CN111931353A - Scattered field solving method applied to simulated FSS structure - Google Patents

Scattered field solving method applied to simulated FSS structure Download PDF

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CN111931353A
CN111931353A CN202010687317.2A CN202010687317A CN111931353A CN 111931353 A CN111931353 A CN 111931353A CN 202010687317 A CN202010687317 A CN 202010687317A CN 111931353 A CN111931353 A CN 111931353A
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何十全
张天成
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University of Electronic Science and Technology of China
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Abstract

Compared with the traditional VSIE method for performing scattered field solution on a triangular-tetrahedral and RWG-SWG basis function, the method solves the problem of frequency offset, can greatly reduce the unknown quantity and the memory overhead of calculation, and has higher solution efficiency. The invention can more efficiently and accurately guide the design of the FSS structure, is beneficial to the simulation of large-scale arrays, and has practical engineering application value.

Description

Scattered field solving method applied to simulated FSS structure
Technical Field
The invention relates to the field of electromagnetism, in particular to a scattered field solving method applied to a simulated FSS structure.
Background
Frequency Selective Surface (FSS) is very widely used in stealth technology, and is a two-dimensional periodic structure, essentially a spatial filter, consisting of a periodic arrangement of resonant metal elements. The FSS has selective permeability to electromagnetic waves of different frequencies, and is generally divided into a patch type and an aperture type, which are respectively characterized by band stop and band pass, and the main factors influencing the electromagnetic properties of the FSS include a unit form, an arrangement mode, an incident angle and the like. In practical engineering application, the FSS is usually composed of a metal plate and a dielectric substrate, on one hand, the physical strength of the FSS is increased, and on the other hand, the manufacturing process is relatively easier by adopting a PBC plate or a skin coating processing mode. There are generally two ways of media loading: one is single-side loading, and the metal FSS is attached to one side of the dielectric layer; the other is double-side loading, and dielectric layers are wrapped on two sides of the FSS.
The FSS medium loading forms a metal-medium mixed structure, many scholars carry out deep research on solving methods of the metal-medium mixed structure, mainly including a periodic moment method, a mode matching method, a spectral domain method, an equivalent circuit method and the like, but the methods have strong limitations, only infinite periods or regular FSS structures can be solved, and the methods are not suitable for designing novel complex FSS structures (such as conformality, interdigital, nesting and lamination) and also need to be analyzed by depending on a numerical method. Common numerical methods include differential equation and integral equation methods. The differential equation is suitable for solving some complex fine structures, can accurately simulate the change of a field inside a strip line or a cavity, and is low in complexity, but dispersion errors can be brought when the target electrical size is large, so that the solving precision is poor, and the differential equation is not suitable for calculating the FSS structure. The integral equation is a solution method strictly derived according to Maxwell equation set, is a full-wave analysis method, does not introduce redundant errors, automatically meets far-field boundary conditions, can easily calculate the metal surface current distribution and the radar scattering cross section (RCS) of a scatterer, which are exactly concerned by the FSS structure, so the FSS structure generally adopts the integral equation to carry out numerical solution and analysis.
The most well known of the simulation software of integral equations is FEKO (electron field scattering) of EMSS company, which is commonly used for analyzing the scattering characteristics of complex structures, and the FEKO uses a volume surface integral equation method (VSIE) for metal-medium mixed structures. However, based on a large number of numerical simulation cases, it is found that the FEKO traditional VSIE method has a problem of inaccurate calculation when calculating a metal-medium mixed structure of the FSS, which is mainly characterized in that a frequency response curve shifts to a high frequency, and the frequency shift is obviously intolerable for the FSS, which is a structure very sensitive to frequency variation. And the traditional VSIE method has large memory consumption and low solving efficiency, is difficult to analyze a large-scale FSS array and limits the research process in engineering.
Disclosure of Invention
Aiming at the defects in the prior art, the scattered field solving method applied to the simulated FSS structure solves the frequency offset problem of the traditional VSIE, has less grid number and memory overhead, has higher iterative convergence speed, and is suitable for solving a more complicated electrically large-size FSS structure.
