A subgridding method for the time-domain finite-difference methodto solve Maxwell's equations

A modification to the time-domain finite-difference method (TDFDM) that uses a variable step size is investigated. The entire computational volume is divided into a coarse grid with a large step size. A fine grid with a small step size is introduced only around discontinuities. The corresponding time increments are related to the spatial increments with the same ratio in order to minimize the numerical dispersion. The fields within the coarse and fine grids are found using the TDFDM, while an interpolation in space and time is utilized to calculate the tangential electric field on the coarse-fine grid boundary. This subgridding decreases the required computer memory and therefore expands the capability of the TDFDM. The technique is shown to be numerically stable and does not entail any extra numerical error. The method is applied to the calculation of waveguides and microstrips     

Spectral Lanczos decomposition method for time domain and frequencydomain finite-element solution of Maxwell's equations

A technique that combines the spectral Lanczos decomposition method and the finite-element method is introduced that can be used to solve Maxwell's equations in both the frequency and time domains. The present technique is an implicit, unconditionally stable finite-element time and frequency domain scheme. The Lanczos process is implemented only at the largest time or frequency of interest. The efficiency and effectiveness of this new technique are illustrated by using the numerical example of a three-dimensional dielectric-loaded cavity resonator     

An unstaggered colocated finite-difference scheme for solvingtime-domain Maxwell's equations in curvilinear coordinates

In this paper, we present a new unstaggered colocated finite-difference scheme for solving time-domain Maxwell's equations in a curvilinear coordinate system. All components of the electric and magnetic fields are defined at the same spatial point. A combination of one-sided forward- and backward-difference (FD/BD) operators for the spatial derivatives is used to produce the same order of accuracy as a staggered, central differencing scheme. In the temporal variable, the usual leapfrog integration is used. The computational domain is bounded at the far end by a curvilinear perfectly matched layer (PML). The PML region is terminated with a first-order Engquist-Majda-type absorbing boundary condition (ABC). A comparison is shown with results available in the literature for TE_z scattering by conducting cylinders. Equations are also presented for the three-dimensional (3-D) case     

Efficient time-domain and frequency-domain finite-element solutionof Maxwell's equations using spectral Lanczos decomposition method

An efficient three-dimensional solver for the solution of the electromagnetic fields in both time and frequency domains is described. The proposed method employs the edge-based finite-element method (FEM) to discretize Maxwell's equations. The resultant matrix equation after applying the mass-lumping procedure is solved by the spectral Lanczos decomposition method (SLDM), which is based on the Krylov subspace (Lanczos) approximation of the solution. This technique is, therefore, an implicit unconditionally stable finite-element time and frequency-domain scheme, which requires the implementation of the Lanczos process only at the largest time or frequency of interest. Consequently, a multiple time- and frequency-domain analysis of the electromagnetic fields is achieved in a negligible amount of extra computing time. The efficiency and effectiveness of this new technique are illustrated by using numerical examples of three-dimensional cavity resonators     

The time domain Green's function and propagator for Maxwell's equations

The free space time domain propagator and corresponding dyadic Green's function for Maxwell's differential equations are derived in one-, two-, and three-dimensions using the propagator method. The propagator method reveals terms that contribute in the source region, which to our knowledge have not been previously reported in the literature. It is shown that these terms are necessary to satisfy the initial condition, that the convolution of the Green's function with the field must identically approach the initial field as the time interval approaches zero. It is also shown that without these terms, Huygen's principle cannot be satisfied. To illustrate the value of this Green's function two analytical examples are presented, that of a propagating plane wave and of a radiating point source. An accurate propagator is the key element in the time domain path integral formulation for the electromagnetic field.     

High spatial order finite element method to solve Maxwell's equations in time domain

This paper presents a finite element method with high spatial order for solving the Maxwell equations in the time domain. In the first part, we provide the mathematical background of the method. Then, we discuss the advantages of the new scheme compared to a classical finite-difference time-domain (FDTD) method. Several examples show the advantages of using the new method for different kinds of problems. Comparisons in terms of accuracy and CPU time between this method, the FDTD and the finite-volume time-domain methods are given as well.     

