calculation domain shown, only three local influence points are selected on the x-axis to calculate On the y-axis, three other local points are used to calculate For higher-order partial derivatives, it is necessary to use multiple first-order partial derivative calculations at the observation point.

[0045] (4) The interpolation matrix of the calculated field quantity at each observation point is obtained according to the local radial basis function collocation method. For the calculated field quantity u, the radial basis function expression is:

[0046]

[0047] Among them, x = (x, y, z) is the coordinate of the observation point, x k is the coordinate of the local influence point, α k is the weight function, and φ is the radial basis function. The radial basis function used in deformation calculation is the multi-quadratic surface MQ radial basis function:

[0048]

[0049] Where r = ‖xx k ‖ is the Euclidean distance between the observation point and the local influence point, and c is the shape parameter. Since the value of the shape parameter c is related to the distance between two adjacent points in the lattice, if the distance between the two points is d, the value of c can be arbitrarily between 15d and 10000d.

[0050] make φ′=[φ(‖x-x1‖)φ(‖x-x2‖)φ(‖x-x3‖)], then the interpolation matrix of any point in the field is:

[0051]

[0052] Let the partial differential operator for calculating the field quantity be L, Lφ′=[Lφ(‖x-x1‖)Lφ(‖x-x2‖)Lφ(‖x-x3‖)], then the partial differential form interpolation matrix of the field quantity is expressed as:

[0053]

[0054] (5) Establish the control equation of component deformation and determine the boundary conditions according to the working conditions of the component. Typical elastic mechanics equations and boundary condition equations can be used.

[0055] (6) Construct the equation for calculating the displacement field and calculate the displacement value of each observation point. Substitute the control equations and boundary conditions of the component stress deformation into equations (s.3) and (s.4). For n observation points, 3n equations will be generated. Solving the equations can obtain the displacement field of all observation points: {u x ,u y ,u z}. In specific calculations, the equations can be integrated into a matrix form:

[0056] Φu=F (s.5)

[0057] Where u is a 3n×1 vector whose elements contain the displacements of all the calculated points in the three coordinate directions. F is a 3n×1 vector whose elements are determined according to the control equations and boundary conditions. Φ is a 3n×3n sparse matrix whose non-zero elements are given by equations (s.3) and (s.4). The result determines the solution of equation (s.5), where u is the displacement value of each observation point.

[0058] (7) According to the needs of the project, the displacement value of the observation point is used to calculate the stress field and strain field. Substituting the obtained displacement field into the geometric equation of elastic mechanics can obtain the strain component field. Substituting the strain component into the constitutive equation of elastic mechanics can obtain the stress component field.

[0059] Example

[0060] In this embodiment, a cantilever beam subjected to uniform load is selected, and its material is elastic material, and the elastic modulus E=1.2×10 9 Pa, Poisson's ratio v = 0.2, density ρ = 2500 kg / m 3 The beam size is 1m×4m×1m, Figure 3 It is the cantilever beam size state.

[0061] Set N in the component scope x ×N y ×N z =21×41×21 dot matrix, and determine the local influence point for each observation point. According to the basic idea of ​​radial basis function collocation method, multiple quadratic surface MQ radial basis function is used. Construct an interpolation matrix. Since the dot density in each coordinate axis direction is different, we use Determine the shape parameter in the radial basis function, where c x , c y , c z The shape parameter used to solve the partial derivatives in each coordinate axis direction, d x , d y , d z is the spacing of the dots in each coordinate axis direction.

[0062] The upper surface of the beam is subjected to a uniformly distributed load q = -10 4 Pa. The mathematical expression of its boundary condition is:

[0063]

[0064] The solution equation of the displacement field of the structure component is solved, and the displacement field formed by all observation points is obtained. The following table 1 lists the calculated maximum displacement of the beam and its relative error with the finite element calculation result. Since the sign of the displacement value represents the direction, the symbol Represents the maximum displacement value in each direction, Rel x ,Rel y ,Rel z is the relative error of the maximum displacement value in each direction.

[0065] Table 1

[0066]

[0067] The range of choices for shape parameters is very wide. Figure 4 The relative error results obtained by the embodiments under different shape parameters are shown in the figure. The corresponding Rel value x 、Rel y 、Rel z ,It can be seen from the figure that the shape parameters have very little effect on the calculation error.

[0068] According to engineering needs, the displacement field is substituted into the geometric equation of elastic mechanics to obtain the strain component field. The strain component is substituted into the constitutive equation of elastic mechanics to obtain the stress component field.

