CN113033052B - Electromagnetic rapid numerical modeling method for honeycomb wave-absorbing structure - Google Patents

Electromagnetic rapid numerical modeling method for honeycomb wave-absorbing structure Download PDF

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CN113033052B
CN113033052B CN202110330773.6A CN202110330773A CN113033052B CN 113033052 B CN113033052 B CN 113033052B CN 202110330773 A CN202110330773 A CN 202110330773A CN 113033052 B CN113033052 B CN 113033052B
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杨明林
袁晓伟
盛新庆
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Beijing Institute of Technology BIT
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Abstract

The invention provides an electromagnetic rapid numerical modeling method for a wave-absorbing honeycomb structure, which comprises the following steps of firstly, dividing a model into a full-wave accurate electromagnetic modeling area and a homogeneous equivalent modeling area according to the wave-absorbing honeycomb structure; the walls of the honeycomb in the area of the full-wave accurate electromagnetic modeling area are removed to form a thin-wall central plane, the discontinuity of an electromagnetic field is described by adopting impedance boundary conditions and the boundary conditions are used as discrete boundary conditions of finite elements in the whole area, so that the edge effect of a honeycomb structural plate can be accurately simulated, the honeycomb units in the remaining inner area are equivalent to uniform media by adopting an H-S variational theory and then are subjected to finite element discrete modeling, the homogeneous equivalent modeling greatly simplifies a model structure, the limitation of the geometric shape of the original honeycomb structure is removed, the modeling flexibility is greatly enhanced, meanwhile, normal subdivision density can be used for generating a calculation grid, the number of the grid is reduced, and the calculation efficiency is obviously improved.

Description

Electromagnetic rapid numerical modeling method for honeycomb wave-absorbing structure
Technical Field
The invention belongs to the technical field of electromagnetic calculation, and particularly relates to an electromagnetic rapid numerical modeling method for a honeycomb wave-absorbing structure.
Background
Modern military detection technology and accurate guidance technology develop rapidly, aircrafts in a future battlefield face more severe tests, and stealth technology is highly valued by various countries in order to enhance the overall operational capacity of survival, penetration and the like of weapon systems. The honeycomb wave-absorbing structure is a wave-absorbing material which is light in weight, low in heat conductivity coefficient and excellent in mechanical and electromagnetic properties, and attracts more and more attention.
With the development and progress of electromagnetic calculation methods and high-performance calculation technologies, electromagnetic simulation becomes an important means for assisting stealth design. The rapid simulation and deep analysis of electromagnetic scattering characteristics of structural wave-absorbing materials such as honeycombs and the like are important technical directions for improving the electromagnetic stealth design level of an aircraft. The size of the honeycomb structure unit is small and reaches millimeter level; the honeycomb wall is extremely thin and is in a submillimeter level; and the size of the whole honeycomb wave-absorbing structural part can reach the meter level. The structural characteristics actually form a multi-scale problem which is relatively uniformly distributed, both modeling and calculation are challenging problems in the field of computational electromagnetism, and an effective simulation calculation means is not available.
At present, the electromagnetic simulation of the honeycomb structure is mostly approximated by a homogenization equivalent method. As a typical periodic structure material, the honeycomb wave-absorbing structure can be equivalent to a uniform medium body by a homogenization method, for example, the honeycomb wave-absorbing material can be equivalent to a uniform uniaxial isotropic medium by the homogenization method based on the H-S variation theory, and can achieve better equivalent precision at a lower frequency band.
Recently, an equivalent area decomposition element combining pole technology of an extremely thin structure is provided for a honeycomb wave absorbing structure. Thin-wall equivalent approximation of the honeycomb structure is performed by applying Impedance Boundary (IBC) conditions, and then calculated in a manner that removes complex thin-wall structures as boundary conditions in the finite element calculation region in the dynode. The realization mode can avoid the over-dense mesh subdivision brought by the wall structure of the extremely thin micron-sized honeycomb, and improves the calculation efficiency on the premise of ensuring the calculation precision.
Disadvantages of the equivalent homogenization method: as a structural material, the equivalent electromagnetic parameters of the honeycomb material have dispersibility and anisotropy, when the homogenization method is used for analyzing the equivalent dielectric constant of the honeycomb wave-absorbing material, the influence of specific structural parameters on the material is difficult to accurately reflect, particularly in a higher frequency band, the wavelength is smaller relative to the structural period, and the quasi-static approximate condition is difficult to achieve, so that the equivalent calculation precision is uncontrollable, and particularly under the condition of large deflection angle incidence which is relatively concerned in stealth design, the equivalent precision is poor. In addition, the equivalence of the homogenization method is based on an infinite periodic structure, and the edge effect brought by the finite wave-absorbing cellular board can bring errors to the equivalence result.
