Effective Absorbing Boundary Conditions Research Articles

A new implementation of convolutional PML for second-order elastic wave equation

The perfectly matched layer (PML) boundary condition has been widely used as a very effective absorbing boundary condition for seismic wavefield simulations. Convolutional PML (CPML) achieved by using a complex frequency-shifted stretch function was the latest development to further improve PML’s absorption performance for near-grazing angle incident waves as well as for low-frequency incident waves. However, the mathematical theory of the PML method is derived from the first-order equation, all PML implementations of second-order equations are to introduce auxiliary equations or variables to rewrite original PML equations, which will complicate the implementation. In this article, we propose a simple and efficient CPML implementation method for the second-order elastic wave equation, which directly simulates the second-order CPML equation. The main advantage of this method is that there is no need to introduce auxiliary variables or auxiliary equations to convert the second-order PML equation from the complex coordinate space to the real axis. Compared with the conventional CPML method for the second-order elastic wave equation, it introduces only eight convolution variables. We demonstrate the validity and absorption performance through extensive numerical experiments.

An effective absorbing layer for the boundary condition in acoustic seismic wave simulation

Efficient numerical simulation of seismic wavefields generally involves truncating the Earth model in order to keep computing time and memory requirements down. Absorbing boundary conditions, therefore, are applied to remove the boundary reflections caused by this truncation, thereby allowing for accurate modeling of wavefields. In this paper, we derive an effective absorbing boundary condition for both acoustic and elastic wave simulation, through the simplification of the damping term of the split perfectly matched layer (SPML) boundary condition. This new boundary condition is accurate, cost-effective, and easily implemented, especially for high-performance computing. Stability analysis shows that this boundary condition is effectively as stable as normal (non-absorbing) wave equations for explicit time-stepping finite differences. We found that for full-waveform inversion (FWI), the strengths of the effective absorbing layer—a reduction of the computational and memory cost coupled with a simplistic implementation—significantly outweighs the limitation of incomplete absorption of outgoing waves relative to the SPML. More importantly, we demonstrate that this limitation can easily be overcome through the use of two strategies in FWI, namely variable cell size and model extension thereby fully compensating for the imperfectness of the proposed absorbing boundary condition.

A Time-Domain Propagator Numerical Method for Computational Electromagnetics

A new time-domain numerical method for full-wave electromagnetic scattering and propagation is presented. The propagator method can be described as a numerical integral operator technique for moving the electromagnetic field through numerical space at successive increments in time. The numerical equations are obtained by discretizing electric and magnetic field propagator integrals. A primary feature of the propagator method is that all six electromagnetic-field components are calculated at each numerical grid point and all components are in time synchronization. Constant spatial and time increments are maintained throughout an inhomogeneous numerical space by employing an extrapolation procedure that ensures causality. A simple and effective first-order absorbing boundary condition, described as the null boundary condition, is introduced. Examples, provided in one-, two-, and three-dimensions, include plane wave reflection from and transmission through a planar boundary, and scattering and radar cross sections for canonical dielectric objects.

Reverse Time Migration Using the Pseudospectral Time-Domain Algorithm

We propose a pre-stack reverse time migration (RTM) seismic imaging method using the pseudospectral time-domain (PSTD) algorithm. Traditional pseudospectral method uses the fast Fourier transform (FFT) algorithm to calculate the spatial derivatives, but is limited by the wraparound effect due to the periodicity assumed in the FFT. The PSTD algorithm combines the pseudospectral method with a perfectly matched layer (PML) for acoustic waves. PML is a highly effective absorbing boundary condition that can eliminate the wraparound effect. It enables a wide application of the pseudospectral method to complex models. RTM based on the PSTD algorithm has advantages in the computational efficiency compared to traditional methods such as the second-order and high order finite difference time-domain (FDTD) methods. In this work, we implement the PSTD algorithm for acoustic wave equation based RTM. By applying the PSTD-RTM method to various seismic models and comparing it with RTM based on the eighth-order FDTD method, we find that PSTD-RTM method has better performance and saves more than 50% memory. The method is suitable for parallel computation, and has been accelerated by general purpose graphics processing unit.

