CBF-based MFIE consistent discrete electromagnetic scattering analysis method

Abstract

The invention discloses a CBF-based MFIE consistent discrete electromagnetic scattering analysis method. The method comprises the following steps: establishing a magnetic field integral equation for an electromagnetic scattering problem of a metal conductor object; performing triangular surface element discretization on the surface of the object to generate an RWG primary function and a BC primary function; performing block processing on the object, and dividing a primary function into different sub-blocks; solving a sub-matrix equation of each sub-block, generating a characteristic primary function, and replacing an original primary function; constructing a reduced matrix equation by using a feature basis function method; iteratively solving the matrix equation to obtain an equivalent surface current; and solving a far field value of the scattering electric field of the object according to the equivalent surface current. According to the method, the memory and time required for solving the MFIE in the electromagnetic scattering problem are remarkably reduced, and the problem that the precision of a traditional MFIE matrix equation is poor is solved while the good convergence of the MFIE matrix equation is kept.

Description

Technical Field

[0001] The invention belongs to the technical field of electromagnetic scattering analysis, and relates to an efficient method for quickly solving electromagnetic scattering of metal objects, and in particular to an electromagnetic scattering analysis method based on a consistent discretization of a magnetic field integral equation of a characteristic basis function. Background Art

[0002] The electromagnetic scattering problem has always received great attention at home and abroad in military weapons and equipment. The Method of Moments (MoM) is an efficient method for analyzing the electromagnetic scattering characteristics of objects. Under the framework of the MoM, there are two most classic integral equations for solving the electromagnetic scattering problem of metal objects, one is the Electric Field Integral Equation (EFIE) and the other is the Magnetic Field Integral Equation (MFIE). The electric field integral equation has two most significant characteristics, one is the advantage of high accuracy and the other is the disadvantage of slow convergence; this disadvantage is particularly obvious when the discrete density of the target object grid increases, and it is often impossible to solve because the convergence threshold cannot be reached. In contrast to the electric field integral equation, the convergence of the magnetic field integral equation is quite excellent, but its accuracy is quite poor. Therefore, people hope to improve the magnetic field integral equation and deal with its poor accuracy.

[0003] Scholars have conducted many studies on the problem of poor accuracy of magnetic field integral equations. Some scholars believe that correcting the solid angle factor can improve accuracy; some scholars believe that replacing the basis function with a line-line basis function can improve accuracy; some scholars believe that a basis function with consistent curl should be used; some scholars believe that the accuracy of the impedance element of the magnetic field integral equation MFIE should be improved, and then many singular value extraction techniques have emerged based on this concept; some scholars believe that the unit operator discretization error in MFIE is large, and proposed a technique to improve the unit operator discretization. Recently, some scholars proposed that using the rotated BC (Buffa-Christiansen) basis function as a detection function to detect MFIE can improve the numerical accuracy of MFIE. This method can not only reduce the unit operator discretization error, but also its dual detection properties can explain the high accuracy of traditional EFIE and the poor accuracy of traditional MFIE.

[0004] Although the use of BC basis functions can improve the accuracy of the magnetic field integral equation, the number of unknowns of this technology has not been reduced. Since its memory complexity increases with the square of the number of unknowns, when the number of unknowns increases, the memory required to solve the matrix equation increases in the form of a square. However, computing resources are always limited. When encountering large electrical problems that need to be solved, the lack of computing resources makes it completely unable to handle such problems. Summary of the invention

[0005] In view of the above problems, the present invention proposes a method for electromagnetic scattering analysis based on a consistent discretization of the magnetic field integral equation with characteristic basis functions, which can solve the electromagnetic scattering characteristics of metal objects in a more efficient form. This new efficient and fast electromagnetic scattering solution method uses characteristic basis functions, which can greatly reduce the number of unknown quantities, thereby reducing the time and memory required for electromagnetic scattering analysis. At the same time, the good convergence of the magnetic field integral equation is retained, and the convergence will not deteriorate due to the increase in the mesh discretization density of the object.

[0006] The technical solution of the present invention is: comprising the following steps:

[0007] Step 1: For the electromagnetic scattering problem of metal objects, the magnetic field integral equation MFIE is established according to the magnetic field boundary conditions of the conductor.

[0008] Step 2: Divide all surfaces of the object into triangular meshes to obtain the initial mesh, and generate RWG (Rao-Wilton-Glisson) basis functions on the initial mesh.

