Cable electric field intensity calculation-oriented meshless node generation method

Abstract

The invention discloses a meshless node generation method for cable electric field intensity calculation, and belongs to the field of meshless node generation methods. The generation method comprises the following steps: (1) establishing a background grid by taking solved boundary information and model material information as model parameters; generating an initial node based on the background grid; (2) calculating the field quantity value of the initial node in the step (1), judging the difference value between the number of iterations and the maximum number of iterations, and stopping the algorithm when the number of iterations is greater than the maximum number of iterations; when the number of iterations is smaller than or equal to the maximum number of iterations, performing the next step; (3) calculating a node error of the initial node, and marking a low-precision node of which the error is greater than an error threshold value; and adding a refined node near the low-precision node, taking the refined node and the initial node as final nodes, and repeating the step (2) until the algorithm is stopped. According to the method, a high-precision calculation result can be obtained through a small number of nodes, so that the calculation precision and the convergence speed of the meshless method are improved.

Description

Technical Field

[0001] The present invention relates to the field of gridless node generation methods, and more specifically, to a gridless node generation method for cable electric field strength calculation. Background Art

[0002] With the continuous improvement of urban land development level and power load density, land for industrial development and municipal facilities is becoming increasingly tight. In order to adapt to the characteristics of high urban load density, complex system wiring, and to take into account the needs of urban appearance, urban power grids often choose to use high-voltage and ultra-high-voltage cables for power grid construction. The use of high voltage and ultra-high voltage places higher requirements on the insulation performance of cables. The electric field analysis of power cables plays a vital role in the evaluation of cable insulation performance. According to the electric field distribution in the power cable, the cable structure, shielding layer material, insulation layer material, etc. can be designed in size and optimized in performance to achieve the goal of long-term safe and economical operation of the power grid.

[0003] At present, the common cable electric field analysis methods are the electric field strength analysis and calculation methods based on the finite element method and the meshless method. Among them, the finite element method requires the solution domain to be divided, and the quality of the divided mesh has a great influence on the speed and accuracy of the calculation; when the shape of the solution domain is complex, it is difficult for the finite element method to obtain a satisfactory divided mesh. The meshless method uses nodes to discretize the solution domain, is not limited by the mesh quality, and is suitable for solving complex-shaped equipment. The nodes of the meshless method can better cover the collective shape of the cable, especially for areas with complex or irregular shapes. The meshless method can arrange nodes more flexibly to adapt to complex geometric structures. Therefore, the number and position of nodes are the key factors that determine the speed and accuracy of meshless method calculations.

[0004] However, the current gridless method has the following major defects in node generation: (1) There is no reasonable node generation criterion; if too many nodes are generated, the calculation speed is slow; if too few nodes are generated, the calculation accuracy is low; (2) There is no reasonable node refinement criterion; where to add nodes and how many nodes to add to improve the calculation accuracy have not been well solved.

[0005] Therefore, there is an urgent need to provide a method for generating meshless nodes, which can improve the calculation accuracy and speed of the meshless method. Summary of the invention

[0006] In view of the problems existing in the prior art, the purpose of the present invention is to provide a two-step generation method for meshless nodes, the first step of which is an initial node generation method based on a background grid, and the second step is an error-based refinement node generation method. This generation method not only improves the node generation rate of the meshless method, but also improves the calculation accuracy and convergence speed of the meshless method using the node generation method, thereby providing technical support for cable electric field analysis.

[0007] To solve the above problems, the present invention adopts the following technical solutions.

[0008] A meshless node generation method for cable electric field strength calculation is provided. The generation method is as follows: (1) a background grid is established by taking boundary information and model material information as model parameters; then initial nodes are generated based on the background grid;

[0009] (2) Calculate the field value of the initial node in step (1), and determine the difference between the number of iterations and the maximum number of iterations. When the number of iterations is greater than the maximum number of iterations, stop the algorithm; when the number of iterations is less than or equal to the maximum number of iterations, proceed to the next step;

[0010] (3) Calculate the node error of the initial node and mark the low-precision nodes whose errors are greater than the error threshold; add refined nodes near the low-precision nodes, and use the refined nodes and the initial nodes as the final nodes. Repeat step (2) until the algorithm stops.