An effective absorbing boundary condition for linear long-wave and linear dispersive-wave tsunami simulations

We numerically simulated the propagation of tsunami waves with finite difference methods by using perfectly matched layer (PML) boundary conditions to effectively eliminate artificial reflections from model boundaries. The PML method damps the tsunami height and velocity of seawater only in directions perpendicular to the boundary. Although the additional terms required to implement the PML conditions make the use of the PML technique difficult for linear dispersive tsunami waves, we have proposed an empirical extension of the PML method for modeling dispersive tsunami waves. Even for heterogeneous, realistic bathymetries, numerical tests demonstrated that the PML boundary condition dramatically decreased artificial reflections from model boundaries compared to the use of traditional boundary conditions. The use of PML boundary conditions for numerical modeling of tsunamis is especially useful because it facilitates use of the later phases of tsunamis that would otherwise be compromised by artifacts caused by reflections from model boundaries.

Accurate absorbing boundary conditions for anisotropic elastic media. Part 2: Untilted non-elliptic anisotropy

With the ultimate goal of devising effective absorbing boundary conditions (ABCs) for general elastic media, we investigate the accuracy aspects of local ABCs designed for untilted non-elliptic anisotropy in the frequency domain (time-harmonic analysis). While simple space–time transformations are available to treat the wavemodes with opposing phase and group velocities present in elliptic anisotropic media, no such transformations are known to exist for the case of non-elliptic anisotropy. In this paper, we use the concept of layer groupings along with an unconventional stretching of the finite element mesh to guarantee the accuracy of local ABCs designed to treat all propagating wavemodes, even those with opposing phase and group velocities. The local ABC used here is the perfectly matched discrete layer (PMDL) which is a simple variant of perfectly matched layers (PMLs) that is also equivalent to rational approximation-based local ABCs (rational ABCs); it inherits the straightforward approximation properties of rational ABCs along with the versatility of PML. The approximation properties of PMDL quantified through its reflection matrix allow us to (a) show that it is impossible to design an accurate PMDL with wavenumber-independent parameters, (b) theoretically demonstrate the ability of wavenumber-dependent parameters to ensure accuracy, and finally (c) design a practical though unconventional stretching of the finite element PMDL mesh that facilitates the implementation of wavenumber-dependent parameters. The validity of this work is demonstrated through a series of numerical experiments.

Accurate absorbing boundary conditions for anisotropic elastic media. Part 1: Elliptic anisotropy

With the ultimate goal of devising effective absorbing boundary conditions (ABCs) for heterogeneous anisotropic elastic media, we investigate the accuracy aspects of local ABCs designed for tilted elliptic anisotropy in the frequency domain (time-harmonic case). Such media support both anti-plane and in-plane wavemodes with opposing signs of phase and group velocities (cpxcgx<0) that have long posed a significant challenge to the design of accurate (and stable) local ABCs. By first considering the simpler case of scalar anti-plane waves, we show that it is possible to overcome the challenges posed by cpxcgx<0 by simply utilizing the inevitable reflections occurring at the truncation boundaries. This understanding helps us to explain the ability of a recently developed local ABC – the perfectly matched discrete layer (PMDL) – to handle the challenges posed by cpxcgx<0without the need of intervening space–time transformations. PMDL is a simple variant of perfectly matched layers (PML) that is also equivalent to rational approximation-based local ABCs (rational ABCs); it inherits the straightforward approximation properties of rational ABCs along with the versatility of PML. The approximation properties of PMDL quantified through its reflection matrix is used to derive simple bounds on the PMDL parameters necessary for the accurate absorption of all outgoing anti-plane and in-plane wavemodes – including those with cpxcgx<0. Beyond the previously derived bound on the real parameters of PMDL sufficient for the absorption of outgoing propagating anti-plane wavemodes, we present bounds on the complex parameters of PMDL necessary for the absorption of outgoing propagating and evanescent wavemodes for both anti-plane and coupled in-plane pressure and shear waves. The validity of this work is demonstrated through a series of numerical experiments.

