CN104699879B

CN104699879B - The multiple rotary equivalent simulation method of Complex multi-target electromagnetic scattering

Info

Publication number: CN104699879B
Application number: CN201310666685.9A
Authority: CN
Inventors: 陈如山; 樊振宏; 丁大志; 苏婷
Original assignee: Nanjing University of Science and Technology
Current assignee: Nanjing University of Science and Technology
Priority date: 2013-12-10
Filing date: 2013-12-10
Publication date: 2018-04-03
Anticipated expiration: 2033-12-10
Also published as: CN104699879A

Abstract

The invention discloses a kind of multiple rotary equivalent simulation method of Complex multi-target electromagnetic scattering.Step is as follows：The grid model of each sub-goal is established, obtains corresponding subregion；In the corresponding local coordinate system of each sub-regions, the collision matrix based on curved surface quadrangle high-order hierarchical basis functions and triangle RWG basic functions between transfer matrix and cylinder and the cylinder inside specific item standard type based on rotationally symmetric body basic function between sphere and cylinder is determined；The transmission matrix based on rotationally symmetric body basic function, the coefficient transition matrix based on curved surface quadrangle high-order hierarchical basis functions, the coefficient transition matrix based on rotationally symmetric body basic function between sphere are determined, searches cylinder top-surface camber quadrangle numbering and the corresponding relation between bus segment number；Driving source is set, and incidence wave is uniform plane wave；The equivalent electromagnetic current on all subregion sphere is determined, finally gives Radar Cross Section.The present invention can efficiently analyze the electromagnetic scattering of Complex multi-target, be significant.

Description

The multiple rotary equivalent simulation method of Complex multi-target electromagnetic scattering

Technical field

The present invention relates to electromagnetic simulation technique field, particularly a kind of multiple rotary of Complex multi-target electromagnetic scattering is equivalent Emulation mode.

Background technology

Often need to analyze Complex multi-target electromagnetic scattering problems, such as cloud in Practical Project.Quick analysis calculates this The electromagnetic scattering of a little multiple target bodies is the important research direction in the fields such as radar detection, target identification.Rotationally symmetric body moment method (M.Andreasen,"Scattering from bodies of revolution,"Antennas and Propagation, IEEE Transactions on, vol.13, pp.409-418,1965.) it can efficiently analyze the electricity of all kinds of rotationally symmetric bodies Magnetic scattering, compared to traditional based on RWG basic functions（Rao, S.Wilton and D.Glisson, A., " Electromagnetic scattering by surfaces of arbitrary shape ", IEEE Transactions on,vol.30,pp.303-310,1982）The analysis method memory consumption of subdivision modeling and calculating time will be many less.So And most of target and without rotational symmetry characteristic but there is class rotational symmetry characteristic, such as guided missile model in reality.Guided missile Bomb body rotational symmetry, if considering, empennage, bolt etc. just destroys original rotational symmetry characteristic, it is impossible to directly using rotational symmetry Body moment Method Analysis.In addition, rotationally symmetric body moment method can not efficiently analyze multiple not coaxial rotational symmetry volume scatterings Scattering problems.Moment method based on curved surface quadrangle high-order hierarchical basis functions compares the moment method unknown quantity based on RWG basic functions Less, precision is high.But difficulty is generated to irregular model meshes, matrix condition is poor.

The content of the invention

It is an object of the invention to provide a kind of efficient quick, the multiple rotation of the Complex multi-target electromagnetic scattering of flexibility and reliability Turn equivalent simulation method.

Realizing the technical solution of the object of the invention is：A kind of multiple rotary of Complex multi-target electromagnetic scattering etc. imitates True method, the Complex multi-target include at least two complex targets, and step is as follows：

1st step, establish the grid model of each sub-goal：Established outside each sub-goal and just surround the sub-goal completely Cylinder, the sphere for just surrounding the cylinder completely is established outside each cylinder, obtains subregion corresponding with each sub-goal；

2nd step, in the corresponding local coordinate system of each sub-regions, determine to be based on rotationally symmetric body between sphere and cylinder Inside the transfer matrix and cylinder of basic function and the cylinder curved surface quadrangle high-order hierarchical basis functions are based between specific item standard type With the collision matrix of triangle RWG basic functions；

3rd step, determine the transmission matrix based on rotationally symmetric body basic function between sphere；

4th step, it is determined that the coefficient transition matrix based on curved surface quadrangle high-order hierarchical basis functions, based on rotationally symmetric body base The coefficient transition matrix of function, search the corresponding relation between cylinder top-surface camber quadrangle numbering and bus segment number；

5th step, driving source is set, and incidence wave is uniform plane wave；

6th step, equation group is established, solve equation group and obtain the equivalent electromagnetic current on each sub-regions sphere；

7th step, scattered field is determined by the equivalent electromagnetic current on each sub-regions sphere, obtains Radar Cross Section.

Compared with prior art, its remarkable result is the present invention：（1）The introducing analysis of rotationally symmetric body moment method is not had The problem of rotational symmetry characteristic；（2）By the method and basic function coefficient conversion method of axis rotation method system, curved surface is established Bridge between quadrangle high order MoM, rotationally symmetric body moment method, RWG basic function moment methods, so as to reference to them The respective same problem of benefit analysis, it is a flexible Domain Decomposition Method；（3）Method by establishing two layers of equivalent face So that the quantity of the curved surface quadrilateral mesh on equivalent face reduces, the high efficiency of rotationally symmetric body moment method preferably make use of；（4）The introducing of rotationally symmetric body moment method reduces the multilevel fast multipole method layering number based on point domain basic function moment method Amount, so as to avoid the defects of multilevel fast multipole is high-rise time-consuming.

Brief description of the drawings

Fig. 1 is the structural representation of Complex multi-target scattering model of the present invention.

Fig. 2 is the flow chart of the multiple rotary equivalent simulation method of Complex multi-target electromagnetic scattering of the present invention.

Fig. 3 is the illustraton of model of cloud in the inventive method.

Fig. 4 is the schematic diagram of two layers of equivalent face of each sub-regions in the inventive method.

Fig. 5 is three kinds of subdivision cell schematics in the inventive method, wherein (a) is sphere and circumference bus subdivision, (b) is Cylinder quadrangle grids, (c) are sub-goal triangulation.

Fig. 6 is three kinds of basic function schematic diagrames in the inventive method, wherein (a) is RWG basic functions, (b) is Based on Triangle Basis, (c) it is the curved surface quadrangle in the former space of high-order hierarchical basis functions（It is left）With the quadrangle in parameter coordinate system（It is right）.

Fig. 7 is the relation schematic diagram between each local coordinate system in the inventive method.

