CN107944113A - A kind of method for calculating three-dimensional high-speed translation Electromagnetic Scattering of Target field - Google Patents
A kind of method for calculating three-dimensional high-speed translation Electromagnetic Scattering of Target field Download PDFInfo
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Abstract
The present invention provides a kind of method for calculating three-dimensional high-speed translation Electromagnetic Scattering of Target field, it is related to Electromagnetic Calculation field, the present invention is using amygdaloid body as target, design is established relative to the static kinetic coordinate system of target, amygdaloid body grid model is generated, freely choosing for time step and spatial mesh size is realized, then introduces incidence wave in relative motion coordinate system, the electromagnetic scatter fields in relative motion coordinate system, the electromagnetic scatter fields being finally converted under rest frame are calculated with FDTD.Present invention introduces relative to the static kinetic coordinate system of amygdaloid body target, incidence wave is introduced, includes amplitude distribution, frequency distribution and phase distribution；Solve the transformational relation between two coordinate systems, using these transformational relations, obtained electromagnetic scatter fields value of the amygdaloid body target under high speed translation state；The advantages of the method for the present invention is novel, simple to operation, three dimensions complex targets high speed translation electromagnetic scatter fields more suitable for processing.
Description
Technical Field
The invention relates to the field of electromagnetic field calculation, in particular to a method for calculating an electromagnetic scattering field.
Background
With the flying speed of aviation devices becoming faster and faster, electromagnetic research on high-speed moving objects is becoming a hot problem. However, with the high-speed movement of the object, the electric field scattered field of the object in the moving state is also changed according to the theory of relativity.
A numerical method for calculating electromagnetic scattering of a moving object begins to appear, and up to now, some scholars propose a numerical method for electromagnetic scattering of a moving object, such as an equivalent surface current method, a moving boundary method, and the like. The traditional method has the defects of coordinate system limitation, single boundary condition or single solving result or incapability of adapting to complex and various shapes of three-dimensional targets and the like, and is further not suitable for electromagnetic field value research of high-speed or ultra-high-speed moving objects. Therefore, it is very necessary to design a technology capable of solving the problem of the high-speed translation electromagnetic field value of the complex target.
Disclosure of Invention
In order to overcome the defects of the prior art, the invention provides a technical scheme for calculating the electromagnetic scattering field of a three-dimensional high-speed translation target. The invention takes the almond body as a target, and designs and establishes a motion coordinate system which is static relative to the target. And (2) generating an almond body grid model by using Fortran language processing, realizing free selection of time step length and space step length, introducing incident waves into a relative motion coordinate system, calculating an electromagnetic scattering field in the relative motion coordinate system by using FDTD, and finally converting the electromagnetic scattering field into an electromagnetic scattering field in a static coordinate system.
The technical scheme adopted by the invention for solving the technical problem comprises the following steps:
the method comprises the following steps: establishing a motion coordinate system K 'which is static relative to the high-speed translation target, and determining the characteristics of the motion coordinate system K';
step 1.1: establishing a relative motion coordinate system K ', wherein the relative motion coordinate system K ' is static relative to a target of high-speed translation, namely the relative motion coordinate system K ' has the same motion speed and motion direction with the target;
step 1.2: establishing a relation between space step lengths delta x ', delta y', delta z 'relative to the moving coordinate system K' and space step lengths delta x, delta y and delta z of the static coordinate system K, and a relation between time step length delta t 'relative to the moving coordinate system K' and delta t of the static coordinate system; the target has the same speed magnitude v and motion angle theta 'as the relative motion coordinate system K'v,Wherein theta'vRepresenting the included angle between the motion direction and the positive semi-axis of the z axis, ranging from 0 to 180 degrees,representing the included angle between the projection of the motion direction to the xoy plane and the positive half shaft of the x axis, the range is 0-360 degrees, and the angle isThe spatial step size Δ x ', Δ y', Δ z 'and Δ x, Δ y, Δ z relationships, and the temporal step size Δ t' and Δ t relationships can be obtained according to equation (1):
where c is the speed of light, x, y, z and t are the spatial and temporal components in the stationary coordinate system K, x ', y ', z ' and t ' are the spatial and temporal components in the relative motion coordinate system K ', v is the target motion speed, and the components of the speed in the x, y and z directions are respectivelyvz＝vcosθvIn the formula
Step 1.3: according to Courant stability conditionsDetermining the space step length delta x ', delta y', delta z 'and the time step length delta t' in the relative motion coordinate system K ', and thus finishing establishing the motion coordinate system K' which is static relative to the high-speed translation target;
step two: establishing an almond body grid model;
step 2.1: the method is characterized in that a computer is used for modeling, an almond body is selected as a three-dimensional high-speed translation target, and a control equation for establishing a mathematical model is shown as a formula (2):
wherein psi is an angle quantity for controlling the trajectory of the shape of the almond body in the yoz plane, and the value range-pi is not less than psi is not more than pi; d is the total length of the almond body along the x direction; a is1And a2The positive half shaft and the negative half shaft of the almond body respectively account for the proportionality coefficient of the total length in the x direction and satisfy a1+a2＝1；b1And b2The coefficients for controlling the length of the almond body in the x negative half axis and the x positive half axis in the y direction and the z direction respectively; n is a reduction coefficient for controlling the length of the almond body in the z direction, and n is an integer greater than zero; c. C1And c2For controlling the reduction coefficient of the length of the x positive half shaft of the almond body along the x direction, the requirement is metAnd 0 < c1< 1, thereby obtaining a positive x-axis halfForming a tip, b1And b2Satisfy | b1-b2(1-c1)|＜10-5；