A comparison of transient solutions of Maxwell's equations to thatof the modified Maxwell's equations

In 1986 H.F. Harmuth introduced a modification of Maxwell's equations to study the propagation of transient electric and magnetic field strengths in lossy media. Opponents of this modification of Maxwell's equations have claimed and attempted to demonstrate that Maxwell's equations in their known forms can correctly be solved, for example by the Laplace transformation method, to obtain solutions of transient electric and associated magnetic field strengths in lossy media without encountering any difficulties. This work presents detailed computer plots of Harmuth's transient solutions of the modified Maxwell's equations and that of Maxwell's equations solved by the Laplace transformation characteristic for the two solutions, which indicate that they are not the same. It is shown that Harmuth's procedure results in physically more plausible solutions     

A vector diagram of Maxwell's equations

A vector diagram illustrating Maxwell's equations is derived. Upon equating various components of vectors in the diagram, a number of common relationships between field quantities are obtained. The potentials and their relationships to field quantities may also be represented. Justification of the procedure used to construct the diagram is based on Fourier transformation of Maxwell's equations. While the diagram is mostly of interest for its novelty, it has found use in clarifying certain gauge choices in dealing with electromagnetic potentials     

Finite element implementation of Maxwell's equations for image reconstruction in electrical impedance tomography

Traditionally, image reconstruction in electrical impedance tomography (EIT) has been based on Laplace's equation. However, at high frequencies the coupling between electric and magnetic fields requires solution of the full Maxwell equations. In this paper, a formulation is presented in terms of the Maxwell equations expressed in scalar and vector potentials. The approach leads to boundary conditions that naturally align with the quantities measured by EIT instrumentation. A two-dimensional implementation for image reconstruction from EIT data is realized. The effect of frequency on the field distribution is illustrated using the high-frequency model and is compared with Laplace solutions. Numerical simulations and experimental results are also presented to illustrate image reconstruction over a range of frequencies using the new implementation. The results show that scalar/vector potential reconstruction produces images which are essentially indistinguishable from a Laplace algorithm for frequencies below 1 MHz but superior at frequencies reaching 10 MHz.     

A domain decomposition method for the vector wave equation

A nonoverlapping domain decomposition method (DDM) is presented for the finite-element (FE) solution of electromagnetic scattering problems by inhomogeneous three-dimensional (3-D) bodies. The computational domain is partitioned into concentric subdomains on the interfaces of which conformal vector transmission conditions are prescribed and that can be implemented in the inhomogeneous part. The DDM is numerically implemented when a conformal vector absorbing boundary condition (ABC) is utilized on the outer boundary terminating the FE mesh, while employing the standard edge-based FE formulation. Then, numerical experiments are performed on a sphere and a cone sphere that emphasize the advantages of this technique in terms of memory storage and computing times, especially when the total number of unknowns is very large. Also, these numerical experiments serve as a severe test for the performances of the ABC     

一种分析多目标散射问题的区域分解方法

提出了一种用于分析复杂多目标散射问题的区域分解方法. 在该方法中,每个目标作为一个独立的计算区域采用矢量有限元方法进行分析;各个区域之间通过基于格林函数的边界积分方程进行耦合;所得到的耦合矩阵方程采用基于Foldy-Lax多径散射方程的特征基函数方法进行求解. 由于矢量有限元方法的灵活性,该区域分解方法特别适合于求解多个具有相同结构复杂目标的散射问题. 数值算例验证了该方法的准确性和处理复杂多目标散射问题的能力.     

分析波导问题的松弛迭代区域分解法

针对基于Schwarz交替法的选代区域分解法,在分析波导问题时遇到的不收敛的困难,该文从实际场分布出发,在划分区域的虚拟边界上给出了连接子域的吸收虚拟边界条件,用以保证相邻子域间的波传播,从而构建了一种能够分析波导问题的收敛的迭代区域分解法。在此基础上进一步引入松弛算法,用于加快迭代收敛速度。数值计算结果表明了该方法的有效性。     

A local mesh refinement algorithm for the time domain-finitedifference method using Maxwell's curl equations

An efficient local mesh refinement algorithm, subdividing a computational domain to resolve fine dimensions in a time-domain-finite-difference (TD-FD) space-time grid structure, is discussed. At a discontinuous coarse-fine mesh interface, the boundary conditions for the tangential and normal field components are enforced for a smooth transition of highly nonuniform held quantities     

The finite-difference time-domain (FDTD) and the finite-volumetime-domain (FVTD) methods in solving Maxwell's equations