Fig. 8 is corresponding radar scattering coefficient simulation result figure in the embodiment of the present invention 1.

Embodiment

With reference to Fig. 1~7, imitated with multiple rotary of specific five missile electromagnetic scatterings of Complex multi-target model etc. Exemplified by true, the present invention is further elaborated.

With reference to Fig. 1, the structural representation of Complex multi-target scattering model of the present invention, to the more of Complex multi-target electromagnetic scattering Secondary rotation equivalent simulation method flow is as shown in Fig. 2 the Complex multi-target includes at least two complex target such as Fig. 3 institutes Show, comprise the following steps that：

As shown in Figures 4 and 5, using the discrete each sub-goal surface of triangular element；Being established outside each sub-goal can be with The cylinder of the sub-goal is just surrounded completely, it is discrete to periphery progress using curved surface quadrilateral units, while use line segment The bus of discrete cylinder；The just complete sphere for surrounding the face of cylinder is established outside each cylinder, uses the mother of the discrete sphere of line segment Line, global coordinate system and local coordinate system are established, it is an independent subregion to define the region surrounded by sphere；

On subregion i：

（2.1）The definition of three kinds of basic functions is provided first, as shown in Figure 6：

（a）RWG basic functions are defined on triangle surface subdivision unit

Wherein, n is that RWG basic functions number nth bar side in as triangle subdivision unit, l_nFor the length on nth bar side, Represent upper triangle corresponding to nth bar sideArea,Represent lower triangle corresponding to nth bar sideArea,Table Show that r points point to free summit in upper triangle corresponding to nth bar side（Remove two remaining triangular apex in summit of trimming n）'s Direction vector,The direction vector of free summit sensing r points in lower triangle corresponding to nth bar side is represented, as shown in Fig. 6 (a).

Then the electromagnetic current on scattering object surface is with RWG base function expansions：

Wherein, J (r) represents the electric current of scattering object surface any point r points, and M (r) represents the magnetic current of r points,Represent electricity Stream J (r) uses RWG basic functionsExpansion corresponds to the expansion coefficient of n-th of basic function,Represent that magnetic current M (r) is used RWG basic functionsExpansion corresponds to the expansion coefficient of n-th of basic function, N^rwgRepresent that equivalent cylinder uses RWG basic function exhibitions Open the number of rear unknown quantity.

（b）The rotationally symmetric body basic function on rotationally symmetric body bus section, busbar section

The basic function is in generatrix directionUpper is triangular basis functional form, in circumferential directionUpper is exponential function form：

Wherein subscript BoR represents rotationally symmetric body, T_n'(t) Based on Triangle Basis is represented, is one-dimensional local base function, T_n'(t) It is defined on two subdivision being connected line segments, this two lines section is referred to as leading portion and back segment respectively for we, and its expression formula is T_n'(t) Based on Triangle Basis is represented, as shown in Fig. 6 (b), its expression formula is：

Wherein, n' numbers for BoR Based on Triangle Basis, and t represents the generatrix direction component of r points；ρ (r) represents r points to rotation pair Claim the vertical range of body rotary shaft；φ represents the circumferential angle of r points；e^jαφRepresent that Fourier expansion corresponds to the α pattern Exponential term；Represent the generatrix direction of r points；Represent the circumferential direction of r points；N^BoRRepresent that rotationally symmetric body basic function is corresponding Unknown quantity number；The starting point tangential component of leading portion corresponding to n-th of Based on Triangle Basis is represented,Represent n-th of triangle The terminal tangential component of leading portion corresponding to basic function is the starting point tangential component of back segment,Represent that n-th of Based on Triangle Basis is corresponding Back segment terminal tangential component；Δ_nRepresent the length of leading portion, Δ_n+1Represent the length of back segment；

WhereinRepresent that electric current J (r) uses rotationally symmetric body basic functionExpansion corresponds to the of the α pattern N' basic function generatrix directionExpansion coefficient；Represent that electric current J (r) uses rotationally symmetric body basic functionExpansion It is circumferential corresponding to the n-th ' individual basic function of the α patternThe expansion coefficient in direction；Represent that magnetic current M (r) uses rotation pair Claim body basic functionExpansion corresponds to the n-th ' individual basic function generatrix direction of the α patternExpansion coefficient；Represent Magnetic current M (r) uses rotationally symmetric body basic functionThe n-th ' individual basic function that expansion corresponds to the α pattern is circumferentialDirection Expansion coefficient.

（c）High-order hierarchical basis functions are defined on curved surface quad patch subdivision unit

During with curved surface quadrilateral units come discrete object surface, the span in the both direction of curved surface quadrilateral units It is -1≤u, v≤1, as shown in Fig. 6 (c), electric current, the magnetic current of body surface are represented by J=a_uJ^u+a_vJ^v, M=a_uM^u+a_vM^v, WhereinFor unit direction vector.

By taking electric current as an example, the expression of high-order hierarchical basis functions is as follows：

Multinomial in above formulaWithIt is the accurate orthogonal Legnedre polynomial of amendment；P_n(u), P_n(v) it is to strangle Allow moral multinomial；For scale factor：

In formula, J (u, v) is curved surface Jacobi's factor, J (u, v)=| a_u×a_v|,The high-order that will exactly solve Hierarchical basis functions unknowm coefficient；M^u、M^vRepresent to deploy exponent number along the electric current on electric current u and v both directions respectively.

（2.2）Establish local coordinate system x_i'y_i'z_i', as shown in fig. 7, determined in the local coordinate system equivalent cylinder and Collision matrix S between sub-goal_cp, cylinder to sphere transfer matrix Z_sc, sphere to cylinder transfer matrix Z_cs：

(a) spinning solution and spin matrix between local coordinate system

If global coordinate system is xyz, respectively with the central point O of each sub-regions_iFor origin, establish parallel to world coordinates The local coordinate system x of system_iy_iz_i, by a reference point P in space by rotating another local coordinate system needed, It is designated as x_i ^py_i ^pz_i ^pIf the coordinate of reference point is (x₀,y₀,z₀), then：

Wherein, θ^rotAnd φ^rotFor rotary reference angle, according to θ^rotAnd φ^rot, according to following two step by local coordinate system x_iy_iz_iRotation obtains local rectangular coordinate system x_i ^py_i ^pz_i ^p：

1. with local coordinate system x_iy_iz_iZ-axis be rotary shaft, turn clockwiseSo that local coordinate system x_iy_iz_i Y-axis pass through subpoint P' of the P points in xoy planes, obtain local coordinate system x_i ^py_i ^pz_i ^pX-axis position；

2. turned clockwise θ using the x-axis that obtains in 1. as rotary shaft^rotSo that local coordinate system x_iy_iz_iZ-axis pass through P Point, obtain local coordinate system x_i ^py_i ^pz_i ^pZ-axis；