[0011] Further, S1: solve the initial node generation step l step , the initial node generation step length l step The calculation formula is as follows:

[0012]

[0013] Among them, l step is the initial node generation step size, w is the width of the initial background grid; n is the number of nodes between two vertices on one side of the grid;

[0014] S2: n starts iteration from zero, l0 is the maximum step length defined by the designer; determine the initial node to generate the step length l step With the size of l0, when l step Greater than l0, n=n+1, repeat step S1; repeat until l step ≤l0, output l step ;

[0015] S3: Generate step length l based on the initial node obtained in step S2 step , generate the initial node.

[0016] Furthermore, the method for marking low-precision nodes in step (3) is as follows:

[0017] Step 1: Calculate the node error and use the maximum node error in each background grid as the error value of the background grid; specify the error value of the second highest background grid obtained in the current iteration as the error threshold;

[0018] Step 2: Mark all background grids whose error values ​​are greater than the error threshold;

[0019] Step 3: In the background grid marked in step 2, compare the node error and the error threshold, and mark the nodes whose errors are greater than the error threshold. The marked nodes are low-precision nodes.

[0020] Furthermore, the calculation method of the node error is as follows:

[0021] Use radial point interpolation to calculate the field value of high-density nodes and the field value of the current node; and bring them into the formula to calculate the node error The calculation formula is as follows:

[0022]

[0023] In the formula, is the node error, Φ h is the field value of the high-density node, and Φc is the field value of the current node.

[0024] Furthermore, the high-density node calculation method requires the maximum initial step length defined by the designer. The maximum initial step size defined by the designer The calculation formula is:

[0025]

[0026] in, A maximum initial step size defined for designers of high-density node computation methods; The maximum step size defined by the designer for the current node.

[0027] Furthermore, in step (3), the methods of adding refined nodes near low-precision nodes are divided into four categories:

[0028] (a) The marked grid contains only one low-precision node: Use the nearest neighbor search to find the node closest to the marked low-precision node as the neighboring node; take the midpoint between the marked low-precision node and the neighboring node as the refined node;

[0029] (b) The marked grid contains two low-precision nodes: the midpoint of the two marked nodes is taken as the refined node;

[0030] (c) The marked grid contains three low-precision nodes: if the marked nodes can form a triangle, the centroid of the triangle is taken as the refinement node; if the marked nodes cannot form a triangle, the midpoint of the marked node is taken as the refinement node;

[0031] (d) The marked grid contains more than 3 low-precision nodes: select any marked low-precision node as the starting point, and use nearest neighbor search to find the 2 nodes closest to it; use method (c) until the number of remaining marked low-precision nodes is less than 3; then use method (a) or method (b) to traverse all marked low-precision nodes in the marked grid.

[0032] Furthermore, in the method (a), if the number of nodes closest to the low-precision node is greater than one, one of them is randomly selected as a neighboring node.

[0033] Compared with the prior art, the advantages of the present invention are:

[0034] This scheme arranges the number and position of nodes of the meshless method appropriately through the generation method of meshless nodes, and can obtain high-precision calculation results through a small number of nodes, thereby improving the calculation accuracy and convergence speed of the meshless method, and providing technical support for cable electric field analysis. BRIEF DESCRIPTION OF THE DRAWINGS

[0035] Figure 1 It is a general flow chart of the two-step generation method of the gridless node of the present invention;

[0036] Figure 2 It is a schematic diagram of the error-based refinement point generation method of the present invention;

[0037] Figure 3 The background grid, initial nodes and final node distribution diagram of the two-dimensional metal box problem of the present invention;

[0038] Figure 4 The electric field error distribution diagram of the two-dimensional metal box problem of the present invention;

[0039] Figure 5 It is a potential cloud diagram of the two-dimensional metal box problem of the present invention;

[0040] Figure 6 The cable background grid and node distribution diagram of the present invention;

[0041] Figure 7 A potential cloud diagram of the power cable of the present invention;

[0042] Figure 8 This is the electric field cloud diagram of the power cable of the present invention. DETAILED DESCRIPTION

[0043] Embodiment 1:

[0044] 1. A meshless node generation method for cable electric field strength calculation

[0045] See also Figure 1-Figure 2 A two-step method for generating gridless nodes includes a method for generating initial nodes based on a background grid and a method for generating refined nodes based on errors; the specific method for generating gridless nodes is as follows:

[0046] Step 1: Set model parameters including solution domain boundary information, model material information, etc.; set the initial background grid width w and the maximum step length l0 defined by the designer.

[0047] Step 2: Use the value determined in step 1 to calculate the initial node generation step length l step The value of is used to determine the number of initial nodes to be generated.