The mathematical model is established according to the formula (2), and the size range of the almond body in the x direction is x epsilon (a)1×d,a2X d), the size range of the y direction is y epsilon (-b)1×d,b1X d) in the z-direction
Step 2.2: according to the size range of the almond body in the x, y and z directions and the size of the space step length of the relative motion coordinate system K' in the step 2.1, the grid number Is rounded by integer digits, the scattering boundary, the absorption boundary, the connection boundary and the total field boundary of the three-dimensional space are determined, wherein the scattering boundary belongs to the class i (Is min, Is max); j ∈ (Js min, Js max); k ∈ (Ks min, Ksmax); the ranges respectively representing the scattering boundaries in the x direction are (Is min, Is max), i.e. the minimum value and the maximum value in the x direction; the range of the scattering boundary in the y direction is (Js min, Js max), i.e. the minimum and maximum values in the y direction; the range of the scattering boundary in the z direction is (Ks min, Ks max), i.e., the minimum and maximum values in the z direction;
step 2.3: converting the absolute size into a grid number, if the position of a certain grid (i, j, k) in the space satisfies the mathematical model in step 2.1, that is, a grid with coordinates (i, j, k), the actual coordinates of the grid are (x, y, z) — (i × Δ x, j × Δ y, k × Δ z), where x, y, z respectively represent the actual position coordinate values of the grid, Δ x, Δ y, Δ z respectively represent the space step length, i, j, k are the grid number of the grid in the x, y, z direction, respectively, substituting the coordinate values (x, y, z) into the mathematical model in step 2.1, and if the values of x, y, z satisfy the relational expression of formula (2), the grid is inside the almond body, that is set as a metal; if the values of x, y and z do not meet the relational expression of the formula (2), the grid is outside the almond body and is set as air, and the rest is done in sequence, so that an almond body model in the space can be obtained, and the almond body model is divided into cuboid cellular models according to the grid, so that planar model data in the txt format is obtained;
step three: introducing an incident plane wave into a relative motion coordinate system K';
step 3.1: determining incident wave under a static coordinate system K, and defining amplitude, frequency and incident angle thetai，θiRepresenting the included angle between the incident direction and the positive half shaft of the z axis, ranging from 0 to 180 degrees,representing the included angle between the projection of the incident direction to the xoy plane and the positive half shaft of the x axis, ranging from 0 degrees to 360 degrees, and the incident wave satisfies the conditionWherein,is the electric field strength of the incident wave, E0Is the initial value of amplitude, t is time, omegaiIs the angular frequency, equal to 2 pi f, f is the frequency of the incident wave, kiIs wave number, equal to ωi/c；
Step 3.2: obtaining the amplitude E of the incident wave in the relative motion coordinate system K' and the static coordinate system K by using Lorentz transformation0And E'0Frequencies f and f', and angle of incidence θi,And theta'i,Introducing the incident wave into a relative motion coordinate system K' which is static relative to the almond body target at the total field boundary;
step four: calculating by using an FDTD algorithm to obtain electric field and magnetic field data in a motion coordinate system K';
step 4.1: inputting the running time step number time stop;
step 4.2: the moving speed v and the moving angle theta of the almond body are input'v,
Step 4.3: incident angle theta of input incident wavei,And converting it to theta'i,
Step 4.4: differential calculation of a three-dimensional FDTD algorithm is applied, and the electric field and the magnetic field are different by half time step, so that iterative solution is carried out on time until the time step number is finished from 1 to time stop, and the space electric field at each moment is obtainedAnd a magnetic fieldThe iterative formula is as follows:
wherein Ande ' representing electric field values of the grid node (i ', j ', K ') in the relative motion coordinate system K ' at time n +1 and time n ', respectively 'xThe components of the first and second images are,andare respectively represented inH ' of magnetic field values in relative motion coordinate system K ' of time mesh nodes (i ', j ', K ') and (i ', j ' -1, K ') 'zThe components of the first and second images are,andare respectively represented inH ' of magnetic field values in relative motion coordinate system K ' of time mesh nodes (i ', j ', K ') and (i ', j ', K ' -1) 'yThe components, ε (m) is the dielectric coefficient at spatial position m, σ (m) is the conductivity at spatial position m;
step five: converting the electric field E and the magnetic field H obtained in the step 4.4 into a static coordinate system K, and interpolating to obtain the high-speed translation electromagnetic scattering field of the almond body, wherein the detailed steps are as follows:
step 5.1: transforming the electromagnetic field value under the relative motion coordinate system K' into a static coordinate system K by combining Maxwell equation and Lorentz transformation; according to field value change Being parallel components with respect to the electric field in the motion coordinate system K',being the perpendicular component of the electric field in the relative motion coordinate system K',is the parallel component of the magnetic field and,is the vertical component of the magnetic field;
step 5.2: due to the influence of speed, the grid sizes of the relative motion coordinate system K ' and the stationary coordinate system K are not consistent, the electromagnetic field values in the stationary coordinate system K obtained in step 5.1 are converted from the field values in the relative motion coordinate system K ', the grid sizes corresponding to the field values are δ ' in the relative motion coordinate system K ', therefore, the grid sizes in the stationary coordinate system K are to be re-divided, that is, in the stationary coordinate system K, the electromagnetic field values of each spatial grid in the stationary coordinate system K are obtained by interpolation according to Δ x ═ Δ y ═ Δ z ═ Δ ' and then the grids are to be re-divided, three-dimensional interpolation is performed, a matlab engine is called by using format, the matlab contains functions of three-dimensional interpolation, namely, mesh 3, and the interpolation mode is linear, that is, linear interpolation is to obtain the electromagnetic field values;
step 5.3: and 5.2, obtaining the frequency distribution of the electromagnetic field value obtained in the step 5.2 by adopting an FFT algorithm, namely extracting the amplitude and the phase of the electromagnetic field value simultaneously.
The calculation aiming at the three-dimensional high-speed translation target comprises but not limited to a cube, a cuboid, a sphere, a cylinder, a cone, a prism and a round table, and the combination of the cube, the cuboid, the sphere, the cylinder, the cone, the prism and the round table; also includes conical spheres and polyhedrons.