According to arbitrfary point Q in space after being rotated with upper type in local coordinate system x_iy_iz_iAnd x_i ^py_i ^pz_i ^pIn coordinate（x_qi, y_qi,z_qi）With（x_qi ^p,y_qi ^p,z_qi ^p）Between relation be：

Wherein coordinate system x_iy_iz_iTo x_i ^py_i ^pz_i ^pSpin matrix R^SLFor：

Coordinate system x_i ^py_i ^pz_i ^pTo x_iy_iz_iSpin matrix R^LSFor：

If reference point falls in local coordinate system x_iy_iz_iZ-axis on, then need specify φ^rot, work as φ^rotRepresented not at=90 ° Need to rotate；

(b) local coordinate system x is established_i'y_i'z_i', determined in the local coordinate system between equivalent cylinder and sub-goal Collision matrix S^cp：

In following formula, footmark c represents the face of cylinder, and s represents sphere, and p represents internal sub-goal surface；Cs represents that sphere s is relative Analogize in face of cylinder c, sc, cp, pc, pp, ss；

It is reference point P to select any point on column rotating shaft, and local coordinate system x is obtained according to the spinning solution in (a)_i' y_i'z_i', note spin matrix isWithThe collision matrix S between equivalent cylinder and sub-goal is determined in the coordinate system^cp, Electromagnetic current on its central column face, which uses, is based on curved surface quadrangle high-order hierarchical basis functionsExpansion, and the electric current on sub-goal The RWG base function expansions based on triangle are then used, expression is：

S^cp=Z^cp[Z^pp]^-1Z^pc（14）

Wherein：

In formula, [Z^pp]^-1For [Z^pp] inverse matrix；M, n are the numbering of basic function；

Define integro-differential operator L, K：

Wherein

Wherein x represents electric current J or magnetic current M, A (x) represent vector position, and ψ (x) represents scalar potential, and C (x) represents curl field, pv Principal value integral is represented, η is plane wave impedance,For unit normal vector, g represents source point r' to site r Green's function, and it is expressed Formula is：

(c) in local coordinate system x_i'y_i'z_i' in determine cylinder to sphere transfer matrix Z^sc, sphere to cylinder transmission Matrix Z^cs：

Electromagnetic current on cylinder uses BoR basic functionsDeploy, the electromagnetic current on sphere uses BoR basic functionsExpansion, r are site, and r' is source point, transfer matrix Z^scIt is caused equivalent on sphere that source on cylinder is described Source, its expression formula are：

Transfer matrix Z^csThe caused equivalent source on cylinder of the source on sphere is described, is similarly obtained：

Equivalent electromagnetic current on each sphere can encourage on the sphere on other subregions it is new wait it is incident imitate it is electric Magnetic current.If there is equivalent electromagnetic current J on equivalent sphere j_i,M_i, then the source caused new incident equivalent electromagnetic current on sphere i J_j',M_j', the effect can be by transmission matrix T_ijDescription：

In order to use rotationally symmetric body moment method to calculate transmission matrix, it is necessary to which establishing one make it that two spheres are coaxial Local coordinate system x_ij"y_ij"z_ij", as shown in Figure 7.Local coordinate system now to be rotated is x_iy_iz_i, reference point elects sphere as J centre of sphere O_j, according to step（2.2）In (a) obtain coordinate system x_ij"y_ij"z_ij", remember that corresponding spin matrix isWithIn local coordinate system x_ij"y_ij"z_ij" in, transmission matrixFormula it is as follows：

（a）Coefficient transition matrix：

The present invention has used different local coordinate systems to describe same group of electromagnetic current coefficient for same sphere, for same One cylinder has used two kinds of basic functions to describe same group of electromagnetic current coefficient, it would therefore be desirable to carry out turning for basic function coefficient Change；

By taking electric current as an example, if the electric current of each point is, it is known that be designated as J (r) on a face, f (r) is basic function, f^*(r) it is the base The test function of function：

Wherein N_fFor the number of basic function, e_nFor the coefficient of basic function, then coefficient vector e can be tried to achieve by following formula：

[U]^-1For coefficient solution matrix.It is rotationally symmetric body basic function f to take f (r) respectively^BoRAnd high-order hierarchical basis functions (r) f^HO(r) rotationally symmetric body basic function coefficient solution matrix [U corresponding to, obtaining^BoR]^-1Square is solved with high-order hierarchical basis functions coefficient Battle array [U^HO]^-1。

（b）Search the corresponding relation between cylinder top-surface camber quadrangle numbering and bus segment number：

When known high-order hierarchical basis functions coefficient seeks rotationally symmetric body basic function coefficient, required to look up when calculating vectorial w And the numbering of curved surface quadrangle corresponding to Gauss point is recorded, so as to rapid translating basic function coefficient.The process of lookup is as follows：

It is grouped first by Octree, calculates the group where Gauss point；Then the Gauss point is searched out by preset angle configuration Corresponding curved surface quadrangle numbering；Respectively along u in the parameter coordinate system of the curved surface quadrangle, v both directions are in [- 1,1] Sampling, calculates these coordinates under former coordinate system, the parameter coordinate closest to the point of Gauss point is recorded, to calculate this Electromagnetic current of the point under parameter coordinate system.

5th step, driving source is set, and incidence wave is uniform plane wave, draws the equivalent incoming electromagnetic stream on sphere；

Uniform plane wave is radiated on the sphere of each sub-regions：

Wherein, E^incFor incident electric fields, H^incFor incident magnetic,It is normal to propagate for plane wave propagation direction unit vector, k Number,Represent direction of an electric field unit vector.

WhereinRepresent rotationally symmetric body basic function coefficient corresponding to the incoming electromagnetic stream of ith zone；

By field boundary condition and the null field principle of equal effects equation is established on each equivalent sphere：

Wherein, i=1 ..., N_obj, j=1 ..., N_obj,i≠j；N_objTotal subregion number is represented, i, j represent that subregion is compiled Number,Represent rotationally symmetric body basic function electromagnetic current coefficient by local coordinate system x_iy_iz_iIt is transformed into local coordinate system x_i'y_i' z_i' in process；Represent to be transformed into high-order hierarchical basis functions coefficient by rotationally symmetric body basic function electromagnetic current coefficient Process；Expression is transformed into the process of rotationally symmetric body basic function coefficient by high-order hierarchical basis functions electromagnetic current coefficient；Represent rotationally symmetric body basic function electromagnetic current coefficient by local coordinate system x_i'y_i'z_i' it is transformed into local coordinate system x_iy_iz_i In process；Represent by sphere j by rotationally symmetric body basic function electromagnetic current coefficient by local coordinate system x_jy_jz_jIt is transformed into Local coordinate system x_ij"y_ij"z_ij" in process；Represent by sphere i by rotationally symmetric body basic function electromagnetic current coefficient by office Portion coordinate system x_ij"y_ij"z_ij" it is transformed into local coordinate system x_iy_iz_iIn process.