The ultra-thin structure equivalent type area decomposition element combining technology has the following defects: this method still requires the accuracy of establishing the regular hexagonal honeycomb structure accurately with the correctness imposed by the IBC boundary conditions when modeling. Therefore, the size of the tetrahedral unit generated after the model is subdivided cannot be larger than the radius of the circumscribed circle of the honeycomb unit, and the flexibility of mesh subdivision is greatly limited. For example, for an AC-NH industrial grade aramid paper honeycomb widely used in the field of aviation, the minimum unit size is only 1.83mm, and in a lower frequency band, the average mesh division density generated by division may far exceed the accuracy required by calculation, and the simulation efficiency is drastically reduced. In addition, for the wave-absorbing honeycomb structural member with the size of a few meters in actual engineering, the modeling and subdivision of all units of the whole honeycomb structure are difficult to perform.
Disclosure of Invention
In view of the above, the invention aims to provide an electromagnetic rapid numerical modeling method for a honeycomb wave-absorbing structure, which can accurately and efficiently perform hybrid analysis modeling.
A method for electromagnetic numerical modeling of a honeycomb wave-absorbing structure comprises the following steps:
step 1, dividing the whole calculation area into a finite element calculation area and a boundary element calculation area; the wave-absorbing structure comprises a finite element calculation area, a boundary element calculation area and a wave-absorbing structure, wherein the finite element calculation area is a three-dimensional model entity part of the wave-absorbing structure, and the boundary element calculation area is a closed curved surface surrounding the finite element calculation area in the wave-absorbing structure;
then dividing the element-limited calculation area into a full-wave accurate electromagnetic modeling area and a homogeneous equivalent modeling area; the full-wave precise electromagnetic modeling area is an annular area with a set width outside the honeycomb wave-absorbing structure; each honeycomb unit in the accurate electromagnetic modeling area is subjected to entity modeling, the boundary of each honeycomb unit is the outer surface of the wall of the original honeycomb, and the homogeneous equivalent modeling area is the middle part of the accurate electromagnetic modeling area subjected to wave absorption structure full wave removal of the honeycomb wave absorption structure and is replaced by a uniform solid medium, so that a mixed equivalent honeycomb model is finally obtained;
step two: dividing the homogeneous equivalent modeling area by using a tetrahedral mesh, and defining a contact surface of the homogeneous equivalent modeling area and the solid hexagonal prism at the innermost side of the full-wave accurate electromagnetic modeling area as a finite element interface; a solid hexagonal prism structure in the full-wave precise electromagnetic modeling area is also subdivided by using a tetrahedral mesh; then, performing triangular mesh subdivision on the boundary element calculation area;
and step three, calculating the model split in the step two to obtain the magnetic field distribution.
Preferably, the specific method of the third step is as follows:
adopting a complete second-order Robin transmission condition at a finite element interface of the full-wave accurate modeling region and the homogeneous equivalent modeling region;
adopting a first-order Robin transmission condition on an interface between the finite element calculation region and the boundary element calculation region;
simulating an electromagnetic field of the homogeneous equivalent modeling area through finite element functional variational simulation;
in the full-wave accurate modeling area, the impedance boundary condition of the electromagnetic field discontinuity condition at two ends of the outer surface of each honeycomb unit is equivalent;
simulating the field of the full-wave precise electromagnetic modeling area through functional variation;
describing the boundary condition of the boundary element calculation area by adopting a joint integral equation;
and solving to obtain the magnetic field distribution of the model.
Preferably, the fully second-order Robin transmission condition adopted at the finite element interface between the full-wave accurate modeling region and the homogeneous equivalent modeling region is specifically as follows:
Figure BDA0002994464740000031
wherein the content of the first and second substances,
Figure BDA0002994464740000032
Figure BDA0002994464740000033
Figure BDA0002994464740000034
representing finite element interfaces Γ mn The upper minimum grid edge size is the smallest,
Figure BDA0002994464740000035
is the unit of an imaginary number,
Figure BDA0002994464740000036
is the wave number of free space, wherein 0 And ε 0 Respectively represent permeability and dielectric constant in air; ω ═ 2 π f denotes the angular frequency, f denotes the calculated operating frequency;
Figure BDA0002994464740000037
and
Figure BDA0002994464740000038
the distribution represents the current and electric field at the interface, i ═ m, n, and m and n represent the full-wave accurate electromagnetic modeling region and the homogeneous equivalent modeling region, respectively.
