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     

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     

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     

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     

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     

A new absorbing boundary condition structure for waveguide analysis

The existence of evanescent waves and waves near cutoff frequencies limits the accuracy of the fields computed in waveguides using the finite-difference time-domain method, and prompted several researchers to design complicated boundary conditions, including combinations of perfectly matched layers and Higdon's higher order absorbing boundary conditions (ABCs). Instead, we employ a terminating structure in which the lateral walls are made absorbing in addition to the longitudinal walls. The undesirable lateral waves at the normal boundary interface are slowed down and effectively attenuated in the lateral walls, while the propagating waves are absorbed in the longitudinal walls. Numerical calculations for pulse excitation of a rectangular waveguide, using the simple Mur's first-order ABC, demonstrate the usefulness of the method     

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     

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     

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     

Solution of the time-dependent Schrdinger equation with absorbing boundary conditions

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 Schr dinger 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 perfectl...     

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     

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.     

Solution of the time-dependent Schr(o)dinger equation with absorbing boundary conditions

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 Schr(o)dinger 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 perfor-mances 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.     

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     

Absorbing boundary conditions for convex object-conformableboundaries

Absorbing boundary conditions (ABCs) are developed that can be applied on object-conformable outer boundaries. The new ABCs are based on the local enforcement of the Nth order Bayliss-Turkel boundary conditions where a scattering center is defined for each outer boundary node. A demonstration of the effectiveness of the new construction is provided by considering representative numerical experiments using the finite-elements method. Results show that the new ABCs provide accuracy that compares very favorably with the method of moments solution     

Absorbing boundary conditions for the three-dimensional vector waveequation

Absorbing boundary conditions (ABCs) are constructed for the finite-element solution of the three-dimensional (3-D) vector wave equations. Applied on spherical outer boundaries, the new operators are derived by first representing the scalar components of the field in a series of powers 1/τ. Then the Bayliss-Turkel (1980) boundary operators are enforced on scalar field components, which are tangential to the outer boundary. Unlike previous boundary condition constructions, the new scheme makes possible the implementation of operators of order three or higher, thus increasing the accuracy potential of analytic ABCs     

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     

A new FEM approach for open boundary Laplace's problem

An efficient improved finite element method (FEM) is presented for electromagnetic Laplace's problems with open boundary. The whole infinite domain is divided into a set of infinite elements instead of ordinary finite elements. Since a special FEM discretization and FEM solving procedure are used, it can not only take much less computer memory than that the conventional FEM needs, but also avoid the calculation error introduced by the truncated boundary or absorbing boundary condition used in conventional FEM     

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.