[0009] Step 3: Perform barycentric encryption on the initial mesh. Connect the three vertices of each triangle on the initial mesh to the midpoints of its opposite sides respectively, divide the triangle into six, and obtain six small triangles. The mesh composed of these small triangles is called a barycentric encrypted mesh. Establish the BC basis function on the mesh obtained after barycentric encryption.

[0010] Step 4: Divide the object into blocks and divide the RWG basis function and BC basis function into different sub-blocks.

[0011] In step 4, there is no strict rule on how to divide the object into blocks. The appropriate block structure is selected according to the actual situation. Commonly used block structures in practical applications are: binary tree structure, quadtree structure and octree structure. For three-dimensional target problems, a commonly used block structure is octree structure.

[0012] Step 5: Generate characteristic basis functions for all sub-blocks as new basis functions.

[0013] This step can be considered as a sub-problem from a certain perspective. According to the basis functions in the block, the impedance matrix equation of the block is constructed. Two mutually orthogonal polarized plane waves are used for illumination at the sampling incident angles sampled on the entire sphere to obtain the BC basis function and RWG basis function current solutions under these illuminations; then the Gram matrix connecting the two current solutions is subjected to singular value decomposition (SVD), and finally the characteristic basis coefficients are constructed.

[0014] Step 6: Compress the initial impedance matrix using the characteristic basis functions and construct the reduced impedance matrix equation.

[0015] Step 7: Iteratively solve the reduced matrix equation to obtain the equivalent surface current. Based on the obtained equivalent surface current solution, the far-field value of the target scattered electric field is calculated, and the radar cross section (RCS) of the object is further calculated.

[0016] The method of the present invention performs efficient electromagnetic scattering analysis based on magnetic field integral equations and characteristic basis functions, and is applicable to electromagnetic scattering analysis of metal objects. The radar scattering characteristics of the object obtained by solving the problem can be used in the stealth design process of military targets and target detection of radar systems. The method of the present invention significantly reduces the computational memory and time required for solving electromagnetic scattering by traditional MFIE, improves the problem of poor accuracy of traditional MFIE matrix equations while maintaining good convergence of MFIE matrix equations, and is applicable to solving electromagnetic scattering of medium and large targets.

[0017] The present invention has the following beneficial effects:

[0018] 1. Fewer unknowns: The characteristic basis function method is used to generate characteristic basis functions, which are used as new basis functions to replace the original RWG basis functions, reducing the number of unknowns and thus reducing memory requirements.

[0019] 2. High precision: Since the characteristic basis function is a linear combination of the RWG basis function or the BC basis function, the related properties of these basis functions in the mathematical space are not changed, and the dual detection scheme is retained. Therefore, the present invention maintains the original high precision advantage of the MFIE improved by using the BC basis function.

[0020] 3. Excellent iterative convergence: MFIE belongs to the second kind of Fredholm integral equation. If this kind of equation can be discretized well, a well-converged integral equation can be obtained. The Gram matrix of the characteristic basis function has good properties and can discretize the unit operator in MFIE well, maintaining good convergence. BRIEF DESCRIPTION OF THE DRAWINGS

[0021] Figure 1 It is a schematic diagram of performing centroid encryption on triangles on an initial mesh in an embodiment of the present invention.

[0022] Figure 2 It is a schematic diagram of performing octree partitioning on an object in an embodiment of the present invention.

[0023] Figure 3 It is a two-station RCS result diagram of an amygdala in an embodiment of the present invention.

[0024] Figure 4This is a single-station RCS result diagram of an amygdala in an embodiment of the present invention. DETAILED DESCRIPTION

[0025] In order to clearly illustrate the technical features of the present invention, the technical solution of the present invention is described in detail below through specific implementation methods and in conjunction with the accompanying drawings.

[0026] The present invention provides an electromagnetic scattering analysis method based on a consistent discretization of a magnetic field integral equation of a characteristic basis function, comprising the following steps:

[0027] Step 1: For the electromagnetic scattering problem of metal objects, the magnetic field integral equation MFIE is established based on the magnetic field boundary conditions of the conductor. The expression of the magnetic field integral equation MFIE is:

[0028]

[0029] Where J is the equivalent surface current, is the external normal, S is the surface of the target object, H ine is the incident magnetic field, r is the field point; K PV is the integral operator, I is the unit operator, and the expression is:

[0030]

[0031] I{X(r)}=X(r) (3)

[0032] Where r′ is the source point, PV represents the Cauchy principal value integral, X(r) represents the function at the field point, X(r′) represents the function at the source point, and G(r, r′) is the Green function of free space, which is expressed as

[0033]

[0034] Among them, j is the imaginary unit, k 0 is the wave number in free space.

