Perfectly Matched Layer Formulation Research Articles

2D and 3D modeling of wave propagation in cold magnetized plasma near the Tore Supra ICRH antenna relying on the perfecly matched layer technique

A novel method to simulate ion cyclotron wave coupling in the edge of a tokamak plasma with the finite element technique is presented. It is applied in the commercial software COMSOL Multiphysics. Its main features include the perfectly matched layer (PML) technique to emulate radiating boundary conditions beyond a critical cutoff layer for the fast wave (FW), full-wave propagation across the inhomogeneous cold peripheral plasma and a detailed description of the wave launcher geometry. The PML technique, while widely used in numerical simulations of wave propagation, has scarcely been used for magnetized plasmas, due to specificities of this gyrotropic material. A versatile PML formulation, valid for full dielectric tensors, is summarized and interpreted as wave propagation in an artificial medium. The behavior of this technique has been checked for plane waves on homogeneous plasmas. Wave reflection has been quantified and compared to analytical predictions. An incompatibility issue for adapting the PML for forward (FW) and backward (slow wave (SW)) propagating waves simultaneously has been evidenced. In a tokamak plasma, this critical issue is overcome by taking advantage of the inhomogeneous density profile to reflect the SW before it reaches the PML. The simulated coupling properties of a Tore Supra ion cyclotron resonance heating (ICRH) antenna have been compared to experimental values in a situation of good single-pass absorption. The necessary antenna elements to include in the geometry to recover the coupling properties obtained experimentally are also discussed.

Application of perfectly matched layer for scalar arbitrarily wide-angle wave equations

Arbitrarily wide-angle wave equation (AWWE) is a space domain, high-order one-way wave equation (OWWE). Its accuracy can be arbitrarily increased, and it is amenable to easy numerical implementation. Those properties make it outstanding among the existing OWWEs and further enable it to be a desirable tool for migration. We extend the perfectly matched layer (PML) to 3D scalar AWWE to provide a good approach to suppress artifacts arising at truncation boundaries. We follow the concept of complex coordinate stretching, and the derivation procedure of PML for AWWE is straightforward. An existing finite-difference scheme is adopted to fit the split PML formulation and its stability is observed through numerical examples. The performance of the developed PML condition is compared with two different wave-equation based absorbing boundary conditions. Numerical results illustrate that the PML condition used in AWWE propagator can effectively absorb the propagating waves and evanescent waves at a price of limited additional computation cost.

Efficient DSP-Higher-Order PML Formulations for the Metal Plate Buried in Dispersive Soil Half-Space Problem

An efficient and unsplit-field implementation of the higher order perfectly matched layer (PML) based on the stretched coordinate PML (SC-PML) formulations and digital signal processing (DSP) techniques is proposed to truncate the finite-difference time-domain (FDTD) lattices. The higher order PML has the advantages of both the conventional PML and the complex frequency-shifted PML (CFS-PML) in terms of absorbing performances. A 3-D FDTD simulation of the metal plate buried in dispersive soil half-space problem has been carried out to validate these formulations. It is shown that the proposed PML formulations with the higher order scheme are efficient in terms of attenuating both the low-frequency propagating waves and evanescent waves and reducing late-time reflections.

A symmetric hybrid formulation for transient wave simulations in PML-truncated heterogeneous media

The numerical simulation of wave motion in arbitrarily heterogeneous semi-infinite media requires the truncation of the semi-infinite extent of the domain to yield a finite computational domain. In the presence of heterogeneity, the domain truncation is best accomplished via the introduction of perfectly-matched-layers (PMLs) at the truncation surface.By and large, most PML formulations treat in an identical manner both the interior domain and the PML buffer zone. By construction, the complex-coordinate-stretched equations used to introduce the PML, also serve to describe the interior domain, where they reduce to the original, unstretched, system of governing equations. Such a unified treatment, however, results in increased computational cost.In this development, we discuss a hybrid formulation that leads to a mixed form within the PML, coupled with a standard displacement-only form for the interior domain, both of which are second-order in time. We discuss the formulation and the numerical implementation using finite elements in the context of a standard Galerkin scheme that yields fully symmetric discrete forms, and results in optimal computational cost. We show that existing displacement-based codes for interior domains can be easily modified to accommodate PMLs as a means of domain truncation.We report on numerical results demonstrating the stability, efficacy, and cost-effectiveness of the hybrid formulation.