Use the above-mentioned equation of GMRES iteratives（27）, it is 1.0e-3 that precision is blocked in general setting.

Equivalent scattering electromagnetic current coefficient on each equivalent sphere is it has been determined that following equation is true in global coordinate system xyz Determine Radar Cross Section σ：

Wherein E^sFor far field scattered field, there is the equivalent electromagnetic current solved to be calculated, EⁱRepresent far field in-field.

Embodiment 1

According to the multiple rotary equivalent simulation method of Complex multi-target electromagnetic scattering of the present invention, we are to a reality The electromagnetic scattering of one group of guided missile is emulated in the free space of border, as shown in Figure 3.The each posture of guided missile in free space is each Differ, but they have duplicate construction.The center of each sub-regions is：（0,0,0）,（- 2m, -4m, 10m）,（- 7m, 0, -2m）,（7m, 2m, 0）,（5m, 4m, 10m）, it is with reference to point coordinates（0,0,1m）_φ^rot=90 °,（0,1m, 2.7474m）,（0,0m, 1m）_φ^rot=50 °,（0,0m, 1.732m）_φ^rot=40°.Plane wave incidence, incident angle are（θ^inc =0 °, φ^inc=0 °）.As Fig. 5 establishes equivalent cylinder and sphere the subdivision information of regional respectively.By the inventive method, Radar Cross Section accurately is calculated, as shown in Figure 8.

In summary, the multiple rotary equivalent simulation method of Complex multi-target electromagnetic scattering of the present invention, utilizes Region Decomposition Thought propose the equivalent emulation mode of multiple rotary, this method can inherit the high-efficient characteristic of rotationally symmetric body moment method, And can analyzes the electromagnetic scattering problems of general Complex multi-target problem.

Claims

1. a kind of multiple rotary equivalent simulation method of Complex multi-target electromagnetic scattering, it is characterised in that the Complex multi-target It is as follows including at least two complex targets, step：

1st step, establish the grid model of each sub-goal：The circle for just surrounding the sub-goal completely is established outside each sub-goal Post, the sphere for just surrounding the cylinder completely is established outside each cylinder, obtain subregion corresponding with each sub-goal；

2nd step, in the corresponding local coordinate system of each sub-regions, determine to be based on rotationally symmetric body base letter between sphere and cylinder Inside several transfer matrixes and cylinder and the cylinder curved surface quadrangle high-order hierarchical basis functions and three are based between specific item standard type The collision matrix of angular RWG basic functions；

4th step, it is determined that the coefficient transition matrix based on curved surface quadrangle high-order hierarchical basis functions, based on rotationally symmetric body basic function Coefficient transition matrix, search the corresponding relation between cylinder top-surface camber quadrangle numbering and bus segment number；

5th step, driving source is set, and incidence wave is uniform plane wave；

2. the multiple rotary equivalent simulation method of Complex multi-target electromagnetic scattering according to claim 1, it is characterised in that The grid model of each sub-goal is established described in 1st step, detailed process is as follows：

(1.1) using the discrete each sub-goal surface of triangular element；

(1.2) cylinder for just surrounding the sub-goal completely is established outside each sub-goal, using curved surface quadrilateral units to circle Post surface carries out bus that is discrete, while using the discrete cylinder of line segment；

(1.3) sphere for just surrounding the cylinder completely is established outside each cylinder, using the bus of the discrete sphere of line segment, is established Global coordinate system and local coordinate system, it is an independent subregion to define the region surrounded by sphere.

3. the multiple rotary equivalent simulation method of Complex multi-target electromagnetic scattering according to claim 1, it is characterised in that Described in 2nd step in the corresponding local coordinate system of each sub-regions, determine to be based on rotationally symmetric body base between sphere and cylinder Inside the transfer matrix and cylinder of function and the cylinder between specific item standard type based on curved surface quadrangle high-order hierarchical basis functions and The collision matrix of triangle RWG basic functions, detailed process are as follows：

The definition of (2.1) three kinds of basic functions：

RWG basic functions are defined on triangle surface subdivision unitThe definition rotation pair on rotationally symmetric body bus section, busbar section Claim body basic functionHigh-order hierarchical basis functions are defined on curved surface quad patch subdivision unit

(2.2) local coordinate system x is established_i'y_i'z_i', dissipating between equivalent cylinder and sub-goal is determined in the local coordinate system Penetrate matrix S_cp, cylinder to sphere transfer matrix Z_sc, sphere to cylinder transfer matrix Z_cs：

(a) spinning solution and spin matrix between local coordinate system

If global coordinate system is xyz, respectively with the central point O of each sub-regions_iFor origin, establish parallel to global coordinate system Local coordinate system x_iy_iz_i, by a reference point P in space by rotating another local coordinate system needed, it is designated as x_i ^py_i ^pz_i ^pIf the coordinate of reference point is (x₀,y₀,z₀), then：

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Wherein, θ^rotAnd φ^rotFor rotary reference angle, according to θ^rotAnd φ^rot, according to following two step by local coordinate system x_iy_iz_i Rotation obtains local rectangular coordinate system x_i ^py_i ^pz_i ^p：

1. with local coordinate system x_iy_iz_iZ-axis be rotary shaft, turn clockwiseSo that local coordinate system x_iy_iz_iY Axle passes through subpoint P' of the P points in xoy planes, obtains local coordinate system x_i ^py_i ^pz_i ^pX-axis position；

2. turned clockwise θ using the x-axis that obtains in 1. as rotary shaft^rotSo that local coordinate system x_iy_iz_iZ-axis pass through P points, obtain To local coordinate system x_i ^py_i ^pz_i ^pZ-axis；

According to arbitrfary point Q in space after being rotated with upper type in local coordinate system x_iy_iz_iAnd x_i ^py_i ^pz_i ^pIn coordinate (x_qi,y_qi, z_qi) and (x_qi ^p,y_qi ^p,z_qi ^p) between relation be：