[0069] Result analysis: The present invention can quickly and effectively calculate the displacement, stress and strain field of components. Compared with finite element data, the calculation method can achieve high-precision calculation of deformation results. Since the selection of local influence points in different directions is completely independent, different lattice densities are selected in the length and width directions for components with large length-to-width ratios such as rods and plates, which reduces the number of required calculation points and greatly improves the calculation efficiency. This efficiency improvement will become more obvious as the aspect ratio increases. This calculation method only uses 3 local influence points for deformation calculation, which greatly reduces the influence of shape parameters on calculation accuracy. The range of possible values ​​of shape parameters is very wide, which greatly improves the stability of deformation calculation.

[0070] The above description is only an embodiment of the present invention and does not limit the present invention in any form. Therefore, any simple modification, equivalent transformation or decoration made according to the technical essence of the present invention without departing from the content of the present invention is still within the scope of patent protection of the present invention.

Claims

1. A method for calculating deformation of components with large aspect ratio based on local radial basis function collocation method, characterized by: The steps include: (1) Obtain the mechanical parameters of the component under investigation, including elastic modulus E, Poisson's ratio v, and density ρ; (2) Configure a matrix of observation points inside and on the boundary of the component under investigation, and arrange the configured matrix along the coordinate axis; use N x ,N y ,N z Represents the number of point arrays along the three coordinate axes, so the total number of observation points is n = N x ×N y ×N z ; (3) Determine the local influence point for each observation point in the lattice, and use the local influence point to interpolate and calculate the first-order partial derivative value of the field quantity u at the corresponding observation point; The local influence point selects the three closest points on the same line as the observation point according to the direction of the partial derivative; that is, at the observation point (x i ,y i ,z i ), calculate the partial derivative of the field quantity u When , select the local influence point along the x direction; similarly, calculate and When , points are selected along the y and z directions respectively; at this time, the same observation point should have three groups of local influence points in different directions; (4) Obtaining the interpolation matrix of the calculated field quantity u at each observation point according to the local radial basis function collocation method; (5) Establish the control equation of component deformation and determine the boundary conditions according to the working conditions of the component; (6) Substitute the control equations and boundary conditions into the corresponding interpolation matrix to construct a set of equations for calculating the displacement field and calculate the displacement value of each observation point; (7) According to engineering needs, the displacement values ​​of the observation points are used to calculate the stress field and strain field; the obtained displacement field is substituted into the geometric equation to obtain the strain component field; the strain component is substituted into the constitutive equation to obtain the stress component field.

2. The method for calculating deformation of a component with a large aspect ratio based on a local radial basis function collocation method according to claim 1 is characterized by: In step (4), the field quantity u is interpolated and expressed as: Among them, x = (x, y, z) is the coordinate of the observation point, x k is the coordinate of the local influence point, α k is the weight function, φ is the radial basis function; the radial basis function used in deformation calculation is the multi-quadratic surface MQ radial basis function: Where r = ‖xx k ‖ is the Euclidean distance between the observation point and the local influence point, and c is the shape parameter; if the distance between the two points is d, the value of c can be arbitrarily between 15d and 10000d; make φ′=[φ(‖x-x1‖)φ(‖x-x2‖)φ(‖x-x3‖)], then the interpolation matrix of the field quantity u at any point in the field is: Let the partial differential operator for calculating the field quantity be L, Lφ′=[Lφ(‖x-x1‖)Lφ(‖x-x2‖)Lφ(‖x-x3‖)], and the partial differential form interpolation matrix of the field quantity u is:

3. The method for calculating deformation of a component with a large aspect ratio based on a local radial basis function collocation method according to claim 1 is characterized by: In step (5), the control equation for component deformation is selected as the elastic control equation: in, is the Laplace operator, and the equation contains three displacement field quantities: {u x ,u y ,u z }, are the components of displacement in the three coordinate axis directions; in addition, the displacement boundary conditions of the component are expressed as: in is the known displacement component of the boundary; the stress boundary condition of the component is expressed as: Among them, {T x , T y , T z } are the known stress components on the boundary, {l x , l y , l z } is the cosine value of the angle between the boundary normal vector and each coordinate axis; {σ x , σ y , σ z } and τ xy =τ yx , τ yz =τ zy , τ xz =τ zx are the normal stress and shear stress at the boundary point.

Images