A weak form of the conjugate gradient FFT method fortwo-dimensional TE scattering problems

The problem of two-dimensional scattering of a transversal electric polarized wave, by a dielectric object is formulated in terms of a hypersingular integral equation, in which a grad-div operator acts on a vector potential. The vector potential is a spatial convolution of the free-space Green's function and the contrast source over the domain of interest. A weak form of the integral equation for the unknown electric flux density is obtained by testing it with rooftop functions. The vector potential is expanded in a sequence of the rooftop functions and the grad-div operator is integrated analytically over the dielectric object domain only. The method shows excellent numerical performance     

A weak form of the conjugate gradient FFT method for plate problems

A number of electromagnetic field problems for planar structures can be formulated in terms of a hypersingular integral equation, in which a grad-div operator acts on a vector potential. The vector potential is a convolution of the free-space Green's function and some surface current density over the domain of interest. A weak form of this integral equation is obtained by testing it with subdomain basis functions defined over the plate domain only. As a next step, the vector potential is expanded in a sequence of subdomain basis functions and the grad-div operator is integrated analytically over the plate domain only. For the problem of electromagnetic scattering by a plate, the method shows excellent numerical performance. The numerical difficulties encountered in some previous conjugate gradient fast Fourier transform (CGFFT) methods have been eliminated     

Preconditioned conjugate gradient methods for solving electromagnetic problems

It is shown that appropriately chosen preconditioners can significantly improve the convergence rate of the conjugate gradient (CG) algorithm as applied to electromagnetic problems. Preconditioners are constructed for the problems of scattering from frequency selective surfaces and scattering from infinite cylinders.     

预条件共轭梯度法在辐射和散射问题中的应用

用矩量法求解一些辐射和散射问题 ,如线天线辐射和线状体散射等问题时 ,可以产生一个 Toeplitz线性方程组 ,采用预条件共轭梯度法 (PCG)与快速富里叶变换 (FFT)的结合方法 (PCGFFT)来求解该方程组 ,其中预条件器采用 T.Chan的优化循环预条件器。使用 PCGFFT算法 ,可有效地节省内存 ,提高了计算速度。为说明其有效性 ,将 PCGFFT算法与 CGFFT算法以及 Levinson递推算法进行了对比。     

Convergence of the conjugate gradient method when applied to matrix equations representing electromagnetic scattering problems

An iterative procedure based on the conjugate gradient method is used to solve a variety of matrix equations representing electromagnetic scattering problems, in an attempt to characterize the typical rate of convergence of that method. It is found that this rate depends on the cell density per wavelength used in the discretization, the presence of symmetries in the solution, and the degree to which mixed cell sizes are used in the models. Assuming cell densities used in the discretization are in the range of ten per linear wavelength, the iterative algorithm typically requiresN/4toN/2steps to converge to necessary accuracy, whereNis the order of the matrix under consideration.     

An iterative method for solving scattering problems

An iterative method is developed for computing the current induced by plane wave excitation on conducting bodies of arbitrary shape. In this method, the scattering body is divided into lit- and shadow-side regions separated by the geometric optics boundary. The induced current at any point on the surface of the scatterer is expressed as the sum of an approximate optics current and a correction current. Both of these currents are computed by iteration for the lit and shadow regions separately. The general theory is presented and applied to the problems of scattering from a two-dimensional cylinder of circular and square cross sections. The results are compared with the method of moments and good agreement is obtained. This method does not give erroneous results at internal resonances of the scatterer, does not suffer from computer storage problems and can be extended to nonperfect conductors as well as to three-dimensional bodies.     

解决电磁散射问题的现代高频技术

电大尺寸散射体的高频散射高效算法问题是一个重要而具有挑战性的问题.本文首先对解决电磁散射问题的现代高频方法的研究进展进行了综述.其次,对高频散射场的高振荡特性,提出了数值最速下降路径法来计算高频散射场.最后,综合讨论了高频驻相点贡献凯勒锥以及爬行波场.     