The method has the advantages that the Fortran language is adopted for carrying out grid modeling, the moving coordinate system which is static relative to the almond body target is introduced, the incident wave is introduced, the field value in the moving coordinate system which is static relative to the almond body target is obtained through calculation, then field value transformation and interpolation are carried out, and finally the electromagnetic scattering field in the static coordinate system is obtained, wherein the electromagnetic scattering field comprises amplitude distribution, frequency distribution and phase distribution; the method solves the conversion relation between two coordinate systems, and obtains the electromagnetic scattering field value of the almond body target in a high-speed translation state by utilizing the conversion relation; the method is novel, does not relate to mathematical formulas which must be adopted by other numerical methods such as Green function, matrix, progressive function, basis function and the like, is simple and easy to operate, and is suitable for processing the advantages of more three-dimensional space complex target high-speed translation electromagnetic scattering fields.
Drawings
Fig. 1 is a schematic diagram of a motion coordinate system K' of the present invention that is stationary with respect to a translational target.
FIG. 2 is a cross-sectional display of the almond body modeling of the present invention.
Fig. 3 is a boundary diagram of the present invention.
FIG. 4 is a diagram of the computational process of the present invention.
FIG. 5 is a schematic illustration of interpolation according to the present invention.
FIG. 6 is an electric field distribution diagram of the XOY cross section of the present invention.
Fig. 7 is a plot of the frequency distribution of the XOY cross section of the present invention.
Fig. 8 is a phase profile of the XOY cross section of the present invention.
Detailed Description
The invention is further illustrated with reference to the following figures and examples.
A method for calculating an electromagnetic scattering field of a three-dimensional high-speed translation target comprises the following steps:
the method comprises the following steps: establishing a motion coordinate system K 'which is static relative to the high-speed translation target, and determining the characteristics of the motion coordinate system K';
step 1.1: establishing a relative motion coordinate system K ', wherein the relative motion coordinate system K ' is static relative to a target of high-speed translation, namely the relative motion coordinate system K ' has the same motion speed and motion direction with the target;
step 1.2: establishing a relation between space step lengths delta x ', delta y', delta z 'relative to the moving coordinate system K' and space step lengths delta x, delta y and delta z of the static coordinate system K, and a relation between time step length delta t 'relative to the moving coordinate system K' and delta t of the static coordinate system; the target has the same speed magnitude v and motion angle theta 'as the relative motion coordinate system K'v,Wherein theta'vRepresenting the included angle between the motion direction and the positive semi-axis of the z axis, ranging from 0 to 180 degrees,representing the included angle between the projection of the motion direction to the xoy plane and the positive half shaft of the x axis, the range is 0-360 degrees, and the angle isThe spatial step size Δ x ', Δ y', Δ z 'and Δ x, Δ y, Δ z relationships, and the temporal step size Δ t' and Δ t relationships can be obtained according to equation (1):
where c is the speed of light, x, y, z and t are the spatial and temporal components in the stationary coordinate system K, x ', y ', z ' and t ' are the spatial and temporal components in the relative motion coordinate system K ', v is the target motion speed, and the components of the speed in the x, y and z directions are respectivelyvz＝vcosθvIn the formula
Step 1.3: according to Courant stability conditionsDetermining the space step length delta x ', delta y', delta z 'and the time step length delta t' in the relative motion coordinate system K ', and thus finishing establishing the motion coordinate system K' which is static relative to the high-speed translation target;
as shown in FIG. 1, a motion coordinate system K ' which is relatively static with respect to a high-speed translation target is shown, and the target has the same speed magnitude v and the same motion angle theta ' as the coordinate system 'v,Where v is 0.1c, where the angleThe relationship between the space step Δ x ', Δ y', Δ z 'and Δ x, Δ y, Δ z, and the relationship between the time step Δ t' and Δ t can be obtained according to the formula (1), and the space step Δ x ', Δ y', Δ z ═ λ '/120, and the time step Δ t ═ Δ x'/2 c are taken according to the Courant stability condition.
Step two: establishing an almond body grid model;
step 2.1: the method is characterized in that a Fortran language is used for modeling, an almond body is selected as a three-dimensional high-speed translation target, and a control equation for establishing a mathematical model is shown as a formula (2):
wherein psi is an angle quantity for controlling the trajectory of the shape of the almond body in the yoz plane, and the value range-pi is not less than psi is not more than pi; d is the total length of the almond body along the x direction; a is1And a2The positive half shaft and the negative half shaft of the almond body respectively account for the proportionality coefficient of the total length in the x direction and satisfy a1+a2＝1；b1And b2The coefficients for controlling the length of the almond body in the x negative half axis and the x positive half axis in the y direction and the z direction respectively; n is a contraction for controlling the length of the almond body in the z directionSubtracting the coefficient, and n is an integer greater than zero; c. C1And c2For controlling the reduction coefficient of the length of the x positive half shaft of the almond body along the x direction, the requirement is metAnd 0 < c1< 1, thereby forming a tip at the x positive half axis; in order to ensure that the left and right parts of the model are coincident in the plane x being 0, b1And b2Need to satisfy | b1-b2(1-c1)|＜10-5；
In the invention, take a1＝-0.416667，a2＝0.58333，b1＝0.193333，b2＝4.83345，c1＝0.96，c2＝2.08335，d＝0.252374，n＝3；
The mathematical model is established according to the formula (2), and the size range of the almond body in the x direction is x epsilon (a)1×d,a2X d) (-0.105156,0.147217), and the size range in the y direction is y e (-b)1×d,b1X d) (-0.04879,0.04879) and the dimension in the z direction is in the range