In order to achieve the purpose of the invention, the invention adopts the technical scheme that: a scattered field solving method applied to a simulated FSS structure comprises the following steps:
s1, establishing an FSS geometric model according to the position of the FSS geometric model and the material parameters of the medium body;
s2, setting simulation frequency, excitation and scattering field RCS parameters required by simulation;
s3, establishing a volume-surface integral equation VSIE according to the electromagnetic field theory and the simulation frequency;
s4, meshing the FSS geometric model according to the excited wavelength to obtain a split mesh;
s5, standardizing each sub-grid in the split grids, and establishing a quasi-orthogonal basis function for each standard sub-grid;
s6, carrying out discretization processing on the volume-area integral equation VSIE by adopting a quasi-orthogonal basis function, and carrying out matrixing processing on the discretized volume-area integral equation VSIE to obtain a matrix equation;
s7, solving a matrix equation by adopting an iterative method, and reducing the complexity of multiplication of moment vectors by adopting a multi-layer fast multipole algorithm to obtain current distribution on a sub-grid;
and S8, integrating the currents of all the sub-grids according to the set parameters of the scattered field RCS, and solving the scattered field RCS of each angle on the FSS geometric model.
Further, the parameters excited in step S2 include: incident angle theta and angle under spherical coordinate system
Figure BDA0002588046490000031
Amplitude and polarization mode; the parameters of the fringe field RCS include: range of angle theta, interval of theta, angle in spherical coordinate system
Figure BDA0002588046490000032
A range of
Figure BDA0002588046490000033
The interval of (c).
Further, the volume integral equation VSIE in step S3 is:
Figure BDA0002588046490000034
E(r)＝Einc(r)+Esca(r)r∈V
Figure BDA0002588046490000035
Figure BDA0002588046490000036
Figure BDA0002588046490000037
wherein the content of the first and second substances,
Figure BDA0002588046490000038
in a first outer normal direction, the first outer normal direction,
Figure BDA0002588046490000039
in a second external normal direction, Einc(r) is the incident field, Esca(r) is the scattered field, r is the field point vector, "·" is the point product, S is the metal surface of the FSS geometric model, V is the dielectric body, E (r) is the total electric field,
Figure BDA00025880464900000310
the field generated for the equivalent source of the metal plane,
Figure BDA00025880464900000311
a field generated by a dielectric equivalent source, i being an imaginary unit, k0Is wave number, η0377 omega is free space wave impedance, r' is source point vector, Jpec(r') a metal surface current,
Figure BDA00025880464900000312
in order to be the first vector differential operator,
Figure BDA00025880464900000313
as a second vector differential operator, G (r, r)′) Is a Green function in free space, ω is the phase constant, χ (r)′) For media contrast, D (r') is the electrical displacement vector,r(r') is a dielectric constant.
Further, the principle of meshing the FSS geometric model in step S4 is as follows: the method comprises the following steps of (1) dividing a metal surface by adopting a quadrilateral surface grid, and dividing a medium body by adopting a hexahedral body grid to obtain a quadrilateral grid and a hexahedral grid, namely a divided grid; the size of the cells of the quadrilateral mesh and the hexahedral mesh is less than 1/10 of an excited wavelength, the metal arm on the FSS geometric model at least comprises 3-5 meshes, the edge line of the meshes is coincident with or close to the edge of the FSS geometric model, and the quadrilateral mesh and the hexahedral mesh share the same node on the interface of the metal surface and the dielectric body.
Further, step S5 includes the steps of:
s51, standardizing the quadrilateral grids and the hexahedral grids to obtain square grids and cubic grids;
and S52, establishing quasi-orthogonal basis functions for each square grid and each square grid.