[0048] Initial node generation step length l step The calculation method of is as follows:

[0049] Based on the initial background grid width w of the solution domain and the number of nodes n between two vertices on one side of the grid, the initial node generation step length l is calculated. step ; The calculation formula is as follows:

[0050]

[0051] Among them, l step is the initial node generation step; w is the initial background mesh width; n is the number of nodes between two vertices on one side of the mesh, n starts from zero and increases in each iteration until l step ≤l0, output l step ; l0 is the maximum step size defined by the designer.

[0052] Step 3: Based on the final calculation of step 2, step , generate the initial node.

[0053] Step 4: Calculate the field value of the initial node using the radial point interpolation method (RPIM) The radial point interpolation method is used to calculate the field value of high-density nodes

[0054] The relationship between the maximum initial step length defined by the designer of the high-density node calculation method for calculating the field value of the high-density node and the maximum initial step length of the present invention is:

[0055] in, represents the maximum initial step size defined by the designer of the high-density node calculation method; Represents the maximum initial step size defined by the designer of the present invention.

[0056] The calculation method for calculating the field magnitude using radial point interpolation is as follows:

[0057] The model of the problem to be solved is established based on the Maxwell equations. The cable electric field strength calculation problem is reduced to a boundary value problem and solved. The solution formula is shown in formula (1).

[0058]

[0059] in, and is an integral or differential operator, f and g are known functions, is the node position vector, Ω and r are the solution domain and boundary. is the distribution function of the field quantity to be solved in the solution domain Ω. Since this paper adopts the radial point interpolation method based on polynomial basis functions, It is estimated by formula (2).

[0060]

[0061] in, is the field quantity distribution function An approximate function of i and b j is the coefficient matrix, p j m is an element of the polynomial. is the radial basis function, which is related to the node positions and number of nodes in the solution domain Ω.

[0062] Said The calculation formula is shown in formula (3):

[0063]

[0064] in, is the position vector of the specified central node, and c is the shape function of the radial basis function.

[0065] Combining equation (1)-equation (3) we can get equation (4):

[0066]

[0067] Solve for ξ in equation (4), and then substitute ξ into equation (2) to obtain the approximate value of the field quantity distribution function: That is, the field value of the initial node is obtained

[0068] Step 5: Determine whether the number of iterations i of the node has reached the maximum number of iterations imax (which can be any integer greater than 1). If the number of iterations i is greater than the maximum number of iterations imax, stop the algorithm; if it is less than or equal to the maximum number of iterations imax, proceed to step 6.

[0069] Step 6: Calculate the node error based on the field value calculated in step 4, set the error threshold, and use the dynamic error threshold to mark the low-precision nodes.

[0070] (a) The calculation method is as follows: Based on the field value of the high-density node calculated in step 4 and the field value of the initial node, calculate the node error of the same node

[0071] The node error The calculation equation is as follows:

[0072]

[0073] in, is the node error, Φ h is the calculation result of the current node obtained using the high-density node calculation method, Φ c The calculation result of the current node.

[0074] (b) selecting the maximum error of the nodes in each background grid to represent the error value of the background grid; comparing the error values ​​of all background grids obtained in the current iteration, and specifying the error value of the second highest background grid obtained as the error threshold;

[0075] (c) comparing the error value of each background grid obtained in step (b) with the error threshold, and marking the background grid whose error value is greater than the error threshold;

[0076] (d) In the marked background grid, compare the error value of the node with the error threshold. If the error value of the node is greater than the error threshold, mark the node. The marked node is a low-precision node.

[0077] Step 7: Generate refined nodes around the low-precision nodes, and output new meshless nodes composed of the initial nodes and the refined nodes to step 4 for a loop.

[0078] The schematic diagram of the error-based refinement point generation method is as follows: Figure 2 As shown in the figure, the red marked points are low-precision nodes, that is, marked nodes; × is the newly generated refined nodes.

[0079] There are four types of methods for generating refinement nodes, as shown below:

[0080] Type I: The marked background grid contains only one marked node: Use the nearest neighbor search to find the node closest to the marked node as the neighboring node. If the number of nodes closest to the marked node is greater than one, randomly select one of them as the neighboring node. Take the midpoint between the marked node and its neighboring nodes as the new refinement node, such as Figure 2 As shown in part (a).

[0081] Type II: The marked background grid contains two marked nodes. The midpoint of the two marked nodes is taken as the new refinement node, such as Figure 2 (b) as shown.

