Perfectly Matched Layer Model Research Articles

Analysis and application of a time-domain finite element method for the Drude metamaterial perfectly matched layer model

In this paper, we develop a finite element method (FEM) to solve the Drude metamaterial perfectly matched layer (PML) model by edge elements. Both the stability analysis and error estimate for the scheme are established. To our best knowledge, this is the first paper on developing and analyzing a FEM for this Drude PML model. Numerical results are presented to justify the theoretical error estimate and demonstrate the effectiveness of this PML in absorbing outgoing waves in the Drude metamaterial.

Developing and analyzing an explicit unconditionally stable finite element scheme for an equivalent Bérenger’s PML model

The original Bérenger’s perfectly matched layer (PML) was quite effective in simulating wave propagation problem in unbounded domains. But its stability is very challenging to prove. Later, some equivalent PML models were developed by Bécache and Joly [ESAIM: M2AN 36 (2002) 87–119] and their stabilities were established. Hence studying and developing efficicent numerical methods for solving those equivalent PML models are needed and interesting. Here we propose a novel explicit unconditionally stable finite element scheme to solve an equivalent Bérenger’s PML model. Both the stability and convergence analysis are proved for the proposed scheme. Numerical results justifying the theoretical analysis are presented. We also demonstrate the effectiveness of this PML in simulating wave propagation in the free space. To our best knowledge, this is the first explicit unconditionally stable finite element scheme developed for this PML model.

Analysis and application of a spatial fourth-order finite difference scheme for the Ziolkowski's PML model

Perfectly Matched Layer (PML) technique is an effective tool to reduce the unbounded wave propagation problem to a bounded domain problem. In this paper, we develop and analyze a fourth-order in space finite difference scheme for solving the Ziolkowski's PML model. Though fourth-order in space schemes have been widely used in solving Maxwell's equations, few papers provide a rigorous convergence analysis. One of the major contributions here is to fill this gap by proving the fourth-order pointwise convergence of the scheme. Numerical results are presented to justify our analysis, and demonstrate the effectiveness of this PML model in absorbing outgoing waves.

Discontinuous Galerkin discretizations and analysis for the Cohen–Monk PML model

We investigate the two-dimensional (2-D) perfectly matched layer (PML) models reformulated from the 3-D PML model originally developed by Cohen and Monk in 1999. We propose the discontinuous Galerkin methods for solving both 2-D TMz and TEz models. We establish the proofs of the stability and error estimate for the proposed schemes. Finally, numerical results are presented to demonstrate the accuracy and performance of our method.

Development and analysis of a new finite element method for the Cohen–Monk PML model

This work deals with the Cohen–Monk Perfectly Matched Layer (PML) model. We first carry out the stability analysis of its equivalent form. Then we propose and analyse a finite element scheme for solving this equivalent PML model. Discrete stability and optimal error estimate are established. Numerical results are presented to justify the analysis and effectiveness of this PML model. This paper presents the first mathematical analysis for this PML model and the corresponding numerical analysis for the proposed finite element scheme.

A stable discontinuous Galerkin method for the perfectly matched layer for elastodynamics in first order form

We present a stable discontinuous Galerkin (DG) method with a perfectly matched layer (PML) for three and two space dimensional linear elastodynamics, in velocity-stress formulation, subject to well-posed linear boundary conditions. First, we consider the elastodynamics equation, in a cuboidal domain, and derive an unsplit PML truncating the domain using complex coordinate stretching. The hyperbolic structure of the underlying system enables the construction of continuous energy estimates, in the time domain for the elastic wave equation, and in the Laplace space for a sequence of PML model problems, with variations in one, two and three space dimensions, respectively. They correspond to PMLs normal to boundary faces, along edges and in corners. Second, we develop a DG numerical method for the linear elastodynamics equation using physically motivated numerical flux and penalty parameters, which are compatible with all well-posed, internal and external, boundary conditions. When the PML damping vanishes in all directions, by construction, our choice of penalty parameters yield an upwind scheme and a discrete energy estimate analogous to the continuous energy estimate. Third, to ensure numerical stability of the discretization when PML damping is present, it is necessary to extend the numerical DG fluxes, and the numerical inter-element and boundary procedures, to the PML auxiliary differential equations. This is crucial for deriving discrete energy estimates analogous to the continuous energy estimates. Numerical solutions are evolved in time using the high order arbitrary derivative (ADER) time stepping scheme of the same order of accuracy with the spatial discretization. By combining the DG spatial approximation with the high order ADER time stepping scheme and the accuracy of the PML we obtain an arbitrarily high-order accurate wave propagation solver in the time domain. Numerical experiments are presented in two and three space dimensions corroborating the theoretical results.