Wherein coordinate system x_iy_iz_iTo x_i ^py_i ^pz_i ^pSpin matrix R^SLFor：

<mrow> <msup> <mi>R</mi> <mrow> <mi>S</mi> <mi>L</mi> </mrow> </msup> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msup> <mi>cos&phi;</mi> <mrow> <mi>r</mi> <mi>o</mi> <mi>t</mi> </mrow> </msup> </mrow> </mtd> <mtd> <mrow> <mo>-</mo> <msup> <mi>cos&phi;</mi> <mrow> <mi>r</mi> <mi>o</mi> <mi>t</mi> </mrow> </msup> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <msup> <mi>cos&theta;</mi> <mrow> <mi>r</mi> <mi>o</mi> <mi>t</mi> </mrow> </msup> <msup> <mi>cos&phi;</mi> <mrow> <mi>r</mi> <mi>o</mi> <mi>t</mi> </mrow> </msup> </mrow> </mtd> <mtd> <mrow> <msup> <mi>cos&theta;</mi> <mrow> <mi>r</mi> <mi>o</mi> <mi>t</mi> </mrow> </msup> <msup> <mi>sin&phi;</mi> <mrow> <mi>r</mi> <mi>o</mi> <mi>t</mi> </mrow> </msup> </mrow> </mtd> <mtd> <mrow> <mo>-</mo> <msup> <mi>sin&theta;</mi> <mrow> <mi>r</mi> <mi>o</mi> <mi>t</mi> </mrow> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msup> <mi>sin&theta;</mi> <mrow> <mi>r</mi> <mi>o</mi> <mi>t</mi> </mrow> </msup> <msup> <mi>cos&phi;</mi> <mrow> <mi>r</mi> <mi>o</mi> <mi>t</mi> </mrow> </msup> </mrow> </mtd> <mtd> <mrow> <msup> <mi>sin&theta;</mi> <mrow> <mi>r</mi> <mi>o</mi> <mi>t</mi> </mrow> </msup> <msup> <mi>sin&phi;</mi> <mrow> <mi>r</mi> <mi>o</mi> <mi>t</mi> </mrow> </msup> </mrow> </mtd> <mtd> <mrow> <msup> <mi>cos&theta;</mi> <mrow> <mi>r</mi> <mi>o</mi> <mi>t</mi> </mrow> </msup> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow>

Coordinate system x_i ^py_i ^pz_i ^pTo x_iy_iz_iSpin matrix R^LSFor：

<mrow> <msup> <mi>R</mi> <mrow> <mi>L</mi> <mi>S</mi> </mrow> </msup> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msup> <mi>sin&phi;</mi> <mrow> <mi>r</mi> <mi>o</mi> <mi>t</mi> </mrow> </msup> </mrow> </mtd> <mtd> <mrow> <msup> <mi>cos&theta;</mi> <mrow> <mi>r</mi> <mi>o</mi> <mi>t</mi> </mrow> </msup> <msup> <mi>cos&phi;</mi> <mrow> <mi>r</mi> <mi>o</mi> <mi>t</mi> </mrow> </msup> </mrow> </mtd> <mtd> <mrow> <msup> <mi>sin&theta;</mi> <mrow> <mi>r</mi> <mi>o</mi> <mi>t</mi> </mrow> </msup> <msup> <mi>cos&phi;</mi> <mrow> <mi>r</mi> <mi>o</mi> <mi>t</mi> </mrow> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <msup> <mi>cos&phi;</mi> <mrow> <mi>r</mi> <mi>o</mi> <mi>t</mi> </mrow> </msup> </mrow> </mtd> <mtd> <mrow> <msup> <mi>cos&theta;</mi> <mrow> <mi>r</mi> <mi>o</mi> <mi>t</mi> </mrow> </msup> <msup> <mi>sin&phi;</mi> <mrow> <mi>r</mi> <mi>o</mi> <mi>t</mi> </mrow> </msup> </mrow> </mtd> <mtd> <mrow> <msup> <mi>sin&theta;</mi> <mrow> <mi>r</mi> <mi>o</mi> <mi>t</mi> </mrow> </msup> <msup> <mi>sin&phi;</mi> <mrow> <mi>r</mi> <mi>o</mi> <mi>t</mi> </mrow> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>-</mo> <msup> <mi>sin&theta;</mi> <mrow> <mi>r</mi> <mi>o</mi> <mi>t</mi> </mrow> </msup> </mrow> </mtd> <mtd> <mrow> <msup> <mi>cos&theta;</mi> <mrow> <mi>r</mi> <mi>o</mi> <mi>t</mi> </mrow> </msup> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow>

If reference point falls in local coordinate system x_iy_iz_iZ-axis on, then need specify φ^rot, work as φ^rotBeing represented at=90 ° need not Rotation；

(b) local coordinate system x is established_i'y_i'z_i', the scattering between equivalent cylinder and sub-goal is determined in the local coordinate system Matrix S^cp：

In following formula, footmark c represents the face of cylinder, and s represents sphere, and p represents internal sub-goal surface；Cs represents sphere s relative to circle Cylinder c, sc, cp, pc, pp, ss analogize；

It is reference point P to select any point on column rotating shaft, and local coordinate system x is obtained according to the spinning solution in (a)_i'y_i' z_i', note spin matrix isWithThe collision matrix S between equivalent cylinder and sub-goal is determined in the coordinate system^cp, its Electromagnetic current on central column face, which uses, is based on curved surface quadrangle high-order hierarchical basis functionsExpansion, and the electric current on sub-goal is then Using the RWG base function expansions based on triangle, expression is：

S^cp=Z^cp[Z^pp]^-1Z^pc (6)

