On Optimal Finite-Difference Approximation of PML

Theoretical and quantitative evaluation of hybrid PML-ABCs for seismic wave simulation

The absorbing boundary condition (ABC) and the perfectly matched layer (PML) are employed to suppress the numerical reflections in wave propagation methods. The PML has earlier been shown to fail at the grazing angle propagation of beam. The ABC is preferred over the PML where refractive index is not an analytic function. We have explored the performance of the ABC in reflection-less absorption of non-paraxial propagation of beams and have compared it against the PML. We have considered an example of the propagation of Gaussian beams through a homogeneous medium terminated with ABC or PML. We have used the collocation method for propagation. We have found that both the boundary conditions are comparable in performance. The ABC performed relatively better than the PML for beams at smaller angles. However, the performance of the PML is better than that of the ABC at larger tilt angles.

For wave propagation problems in infinite-domains, truncation of the far-field by an artificial boundary is essential to define a bounded domain of interest; resultant difficulties are the radiation condition and generation of incident waves. Instead of the radiation condition, an absorbing boundary condition (ABC) is implemented in an inhomogeneous exterior domain, based on the perfectly matched layer (PML) technique. Complex coordinate stretching, a key PML concept, analytically continues a governing equation in the domain of interest into a PML equation as an anisotropic absorbing boundary in the exterior domain; a system of equivalent equations governs waves in the whole computational domain, and thus the PML is reflectionless at the artificial boundary, an interface between the PML and the domain of interest. For generation of incident waves, wave-maker modeling is proposed based on an idea of plain topography and wave-ray tracing. A source function along the interface is a curvilinear integral in a weak form for the governing equation. Finite element examples for wave propagation in one-dimension verify ability and efficiency of techniques with PML ABCs; besides, those for wave scattering by a circular cylinder on a slope show effectiveness of the wave-maker modeling.

Advances in Computational Aeroacoustics (CAA) depend critically on the availability of accurate, nondispersive, least dissipative computation algorithm as well as high quality numerical boundary treatments. This paper focuses on the Perfectly Matched Layer (PML) for external boundaries in CAA. To achieve low dissipation and dispersion errors, Dispersion-Relation-Preserving (DRP) Schemes are used for spatial discretization of the acoustic equations. The classical fourth-order Runge-Kutta time scheme is applied to the acoustic equations for time discretization. Four cases are given to illustrate the 2D PML equations for the linearized/nonlinear Euler equations in Cartesian coordinates and Cylindrical coordinates. The results show that the PML is effective as absorbing boundary condition. Those are basis for PML in actual computations of acoustic problems.

The finite-difference time-domain (FDTD) method in conjunction with the perfectly matched layers (PML) absorbing boundary condition (ABC) has been a popular choice for the simulation of ground penetrating radars (GPRs) due to the following reasons: FDTD is suitable for modeling inhomogeneities. FDTD is also suitable for the computation of time-dependent radar signals. PML is a very effective ABC that renders the FDTD method quite efficient. In this paper, in addition to the conventional usage of PML to truncate the FDTD computational domain, we report on how the PML can also be used to simulate an absorbing material inside the FDTD computational domain.

We extend Berenger's perfectly matched layers (PML) to conductive media. A finite-difference-time-domain (FDTD) algorithm with PML as an absorbing boundary condition is developed for solutions of Maxwell's equations in inhomogeneous, conductive media. For a perfectly matched layer in a conductive medium, an additional term involving the time-integrated electric field has to be introduced to account for the coupling between the loss from the PML and the normal conduction loss. This absorbing boundary condition is proven to be highly effective for the absorption of outgoing waves at the computational edge even when a dipping interface intersects the outer boundary. The algorithm is validated by analytical solutions, and is also compared with Liao's absorbing boundary condition. Numerical results for subsurface radar measurements are shown to demonstrate the applications of this method. © 1997 John Wiley & Sons, Inc. Microwave Opt Technol Lett14, 134–137, 1997

ABSTRACTAs a complementary way to traditional wave-equation-based forward modeling methods, lattice spring model (LSM) is introduced into seismology for wavefield modeling owing to its remarkable stability, high-calculation accuracy, and flexibility in choosing simulation meshes, and so forth. The LSM simulates seismic-wave propagation from a micromechanics perspective, thus enjoying comprehensive characterization of elastic dynamics in complex media. Incorporating an absorbing boundary condition (ABC) is necessary for wavefield modeling to avoid the artificial reflections caused by truncated boundaries. To the best of our knowledge, the perfectly matched layer (PML) method has been a routine ABC in the wave-equation-based numerical modeling of wave physics. However, it has not been used in the nonwave-equation-based LSM simulations. In this work, we want to apply PML to LSM to attenuate the boundary reflections. We divide the whole simulation region into PML region and inner region, PML region surrounds the inner region. To incorporate PML to LSM, we establish elastic-wave equations corresponding to LSM. The simulation in the PML region is conducted using the established wave equations and the simulation in the inner region is conducted using LSM. Three simulation examples show that the PML scheme is effective and outperforms Gaussian ABC.