Comments on “An improved pulse-basis conjugate gradient FFTmethod for the thin conducting plate problem” [with reply]

Peter and Zwamborn comment on the paper by Tran and McCowen (IEEE Trans. Antenn. Propag., vol.41, p.185-90, 1993) which presents pulse-basis expansion functions and Dirac δ testing functions within a conjugate gradient FFT (CGFFT) formulation. In the conclusions, Tran and McCowen state: “Although these spurious effects do not appear in the CGFFT formulation using the rooftop basis function proposed by Zwamborn and van den Berg (1991), there are more computational costs associated with their method than with ours.” Peter and Zwamborn disagree with this statement. A reply is given by Tran and McCowen     

A constrained conjugate gradient method for solving the magnetic field boundary integral equation

It is well-known that electromagnetic solutions of boundary integral equations for perfectly electrically conducting scatterers are nonunique for those frequencies which correspond to interior resonances of the scatterer. In this paper a simple and efficient computational method is developed, in which the interior integral representations, required to hold on an interior closed surface, are used as a sufficient constraint to restore uniqueness. We use the interior equations together with the second kind magnetic field integral equation, so that the ill-posedness of the interior equations does not give a problem. We develop a constrained conjugate gradient method that minimizes a cost functional consisting of two terms. The first term is the error norm with respect to the magnetic field boundary integral equation, while the second term is the error norm with respect to the interior equations over a closed interior surface, which is chosen as small as possible. Some numerical examples show the robustness and efficiency of the pertaining computational procedure.     

On the implementation of the conjugate gradient Fourier transformmethod for scattering by planar plates

The accuracy of two conjugate gradient fast Fourier transform formulations for computing the electromagnetic scattering by resistive plates of an arbitrary periphery is discussed. One of the formulations is based on a discretization of the integral equations prior to the introduction of the Fourier transform, whereas the other is based on a similar discretization after the introduction of the Fourier transform. The efficiency and accuracy of these formulations are examined by comparison with measured data for rectangular and nonrectangular plates. The latter method is found to provide a more accurate computation of the plate scattering by eliminating aliasing errors (other than those due to undersampling). It is also found to be substantially more efficient. Its greatest advantage is realized when solving large systems generated by convolutional operators not yielding Toeplitz matrices, as is the case with plates having nonuniform resistivity     

The application of the conjugate gradient method for the solution of electromagnetic scattering from arbitrarily oriented wire antennas

The method of conjugate gradients is applied to the analysis of radiation from thin-wire antennas. With this iterative technique, it is possible to solve electrically large arbitrarily oriented wire structures without storing any matrices as is conventionally done in the method of moments. The basic difference between the proposed method and Galerkin's method, for the same expansion functions, is that for the iterative technique we are solving a least squares problem. Hence, as the order of the approximation is increased, the proposed technique guarantees a monotonic decrease of the least squared error (parallel AI - Yparallel^{2}), whereas Galerkin's method does not. Even though the method converges for any initial guess, a good one may significantly reduce the time of computation. Also, explicit error formulas are given for the rate of convergence of this method. Hence, any problem can be solved to a prespecified degree of accuracy. It is shown that the method has the advantage of a direct solution as the final solution is obtained in a finite number of steps. The method is also suitable for solving singular operator equations in which case the method monotonically converges to the least squares solution with minimum norm. Numerical results are presented for the thin-wire antennas and are compared with the solution obtained by the method of moments.     

The use of vector transforms in solving some electromagnetic scattering problems

It is shown that there exists a class of vector transforms and vector series expansions that are independent of their scalar counterparts. They can be used to simplify the solutions of a class of vector electromagnetic scattering problems. The applications are shown of vector Fourier, Hankel, and Mathieu transforms and series expansions to the formal solutions of scattering by rectangular, circular and elliptical disks and open-ended waveguides. The problems solved are canonical problems, however, more complex problems for applications in microwaves, microstrip integrated circuits, geophysical probing, etc. can be solved in a similar fashion.     

Unimoment method of solving antenna and scattering problems

It has been shown by this investigator and numerous others [6], [7], [8] that exterior boundary value problems involving localized inhomogeneous media are most conveniently solved using finite difference or finite element techniques together with integral equations or harmonic expansions, which satisfy the radiation conditions. The methods result in large matrices that are partly full and partly sparse; and methods to solve them, such as iteration or banded matrix methods are not very satisfactory. The unimoment method alleviates the difficulties by decoupling exterior problems from the interior boundary value problems. This is done by solving the interior problem many times so thatNlinearly independent solutions are generated. The continuity conditions are then enforced by a linear combination of theNindependent solutions, which may be done by solving much smaller matrices. Methods of generating solutions of the interior problems are discussed.     