Wherein：

<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mrow> <mo>&lsqb;</mo> <msup> <mi>Z</mi> <mrow> <mi>c</mi> <mi>p</mi> </mrow> </msup> <mo>&rsqb;</mo> </mrow> <mrow> <mi>m</mi> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mo><</mo> <msubsup> <mi>f</mi> <mi>m</mi> <mrow> <mi>H</mi> <mi>O</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> <mo>,</mo> <mo>-</mo> <mover> <mi>n</mi> <mo>^</mo> </mover> <mo>&times;</mo> <msub> <mi>K</mi> <mrow> <mi>c</mi> <mi>p</mi> </mrow> </msub> <mo>,</mo> <msubsup> <mi>f</mi> <mi>n</mi> <mrow> <mi>R</mi> <mi>W</mi> <mi>G</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <msup> <mi>r</mi> <mo>&prime;</mo> </msup> <mo>)</mo> </mrow> <mo>></mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo><</mo> <msubsup> <mi>f</mi> <mi>m</mi> <mrow> <mi>H</mi> <mi>O</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> <mo>,</mo> <mo>-</mo> <mfrac> <mn>1</mn> <mi>&eta;</mi> </mfrac> <mover> <mi>n</mi> <mo>^</mo> </mover> <mo>&times;</mo> <msub> <mi>L</mi> <mrow> <mi>c</mi> <mi>p</mi> </mrow> </msub> <mo>,</mo> <msubsup> <mi>f</mi> <mi>n</mi> <mrow> <mi>R</mi> <mi>W</mi> <mi>G</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <msup> <mi>r</mi> <mo>&prime;</mo> </msup> <mo>)</mo> </mrow> <mo>></mo> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mrow> <mo>&lsqb;</mo> <msup> <mi>Z</mi> <mrow> <mi>p</mi> <mi>p</mi> </mrow> </msup> <mo>&rsqb;</mo> </mrow> <mrow> <mi>m</mi> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mo>&lsqb;</mo> <mo><</mo> <msubsup> <mi>f</mi> <mi>m</mi> <mrow> <mi>R</mi> <mi>W</mi> <mi>G</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> <mo>,</mo> <msub> <mi>L</mi> <mrow> <mi>p</mi> <mi>p</mi> </mrow> </msub> <mo>,</mo> <msubsup> <mi>f</mi> <mi>n</mi> <mrow> <mi>R</mi> <mi>W</mi> <mi>G</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <msup> <mi>r</mi> <mo>&prime;</mo> </msup> <mo>)</mo> </mrow> <mo>></mo> <mo>&rsqb;</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mrow> <mo>&lsqb;</mo> <msup> <mi>Z</mi> <mrow> <mi>p</mi> <mi>c</mi> </mrow> </msup> <mo>&rsqb;</mo> </mrow> <mrow> <mi>m</mi> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mo>&lsqb;</mo> <mtable> <mtr> <mtd> <mrow> <mo><</mo> <msubsup> <mi>f</mi> <mi>m</mi> <mrow> <mi>R</mi> <mi>W</mi> <mi>G</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> <mo>,</mo> <msub> <mi>L</mi> <mrow> <mi>p</mi> <mi>c</mi> </mrow> </msub> <mo>,</mo> <msubsup> <mi>f</mi> <mi>n</mi> <mrow> <mi>H</mi> <mi>O</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <msup> <mi>r</mi> <mo>&prime;</mo> </msup> <mo>)</mo> </mrow> <mo>></mo> </mrow> </mtd> <mtd> <mrow> <mo><</mo> <msubsup> <mi>f</mi> <mi>m</mi> <mrow> <mi>R</mi> <mi>W</mi> <mi>G</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> <mo>,</mo> <msub> <mi>&eta;K</mi> <mrow> <mi>p</mi> <mi>c</mi> </mrow> </msub> <mo>,</mo> <msubsup> <mi>f</mi> <mi>n</mi> <mrow> <mi>H</mi> <mi>O</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <msup> <mi>r</mi> <mo>&prime;</mo> </msup> <mo>)</mo> </mrow> <mo>></mo> </mrow> </mtd> </mtr> </mtable> <mo>&rsqb;</mo> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow>

Define integro-differential operator L, K：

Wherein

<mrow> <mtable> <mtr> <mtd> <mrow> <mi>A</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <munder> <mo>&Integral;</mo> <mrow> <mo>&part;</mo> <mi>&Omega;</mi> </mrow> </munder> <msup> <mi>xgds</mi> <mo>&prime;</mo> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>&psi;</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <munder> <mo>&Integral;</mo> <mrow> <mo>&part;</mo> <mi>&Omega;</mi> </mrow> </munder> <mrow> <mo>(</mo> <mo>&dtri;</mo> <mo>&CenterDot;</mo> <mi>x</mi> <mo>)</mo> </mrow> <msup> <mi>gds</mi> <mo>&prime;</mo> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>C</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>p</mi> <mi>v</mi> <munder> <mo>&Integral;</mo> <mrow> <mo>&part;</mo> <mi>&Omega;</mi> </mrow> </munder> <mi>x</mi> <mo>&times;</mo> <msup> <mo>&dtri;</mo> <mo>&prime;</mo> </msup> <msup> <mi>gds</mi> <mo>&prime;</mo> </msup> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>9</mn> <mo>)</mo> </mrow> </mrow>

Wherein x represents electric current J or magnetic current M, A (x) represent vector position, and ψ (x) represents scalar potential, and C (x) represents curl field, and pv is represented Principal value integral, η are plane wave impedance,For unit normal vector, g represents source point r' to site r Green's function, and its expression formula is：

<mrow> <mi>g</mi> <mrow> <mo>(</mo> <mi>r</mi> <mo>,</mo> <msup> <mi>r</mi> <mo>&prime;</mo> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>j</mi> <mi>k</mi> <mrow> <mo>|</mo> <mrow> <mi>r</mi> <mo>-</mo> <msup> <mi>r</mi> <mo>&prime;</mo> </msup> </mrow> <mo>|</mo> </mrow> </mrow> </msup> <mrow> <mo>|</mo> <mrow> <mi>r</mi> <mo>-</mo> <msup> <mi>r</mi> <mo>&prime;</mo> </msup> </mrow> <mo>|</mo> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>10</mn> <mo>)</mo> </mrow> </mrow>

(c) in local coordinate system x_i'y_i'z_i' in determine cylinder to sphere transfer matrix Z^sc, sphere to cylinder transfer matrix Z^cs：

Electromagnetic current on cylinder uses BoR basic functionsDeploy, the electromagnetic current on sphere uses BoR basic functions Expansion, r are site, and r' is source point, transfer matrix Z^scSource on cylinder caused equivalent source on sphere is described, it is expressed Formula is：