Preferably, the interface between the outer surface of the finite element calculation region and the boundary element calculation region adopts a first-order Robin transmission condition, which specifically comprises:
Figure BDA0002994464740000039
Figure BDA00029944647400000310
Figure BDA00029944647400000311
and
Figure BDA00029944647400000312
representing the outer surface of a finite element computation region
Figure BDA00029944647400000313
Current and electric field on;
Figure BDA00029944647400000314
and
Figure BDA00029944647400000315
to representBoundary element calculation area surface
Figure BDA00029944647400000316
Current and electric field.
Preferably, the electromagnetic field of the homogeneous equivalent modeling area is modeled as follows through finite element functional variational modeling:
Figure BDA00029944647400000317
wherein Z is 0 For free space impedance, omega is used inside the homogeneous equivalent modeling region m Indicating that dV represents unit volume and dS corresponds to unit area of the integration surface; e m Represents omega m The electric field in (1) is,
Figure BDA00029944647400000318
and
Figure BDA00029944647400000319
respectively represent gamma m And
Figure BDA00029944647400000320
the magnetic field on the surface of the wafer,
Figure BDA00029944647400000321
and
Figure BDA00029944647400000322
respectively represent gamma m And
Figure BDA00029944647400000323
normal on the face; epsilon m,r Represents omega m Relative dielectric constant of (1).
Preferably, the impedance boundary condition is equivalent to:
Figure BDA00029944647400000324
wherein the content of the first and second substances,
Figure BDA0002994464740000041
is a unit normal vector; z is a radical of formula s ＝jω(ε r -ε 0 )d，ε r Is the dielectric constant of the honeycomb wall, d is the thickness of the honeycomb wall before equivalence; e + Representing the electric field on the inner wall of the honeycomb cell before equivalence, E - Representing the electric field on the outer wall of the honeycomb cell before equivalence, and E representing the electric field in the honeycomb wall before equivalence; h + Representing the magnetic field on the inner wall of the cell before equivalence, H - Representing the magnetic field on the outer wall of the cell before equivalence.
Preferably, the field of the full-wave accurate electromagnetic modeling area can be simulated by functional variational components as follows:
Figure BDA0002994464740000042
wherein, gamma is k Mesh information representing the cell walls within the full-wave accurate electromagnetic modeling area,
Figure BDA0002994464740000043
is expressed as gamma k Normal to the face.
Preferably, the boundary condition of the boundary element calculation region is described by using a joint integral equation as follows:
Figure BDA0002994464740000044
wherein the content of the first and second substances,
Figure BDA0002994464740000045
E s and H s Calculating electric and magnetic fields on the surface of the region, E, for the boundary elements, respectively inc And
Figure BDA0002994464740000046
respectively representing the incident electric and magnetic fields,
Figure BDA0002994464740000047
represents the normal to the surface;
Figure BDA0002994464740000048
and
Figure BDA0002994464740000049
the operator is an operator for taking the tangential component of the acted surface, the direction of the operator is consistent with the acted quantity, and the direction of the operator is vertical to the acted quantity;
Figure BDA00029944647400000410
and
Figure BDA00029944647400000411
is an integral differential operator, wherein
Figure BDA00029944647400000412
Has been removed:
Figure BDA00029944647400000413
Figure BDA00029944647400000414
wherein X (r ') is surface vector current or surface vector magnetic current, r is field point, r' is source point, and is arbitrary parameter, G 0 (r, r ') is a green's function,
Figure BDA00029944647400000415
represents the gradient to r;
Figure BDA00029944647400000416
representing the gradient to r'.
Preferably, the solving method comprises:
simultaneous equations (9), (14) and (16) to obtain finite element equations of the calculation region; and then combining equations (12), (13) and (17) to obtain a total equation of the unknown numbers of the internal electric field, the interface electric field and the external surface electric field of the equivalent region, and solving to obtain the magnetic field distribution.
Preferably, the dielectric constant of the homogeneous equivalent modeling region is calculated by:
the dielectric constant in the z direction is:
ε z ＝gε a +(1-g)ε 0 (2)
dielectric constant of honeycomb medium in x and y directions of
Figure BDA0002994464740000051
Figure BDA0002994464740000052
ε 0 Is the dielectric constant of air, epsilon a Dielectric constant of the honeycomb medium, fill factor thereof
Figure BDA0002994464740000053
Where t is the distance between the parallel edges of the cell inner wall and p is the distance between the parallel edges of the cell outer wall.
Preferably, the set width is a width of 8 cells.
