A theoretical study of numerical absorbing boundary conditions

Electromagnetic field computation involving inhomogeneous, arbitrarily-shaped objects may be carried out conveniently by using partial differential equation techniques, e.g., the finite element method (FEM). When solving open region problems using these techniques, it becomes necessary to enclose the scatterer with an outer boundary on which an absorbing boundary condition (ABC) is applied, and analytically-derived ABCs, e.g., the Bayliss-Gunzburger-Turkel and Engquist-Majda boundary conditions have been used extensively for this purpose. Numerical absorbing boundary conditions (NABCs) have been proposed as alternatives to analytical ABCs, and they are based upon a numerically-derived relationship that links the values of the field at the boundary nodes to those at the neighboring nodes. In the paper the authors demonstrate, analytically, that these NABCs become equivalent to many of the existing analytical ABCs in the limit as the cell size tends to zero. In addition, one can evaluate the numerical efficiency of these NABCs by using as an indicator the reflection coefficient for plane and cylindrical waves incident upon an arbitrary boundary     

Numerical implementation of second- and third-order conformalabsorbing boundary conditions for the vector-wave equation

A rigorous implementation, in an edge-based finite-element formulation of second- and third-order conformal absorbing boundary conditions (ABCs) is presented for the solution of three-dimensional (3-D) scattering problems when the boundary S terminating the mesh is the surface of a parallelepiped. A special treatment is provided for the singularities (edges) of S. A systematic numerical study is carried out that compares the performances of these ABCs with those of the standard zero-order ABC, as well as of a more simple, though less rigorous, implementation of the second-order ABC. When S is separated from the scatterer by only one or two layers of elements, the numerical results that are presented demonstrate the good level of numerical accuracy achieved when the second- and third-order ABCs are employed and the singularities of S are appropriately dealt with     

Vector one-way wave absorbing boundary conditions for FEMapplications

In the paper a derivation is presented which leads to a new and general class of vector absorbing boundary conditions (ABCs) for use with the finite element method (FEM). The derivation is based on a vector one-way wave equation and a polynomial approximation of the vector radical. It is shown that wide-angle absorbing boundary conditions, as proposed in Halpern and Trefethen (1988) for optimal absorption of out-going waves, can be obtained in vector form. Vector plane waves are used to evaluate the accuracy and the reflection performance of these boundary conditions in a wide range of incidence angles. The implementation of the vector ABCs in a FEM formulation is also provided to show how up to the fifth-order absorbing accuracy can be achieved with derivatives only up to the second-order. A possible formulation is described which not only yields a third-order accuracy with first-order derivatives, but also retains the symmetry of the FEM matrix     

A numerical absorbing boundary condition for finite difference andfinite element analysis of open periodic structures

In this paper we present a novel approach to deriving local boundary conditions, that can be employed in conjunction with the Finite Difference/Finite Element Methods (FD/FEM) to solve electromagnetic scattering and radiation problems involving periodic structures. The key step in this approach is to derive linear relationships that link the value of the field at a boundary grid point to those at the neighboring points. These linear relationships are identically satisfied not only by all of the propagating Floquet modes but by a few of the leading evanescent ones as well. They can thus be used in lieu of absorbing boundary conditions (ABCs) in place of the usual FD/FEM equations for the boundary points. Guidelines for selecting the orders of the evanescent Floquet modes to be absorbed are given in the paper. The present approach not only provides a simple way to derive an accurate boundary condition for mesh truncation, but also preserves the banded structure of the FD/FEM matrices. The accuracy of the proposed method is verified by using an internal check and by comparing the numerical results with the analytic solution for perfectly conducting strip gratings     

Total-field absorbing boundary conditions for the time-domainelectromagnetic field equations

A method is described for generating absorbing boundary conditions (ABCs) that can be applied to the total fields rather than the usual scattered fields. As compared with the traditional use of ABCs for total-field formulations, this method has the advantages that it does not require the introduction of a mathematical connection surface between the total-field region and the scattered-field region; the total field is computed in the entire domain of computation. The incident field is accounted for by augmenting the ABC used. The resulting code is much simpler than one using ABCs for scattered fields together with a connection surface and the numerical results are much more easily interpreted since they consist of total fields only     

Second-order absorbing boundary conditions for marching-on-in-order scheme

In this letter, we derive second-order absorbing boundary conditions (ABCs) for a marching-on-in-order scheme, which is a time-domain method with weighted Laugerre polynomials and free of stability constraint. This method does not have to deal with time steps, and may be computationally much more efficient than conventional finite-difference time-domain (FDTD)methods with too many time steps to complete a solution. Starting from the three-dimensional (3-D) case of the marching-on-in-order scheme, we deduce a second-order ABC and apply it to a 3-D microstrip line example. The results show the second-order ABC performs better than the first-order one for guided wave problems.     

