High-order Absorbing Boundary Conditions Research Articles

High-order local absorbing conditions for the wave equation: Extensions and improvements

The solution of the time-dependent wave equation in an unbounded domain is considered. An artificial boundary B is introduced which encloses a finite computational domain. On B an absorbing boundary condition (ABC) is imposed. A formulation of local high-order ABCs recently proposed by Hagstrom and Warburton and based on a modification of the Higdon ABCs, is further developed and extended in a number of ways. First, the ABC is analyzed in new ways and important information is extracted from this analysis. Second, The ABCs are extended to the case of a dispersive medium, for which the Klein–Gordon wave equation governs. Third, the case of a stratified medium is considered and the way to apply the ABCs to this case is explained. Fourth, the ABCs are extended to take into account evanescent modes in the exact solution. The analysis is applied throughout this paper to two-dimensional wave guides. Two numerical algorithms incorporating these ABCs are considered: a standard semi-discrete finite element formulation in space followed by time-stepping, and a high-order finite difference discretization in space and time. Numerical examples are provided to demonstrate the performance of the extended ABCs using these two methods.

LOCAL HIGH-ORDER ABSORBING BOUNDARY CONDITIONS FOR TIME-DEPENDENT WAVES IN GUIDES

The scalar wave equation in a two-dimensional semi-infinite wave guide is considered. The recently proposed Hagstrom–Warburton (H–W) local high-order absorbing boundary conditions (ABCs), which are based on a modification of the Higdon ABCs, are presented in this context. The P-order ABC involves the free parameters 0 < aj ≤ 1, for j = 0, 1, …, P, which have to be chosen. The choice aj = 1 for all j is shown to be satisfactory, in general, although not necessarily optimal. The optimal choice of the parameters is discussed via both theoretical analysis and numerical experiments. In addition, an adaptive scheme which controls the time-varying values of P and aj is presented and tested.

Padded continued fraction absorbing boundary conditions for dispersive waves

Continued fraction absorbing boundary conditions (CFABCs) are new arbitrarily high-order local absorbing boundary conditions that are highly effective in simulating wave propagation in unbounded domains. The current versions of CFABCs are developed for the non-dispersive acoustic wave equation with convex polygonal computational domains. In this paper, the CFABCs are modified through augmentation of special padding elements, and are effective for absorbing evanescent waves occurring in dispersive wave problems while retaining their absorption properties for propagating waves. The padded CFABCs for dispersive wave equations result in fourth order evolution equations, which are solved using an efficient combination of Crank Nicholson method, Newmark time-stepping scheme, and operator splitting ideas. The effectiveness of the padded CFABCs and their implementation is illustrated through numerical examples with varying levels of dispersion.

Finite element formulation with high-order absorbing boundary conditions for time-dependent waves

The Hagstrom–Warburton high-order absorbing boundary conditions (ABCs) are considered. They are based on a high-order form of the Higdon ABCs using auxiliary variables and constitute a modification of the previously proposed Givoli–Neta ABCs. Here the Hagstrom–Warburton ABCs, which were originally used in a finite difference scheme, are incorporated into a finite element formulation. Exterior time-dependent problems are considered with rectangular computational domains. Special corner conditions are used in conjunction with the ABCs to make the truncated problem well-posed. The properties of the Hagstrom–Warburton and Givoli–Neta formulations are compared, and the relations between the two formulations are established. Numerical examples demonstrate the performance of the Hagstrom–Warburton finite element scheme.

Discrete Absorbing Boundary Conditions for Schrödinger-Type Equations. Construction and Error Analysis

When a partial differential equation in an unbounded domain is solved numerically, it is necessary to introduce artificial boundary conditions. In this paper, a general class of absorbing boundary conditions is constructed for one-dimensional Schrödinger-type equations discretized in space by finite differences. For this, rational approximations to the transparent boundary conditions are used. We study the simplest case in detail, obtaining an estimate for the full discrete error and showing that the discrete problem is weakly unstable. Moreover, we show numerically that the discrete problems associated to higher order absorbing boundary conditions are more unstable. Several numerical experiments confirm the results previously obtained.

CONTINUED-FRACTION ABSORBING BOUNDARY CONDITIONS FOR THE WAVE EQUATION

Absorbing boundary conditions are generally required for numerical modeling of wave phenomena in unbounded domains. Local absorbing boundary conditions are generally preferred for transient analysis because of their computational efficiency. However, their accuracy is severely limited because the more accurate high-order boundary conditions cannot be implemented easily. In this paper, a new arbitrarily high-order absorbing boundary condition based on continued fraction approximation is presented. Unlike the existing boundary conditions, this one does not contain high-order derivatives, thus making it amenable to implementation in conventional C0 finite element and finite difference methods. The superior numerical properties and implementation aspects of this boundary condition are discussed. Numerical examples are presented to illustrate the performance of these new high-order boundary condition.

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.

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.

SLIM: a slender technique for unbounded-field problems

Electrostatic- and electromagnetic-field problems in unbounded regions are often solved using finite differences (FD's) or finite elements (FE) combined with approximate boundary conditions. Inversion of the sparse FD or FE matrix is then required. First-order or higher order absorbing boundary conditions may be used, or one can use more accurate boundary conditions obtained by the measured equation of invariance (MEI) or by iteration. The more accurate boundary conditions are helpful because they permit reduction of the size of the mesh and, thus, the number of unknowns. In this paper, we show that the process can be carried to a maximally simple limit in which the mesh is reduced to a single layer and the matrix-inversion step disappears entirely. This results in the single-layer iterative method (SLIM), an unusually simple technique for unbounded-field problems. Computational experiments demonstrate the effectiveness of SLIM in electrostatics, and also in electrodynamic examples, such as scattering of TM plane waves from a perfectly conducting cylinder. The technique is most likely to be useful in large or complex problems where simplification is helpful, or in repetitive calculations such as scattering of radiation from many angles of incidence.