[0035] Step 2: Divide all surfaces of the object into triangular meshes to obtain the initial mesh, and generate RWG basis functions on the initial mesh.

[0036] Step 3: Perform barycentric encryption on the initial grid and establish BC basis functions on the grid obtained after barycentric encryption.

[0037] Figure 1 A method for barycentric encryption of triangles on the initial mesh is given. The three vertices of each triangle on the initial mesh are connected to the midpoints of its opposite sides, and the triangle is divided into six to obtain six small triangles. The mesh composed of these small triangles is called a barycentric encrypted mesh.

[0038] Step 4: Divide the object into blocks and divide the RWG basis functions and BC basis functions into different sub-blocks according to the octree structure.

[0039] Consider an object in three-dimensional space, such as Figure 2 As shown, the object is divided into blocks in the form of an octree, and the RWG basis functions and BC basis functions on the surface of the object are classified into corresponding sub-blocks according to their positions in the three-dimensional space.

[0040] Step 5: Generate characteristic basis functions for all sub-blocks as new basis functions.

[0041] According to the RWG basis function and BC basis function in the block, the electric field integral equation (EFIE) impedance matrix equation of the block is constructed using the moment method, and the incident angle of the entire sphere is sampled. At these sampled incident angles, two mutually orthogonal polarized plane waves are used for irradiation to obtain the current solutions under these irradiations. Without loss of generality, taking the mth block as an example, the RWG matrix equation and BC matrix equation of the mth subproblem are expressed as

[0042]

[0043] in, is the L-operator RWG impedance matrix of the mth block, is the incident electric field excitation of the RWG of the mth block, is the RWG current coefficient matrix to be determined for the mth block, is the L-operator BC impedance matrix of the mth block, is the incident electric field excitation of the BC of the mth block, is the BC current coefficient matrix of the mth block. The L operator expression is

[0044]

[0045] Directly solving these two matrix equations yields the current solution and The Gram matrix G′ connecting these two current solutions m The expression is

[0046]

[0047] The superscript H represents the conjugate transpose of the matrix, G m is the Gram matrix connecting the rotated BC basis function and the RWG basis function in the mth block. Perform a singular value decomposition (SVD) on the Gram matrix connecting these two current solutions.

[0048]

[0049] Among them, U m and V m is a unitary matrix, D m is a diagonal matrix whose diagonal elements are G′ m The singular values ​​of the unitary matrix U are retained according to the threshold of the singular value decomposition m and V m The first k m Column, get and Accordingly, is to keep the first k m The diagonal matrix of singular values, is a diagonal matrix whose diagonal elements are Then, the pair of eigenvalues ​​of the mth block is and The expressions are

[0050]

[0051] Arrange the characteristic basis coefficients of all blocks diagonally to obtain the characteristic basis function of the entire problem. Its characteristic basis coefficient matrix C BC and C RWG It is expressed as:

[0052]

[0053] Where B represents the number of blocks.

[0054] Step 6: Use the characteristic basis functions to compress the initial impedance matrix and construct the reduced impedance matrix equation. Use the moment method to discretize the surface integral equation and construct the reduced impedance matrix equation.

[0055] The reduced matrix equation of MFIE is expressed as

[0056] Z C I C =V C (14)

[0057] Among them, Z C is the characteristic basis function C BC and C RWG As the impedance matrix of the detection function and basis function, V C is the characteristic basis function C BC Incident magnetic field excitation as a function of detection, I C is the required characteristic basis current coefficient matrix of the whole problem; the expressions of impedance matrix and incident magnetic field excitation are

[0058] Z C =(CBC ) H ZC RWG (15)

[0059] V C =(C BC ) H V (16)

[0060] Where Z is the impedance matrix of the magnetic field integral equation MFIE with the BC basis function and the RWG basis function as the detection function and basis function respectively, and V is the incident magnetic field excitation of the BC basis function as the detection function; when the characteristic basis current solution I is obtained C After that, the final current solution I is given by the following formula RWG

[0061] I RWG =C RWG I C (17)

[0062] Current vector I RWG The coefficient a of the RWG basis function n The composition is expressed as

[0063] I RWG =(a 1 , a 2 , …, a N ) T (18)

[0064] Where N is the total number of RWG basis functions and the superscript T represents the transpose of the vector.

