CN113792257A - Electromagnetic scattering solving method based on MBRWG and grid adaptive encryption - Google Patents
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Abstract
The invention discloses an electromagnetic scattering solving method based on MBRWG and mesh self-adaptive encryption, which comprises the steps of firstly subdividing all triangles of an initial mesh, then projecting the current coefficient of the initial mesh onto the subdivided mesh, obtaining the voltage vector of the subdivided mesh through Galois detection calculation, then comparing the voltage vector obtained by directly irradiating the voltage vector obtained by projection calculation and a plane wave on the subdivided mesh, marking the triangular mesh with larger voltage vector error, finally carrying out local mesh encryption on the part with larger error according to a marked graph, and processing a non-conformal mesh by using a plurality of RWG (MBRWG) basis functions. When the method is applied to solving the electromagnetic scattering problem of the multi-scale target, the grid number and the calculation precision can be well balanced.
Description
Technical Field
The invention relates to an electromagnetic scattering solving method based on MBRWG and grid adaptive encryption, which is suitable for analyzing a multi-scale problem.
Background
Grid quality is critical to accurately solving many problems. Generally speaking, the accuracy of the solution is proportional to the density of the grid, but the solution efficiency is reduced due to the excessive density of the grid, so that the trade-off between the grid density and the calculation cost is required. It is a challenge how to achieve the desired accuracy for multi-scale targets. Adaptive partial encryption has two key steps: 1. error detection, 2, local encryption.
For the surface-area equation-based approach, local a posteriori error estimation techniques have been applied to identify high error regions in the case of simple geometries. In addition, Ubeda et al propose a geometry-based mesh encryption method, which is mainly used to encrypt sharp edges to improve the solution accuracy. Recently, Vasquez et al have proposed an adaptive encryption method for complex three-dimensional surface equations. The method improves the overall precision by encrypting the grid with larger residual errors.
In order to solve the non-conformal grids (including grids with different sizes and unmatched nodes) after adaptive encryption, the Vasquez method adopts a discontinuous galileo technology based on Half RWG (Half RWG, HRWG) basis functions. However, the discontinuous Galois field technique is introduced, so that the number of iteration steps of the impedance matrix iterative solution is increased, and the iterative solution efficiency is influenced. In addition, the penalty term and its coefficient in the discontinuous galileo technique need to be selected by experience, which is inconvenient for use.
Disclosure of Invention
The purpose of the invention is as follows: by using a Multi-Branch basis function (MBRWG) basis function and a mesh with large local encryption residuals, a good balance of accuracy and efficiency can be achieved. The method can reduce the number of grids as much as possible under the condition of ensuring the precision, and is suitable for solving the multi-scale problem. When processing non-conformal grid parts, the invention replaces HRWG with MBRWG, which can not only avoid selecting punishment item, but also reduce the number of unknown quantity. Compared with the HRWG, the MBRWG does not destroy the continuity of the current and has better iterative convergence characteristics.