Step 2.2: according to the size range of the almond body in the x, y and z directions and the size of the space step length of the relative motion coordinate system K' in the step 2.1, the grid number Is rounded by integer digits, the scattering boundary, the absorption boundary, the connection boundary and the total field boundary of the three-dimensional space are determined, wherein the scattering boundary belongs to the class i (Is min, Is max); j ∈ (Js min, Js max); k ∈ (Ks min, Ksmax); the ranges respectively representing the scattering boundaries in the x direction are (Is min, Is max), i.e. the minimum value and the maximum value in the x direction; the range of the scattering boundary in the y direction is (Js min, Js max), i.e. the minimum and maximum values in the y direction; the range of the scattering boundary in the z direction is (Ks min, Ks max), i.e., the minimum and maximum values in the z direction;
as shown in fig. 3, absorption, scattering and total field boundaries are specified, and the values are as follows:
| Imin,Imax | Jmin,Jmax | Kmin,Kmax | |
| absorption boundary | -112,129 | -90,90 | -76,76 |
| Scattering boundary | -42,59 | -20,20 | -6,6 |
| Total field boundary | -62,79 | -40,40 | -26,26 |
Step 2.3: converting the absolute size into a grid number, if the position of a certain grid (i, j, k) in the space satisfies the mathematical model in step 2.1, that is, a grid with coordinates (i, j, k), the actual coordinates of the grid are (x, y, z) — (i × Δ x, j × Δ y, k × Δ z), where x, y, z respectively represent the actual position coordinate values of the grid, Δ x, Δ y, Δ z respectively represent the space step length, i, j, k are the grid number of the grid in the x, y, z direction, respectively, substituting the coordinate values (x, y, z) into the mathematical model in step 2.1, and if the values of x, y, z satisfy the relational expression of formula (2), the grid is inside the almond body, that is set as a metal; if the values of x, y and z do not meet the relational expression of the formula (2), the grid is outside the almond body and is set as air, and the rest is done in sequence, so that an almond body model in the space can be obtained, and the almond body model is divided into cuboid cellular models according to the grid, so that planar model data in the txt format is obtained;
step three: introducing an incident plane wave into a relative motion coordinate system K';
step 3.1: determining incident wave under a static coordinate system K, and defining amplitude, frequency and incident angle thetai,θiRepresenting the included angle between the incident direction and the positive half shaft of the z axis, ranging from 0 to 180 degrees,representing the included angle between the projection of the incident direction to the xoy plane and the positive half shaft of the x axis, ranging from 0 degrees to 360 degrees, and the incident wave satisfies the conditionWherein,is the electric field strength of the incident wave, E0Is the initial value of amplitude, t is time, omegaiIs the angular frequency, equal to 2 pi f, f is the frequency of the incident wave, kiIs wave number, equal to ωi/c；
The invention introduces the incident wave at the boundary of the total field, and specifies the amplitude E of the incident wave in a coordinate system K01, frequency f 1GHz, angleObtaining an amplitude E ' by referring parameters of incident waves in K ' to formula (1) and phase invariance '01.106, frequency f 1.105GHz, and angle of incidence
Step 3.2: obtaining the amplitude E of the incident wave in the relative motion coordinate system K' and the static coordinate system K by using Lorentz transformation0And E'0Frequencies f and f', and angle of incidence θi,And theta'i,Introducing the incident wave into a relative motion coordinate system K' which is static relative to the almond body target at the total field boundary;
step four: calculating by using an FDTD algorithm to obtain electric field and magnetic field data in a motion coordinate system K';
step 4.1: inputting the running time step number time stop;
step 4.2: the moving speed v and the moving angle theta of the almond body are input'v,
Step 4.3: incident angle theta of input incident wavei,And converting it to theta'i,
Step 4.4: differential calculation of a three-dimensional FDTD algorithm is applied, and the electric field and the magnetic field are different by half time step, so that iterative solution is carried out on time until the time step number is finished from 1 to time stop, and the space electric field at each moment is obtainedAnd a magnetic fieldThe iterative formula is as follows:
wherein Ande ' representing electric field values of the grid node (i ', j ', K ') in the relative motion coordinate system K ' at time n +1 and time n ', respectively 'xThe components of the first and second images are,andare respectively represented inH ' of magnetic field values in relative motion coordinate system K ' of time mesh nodes (i ', j ', K ') and (i ', j ' -1, K ') 'zThe components of the first and second images are,andare respectively represented inH ' of magnetic field values in relative motion coordinate system K ' of time mesh nodes (i ', j ', K ') and (i ', j ', K ' -1) 'yThe components, ε (m) is the dielectric coefficient at spatial position m, σ (m) is the conductivity at spatial position m;
as shown in fig. 4, a flowchart illustrating a calculation process is first modeled and initialized to Δ x ', Δ y ', Δ z ', Δ t ' in the motion coordinate system K ', and then the time step is input to 8692, and each angle parameter is iteratively calculated with the time step using formula (3);
step five: converting the electric field E and the magnetic field H obtained in the step 4.4 into a static coordinate system K, and interpolating to obtain the high-speed translation electromagnetic scattering field of the almond body, wherein the detailed steps are as follows:
step 5.1: transforming the electromagnetic field value under the relative motion coordinate system K' into a static coordinate system K by combining Maxwell equation and Lorentz transformation; according to field value change Being parallel components with respect to the electric field in the motion coordinate system K',being the perpendicular component of the electric field in the relative motion coordinate system K',is the parallel component of the magnetic field and,is the vertical component of the magnetic field;
step 5.2: due to the influence of speed, the grid sizes of the relative motion coordinate system K ' and the stationary coordinate system K are not consistent, the electromagnetic field values in the stationary coordinate system K obtained in step 5.1 are converted from the field values in the relative motion coordinate system K ', the grid sizes corresponding to the field values are δ ' in the relative motion coordinate system K ', therefore, the grid sizes in the stationary coordinate system K are to be re-divided, that is, in the stationary coordinate system K, the electromagnetic field values of each spatial grid in the stationary coordinate system K are obtained by interpolation according to Δ x ═ Δ y ═ Δ z ═ Δ ' and then the grids are to be re-divided, three-dimensional interpolation is performed, a matlab engine is called by using format, the matlab contains functions of three-dimensional interpolation, namely, mesh 3, and the interpolation mode is linear, that is, linear interpolation is to obtain the electromagnetic field values;
step 5.3: and 5.2, obtaining the frequency distribution of the electromagnetic field value obtained in the step 5.2 by adopting an FFT algorithm, namely extracting the amplitude and the phase of the electromagnetic field value simultaneously.