Further, the quasi-orthogonal basis functions of the square grid in step S52 are:
Figure BDA0002588046490000041
T0,0＝u′v′
T1,0＝uv′
T0,1＝u′v
T1,1＝uv
u′＝1-u，v′＝1-v
the quasi-orthogonal basis function established by the square grid is as follows:
Figure BDA0002588046490000042
T0,0,0＝u′v′w′T0,0,1＝u′v′w
T0,1,0＝u′vw′T0,1,1＝u′vw
T1,0,0＝uv′w′T1,0,1＝uv′w
T1,1,0＝uvw′T1,1,1＝uvw
u′＝1-u，v′＝1-v，w′＝1-w
wherein f isquad(r) is the quasi-orthogonal basis function of the square grid, S' is the area of the planar grid, ri,jAs field point vectors, Ti,jIs a position differential, T0,0,T1,0,T0,1,T1,1Local Cartesian coordinate vectors of corresponding points of the square grids respectively, u is a horizontal coordinate value, upsilon is a vertical coordinate value, and fhex(r) is the quasi-orthogonal basis function of the cubic grid, V' is the area of the cubic grid, ri,j,kAs field point vectors, Ti,j,kIs a position differential, T0,0,0,T0,1,0,T1,0,0,T1,1,0,T0,0,1,T0,1,1,T1,0,1,T1,1,1The vector quantities are local Cartesian coordinate vectors of corresponding points of the cubic grid respectively, w is a vertical coordinate numerical value, u ' is an abscissa parameter, v ' is an ordinate parameter, and w ' is a vertical coordinate parameter.
Further, step S6 includes the following substeps:
s61, carrying out discrete processing on the volume surface integral equation VSIE by adopting a quasi-orthogonal basis function, and expanding the surface current on the FSS geometric model by the quasi-orthogonal basis function to obtain an expression of discrete current:
Figure BDA0002588046490000051
Figure BDA0002588046490000052
wherein i is an imaginary unit, k0Is wave number, η0377 Ω is the free space wave impedance, Jpec(r') metal surface current, n is the number of quasi-orthogonal basis functions, fn(r') is the nth quasi-orthogonal basis function, N is the total number of quasi-orthogonal basis functions, unThe coefficient of the expansion current of the nth quasi-orthogonal basis function is S', a surface grid area, a volume grid area, a phase constant and a potential displacement vector;
s62, according to the expression of the discrete current, carrying out matrixing processing on the discrete volume surface integral equation VSIE by adopting a Galileo method to obtain a matrix equation:
[Z]{u}＝{V}
Figure BDA0002588046490000061
Vm＝-∫fm(r)Einc(r)dr
wherein [ Z ]]For impedance matrix, { u } is vector to be solved, { V } is excitation vector, r is field point vector, r' is source point vector, m, n are numbers of quasi-orthogonal basis functions, ZmnIs [ Z ]]Of the m-th row and the n-th column, VmM-th element of { V }, fm(r) is the mth quasi-orthogonal basis function, fn(r') is the nth quasi-orthogonal basis function, k0In terms of the wave number, the number of waves,
Figure BDA0002588046490000062
in order to be the first vector differential operator,
Figure BDA0002588046490000063
in order to be the second vector differential operator,
Figure BDA0002588046490000064
for the second outer normal direction, "·" is a dot product, G (r, r ') is a Green function in free space, χ (r') is the medium contrast, EincAnd (r) is an incident field, and an integration area is determined by a grid to which r and r' belong.
Further, in step S7, the iterative method uses a generalized minimum residual algorithm, the convergence residual is set to 0.01, and after solving the solution vector { u }, the current distribution of all sub-grids is obtained.
In conclusion, the beneficial effects of the invention are as follows:
(1) there is no frequency offset problem. Because the normal boundary condition of the electric displacement vector is implicit in the moment method solving process of the VSIE, the frequency deviation of the VSIE is caused by inaccurate description of the tangential component of the electric field, namely the tangential electric field of the metal surface is zero. Because the tetrahedron is not perfectly perpendicular to the metal surface in the conventional VSIE method, the electric field described by the linear combination of the SWG basis functions still has a slight tangential component near the metal surface, eventually causing a frequency offset. In the VSIE method, the hexahedral mesh is basically vertical to the metal surface, so that a tiny tangential component does not exist basically, so that the tangential boundary condition at the metal-medium interface is easier to meet, the final solution precision of the VSIE is ensured, and the effect of eliminating the frequency offset is achieved.
(2) The number of grids and memory overhead is reduced. Because FSS is generally a planar or slightly curvature-varying structure, the quadrilateral-hexahedral mesh of the present invention can simulate its geometry well. Compared with the traditional triangle-tetrahedron mesh, under the same mesh size, one quadrangle and hexahedron can replace two triangles and three tetrahedrons respectively, so the mesh number can be greatly reduced, thereby reducing the calculation amount and the calculation complexity, and particularly showing that the peak memory consumption is obviously reduced. The VSIE method is beneficial to saving limited computer resources and simplifying the analysis difficulty of the electrically large multi-grid FSS.