Enhanced PML Based on the Long Short Term Memory Network for the FDTD Method

This paper proposes a new absorbing boundary condition (ABC) computation approach based on the deep learning technique. Benefited from the sequence dependence feature, the Long Short-Term Memory (LSTM) network is employed to replace the conventional perfectly matched layer (PML) ABC for the Finite-Difference Time-Domain (FDTD) solving process. The newly proposed LSTM based PML model is trained by the electromagnetic field data on the interface of the conventional PML. Different from the conventional PML, the newly proposed model only needs one cell layer as the boundary. Hence, the newly proposed method conveniently reduces both the algorithm’s complexity and the area of computation domain of FDTD. Additionally, the newly proposed LSTM based PML model can achieve higher accuracy than the conventional artificial neural network (ANN) based PML, thanks to the sequence dependence feature of the LSTM networks. Numerical examples have illustrated the capability and the accuracy of the proposed LSTM model. The results illustrate that the new method can be compatibly embedded into the FDTD solving process with the high accuracy.

Development and analysis of both finite element and fourth-order in space finite difference methods for an equivalent Berenger's PML model

This paper deals with an equivalent Berenger's Perfectly Matched Layer (PML) model. We first develop a finite element scheme using edge elements to solve this model. We prove a discrete stability of this method, which inherits the stability obtained in the continuous case. Then we propose a fourth-order in space finite difference scheme for solving this PML model. Numerical stability similar to the continuous stability and the optimal error estimate are established for the difference scheme. Here only second order time discretizations are considered for both schemes. Finally, numerical results are presented to justify our analysis and demonstrate the effectiveness of this PML model for absorbing impinging waves.

Miniaturized Anechoic Chamber Constructed Based on an Inhomogeneous PML Model

In electromagnetics, anechoic chambers are used to obtain artificial analogs of free space. Such chambers are electrically large due to the inherent angle-dependent nature of absorbing materials. Existing models of perfectly matched layer (PML) imply that angle-independent reflectionless absorptions can hardly be achieved without involving gain media. In this paper, we mathematically derived an unconventional PML model, based on which the angle-dependent nature of absorbing materials can be manipulated to construct electrically small anechoic chambers. As an example, a semiclosed cylindrical chamber was experimentally implemented with an inhomogeneous artificial surface as thin as 0.012 wavelength. The inner diameter of this chamber is six wavelengths. The spatial voltage standing wave ratio (VSWR) experimentally measured inside this chamber complies with the theoretical analysis and full-wave simulation. With the aid of the dielectric 3-D printing technology, high-performance, microsized anechoic chambers can be constructed for scientific research and technology development in the future.

Mathematical analysis of Ziolkowski’s PML model with application for wave propagation in metamaterials

In this paper we investigate one Perfectly Matched Layer (PML) model proposed by Ziolkowski in 1999. Various schemes for solving this PML model have been developed and have been shown to be effective in absorbing outgoing waves when the wave propagation in unbounded domain problem is reduced to a bounded domain problem. However, a rigorous analysis of this model is lacking. In this paper we establish the stability of this PML model and propose a fully-discrete finite element scheme to solve this model with edge elements. Discrete stability and optimal error estimate of this scheme are proved. Numerical results justifying the analysis and demonstrating the effectiveness of this PML model are presented.

A Modified PML Acoustic Wave Equation

In this paper, we consider a two-dimensional acoustic wave equation in an unbounded domain and introduce a modified model of the classical un-split perfectly matched layer (PML). We apply a regularization technique to a lower order regularity term employed in the auxiliary variable in the classical PML model. In addition, we propose a staggered finite difference method for discretizing the regularized system. The regularized system and numerical solution are analyzed in terms of the well-posedness and stability with the standard Galerkin method and von Neumann stability analysis, respectively. In particular, the existence and uniqueness of the solution for the regularized system are proved and the Courant-Friedrichs-Lewy (CFL) condition of the staggered finite difference method is determined. To support the theoretical results, we demonstrate a non-reflection property of acoustic waves in the layers.