Remarks on the stability of Cartesian PMLs in corners

This work is a contribution to the understanding of the question of stability of Perfectly Matched Layers (PMLs) in corners, at continuous and discrete levels. First, stability results are presented for the Cartesian PMLs associated to a general first-order hyperbolic system. Then, in the context of the pressure–velocity formulation of the acoustic wave propagation, an unsplit PML formulation is discretized with spectral mixed finite elements in space and finite differences in time. It is shown, through the stability analysis of two different schemes, how a bad choice of the time discretization can deteriorate the CFL stability condition. Some numerical results are finally presented to illustrate these stability results.

Perfectly matched layers for frequency-domain integral equation acoustic scattering problems

Simulations of acoustic wavefields in inhomogeneous media are always performed on finite numerical domains. If contrasts actually extend over the domain boundaries of the numerical volume, unwanted, non-physical reflections from the boundaries will occur. One technique to suppress these reflections is to attenuate them in a locally reflectionless absorbing boundary layer enclosing the spatial computational domain, a perfectly matched layer (PML). This technique is commonly applied in time-domain simulation methods like finite element methods or finite-difference time-domain, but has not been applied to the integral equation method. In this paper, a PML formulation for the three-dimensional frequency-domain integral-equation-based acoustic scattering problem is derived. Three-dimensional acoustic scattering configurations are used to test the PML formulation. The results demonstrate that strong attenuation (a factor of 200 in amplitude) of the scattered pressure field is achieved for thin layers with a thickness of less than a wavelength, and that the PMLs themselves are virtually reflectionless. In addition, it is shown that the integral equation method, both with and without PMLs, accurately reproduces pressure fields by comparing the obtained results with analytical solutions.

An efficient finite element time‐domain formulation for the elastic second‐order wave equation: A non‐split complex frequency shifted convolutional PML

AbstractThe perfectly matched layer (PML) technique has demonstrated very high efficiency as absorbing boundary condition for the elastic wave equation recast as a first‐order system in velocity and stress in attenuating non‐grazing bulk and surface waves. This paper develops a novel convolutional PML formulation based on the second‐order wave equation with displacements as the only unknowns to annihilate spurious reflections from near‐grazing waves. The derived variational form allows for the use of e.g. finite element and the spectral element methods as spatial discretization schemes. A recursive convolution update scheme of second‐order accuracy is employed such that highly stable, effective time integration with the Newmark‐beta (implicit and explicit with mass lumping) method is achieved. The implementation requires minor modifications of existing displacement‐based finite element software, and the stability and efficiency of the proposed formulation is verified by relevant two‐dimensional benchmarks that accommodate bulk and surface waves. Copyright © 2011 John Wiley & Sons, Ltd.

PML absorbing boundary condition for the integral‐based high‐order FV24 FDTD algorithm

AbstractAn integral equations‐based perfectly matched layers (PML) implementation is presented for the highly phase‐coherent FV24 finite‐difference time‐domain (FDTD) algorithm. The implementation allows including field values off the grid axes in the split‐field PML formulation conserving in the process the continuity and phase coherency of the FV24 algorithm when modeling absorbing boundary conditions (ABCs). It also eliminates the need for cumbersome subgridded low‐order FDTD subregions that until now were required to model PML ABCs within integral‐based high‐order FDTD simulations. The developed approach was numerically tested and found to match the PML behavior of the standard FDTD method at normal wave incidence on ABC boundaries and exceeds it at highly oblique wave incidence. This development serves to improve the capability and practicality of the computationally efficient FV24 algorithm when modeling electrically large structures in 3‐D space. Copyright © 2010 John Wiley & Sons, Ltd.