(3) The solving efficiency is higher. The impedance matrix needs to be solved iteratively, and the speed of iteration mainly depends on the characteristics of the iterative method and the impedance matrix. The iteration method selects the generalized minimum residual error algorithm with better convergence; in addition, because the edges of the quadrilateral-hexahedral mesh are basically vertical, and different currents described by the quasi-orthogonal basis functions are basically orthogonal to each other, the generated impedance matrix has good performance, the condition number of the matrix is small, and the equation has good convergence characteristics. Compared with the traditional VSIE method, the VSIE method can save more operation time and improve the working efficiency of engineering researchers.
Drawings
FIG. 1 is a flow chart of a fringe field solution method applied to a simulated FSS structure;
FIG. 2 is a diagram of a standard grid and quasi-orthogonal basis functions;
FIG. 3 is a schematic diagram of a grid type;
FIG. 4 is a 2x2 FSS geometric model diagram;
FIG. 5 is a graph comparing frequency response curves for different methods;
FIG. 6 is a comparison of iterative convergence for different methods;
FIG. 7 is a comparison of computational overhead for different methods.
Detailed Description
The following description of the embodiments of the present invention is provided to facilitate the understanding of the present invention by those skilled in the art, but it should be understood that the present invention is not limited to the scope of the embodiments, and it will be apparent to those skilled in the art that various changes may be made without departing from the spirit and scope of the invention as defined and defined in the appended claims, and all matters produced by the invention using the inventive concept are protected.
As shown in fig. 1, a method for solving a scattered field applied to a simulated FSS structure includes the following steps:
s1, establishing the FSS geometric model according to the position of the FSS geometric model and the material parameters of the dielectric body, wherein the material parameters of the dielectric body comprise: dielectric constant and loss tangent, the FSS geometric model includes: the metal surface and the dielectric body;
s2, setting simulation frequency, excitation and scattering field RCS parameters required by simulation, and determining an incident angle, an angle interval and a polarization mode if the simulation is plane wave excitation;
the parameters excited in step S2 include: incident angle theta and angle under spherical coordinate system
Figure BDA0002588046490000081
Amplitude and polarization mode; the parameters of the fringe field RCS include: range of angle theta, interval of theta, angle in spherical coordinate system
Figure BDA0002588046490000082
A range of
Figure BDA0002588046490000083
The interval of (c).
S3, establishing a volume-surface integral equation VSIE according to the electromagnetic field theory and the simulation frequency;
the volume integral equation VSIE in step S3 is:
Figure BDA0002588046490000084
E(r)＝Einc(r)+Esca(r)∈V
Figure BDA0002588046490000085
Figure BDA0002588046490000086
Figure BDA0002588046490000087
wherein the content of the first and second substances,
Figure BDA0002588046490000088
in a first outer normal direction, the first outer normal direction,
Figure BDA0002588046490000089
in a second external normal direction, Einc(r) is the incident field, Esca(r) is the scattered field, r is the field point vector, "·" is the point product, S is the metal surface of the FSS geometric model, V is the dielectric body, E (r) is the total electric field,
Figure BDA00025880464900000810
the field generated for the equivalent source of the metal plane,
Figure BDA00025880464900000811
a field generated by a dielectric equivalent source, i being an imaginary unit, k0Is wave number, η0377 omega is free space wave impedance, r' is source point vector, Jpec(r') a metal surface current,
Figure BDA0002588046490000091
in order to be the first vector differential operator,
Figure BDA0002588046490000092
for the second vector differential operator, G (r, r ') is the Green's function in free space, ω is the phase constant, x (r)′) For media contrast, D (r') is the electrical displacement vector,r(r') is a dielectric constant.
S4, meshing the FSS geometric model according to the excited wavelength to obtain a split mesh;
the principle of meshing the FSS geometric model in step S4 is as follows: the method comprises the following steps of (1) dividing a metal surface by adopting a quadrilateral surface grid, and dividing a medium body by adopting a hexahedral body grid to obtain a quadrilateral grid and a hexahedral grid, namely a divided grid; the size of the cells of the quadrilateral mesh and the hexahedral mesh is less than 1/10 of an excited wavelength, the metal arm on the FSS geometric model at least comprises 3-5 meshes, the edge line of the meshes is coincident with or close to the edge of the FSS geometric model, and the quadrilateral mesh and the hexahedral mesh share the same node on the interface of the metal surface and the dielectric body.
