Order Absorbing Boundary Condition Research Articles

A finite element scheme with a high order absorbing boundary condition for elastodynamics

A high-order absorbing boundary condition (ABC) is devised on an artificial boundary for time-dependent elastic waves in unbounded domains. The configuration considered is that of a two-dimensional elastic waveguide. In the exterior domain, the unbounded elastic medium is assumed to be isotropic and homogeneous. The proposed ABC is an extension of the Hagstrom–Warburton ABC which was originally designed for acoustic waves, and is applied directly to the displacement field. The order of the ABC determines its accuracy and can be chosen to be arbitrarily high. The initial boundary value problem including this ABC is written in second-order form, which is convenient for geophysical finite element (FE) analysis. A special variational formulation is constructed which incorporates the ABC. A standard FE discretization is used in space, and a Newmark-type scheme is used for time-stepping. A long-time instability is observed, but simple means are shown to dramatically postpone its onset so as to make it harmless during the simulation time of interest. Numerical experiments demonstrate the performance of the scheme.

ℎ𝑝-Discontinuous Galerkin methods for the Helmholtz equation with large wave number

In this paper we develop and analyze some interior penalty h p hp -discontinuous Galerkin ( h p hp -DG) methods for the Helmholtz equation with first order absorbing boundary condition in two and three dimensions. The proposed h p hp -DG methods are defined using a sesquilinear form which is not only mesh-dependent (or h h -dependent) but also degree-dependent (or p p -dependent). In addition, the sesquilinear form contains penalty terms which not only penalize the jumps of the function values across the element edges but also the jumps of the first order tangential derivatives as well as jumps of all normal derivatives up to order p p . Furthermore, to ensure the stability, the penalty parameters are taken as complex numbers with positive imaginary parts, so essentially and practically no constraint is imposed on the penalty parameters. It is proved that the proposed h p hp -discontinuous Galerkin methods are stable (hence, well-posed) without any mesh constraint. For each fixed wave number k k , sub-optimal order (with respect to h h and p p ) error estimates in the broken H 1 H^1 -norm and the L 2 L^2 -norm are derived without any mesh constraint. The error estimates as well as the stability estimates are improved to optimal order under the mesh condition k 3 h 2 p − 2 ≤ C 0 k^3h^2p^{-2}\le C_0 by utilizing these stability and error estimates and using a stability-error iterative procedure, where C 0 C_0 is some constant independent of k k , h h , p p , and the penalty parameters. To overcome the difficulty caused by strong indefiniteness (and non-Hermitian nature) of the Helmholtz problems in the stability analysis for numerical solutions, our main ideas for stability analysis are to make use of a local version of the Rellich identity (for the Laplacian) and to mimic the stability analysis for the PDE solutions given in [19, 20, 33], which enable us to derive stability estimates and error bounds with explicit dependence on the mesh size h h , the polynomial degree p p , the wave number k k , as well as all the penalty parameters for the numerical solutions.

Finite Element Analysis of Electromagnetic Scattering Using $p$-Adaption and an Iterative Absorbing Boundary Condition

Solving scattering problems using the finite element method (FEM) introduces two sources of error: from the discretization and from the truncation boundary. Here both errors are addressed in an iterative, balanced way. For the discretization error, p-adaption is used. An a posteriori error indicator automatically determines which elements should be increased in order to reduce and equi-distribute the discretization error. For the boundary error, an iterative absorbing boundary condition is applied. The overall scheme starts with low order polynomials and a first order absorbing boundary condition and progressively improves the quality of the solution by a combination of p-adaption and updating the boundary condition, so that both the discretization and the boundary errors are similar at each stage. Results are presented to show the reduction of computation time when p-adaption is used as opposed to an earlier approach of uniformly increasing the element orders.

