Unsplit Perfectly Matched Layer Research Articles

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.

An Efficient Unsplit Perfectly Matched Layer for Finite-Element Time-Domain Modeling of Elastodynamics in Cylindrical Coordinates

The truncation of an unbounded medium into a finite domain and the elimination of reflections from artificial medium boundaries is important in numerical simulations of elastodynamic equations. The perfectly matched layer (PML) technique as a novel absorbing boundary has demonstrated very high efficiency for elastic wave equation models. Based on the stretched coordinate concept, an efficient and unsplit-field perfectly matched layer equation for elastic waves is formulated in a cylindrical coordinate system. By introducing integrated complex variables for radial direction and auxiliary functions, the PML formulation is extended in cylindrical coordinates based on the second-order elastic wave equation with displacements as basic unknowns. Finite-element time-domain modeling of the second-order unsplit PML for the elastodynamic equation is presented, which is a standard displacement-based formulation for the computational domain including a PML region. The formulas for the special cases with 2-D axisymmetric coordinates and polar coordinates are also given. The efficiency of the proposed formulations is illustrated by a numerical example using the finite-element method.

Perfectly matched absorbing layer for modelling transient wave propagation in heterogeneous poroelastic media

AbstractThe perfectly matched layer (PML) has been demonstrated to be an efficient absorbing boundary for near-field wave simulation. For heterogeneous media, the property of the PML needs to be carefully specified to avoid numerical instability and artificial reflection because part of it lies at the discontinuous interface. Coupled acoustic-poroelastic (A-P) media or coupled elastic-poroelastic (E-P) media often arise in the field of geophysics. However, PMLs that appropriately terminate these heterogeneous poroelastic media are still lacking. The main purpose of this paper is to explore the application of unsplit PMLs for transient wave modeling in infinite, heterogeneous, coupled A-P media or coupled E-P media. To this end, a consistent derivation of memory-efficient PML formulations for the second-order Biot's equations, elastic wave equations and acoustic wave equations is performed based on complex coordinate transformation using auxiliary differential equations. Furthermore, the interface boundary conditions inside the absorbing layer are rigorously derived for the considered A-P and E-P cases. Finally, the weak form of PML formulations for coupled poroelastic problems is presented. The finite element method is used to validate the proposed PML based on several two-dimensional benchmarks. The accuracy and stability of weak PML formulations are investigated. In particular, for coupled acoustic-poroelastic PML, two extreme (open-pore and sealed-pore) interface conditions are considered and PML results are compared with known analytical solutions. This study demonstrates the ability of the PML to effectively eliminate outgoing bulk waves and surface waves in coupled poroelastic media.

On energy stable discontinuous Galerkin spectral element approximations of the perfectly matched layer for the wave equation

In this paper, we develop a provably energy stable discontinuous Galerkin spectral element method (DGSEM), approximating the perfectly matched layer (PML) for the three and two space dimensional (3D and 2D) linear acoustic wave equations, in first order form, subject to well-posed linear boundary conditions. First, using the well-known complex coordinate stretching, we derive an efficient un-split modal PML for the 3D acoustic wave equation, truncating a cuboidal computational domain. Second, we prove asymptotic stability of the continuous PML by deriving energy estimates in the Laplace space, for the 3D PML in a heterogeneous acoustic medium, assuming piece-wise constant PML damping. In the time-domain, the energy estimate translates to a bound for the solutions in terms of the initial data. Third, we develop a DGSEM for the wave equation using physically motivated numerical flux, with penalty weights, which are compatible with all well-posed, internal and external, boundary conditions. When the PML damping vanishes, by construction, our choice of penalty parameters yields an upwind scheme and a discrete energy estimate analogous to the continuous energy estimate. Fourth, to ensure numerical stability of the discretization when PML is present, it is necessary to systematically extend the numerical fluxes, and the inter-element and boundary procedures, to the PML auxiliary differential equations. This is critical for deriving discrete energy estimates analogous to the continuous energy estimates. Finally, we propose a procedure to compute PML damping coefficients such that the PML error converges to zero, at the optimal convergence rate of the underlying numerical method. Numerical solutions are evolved in time using the high order Taylor-type time stepping scheme of the same order of accuracy of the spatial discretization. By combining the DGSEM spatial approximation with the high order Taylor-type time stepping scheme and the accuracy of the PML we obtain an arbitrarily accurate wave propagation solver in the time domain. Numerical experiments are presented in 2D and 3D corroborating the theoretical results.