S5, standardizing each sub-grid in the split grids, and establishing a quasi-orthogonal basis function for each standard sub-grid;
step S5 includes the following steps:
s51, standardizing the quadrilateral grids and the hexahedral grids to obtain square grids and cubic grids;
each quadrilateral grid and each hexahedral grid in the space are converted into standard grids (squares and cubes) with the side length of 1, one vertex is taken as an origin, edges adjacent to the origin are taken as coordinate axes, a local coordinate system is established on the standard geometry, and the local coordinate system and the initial space coordinate system can be converted into each other through coordinate ratio transformation.
S52, establishing quasi-orthogonal basis functions for each of the square grid and the square grid, as shown in fig. 2.
The quasi-orthogonal basis functions of the square grid in step S52 are:
Figure BDA0002588046490000093
T0,0＝u′v′
T1,0＝uv′
T0,1＝u′v
T1,1＝uv
u′＝1-u，v′＝1-v
the quasi-orthogonal basis function established by the square grid is as follows:
Figure BDA0002588046490000101
T0,0,0＝u′v′w′T0,0,1＝u′v′w
T0,1,0＝u′vw′T0,1,1＝u′vw
T1,0,0＝uv′w′T1,0,1＝uv′w
T1,1,0＝uvw′T1,1,1＝uvw
u′＝1-u，v′＝1-v，w′＝1-w
wherein f isquad(r) is the quasi-orthogonal basis function of the square grid, S' is the area of the planar grid, ri,jAs field point vectors, Ti,jIs a position differential, T0,0,T1,0,T0,1,T1,1Local Cartesian coordinate vectors of corresponding points of the square grids respectively, u is a horizontal coordinate value, upsilon is a vertical coordinate value, and fhex(r) is the quasi-orthogonal basis function of the cubic grid, V' is the area of the cubic grid, ri,j,kAs field point vectors, Ti,j,kIs a position differential, T0,0,0,T0,1,0,T1,0,0,T1,1,0,T0,0,1,T0,1,1,T1,0,1,T1,1,1The vector quantities are local Cartesian coordinate vectors of corresponding points of the cubic grid respectively, w is a vertical coordinate numerical value, u ' is an abscissa parameter, v ' is an ordinate parameter, and w ' is a vertical coordinate parameter.
S6, carrying out discretization processing on the volume-area integral equation VSIE by adopting a quasi-orthogonal basis function, and carrying out matrixing processing on the discretized volume-area integral equation VSIE to obtain a matrix equation;
step S6 includes the following substeps:
s61, carrying out discrete processing on the volume surface integral equation VSIE by adopting a quasi-orthogonal basis function, and expanding the surface current on the FSS geometric model by the quasi-orthogonal basis function to obtain an expression of discrete current:
Figure BDA0002588046490000111
Figure BDA0002588046490000112
wherein i is an imaginary unit, k0Is wave number, η0377 Ω is the free space wave impedance, Jpec(r') metal surface current, n is the number of quasi-orthogonal basis functions, fn(r') is the nth quasi-orthogonal basis function, N is the total number of quasi-orthogonal basis functions, unCoefficient of the spreading current, S, for the nth quasi-orthogonal basis function′Is a surface grid region, and V' is a body gridGrid region, omega is phase constant, D (r') is electric displacement vector;
s62, according to the expression of the discrete current, carrying out matrixing processing on the discrete volume surface integral equation VSIE by adopting a Galileo method to obtain a matrix equation:
[Z]{u}＝{V}
Figure BDA0002588046490000113
Vm＝-∫fm(r)Einc(r)dr
wherein [ Z ]]For impedance matrix, { u } is vector to be solved, { V } is excitation vector, r is field point vector, r' is source point vector, m, n are numbers of quasi-orthogonal basis functions, ZmnIs [ Z ]]Of the m-th row and the n-th column, VmM-th element of { V }, fm(r) is the mth quasi-orthogonal basis function, fn(r') is the nth quasi-orthogonal basis function, k0In terms of the wave number, the number of waves,
Figure BDA0002588046490000114
in order to be the first vector differential operator,
Figure BDA0002588046490000115
in order to be the second vector differential operator,
Figure BDA0002588046490000116
for the second outer normal direction, "·" is a dot product, G (r, r ') is a Green function in free space, χ (r') is the medium contrast, EincAnd (r) is an incident field, and an integration area is determined by a grid to which r and r' belong.