Machine-Learning-Based PML for the FDTD Method

In this letter, a novel absorbing boundary condition (ABC) computation method for finite-difference time-domain (FDTD) is proposed based on the machine learning approach. The hyperbolic tangent basis function (HTBF) neural network is introduced to replace traditional perfectly matched layer (PML) ABC during the FDTD solving process. The field data on the interface of conventional PML are employed to train HTBF-based PML model. Compared to the conventional approach, the novel method greatly decreases the size of a computation domain and the computation complexity of FDTD because the new model only involves the one-cell boundary layer. Numerical examples are provided to benchmark the performance of the proposed method. The results demonstrate that the newly proposed method could replace conventional PML and could be integrated into FDTD solving process with satisfactory accuracy and compatibility to FDTD. According to our knowledge, this proposed model combined artificial neural network (ANN) model is an unreported new approach based on a machine learning based for FDTD.

Numerical considerations on the modeling of source and boundary conditions for the frequency domain visco-acoustic wave equation solution

Seismic modeling is an important step in the process used for imaging Earth subsurface. Current applications require accurate models associated with solutions of the wave propagation equation in real media. Unfortunately, it is common not to find in the technical literature deep discussions on the impact of specific details associated with the physical modeling of some crucial ingredients of the process, such as seismic source term and boundary conditions. In this paper, we discuss some issues related to the modeling of wave propagation in visco-acoustic media using finite differences. We focus our attention on two major elements of the modeling problem that are associated to the source term and the boundary conditions. We show that the source term can be modeled using a scale parameter that controls the spread of energy and shows that this parameter is a function of frequency and position of the source. As to boundary conditions, we show that Perfectly Matched Layer (PML) parameters are also frequency dependent. For both cases, seismic source scale parameter and PML model parameters we provide values and functions that optimize the performance of the approach for problems where visco-acoustic wave propagation is required. Frequency domain Full Waveform Inversion (FWI), or Reverse Time Migration (RTM) processes that depend fundamentally on the appropriate modeling of the wave-field are potential fields of application of these results.

Application of perfectly matched layer to soil-foundation interaction analysis

Despite of the limitation in modeling infinite space, the finite element method (FEM) is one of the most used tools to numerically study the geotechnical problems regarding the capacity of simulating different geometries, conditions and material behaviors. A kind of absorbing layer named perfectly matched layer (PML) has been applied to modeling the radiation damping using FEM, which makes the dynamic analysis of soil-structure interaction more accurate. The PML is capable of absorbing incident waves under any angle and frequency, ensuring them to pass through the model boundaries without reflection. In this context, a new FEM program has been written and the PML formula has been implemented by rewriting the dynamic equation of motion and deriving new properties for the quadrilateral elements. The analysis of soil-foundation interaction by applying the PML is validated by the evaluation of impedance/compliance functions for different ground conditions. The results obtained from the PML model match the extended mesh results, even though the domain is small enough that other types of absorbing boundaries can reflect waves back to the foundation. The mechanism of the wave propagation in the region shows that the forced vibrations can be fully absorbed and damped by the boundaries surrounded by PMLs which is the role of radiation damping in FEM modeling.

Analysis and application of an equivalent Berenger’s PML model

Abstract The perfectly matched layer (PML) is a technique initially proposed by Berenger for solving unbounded electromagnetic problems with the finite-difference time-domain method. In this work, we first formulate an equivalent PML model from the original Berenger PML model in the corner region, and then establish its stability. We further develop a discontinuous Galerkin method to solve this PML model, and discrete stability similar to the continuous case is proved. To demonstrate the absorbing property of this PML model, we apply it to simulate wave propagation in metamaterials.