Efficiency of perfectly matched layers for seismic wave modeling in second-order viscoelastic equations

Summary In order to improve the perfectly matched layer (PML) efficiency in viscoelastic media, we first propose a split multi-axial PML (M-PML) and an unsplit convolutional PML (C-PML) in the second-order viscoelastic wave equations with the displacement as the only unknown. The advantage of these formulations is that it is easy and efficient to revise the existing codes of the second-order spectral element method (SEM) or finite-element method (FEM) with absorbing boundaries in a uniform equation, as well as more economical than the auxiliary differential equations PML. Three models which are easily suffered from late time instabilities are considered to validate our approaches. Through comparison the M-PML with C-PML efficiency of absorption and stability for long time simulation, it can be concluded that: (1) for an isotropic viscoelastic medium with high Poisson's ratio, the C-PML will be a sufficient choice for long time simulation because of its weak reflections and superior stability; (2) unlike the M-PML with high-order damping profile, the M-PML with second-order damping profile loses its stability in long time simulation for an isotropic viscoelastic medium; (3) in an anisotropic viscoelastic medium, the C-PML suffers from instabilities, while the M-PML with second-order damping profile can be a better choice for its superior stability and more acceptable weak reflections than the M-PML with high-order damping profile. The comparative analysis of the developed methods offers meaningful significance for long time seismic wave modeling in second-order viscoelastic wave equations.

Efficient Unsplit Perfectly Matched Layers for Finite-Element Time-Domain Modeling of Elastodynamics

AbstractThe perfectly matched layer (PML) is a highly efficient absorbing boundary used for the numerical modeling of an elastic wave equation on an unbounded domain. In this work, the authors are concerned with a second-order unsplit PML for transient elastodymanic problems in a semi-plane medium with finite-element approximations. First, based on the concept of stretched coordinates, an efficient unsplit PML formulation is proposed without higher derivatives. Then a finite-element time-domain scheme of a second-order PML in a displacement formulation is developed, in which the Galerkin method is used in space discretization and a Newmark-type scheme is employed for time stepping. Inside the absorbing layer, only one auxiliary vector is required. Hence, the scheme is cheap to implement and easily coupled with standard finite-element methods. Finally, the accuracy and efficiency of the present unsplit PML is demonstrated in numerical examples with a finite-element time-domain scheme.

Simulation and Phantom Study of the Acoustic Nonlinear B/A Parameter for Biological Tissues by Using Comparative Method

The nonlinearity parameter B/A is an important indicator of the nonlinear acoustical effects for biological tissues. This paper investigates nonlinear B/A parameter for biological tissues by using comparative method. The investigations are carried out in both computer simulation and phantom study. In simulation study, Pseudo-spectral time domain (PSTD) method for second-order acoustic equation with a novel un-split Perfectly Matched Layer is proposed to improve the simulation efficiency and meanwhile maintain high accuracy. In phantom study, an experimental platform based on the Ritec SNAP-5000 system is designed. Experiments in simulation and phantom study achieve high estimation accuracy for B/A with percentage error less than 3%. Experimental results show that the comparative method can provide accurate B/A measurement for biological tissues.

A novel unsplit perfectly matched layer for the second-order acoustic wave equation

When solving acoustic field equations by using numerical approximation technique, absorbing boundary conditions (ABCs) are widely used to truncate the simulation to a finite space. The perfectly matched layer (PML) technique has exhibited excellent absorbing efficiency as an ABC for the acoustic wave equation formulated as a first-order system. However, as the PML was originally designed for the first-order equation system, it cannot be applied to the second-order equation system directly. In this article, we aim to extend the unsplit PML to the second-order equation system. We developed an efficient unsplit implementation of PML for the second-order acoustic wave equation based on an auxiliary-differential-equation (ADE) scheme. The proposed method can benefit to the use of PML in simulations based on second-order equations. Compared with the existing PMLs, it has simpler implementation and requires less extra storage. Numerical results from finite-difference time-domain models are provided to illustrate the validity of the approach.