The calculation aiming at the three-dimensional high-speed translation target comprises but not limited to a cube, a cuboid, a sphere, a cylinder, a cone, a prism and a round table, and the combination of the cube, the cuboid, the sphere, the cylinder, the cone, the prism and the round table; also includes conical spheres and polyhedrons.
As shown in fig. 5, during the interpolation, the coordinate system K will be re-interpolated in a format where Δ x ═ Δ y ═ Δ z ═ δ ', where δ ' is a discrete interval in the coordinate system K ';
as shown in fig. 6, after a txt data file is obtained, origin drawing software is opened, a book1 is closed, new matrix is selected, and the data file is imported, so that not only can an electric field value distribution diagram of an XOY section be obtained, but also a field value distribution of any section in space, namely an electromagnetic scattering field of an almond body target, can be obtained;
as shown in fig. 7, by performing an FFT algorithm on the field values, the frequency distribution of the XOY plane of the high-speed translation target of the almond body is also obtained, the forward scattering is greater than the backward scattering, and at the same time, the frequency value of any point in space can be obtained;
as shown in fig. 8, the phase of the FFT-processed data is extracted to obtain a phase distribution map of the XOY plane of the high-speed translation target of the almond body.
Claims (2)
1. A method for calculating an electromagnetic scattering field of a three-dimensional high-speed translation target is characterized by comprising the following steps:
the method comprises the following steps: establishing a motion coordinate system K 'which is static relative to the high-speed translation target, and determining the characteristics of the motion coordinate system K';
step 1.1: establishing a relative motion coordinate system K ', wherein the relative motion coordinate system K ' is static relative to a target of high-speed translation, namely the relative motion coordinate system K ' has the same motion speed and motion direction with the target;
step 1.2: establishing a spatial step size of a relative motion coordinate system KThe relation between Δ x ', Δ y ', Δ z ' and the spatial step Δ x, Δ y, Δ z of the stationary coordinate system K, and the relation between the time step Δ t ' of the relative motion coordinate system K ' and the Δ t of the stationary coordinate system; the target has the same speed magnitude v and motion angle theta 'as the relative motion coordinate system K'v,Wherein theta'vRepresenting the included angle between the motion direction and the positive semi-axis of the z axis, ranging from 0 to 180 degrees,representing the included angle between the projection of the motion direction to the xoy plane and the positive half shaft of the x axis, the range is 0-360 degrees, and the angle isThe spatial step size Δ x ', Δ y', Δ z 'and Δ x, Δ y, Δ z relationships, and the temporal step size Δ t' and Δ t relationships can be obtained according to equation (1):
<mrow> <mfenced open = "(" close = ")"> <mtable> <mtr> <mtd> <mrow> <mi>c</mi> <mi>t</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mfenced open = "(" close = ")"> <mtable> <mtr> <mtd> <mi>&gamma;</mi> </mtd> <mtd> <mrow> <mi>&gamma;</mi> <mfrac> <msub> <mi>v</mi> <mi>x</mi> </msub> <mi>c</mi> </mfrac> </mrow> </mtd> <mtd> <mrow> <mi>&gamma;</mi> <mfrac> <msub> <mi>v</mi> <mi>y</mi> </msub> <mi>c</mi> </mfrac> </mrow> </mtd> <mtd> <mrow> <mi>&gamma;</mi> <mfrac> <msub> <mi>v</mi> <mi>z</mi> </msub> <mi>c</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>&gamma;</mi> <mfrac> <msub> <mi>v</mi> <mi>x</mi> </msub> <mi>c</mi> </mfrac> </mrow> </mtd> <mtd> <mrow> <mn>1</mn> <mo>+</mo> <mfrac> <mrow> <msup> <msub> <mi>v</mi> <mi>x</mi> </msub> <mn>2</mn> </msup> </mrow> <msup> <mi>v</mi> <mn>2</mn> </msup> </mfrac> <mrow> <mo>(</mo> <mi>&gamma;</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mfrac> <mrow> <msub> <mi>v</mi> <mi>x</mi> </msub> <msub> <mi>v</mi> <mi>y</mi> </msub> </mrow> <msup> <mi>v</mi> <mn>2</mn> </msup> </mfrac> <mrow> <mo>(</mo> <mi>&gamma;</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mfrac> <mrow> <msub> <mi>v</mi> <mi>z</mi> </msub> <msub> <mi>v</mi> <mi>x</mi> </msub> </mrow> <msup> <mi>v</mi> <mn>2</mn> </msup> </mfrac> <mrow> <mo>(</mo> <mi>&gamma;</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>&gamma;</mi> <mfrac> <msub> <mi>v</mi> <mi>y</mi> </msub> <mi>c</mi> </mfrac> </mrow> </mtd> <mtd> <mrow> <mfrac> <mrow> <msub> <mi>v</mi> <mi>x</mi> </msub> <msub> <mi>v</mi> <mi>y</mi> </msub> </mrow> <msup> <mi>v</mi> <mn>2</mn> </msup> </mfrac> <mrow> <mo>(</mo> <mi>&gamma;</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mn>1</mn> <mo>+</mo> <mfrac> <mrow> <msup> <msub> <mi>v</mi> <mi>y</mi> </msub> <mn>2</mn> </msup> </mrow> <msup> <mi>v</mi> <mn>2</mn> </msup> </mfrac> <mrow> <mo>(</mo> <mi>&gamma;</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mfrac> <mrow> <msub> <mi>v</mi> <mi>y</mi> </msub> <msub> <mi>v</mi> <mi>z</mi> </msub> </mrow> <msup> <mi>v</mi> <mn>2</mn> </msup> </mfrac> <mrow> <mo>(</mo> <mi>&gamma;</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>&gamma;</mi> <mfrac> <msub> <mi>v</mi> <mi>z</mi> </msub> <mi>c</mi> </mfrac> </mrow> </mtd> <mtd> <mrow> <mfrac> <mrow> <msub> <mi>v</mi> <mi>z</mi> </msub> <msub> <mi>v</mi> <mi>x</mi> </msub> </mrow> <msup> <mi>v</mi> <mn>2</mn> </msup> </mfrac> <mrow> <mo>(</mo> <mi>&gamma;</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mfrac> <mrow> <msub> <mi>v</mi> <mi>y</mi> </msub> <msub> <mi>v</mi> <mi>z</mi> </msub> </mrow> <msup> <mi>v</mi> <mn>2</mn> </msup> </mfrac> <mrow> <mo>(</mo> <mi>&gamma;</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mn>1</mn> <mo>+</mo> <mfrac> <mrow> <msup> <msub> <mi>v</mi> <mi>z</mi> </msub> <mn>2</mn> </msup> </mrow> <msup> <mi>v</mi> <mn>2</mn> </msup> </mfrac> <mrow> <mo>(</mo> <mi>&gamma;</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "(" close = ")"> <mtable> <mtr> <mtd> <mrow> <msup> <mi>ct</mi> <mo>&prime;</mo> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <msup> <mi>x</mi> <mo>&prime;</mo> </msup> </mtd> </mtr> <mtr> <mtd> <msup> <mi>y</mi> <mo>&prime;</mo> </msup> </mtd> </mtr> <mtr> <mtd> <msup> <mi>z</mi> <mo>&prime;</mo> </msup> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow>