The invention has the following beneficial effects:
the invention provides an electromagnetic rapid numerical modeling method for a wave-absorbing honeycomb structure, which comprises the following steps of firstly, dividing a model into a full-wave accurate electromagnetic modeling area and a homogeneous equivalent modeling area according to the wave-absorbing honeycomb structure; the walls of the honeycomb in the area of the full-wave accurate electromagnetic modeling area are removed to form a thin-wall central plane, the discontinuity of an electromagnetic field is described by adopting an impedance boundary condition and the impedance boundary condition is used as a boundary condition for the discrete finite element of the whole area, so that the edge effect of a honeycomb structural plate can be accurately simulated, the honeycomb units in the remaining inner area are equivalent to a uniform medium by adopting an H-S variational theory and then are subjected to finite element discrete modeling, the homogeneous equivalent modeling greatly simplifies a model structure, the limitation of the geometric shape of the original honeycomb structure is removed, the modeling flexibility is greatly enhanced, meanwhile, normal subdivision density can be used for generating a calculation grid, the grid number is reduced, and the calculation efficiency is obviously improved; the continuity of an on-plane tangential electromagnetic field and an on-line normal electromagnetic flow is ensured by adopting a Robin transmission condition on an interface of a full-wave precise electromagnetic modeling area and a homogeneous equivalent modeling area, the sub-areas are connected together according to a non-conformal area decomposition finite element technology, and the whole finite element calculation area is cut off by adopting an integral equation method, so that the calculation accuracy is ensured; and the size of the tetrahedral mesh is uniform during subdivision, the mesh quality is good, and the calculation precision is high.
Drawings
FIG. 1 is a schematic diagram of coordinates of a wave-absorbing honeycomb structure;
FIG. 2 is a schematic diagram of an equivalent cellular board subdivision model;
FIG. 3 is a schematic diagram of a hybrid homogeneous method-impedance boundary equivalent cellular model;
FIG. 4 is a graph comparing the scattering results of the method of the present invention with Impedance Boundary (IBC), Homogenization (HS);
FIG. 5 is a graph comparing the scattering results of the method of the present invention with Impedance Boundary (IBC), Homogenization (HS);
FIG. 6 is a graph comparing the scattering results of the method of the present invention with Impedance Boundary (IBC), Homogenization (HS);
FIG. 7 is a graph comparing the scattering results of the method of the present invention with Impedance Boundary (IBC), Homogenization (HS);
FIG. 8 is a graph comparing the scattering results of the method of the present invention with Impedance Boundary (IBC), Homogenization (HS);
FIG. 9 is a graph comparing the scattering results of the method of the present invention with Impedance Boundary (IBC), Homogenization (HS);
FIG. 10 is a graph comparing the scattering results of the method of the present invention with Impedance Boundary (IBC), Homogenization (HS);
FIG. 11 is a graph comparing scattering results of the method of the present invention with Impedance Boundary (IBC) and Homogenization (HS).
Detailed Description
The invention is described in detail below by way of example with reference to the accompanying drawings.
The invention provides a hybrid accurate modeling and homogeneous equivalent rapid numerical modeling method for a honeycomb wave-absorbing structure, which specifically comprises the following steps:
according to the synthetic element polar theory, the whole calculation area is divided into a Finite Element (FE) calculation area and a boundary element (BI) calculation area. The finite element calculation area is a solid part of the three-dimensional model, and the boundary element calculation area is a closed curved surface surrounding the finite element calculation area. In the invention, the wave-absorbing honeycomb solid structure is a finite element calculation region, and the outer surface surrounding the honeycomb solid structure is a boundary element calculation region.
The method comprises the following steps: and (4) carrying out equivalent treatment on the actual wave-absorbing honeycomb structure, and then modeling. The model is first divided into a full-wave accurate electromagnetic modeling region and a homogeneous equivalent modeling region. The full-wave accurate electromagnetic modeling area can accurately simulate the edge effect caused by the limited size. Specifically, 8 layers of honeycomb units outside the honeycomb plate are selected to form an annular area which is a precise electromagnetic modeling area, and the rest areas are homogeneous equivalent modeling areas. And after partitioning, carrying out entity modeling on the wave-absorbing cellular board by using modeling software such as CATIA (computer-graphics aided three-dimensional Interactive application system), ANSYS (ANSYS) and the like or manually. Each honeycomb unit of the full-wave accurate electromagnetic modeling area needs to be subjected to solid modeling, and the boundary of each unit is the outer surface of the original honeycomb wall, namely the hexagonal prism unit structure shown in fig. 2. The structure of the middle part of the honeycomb plate model is replaced by a uniform solid medium through boolean operations such as cutting, combining and the like, and a hybrid equivalent honeycomb model is obtained, as shown in fig. 3.