3-D frequency-domain seismic wave modelling in heterogeneous, anisotropic media using a Gaussian quadrature grid approach

SUMMARY We present an extension of the 3-D spectral element method (SEM), called the Gaussian quadrature grid (GQG) approach, to simulate in the frequency-domain seismic waves in 3-D heterogeneous anisotropic media involving a complex free-surface topography and/or sub-surface geometry. It differs from the conventional SEM in two ways. The first is the replacement of the hexahedral element mesh with 3-D Gaussian quadrature abscissae to directly sample the physical properties or model parameters. This gives a point-gridded model which more exactly and easily matches the free-surface topography and/or any sub-surface interfaces. It does not require that the topography be highly smooth, a condition required in the curved finite difference method and the spectral method. The second is the derivation of a complex-valued elastic tensor expression for the perfectly matched layer (PML) model parameters for a general anisotropic medium, whose imaginary parts are determined by the PML formulation rather than having to choose a specific class of viscoelastic material. Furthermore, the new formulation is much simpler than the time-domain-oriented PML implementation. The specified imaginary parts of the density and elastic moduli are valid for arbitrary anisotropic media. We give two numerical solutions in full-space homogeneous, isotropic and anisotropic media, respectively, and compare them with the analytical solutions, as well as show the excellent effectiveness of the PML model parameters. In addition, we perform numerical simulations for 3-D seismic waves in a heterogeneous, anisotropic model incorporating a free-surface ridge topography and validate the results against the 2.5-D modelling solution, and demonstrate the capability of the approach to handle realistic situations.

Mixed perfectly-matched-layers for direct transient analysis in 2D elastic heterogeneous media

We are concerned with elastic waves arising in plane-strain problems in an elastic semi-infinite arbitrarily heterogeneous medium. Specifically, we discuss the development of a new mixed displacement–stress formulation for forward elastic wave simulations in perfectly-matched-layer (PML)-truncated heterogeneous media. To date, most PML formulations split the displacement and stress fields, resulting in non-physical components for each field. In this work, we favor unsplit schemes, primarily for the relative ease by which the resulting forms can be incorporated into existing codes, the ease by which the resulting semi-discrete forms can be integrated in time, and the ease by which they can be used in adjoint formulations arising in inverse problems, contrary to most past and current developments. We start by following classical lines, and apply complex-coordinate-stretching to the governing equations in the frequency domain, while retaining both displacements and stress quantities as unknowns. With the aid of auxiliary variables the resulting mixed form is rendered second-order in time, thereby allowing the use of standard time integration schemes. We report on numerical simulations demonstrating the stability and efficacy of the approach.

Modelling Seismic Oceanography Experiments by Using First- and Second-Order Complex Frequency Shifted Perfectly Matched Layers

This work investigates the ability of modelling seismic oceanography experiments by using underwater acoustic propagation equations. Seismic oceanography tries to retrieve the fine structure of the ocean water masses by processing the acoustic waves reflected in the low-contrast interfaces of fronts, eddies, internal waves or thermohaline intrusions. Since the reflectivity of such interfaces is of order 10 ―3 ―10 ―4 , the absorption capability of the numerical boundaries becomes crucial. Complex Frequency Shifted offers a better alternative to classical Perfectly Matched Layer formulation, but has not yet been extended to acoustic equations. Here, first- and second-order Complex Frequency Shifted Perfectly Matched Layers equations are proposed which can provide reflection coefficients of order 10 ―5 . Therefore, a numerical Finite-Difference Time-Domain (FDTD) scheme combined with the proposed CFS-PML equations is able to model such experiments.