S7, solving a matrix equation by adopting an iterative method, and reducing the complexity of multiplication of moment vectors by adopting a multi-layer fast multipole algorithm to obtain current distribution on a sub-grid;
in the step S7, the iterative method adopts a generalized minimum residual error algorithm, the convergence residual error is set to be 0.01, and the current distribution of all sub-grids is obtained after the vector { u } is solved.
And S8, integrating the currents of all the sub-grids according to the set parameters of the scattered field RCS, and solving the scattered field RCS of each angle on the FSS geometric model.
In conclusion, the beneficial effects of the invention are as follows:
(1) there is no frequency offset problem. Because the normal boundary condition of the electric displacement vector is implicit in the moment method solving process of the VSIE, the frequency deviation of the VSIE is caused by inaccurate description of the tangential component of the electric field, namely the tangential electric field of the metal surface is zero. Because the tetrahedron is not perfectly perpendicular to the metal surface in the conventional VSIE method, the electric field described by the linear combination of the SWG basis functions still has a slight tangential component near the metal surface, eventually causing a frequency offset. In the VSIE method, the hexahedral mesh is originally approximately vertical to the metal surface, as shown in FIG. 3, so that a tiny tangential component does not exist basically, tangential boundary conditions at the metal-medium interface are easier to meet, the final solution precision of the VSIE is ensured, and the effect of eliminating frequency offset is achieved.
(2) The number of grids and memory overhead is reduced. Because FSS is generally a planar or slightly curvature-varying structure, the quadrilateral-hexahedral mesh of the present invention can simulate its geometry well. Compared with the traditional triangle-tetrahedron mesh, under the same mesh size, one quadrangle and hexahedron can replace two triangles and three tetrahedrons respectively, so the mesh number can be greatly reduced, thereby reducing the calculation amount and the calculation complexity, and particularly showing that the peak memory consumption is obviously reduced. The VSIE method is beneficial to saving limited computer resources and simplifying the analysis difficulty of the electrically large multi-grid FSS.
(3) The solving efficiency is higher. The impedance matrix needs to be solved iteratively, and the speed of iteration mainly depends on the characteristics of the iterative method and the impedance matrix. The iteration method selects the generalized minimum residual error algorithm with better convergence; in addition, because the edges of the quadrilateral-hexahedral mesh are basically vertical, and different currents described by the quasi-orthogonal basis functions are basically orthogonal to each other, the generated impedance matrix has good performance, the condition number of the matrix is small, and the equation has good convergence characteristics. Compared with the traditional VSIE method, the VSIE method can save more operation time and improve the working efficiency of engineering researchers.
In this embodiment, a 2 × 2 cross-shaped aperture FSS structure with a resonant frequency around 10GHz is analyzed, the dielectric constant of the medium is 2.8, the size is as shown in fig. 4 (unit mm), the plane waves are vertically incident, and the mesh size of the subdivision is 0.4 mm. 56 frequency points are arranged in total, the interval between 5.0-9.0GHz and 11.0-15.0GHz is 0.5, the interval between 9.0-9.6GHz and 10.4-11.0GHz is 0.1, the interval between 9.6-10.4GHz is 0.03, and the closer to the resonant frequency, the denser the frequency points are, so as to accurately simulate the change of frequency response. The backward RCS of the FSS structure is solved by respectively adopting the VSIE method (quadrilateral-hexahedron) of the invention and the traditional VSIE method (triangular-tetrahedron).