Discontinuous Galerkin Methods for the Helmholtz Equation with Large Wave Number

This paper develops and analyzes some interior penalty discontinuous Galerkin (IPDG) methods using piecewise linear polynomials for the Helmholtz equation with the first order absorbing boundary condition in two and three dimensions. It is proved that the proposed discontinuous Galerkin methods are stable (hence well-posed) without any mesh constraint. For each fixed wave number $k$, optimal order (with respect to $h$) error estimate in the broken $H^1$-norm and suboptimal order estimate in the $L^2$-norm are derived without any mesh constraint. The latter estimate improves to optimal order when the mesh size $h$ is restricted to the preasymptotic regime (i.e., $k^2 h \gtrsim 1$). Numerical experiments are also presented to gauge the theoretical result and to numerically examine the pollution effect (with respect to $k$) in the error bounds. The novelties of the proposed IPDG methods include the following: First, the methods penalize not only the jumps of the function values across the element edges but also the jumps of the normal and tangential derivatives; second, the penalty parameters are taken as complex numbers of positive imaginary parts, so essentially and practically no constraint is imposed on the penalty parameters. Since the Helmholtz problem is a non-Hermitian and indefinite linear problem, as expected, the crucial and the most difficult part of the whole analysis is to establish the stability estimates (i.e., a priori estimates) for the numerical solutions. To this end, the cruxes of our analysis are to establish and to make use of a local version of the Rellich identity (for the Laplacian) and to mimic the stability analysis for the PDE solutions given in [P. Cummings, Analysis of Finite Element Based Numerical Methods for Acoustic Waves, Elastic Waves and Fluid-Solid Interactions in the Frequency Domain, Ph.D. thesis, The University of Tennessee, Knoxville, TN, 2001], [P. Cummings and X. Feng, Math. Models Methods Appl. Sci., 16 (2006), pp. 139-160], [U. Hetmaniuk, Commun. Math. Sci., 5 (2007), pp. 665-678].

Optimal Discontinuous Galerkin Methods for the Acoustic Wave Equation in Higher Dimensions

In this paper, we develop and analyze a new class of discontinuous Galerkin (DG) methods for the acoustic wave equation in mixed form. Traditional mixed finite element (FE) methods produce energy conserving schemes, but these schemes are implicit, making the time-stepping inefficient. Standard DG methods give explicit schemes, but these approaches are typically dissipative or suboptimally convergent, depending on the choice of numerical fluxes. Our new method can be seen as a compromise between these two kinds of techniques, in the way that it is both explicit and energy conserving, locally and globally. Moreover, it can be seen as a generalized version of the Raviart–Thomas FE method and the finite volume method. Stability and convergence of the new method are rigorously analyzed, and we have shown that the method is optimally convergent. Furthermore, in order to apply the new method for unbounded domains, we apply our new method with the first order absorbing boundary condition. The stability of the resulting numerical scheme is analyzed.

Lacunae based stabilization of PMLs

Perfectly matched layers (PMLs) are used for the numerical solution of wave propagation problems on unbounded regions. They surround the finite computational domain (obtained by truncation) and are designed to attenuate and completely absorb all the outgoing waves while producing no reflections from the interface between the domain and the layer. PMLs have demonstrated excellent performance for many applications. However, they have also been found prone to instabilities that manifest themselves when the simulation time is long. Hereafter, we propose a modification that stabilizes any PML applied to a hyperbolic partial differential equation/system that satisfies the Huygens’ principle (such as the 3D d’Alembert equation or Maxwell’s equations in vacuum). The modification makes use of the presence of lacunae in the corresponding solutions and allows us to establish a temporally uniform error bound for arbitrarily long-time intervals. At the same time, it does not change the original PML equations. Hence, the matching properties of the layer, as well as any other properties deemed important, are fully preserved. We also emphasize that besides the aforementioned PML instabilities per se, the methodology can be used to cure any other undesirable long-term computational phenomenon, such as the accuracy loss of low order absorbing boundary conditions.

