An Integral Equation Mesh Adaptive Refinement Method Based on the Estimation of Non-Conformal Mesh Current Discontinuity

Abstract

The present invention belongs to the field of electromagnetic simulation technology, and provides an integral equation mesh adaptive encryption method based on non-conformal grid current discontinuity estimation for efficiently obtaining high-precision simulation results. First, the present invention discretizes the surface of the model using conformal or non-conformal triangular meshes, defines the MultiBranch-RWG basis functions on the discretized mesh elements, establishes the electric field integral equation, and solves to obtain the surface current distribution. Then, according to the surface current distribution, the current discontinuity error of each mesh element is calculated using the current discontinuity error estimation method. Finally, the current discontinuity errors of all mesh elements are traversed. When the current discontinuity error of any mesh element is greater than the preset threshold, adaptive encryption is performed, and the mesh after adaptive encryption is used as the input for iteration; otherwise, the mesh information is output. The present invention can achieve higher precision with less computational cost, has more flexible mesh generation, and is more suitable for complex targets.

Description

Technical Field

[0001] The present invention belongs to the field of electromagnetic simulation technology, and specifically provides an integral equation mesh adaptive encryption method based on non-conformal grid current discontinuity estimation. Background Art

[0002] In computational electromagnetics, the realization of fast and accurate analysis of conductor targets depends on many factors, and grid quality is one of the key factors; generally, denser grids can achieve better accuracy, but they also bring higher computational burdens. Grid adaptive encryption technology expects to achieve a balance between grid density and computational cost. The core of grid adaptive encryption technology is the error estimator, which can judge which grid cells have larger errors; among the existing error estimators, current discontinuity estimation is an error estimation method with low computational cost and high computational accuracy, but the existing technology is only applicable to conformal grids. For example, the literature: S.K. Kim and A.F. Peterson, "Evaluation of Local Error Estimators for the RWG-Based EFIE," in IEEE Transactions on Antennas and Propagation, vol. 66, no. 2, pp. 819-826, Feb. 2018. However, when analyzing multi-scale models, the generation of conformal grids is more difficult, and the unknowns generated are often much more than those of non-conformal grids; therefore, the present invention provides an integral equation mesh adaptive encryption method based on non-conformal grid current discontinuity estimation for multi-scale models. Summary of the Invention

[0003] The purpose of the present invention is to provide an integral equation mesh adaptive encryption method based on non-conformal grid current discontinuity estimation, which is used to efficiently obtain high-precision simulation results and achieve the highest possible computational accuracy at the lowest possible computational cost.

[0004] To achieve the above purpose, the technical solution adopted by the present invention is as follows:

[0005] An integral equation mesh adaptive encryption method based on non-conformal grid current discontinuity estimation, characterized by comprising the following steps:

[0006] Step 1. Discretize the model surface using conformal or non-conformal triangular grids;

[0007] Step 2. Define the MultiBranch-RWG basis function on the discretized grid cells, establish the electric field integral equation, solve to obtain the current coefficients, and then calculate the surface current distribution from the current coefficients;

[0008] Step 3. According to the surface current distribution, the current discontinuity error of each grid unit is calculated using the current discontinuity error estimation method;

[0009] Step 4. Traverse the current discontinuity errors of all grid cells. When the current discontinuity error of any grid cell is greater than the preset threshold, perform adaptive encryption, and use the adaptively encrypted grid as input, return to step 2 for iteration; otherwise, output the grid information.

[0010] Furthermore, the specific process of step 2 is:

[0011] The surface current J of the equivalent surface is discretized using the MultiBranch-RWG basis function as follows: Among them, a n is the current coefficient, f n is the MultiBranch-RWG basis function, N is the total number of basis functions, n = 1, 2, ... N; the electric field integral equation established according to the discrete form of the surface current J is:

[0012] [Z mn ][J n ]=[b m ]

[0013]

[0014] Where r is the field point, r′ is the source point, G(r,r′) represents the Green’s function, S n is the MultiBranch-RWG basis function f n The domain of S m is the MultiBranch-RWG basis function f m The domain of E inc is the incident electric field, k is the wave number, ω is the incident wave angular frequency, μ0 is the free space permeability, and j is the imaginary unit;

[0015] The electric field integral equation obtained by discretization is solved iteratively to obtain the current coefficient {a n} 1≤n≤N , and then the surface current distribution of the equivalent surface is obtained.