Higher order impedance and absorbing boundary conditions

Traditionally, generalized impedance boundary conditions (GIBCs) have been used to model dielectrics and coated surfaces, and absorbing boundary conditions (ABCs) have been used to simulate nonreflecting surfaces. The two types have the same mathematical form and, in most instances, a higher order condition involving higher order field derivatives has a better accuracy. We demonstrate that there is a close connection between the two and this enables us to use a systematic method which is available for generating GIBCs of any desired order to derive new two- and three-dimensional ABCs. The method is applicable to curvilinear/doubly-curved surfaces and examples are given. Finally, curves are presented that quantify the accuracy of two-dimensional ABCs up to the fourth order, and show how higher order ABCs can improve the efficiency of large scale partial differential equation (PDE) solutions     

Optimization of nonlinear dispersive APML ABC for the FDTD analysis of optical solitons

We have investigated the parameter optimization for the nonlinear dispersive anisotropic perfectly matched layer (A-PML) absorbing boundary conditions (ABCs) for the two- and the three-dimensional (2D and 3D) finite-difference time-domain (FDTD) analyses of optical soliton propagation. The proposed PML is applied to the FDTD method of the standard and the high-spatial-order schemes. We first searched for the optimum values of the loss factor, permittivity, and the order of polynomial grading for particular numbers of APML layers in a two-dimensional (2-D) setting with Kerr and the Raman nonlinearity and Lorentz dispersion, and then we applied the optimized APML to a full three-dimensional (3-D) analysis of nonlinear optical pulse propagation in a glass substrate. An optical pulse of spatial and temporal soliton profile has been launched with sufficient intensity of electric field to yield a soliton pulse, and a reflection of -60dB has been typically obtained both for the 2-D and the 3-D cases.     

Implementation of anisotropic PML for frequency domain FEM solution of discontinuity in dielectric waveguide

The anisotropic Perfectly Matched Layer(PML) absorbing boundary condition is implemented in a 2-D finite element formulation to solve dielectric waveguide discontinuity problems. The choice of parameters of anisotropic PML has been investigated. Using the boundary truncating technique, the solution process of Finite-Element Method (FEM) has been greatly simplified compared with other hybrid methods. The required computational resources have also significantly declined since the anisotropic PML interface can be placed much closer to the scatterer compared to other well known artificial boundary.     

Stability of absorbing boundary conditions

Higher order absorbing boundary conditions (ABCs) exhibit instabilities that can be detrimental to a wide class of finite-difference time-domain (FDTD) open-region simulations. Earlier works attributed the cause of instabilities to the intrinsic construction or makeup of the ABCs, and consequently to the pole-zero distribution of the transfer function that characterizes the boundary condition. We investigate the cause of instability, We focus on axial boundary conditions such as Higdon (1986, 1990), Bayliss-Turkel (1980), and Liao, and show through an empirical study that these ABCs are not intrinsically unstable in their original unmodified forms. Furthermore, we show that the instability typically observed in FDTD open-region simulations is caused by an artifact of the rectangular computational domain, contrary to previously conjectured hypotheses or theories. These findings will have strong implications that can aid in the construction of stable FDTD schemes     

A new 2-D model for the potential distribution and threshold voltage of fully depleted short-channel Si-SOI MESFETs

A new two-dimensional (2-D) analytical model for the threshold voltage of a fully depleted short-channel Si-MESFETs fabricated on the silicon-on-insulator (SOI) has been presented in this paper. The 2-D potential distribution functions in the active layer of the device is approximated as a parabolic function and the 2-D Poisson's equation has been solved with suitable boundary conditions to obtain the bottom potential at the Si/oxide layer interface. The calculations have been carried out for both uniform and nonuniform doping profiles in two dimensions. The minimum bottom potential is used to monitor the drain-induced barrier lowering effect and consequently an analytical expression for the threshold voltage of the device has been derived. The numerical results for the bottom potential and threshold voltage considering a wide range of device parameters have also been presented. The model has been compared with the simulated results obtained by using the ATLAS Device Simulation Software to show the validity of the proposed model. For uniform doping profile, the numerical results have also been compared with the reported data in the literature and a good agreement is observed among the three. The proposed model is simple and easy to understand the behavior of the fully depleted short-channel SOI-MESFETs as compared to the other models reported in the literature.     