[0065] Step 7: Iteratively solve the reduced matrix equation to obtain the equivalent surface current; based on the obtained equivalent surface current solution, calculate the far-field value of the target scattered electric field, and further calculate the radar scattering cross section RCS of the object.

[0066] After using the iterative solver to solve the MFIE reduction matrix equation to obtain the current coefficient, the final equivalent surface current J(r) is expressed as:

[0067]

[0068] Among them, f n (r) is the nth RWG basis function, a n is the current coefficient corresponding to the nth RWG basis function.

[0069] According to the equivalent surface current solution, the far-field value E of the target scattered electric field is solved. sca :

[0070] E sca (r) = Z 0L{J(r′)} (20)

[0071] Further calculation is performed to obtain the radar cross section RCS of the target, which is represented by the symbol σ and the calculation expression is:

[0072]

[0073] The unit of σ is m. 2 , E inc is the incident electric field.

[0074] Example 1

[0075] This embodiment uses an almond with a maximum length of 4.9 m to verify the advantages of the present invention in terms of memory and accuracy.

[0076] Figure 3 The bi-station RCS results of the present invention (abbreviated as prop.method) and the traditional magnetic field integral equation MFIE and the electric field integral equation (EFIE) are given as numerical reference solutions. Figure 4 The single-station RCS results are given, and it can be seen that the results of the present invention are more accurate than the traditional magnetic field integral equation (MFIE).

[0077] Table 1 shows the comparison of the number of unknowns, memory usage, average number of iterations and single-station solution time of the traditional MFIE, the MFIE with the rotated BC basis function as the detection function (nBC-tested MFIE) and the present invention (prop.method) when solving the single-station problem. It can be seen that the present invention has fewer unknowns than the traditional MFIE and the nBC-tested MFIE, and therefore occupies less computer memory, which reflects the great advantage of the present invention in terms of computing resource requirements. Since the system matrix length in the present invention is shorter, the matrix-vector product in the iterative solution process has a short calculation length, and the number of iterations required to reach convergence is less than that of the other two methods, so the iterative solution time is greatly reduced, which reflects the great advantage of the present invention in terms of computing time.

[0078] Table 1

[0079]

[0080] There are many specific implementation ways of the present invention. The above is only the preferred implementation mode of the present invention. It should be pointed out that for ordinary technicians in this technical field, several improvements can be made without departing from the principle of the present invention. These improvements should also be regarded as the protection scope of the present invention.

Claims

1. A method for electromagnetic scattering analysis based on consistent discretization of magnetic field integral equations of characteristic basis functions, characterized in that: The steps include: Step 1: For the electromagnetic scattering problem of metal objects, the magnetic field integral equation MFIE is established according to the magnetic field boundary conditions of the conductor; Step 2: Divide all surfaces of the object into triangular meshes to obtain the initial mesh, and generate RWG basis functions on the initial mesh; Step 3: Perform gravity encryption on the initial mesh, connect the three vertices of each triangle on the initial mesh to the midpoints of its opposite sides, divide the triangle into six, and obtain six small triangles; The mesh composed of these small triangles is called the barycentric encrypted mesh; the BC basis function is established on the mesh obtained after barycentric encryption; Step 4: Divide the object into blocks and divide the RWG basis function and BC basis function into different sub-blocks; Step 5: Generate characteristic basis functions for all sub-blocks as new basis functions; Step 6: Use the characteristic basis function to compress the initial impedance matrix and construct the reduced impedance matrix equation; Step 7: Iteratively solve the reduced impedance matrix equation to obtain the equivalent surface current. Based on the obtained equivalent surface current, calculate the far-field value of the target scattered electric field, and further calculate the radar scattering cross section of the object.

2. The electromagnetic scattering analysis method according to claim 1, characterized in that: In step 1, the expression of the magnetic field integral equation MFIE is Where J is the equivalent surface current, is the outward normal, is the target surface, H inc is the incident magnetic field, r is the field point; K PV is the integral operator, I is the unit operator, and the expression is: I{X(r)}=X(r) (3) Where r′ is the source point, PV represents the Cauchy principal value integral, X(r) represents the function at the field point, X(r′) represents the function at the source point, and G(r, r′) is the Green function of free space, which is expressed as Here, j is the imaginary unit and k0 is the wave number in free space.

