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。     

Solution to some electromagnetic problems using fast Fourier transform with conjugate gradient method

Using the fast Fourier transform (FFT) to compute the convolution integrals that appear in the conjugate-gradient method (CGM), an efficient numerical procedure to solve electromagnetic problems is obtained. In comparison with the method of moments (MM), the proposed FFT-CGM avoids the storage of large matrices and reduces the computer time by orders of magnitude.     

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     

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.     

Calculation of electromagnetic scattering from a dielectric cylinder using the conjugate gradient method and FFT

The scattering problem of an axially uniform dielectric cylinder is formulated in terms of the electric field integral equation, where the cylinder is of general cross-sectional shape, inhomogeneity, and anisotropy, and the incident field is arbitrary. Using the pulse-function expansion and the point-matching technique, the integral equation is reduced to a system of simultaneous equations. Then, a published procedure for solving the system using the conjugate gradient method and the fast Fourier transform (FFT) is generalized to the case of oblique-incidence scattering.     

Conjugate gradient method applied to inverse scattering problem

A new reconstruction algorithm for diffraction tomography is presented. The algorithm is based on the minimization of a functional which is defined as the norm of the discrepancy between the measured scattering amplitude and the calculated one for an estimated object function. By using the conjugate gradient method to minimize the functional, one can derive an iterative formula for getting the object function. Numerical results for some two-dimensional scatterers show that the algorithm is very effective in reconstructing refractive index distributions to which the first-order Born approximation can not be applied. In addition, the number of iterations is reduced by using a priori information about the outer boundary of the objects. Furthermore, the method is not so sensitive to the presence of noise in the scattered field data     

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     

基于频域广义传输矩阵方法的电磁散射特性分析

多尺度复杂电子系统的电磁场问题难以用单一的计算电磁学方法进行高效数值计算.基于区域分解方法和惠更斯等效原理,提出了频域广义传输矩阵（generalized transition matrix,GTM）方法:将系统分解为多个子模块,通过电场积分方程（electric field integreal equation,EFIE）把各个子模块的电磁特性进行提炼,再考虑所有子模块之间的电磁耦合,计算系统整体电磁场分布.GTM方法把多尺度问题转化为尺度相对比较单一的问题进行处理,在分析各种复合结构、非均匀各向异性介质、大型相控阵天线等电磁散射特性时,提供了灵活的解决方案.论文给出了GTM在手征介质、开口腔体以及Vivaldi相控阵天线电磁特性分析中的应用算例,当未知量个数压缩到原来的十分之一时,GTM计算结果与直接用矩量法（methed of moment,MoM）求解的计算结果非常吻合.GTM可以简洁地表示目标问题的电磁散射特征,与传统MoM相比,大幅度减少了基函数的数量,具有较高的计算精度和效率.     

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.     

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     

Effect of sampling rate on the conjugate gradient method applied tosignal extrapolation

A different response to changes in sampling rate has been observed for conjugate gradients as opposed to SVD (singular-value decomposition) applied to signal extrapolation. The conjugate gradient solution is described in terms of the SVD, which leads to an explanation of the behavior of the conjugate gradient method in this context. Formulating the conjugate gradient method as a spectral filtering method and analyzing it in terms of the spectral filter functions generated as the iteration progresses have made it possible to explain the difference in the effect of sampling rate changes on the conjugate gradients approach to signal extrapolation as compared to the truncated SVD approach discussed by Sullivan and Liu (1984)     

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.     

The three dimensional weak form of the conjugate gradient FFTmethod for solving scattering problems

The problem of electromagnetic scattering by a three-dimensional dielectric object 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 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 relevant unknown quantity is obtained by testing it with appropriate testing functions. The vector potential is then expanded in a sequence of the appropriate expansion functions and the grad-div operator is integrated analytically over the scattering object domain only. A weak form of the singular Green's function has been used by introducing its spherical mean. As a result, the spatial convolution can be carried out numerically using a trapezoidal integration rule. This 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     

分段积分共形算法在电磁散射问题中的应用

提出一种分段积分共形时域有限差分算法(PI-CFDTD),用于计算电磁散射问题。首先采用邻近网格电场场量插值表示变形网格中沿线电场场量,再以分段积分代替传统电场沿线积分求解,减小了阶梯误差,提高了电场环路积分的计算精度。推导出两种不同类型变形网格的电场环路积分公式,并对PI-CFDTD算法的稳定性进行研究,归纳得到应用原则。以金属方形平板和金属圆形平板作为算例进行验证,通过与传统时域有限差分法(FDTD)、传统共形时域有限差分法(CFDTD)以及矩量法(MoM)进行比较,表明PI-CFDTD计算精度更高。     

Three-dimensional biorthogonal multiresolution time-domain method and its application to electromagnetic scattering problems

A three-dimensional (3-D) multiresolution time-domain (MRTD) analysis is presented based on a biorthogonal-wavelet expansion, with application to electromagnetic-scattering problems. We employ the Cohen-Daubechies-Feauveau (CDF) biorthogonal wavelet basis, characterized by the maximum number of vanishing moments for a given support. We utilize wavelets and scaling functions of compact support, yielding update equations involving a small number of proximate field components. A detailed analysis is presented on algorithm implementation, with example numerical results compared to data computed via the conventional finite-difference time-domain (FDTD) method. It is demonstrated that for 3-D scattering problems the CDF-based MRTD often provides significant computational savings (in computer memory and run time) relative to FDTD, while retaining numerical accuracy.     

Finite-element solution to electromagnetic scattering problems bymeans of the Robin boundary condition iteration method

A new numerical method, called the Robin boundary condition iteration (RBCI), is proposed for the finite-element (FE) solution of electromagnetic scattering problems in open boundary domains. The unbounded domain is truncated to a bounded one by means of a fictitious boundary that contains the scatterer and on which a suitable nonhomogeneous Robin (mixed) boundary condition is assumed for the Helmholtz equation in the bounded domain. The Robin condition is expressed by means of an integral formula (the second Green identity) in terms of the field in the interior of the bounded domain, with the integration surface being a surface strictly enclosed by the truncation boundary. The discretized differential and integral equations are then coupled together to solve the problem. The formulation is completely immune from the well-known interior resonance problems. A simple and effective iterative solving scheme is described. Examples are also provided to validate RBCI and compare it with other methods     

Computer algebra systems applied to some electromagnetic problems

This paper deals with computer algebra systems and explains how these systems are likely to be extremely relevant in addressing electromagnetic problems. After a historical survey, it discusses the various off-the-shelf systems and their features. We suggest how given problems can be formalized and analysed using symbolic algebra systems. Several applied examples are given, with an important bibliography. We study the flat and conformai microstrip antennas : Green’s functions, solutions of the characteristic equations, and so on. We give many informations on the computer algebra systems on the Internet network.     

The complex bi-conjugate gradient solver applied to largeelectromagnetic scattering problems, computational costs, and costscalings

The complex bi-conjugate gradient iterative method is applied to an isoparametric boundary integral equation formulation for frequency-domain electromagnetic scattering problems. It is demonstrated to work well on large and geometrically complex examples, including a 20 wavelength slender dipole, the NASA almond, and a resonant cavity. On such problems, with asymmetric curvilinear irregular meshes and nontrivial geometries, the number of iterations required seems to increase rather more than linearly with body size, indicating an overall ~sixth power cost scaling. This scaling is essentially as for direct methods, but with costs still a small fraction of the direct approach. A method is proposed for the selection of a termination condition designed to avoid seeking the approximate answer too precisely; it typically permits a further halving of costs