In order to achieve the purpose, the technical scheme of the invention is realized as follows:
an electromagnetic scattering solving method based on MBRWG and grid adaptive encryption is characterized by comprising the following steps:
step 1: establishing a surface area partial equation for scattering calculation aiming at the electromagnetic scattering problem of a conductor target, then dispersing the surface of the conductor by using triangular meshes, defining RWG basis functions on each adjacent triangular mesh pair, dispersing a surface integral equation by using the defined RWG basis functions and a moment method, calculating by using a traditional moment method to obtain an impedance matrix, and obtaining an initial current coefficient by iteratively solving the impedance matrix;
step 2: carrying out multilayer binary grid subdivision on the initial grid to obtain multilayer fully-encrypted grids, and calculating the inclusion relation and the projection coefficient of RWG basis functions between every two adjacent layers of grids;
and 3, step 3: utilizing the projection coefficient of the RWG basis function between every two adjacent layers of grids obtained in the step 2 to project the initial current coefficient obtained in the step 1 downwards layer by layer until the bottommost layer of fully-encrypted grids are obtained; subtracting a voltage vector obtained by multiplying a current coefficient of the bottommost layer full-encryption grid by an impedance matrix from a voltage vector obtained by irradiating the bottommost layer full-encryption grid by a plane wave to obtain an error vector; normalizing the error on the bottommost full-encryption grid to obtain a relative error;
and 4, step 4: comparing the absolute value of the relative error obtained in the step 3 with a set error threshold, marking two triangular meshes of the related bottommost RWG when the absolute value of the relative error is larger than the set error threshold to obtain a mesh mark map of the bottommost fully-encrypted mesh, subdividing a father-child mesh into four child triangular meshes according to the mesh mark map of the bottommost fully-encrypted mesh and a parent-child mesh relation caused by a binary subdivision mesh method, and projecting layer by layer upwards until the initial mesh to obtain the triangular mesh mark map of each layer of fully-encrypted mesh;
and 5, step 5: according to the triangular mesh mark diagram of each layer of fully-encrypted mesh obtained in the step 4, local encryption is performed layer by layer from the initial mesh, namely binary mesh subdivision is performed on the triangular mesh marked by each layer of mesh to obtain a final local encryption mesh diagram; defining a multi-branch basis function, namely a non-conformal grid caused by local encryption processed by the MBRWG, calculating by using a mixed basis function based on the RWG and the MBRWG and a moment method to obtain an impedance matrix, obtaining a current coefficient of the encrypted grid by iteratively solving the impedance matrix, and obtaining an RCS result by current calculation.
The invention has the following beneficial effects:
1. the numerical precision is controllable: the invention can adaptively encrypt the grid with larger residual error, so that the grid quality can be improved subsequently through codes even if the initial grid quality is not satisfactory. By local adaptive encryption, under the condition of ensuring numerical accuracy, the unknown quantity is reduced as much as possible, and the grid quantity and the calculation accuracy can be well balanced.
2. Is suitable for processing multi-scale problems: when the target is a multi-scale structure, the fine structure of the target can be encrypted to improve the overall accuracy.
3. Good astringency: since the invention can generate non-conformal grids in the local encryption process, the current continuity at the non-conformal part can be ensured by adopting the MBRWG basis function to replace the traditional HRWG basis function. An MBRWG is a positive triangular mesh corresponding to multiple negative triangular meshes, as opposed to a positive triangular mesh corresponding to a negative triangular mesh of a conventional RWG. The MBRWG connects regions of non-uniform mesh size while ensuring that no charge accumulates at the non-conformality. Using MBRWG may both reduce the number of unknowns and increase the iterative solution speed compared to using half rwg (hrwg) to process a non-conformal mesh.
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FIG. 1 is a schematic diagram of a mesh subdivision process according to an embodiment of the present invention;
FIG. 2 is a schematic illustration of a marking process according to an embodiment of the present invention;
FIG. 3 is a schematic diagram of an encryption process according to an embodiment of the present invention;
FIG. 4 is a schematic view of an embodiment of the present invention RWG internally subdivided into eight new RWGs;
FIG. 5 shows a multi-branch base function MBRWG base function occurring naturally in the local encryption process according to an embodiment of the present invention;
FIG. 6 is a process of two-level full encryption of a cube in an embodiment of the invention;
FIG. 7 is a diagram of an embodiment of the present invention for labeling triangles of a subdivision of a cube based on an error map;
FIG. 8 is a process of subdividing a mesh marker map to project markers upward layer by layer in accordance with an embodiment of the present invention;
FIG. 9 is a cube two-layer partial encryption process according to an embodiment of the present invention;
FIG. 10 is the result of the partial encryption of a boat in accordance with an embodiment of the present invention;
FIG. 11 is a comparison of RCS results and FEKO results before and after partial encryption of a boat in accordance with an embodiment of the present invention.