[0082] Type III: The marked background grid contains 3 marked nodes. Check whether the marked nodes can form a triangle. If not, take the midpoint of the marked node as the new refinement node, such as Figure 2 (c) If a triangle can be formed, the centroid of the triangle is taken as the new refinement node, as shown in Figure 2 (d) as shown.

[0083] Type IV: If the marked background grid contains more than 3 marked nodes, select any one of the marked nodes as the starting point, use the neighborhood search to find the two nodes closest to it, and implement the type III operation until the number of all remaining marked nodes is less than 3; then implement the type I or type II operation until all marked nodes in the refined area are traversed.

[0084] 2. Calculation of the electric field and potential of the metal box

[0085] The proposed method is used to calculate the electric field and potential distribution of a two-dimensional metal box. The metal box is 3 meters long and 1 meter wide.

[0086] Step 1: Take the metal box as the model parameter, set the background grid width w to 1m, and the maximum step length l0 defined by the designer to 1 / 3m. Substitute it into (Formula 1) to solve the initial node generation step length l step , and generate step length l according to the initial node step Generate the initial node.

[0087] Step 2: Use radial point interpolation method (RPIM) to calculate the field value of the initial node in step 1 and use radial point interpolation method to calculate the field value of high-density nodes; when the number of iterations i is less than the maximum number of iterations i max , use (Formula 2) to calculate the node error and determine the error threshold. When the number of iterations i is greater than the maximum number of iterations i max , stop the algorithm.

[0088] Step 3: Mark the background grid of the error threshold, and then mark the nodes in the marked background grid whose node errors are greater than the error threshold. The marked nodes here are low-precision nodes; add refined nodes near the low-precision nodes according to the rules and output new grid-free nodes composed of refined nodes and initial nodes; then repeat step 2 until the algorithm stops.

[0089] Figure 3 The background grid, initial and final node distribution are shown. Figure 3 As shown in (a), 3 background grids are used and 40 initial nodes are obtained using the initial node generation method. Figure 3 In (b), red dots indicate marked low-precision nodes, while × indicates nodes added through iterations. Figure 4 A plot showing the electric field error distribution for the 2D metal box problem. Figure 5 The potential cloud diagram of the two-dimensional metal box problem obtained by the promotion method is shown, indicating that the proposed method can obtain the potential distribution in the computational domain. In solving the two-dimensional metal box problem, the comparison results of the meshless node generation method of this scheme and other meshless node generation methods are shown in Table 1.

[0090] Table 1 Comparison results between the gridless node generation method of this scheme and other gridless node generation methods

[0091] algorithm Maximum electric field error Number of nodes Hybrid method 9.80e-3 40 RBF method 7.65e-3 110 Proposed method 4.95e-3 43

[0092] As can be seen from Table 1, compared with the electric field strength calculation results of the meshless node generation method provided by the Hybrid method, the number of nodes generated by the meshless node generation method proposed in this scheme is slightly larger than the number of nodes of the Hybrid method, but the maximum value of the electric field error of this scheme is much smaller than the maximum value of the electric field error calculated by the Hybrid method. Compared with the node data and the maximum value of the electric field error generated by the meshless node proposed by the RBF method, the number of nodes and the maximum value of the electric field error obtained by this scheme are both smaller than those of the RBF method. These data show that the meshless node generation method provided by this scheme can greatly improve the calculation accuracy while increasing the number of nodes.

[0093] 3. Electric field calculation of power cable

[0094] The electric field and potential of the power cable section are calculated using the proposed method. The radius of the cable is 51mm. From the inside to the outside, it consists of a copper conductor, an insulating layer, a semiconductor, a metal sheath, and an outer sheath. An 11kV voltage is applied to the copper conductor. Due to the symmetrical structure, a 1 / 4 model of the cable is constructed. The background grid is set as a sector ring with a sector angle of 15°. The width w of the grid is the thickness of the material, and l0 is 2mm. The above algorithm is used to calculate the distribution of the electric field strength in the cable.

[0095] The background grid and node distribution of the cable are shown in the figure below. Figure 6 As shown; the potential cloud diagram of the cable is as follows Figure 7 As shown; the electric field cloud diagram of the cable is as follows Figure 8 shown.

[0096] The gridless node generation method of this scheme is compared with the traditional grid node generation method. The comparison results are shown in Table 2.