Application of the Nearly Perfectly Matched Layer to Seismic-Wave Propagation Modeling in Elastic Anisotropic Media

Seismic anisotropy has become a main research topic in both seismic data processing and interpretation (e.g., migration and impedance inversion). As an important part of seismic anisotropy, numerical modeling can help us better understand seismic responses when seismic waves propagate through the interior of the Earth. For the numerical implementation of finite-difference time-domain algorithms, boundary conditions are needed to suppress the spurious reflections from the artificially truncated edges of the discrete model. Here, the nearly perfectly matched layer (NPML) technique is applied to a numerical simulation of seismic-wave propagation in a 2D elastic anisotropic medium, which has been proven to be a very effective absorbing boundary condition in electromagnetic, acoustic, and elastic isotropic media. The equivalence of the wave absorbing performance between the NPML and the standard perfectly matched layer (PML) for elastic wave modeling is theoretically proven in this paper. To check the absorbing ability of the NPML in the cases of grazing incidence and longer simulations (up to 200 s), we choose an elongated anisotropic model in our test. Test results indicate that the NPML has essentially the same computational accuracy as the standard PML but with higher computational savings.

A Magnetic Field-Independent Absorbing Boundary Condition for Magnetized Cold Plasma

An effective absorbing boundary condition (ABC) based on the second-order approximations of Engquist and Majda's wave equations is presented for terminating three-dimensional finite-difference time-domain (FDTD) models employing the E-J collocated magnetized cold plasma algorithm. Numerical tests demonstrate that this ABC can effectively terminate magnetized plasma models involving arbitrary and spatially varying directions of the background magnetic field, a capability that is not provided by the recently developed KPML or NIMPML for magnetized cold plasma. This new capability will permit implementation of the Engquist and Majda-based ABC of this letter into localized and particularly global FDTD models of the Earth-ionosphere system involving varying directions of the magnetic field throughout the grid. An accuracy and stability analysis of this ABC is presented. Reflection errors are shown to be consistently less than 5% for various magnetic field directions and for magnetized plasmas having different electron densities by incorporating the phase velocity calculated at the center frequency of the source spectrum into the ABC formulation.

Absorbing boundary conditions for scalar waves in anisotropic media. Part 1: Time harmonic modeling

With the ultimate goal of devising effective absorbing boundary conditions (ABCs) for general anisotropic media, we investigate the accuracy aspects of local ABCs designed for the scalar anisotropic wave equation in the frequency domain (time harmonic case). The ABC analyzed in this paper is the perfectly matched discrete layers (PMDL). PMDL is a simple variant of perfectly matched layers (PML) and is equivalent to rational approximation-based local ABCs. Specifically, we derive a sufficient condition for PMDL to accurately absorb wave modes with outgoing group velocities and this condition turns out to be a simple bound on the PMDL parameters. The reflection coefficient derived in this paper clearly reveals that the PMDL absorption is based on group velocities, and not phase velocities, and hence a PMDL can be designed to correctly identify and accurately absorb all outgoing wave modes (even those with opposing signs of phase and group velocities). The validity of the sufficient condition is demonstrated through a series of frequency domain simulations. In part 2 of this paper [S. Savadatti, M.N. Guddati, Absorbing boundary conditions for scalar waves in anisotropic media. Part 2: Time-dependent modeling, J. Comput. Phys. (2010), doi:10.1016/j.jcp.2010.05.017], the accuracy condition presented here is shown to govern both the well-posedness and accuracy aspects of PMDL designed for transient (time-dependent) modeling of scalar waves in anisotropic media.

Absorbing boundary conditions for scalar waves in anisotropic media. Part 2: Time-dependent modeling

With the ultimate goal of devising effective absorbing boundary conditions (ABCs) for general anisotropic media, we investigate the well-posedness and accuracy aspects of local ABCs designed for the transient modeling of the scalar anisotropic wave equation. The ABC analyzed in this paper is the perfectly matched discrete layers (PMDL), a simple variant of perfectly matched layers (PML) that is also equivalent to rational approximation based ABCs. Specifically, we derive the necessary and sufficient condition for the well-posedness of the initial boundary value problem (IBVP) obtained by coupling an interior and a PMDL ABC. The derivation of the reflection coefficient presented in a companion paper (S. Savadatti, M.N. Guddati, J. Comput. Phys., 2010, doi:10.1016/j.jcp.2010.05.018) has shown that PMDL can correctly identify and accurately absorb outgoing waves with opposing signs of group and phase velocities provided the PMDL layer lengths satisfy a certain bound. Utilizing the well-posedness theory developed by Kreiss for general hyperbolic IBVPs, and the well-posedness conditions for ABCs derived by Trefethen and Halpern for isotropic acoustics, we show that this bound on layer lengths also ensures well-posedness. The time discretized form of PMDL is also shown to be theoretically stable and some instability related to finite precision arithmetic is discussed.