Implementation and Derivation of Conformal Absorbing Boundary Conditions for the Vector Wave Equation

A rigorous implementation, in an edge-based finite element formulation, of a second-order absorbing boundary condition (ABC) is presented for the solution of the 3-D scattering problem when the boundary S terminating the mesh is conformal to the object. Numerical experiments performed on a cone-sphere and a cylinder with hemispherical caps are presented that justify the use of this ABC. A high-order ABC is derived in the locally planar approximation, and its performances are evaluated for the analytical case of the sphere, which serves to address a number of important issues pertaining, e.g., to the accuracy with which the creeping waves are taken into account when an ABC is utilized.

Exact implementation of higher order Bayliss-Turkel absorbing boundary operators in finite-element simulation

A simple yet powerful scheme is employed to incorporate Bayliss-Turkel absorbing boundary conditions (ABCs) of any order into the finite-element simulation of open-region radiation problems. Unlike previous attempts to apply higher order ABC's in finite elements, the new implementation is exact. The exact implementation is made possible by incorporating normal derivatives in the ABC formulation through direct algebraic substitution. This scheme is found to offer enhanced accuracy while negligibly affecting the sparsity of the system matrix.

Stable FDTD solutions with higher-order absorbing boundary conditions

One-way wave equation type boundary operators can produce instabilities in finite-difference-time-domain (FDTD) simulations. Such instabilities become acute when the order of the operator is three or higher. Previous efforts to stabilize these operators sacrificed either efficiency or accuracy. This Letter presents a simple scheme that produces stable FDTD solutions by using double-precision arithmetic over a fraction of the domain while using damping coefficients that are small enough to maintain accuracy, especially over the lower frequencies. This procedure gives accurate solutions while only marginally increasing the memory overhead. © 1997 John Wiley & Sons, Inc. Microwave Opt Technol Lett 15: 132–134, 1997.

A comparative study of the Berenger perfectly matched layer, the superabsorption technique and several higher-order ABC's for the FDTD algorithm in two and three dimensional problems

A new comparison of higher-order Engquist & Majda (1977), Higdon (1986), and Liao (1992) absorbing boundary conditions (ABCs) and the superabsorption method with the Berenger (see J. Comput. Physics, vol.114, p.185-200, 1994) perfectly matched layer (PML), is conducted. Reflection errors are calculated both globally and locally in 2-D and especially in new 3-D FDTD simulations in the presence or absence of a perfectly conducting scatterer. Numerical results indicate that the PML is by far superior to all other ABCs, especially when their layers, are implemented. It is also shown that the 5th-order Liao ABC incorporated in the superabsorption technique, yields a scheme which is almost equivalent to the 4-layer PML when referring to the induced local error. Absorption performance of the new boundary conditions obtained from the combination of the PML with the 2nd and 3rd-order Liao ABCs is, finally, evaluated.

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 comparison of the Berenger perfectly matched layer and the Lindman higher-order ABC's for the FDTD method

Higher-order absorbing boundary conditions are compared to the recently introduced Berenger perfectly matched layer (PML) absorbing boundary conditions (ABC). Reflections caused by the ABC's are examined in both the time and frequency domains for the case of a line source radiating in a finite computational domain. It is shown that the PML ABC significantly reduces reflections from the truncation of the computational grid when compared to 7th order Lindman ABC's. Also, except for at low frequencies, higher-order absorbing boundary conditions are no better than 2nd order Mur absorbing boundaries. >

The maximal accuracy of stable difference schemes for the wave equation

We consider three time-level difference schemes, symmetric in time and space, for the solution of the wave equation,u tt =c2u xx , given by $$\sum\limits_{j = - S}^S {b_j U_{n + 1,m + j} + } \sum\limits_{j = - S}^S {a_j U_{n,m + j} + } \sum\limits_{j = - S}^S {b_j U_{n - 1,m + j} } = 0.$$ It has already been proved that the maximal order of accuracyp of such schemes is given byp ≤ 2(s + S). In this paper we show that the requirement of stability does not reduce this maximal order for any choice of the pair (s, S). The result is proved by introducing an order star on the Riemann surface of the algebraic function associated with the scheme. Furthermore, Pade schemes, withS = 0,s > 0, ands = 0,S > 0, are proved to be stable for 0 < μ < 1, where μ is the Courant number. These schemes can be implemented with high-order absorbing boundary conditions without reducing the range of μ for which stable solutions are obtained.

An element for the analysis of transient exterior fluid-structure interaction problems using the FEM

A finite element-based procedure is presented for the solution directly in the time domain of transient problems involving structures submerged in an infinite acoustic fluid. A central component of the methodology employed herein, and presented in greater detail elsewhere, is a novel element that arises upon discretization of a high-order absorbing boundary condition introduced in the formulation of the fluid-structure interaction problem in order to render the computational domain finite. The new element is local in both time and space and is completely defined by a pair of symmetric stiffness and damping matrices. The familiar form of the discritized equations of motion for the structure is retained with its symmetry and sparseness intact. Standard temporal integration techniques can then be used for the solution of the equations. In this paper we present the methodology in a two-dimensional setting, together with numerical examples involving both circular and non-circular shells. Although the focus is on the time domain, the methodology is equally applicable in the frequency domain, thus providing a unified and efficient tool for the treatment of the exterior structural acoustics problem.

Higher order absorbing boundary conditions for the finite-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.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>