where c is the speed of light, x, y, z and t are the spatial and temporal components in the stationary coordinate system K, x ', y ', z ' and t ' are the spatial and temporal components in the relative motion coordinate system K ', v is the target motion speed, and the components of the speed in the x, y and z directions are respectivelyvz＝vcosθvIn the formula
Step 1.3: according to Courant stability conditionsDetermining the space step length delta x ', delta y', delta z 'and the time step length delta t' in the relative motion coordinate system K ', and thus finishing establishing the motion coordinate system K' which is static relative to the high-speed translation target;
step two: establishing an almond body grid model;
step 2.1: the method is characterized in that a computer is used for modeling, an almond body is selected as a three-dimensional high-speed translation target, and a control equation for establishing a mathematical model is shown as a formula (2):
<mrow> <mtable> <mtr> <mtd> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <mo>-</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> <mi>d</mi> <mo><</mo> <mi>x</mi> <mo><</mo> <mn>0</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>y</mi> <mo>=</mo> <msub> <mi>b</mi> <mn>1</mn> </msub> <mi>d</mi> <msqrt> <mrow> <mn>1</mn> <mo>-</mo> <msup> <mrow> <mo>(</mo> <mfrac> <mi>x</mi> <mrow> <msub> <mi>a</mi> <mn>1</mn> </msub> <mi>d</mi> </mrow> </mfrac> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> <mi>cos</mi> <mi>&psi;</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>z</mi> <mo>=</mo> <mfrac> <msub> <mi>b</mi> <mn>1</mn> </msub> <mi>n</mi> </mfrac> <mi>d</mi> <msqrt> <mrow> <mn>1</mn> <mo>-</mo> <msup> <mrow> <mo>(</mo> <mfrac> <mi>x</mi> <mrow> <msub> <mi>a</mi> <mn>1</mn> </msub> <mi>d</mi> </mrow> </mfrac> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> <mi>sin</mi> <mi>&psi;</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> </mtd> <mtd> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <mn>0</mn> <mo><</mo> <mi>x</mi> <mo><</mo> <msub> <mi>a</mi> <mn>2</mn> </msub> <mi>d</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>y</mi> <mo>=</mo> <msub> <mi>b</mi> <mn>2</mn> </msub> <mi>d</mi> <mo>&lsqb;</mo> <msqrt> <mrow> <mn>1</mn> <mo>-</mo> <msup> <mrow> <mo>(</mo> <mfrac> <mi>x</mi> <mrow> <msub> <mi>c</mi> <mn>2</mn> </msub> <mi>d</mi> </mrow> </mfrac> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> <mo>-</mo> <msub> <mi>c</mi> <mn>1</mn> </msub> <mo>&rsqb;</mo> <mi>cos</mi> <mi>&psi;</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>z</mi> <mo>=</mo> <mfrac> <msub> <mi>b</mi> <mn>2</mn> </msub> <mi>n</mi> </mfrac> <mi>d</mi> <mo>&lsqb;</mo> <msqrt> <mrow> <mn>1</mn> <mo>-</mo> <msup> <mrow> <mo>(</mo> <mfrac> <mi>x</mi> <mrow> <msub> <mi>c</mi> <mn>2</mn> </msub> <mi>d</mi> </mrow> </mfrac> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> <mo>-</mo> <msub> <mi>c</mi> <mn>1</mn> </msub> <mo>&rsqb;</mo> <mi>sin</mi> <mi>&psi;</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow>
wherein psi is an angle quantity for controlling the trajectory of the shape of the almond body in the yoz plane, and the value range-pi is not less than psi is not more than pi; d is the total length of the almond body along the x direction; a is1And a2The positive half shaft and the negative half shaft of the almond body respectively account for the proportionality coefficient of the total length in the x direction and satisfy a1+a2＝1；b1And b2The coefficients for controlling the length of the almond body in the x negative half axis and the x positive half axis in the y direction and the z direction respectively; n is a reduction coefficient for controlling the length of the almond body in the z direction, and n is an integer greater than zero; c. C1And c2For controlling the reduction coefficient of the length of the x positive half shaft of the almond body along the x direction, the requirement is metAnd 0 < c1< 1, thereby forming a tip at the x positive half axis, b1And b2Satisfy | b1-b2(1-c1)|＜10-5；
The mathematical model is established according to the formula (2), and the size range of the almond body in the x direction is x epsilon (a)1×d,a2X d), the size range of the y direction is y epsilon (-b)1×d,b1X d) in the z-direction
Step 2.2: according to the size range of the almond body in the x, y and z directions and the size of the space step length of the relative motion coordinate system K' in the step 2.1, the grid number is rounded by integer digits, the scattering boundary, the absorption boundary, the connection boundary and the total field boundary of the three-dimensional space are determined, wherein the scattering boundary belongs to the i e (Ismin, Ismax); j ∈ (Jsmin, Jsmax); k ∈ (Ksmn, Ksmax); the ranges representing the scattering boundaries in the x-direction are (Ismin, Ismax), i.e. the minimum and maximum values in the x-direction, respectively; the range of the scattering boundary in the y direction is (Jsmin, Jsmax), i.e. the minimum and maximum values in the y direction; the range of the scattering boundary in the z direction is (Ksmin, Ksmax), i.e. the minimum and maximum values in the z direction;