[0016] Furthermore, the specific process of step 3 is as follows:

[0017] Calculate the current discontinuity error of the ith unit:

[0018]

[0019] Among them, Error i represents the current discontinuity error of the i-th unit, ε (i,m)Denote the current discontinuity error of the m-th edge of the i-th element. Denote the maximum value of the tangential component of the surface current density at the midpoints of all mesh edges, A i Denote the area of the i-th element.

[0020] Furthermore, in step 3, the calculation process of the current discontinuity error for each edge in the element is as follows:

[0021] Mark the MultiBranch-RWG basis function corresponding to the m-th edge of the i-th element as f (i,m) , if the m-th edge of the i-th element is the positive triangular edge of f (i,m) , then ε (i,m) is expressed as:

[0022]

[0023] where, denotes the unit vector tangent to the m-th edge of the i-th element, Q (i,m) denotes the number of negative triangles of the MultiBranch-RWG basis function f (i,m) , r q denotes the midpoint of the overlapping edge on the q-th negative triangle of the MultiBranch-RWG basis function f (i,m) , denotes the current calculated based on the i-th element at r q , denotes the current calculated based on the q-th negative triangle at r q ;

[0024] If the m-th edge of the i-th element is the negative triangular edge of f (i,m) , then ε (i,m) is expressed as:

[0025]

[0026] where, denotes the current calculated based on the i-th element at r q , denotes the current calculated based on the positive triangle at r q .

[0027] Furthermore, in step 4, the specific process of adaptive encryption is as follows: Traverse the current discontinuity errors of all mesh elements, select the top 30% of the mesh elements with the largest errors, and perform midpoint encryption on each selected mesh.

[0028] Further, in step 4, the specific process of midpoint encryption is as follows: Connect the midpoints of the three sides of the triangular element to be encrypted to divide the triangular element to be encrypted into four triangular elements equally.

[0029] Based on the above technical solution, the beneficial effects of the present invention are as follows:

[0030] The present invention provides an integral equation grid adaptive encryption method based on non-conformal grid current discontinuity estimation for efficiently obtaining high-precision simulation results; the computational cost of this method is much smaller than that of the traditional grid adaptive encryption method based on residual estimation. Compared with the traditional integral equation grid adaptive encryption method based on conformal grid current discontinuity estimation, its grid generation is more flexible and more suitable for complex targets. Description of the Drawings

[0031] Figure 1 It is the initial grid diagram of the aircraft model in the embodiment of the present invention.

[0032] Figure 2 It is the final grid diagram after the adaptive encryption in the embodiment of the present invention.

[0033] Figure 3 It is the grid error diagram of each iteration in the adaptive encryption process in the embodiment of the present invention.

[0034] Figure 4 It is the scattered near-field diagram of each iteration in the adaptive encryption process and the scattered near-field diagram of the reference result in the embodiment of the present invention.

[0035] Figure 5 It is the schematic diagram of the principle of grid element encryption in the embodiment of the present invention. Detailed Embodiment

[0036] To make the purpose, technical solution and beneficial effects of the present invention clearer and more understandable, the present invention will be further described in detail below with reference to the drawings and embodiments.

[0037] This embodiment provides an integral equation grid adaptive encryption method based on non-conformal grid current discontinuity estimation. Taking the electromagnetic scattering of an aircraft model as an example, it includes the following steps:

[0038] Step 1. Discretize the model surface using conformal or non-conformal triangular grids;

[0039] Step 2. Define the MultiBranch-RWG basis function on the discretized grid elements, establish the electric field integral equation, solve to obtain the current coefficients, and then calculate the surface current distribution from the current coefficients; the specific process is as follows:

[0040] Traverse each edge of all triangular meshes to determine whether there are two triangular edges that completely coincide or one triangular edge completely contains several other triangular edges; if there are two triangular edges that completely coincide, define the RWG basis function within the two triangles sharing this edge; if one triangular edge completely contains several other triangular edges, define the MultiBranch-RWG basis function within these triangles. The former corresponding triangles are equilateral triangles, and the latter corresponding triangles are negative triangles; actually, the RWG basis function is a special case of the MultiBranch-RWG basis function when there is only one negative element. Therefore, in the following text, the meaning of the MultiBranch-RWG basis function includes the RWG basis function;

[0041] According to the MultiBranch-RWG basis function, establish the electric field integral equation for the current coefficients to be solved, specifically:

[0042] Discretely express the surface current J on the equivalent surface using the MultiBranch-RWG basis function as: where, a n is the current coefficient, f n is the MultiBranch-RWG basis function, N is the total number of basis functions, and n = 1, 2,... N; the electric field integral equation established according to the discrete form of the surface current J is:

[0043] [Z mn [J n = [b m

[0044]

[0045] where, r is the field point, r′ is the source point, G(r, r′) represents the Green's function, S n is the domain of the MultiBranch-RWG basis function f n , S m is the domain of the MultiBranch-RWG basis function f m ; k is the wave number, "·" represents the vector dot product, and "▽·" and "▽′·" both represent the gradient operator (the superscript "'" is only used to distinguish the objects of action); E inc is the incident electric field, ω is the incident wave angular frequency, μ0 is the magnetic permeability of free space, ε0 is the conductivity of free space, and j is the imaginary unit;

[0046] Iteratively solve the discretized electric field integral equation to obtain the current coefficients {a n}, and then obtain the surface current distribution on the equivalent surface; 1≤n≤N

[0047] ​​Step 3. According to the surface current distribution, use the current discontinuity error estimation method to calculate the current discontinuity error of each grid cell; the specific process is as follows:

[0048] Taking the calculation of the current discontinuity error of the i-th cell as an example, the current discontinuity error is expressed as:

[0049]

[0050] where Error i represents the current discontinuity error of the i-th cell, and ε (i,m) represents the current discontinuity error of the m-th side of the i-th cell, represents the maximum value of the tangential component of the surface current density at the midpoints of all grid edges, and A i represents the area of the i-th cell;

[0051] Furthermore, mark the MultiBranch-RWG basis function corresponding to the m-th side of the i-th cell as f (i,m) , if the m-th side of the i-th cell is the positive triangular side of f (i,m) , then ε (i,m) is expressed as:

[0052]

[0053] where, represents the unit vector tangent to the m-th side of the i-th cell, and Q (i,m) represents the number of negative triangles of the MultiBranch-RWG basis function f (i,m) (if corresponding to the RWG basis function, then Q (i,m) = 1), r q represents the midpoint of the overlapping edge on the q-th negative triangle of the MultiBranch-RWG basis function f (i,m) , represents the current calculated based on the i-th cell at r q , and represents the current calculated based on the q-th negative triangle at r q ;

[0054] If the m-th side of the i-th cell is the negative triangular side of f (i,m) , then ε (i,m) is expressed as:

[0055]

[0056] where, represents the current calculated based on the i-th cell at r q , and represents rq The current calculated based on an equilateral triangle.

[0057] In the present invention, by using the MultiBranch-RWG basis function, only the normal continuity of the current at the common edge is forcibly guaranteed, without forcibly requiring the tangential continuity, and the characteristic that the current discontinuity is strong where the current changes rapidly on the object surface. The discontinuity error is used as an identifier to indicate whether the grid needs to be refined;

[0058] Step 4. Traverse the current discontinuity errors of all grid cells. When the current discontinuity error of any grid cell is greater than a preset threshold, perform adaptive refinement, and use the grid after adaptive refinement as the input, and return to Step 2 for iteration; otherwise, output the grid information;

[0059] The specific process of the adaptive refinement is as follows: Traverse the current discontinuity errors of all grid cells, select the top 30% of the grid cells with the largest errors, and perform midpoint refinement on each selected grid;

[0060] The specific process of the midpoint refinement is as follows: Connect the midpoints of the three sides of the triangle unit to be refined, and divide the triangle unit to be refined into four triangle units, as Figure 5 shown.

[0061] More specifically, in this embodiment, the initial grid of the metal aircraft model is as Figure 1 shown. The length of the aircraft is 17.9 m, the width is 12 m, the height is 5.7 m, the calculation frequency is 300 MHz, the initial grid cell size is about 0.2λ, and the error threshold for the grid adaptive refinement process is set to 5×10 -4 ; According to the grid information, define the MultiBranch-RWG basis function on the discretized triangular grid. The total number of basis functions on the initial grid is 8301; The grid adaptive refinement terminates after 4 iterations. The grid error diagrams for each iteration during the adaptive refinement process are as Figure 3 shown. The finally obtained grid is as Figure 2 shown. The total number of unknowns is 128586. Taking the calculation result of the reference grid with a cell size of 0.0375λ as a comparative example, the number of unknowns in the reference grid is 531264; To prove the effectiveness of the present invention, use the scattering near-field calculation formula to calculate Figure 2 the scattering near-field at the red rectangle in Figure 4As shown, it can be seen from the figure that the scattered near field calculated by the final mesh of the adaptive encryption of the present invention is almost the same as the scattered near field calculated by the reference mesh, but the unknowns between them differ by nearly 3 times. This proves that the integral equation mesh adaptive encryption method proposed by the present invention based on the non-conformal grid current discontinuity estimation can achieve higher accuracy with fewer unknowns.