Detailed description of the preferred embodiments
The implementation of the technical scheme is further described in detail by taking two-layer local encryption as an example and combining the accompanying drawings:
the first step is as follows: establishing a surface area partial equation for scattering calculation aiming at the electromagnetic scattering problem of a conductor target, then dispersing the surface of the conductor by using triangular meshes, defining RWG basis functions on each adjacent triangular mesh pair, dispersing a surface integral equation by using the defined RWG basis functions and a moment method, calculating by using a traditional moment method to obtain an impedance matrix, and obtaining an initial current coefficient I by iteratively solving the impedance matrix;
the second step is that: and carrying out binary grid subdivision on each triangle of the initial grid to obtain two layers of fully-encrypted grids, and calculating the inclusion relation and the projection coefficient of the RWG basis function between two adjacent layers of grids (namely the coefficient of the RWG basis function between two adjacent layers of grids).
The binary mesh subdivision process is as shown in fig. 1, taking the midpoints of three sides of a triangle of the initial mesh, and subdividing the original triangle into four small triangles through pairwise connection. At this time, the RWG areas originally defined on the initial mesh are redefined into eight new RWGs after the first encryption, as shown in fig. 4. Based on the same divergence of the basis functions of the same region, we can obtain equation (4).
Wherein A is+And A-Is the area of the positive and negative triangular meshes of the RWG, AiIs the first after subdivisioni area of small triangle, aiIs the projection coefficient of the ith RWG, liIs the length of the common edge of the RWG. Solving equation (4) yields the following solution for the coefficients:
wherein α ═ a2l2a3l3Since the coefficient is an unknown quantity, the value of α is calculated by the following formula
Wherein
Is the unit normal vector of the RWG common edge, RjIs the layer j RWG basis function.
RWG basis function RWG defined on the j-th layer full-encryption grid through formula (5)jProjected coefficients to eight RWG basis functions RWG of the j +1 th layerj+1The above step (1);
wherein
Is the mth RWG basis function, a, defined on the jth layer of the fully-encrypted meshiIs the coefficient of the projection of the light,
is the RWG basis function defined on the layer j +1 encryption grid of the same region.
The same operation is performed on the first layer full encryption mesh, so that the second layer full encryption mesh and the projection relationship of the RWG basis functions between the first layer and the second layer can be obtained, and the bottommost layer full encryption mesh is called as a subdivision mesh, as shown in fig. 6.
The third step: and (4) utilizing the inclusion relation and the projection coefficient of the RWG basis function between every two adjacent layers of grids obtained in the second step, and downwardly projecting the initial current coefficient obtained in the first step layer by layer until the bottom layer is a fully-encrypted grid.
Of note is each RWGj+1Belong to three RWGsjSo each RWGj+1The current coefficient of (A) is three times RWGjThe current coefficient projection of (a). By recording each RWGjEight RWGs ofj+1And the coefficients thereof, the projection relationship between the two layers of fully-encrypted meshes can be obtained.
Obtaining current coefficient I of subdivided gridRThen, the impedance matrix Z of the subdivided grid is calculated using equation (1)RAnd the current coefficient I obtained by projection is compared with the current coefficient I obtained by projectionRMultiplying to obtain a voltage vector VR；
ZRIR＝VR (1)
Then irradiating the bottommost fully-encrypted grid by using plane waves to obtain another group of voltage vectors ViSubtracting the two groups of voltage vectors according to the formula (2) to obtain an error vector ER；
ER＝VR-Vi (2)
Obtaining a relative error element by normalizing the error by the following RWG basis function relative error calculation formula (3)
Wherein,
representing the mth RWG basis function of the l-th layer (i.e., the bottommost fully-encrypted mesh, the subdivided mesh), according to the relative error of each RWG basis function of the subdivided mesh
Obtaining a relative voltage error map of a subdivided grid
As shown in fig. 7 (a).