Step two: and (4) subdividing the hybrid equivalent cellular model. The central solid medium in the hybrid model is split apart from the surrounding solid hexagonal prism structure. The central solid square structure is subdivided by using a tetrahedral mesh, and the contact surface of the square and the solid hexagonal prism is set as an interface. The surrounding solid hexagonal prism structure is also subdivided using tetrahedral meshes, and the mesh information of the hexagonal prism sides needs to be preserved while the face in contact with the central square structure is set as the interface. After the two divisions, mesh information calculated by Finite Element (FE) of two equivalent areas is obtained. Then, performing triangular mesh subdivision on the outer surface of the whole hybrid model to obtain mesh information calculated by a boundary element (BI) of the hybrid model;
step three: and carrying out homogeneous equivalence on a homogeneous equivalent modeling area of the wave-absorbing honeycomb structure by using a homogeneous method based on an H-S theory to obtain equivalent dielectric parameters. The honeycomb periodic structure composite material is a two-phase medium, and the dielectric constant of the periodic structure material is expressed by a tensor form:
Figure BDA0002994464740000071
for the wave-absorbing honeycomb structure, a coordinate system is established, as shown in FIG. 1, the direction perpendicular to the honeycomb wall is x, the direction parallel to the honeycomb wall is y, the direction parallel to the cells is z, for a regular hexagonal honeycomb, the periods of the x and y directions are all set as a, and the dielectric constant of air is epsilon 0 Dielectric constant of honeycomb medium is epsilon a Fill factor of
Figure BDA0002994464740000072
Where t is the distance between the parallel sides of the cell inner wall and p is the distance between the parallel sides of the cell outer wall, then the dielectric constant in the z direction is:
ε z ＝gε a +(1-g)ε 0 (2)
for the dielectric constants in the x and y directions, one can derive from the variational theory:
Figure BDA0002994464740000073
Figure BDA0002994464740000074
has epsilon for the cellular periodic structure x ＝ε y ＝ε ⊥ Namely:
Figure BDA0002994464740000075
Figure BDA0002994464740000076
from epsilon 0 ＜ε ⊥ ＜ε a It can be further calculated that:
Figure BDA0002994464740000077
Figure BDA0002994464740000078
the value range of the equivalent dielectric parameter of the periodic composite wave-absorbing material, namely the upper limit equivalent value can be determined according to the formula
Figure BDA0002994464740000079
And lower equivalent value
Figure BDA00029944647400000710
When the periodic structure unit cell surrounds a medium with small electromagnetic parameters by a medium with large electromagnetic parameters, an H-S upper bound equivalent formula is used. It can be seen from fig. 1 that the honeycomb material is formed by surrounding air columns with honeycomb walls, and therefore the equivalent dielectric constant of the wave-absorbing honeycomb material can be obtained by using the upper equivalent formula (8) and the formula (2).
Step four: and the continuity of the in-plane tangential electromagnetic field and the in-line normal electromagnetic flow is ensured by adopting Robin transmission conditions on the interface of two different equivalent regions and the interface of FE-BI.
Adopting complete second-order Robin transmission conditions at finite element interfaces (FE-FE) of the accurate modeling region and the homogeneous equivalent modeling region:
Figure BDA0002994464740000081
wherein the content of the first and second substances,
Figure BDA0002994464740000082
Figure BDA0002994464740000083
Figure BDA0002994464740000084
representing finite element interfaces Γ mn The upper minimum grid edge size is the smallest,
Figure BDA0002994464740000085
is the unit of an imaginary number,
Figure BDA0002994464740000086
is the wave number of free space, wherein 0 And ε 0 Respectively represent permeability and dielectric constant in air; ω ═ 2 π f denotes the angular frequency, f denotes the calculated operating frequency;
Figure BDA0002994464740000087
and
Figure BDA0002994464740000088
the distribution represents the current and electric field at the interface, i ═ m, n, and m and n represent the full-wave accurate electromagnetic modeling region and the homogeneous equivalent modeling region, respectively.
First order Robin transmission conditions are employed at finite element outer surface-boundary interfaces (FE-BI) of the exact modeling region and the homogeneous equivalent modeling region:
Figure BDA0002994464740000089
Figure BDA00029944647400000810
Figure BDA00029944647400000811
and
Figure BDA00029944647400000812
representing the outer surface of a finite element computation region
Figure BDA00029944647400000813
Current and electric field on;
Figure BDA00029944647400000814
and
Figure BDA00029944647400000815
representing bounding element computation region surfaces
Figure BDA00029944647400000816
Current and electric field.
Step five: the electromagnetic field of the homogeneous equivalent modeling region can be modeled by finite element functional variational modeling as:
Figure BDA00029944647400000817
wherein Z is 0 For free space impedance, omega is used inside the homogeneous equivalent modeling region m Indicating that dV represents unit volume and dS corresponds to unit area of the integration surface; e m Represents omega m In the electric field of (a) is,
Figure BDA00029944647400000818
and
Figure BDA00029944647400000819
respectively representing gamma m And
Figure BDA00029944647400000820
the magnetic field on the surface of the wafer,
Figure BDA00029944647400000821
and
Figure BDA00029944647400000822
respectively representing gamma m And
Figure BDA00029944647400000823
normal on the face; epsilon m,r And mu m,r Each represents omega m Relative permittivity and permeability, ∈ m,r Determined from the formulas (2) and (8) < mu > m,r Is set to 1.