Perfectly matched layer on curvilinear grid for the second-order seismic acoustic wave equation

A curvilinear-grid perfectly matched layer (PML) absorbing boundary condition for the second-order seismic acoustic wave equation is presented in this paper. The rectangular grids are transformed into curvilinear grids by using a mathematical mapping to fit the curvilinear boundary, and the original wave equation is reformulated under the curvilinear coordinate system. Based on the reformulated wave equation, theoretical expressions and analysis of the curvilinear-grid PML are given. Furthermore, PML model 1 with symmetric form and PML model 2 with asymmetric form are derived from the same acoustic wave equation. By combination with the finite difference (FD) method, these two models are applied to seismic wave modelling with surface topography. The results show that the absorption effect of these two models discretised by the same second-order time difference and second-order space difference are different, and the symmetric-form PML yields better modelling results than the asymmetric-form.

2.5-D frequency-domain seismic wave modeling in heterogeneous, anisotropic media using a Gaussian quadrature grid technique

We present an extension of the spectral element method (SEM), called the Gaussian quadrature grid technique, for 2.5-D frequency-domain seismic wave modeling in heterogeneous, anisotropic media having arbitrary free-surface topography. The technique has two new features. First, it employs a point-gridded model sampled by irregular Gaussian quadrature abscissa rather than a hexahedral-element mesh so as to simplify the procedures of matching the free-surface topography and the subsurface interface geometry. Furthermore, it offers flexibilities in the subdomain sizes, the Gaussian quadrature scheme and orders employed, and the number of subdomain abscissae in terms of the model geological characteristics. Second, we have incorporated a simple implementation of the 2.5-D perfectly matched layer (PML) technique to suppress the reflections from the artificial boundaries. We show that removing the artificial reflections in arbitrary anisotropic media can be achieved by simply employing the so-called “PML model parameters,” which are specified complex density and elastic moduli obtained by the PML theory. The two features do not compromise the main advantages of the SEM, such as the high accuracy, the sparse system matrix, and wide applicability for arbitrary geological models. We numerically show the excellent effects of the PML model parameters and the capability of the presented method by means of homogeneous, isotropic, and anisotropic models that have analytic solutions for verification. In addition, we demonstrate performance on homogeneous and heterogeneous, anisotropic models, both having an undulating free-surface topography. The model results yield global views of the vectorial wavefields in the frequency domain.

Irregular perfectly matched layers for 3D elastic wave modeling

We have developed a perfectly matched layer (PML) absorbing boundary condition that can be imposed along an arbitrary geometric boundary in 3D elastic wave modeling. The scheme is developed by using the local coordinate system-based PML splitting equations and integral approach of the PML equations under a discretization of tetrahedral grids. However, no explicit coordinate transformations arise. The local coordinate system-based PML splitting equations make it possible to decay incident waves around the direction normal to the irregular geometric boundaries, instead of a coordinate axis direction. Based on the resulting 3D irregular PML model, we can flexibly construct the computational domain with smaller nodes by cutting uninterested zones. This results in significant reductions in computational cost and memory requirements for 3D simulations. By building a smooth artificial boundary, the irregular PML model can avoid the respective treatments to the edges and corners of the artificial boundaries. Also, the irregular PML model may reduce the grazing incidence that makes the PML model less efficient by changing the shapes of the artificial boundaries. Numerical testing was used to demonstrate the performance of the irregular PML model.

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.

A TIME DOMAIN METHOD FOR MODELING VISCOACOUSTIC WAVE PROPAGATION

In many applications, and in particular in seismology, realistic propagation media disperse and attenuate waves. This dissipative behavior can be taken into account by using a viscoacoustic propagation model, which incorporates a complex and frequency-dependent viscoacoustic modulus in the constitutive relation. The main difficulty then lies in finding an efficient way to discretize the constitutive equation as it becomes a convolution integral in the time domain. To overcome this difficulty the usual approach consists in approximating the viscoacoustic modulus by a low-order rational function of frequency. We use here such an approximation and show how it can be incorporated in the velocity-pressure formulation for viscoacoustic waves. This formulation is coupled with the fictitious domain method which permits us to model efficiently diffraction by objects of complicated geometry and with the Perfectly Matched Layer Model which allows us to model wave propagation in unbounded domains. The space discretization of the problem is based on a mixed finite element method and for the discretization in time a 2nd order centered finite difference scheme is employed. Several numerical examples illustrate the efficiency of the method.