Performance of convolutional perfectly matched layers for pseudospectral time domain poroviscoelastic schemes

This paper presents an implementation of the convolutional perfectly matched layer for the velocity–stress formulation of poroviscoelastic equations in anisotropic media, with high-order time and space formulations. After its introduction by Bérenger in 1994 for Maxwell's equation, the perfectly matched layer (PML) quickly became the standard approach to absorb outgoing waves on a numerical mesh. Subsequent developments include generalization and formulation in terms of a convolutional operator and auxiliary differential equations that improve efficiency and allow for the implementation of high-order time integration schemes. In this work, a fourth-order Runge–Kutta scheme is employed. Also, with the convolutional formulation, the wavefields need not be split as in the original formalism. Such unsplit PMLs allow for the implementation with a pseudospectral operator as presented in this paper. Although efforts have been deployed for the optimization of PML parameters, such an approach is not always tractable, especially for seismic wave propagation because different types of waves interact with the medium. Hence, the performance of the implementation is evaluated through a series of numerical experiments. Tests are performed for waves impinging on the PML interface both at normal and grazing incidence, for both isotropic and anisotropic media. The results highlight the advantage of the convolutional formulation for waves at grazing incidence and show that anisotropic media require a larger number of absorbing layers to achieve performances comparable to those obtained in isotropic media.

Distinguishing correct from incorrect PML proposals and a corrected unsplit PML for anisotropic, dispersive media

We show that some previous proposals for perfectly matched layer (PML) absorbers in anisotropic media or for waveguides at oblique incidence are not, in fact true PMLs; in previous work we similarly showed a failure of several PML proposals for periodic media (photonic crystals). We therefore argue that a more careful validation scheme is required for PML proposals, in contrast to past authors who have typically checked only that reflections are small for a fixed resolution, and suggest a simple validation scheme that can be readily applied to any PML proposal regardless of derivation or implementation. We demonstrate this test for a corrected, unsplit-field PML valid for anisotropic, dispersive media, implemented in both planewave-expansion and finite-difference time-domain (FDTD) methods.

A variational formulation of a stabilized unsplit convolutional perfectly matched layer for the isotropic or anisotropic seismic wave equation

In the context of the numerical simulation of seismic wave propagation, the perfectly matched layer (PML) absorbing boundary condition has proven to be efficient to absorb surface waves as well as body waves with non grazing incidence. But unfortunately the classical discrete PML generates spurious modes traveling and growing along the absorbing layers in the case of waves impinging the boundary at grazing incidence. This is significant in the case of thin mesh slices, or in the case of sources located close to the absorbing boundaries or receivers located at large offset. In previous work we derived an unsplit convolutional PML (CPML) for staggered-grid finite-difference integration schemes to improve the efficiency of the PML at grazing incidence for seismic wave propagation. In this article we derive a variational formulation of this CPML method for the seismic wave equation and validate it using the spectral-element method based on a hybrid first/secondorder time integration scheme. Using the Newmark time marching scheme, we underline the fact that a velocity-stress formulation in the PML and a second-order displacement formulation in the inner computational domain match perfectly at the entrance of the absorbing layer. The main difference between our unsplit CPML and the split GFPML formulation of Festa and Vilotte (2005) lies in the fact that memory storage of CPML is reduced by 35% in 2D and 44% in 3D. Furthermore the CPML can be stabilized by correcting the damping profiles in the PML layer in the anisotropic case. We show benchmarks for 2D heterogeneous thin slices in the presence of a free surface and in anisotropic cases that are intrinsically unstable if no stabilization of the PML is used. Keyword: Perfectly matched layers, Spectral elements, Boundary conditions, 1 Universite de Pau et des Pays de l’Adour, CNRS and INRIA Magique-3D. Laboratoire de Modelisation et Imagerie en Geosciences UMR 5212, Avenue de l’Universite, 64013 Pau cedex, France. E-mail:roland.martin@univ-pau.fr, dimitri.komatitsch@univ-pau.fr 2 Institut universitaire de France, 103 boulevard Saint-Michel, 75005 Paris, France. 3 Department of Electrical and Computer Engineering, University of Kentucky, Lexington, KY 40506-0046, USA. E-mail:gedney@engr.uky.edu A Variational Formulation of a Stabilized Unsplit Layer 275

A High-Order Time and Space Formulation of the Unsplit Perfectly Matched Layer for the Seismic Wave Equation Using Auxiliary Differential Equations (ADE-PML)

A High-Order Time and Space Formulation of the Unsplit Perfectly Matched Layer for the Seismic Wave Equation Using Auxiliary Differential Equations (ADE-PML)

An unsplit convolutional perfectly matched layer technique improved at grazing incidence for the viscoelastic wave equation