[0097] Table 2 Comparison results between the proposed gridless node generation method and the traditional gridless node generation method

[0098] algorithm Maximum absolute error of electric field (%) Number of nodes Calculation time(s) RBF method 7.1 647 53.35 Proposed method 5.4 657 24.12

[0099] As shown in Table 2, in the process of calculating the cable electric field, the number of nodes generated by the method for generating gridless nodes provided by this solution is greater than that of the traditional method for generating gridless nodes. However, the maximum absolute error of the electric field of this solution is less than that of the electric field calculated by the RBF method. In addition, the calculation time of this solution is less than that of the RBF method. These data show that the method proposed in this paper has high accuracy and also has advantages in calculation speed.

Claims

1. A meshless node generation method for cable electric field strength calculation, characterized in that: The generation method is as follows: (1) Establishing a background grid using the boundary information and model material information as model parameters; then generating initial nodes based on the background grid; (2) Calculate the field value of the initial node in step (1), and determine the difference between the number of iterations and the maximum number of iterations. When the number of iterations is greater than the maximum number of iterations, stop the algorithm; when the number of iterations is less than or equal to the maximum number of iterations, proceed to the next step; (3) Calculate the node error of the initial node and mark the low-precision nodes whose errors are greater than the error threshold; Add refinement nodes near the low-precision nodes, take the refinement nodes and the initial nodes as the final nodes, and repeat step (2) until the algorithm stops.

2. A method for generating gridless nodes for cable electric field strength calculation according to claim 1, characterized in that: In step (1), the method for generating initial nodes based on the background grid is as follows: S1: Solve the initial node generation step l step , the initial node generation step length l step The calculation formula is as follows: Among them, l step is the initial node generation step size, w is the width of the initial background grid; n is the number of nodes between two vertices on one side of the grid; S2: n starts iteration from zero, l0 is the maximum step length defined by the designer; determine the initial node to generate the step length l step With the size of l0, when l step Greater than l0, n=n+1, repeat step S1; repeat until l step ≤l0, output l step ; S3: Generate step length l based on the initial node obtained in step S2 step , generate the initial node.

3. A method for generating gridless nodes for cable electric field strength calculation according to claim 1, characterized in that: The method of marking low-precision nodes in step (3) is as follows: Step 1: Calculate the node error and use the maximum node error in each background grid as the error value of the background grid; specify the error value of the second highest background grid obtained in the current iteration as the error threshold; Step 2: Mark all background grids whose error values ​​are greater than the error threshold; Step 3: In the background grid marked in step 2, compare the node error and the error threshold, and mark the nodes whose errors are greater than the error threshold. The marked nodes are low-precision nodes.

4. A method for generating gridless nodes for cable electric field strength calculation according to claim 1, characterized in that: The calculation method of the node error is as follows: Use radial point interpolation to calculate the field value of high-density nodes and the field value of the current node; and bring them into the formula to calculate the node error The calculation formula is as follows: in, is the node error, Φ h is the field value of the high-density node, and Φc is the field value of the current node.

5. A method for generating gridless nodes for calculating cable electric field strength according to claim 4, characterized in that: In the process of calculating the field value of the high-density node, the high-density node calculation method needs to calculate the maximum initial step length defined by the designer. The maximum initial step size defined by the designer The calculation formula is: in, A maximum initial step size defined for designers of high-density node computation methods; The maximum step size defined by the designer for the current node.

6. The method for generating meshless nodes for calculating cable electric field strength according to claim 1, characterized in that: In step (3), the methods of adding refined nodes near low-precision nodes are divided into four categories: (a) The marked grid contains only one low-precision node: Use the nearest neighbor search to find the node closest to the marked low-precision node as the neighboring node; take the midpoint between the marked low-precision node and the neighboring node as the refined node; (b) The marked grid contains two low-precision nodes: the midpoint of the two marked nodes is taken as the refined node; (c) The marked grid contains three low-precision nodes: if the marked nodes can form a triangle, the centroid of the triangle is taken as the refinement node; if the marked nodes cannot form a triangle, the midpoint of the marked nodes is taken as the refinement node; (d) The marked grid contains more than 3 low-precision nodes: select any marked low-precision node as the starting point, and use the nearest neighbor search to find the 2 nodes closest to it; use the method (c) until the number of remaining marked low-precision nodes is less than 3; Then, the operation of method (a) or method (b) is used to traverse all the marked low-precision nodes in the marked grid.

7. A method for generating gridless nodes for cable electric field strength calculation according to claim 6, characterized in that: In the method (a), if the number of nodes closest to the low-precision node is greater than one, one of them is randomly selected as a neighboring node.

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