step 2.3: converting the absolute size into a grid number, if the position of a certain grid (i, j, k) in the space satisfies the mathematical model in step 2.1, that is, a grid with coordinates (i, j, k), the actual coordinates of the grid are (x, y, z) — (i × Δ x, j × Δ y, k × Δ z), where x, y, z respectively represent the actual position coordinate values of the grid, Δ x, Δ y, Δ z respectively represent the space step length, i, j, k are the grid number of the grid in the x, y, z direction, respectively, substituting the coordinate values (x, y, z) into the mathematical model in step 2.1, and if the values of x, y, z satisfy the relational expression of formula (2), the grid is inside the almond body, that is set as a metal; if the values of x, y and z do not meet the relational expression of the formula (2), the grid is outside the almond body and is set as air, and the rest is done in sequence, so that an almond body model in the space can be obtained, and the almond body model is divided into cuboid cellular models according to the grid, so that planar model data in the txt format is obtained;
step three: introducing an incident plane wave into a relative motion coordinate system K';
step 3.1: determining incident wave under a static coordinate system K, and defining amplitude, frequency and incident angle thetai,θiRepresenting the included angle between the incident direction and the positive half shaft of the z axis, ranging from 0 to 180 degrees,representing the included angle between the projection of the incident direction to the xoy plane and the positive half shaft of the x axis, ranging from 0 degrees to 360 degrees, and the incident wave satisfies the conditionWherein,is the electric field strength of the incident wave, E0Is the initial value of amplitude, t is time, omegaiIs the angular frequency, equal to 2 pi f, f is the frequency of the incident wave, kiIs wave numberIs equal to omegai/c；
Step 3.2: obtaining the amplitude E of the incident wave in the relative motion coordinate system K' and the static coordinate system K by using Lorentz transformation0And E'0Frequencies f and f', and angle of incidence θi,And theta'i,Introducing the incident wave into a relative motion coordinate system K' which is static relative to the almond body target at the total field boundary;
step four: calculating by using an FDTD algorithm to obtain electric field and magnetic field data in a motion coordinate system K';
step 4.1: inputting the running time step number time stop;
step 4.2: the moving speed v and the moving angle theta of the almond body are input'v,
Step 4.3: incident angle theta of input incident wavei,And converting it to theta'i,
Step 4.4: differential calculation of a three-dimensional FDTD algorithm is applied, and the electric field and the magnetic field are different by half time step, so that iterative solution is carried out on time until the time step number is finished from 1 to time stop, and the space electric field at each moment is obtainedAnd a magnetic fieldThe iterative formula is as follows:
<mrow> <mtable> <mtr> <mtd> <mrow> <msubsup> <mi>E</mi> <mi>x</mi> <mrow> <mo>&prime;</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mrow> <mo>(</mo> <msup> <mi>i</mi> <mo>&prime;</mo> </msup> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>,</mo> <msup> <mi>j</mi> <mo>&prime;</mo> </msup> <mo>,</mo> <msup> <mi>k</mi> <mo>&prime;</mo> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mi>C</mi> <mi>A</mi> <mrow> <mo>(</mo> <mi>m</mi> <mo>)</mo> </mrow> <mo>&CenterDot;</mo> <msubsup> <mi>E</mi> <mi>x</mi> <mrow> <mo>&prime;</mo> <mi>n</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <msup> <mi>i</mi> <mo>&prime;</mo> </msup> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>,</mo> <msup> <mi>j</mi> <mo>&prime;</mo> </msup> <mo>,</mo> <msup> <mi>k</mi> <mo>&prime;</mo> </msup> <mo>)</mo> </mrow> <mo>+</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>C</mi> <mi>B</mi> <mrow> <mo>(</mo> <mi>m</mi> <mo>)</mo> </mrow> <mo>&CenterDot;</mo> <mo>&lsqb;</mo> <mfrac> <mrow> <msubsup> <mi>H</mi> <mi>z</mi> <mrow> <mo>&prime;</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msubsup> <mrow> <mo>(</mo> <msup> <mi>i</mi> <mo>&prime;</mo> </msup> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>,</mo> <msup> <mi>j</mi> <mo>&prime;</mo> </msup> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>,</mo> <msup> <mi>k</mi> <mo>&prime;</mo> </msup> <mo>)</mo> </mrow> <mo>-</mo> <msubsup> <mi>H</mi> <mi>z</mi> <mrow> <mo>&prime;</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msubsup> <mrow> <mo>(</mo> <msup> <mi>i</mi> <mo>&prime;</mo> </msup> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>,</mo> <msup> <mi>j</mi> <mo>&prime;</mo> </msup> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>,</mo> <msup> <mi>k</mi> <mo>&prime;</mo> </msup> <mo>)</mo> </mrow> </mrow> <mrow> <msup> <mi>&Delta;y</mi> <mo>&prime;</mo> </msup> </mrow> </mfrac> <mo>-</mo> <mfrac> <mrow> <msubsup> <mi>H</mi> <mi>y</mi> <mrow> <mo>&prime;</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msubsup> <mrow> <mo>(</mo> <msup> <mi>i</mi> <mo>&prime;</mo> </msup> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>,</mo> <msup> <mi>j</mi> <mo>&prime;</mo> </msup> <mo>,</mo> <msup> <mi>k</mi> <mo>&prime;</mo> </msup> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>)</mo> </mrow> <mo>-</mo> <msubsup> <mi>H</mi> <mi>y</mi> <mrow> <mo>&prime;</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msubsup> <mrow> <mo>(</mo> <msup> <mi>i</mi> <mo>&prime;</mo> </msup> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>,</mo> <msup> <mi>j</mi> <mo>&prime;</mo> </msup> <mo>,</mo> <msup> <mi>k</mi> <mo>&prime;</mo> </msup> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>)</mo> </mrow> </mrow> <mrow> <msup> <mi>&Delta;z</mi> <mo>&prime;</mo> </msup> </mrow> </mfrac> <mo>&rsqb;</mo> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>