<mrow> <msub> <mrow> <mo>&lsqb;</mo> <msup> <mi>Z</mi> <mrow> <mi>s</mi> <mi>c</mi> </mrow> </msup> <mo>&rsqb;</mo> </mrow> <mrow> <mi>m</mi> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mo><</mo> <msubsup> <mi>f</mi> <mrow> <mi>s</mi> <mo>,</mo> <mi>m</mi> </mrow> <mrow> <mi>B</mi> <mi>o</mi> <mi>R</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> <mo>,</mo> <mo>-</mo> <mover> <mi>n</mi> <mo>^</mo> </mover> <mo>&times;</mo> <msub> <mi>K</mi> <mrow> <mi>s</mi> <mi>c</mi> </mrow> </msub> <mo>,</mo> <msubsup> <mi>f</mi> <mrow> <mi>c</mi> <mo>,</mo> <mi>n</mi> </mrow> <mrow> <mi>B</mi> <mi>o</mi> <mi>R</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <msup> <mi>r</mi> <mo>&prime;</mo> </msup> <mo>)</mo> </mrow> <mo>></mo> </mrow> </mtd> <mtd> <mrow> <mo><</mo> <msubsup> <mi>f</mi> <mrow> <mi>s</mi> <mo>,</mo> <mi>m</mi> </mrow> <mrow> <mi>B</mi> <mi>o</mi> <mi>R</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> <mo>,</mo> <mfrac> <mn>1</mn> <mi>&eta;</mi> </mfrac> <mover> <mi>n</mi> <mo>^</mo> </mover> <mo>&times;</mo> <msub> <mi>L</mi> <mrow> <mi>s</mi> <mi>c</mi> </mrow> </msub> <mo>,</mo> <msubsup> <mi>f</mi> <mrow> <mi>c</mi> <mo>,</mo> <mi>n</mi> </mrow> <mrow> <mi>B</mi> <mi>o</mi> <mi>R</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <msup> <mi>r</mi> <mo>&prime;</mo> </msup> <mo>)</mo> </mrow> <mo>></mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo><</mo> <msubsup> <mi>f</mi> <mrow> <mi>s</mi> <mo>,</mo> <mi>m</mi> </mrow> <mrow> <mi>B</mi> <mi>o</mi> <mi>R</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> <mo>,</mo> <mo>-</mo> <mfrac> <mn>1</mn> <mi>&eta;</mi> </mfrac> <mover> <mi>n</mi> <mo>^</mo> </mover> <mo>&times;</mo> <msub> <mi>L</mi> <mrow> <mi>s</mi> <mi>c</mi> </mrow> </msub> <mo>,</mo> <msubsup> <mi>f</mi> <mrow> <mi>c</mi> <mo>,</mo> <mi>n</mi> </mrow> <mrow> <mi>B</mi> <mi>o</mi> <mi>R</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <msup> <mi>r</mi> <mo>&prime;</mo> </msup> <mo>)</mo> </mrow> <mo>></mo> </mrow> </mtd> <mtd> <mrow> <mo><</mo> <msubsup> <mi>f</mi> <mrow> <mi>s</mi> <mo>,</mo> <mi>m</mi> </mrow> <mrow> <mi>B</mi> <mi>o</mi> <mi>R</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> <mo>,</mo> <mo>-</mo> <mover> <mi>n</mi> <mo>^</mo> </mover> <mo>&times;</mo> <msub> <mi>K</mi> <mrow> <mi>s</mi> <mi>c</mi> </mrow> </msub> <mo>,</mo> <msubsup> <mi>f</mi> <mrow> <mi>c</mi> <mo>,</mo> <mi>n</mi> </mrow> <mrow> <mi>B</mi> <mi>o</mi> <mi>R</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <msup> <mi>r</mi> <mo>&prime;</mo> </msup> <mo>)</mo> </mrow> <mo>></mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>11</mn> <mo>)</mo> </mrow> </mrow>

<mrow> <msub> <mrow> <mo>&lsqb;</mo> <msup> <mi>Z</mi> <mrow> <mi>c</mi> <mi>s</mi> </mrow> </msup> <mo>&rsqb;</mo> </mrow> <mrow> <mi>m</mi> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mo><</mo> <msubsup> <mi>f</mi> <mrow> <mi>c</mi> <mo>,</mo> <mi>m</mi> </mrow> <mrow> <mi>B</mi> <mi>o</mi> <mi>R</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> <mo>,</mo> <mo>-</mo> <mover> <mi>n</mi> <mo>^</mo> </mover> <mo>&times;</mo> <msub> <mi>K</mi> <mrow> <mi>c</mi> <mi>s</mi> </mrow> </msub> <mo>,</mo> <msubsup> <mi>f</mi> <mrow> <mi>s</mi> <mo>,</mo> <mi>n</mi> </mrow> <mrow> <mi>B</mi> <mi>o</mi> <mi>R</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <msup> <mi>r</mi> <mo>&prime;</mo> </msup> <mo>)</mo> </mrow> <mo>></mo> </mrow> </mtd> <mtd> <mrow> <mo><</mo> <msubsup> <mi>f</mi> <mrow> <mi>c</mi> <mo>,</mo> <mi>m</mi> </mrow> <mrow> <mi>B</mi> <mi>o</mi> <mi>R</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> <mo>,</mo> <mfrac> <mn>1</mn> <mi>&eta;</mi> </mfrac> <mover> <mi>n</mi> <mo>^</mo> </mover> <mo>&times;</mo> <msub> <mi>L</mi> <mrow> <mi>c</mi> <mi>s</mi> </mrow> </msub> <mo>,</mo> <msubsup> <mi>f</mi> <mrow> <mi>s</mi> <mo>,</mo> <mi>n</mi> </mrow> <mrow> <mi>B</mi> <mi>o</mi> <mi>R</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <msup> <mi>r</mi> <mo>&prime;</mo> </msup> <mo>)</mo> </mrow> <mo>></mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo><</mo> <msubsup> <mi>f</mi> <mrow> <mi>c</mi> <mo>,</mo> <mi>m</mi> </mrow> <mrow> <mi>B</mi> <mi>o</mi> <mi>R</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> <mo>,</mo> <mo>-</mo> <mfrac> <mn>1</mn> <mi>&eta;</mi> </mfrac> <mover> <mi>n</mi> <mo>^</mo> </mover> <mo>&times;</mo> <msub> <mi>L</mi> <mrow> <mi>c</mi> <mi>s</mi> </mrow> </msub> <mo>,</mo> <msubsup> <mi>f</mi> <mrow> <mi>s</mi> <mo>,</mo> <mi>n</mi> </mrow> <mrow> <mi>B</mi> <mi>o</mi> <mi>R</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <msup> <mi>r</mi> <mo>&prime;</mo> </msup> <mo>)</mo> </mrow> <mo>></mo> </mrow> </mtd> <mtd> <mrow> <mo><</mo> <msubsup> <mi>f</mi> <mrow> <mi>c</mi> <mo>,</mo> <mi>m</mi> </mrow> <mrow> <mi>B</mi> <mi>o</mi> <mi>R</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> <mo>,</mo> <mo>-</mo> <mover> <mi>n</mi> <mo>^</mo> </mover> <mo>&times;</mo> <msub> <mi>K</mi> <mrow> <mi>c</mi> <mi>s</mi> </mrow> </msub> <mo>,</mo> <msubsup> <mi>f</mi> <mrow> <mi>s</mi> <mo>,</mo> <mi>n</mi> </mrow> <mrow> <mi>B</mi> <mi>o</mi> <mi>R</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <msup> <mi>r</mi> <mo>&prime;</mo> </msup> <mo>)</mo> </mrow> <mo>></mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>12</mn> <mo>)</mo> </mrow> <mo>.</mo> </mrow>

4. the multiple rotary equivalent simulation method of Complex multi-target electromagnetic scattering according to claim 3, it is characterised in that Transmission matrix based on rotationally symmetric body basic function between determination sphere described in 3rd step, detailed process are as follows：