PML absorbing boundary conditions for the linearized and nonlinear Euler equations in the case of oblique mean flow

AbstractFor the case of uniform mean flow in an arbitrary direction, perfectly matched layer (PML) absorbing boundary conditions are presented for both the linearized and nonlinear Euler equations. Although linear perfectly matched side layers with an oblique mean flow have been studied in previous works, we propose in the present paper a construction of corner layer equations that are dynamically stable. Stability issues are investigated by examining the dispersion relations of linear waves supported by the corner layer equations. For increased efficiency, a pseudo mean flow is included in the derivation of the PML equations for the nonlinear case. Numerical examples are given to support the validity of the proposed equations. Specifically, the linear PML formulation is tested for the case of acoustic, vorticity, and entropy waves traveling with an oblique mean flow. The nonlinear formulation is tested with an isentropic vortex moving diagonally with a constant velocity. Copyright © 2008 John Wiley & Sons, Ltd.

An improved perfectly matched layer for the eigenmode expansion technique

When performing optical simulations for rotationally symmetric geometries using the eigenmode expansion technique, it is necessary to place the geometry under investigation inside a cylinder with perfectly conducting walls. The parasitic reflections at the boundary of the computational domain can be suppressed by introducing a perfectly matched layer (PML) using e.g. complex coordinate stretching of the cylinder radius. However, the traditional PML suffers from an artificial field divergence limiting its usefulness. We show that the choice of a constant cylinder radius leads to mode profiles with exponentially increasing field amplitudes resulting in numerical instability. As a remedy we propose an improved PML based on a mode-dependent cylinder radius and mode profiles with stable field amplitudes. The new PML formulation eliminates the artificial field divergence and ensures numerical stability.

Explicit finite element perfectly matched layer for transient three‐dimensional elastic waves

AbstractThe use of a perfectly matched layer (PML) model is an efficient approach toward the bounded‐domain modelling of wave propagation on unbounded domains. This paper formulates a three‐dimensional PML for elastic waves by building upon previous work by the author and implements it in a displacement‐based finite element setting. The novel contribution of this paper over the previous work is in making this finite element implementation suitable for explicit time integration, thus making it practicable for use in large‐scale three‐dimensional dynamic analyses. An efficient method of calculating the strain terms in the PML is developed in order to take advantage of the lack of the overhead of solving equations at each time step. The PML formulation is studied and validated first for a semi‐infinite bar and then for the classical soil–structure interaction problems of a square flexible footing on a (i) half‐space, (ii) layer on a half‐space and (iii) layer on a rigid base. Numerical results for these problems demonstrate that the PML models produce highly accurate results with small bounded domains and at low computational cost and that these models are long‐time stable, with critical time step sizes similar to those of corresponding fully elastic models. Copyright © 2008 John Wiley & Sons, Ltd.

Perfectly matched layer method in the finite‐difference time‐domain and frequency‐domain calculations

AbstractA complex‐coordinate method known under the guise of the perfectly matched layer (PML) method for treating unbounded domains in computational electrodynamics is related to similar techniques in fluid dynamics and classical quantum theory. It may also find use in electronic‐structure finite‐difference simulations. Straightforward transfer of the PML formulation to other fields does not seem feasible, however, since it is a unique feature of electrodynamics – the natural invariance – that allows analytic trick of complex coordinate scaling to be represented as pure modification of local material parameters within thin layers of space adjacent to the computational domain boundaries, i.e., the PMLs. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Application of the nearly perfectly matched layer in acoustic wave modeling

In this work, we successfully applied an alternative formulation of the perfectly matched layer (PML), the so-called nearly PML (NPML), to acoustic wave propagation modeling. The NPML formulation shows great advantages over the standard complex stretched coordinate PML. The NPML formulation deviates from the standard PML through an inexact variable change, but this fact only affects the wave behavior in the NPML layer, which is outside the region of interest. The equivalence of the wave-absorbing performance between these two PML formulations (the standard complex stretched coordinate PML formulation and the NPML formulation) in 3D Cartesian coordinates for acoustic wave propagation modeling is proved mathematically in this work. In time-domain methods, the advantages of the NPML over the standard PML were explained by both the analytical analysis and the numerical simulations in terms of implementation simplicity and computational efficiency. The computation time saving is up to 17% for the 2D example used in this work. For 3D problems, this computational saving is more significant. After theoretically analyzing the numerical reflections from the NPML and the standard PML, we concluded that these two PML formulations have exactly the same performance, even after spatial discretization. This conclusion is validated by numerical experiment. Finally, we tested the NPML in the Marmousi velocity model and found its wave-absorbing rate is high enough, even for this realistic structure.