The fourth step: setting an error threshold (e.g., 0.1) based on the relative error
Finding out the RWG basis functions larger than the error threshold value on the subdivided grids, marking the two triangular grids of the corresponding RWG basis functions, and finally obtaining a marking map as shown in (b) in FIG. 7.
After the label graph of the subdivided mesh is obtained, the relationship between the parent triangle and the child triangle between two layers of meshes (four child triangle meshes can be obtained after one parent triangle mesh is subjected to binary subdivision) is utilized, then the parent triangle mesh of the previous layer is marked according to the child triangle mesh marked by the current layer (as long as one of the four child triangles of the parent triangle mesh is marked, the child triangle mesh needs to be marked), the triangle mesh of the same region of the previous layer is marked upwards layer by layer until the initial mesh is reached, and the triangle mesh label graph of each layer of the fully-encrypted mesh is obtained, as shown in fig. 8.
Taking two layers as an example, fig. 2 shows the process of labeling (where white is the labeled network, the same applies below). Each layer of triangular meshes corresponds to the next layer of four relatively smaller triangles, and as long as the four smaller triangles are marked, the large triangles to which the small triangles belong also need to be marked.
The fifth step: and according to the triangular mesh mark diagram of each layer of fully-encrypted mesh obtained in the fourth step, local encryption is performed layer by layer from the initial mesh, namely binary mesh subdivision is performed on the triangular mesh marked by each layer of mesh to obtain a final local encryption mesh diagram. The schematic diagram of the encryption process is shown in fig. 3, the labeled graph of the initial mesh is shown in fig. 9 (a), local encryption is performed layer by layer, and binary mesh subdivision is performed on the triangle labeled by the initial mesh to implement the first layer of local encryption, as shown in fig. 9 (b). And then, binary subdivision is carried out on the triangles of the first layer of local encryption grid marks according to the first layer of mark images, so as to obtain a final (c) image of the local encryption grid image 9.
The invention utilizes the MBRWG basis function to process the non-conformal grids appearing in the local encryption process. As shown in fig. 5, a non-conformal mesh occurs during subdivision, and the non-conformal mesh resulting from binary mesh subdivision is naturally suitable for defining the MBRWG basis function. MBRWG, like RWG basis functions, inherently guarantees current continuity, as opposed to HRWG requiring a penalty to deal with charge accumulation problems at grid non-conformality. The definition of the MBRWG basis functions is as follows:
wherein
Is a regular triangle of the MBRWG,
is the ith negative triangle, the number N of the negative trianglesnRegarding the non-conformal mesh specifics, in addition to this, as in the conventional RWG basis function definition: l represents the length of the common side of the RWG basis function, A represents the area of the triangle, and r represents a point on the triangle.
And calculating by using a mixed basis function based on RWG and MBRWG and a moment method to obtain an impedance matrix, obtaining a current coefficient of the local encryption grid by iteratively solving the impedance matrix, and finally obtaining an RCS result through current calculation.
The invention further verifies the improvement of the local encryption grid on the precision through a boat model with the grid size of 0.2 wavelength. As shown in fig. 10, this method is basically encrypted at the edge with large error, and fig. 11 also shows the improvement of the precision. Finally, the data of Table 1 shows the advantages of using MBRWG versus the conventional discontinuous Galois method in terms of unknowns and iteration speed. The solution time refers to the solution time after the local encryption grid is generated.