Evaluation of a Spherical PML for Vector FEM Applications

The implementation and evaluation of a spherical perfectly matched layer (PML) within a Cartesian finite element method context using standard curl-conforming elements is presented in this paper. Results are compared to the long-standing 1st order absorbing boundary condition (ABC) and a new, rigorous implementation of a 2nd order ABC for curl-conforming elements. The 4 and 8 layer spherical PMLs are shown to offer very attractive levels of absorption, with reflections on the order of -60 to -70dB demonstrated. Numerical tests show that the guidelines for Cartesian PML absorbers, in terms of maximum conductivity, also carry over to the spherical PML. The 2nd order ABC is also shown to offer very good performance. Finally, coding issues for both the spherical PML and the analytical ABCs are briefly addressed

Stability analysis of the Mur's absorbing boundary condition in the alternating direction implicit finite-difference method

The stability analysis of the Mur's first order absorbing boundary condition (ABC) in the alternating direction implicit finite-difference time-domain (ADI-FDTD) method is presented. The theoretical stability analysis of this scheme is studied by deriving the amplification matrix. The effect of wave propagation direction on the stability of this scheme is investigated. From the stability analysis, it is found that the Mur's first order ABC in the ADI-FDTD method will be unstable. The instability of the scheme is validated by means of actual numerical simulations.

Unconditionally stable Crank–Nicolson finite-different time-domain method for simulation of three-dimensional microwave circuits

An unconditionally stable Crank–Nicolson finite-difference time-domain (CN-FDTD) algorithm is presented for three-dimensional microwave circuit analysis. First, Mur's first–order absorbing boundary condition is applied this CN-FDTD algorithm. A symmetric successive over relaxation–preconditioned biconjugate-gradient algorithm is also proposed to solve the large sparse matrix equation obtained in the CN-FDTD method. Resonant cavity and several planar microstrip circuits are presented to illustrate the versatility of this technique. Numerical results indicate that with a time-step size excessively larger than the Courant–Friedrich–Levy limit, the accuracy of CN-FDTD is still much higher than that of the alternating-direction implicit FDTD.

SHARP REGULARITY COEFFICIENT ESTIMATES FOR COMPLEX-VALUED ACOUSTIC AND ELASTIC HELMHOLTZ EQUATIONS

We consider complex-valued acoustic and elastic Helmholtz equations with the first order absorbing boundary condition in a star-shaped domain in ℜN for N ≥ 2. It is known that the elliptic regularity coefficients depend on the frequency ω, and have singularities for both zero and infinite frequency. In this paper, we obtain sharp estimates for the coefficients with respect to large frequencies. It is proved that the elliptic regularity coefficients are bounded by first or second order polynomials in ω for large ω. The crux of our analysis is to establish and make use of Rellich identities for the solutions to the acoustic and elastic Helmholtz equations. Our results improve the earlier estimates of Refs. 10 and 11, which were carried out based on layer potential representations of the solutions of the Helmholtz equations.

Implicit spectral element methods and Neumann–Neumann preconditioners for acoustic waves

A numerical approximation of the acoustic wave equation with first order absorbing boundary conditions is considered. The discretization is based on conforming spectral elements in space and implicit finite differences in time. A stability analysis based on the energy method is developed for the fully discrete scheme. The linear system arising at each step is solved by the conjugate gradient method with Balancing Neumann–Neumann preconditioning. Several numerical results illustrate the stability and convergence properties of the approximation schemes, that result spectrally accurate in space and up to second-order in time, while the Neumann–Neumann solver at each time step is scalable and quasi-optimal.