[0062] The above are only the specific embodiments of the present invention. Any feature disclosed in this specification, unless specifically described, can be replaced by other equivalent or similar-purpose alternative features; all the features disclosed, or all the steps in any method or process, except for mutually exclusive features and / or steps, can be combined in any way.

Claims

1. An integral equation mesh adaptive encryption method based on non - conformal grid current discontinuity estimation, characterized in that, Including the following steps: Step 1. Discretize the model surface using conformal or non-conformal triangular meshes; Step 2. Define the MultiBranch-RWG basis functions on the discretized mesh elements, establish the electric field integral equation, solve to obtain the current coefficients, and then calculate the surface current distribution from the current coefficients; Step 3. Calculate the current discontinuity error of each mesh element using the current discontinuity error estimation method based on the surface current distribution; the specific process is as follows: Calculate the current discontinuity error of the i-th element: Among them, Error i represents the current discontinuity error of the i-th cell, ε (i,m) represents the current discontinuity error of the m-th side of the i-th cell, represents the maximum value of the tangential component of the surface current density at the midpoints of all mesh edges, A i represents the area of the i-th cell; The calculation process of the current discontinuity error for each edge in the element is as follows: The MultiBranch-RWG basis function corresponding to the m-th edge of the i-th cell is denoted as f (i,m) , if the m-th edge of the i-th cell is an equilateral triangle edge of f (i,m) , then ε (i,m) is expressed as: Among them, represents the unit vector tangent to the m-th side of the i-th unit, Q (i,m) represents the number of negative triangles of the MultiBranch-RWG basis function f (i,m) , r q represents the midpoint of the common side on the q-th negative triangle of the MultiBranch-RWG basis function f (i,m) ; represents the current calculated based on the i-th unit at r q ; represents the current calculated based on the q-th negative triangle at r q ; If the m-th side of the i-th unit is the negative triangular side of f (i,m) , then ε (i,m) is expressed as: Among them, represents the current calculated based on the i-th unit at r q ; represents the current calculated based on the equilateral triangle at r q ; Step 4. Traverse the current discontinuity errors of all mesh elements. When the current discontinuity error of any mesh element is greater than the preset threshold, perform adaptive refinement, and use the mesh after adaptive refinement as the input, then return to Step 2 for iteration; otherwise, output the mesh information.

2. The integral equation grid adaptive encryption method based on non-conformal grid current discontinuity estimation according to claim 1, characterized in that The specific process of Step 2 is as follows: The surface current J on the equivalent surface is discretely expressed using the MultiBranch-RWG basis function as follows: where a n is the current coefficient, f n is the MultiBranch-RWG basis function, N is the total number of basis functions, and n = 1, 2,... N; The electric field integral equation established based on the discrete form of the surface current J is: [Z mn [J n =[b m ​ where \(r\) is the field point, \(r'\) is the source point, \(G(r,r')\) represents the Green's function, \(S\) n is the domain of the MultiBranch - RWG basis function \(f\) n ; \(S\) m is the domain of the MultiBranch - RWG basis function \(f\) m ; \(E\) inc is the incident electric field, \(k\) is the wave number, \(\omega\) is the angular frequency of the incident wave, \(\mu_0\) is the magnetic permeability of free space, and \(j\) is the imaginary unit. Iteratively solve the discretized electric field integral equation to obtain the current coefficients {a n}, 1≤n≤N and then obtain the surface current distribution of the equivalent surface.

3. The integral equation mesh adaptive encryption method based on non - conformal grid current discontinuity estimation according to claim 1, wherein In Step 4, the specific process of adaptive refinement is as follows: Traverse the current discontinuity errors of all mesh elements, select the top 30% of the mesh elements with the largest errors, and perform midpoint refinement on each selected mesh.

4. The integral equation mesh adaptive encryption method based on non-conformal grid current discontinuity estimation according to claim 3, characterized in that In Step 4, the specific process of midpoint refinement is as follows: Connect the midpoints of the three edges of the triangular element to be refined, and divide the triangular element to be refined into four triangular elements.

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