In the precise modeling area, the original honeycomb thin-wall structure is removed and is equivalent to a face structure, so that the resulting electromagnetic field discontinuity condition at two ends of the face can be equivalent to an Impedance Boundary Condition (IBC):
Figure BDA0002994464740000091
wherein the content of the first and second substances,
Figure BDA0002994464740000092
is a unit normal vector; z is a radical of s ＝jω(ε r -ε 0 )d，ε r Is the dielectric constant of the honeycomb wall, d is the thickness of the honeycomb wall before equivalence; e + Representing the electric field on the inner wall of the honeycomb cell before equivalence, E - Representing the electric field on the outer wall of the honeycomb cell before equivalence, and E representing the electric field in the honeycomb wall before equivalence; h + Representing the magnetic field on the inner wall of the cell before equivalence, H - Representing the magnetic field on the outer wall of the cell before the equivalence.
The field of a full-wave accurate electromagnetic modeling region containing IBC boundary conditions, Robin transmission boundary conditions, can be modeled by functional variational modeling as:
Figure BDA0002994464740000093
wherein, gamma is k Mesh information representing the cell walls within the full-wave accurate electromagnetic modeling region,
Figure BDA0002994464740000094
is expressed as gamma k Normal to the face.
Step six: the boundary conditions of the boundary element calculation region are described as follows by adopting a joint integral equation: (CFIE)
Figure BDA0002994464740000095
Wherein the content of the first and second substances,
Figure BDA0002994464740000096
E s and H s Calculating electric and magnetic fields on the surface of the region, E, for the boundary elements, respectively inc And
Figure BDA0002994464740000097
respectively representing the incident electric and magnetic fields,
Figure BDA0002994464740000098
represents the normal to the surface;
Figure BDA0002994464740000099
and
Figure BDA00029944647400000910
the operator is an operator for taking the tangential component of the acted surface, the direction of the operator is consistent with the acted quantity, and the direction of the operator is vertical to the acted quantity;
Figure BDA00029944647400000911
and
Figure BDA00029944647400000912
is an integral differential operator, wherein
Figure BDA00029944647400000913
Has been removed:
Figure BDA00029944647400000914
Figure BDA0002994464740000101
wherein, X(r ') is surface vector current or surface vector magnetic current, r is field point, r' is source point, is arbitrary parameter, G 0 (r, r ') is a green's function,
Figure BDA0002994464740000102
represents the gradient to r;
Figure BDA0002994464740000103
representing the gradient to r'.
Step seven: equations (9), (14) and (16) are simultaneously established, and finite element equations of the calculation regions are obtained. However, since the electromagnetic field parameters on the outer surface of the finite element region cannot be solved by finite element equations, the unknowns are greater than the equations. And combining the obtained finite element equations with equations (12), (13) and (17) to obtain a total equation of the unknowns of the internal electric field, the interface electric field and the external electric field of the final equivalent region to be solved. The integral equation related full matrix-vector multiplication is accelerated by adopting a multilayer fast multipole technology.
Example (b):
in this embodiment, by using four wave-absorbing cellular board models with different specifications, i.e., a cellular unit with a radius of 1.83mm and a radius of 2.75mm and a single-layer wall thickness of 0.05mm and a single-layer wall thickness of 0.1mm, and by using three methods, i.e., an impedance boundary condition equivalence (IBC) method based on an integrated polarization technology, a homogenization method based on an H-S variation theory, and a mixed impedance boundary-homogenization method equivalence, HH polarization and VV polarization single-station RCS of the wave-absorbing cellular board are calculated at a frequency point of 10GHz, and the results are shown in fig. 4-11, which proves that the equivalence method has good accuracy.
Finally, in order to show the computing power of the method, the grid number of a finite element part and a boundary integral part of an impedance boundary condition equivalence (IBC) method and a mixed impedance boundary-homogeneity method equivalence method is counted, and as shown in tables 1 and 2, the grid number of the honeycomb model can be obviously reduced by the equivalence method, and the computing efficiency is effectively improved.