3. The electromagnetic scattering analysis method based on the consistent discretization of the magnetic field integral equation based on characteristic basis functions according to claim 1 is characterized in that: The fourth step is to process the object in blocks, and divide the RWG basis functions and BC basis functions into different sub-blocks according to the octree structure.

4. The electromagnetic scattering analysis method based on consistent discretization of magnetic field integral equations of characteristic basis functions according to claim 1 is characterized in that: Step 5 is as follows: for all sub-blocks, the electric field integral equation and impedance matrix equation of the block are constructed using the moment method according to the RWG basis function and BC basis function in the block, the incident angle of the entire sphere is sampled, and two mutually orthogonal polarized plane waves are used for irradiation at these sampled incident angles to obtain the current solutions under these irradiations; Taking the mth block as an example, the RWG matrix equation and BC matrix equation of the mth subproblem are expressed as in, is the L-operator RWG impedance matrix of the mth block, is the incident electric field excitation of the RWG of the mth block, is the RWG current coefficient matrix to be determined for the mth block, is the L-operator BC impedance matrix of the mth block, is the incident electric field excitation of the BC of the mth block, is the BC current coefficient matrix of the mth block; the L operator expression is Directly solving these two matrix equations yields the current solution and The Gram matrix G′ connecting these two current solutions m The expression is The superscript H represents the conjugate transpose of the matrix, G m is the Gram matrix connecting the rotated BC basis function and the RWG basis function in the mth block; the Gram matrix connecting these two current solutions is subjected to singular value decomposition Among them, U m and V m is a unitary matrix, D m is a diagonal matrix whose diagonal elements are G′ m The singular values ​​of the unitary matrix U are retained according to the threshold of the singular value decomposition m and V m The first k m Column, get and Accordingly, is to keep the first k m The diagonal matrix of singular values, is a diagonal matrix whose diagonal elements are The arithmetic square root of the diagonal elements in ; then, the pair of characteristic basis functions of the mth block and The expressions are Arrange the characteristic basis coefficients of all blocks diagonally to obtain the characteristic basis function of the entire problem. Its characteristic basis coefficient matrix C BC and C RWG It is expressed as: Where B represents the number of blocks.

5. The electromagnetic scattering analysis method based on consistent discretization of magnetic field integral equations of characteristic basis functions according to claim 1 is characterized in that: The sixth step is to discretize the surface integral equation using the moment method and construct the reduced impedance matrix equation. The reduced matrix equation of MFIE is expressed as z C I C =V C (14) Among them, Z C is the characteristic basis function C BC and C RWG As the impedance matrix of the detection function and basis function, V C is the characteristic basis function C BC Incident magnetic field excitation as a function of detection, I C is the required characteristic basis current coefficient matrix of the whole problem; the expressions of impedance matrix and incident magnetic field excitation are From C =(C BC ) H ZC RWG (15) In C =(C BC ) H In (16) Where Z is the impedance matrix of the magnetic field integral equation MFIE with the BC basis function and the RWG basis function as the detection function and basis function respectively, and V is the incident magnetic field excitation of the BC basis function as the detection function; when the characteristic basis current solution I is obtained C After that, the final current solution I is given by the following formula RWG I RWG =C RWG I C (17) Current vector I RWG The coefficient a of the RWG basis function n The expression is I RWG =(a1,a2,…,a N ) I (18) Where N is the total number of RWG basis functions and the superscript T represents the transpose of the vector.

6. The electromagnetic scattering analysis method based on consistent discretization of magnetic field integral equations of characteristic basis functions according to claim 1 is characterized in that: Step 7 is as follows: After using the iterative solver to solve the MFIE reduction matrix equation to obtain the current coefficient, the final equivalent surface current J(r) is expressed as: Among them, f n (r) is the nth RWG basis function, a n is the current coefficient corresponding to the nth RWG basis function; According to the equivalent surface current solution, the far-field value E of the target scattered electric field is solved. sca : E sca (r)=Z0L{J(r′)} (20) Further calculation is performed to obtain the radar cross section RCS of the target, which is represented by the symbol σ and the calculation expression is: The unit of σ is m. 2 , E inc is the incident electric field.

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