In the context of the simulation of wave propagation, the perfectly matched layer (PML) absorbing boundary layer has proven to be efficient to absorb non-grazing incidence waves. However, the classical discrete PML cannot efficiently absorb waves reaching the absorbing layer at grazing incidence. This is observed, for instance, in the case of thin mesh slices, or in the case of sources located close to the absorbing boundaries or receivers located at large offset. In order to improve the PML efficiency at grazing incidence we derive an unsplit convolutional PML (CPML) for a fourth-order staggered finite-difference numerical scheme applied to the 3-D viscoelastic seismic wave equation. The time marching equations of the standard linear solid mechanisms used do not need to be split and only the memory variables associated with velocity derivatives are stored at each time step. This is important in the case of more than one damping mechanism. Memory storage is reduced by more than 70 per cent in the PML regions in 3-D simulations compared to split PMLs optimized at grazing incidence. We validate the technique based on a benchmark performed in a thin mesh slice.

NUMERICAL SIMULATION OF OPEN WAVEGUIDE CONVERTERS USING FDTD METHOD

We study 3-dimensional asymmetric diffraction problems for waveguide-based electro-dynamic systems, radiating to infinite free space. For calculations we utilize the FDTD (Finite Difference Time Domain) numerical simulation method with the UPML (Unsplit Perfectly Matched Layer) absorbing boundary conditions. This paper states that the FDTD method, in spite of its relatively low calculation speed, has an approved ability of solving certain problems that cannot be solved by the other traditional numerical simulation methods (the method of integral equation, the method of scattering matrix).

Electromagnetic modeling and optimization of spatial power combiners/dividers with hard horns

A generalized scattering matrix (GSM) approach, which utilizes finite-difference time-domain (FDTD) with a modified unsplit PML (perfectly matched layer) and mode matching techniques, is used to analyze spatial power combiners/dividers with hard horn feeds. The simulation results are experimentally verified for a 1:9 spatial power dividing system. Furthermore, the effects of various design parameters on the overall performance of a 1:9 divider is investigated.

A modified unsplit PML formulation for evanescent mode absorption in waveguides

In this letter, a modified unsplit perfectly matched layer (PML) formulation for the absorption of evanescent waves in waveguides is presented. The proposed formulation combines the advantages of the stretched coordinate and unsplit D - E - H PML formulations, which separates PML conductivities from the properties of the physical materials in the FDTD mesh. Results show that the proposed boundary outperforms the traditional unsplit PML in the evanescent region by 20 to 40 dB.

Numerical reflection from FDTD-PMLs: a comparison of the split PML with the unsplit and CFS PMLs

Although perfectly matched layer (PML) absorbing boundary conditions are perfect in theory, an amount of spurious reflection is present in actual computations with the finite-difference time-domain (FDTD) method. This paper compares the reflections produced by three different PMLs, namely the split PML, the unsplit PML, and the recently introduced complex frequency shifted (CFS) PML. It is shown that the reflections from the split and unsplit PMLs are identical, while the CFS PML allows the reflection of evanescent waves to be significantly reduced.

Finite Difference Time Domain (Fdtd) Analysis of an Artificially-Synthesized Absorbing Medium

In this paper, we investigate an artificially-synthesized absorbing medium by using the Finite Difference Time Domain (FDTD) technique. The artificial medium is composed of a doubly-periodic array of lossy electric and magnetic material blocks (i.e., μ and ε media arranged in a checkerboard configuration). We analyze its reflection characteristics for both normal and oblique incidence cases. It is demonstrated that, if properly designed, the reflection characteristics of the checkerboard are considerably superior to those of a uniform ε = μ material of the same thickness. A multi-layer checkerboard is used to truncate a waveguide in the FDTD simulation; the result is comparable to that of an unsplit Perfectly Matched Layer (PML) with the same number of layers.

An unsplit step 3-D PML for use with the FDTD method

An important advance in the use of the finite-difference time-domain (FDTD) method has been the introduction of the perfectly matched layer (PML) to act as the absorbing boundary condition. The initial implementation required the E and H fields to be split. Recent advances have suggested a new unsplit step PML. This paper describes an FDTD implementation of this new unsplit PML in three dimensions, but implements them in the D and H fields. This has the advantage of isolating the PML from the rest of the FDTD computation, but, unlike the split step formulation, requires almost no additional computational resources.