The Perfectly Matched Layer (PML) concept, introduced by Berenger with the aim of synthesizing an absorbing boundary condition (ABC) for the Finite Difference Time Domain (FDTD) method, was recently modified and extended to Finite Element Frequency Domain (FEFD) applications. The modified equations, which neither require the splitting of the field components of interest nor involve negative conductivity parameters, are employed, in this paper, in conjunction with analytic ABC's to obtain a boundary condition (BC) for scattering problems. This BC exhibits an improved performance over a PML medium terminated by a perfect conductor, or an ABC termination alone. >

Because of finite computational domain, an absorbing boundary condition (ABC) is significant in the numerical modelling of wave equations. Liu and Sen (2010 and 2012) proposed a hybrid ABC for time-domain acoustic and elastic modelling. Their scheme has the advantages of small memory, small computation time and nearly perfect absorption. Ren and Liu (2012) extended the scheme to the frequency domain and performed visco-acoustic modelling. To validate the scheme, we develop the hybrid ABC for frequency-domain elastic modelling and compare it with perfectly matched layer (PML). Synthetic data tests demonstrate two ABCs are similarly efficient for high-accuracy modelling with nearly the same computation time and memory, and the hybrid ABC is better than PML for low- or mid-accuracy modelling.

An implementation of a second order unsplit perfectly matched layer (PML) absorbing boundary condition is presented. The formulation is given in the context of seismic modelling and simple models are used to showcase the increased performance. The formulation is presented as a correction procedure to the unchanged governing equations. This approach makes the implementation especially suited for existing FDTD modelling codes. Although, a second order PML might perform better in some cases than the classic or CFS-PML the increased computational requirements and the difficulty in optimising a greater number of PML parameters must be taken into account in deciding the most appropriate PML for a given seismic modelling problem.

A rotated staggered grid finite-difference with the absorbing boundary condition of a perfectly matched layer

The traditional approach of implementing FDTD techniques is to develop different sets of update relations that accommodate for different computational regions, for example, normal FDTD and absorbing boundary condition (ABC) regions. In particular, when a perfectly matched layer (PML) is used for termination of computational grids, we have to deal with a number of field regions in terms of PML faces, edges, and corners which adds a tremendous work load and complexity to the programming environment. This paper aims to develop an efficient approach to analyze and design cylindrical MMICs. In it, we present a unified non-uniform cylindrical finite difference time domain (NU-CFDTD) algorithm in conjunction with an anisotropic perfectly matched layer (APML) ABC in a stretched-coordinate mapping system. The algorithm uses only one set of update equations to handle all computational regions, no matter if they are normal FDTD or PML regions. We determine a non-uniform grid pattern depending on the circuit density for a specific structure to save computational resources, while remaining satisfactory accuracy. In particular, the present algorithm can characterize different material regions by simply assigning different material parameters.

A theoretical approach to obtain optimised profiles of the conductivity for the perfect matched layer (PML) absorbing boundary conditions is presented. The analytical expression recently introduced by the authors to estimate the error caused by the use of the PML backed by a metal plane as the absorbing boundary condition, has been minimised with respect to the conductivity profile following an inverse theory approach. The algorithm uses the Frechet derivative of the theoretical error expression, together with a simple regularisation technique in order to avoid the quasi-singularity intrinsic in the problem formulation. Results show that, following this approach, it is possible to easily obtain the conductivity profiles that render the performances of the absorbing boundary conditions comparable to, or better than, the best performance obtained for just a particular order of a more classic parabolic-like profile.

The measured equation on invariance is applied in time domain (TDMEI) to find the absorbing boundary conditions. The TM radiation of a line source at 1/4 computational domain in a 2-D free space is used as a test example to examine how good the TDMEI is in comparison with conventional absorbing boundary conditions (ABC), such at Mur's (1981) 2nd order ABC and perfectly matched layer (PML), etc. It is found that the TDMEI is more accurate than both ABC and PML for the radiation problem.

Numerical errors encountered when using the perfectly matched layer (PML) absorbing boundary condition with the finite-element method are investigated to discover more efficient implementation schemes. Closed-form expressions for the numerical reflection at an interface between two general biaxial materials are applied to the special case of a PML boundary. Expressions for an anisotropically compressed mesh are then derived, revealing that reflections can be greatly reduced through increasing mesh density only where it is required. Significant improvements over previously reported PML boundaries are demonstrated.