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     

High-performance absorbing boundary conditions for photonic crystalwaveguide simulations

A high-performance absorbing boundary condition is newly developed for the reduction of spurious reflections in photonic crystal (PC) waveguide simulations. The PC waveguide is terminated with a perfectly matched layer (PML) in which the original PC structure remains as is. This PC-based PML works well, compared to a conventional PML, which acts as a homogeneous absorbing medium, simulating a semi-infinite free space and to a distributed-Bragg-reflector waveguide, which was recently developed to reduce reflections from PC waveguide ends, improving a wavenumber matching condition for PC waveguide modes     

Adaptive absorbing boundary conditions in finite-difference timedomain applications for EMC simulations

A special adaptation of the Liao's (1984) absorbing boundary condition (ABC) is presented to optimize the solution of a class of EMI/EMC problems when using the finite-difference time domain (FDTD) method. The Liao ABC is constructed on a per-cell basis to accommodate the most likely direction of incidence on the mesh terminating boundary. Numerical results are presented to demonstrate the validity of the formulation and the improvements over traditional ABC applications     

Retarded time absorbing boundary conditions

Over the past few years simulations of electromagnetic problems in three dimensions using the finite difference time domain (FDTD) method have become increasingly popular. A major problem in such simulations is the truncation of the computational domain. A formulation of this boundary problem using retarded time values of the field inside the computational domain is suggested, and hence the name retarded time absorbing boundary condition (RT-ABC). This formulation allows the boundary to be situated in the near field of the problem and thereby reduces the necessary computational domain, and the present formulation allows error estimates for the numerically calculated fields     

Absorbing boundary conditions applied to model wave propagation inmicrowave integrated-circuits

A general expression of an absorbing boundary condition is presented in this paper to model wave propagation in passive microwave integrated-circuits by the finite-difference time-domain method. Unlike previously developed absorbing boundary conditions which can only absorb propagating waves, this boundary condition can also absorb evanescent waves effectively. The microstrip line is used as an example to demonstrate how to impose this absorbing boundary condition on different outer boundaries of a computation-domain. It is also demonstrated that the numerical stability of this absorbing boundary condition, when it is applied in its high order form, can be maintained by properly selecting its parameters     

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     

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.     

TLM absorbing boundary conditions and microstrip antennasimulations

The authors deal with the application of two different absorbing boundary conditions on a radiating microstrip antenna simulation using the TLM method. A comparative study concerning the respective efficiency of matched terminations and Higdon's conditions is made. For this, we have computed the radiation pattern and the input impedance. The authors have compared these results with those of other authors     

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     

PML absorbing boundary conditions for the characterization of openmicrowave circuit components using multiresolution time-domaintechniques (MRTD)

The multiresolution time domain technique (MRTD) is applied to the modeling of open microwave circuit problems. Open boundaries are simulated by the use of a novel formulation of the perfect matching layer (PML) absorber. The PML is modeled both in split and nonsplit forms and can be brought right on the surface of the planar components. The applicability of the MRTD technique to complex geometries with high efficiency and accuracy in computing the fields at discontinuities is demonstrated through extensive comparisons to conventional finite difference time domain (FDTD). In addition, the numerical reflectivity of the PML absorber is investigated for a variety of cell sizes, some of which are very close to the Nyquist limit (λ/2)     

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     

Higher order absorbing boundary conditions for thefinite-difference time-domain method

Higher-order absorbing boundary conditions are introduced and implemented in a finite-difference time-domain (FDTD) computer code. Reflections caused by the absorbing boundary conditions are examined. For the case of a point source radiating in a finite computational domain, it is shown that the error decreases as the order of approximation of the absorbing boundary condition increases. Fifth-order approximation reduces the normalized reflections to less than 0.2%, whereas the widely used second-order approximation produces about 3% reflections. A method for easy implementation of any order approximation is also presented     

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     

吸收边界条件下含时薛定谔方程的解

The performances of absorbing boundary conditions （ABCs） in four widely used finite difference time domain （FDTD） methods, i.e. explicit, implicit, explicit staggered-time, and Chebyshev methods, for solving the time-dependent Schrodinger equation are assessed and compared. The computation efficiency for each approach is also evaluated. A typical evolution problem of a single Gaussian wave packet is chosen to demonstrate the performances of the four methods combined with ABCs. It is found that ABCs perfectly eliminate reflection in implicit and explicit staggered-time methods. However, small reflection still exists in explicit and Chebyshev methods even though ABCs are applied.     

Maxwellian material-based absorbing boundary conditions for lossymedia in 3-D

A two time-derivative Lorentz material (2TDLM), which has been shown previously to be the correct Maxwellian medium choice to match an absorbing layer to a lossy region, is extended here to a complete absorbing boundary condition (ABC) for three-dimensional (3-D) finite-difference time-domain (FDTD) simulators. The implementation of the lossy 2TDLM (L2TDLM) ABC is presented. It is shown that in contrast to the one-dimensional (1-D) and two-dimensional (2-D) versions, the full 3-D ABC requires a three time-derivative Lorentz material in the edge and corner regions to achieve a rigorous matching of the resulting Maxwellian absorbing layer to the lossy medium. The 3-D ABC implementation thus requires the introduction of an auxiliary field to handle the edge and corner regions to achieve a state-space form of the update equations in the ABC layers. Fully 3-D examples including pulsed dipole radiation and pulsed Gaussian beam propagation in lossless and lossy materials as well as pulse propagation along a microstrip over lossless and lossy materials are included to illustrate the effectiveness of the L2TDLM ABC     

Numerical absorbing boundary conditions for the scalar and vectorwave equations

Electromagnetic field computation may be carried out conveniently by using the finite element method (FEM). When solving open region problems using this technique, it becomes necessary to enclose the scatterer with an outer boundary upon which an absorbing boundary condition (ABC) is applied; analytically-derived ABCs have been used extensively for this purpose. Numerical absorbing boundary conditions (NABCs) have been proposed as alternatives to analytical ABCs. For the two-dimensional (2-D) Helmholtz equation, it has been demonstrated analytically that these NABCs become equivalent to many of the existing analytical ABs in the limit as the cell size tends to zero. In addition, the numerical efficiency of these NABCs has been evaluated by using as an indicator the reflection coefficient for plane and cylindrical waves incident upon an arbitrary boundary. We have extended this procedure to the study of the NABCs derived, for the three-dimensional (3-D) scalar and vector wave equations from the point of view of their numerical implementation in node- and edge-based FEM formulations, respectively     

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     

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     

Higher order formulation of absorbing boundary conditions for finite-difference time-domain method

A simple formulation of absorbing boundary conditions with higher order approximation is proposed for the finite-difference time-domain (FD-TD) method. Although this formulation is based on the third order approximation of the one-way wave equations the authors have succeeded in reducing it to an equation in a form quite similar to the second order approximation.<>     

Comparative evaluation of absorbing boundary conditions usingGreen's functions for layered media

Absorbing boundary conditions are comparatively studied using the Green's functions of the vector and scalar potentials for multilayer geometries and general sources. Since the absorbing boundaries are introduced as additional layers with predefined reflection coefficients into the calculation of the Green's functions, this approach provides an absolute measure of the effectiveness of different absorbing boundaries. The Green's functions are calculated using different reflection coefficients corresponding to different absorbing boundaries and compared to those obtained with no absorbing boundary. It is observed that the perfectly matched layer (PML) is by far the best among the other absorbing boundary conditions whose reflection coefficients are available