A concise conjugate gradient computation of plate problems withmany excitations

One of the major difficulties in the application of the conjugate gradient algorithm for the analysis of electromagnetic scattering problems is the necessity to carry out the calculation separately for each incident wave. In the approach suggested, rather than handling the incident waves directly, a class of possible excitations is represented by a set of strip-type basis functions. For these functions, convergence is predictable and rapid because the majority of the strips are located away from the edges of the scatterer. This choice also facilitates the use of the physical optics approximation as a good initial guess. Once the solutions for all the unit basis functions over the body are known, they can be combined to synthesize the solution for any excitation using the weighting coefficients associated with the expansion of the incident field. Numerical examples are given, and they demonstrate the substantial savings achieved by adopting this approach for the analysis of multiple excitations     

GPU加速混合场积分方程求解导体目标散射问题

利用图形处理单元（GPU）加速混合场积分方程（CFIE）分析导体目标电磁散射问题。较电场积分方程（EFIE）和磁场积分方程（MFIE）,CFIE消除了内谐振问题,并且具有更好的条件数。求解的数值方法为基于 RWG基函数的矩量法（MoM）。所有计算步骤均在 GPU上实现,包括：阻抗元素填充、电压向量填充、矩阵方程的共轭梯度（CG）求解、雷达散射截面（RCS）计算。在保证数值精确度的前提下获得了数十倍的速度提升。     

采用分部外推BCGS-FFT方法快速求解电磁散射问题

为了快速求解电磁散射问题中具有震荡性、奇异性、慢收敛性的索末菲积分,提出了一种利用分部外推算法加速索末菲尾部积分计算,并结合稳定双共轭快速傅里叶变换(stabilized biconjugate gradient fast Fourier transform,BCGS-FFT)算法求解电磁散射问题场分布情况的新方法. 首先给出电场积分方程(electric field integral equation, EFIE)的表达形式,且在求解过程的索末菲积分中应用一种便捷的椭圆积分路径来最小化索末菲积分的震荡性与奇异性,在索末菲尾部积分使用Levin分部外推法来提高积分收敛速度,以此来快速填充并矢格林函数矩阵. 然后对新方法进行了多种数值实验,验证算法的精确度,并对比了新方法与传统BCGS-FFT方法的计算效率,发现在保持相同计算精度的条件下,新方法可节省20%～37%的计算时间. 该方法能应用于复杂散射体嵌入多层空间的电磁散射计算,为快速求解目标区域的电磁散射场提供了一种新的方法.     

The conjugate gradient spectral iterative technique for planarstructures

It is shown that using the spectral iterative technique (SIT) to solve the first-kind integral equation is equivalent to the Neumann iterative solution of a related second-kind integral equation. It is thus shown that SIT only converges when the norm of the operator in the second-kind equation is small enough. Applying a conjugate gradient technique to the second-kind equation results in a convergent iterative scheme. Some representative numerical results show a superiority in the rate of convergence of the conjugate gradient scheme for the second-kind equation (CGSIT-scheme) when compared with the convergence of the conjugate scheme for the original first-kind equation (CG-scheme). The CGSIT-scheme combines the advantages of the conjugate gradient method with those of the spectral iterative technique     

The biconjugate gradient method for electromagnetic scattering

The biconjugate gradient (BCG) method for solving linear systems is shown to be more efficient than the conjugate gradient (CG) method for several examples from electromagnetic scattering. A remedy for the occasional stagnation of the algorithm is proposed. The potential flaw in the BCG algorithm may be avoided when encountered by restarting the algorithm with a perturbed estimate of the solution     

Application of a conjugate gradient FFT method to scattering fromthin planar material plates

The backscatter cross section is calculated for thin material plates with finite electric permittivity, conductivity, and magnetic permeability illuminated by a plane wave. The plates are assumed to be planar with an arbitrary perimeter. The integral equations are formed and solved by a combined conjugate gradient-fast Fourier transform (CG-FFT) method. The CG-FFT method was tested for several geometrics and materials measured and computed backscatter results are compared for a perfectly conducting equilateral triangle plate, a square dielectric and magnetic plate, and a circular dielectric plate. The agreement between measured and computed data is generally good except toward edge-on incidence where several factors cause discrepancies. Accurate approximations to the geometry and far-field integrals become critical near edge-on incidence and, it is postulated that as the incidence angle approaches edge-on, the sampling interval and tolerance should be decreased     

Boundary integrodifferential equations for electromagnetic scattering problems in three dimensions

On the basis of two vector representations of electromagnetic fields we introduce a new system of boundary integrodifferential equations for the solution of scattering problems in three dimensions. The unknowns of this system present two scalar functions, namely, the "" coefficients of Atkinson-Wilcox expansion; electromagnetic field being reconstructed with these functions by means of certain recursive-differential operators. We define an algebraic analog of the equations by expanding unknowns into Fourier series with respect to spherical harmonics. Verification of the approach is done on the basis of the solution of well-known canonical problems.