THE MATRIX FORMULATION OF BOUNDARY INTEGRAL MODELING OF ELASTIC WAVE PROPAGATION IN 2D MULTI-LAYERED MEDIA WITH IRREGULAR INTERFACES

A semi-analytic method based on the propagation matrix formulation of indirect boundary element method to compute response of elastic (and acoustic) waves in multi-layered media with irregular interfaces is presented. The method works recursively starting from the top-most free surface at which a stress-free boundary condition is applied, and the displacement-stress boundary conditions are then subsequently applied at each interface. The basic idea behind this method is the matrix formulation of the propagation matrix (PM) or more recently the reflectivity method as wide used in the geophysics community for the computation of synthetic seismograms in stratified media. The reflected and transmitted wave fields between arbitrary shapes of layers can be computed using the indirect boundary element (BEM) method. Like any standard BEM methods, the primary task of the BEM-based propagation matrix method (thereafter called PM–BEM) is the evaluation of element boundary integral of the Green's function, for which there are standard method that can be adapted. In addition, effective absorbing boundary conditions as used in the finite difference numerical method is adapted in our implementation to suppress the spurious arrivals from the artificial boundaries due to limited model space. To our knowledge, such implementation has not appeared in the literature. Several examples are presented in this paper to demonstrate the effectiveness of this proposed PM–BEM method for modeling elastic waves in media with complex structure.

A compact one‐dimensional modal FDTD method

AbstractThe finite‐difference time‐domain (FDTD) method is an effective technique for computing wideband electrical parameters such as scattering parameters of waveguide structures. in the computations, a known incident is normally required and is usually obtained with a simulation of a long uniform structure. For a three‐dimensional problem, simulation of a long structure can be very memory‐ and CPU time‐intensive. In addition, effective absorbing boundary conditions are needed to effectively terminate the structure even at and below the cutoff frequencies. To address these issues, many one‐dimensional FDTD methods and absorbing schemes were proposed. However, they all have dispersion characteristics different from those of the conventional FDTD method, leading to undesired errors or reflections. In this paper, a new one‐dimensional scheme is developed that has numerical dispersion characteristics very similar to that of the conventional FDTD method. When used as the absorbing boundary condition, it generates reflections of less than −200 dB even at and below the cutoff frequencies for the considered modes. When used to obtain the incident wave, its results have difference of less than −200 dB from that produced by the conventional FDTD method. Copyright © 2007 John Wiley &amp; Sons, Ltd.

EFFICIENT ABC FOR THE FREQUENCY DEPENDENT ALTERNATING DIRECTION-IMPLICIT FDTD METHOD

Ultra WideBand (UWB) signals have a high potential for a wide range of applications. The Finite-Difference Time-Domain (FDTD) method is widely used in the development of UWB technology but has still two main limitations. One is that material parameters are constant over the whole frequency range. Frequency dependent materials can be accommodated by adopting a Debye model. The other is that the minimum simulation time is bound by the Courant-Friedrichs-Lewy (CFL) condition, which can be solved by the application of the Alternating Direction-Implicit (ADI) scheme. The combination of Debye model and ADI scheme in FDTD results in the Frequency-Dependent (FD) ADI-FDTD method. This paper proposes an adaptation of the most effective Absorbing Boundary Condition (ABC), the Complex Frequency-Shift (CFS) Perfectly Matched Layer (PML), to FD-ADI-FDTD. The formulation of CFS-PML for FD-ADI-FDTD is proposed and its performance is assessed. The influence of the Courant Number (CFLN), the media parameters and the number of layers are investigated.