For example, as shown in fig. 5, it can be found that the frequency response curve of the conventional vsee method has a large difference with the accurate value, and the frequency response curve shifts to a high frequency, while the curve of the present invention returns to the vicinity of the accurate value, the mean square error of the two is small, and the result has a high reliability, which indicates that the method solves the problem of frequency shift of the conventional vsee method for solving the FSS structure. Compared with the iterative convergence situation of different methods at 10GHz, as shown in FIG. 6, the VSIE method of the present invention has better convergence, and the number of convergence steps is less than that of the conventional method. Compared with the calculation cost of different methods, as shown in fig. 7, the number of grids and the number of unknowns in the VSIE method of the present invention are small, the memory consumption is low, the calculation time is about one fourth of that of the conventional method, and the solution efficiency is high.

Claims (8)

1. A scattered field solving method applied to a simulated FSS structure is characterized by comprising the following steps:
s1, establishing an FSS geometric model according to the position of the FSS geometric model and the material parameters of the medium body;
s2, setting simulation frequency, excitation and scattering field RCS parameters required by simulation;
s3, establishing a volume-surface integral equation VSIE according to the electromagnetic field theory and the simulation frequency;
s4, meshing the FSS geometric model according to the excited wavelength to obtain a split mesh;
s5, standardizing each sub-grid in the split grids, and establishing a quasi-orthogonal basis function for each standard sub-grid;
s6, carrying out discretization processing on the volume-area integral equation VSIE by adopting a quasi-orthogonal basis function, and carrying out matrixing processing on the discretized volume-area integral equation VSIE to obtain a matrix equation;
s7, solving a matrix equation by adopting an iterative method, and reducing the complexity of multiplication of moment vectors by adopting a multi-layer fast multipole algorithm to obtain current distribution on a sub-grid;
and S8, integrating the currents of all the sub-grids according to the set parameters of the scattered field RCS, and solving the scattered field RCS of each angle on the FSS geometric model.
2. The method for solving the scattered field applied to the simulated FSS structure as claimed in claim 1, wherein the parameters excited in step S2 include: incident angle theta and angle under spherical coordinate system
Figure FDA0002588046480000013
Amplitude and polarization mode; the parameters of the fringe field RCS include: range of angle theta, interval of theta, angle in spherical coordinate system
Figure FDA0002588046480000014
A range of
Figure FDA0002588046480000015
The interval of (c).
3. The method for solving the scattered field applied to the simulated FSS structure as claimed in claim 1, wherein the volume integral equation VSIE in the step S3 is as follows:
Figure FDA0002588046480000011
E(r)＝Einc(r)+Esca(r)r∈V
Figure FDA0002588046480000012
Figure FDA0002588046480000021
Figure FDA0002588046480000022
Figure FDA0002588046480000023
wherein the content of the first and second substances,
Figure FDA0002588046480000024
in a first outer normal direction, the first outer normal direction,
Figure FDA0002588046480000025
in a second external normal direction, Einc(r) is the incident field, Esca(r) is the scattered field, r is the field point vector, "·" is the point product, S is the metal surface of the FSS geometric model, V is the dielectric body, E (r) is the total electric field,
Figure FDA0002588046480000026
the field generated for the equivalent source of the metal plane,
Figure FDA0002588046480000027
a field generated by a dielectric equivalent source, i being an imaginary unit, k0Is wave number, η0377 omega is free space wave impedance, r' is source point vector, Jpec (r') a metal surface current,
Figure FDA0002588046480000028
in order to be the first vector differential operator,
Figure FDA0002588046480000029
for the second vector differential operator, G (r, r ') is the Green's function in free space, ω is the phase constant, χ (r ') is the medium contrast, D (r') is the electric displacement vector,r(r') is a dielectric constant.
4. The method for solving the scattered field applied to the simulated FSS structure as claimed in claim 1, wherein the principle of meshing the FSS geometric model in the step S4 is as follows: the method comprises the following steps of (1) dividing a metal surface by adopting a quadrilateral surface grid, and dividing a medium body by adopting a hexahedral body grid to obtain a quadrilateral grid and a hexahedral grid, namely a divided grid; the size of the cells of the quadrilateral mesh and the hexahedral mesh is less than 1/10 of an excited wavelength, the metal arm on the FSS geometric model at least comprises 3-5 meshes, the edge line of the meshes is coincident with or close to the edge of the FSS geometric model, and the quadrilateral mesh and the hexahedral mesh share the same node on the interface of the metal surface and the dielectric body.