TABLE 1 advantages of MBRWG versus HRWG in unknowns, memory, time, and iteration steps
| Basis functions | Unknown quantity | Impedance matrix Memory (MB) | Solution time(s) | Number of iteration steps |
| RWG+HRWG | 7762+2023 | 731 | 235 | 363 |
| RWG+MBRWG | 7762+626 | 536 | 116 | 97 |
Claims (3)
1. An electromagnetic scattering solving method based on MBRWG and grid adaptive encryption is characterized by comprising the following steps:
step 1: establishing a surface area partial equation for scattering calculation aiming at the electromagnetic scattering problem of a conductor target, then dispersing the surface of the conductor by using triangular meshes, defining RWG basis functions on each adjacent triangular mesh pair, dispersing a surface integral equation by using the defined RWG basis functions and a moment method, calculating by using a traditional moment method to obtain an impedance matrix, and obtaining an initial current coefficient by iteratively solving the impedance matrix;
step 2: carrying out multilayer binary grid subdivision on the initial grid to obtain multilayer fully-encrypted grids, and calculating the inclusion relation and the projection coefficient of RWG basis functions between every two adjacent layers of grids;
and 3, step 3: utilizing the projection coefficient of the RWG basis function between every two adjacent layers of grids obtained in the step 2 to project the initial current coefficient obtained in the step 1 downwards layer by layer until the bottommost layer of fully-encrypted grids are obtained; subtracting a voltage vector obtained by multiplying the current coefficient of the bottommost layer full-encryption grid by the impedance matrix from a voltage vector obtained by irradiating the bottommost layer full-encryption grid by the plane wave to obtain an error; normalizing the error on the bottommost full-encryption grid to obtain a relative error;
and 4, step 4: comparing the absolute value of the relative error obtained in the step 3 with a set error threshold, marking two triangular meshes of the related bottommost RWG when the absolute value of the relative error is larger than the set error threshold to obtain a mesh mark map of the bottommost fully-encrypted mesh, subdividing a father-child mesh into four child triangular meshes according to the mesh mark map of the bottommost fully-encrypted mesh and a parent-child mesh relation caused by a binary subdivision mesh method, and projecting layer by layer upwards until the initial mesh to obtain the triangular mesh mark map of each layer of fully-encrypted mesh;
and 5, step 5: according to the triangular mesh mark diagram of each layer of fully-encrypted mesh obtained in the step 4, local encryption is performed layer by layer from the initial mesh, namely binary mesh subdivision is performed on the triangular mesh marked by each layer of mesh to obtain a final local encryption mesh diagram; defining a multi-branch basis function, namely a non-conformal grid caused by local encryption processed by the MBRWG, calculating by using a mixed basis function based on the RWG and the MBRWG and a moment method to obtain an impedance matrix, obtaining a current coefficient of the encrypted grid by iteratively solving the impedance matrix, and obtaining an RCS result by current calculation.
2. The MBRWG-based locally adaptive encryption method of claim 1, wherein the step 2 comprises the steps of:
step 2-1: taking the midpoints of three sides of the triangle of the previous layer of mesh, and subdividing the triangle into four small triangles through pairwise connection of the midpoints to form the mesh of the current layer;
step 2-2: after the multilayer full encryption is completed, RWG basis functions are defined on the grids, and the inclusion relation and the projection coefficient of the RWG basis functions between every two adjacent layers of grids are solved.
3. The MBRWG-based locally adaptive encryption method of claim 1, wherein the step 3 comprises the steps of:
step 3-1: projecting the initial current coefficient obtained in the step 1 downwards layer by layer according to the projection coefficient of the RWG basis function between every two adjacent layers of grids obtained in the step 2 to obtain a current coefficient I of the bottommost fully-encrypted gridR；
Step 3-2: calculating an impedance matrix Z of the bottommost fully-encrypted meshRAnd the current coefficient I of the bottom full-encryption grid obtained by projection is compared with the current coefficient I of the bottom full-encryption grid obtained by projectionRMultiplying to obtain a voltage vector VR；
ZRIR＝VR (1)
And 3, step 3-3: irradiating the bottommost fully-encrypted grid by using plane waves to obtain another group of voltage vectors ViSubtracting the two groups of voltage vectors to obtain an error vector ER；
ER＝VR-Vi (2)
And 3, step 3-4: to EREach element of the vector
Normalization processing to obtain relative error
The calculation formula is as follows:
wherein
Representing the mth RWG basis function defined on the bottom-most fully encrypted mesh.
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