AN EFFICIENT MODAL FDTD FOR ABSORBING BOUNDARY CONDITIONS AND INCIDENT WAVE GENERATOR IN WAVEGUIDE STRUCTURES

When the finite-difference time-domain method is used to compute waveguide structures, incident waves are needed for calculating electrical parameters (e.g., the scattering parameters), and effective absorbing boundary conditions are required for terminating open waveguide structures. The incident waves are conventionally obtained with inefficient three-dimensional (3D) simulations of long uniform structures, while the absorbing boundary conditions reported so far do not perform well at or below cut-off frequencies. To address the problems, we propose a novel one-dimensional (1D) finite- difference time-domain method in this paper. Unlike the other methods developed so far, the proposed method is derived from the finite-difference time-domain formulation, and therefore has the same numerical characteristics as that of the finite-difference time-domain method. As a result, when used to obtain an incident wave, it produces results almost identical to those produced by the conventional finite- difference time-domain method except computer rounding-off errors. When used as the absorbing boundary condition, it produces reflections of less than −200 dB in entire frequency spectrum including the cut-off frequencies.

Finite difference time domain analysis of cylindrical coplanar waveguide circuits

There is a growing interest in using cylindrical transmission line structures in microwave applications. In this paper, the finite difference time domain (FDTD) method is used to characterize several three-dimensional cylindrical coplanar waveguide (CCPW) geometries. Specifically, a CCPW series stub and a three-section CCPW filter are studied both theoretically and experimentally. Moreover, CCPW gap is characterized and the FDTD results are compared to those obtained using conformal mapping techniques. Effective absorbing boundary conditions are employed to truncate the FDTD mesh at the end walls and the outer radial boundary.

Two-dimensional finite difference time domain analysis of cylindrical transmission lines

Recently, there has been a growing interest in the use of cylindrical transmission lines in a variety of applications. In this paper, the finite difference time domain (FDTD) method is used to calculate the propagation characteristics of cylindrical transmission lines. An efficient two-dimensional (2D) FDTD algorithm is developed by projecting the three-dimensional FDTD cell in the cylindrical coordinates on to the r−φ plane. An effective absorbing boundary condition is employed to truncate the mesh at its outer radial boundary. In addition to a detailed description of the developed 2D-FDTD algorithm, extensive numerical results are derived for different cylindrical transmission lines and compared with data available in the literature. Specifically, the newly proposed cylindrical coplanar waveguide is studied both theoretically and experimentally.

Analysis of cylindrical transmission lines with the finite-difference time-domain method

In this paper, the finite-difference time-domain (FDTD) method is used to calculate the propagation characteristics of cylindrical transmission lines. An efficient two-dimensional FDTD algorithm is developed by projecting the three-dimensional FDTD cell in the cylindrical coordinates onto the r-/spl phi/ plane. An effective absorbing boundary condition is employed to truncate the mesh at its outer radial boundary. Numerical results are derived for different cylindrical transmission lines and compared to data available in the literature. Specifically, the newly proposed cylindrical coplanar waveguide is studied both theoretically and experimentally.

EM scattering analysis of a ferrite cylinder by a FD-FD technique with an effective absorbing boundary condition and MEI

Electromagnetic scattering problem of an arbitrarily shaped ferrite cylinder is analyzed based on the finite difference-frequency domain(FD-FD) method with an effective numerical absorbing boundary condition (ABC) and the measured equation of invariance (MEI) on the terminated boundary. Compared with the method of moments (MoM), both the numerical ABC presented in this paper and the MEI result in dramatic savings in computing time and memory requirement for electrically large objects due to the sparsity of the finite difference equation. The absorbing characteristic of this numerical ABC is demonstrated numerically. The accuracy, memory needs and CPU time of the FD-FD with the numerical ABC or the MEI and the MoM are compared and then result in some important conclusions. Besides, the RCS of some ferrite cylinders are presented.

Use of second order absorbing boundary conditions for the termination of planar waveguides in the FDTD method

The necessity of using effective absorbing boundary conditions (ABCs) to terminate microstrip or other planar waveguides in the FDTD method has long been appreciated but, until recently, little attention has been given in the literature to the best choice of ABC algorithm for this purpose. Although some comparisons of ABCs are now available they, unfortunately, differ in their conclusions. In the Letter the theoretical and numerical behaviour of some published ABCs are compared and the best ABC algorithm for this application is identified.