wherein Andrespectively represent gridsE 'of the electric field values of node (i', j ', K') in relative motion coordinate system K 'at time n +1 and time n'xThe components of the first and second images are,andare respectively represented inH ' of magnetic field values in relative motion coordinate system K ' of time mesh nodes (i ', j ', K ') and (i ', j ' -1, K ') 'zThe components of the first and second images are,andare respectively represented inH ' of magnetic field values in relative motion coordinate system K ' of time mesh nodes (i ', j ', K ') and (i ', j ', K ' -1) 'yThe components, ε (m) is the dielectric coefficient at spatial position m, σ (m) is the conductivity at spatial position m;
step five: converting the electric field E and the magnetic field H obtained in the step 4.4 into a static coordinate system K, and interpolating to obtain the high-speed translation electromagnetic scattering field of the almond body, wherein the detailed steps are as follows:
step 5.1: transforming the electromagnetic field value under the relative motion coordinate system K' into a static coordinate system K by combining Maxwell equation and Lorentz transformation; according to field value change For the flattening of the electric field in the relative motion coordinate system KThe line components are then processed in a row-wise fashion,being the perpendicular component of the electric field in the relative motion coordinate system K',is the parallel component of the magnetic field and,is the vertical component of the magnetic field;
step 5.2: due to the influence of speed, the grid sizes of the relative motion coordinate system K ' and the stationary coordinate system K are not consistent, the electromagnetic field values in the stationary coordinate system K obtained in step 5.1 are converted from the field values in the relative motion coordinate system K ', the grid sizes corresponding to the field values are δ ' in the relative motion coordinate system K ', therefore, the grid sizes in the stationary coordinate system K are to be re-divided, that is, in the stationary coordinate system K, the electromagnetic field values of each spatial grid in the stationary coordinate system K are obtained by interpolation according to Δ x ═ Δ y ═ Δ z ═ Δ ' and then the grids are to be re-divided, three-dimensional interpolation is performed, a matlab engine is called by using format, the matlab contains functions of three-dimensional interpolation, namely, mesh 3, and the interpolation mode is linear, that is, linear interpolation is to obtain the electromagnetic field values;
step 5.3: and 5.2, obtaining the frequency distribution of the electromagnetic field value obtained in the step 5.2 by adopting an FFT algorithm, namely extracting the amplitude and the phase of the electromagnetic field value simultaneously.
2. The method for calculating the electromagnetic scattering field of the three-dimensional high-speed translation target according to claim 1, wherein the method comprises the following steps:
the calculation aiming at the three-dimensional high-speed translation target comprises but not limited to a cube, a cuboid, a sphere, a cylinder, a cone, a prism and a round table, and the combination of the cube, the cuboid, the sphere, the cylinder, the cone, the prism and the round table; also includes conical spheres and polyhedrons.
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| CN112632832A (en) * | 2020-12-28 | 2021-04-09 | 中国科学院电工研究所 | Motion electromagnetic field calculation method based on overlapped grid motion boundary reconstruction |
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| Publication number | Priority date | Publication date | Assignee | Title |
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| CN109359399A (en) * | 2018-10-24 | 2019-02-19 | 西北工业大学 | A kind of Electromagnetic Scattering Characteristics method calculating high speed fine motion class bulb cylinder target |
| CN109359399B (en) * | 2018-10-24 | 2021-04-02 | 西北工业大学 | Method for calculating electromagnetic scattering property of high-speed micro-motion ball-like head cylindrical target |
| CN111337870A (en) * | 2020-04-17 | 2020-06-26 | 中国人民解放军海军装备部驻沈阳地区军事代表局驻大连地区第一军事代表室 | Three-axis electric field correction method based on Euler formula |
| CN111337870B (en) * | 2020-04-17 | 2021-07-06 | 中国人民解放军海军装备部驻沈阳地区军事代表局驻大连地区第一军事代表室 | Three-axis electric field correction method based on Euler formula |
| CN112632832A (en) * | 2020-12-28 | 2021-04-09 | 中国科学院电工研究所 | Motion electromagnetic field calculation method based on overlapped grid motion boundary reconstruction |
| CN112632832B (en) * | 2020-12-28 | 2023-10-27 | 中国科学院电工研究所 | Moving electromagnetic field calculation method based on overlapped grid moving boundary reconstruction |
| CN116720409A (en) * | 2023-06-21 | 2023-09-08 | 安徽大学 | Electromagnetic scattering field calculation method for moving time-varying dispersion medium target |
| CN116720409B (en) * | 2023-06-21 | 2024-03-26 | 安徽大学 | Electromagnetic scattering field calculation method for moving time-varying dispersion medium target |
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