Finite-Element Analysis of Scattering From a Complex BOR Using Spherical Infinite Elements

Recently, a new type of infinite elements (IFEs) was proposed as an accurate and efficient mesh truncation method for the finite element analysis of open–region acoustic scattering. These IFEs are based on the radial expansion of the scattered field and are used to model the scattered field outside a separable surface characterized by a constant radial coordinate (such as a spherical or a spheroidal surface in three dimensions). In this paper, spherical infinite elements are formulated and implemented for the finite element (FE) analysis of time–harmonic electromagnetic scattering from a complex body of revolution (BOR), and their performance is studied by comparison with the finite element solution using conventional absorbing boundary conditions (ABCs). Numerical results are presented to demonstrate that the new type of infinite elements is indeed more accurate than both the first– and second–order ABCs.

On the boundary conditions for EG methods applied to the two‐dimensional wave equation system

AbstractThe subject of the paper is the study of several nonreflecting and reflecting boundary conditions for the evolution Galerkin (EG) methods which are applied for the two‐dimensional wave equation system. Different known tools are used to achieve this aim. Namely, the method of characteristics, the method of extrapolation, the Laplace transformation method, and the perfectly matched layer (PML) method. We show that the absorbing boundary conditions which are based on the use of the Laplace transformation lead to the Engquist‐Majda first and second order absorbing boundary conditions, see [3]. Further, following Bérenger [1] we consider the PML method. We discretize the wave equation system with the leap‐frog scheme inside the PML while the evolution Galerkin schemes are used inside the computational domain. Numerical tests demonstrate that this method produces much less unphysical reflected waves as well as the best results in comparison with other techniques studied in the paper.

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.

A computer aided education tool for electromagnetic scattering phenomena

In this paper, a computer aided education tool to understand electromagnetic scattering phenomena is described. Real solving is performed using the finite element method coupled with a first order absorbing boundary condition. Computed results are displayed as a time animation. Three examples are presented: radiation from a wire (in order to illustrate propagation phenomena), scattering by a lossy dielectric cylinder (in order to compare induced and displacement currents) and scattering by a parallel plate waveguide (in order to understand the cutoff phenomenon).

FEM-ABC and MFCM techniques applied to the solution of 2D scattering problems

The problem of solving open 2D scattering problems using the finite element method (FEM) and the multi-filament current method (MFCM) is addressed in this paper. The second order absorbing boundary condition of Bayliss-Turkel, for noncircular boundaries, and the filament positioning problem in MFCM are investigated. Two problems defined by two perfect cylindric dielectric bodies, with circular and square cross section, illuminated by a transverse magnetic plane wave are studied. Both the procedures investigated to place the filaments are efficient. The results obtained using the FEM, with first and second order triangular elements, are compared to those obtained using MFCW and show good agreement.

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.

Un résultat d'existence et d'unicité pour l'équation de Helmholtz avec conditions aux limites absorbantes d'ordre

In this Note, we prove the existence and uniqueness of the solution to the Helmholtz equation with second order absorbing boundary conditions in a rectangle. The problem is completed by appropriate corner conditions. The formal approach makes use of a regularity result for the solution of the homogeneous problem. To this aim, one has to consider explicitly the singularities of the Laplacian

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.

Generalized Galerkin approximations of elastic waves with absorbing boundary conditions

For the propagation of elastic waves in unbounded domains, absorbing boundary conditions (ABCs) at the fictitious numerical boundaries have been proposed. In this paper we focus on both first- and second-order ABCs in the framework of variational (weak) approximations, like those stemming from Galerkin method (or its variants) for finite element or spectral approximations [1]. In particular, we recover first order conditions as natural (or Neumann) conditions, whereas we propose a penalty residual method for the treatment of second order ABCs. The time discretization is based on implicit backward finite differences, whereas we use spectral Legendre collocation methods set in a variational form for the spatial discretization (treatment of finite element or spectral element approximations is completely similar). Numerical experiments exhibit that the present formulation of second-order ABCs improves the one based on first-order ABCs with regard to both the reduction of the total energy in the computational domain, and the Fourier spectrum of the displacement field at selected points of the elastic medium. A stability analysis is developed for the variational problem in the continuous case both for first- and second-order ABCs. A suitable treatment of ABCs at corners is also proposed.