Local coordinate system to be rotated is x_iy_iz_i, reference point elects sphere j centre of sphere O as_j, (a) in step (2.2) is obtained Coordinate system x_ij"y_ij"z_ij", remember that corresponding spin matrix isWithIn local coordinate system x_ij"y_ij"z_ij" in, between sphere Transmission matrixFormula it is as follows：

<mrow> <msub> <mrow> <mo>&lsqb;</mo> <msubsup> <mi>T</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> <mrow> <mi>s</mi> <mi>s</mi> </mrow> </msubsup> <mo>&rsqb;</mo> </mrow> <mrow> <mi>m</mi> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mo><</mo> <msubsup> <mi>f</mi> <mrow> <mi>s</mi> <mo>,</mo> <mi>m</mi> </mrow> <mrow> <mi>B</mi> <mi>o</mi> <mi>R</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> <mo>,</mo> <mo>-</mo> <mover> <mi>n</mi> <mo>^</mo> </mover> <mo>&times;</mo> <msub> <mi>K</mi> <mrow> <mi>s</mi> <mi>s</mi> </mrow> </msub> <mo>,</mo> <msubsup> <mi>f</mi> <mrow> <mi>s</mi> <mo>,</mo> <mi>n</mi> </mrow> <mrow> <mi>B</mi> <mi>o</mi> <mi>R</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <msup> <mi>r</mi> <mo>&prime;</mo> </msup> <mo>)</mo> </mrow> <mo>></mo> </mrow> </mtd> <mtd> <mrow> <mo><</mo> <msubsup> <mi>f</mi> <mrow> <mi>s</mi> <mo>,</mo> <mi>m</mi> </mrow> <mrow> <mi>B</mi> <mi>o</mi> <mi>R</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> <mo>,</mo> <mfrac> <mn>1</mn> <mi>&eta;</mi> </mfrac> <mover> <mi>n</mi> <mo>^</mo> </mover> <mo>&times;</mo> <msub> <mi>L</mi> <mrow> <mi>s</mi> <mi>s</mi> </mrow> </msub> <mo>,</mo> <msubsup> <mi>f</mi> <mrow> <mi>s</mi> <mo>,</mo> <mi>n</mi> </mrow> <mrow> <mi>B</mi> <mi>o</mi> <mi>R</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <msup> <mi>r</mi> <mo>&prime;</mo> </msup> <mo>)</mo> </mrow> <mo>></mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo><</mo> <msubsup> <mi>f</mi> <mrow> <mi>s</mi> <mo>,</mo> <mi>m</mi> </mrow> <mrow> <mi>B</mi> <mi>o</mi> <mi>R</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> <mo>,</mo> <mo>-</mo> <mfrac> <mn>1</mn> <mi>&eta;</mi> </mfrac> <mover> <mi>n</mi> <mo>^</mo> </mover> <mo>&times;</mo> <msub> <mi>L</mi> <mrow> <mi>s</mi> <mi>s</mi> </mrow> </msub> <mo>,</mo> <msubsup> <mi>f</mi> <mrow> <mi>s</mi> <mo>,</mo> <mi>n</mi> </mrow> <mrow> <mi>B</mi> <mi>o</mi> <mi>R</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <msup> <mi>r</mi> <mo>&prime;</mo> </msup> <mo>)</mo> </mrow> <mo>></mo> </mrow> </mtd> <mtd> <mrow> <mo><</mo> <msubsup> <mi>f</mi> <mrow> <mi>s</mi> <mo>,</mo> <mi>m</mi> </mrow> <mrow> <mi>B</mi> <mi>o</mi> <mi>R</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> <mo>,</mo> <mo>-</mo> <mover> <mi>n</mi> <mo>^</mo> </mover> <mo>&times;</mo> <msub> <mi>K</mi> <mrow> <mi>s</mi> <mi>s</mi> </mrow> </msub> <mo>,</mo> <msubsup> <mi>f</mi> <mrow> <mi>s</mi> <mo>,</mo> <mi>n</mi> </mrow> <mrow> <mi>B</mi> <mi>o</mi> <mi>R</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <msup> <mi>r</mi> <mo>&prime;</mo> </msup> <mo>)</mo> </mrow> <mo>></mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>13</mn> <mo>)</mo> </mrow> <mo>.</mo> </mrow>

5. the multiple rotary equivalent simulation method of Complex multi-target electromagnetic scattering according to claim 1, it is characterised in that The corresponding relation between cylinder top-surface camber quadrangle numbering and bus segment number is searched described in 4th step, i.e., when known high-order is folded When layer basic function coefficient seeks rotationally symmetric body basic function coefficient, search and record rotationally symmetric body basic function integration Gauss point correspondingly Curved surface quadrangle numbering, the process of lookup is as follows：

(4.1) it is grouped first by Octree, calculates the group where Gauss point；

(4.2) and then the Gauss point is searched out by preset angle configuration and corresponds to curved surface quadrangle numbering；

(4.3) respectively along u in the parameter coordinate system of the curved surface quadrangle, v both directions sample in [- 1,1], it is determined that adopting Coordinate of the sampling point under former coordinate system, the parameter coordinate closest to the sampled point of Gauss point is recorded, is determined under parameter coordinate system The electromagnetic current closest to the sampled point of Gauss point.

6. the multiple rotary equivalent simulation method of Complex multi-target electromagnetic scattering according to claim 1, it is characterised in that Equation group is established described in 6th step, equation group is solved and obtains the equivalent electromagnetic current on each sub-regions sphere, detailed process is such as Under：

Wherein, i=1 ..., N_obj, j=1 ..., N_obj,i≠j；N_objTotal subregion number is represented, i, j represent subarea number, Represent rotationally symmetric body basic function electromagnetic current coefficient by local coordinate system x_iy_iz_iIt is transformed into local coordinate system x_i'y_i'z_i' in Process；Expression is transformed into the process of high-order hierarchical basis functions coefficient by rotationally symmetric body basic function electromagnetic current coefficient；Expression is transformed into the process of rotationally symmetric body basic function coefficient by high-order hierarchical basis functions electromagnetic current coefficient；Represent By rotationally symmetric body basic function electromagnetic current coefficient by local coordinate system x_i'y_i'z_i' it is transformed into local coordinate system x_iy_iz_iIn mistake Journey；Represent by sphere j by rotationally symmetric body basic function electromagnetic current coefficient by local coordinate system x_jy_jz_jIt is transformed into local seat Mark system x_ij"y_ij"z_ij" in process；Represent by sphere i by rotationally symmetric body basic function electromagnetic current coefficient by local coordinate It is x_ij"y_ij"z_ij" it is transformed into local coordinate system x_iy_iz_iIn process.