Comparison of performance of absorbing boundary conditions in TLM and FDTD

Both in the transmission line matrix (TLM) and the finite difference time domain (FDTD) methods, absorbing boundary conditions (ABCs) are used to truncate the computational domain for the simulation or open boundary problems. A comparison of the performance of ABCs applied to TLM and FDTD is presented. The results indicate a significant improvement in the performance of an ABC when applied to TLM compared to the performance of the same ABC when applied to FDTD. An explanation for this improvement in the performance is given     

An FDTD model for calculation of gradient-induced eddy currents in MRI system

In most magnetic resonance imaging (MRI) systems, pulsed magnetic gradient fields induce eddy currents in the conducting structures of the superconducting magnet. The eddy currents induced in structures within the cryostat are particularly problematic as they are characterized by long time constants by virtue of the low resistivity of the conductors. This paper presents a three-dimensional (3-D) finite-difference time-domain (FDTD) scheme in cylindrical coordinates for eddy-current calculation in conductors. This model is intended to be part of a complete FDTD model of an MRI system including all RF and low-frequency field generating units and electrical models of the patient. The singularity apparent in the governing equations is removed by using a series expansion method and the conductor-air boundary condition is handled using a variant of the surface impedance concept. The numerical difficulty due to the "asymmetry" of Maxwell equations for low-frequency eddy-current problems is circumvented by taking advantage of the known penetration behavior of the eddy-current fields. A perfectly matched layer absorbing boundary condition in 3-D cylindrical coordinates is also incorporated. The numerical method has been verified against analytical solutions for simple cases. Finally, the algorithm is illustrated by modeling a pulsed field gradient coil system within an MRI magnet system. The results demonstrate that the proposed FDTD scheme can be used to calculate large-scale eddy-current problems in materials with high conductivity at low frequencies.     

Comparative study of absorbing boundary conditions for thetime-domain beam propagation method

Various absorbing boundary conditions (ABCs) are compared in the analysis of the time-domain finite-difference beam propagation method. For a one-dimensional problem, the following ABCs are tested: Higdon's absorbing boundary, Ramahi's complementary operators method (COM), its concurrent version (C-COM) and Berenger's perfectly matched layer (PML). It is found that the second- and third-order C-COMs with three and four boundary cells are comparable to the PMLs with eight and 16 cells, respectively. The effectiveness of the C-COM is also discussed in a two-dimensional problem     

Three-dimensional diffraction by infinite conducting and dielectric wedges using a generalized total-field/scattered-field FDTD formulation

We extend the generalized total-field/scattered-field formulation of the finite-difference time-domain method to permit efficient computational modeling of three-dimensional (3-D) diffraction by infinite conducting and dielectric wedges. This new method allows: 1) sourcing a numerical plane wave having an arbitrary incident angle traveling into, or originating from, a perfectly matched layer absorbing boundary and 2) terminating the infinite wedge inside the perfectly matched layer with negligible reflection. We validate the new method by comparing its results with the analytical diffraction coefficients for an infinite 3-D right-angle perfect electric conductor wedge obtained using the uniform theory of diffraction. Then, we apply the new method to calculate numerical diffraction coefficients for a 3-D infinite right-angle dielectric wedge, covering a wide range of incident and scattering angles. Finally, we show means to compactly store the calculated diffraction coefficients in a manner which permits easy interpolation of the results for arbitrary incidence and observation angles.     

Finite-element time-domain analysis of axisymmetrical radiators

A method of analysis which permits the computation of electromagnetic fields directly in the time domain using the finite element method (FEM) is presented. The method considers coaxially-driven axisymmetrical structures. A new way of implementing absorbing boundary conditions for the coaxial feed is discussed. The method has been successfully used to model coaxially-driven axisymmetrical monopole antennas for a Gaussian pulse excitation. Generally, the results obtained show good agreement with previously published numerical and experimental data     

Conformal absorbing boundary conditions for 3-D problems:derivation and applications

Vector absorbing boundary conditions (ABCs) for doubly curved surfaces are presented, and their applicability to finite elements for scattering calculations are discussed. A performance study of these ABCs is carried out in terms of accuracy and computational requirement, and scattering patterns for several targets are included for validation purposes. It is found that accurate far-field results can be obtained by terminating the finite element mesh a fraction of a wavelength from the scattering structure     

以改进的混合方法预测室外到室内的电波传播

采用了理想匹配吸收层技术改进了了基于衍射几何射线法(GTD)与时域有限差分法(FDTD)的混合方法，改进后的方法使计算结果比基于Mur吸收边界条件的混合法更精确更具有稳定性。该方法可作为一种移动通信系统设计方法的新选择；利用该方法研究了由室外到室内的电波传播预测，给出了计算结果和传播规律性。     

Complex Envelope Crank-Nicolson Nearly PML Algorithm for the Left-handed Material FDTD Simulations

Unconditionally stable complex envelope (CE) absorbing boundary conditions (ABCs) are presented for truncating left handed material (LHM) domains. The proposed algorithm is based on incorporating the Crank Nicolson (CN) scheme into the CE finite difference time domain (FDTD) implementations of the nearly perfectly matched layer (NPML) formulations. The validity of the formulations is shown through numerical example carried out in one dimensional Lorentzian type LHM FDTD domain.