A nonsplit complex frequency-shifted PML based on recursive integration for FDTD modeling of elastic waves

In finite-difference time-domain (FDTD) modeling of elastic waves, absorbing boundary conditions are used to mitigate undesired reflections that can arise at the model’s truncation boundaries. The perfectly matched layer (PML) is generally considered to be the best available absorbing boundary condition. An important but rarely addressed limitation of current PML implementations is that their performance is severely reduced when waves are incident on the PML interface at near-grazing angles. In addition, very low frequency waves as well as evanescent waves could cause spurious reflections at the PML interface. In electromagnetic modeling, similar problems are circumvented by using a complex frequency-shifted stretching function in the PML formulas. However, in elastic-wave modeling using the conventional PML formulation — based on splitting the velocityand stressfields — it is difficult to adopt a complex frequency--shifted stretching function. We present an alternative implemen-tation of a PML that is based on recursive integration and does not require splitting of the velocity and stress fields. Modeling re-sults show that the performance of our implementation using a standard stretching function is identical to that of the convention-al split-field PML. Then we show that the new PML can be modi-fied easily to include the complex frequency-shifted stretching function. Results of models with an elongated domain show that this modification can substantially improve the performance of the PML boundary condition. An efficient implementation of the new PML requires less memory than the conventional split-field PML, and, therefore, is a very attractive alternative to the con-ventional PML. By adopting the complex frequency-shifted stretching function, the PML can accommodate a wide variety of model problems, and hence it is more generic.

Unsplit field implicit PML algorithm for complex envelope dispersive LOD-FDTD simulations

Complex envelope locally one-dimensional (LOD) perfectly matched layer (PML) formulations are presented for truncating dispersive finite-difference time-domain (FDTD) simulations. The formulations are free from the Courant Friedrich Levy stability limit of the explicit FDTD algorithm. A numerical example carried out in two-dimensional linear Lorentz dispersive domain shows that the proposed formulations perform better than the classical LOD PML formulations.

Unconditionally Stable Complex Envelope Wave Equation PML Algorithm for Band Limited FDTD Simulations

Unconditionally stable complex envelope (CE) perfectly matched layer (PML) absorbing boundary conditions (ABCs) are presented for truncating the scalar wave-equation finite difference time domain (WE-FDTD) grids. The formulations are based on incorporating the alternating direction implicit (ADI) scheme into the CE FDTD implementations of the scalar wave-equation derived in the PML region at the domain boundaries. Numerical example carried out in two dimensional domain shows that the proposed formulations are more accurate than the classical ADI scalar wave equation PML formulations when it is used for modelling band limited electromagnetic applications.

Perfectly matched layer implementation for ADI-FDTD in dispersive media

Abstract Alternating direction implicit finite difference time domain (ADI-FDTD) method is unconditionally stable and the maximum time step is not limited by the Courant stability condition, but rather by numerical error. Compared with the conventional FDTD method, the time step of ADI-FDTD can be enlarged arbitrarily and the CPU cost can be reduced. 2D perfectly matched layer (PML) absorbing boundary condition is proposed to truncate computation space for ADI-FDTD in dispersive media using recursive convolution(RC) method and the 2D PML formulations for dispersive media are derived. ADI-FDTD formulations for dispersive media can be obtained from the simplified PML formulations. The scattering of target in dispersive soil is simulated under sine wave and Gaussian pulse excitations and numerical results of ADI-FDTD with PML are compared with FDTD. Good agreement is observed. At the same time the CPU cost for ADI-FDTD is obviously reduced.