TABLE 1 mesh number comparison of two methods for honeycomb with radius of 1.83mm
IBC HS-IBC
FEM 734709 422436
BI 124770 72961
TABLE 2 mesh number comparison of two methods for radius 2.75mm honeycombs
IBC HS-IBC
FEM 1779958 837061
BI 199104 131023
The invention provides an electromagnetic rapid numerical modeling method for a honeycomb wave-absorbing structure. From the angle of numerical calculation, the embodiment shows the calculation results of three equivalent methods for the wave-absorbing honeycomb materials with different specifications, and the accuracy of the method is proved. From the aspect of calculation efficiency, the embodiment shows that the method can obviously reduce the number of grids when calculating the wave-absorbing honeycomb structure and can efficiently calculate the wave-absorbing honeycomb result target.
In summary, the above description is only a preferred embodiment of the present invention, and is not intended to limit the scope of the present invention. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims (11)

1. A method for electromagnetic numerical modeling of a honeycomb wave-absorbing structure is characterized by comprising the following steps:
step 1, dividing the whole calculation area into a finite element calculation area and a boundary element calculation area; the wave-absorbing structure comprises a finite element calculation area, a boundary element calculation area and a wave-absorbing structure, wherein the finite element calculation area is a three-dimensional model entity part of the wave-absorbing structure, and the boundary element calculation area is a closed curved surface surrounding the finite element calculation area in the wave-absorbing structure;
then dividing the element-limited calculation area into a full-wave accurate electromagnetic modeling area and a homogeneous equivalent modeling area; the full-wave precise electromagnetic modeling area is an annular area with a set width outside the honeycomb wave-absorbing structure; each honeycomb unit in the accurate electromagnetic modeling area is subjected to entity modeling, the boundary of each honeycomb unit is the outer surface of the wall of the original honeycomb, and the homogeneous equivalent modeling area is the middle part of the accurate electromagnetic modeling area subjected to wave absorption structure full wave removal of the honeycomb wave absorption structure and is replaced by a uniform solid medium, so that a mixed equivalent honeycomb model is finally obtained;
step two: dividing the homogeneous equivalent modeling area by using a tetrahedral mesh, and defining a contact surface of the homogeneous equivalent modeling area and the solid hexagonal prism at the innermost side of the full-wave accurate electromagnetic modeling area as a finite element interface; a solid hexagonal prism structure in the full-wave precise electromagnetic modeling area is also subdivided by using a tetrahedral mesh; then, performing triangular mesh subdivision on the boundary element calculation area;
and step three, calculating the model split in the step two to obtain the magnetic field distribution.
2. The electromagnetic numerical modeling method for the wave-absorbing honeycomb structure according to claim 1, wherein the specific method in the third step is as follows:
adopting a complete second-order Robin transmission condition at a finite element interface of the full-wave accurate modeling region and the homogeneous equivalent modeling region;
adopting a first-order Robin transmission condition on an interface between the finite element calculation region and the boundary element calculation region;
simulating an electromagnetic field of the homogeneous equivalent modeling area through finite element functional variational simulation;
in the full-wave accurate modeling area, the impedance boundary condition of the electromagnetic field discontinuity condition at two ends of the outer surface of each honeycomb unit is equivalent;
simulating the field of the full-wave precise electromagnetic modeling area through functional variation;
describing the boundary condition of the boundary element calculation area by adopting a joint integral equation;
and solving to obtain the magnetic field distribution of the model.
3. The electromagnetic numerical modeling method for the wave-absorbing honeycomb structure according to claim 2, wherein the completely second-order Robin transmission condition adopted at the finite element interface between the full-wave accurate modeling area and the homogeneous equivalent modeling area is specifically as follows:
Figure FDA0002994464730000011
wherein the content of the first and second substances,
Figure FDA0002994464730000012
Figure FDA0002994464730000021
Figure FDA0002994464730000022
representing finite element interfaces Γ mn The upper minimum size of the mesh edge,
Figure FDA0002994464730000023
is a unit of an imaginary number, and is,
Figure FDA0002994464730000024
is the wave number of free space, wherein 0 And ε 0 Respectively represent permeability and dielectric constant in air; ω ═ 2 π f denotes the angular frequency, f denotes the calculated operating frequency;
Figure FDA0002994464730000025
and
Figure FDA0002994464730000026
the distribution represents the current and electric field at the interface, i ═ m, n, and m and n represent the full-wave accurate electromagnetic modeling region and the homogeneous equivalent modeling region, respectively.
4. The electromagnetic numerical modeling method for the honeycomb wave-absorbing structure according to claim 3, wherein the interface between the outer surface of the finite element calculation region and the boundary element calculation region adopts a first-order Robin transmission condition, which specifically comprises:
Figure FDA0002994464730000027
Figure FDA0002994464730000028
Figure FDA0002994464730000029
and
Figure FDA00029944647300000210
representing the outer surface of a finite element computation region
Figure FDA00029944647300000211
Current and electric field on;
Figure FDA00029944647300000212
and
Figure FDA00029944647300000213
representing a bounding element computation region surface
Figure FDA00029944647300000214
Current and electric field.