The finite-difference time-domain (FDTD) and its current generalizations have been demonstrated to be useful and powerful tools for the calculation of the radar cross section (RCS) of complicated objects, the radiation of antennas in the presence of other structures, and other applications. The mathematical techniques for conformal FDTD have matured; the primary impediments to its implementation are the complex geometries and material properties associated with the problem. Even under these circumstances, FDTD is more flexible to implement because it is based on first principles instead of a clever mathematical trick. This paper gives an account of some new results on conformal FDTD obtained by the authors and their associates at Lockheed Martin Space Company since 1988. The emphasis is on nonsmooth boundary condition simulation     

An efficient implementation of surface impedance boundaryconditions for the finite-difference time-domain method

An efficient way to implement the surface impedance boundary conditions (SIBC) for the finite-difference time-domain (FDTD) method is presented in this paper. Surface impedance boundary conditions are first formulated for a lossy dielectric half-space in the frequency domain. The impedance function of a lossy medium is approximated with a series of first-order rational functions. Then, the resulting time-domain convolution integrals are computed using recursive formulas which are obtained by assuming that the fields are piecewise linear in time. Thus, the recursive formulas derived here are second-order accurate. Unlike a previously published method [7] which requires preprocessing to compute the exponential approximation prior to the FDTD simulation, the preprocessing time is eliminated by performing a rational approximation on the normalized frequency-domain impedance. This approximation is independent of material properties, and the results are tabulated for reference. The implementation of the SIBC for a PEC-backed lossy dielectric shell is also introduced     

Some variational principles for discontinuous Maxwell's equations

In this paper, some variational principles are derived to directly calculate dielectric problems featuring a discontinuous surface. First, a two-field variational principle is deduced from Hamilton's principle for a regular dielectric region. Next, this variational principle is augmented through an involutory transformation, and then a twelve-field variational principle is formulated which generates, as its Euler-Lagrange equations, Maxwell's equations with discontinuous electromagnetic fields.     

High-accuracy FDTD solution of the absorbing wave equation, and conducting Maxwell's equations based on a nonstandard finite-difference model

We previously introduced high-accuracy finite-difference time-domain (FDTD) algorithms based on nonstandard finite differences (NSFD) to solve the nonabsorbing wave equation and the nonconducting Maxwell equations. We now extend our methodology to the absorbing wave equation and the conducting Maxwell equations. We first derive an exact NSFD model of the one-dimensional wave equation, and extend it to construct high-accuracy FDTD algorithms to solve the absorbing wave equation, and the conducting Maxwell's Equations in two and three dimensions. For grid spacing h, and wavelength /spl lambda/, the NSFD solution error is /spl epsiv//spl sim/(h//spl lambda/)/sup 6/ compared with (h//spl lambda/)/sup 2/ for ordinary FDTD algorithms using second-order central finite-differences. This high accuracy is achieved not by using higher-order finite differences but by exploiting the analytical properties of the decaying-harmonic solution basis functions. Besides higher accuracy, in the NSFD algorithms the maximum time step can be somewhat longer than for the ordinary second-order FDTD algorithms.     

求解三维麦克斯韦方程的时域无网格算法研究

发展了用于求解三维麦克斯韦方程的时域无网格算法.算法基于生成的无网格点云,通过泰勒级数展开结合加权最小二乘逼近计算点云中心点上的空间导数,并构造近似黎曼解处理空间离散涉及的通量运算;空间离散后的半离散方程则采用四步Runge-Kutta格式推进求解.结合求解三维麦克斯韦方程,给出了时域无网格算法的具体实施过程,并基于发展的算法,成功地模拟出金属球、立方体及进气道模型等三维散射目标的电磁散射场,获得的雷达散射截面能与理论解、矩量法或精确控制法等结果吻合.     

Solution of Maxwell's equations for log-periodic dipole antennas

A theory of the log-periodic dipole antenna, which is a solution of the antenna boundary-value problem, is presented here. The theory is derived from Maxwell's equations by solving the wave equation in cylindrical coordinates and satisfying all boundary conditions. The theory is not limited to the log-periodic dipole antenna, but can be easily modified and applied to other antenna configurations using parallel linear elements. The radiation coupling between all antenna elements is taken into account; the calculated results show good agreement with the measurements. Current distributions, radiation patterns, and antenna input impedances are considered, and the application of this theory to the problem of optimal log-periodic dipole antennas is presented as well. Such an antenna obtained by numerical computation is discussed in detail.