5. The method for solving the scattered field applied to the simulated FSS structure as recited in claim 4, wherein the step S5 comprises the steps of:
s51, standardizing the quadrilateral grids and the hexahedral grids to obtain square grids and cubic grids;
and S52, establishing quasi-orthogonal basis functions for each square grid and each square grid.
6. The method for solving the scattered field applied to the simulated FSS structure as claimed in claim 5, wherein the quasi-orthogonal basis functions of the square grid in the step S52 are as follows:
Figure FDA0002588046480000031
T0，0＝u′v′
T1，0＝uv′
T0，1＝u′v
T1，1＝uv
u′＝1-u，v′＝1-v
the quasi-orthogonal basis function established by the square grid is as follows:
Figure FDA0002588046480000032
T0，0，0＝u′v′w′T0，0，1＝u′v′w
T0，1，0＝u′vw′T0，1，1＝u′vw
T1，0，0＝uv′w′T1，0，1＝uv′w
T1，1，0＝uvw′T1，1，1＝uvw
u′＝1-u，v′＝1-v，w′＝1-w
wherein f isquad(r) is the quasi-orthogonal basis function of the square grid, S' is the area of the planar grid, ri，jAs field point vectors, Ti，jIs a position differential, T0，0，T1，0，T0，1，T1，1Local Cartesian coordinate vectors of corresponding points of the square grid, u is an abscissa value, v is an ordinate value, fhex(r) is the quasi-orthogonal basis function of the cubic grid, V' is the area of the cubic grid, ri，j，kAs field point vectors, Ti，j，kIs a position differential, T0，0，0，T0，1，0，T1，0，0，T1，1，0，T0，0，1，T0，1，1，T1，0，1，T1，1，1The vector quantities are local Cartesian coordinate vectors of corresponding points of the cubic grid respectively, w is a vertical coordinate numerical value, u ' is an abscissa parameter, v ' is an ordinate parameter, and w ' is a vertical coordinate parameter.
7. The method for solving the scattered field applied to the simulated FSS structure as recited in claim 1, wherein the step S6 comprises the following substeps:
s61, carrying out discrete processing on the volume surface integral equation VSIE by adopting a quasi-orthogonal basis function, and expanding the surface current on the FSS geometric model by the quasi-orthogonal basis function to obtain an expression of discrete current:
Figure FDA0002588046480000041
Figure FDA0002588046480000042
wherein i is an imaginary unit, k0Is wave number, η0377 Ω is the free space wave impedance, Jpec(r') metal surface current, n is the number of quasi-orthogonal basis functions, fn(r') is the nth quasi-orthogonal basis function, N is the total number of quasi-orthogonal basis functions, unThe coefficient of the expansion current of the nth quasi-orthogonal basis function is S', a surface grid area, a volume grid area, a phase constant and a potential displacement vector;
s62, according to the expression of the discrete current, carrying out matrixing processing on the discrete volume surface integral equation VSIE by adopting a Galileo method to obtain a matrix equation:
[z]{u}＝{V}
Figure FDA0002588046480000043
Figure FDA0002588046480000044
Vm＝-∫fm(r)Einc(r)dr
wherein [ z ]]For the impedance matrix, { u } for the vector to be solved, { V } for the excitation vector, rIs field point vector, r' is source point vector, m and n are numbers of quasi-orthogonal basis functions, ZmnIs [ Z ]]Of the m-th row and the n-th column, VmM-th element of { V }, fm(r) is the mth quasi-orthogonal basis function, fn(r') is the nth quasi-orthogonal basis function, k0In terms of the wave number, the number of waves,
Figure FDA0002588046480000045
in order to be the first vector differential operator,
Figure FDA0002588046480000046
in order to be the second vector differential operator,
Figure FDA0002588046480000047
for the second outer normal direction, "·" is a dot product, G (r, r ') is a Green function in free space, χ (r') is the medium contrast, EincAnd (r) is an incident field, and an integration area is determined by a grid to which r and r' belong.
8. The method as claimed in claim 1, wherein the iterative method in step S7 uses a generalized minimum residual error algorithm, the convergence residual error is set to 0.01, and after solving a solution vector { u }, the current distribution of all sub-grids is obtained.
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