5. The electromagnetic numerical modeling method of the honeycomb wave-absorbing structure of claim 4, characterized in that the electromagnetic field of the homogeneous equivalent modeling area is modeled as:
Figure FDA00029944647300000215
wherein, Z 0 For free space impedance, omega is used inside the homogeneous equivalent modeling region m Indicating that dV represents unit volume and dS corresponds to unit area of the integration surface; e m Represents omega m The electric field in (1) is,
Figure FDA00029944647300000216
and
Figure FDA00029944647300000217
respectively representΓ m And
Figure FDA00029944647300000218
the magnetic field on the surface of the wafer,
Figure FDA00029944647300000219
and
Figure FDA00029944647300000220
respectively represent gamma m And
Figure FDA00029944647300000221
normal on the face; epsilon m,r Represents omega m Relative dielectric constant of (2).
6. The electromagnetic numerical modeling method for the honeycomb wave-absorbing structure according to claim 5, characterized in that the impedance boundary conditions are equivalent to:
Figure FDA00029944647300000222
wherein the content of the first and second substances,
Figure FDA00029944647300000223
is a unit normal vector; z is a radical of s ＝jω(ε r -ε 0 )d，ε r Is the dielectric constant of the honeycomb wall, d is the thickness of the honeycomb wall before equivalence; e + Representing the electric field on the inner wall of the honeycomb cell before equivalence, E - Representing the electric field on the outer wall of the honeycomb cell before equivalence, and E representing the electric field in the honeycomb wall before equivalence; h + Representing the magnetic field on the inner wall of the cell before equivalence, H - Representing the magnetic field on the outer wall of the cell before equivalence.
7. The electromagnetic numerical modeling method for the honeycomb wave-absorbing structure according to claim 6, characterized in that the field of the full-wave precise electromagnetic modeling area can be modeled as:
Figure FDA0002994464730000031
wherein, gamma is k Mesh information representing the cell walls within the full-wave accurate electromagnetic modeling area,
Figure FDA0002994464730000032
is expressed as gamma k Normal to the face.
8. The electromagnetic numerical modeling method for the honeycomb wave-absorbing structure according to claim 7, characterized in that the boundary conditions of the boundary element calculation area are described as follows by using a joint integral equation:
Figure FDA0002994464730000033
wherein the content of the first and second substances,
Figure FDA0002994464730000034
E s and H s Calculating the electric and magnetic fields on the surface of the region, E, for the boundary elements, respectively inc And
Figure FDA0002994464730000035
respectively representing the incident electric and magnetic fields,
Figure FDA0002994464730000036
represents the normal to the surface;
Figure FDA0002994464730000037
and
Figure FDA0002994464730000038
the operator is an operator for taking the tangential component of the acted surface, the direction of the operator is consistent with the acted quantity, and the direction of the operator is vertical to the acted quantity;
Figure FDA0002994464730000039
and
Figure FDA00029944647300000310
is an integral differential operator, wherein
Figure FDA00029944647300000311
Has been removed:
Figure FDA00029944647300000312
Figure FDA00029944647300000313
wherein X (r ') is surface vector current or surface vector magnetic current, r is field point, r' is source point, and is arbitrary parameter, G 0 (r, r') is a green function ^ denotes the gradient to r; v represents a gradient to r'.
9. The electromagnetic numerical modeling method for the wave-absorbing honeycomb structure according to claim 8, characterized in that the solving method comprises:
simultaneous equations (9), (14) and (16) to obtain finite element equations of the calculation region; and then combining equations (12), (13) and (17) to obtain a total equation of the unknown numbers of the internal electric field, the interface electric field and the external electric field of the final equivalent region to be solved, and solving to obtain the magnetic field distribution.
10. The electromagnetic numerical modeling method for the honeycomb wave-absorbing structure according to claim 9, wherein the dielectric constant of the homogeneous equivalent modeling area is calculated by:
the dielectric constant in the z direction is:
ε z ＝gε a +(1-g)ε 0 (2)
dielectric constant of honeycomb medium in x and y directions is
Figure FDA0002994464730000041
Figure FDA0002994464730000042
ε 0 Is the dielectric constant of air, epsilon a Dielectric constant of the honeycomb medium, fill factor thereof
Figure FDA0002994464730000043
Where t is the distance between the parallel edges of the cell inner wall and p is the distance between the parallel edges of the cell outer wall.
11. The electromagnetic numerical modeling method for the wave-absorbing honeycomb structure according to claim 1, wherein the set width is the width of 8 honeycomb units.
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