Perfectly Matched Layer Parameters Research Articles

An Adaptive PML Finite Volume Algorithm for the Scattering by Periodic Gratings

In this paper, we studied the finite volume formulations for solving the diffraction grating problem that is truncated by the perfectly matched layer (PML) technique. Based on a reliable the a-posteriori error estimate, an adaptive PML finite volume method is discussed for the numerical approximation of the diffraction grating problem. The PML parameters are obtained numerically by sharp a-posteriori error estimates of the PML finite volume method such as the thickness of the layer and the medium property. It is worth mentioning in the a-posteriori error estimates that we derive the error representation formula and use a [Formula: see text]-orthogonality property of the residual similar to the Galerkin orthogonality used in the finite element method. Furthermore, the lower bound is established to demonstrate the efficiency of the a-posteriori error estimates. Numerical experiments are given to illustrate the accuracy and robustness of our adaptive algorithm.

2D frequency-domain finite-difference acoustic wave modeling using optimized perfectly matched layers

Seismic forward modeling is fundamental in sensitivity analysis and full-waveform inversions. In finite-difference acoustic wavefield simulation, the absorption boundary, especially the perfectly matched layer (PML), is widely used, but the setting of PML parameters is empirical. An optimized complex frequency-shift PML (CFS-PML) for the modeling of acoustic fields has been developed. It refines the selection of parameters for improving the artificial attenuation. The improved CFS-PML boundary condition is applied to a 2D frequency-domain acoustic wave simulation using an optimal 17-point finite-difference scheme. Numerical results are compared with a conventional nine-point scheme in terms of computational time and physical memory consumption. These tests indicate that our CFS-PML absorption boundary can effectively improve numerical accuracy without increasing the computational burden remarkably.

Time- and frequency-domain hybridizable discontinuous Galerkin solvers for the calculation of the Cherenkov radiation

This work proposes novel hybridizable discontinuous Galerkin (HDG) methods, both in the time and in the frequency domain, to accurately compute the Cherenkov radiation emitted by a charged particle travelling in a uniform medium at superluminal speed. The adopted formulations enrich existing HDG approaches for the solution of Maxwell’s equations by including perfectly matched layers (PMLs) to effectively absorb the outgoing waves and Floquet-periodic boundary conditions to connect the boundaries of the computational domain in the direction of the moving charge. A wave propagation problem with smooth solution is used to show the optimal convergence of the HDG variables and the superconvergence of the postprocessed electric field and a second example examines the role of the PML parameters on the absorption of the electromagnetic field. A series of numerical experiments both in 3D and 2D-axisymmetric components show the capability of the proposed methods to faithfully reproduce Cherenkovian effects in different conditions and their high accuracy is confirmed by comparing the numerical results with the Frank–Tamm formula.

Wave propagation in inhomogeneous media via FE/PML method

AbstractThe Perfectly Matched Layer (PML) method is an efficient approach to imposing radiation conditions at the bounded region of interest in case of wave propagation in unbounded domains. This paper presents and validates 3D FE/PML numerical schemes based on two different PML formulations for homogeneous and inhomogeneous geological media exhibiting discrete or continuous inhomogeneity. In the equation of motion for the PML domain the applied stretching behavior is expressed either as complex material properties or as complex coordinates. Both PML formulations are implemented in the FEM and verified against analytical solutions. Three different types of material inhomogeneity are considered: layered half‐space, continuously inhomogeneous half‐space with linear velocity profile and continuously inhomogeneous half‐space with nonlinear velocity profile. Sensitivity analyses are conducted, and the performance of the developed numerical schemes is investigated taking into account a broad variation of the PML parameters. Recommendations are given for the optimal values of the PML parameters for the case of homogeneous and inhomogeneous geological media.

An Efficient Mixed Finite Element Perfectly Matched Layer with Optimal Parameters Selection for Two-Dimensional Time Domain Soil-Structure Interaction Analysis

Perfectly matched layer (PML) is known as one of the best methods to simulate infinite domains in many fields such as soil-structure interaction (SSI). The performance of PML is significantly affected by PML parameters selection. However, the way to select PML parameters still remains unclear. This study proposes a method for PML parameters determination for elastic wave propagation in two-dimensional (2D) media. The scaling and attenuation functions are developed in order to increase the accuracy and effectiveness of the PML. The proposed scheme is applied for a mixed PML in time domain. The finite element method (FEM) formulations of the PML are presented so that it can be easily applied to the existing codes. ABAQUS, a popular FEM code, is used for numerical applications in this study. The proposed PML is imported into ABAQUS by using a user-defined element (UEL) written in Fortran language. Six numerical analyses of SSI are implemented to prove the efficiency of the proposed PML. The numerical analyses cover many realistic problems, including free field, surface structure, and embedded structure problems. The results demonstrate the efficiency of the proposed PML in terms of the accuracy and computational cost.

A sliced-3D approach to finite-difference time-domain modeling by optimizing perfectly matched layers

Finite-difference time-domain forward modeling is often used to gain a more quantitative understanding of the interactions between electromagnetic fields and targets. To undertake full 3D simulations, the computational demands are challenging, so simulations are often undertaken in two dimensions, in which assumptions in the propagation of electromagnetic fields and source type can result in errors. We have developed the concept of a sliced-3D simulation, wherein a thin slice of a 3D domain with strictly 2D geometry is used to minimize computational demands while obtaining synthetic waveforms that contain full 3D propagation effects. This approach requires optimization of perfectly matched layer (PML) boundary condition parameters so as to minimize the errors associated with the source being located close to the boundary, and as a result of grazing-incident angle wave conversion to evanescent energy. We explore the frequency dependence of PML parameters, and we establish a relationship between complex frequency stretching parameters and effective wavelength. The resultant parameter choice is shown to minimize propagation errors in the context of a simple radioglaciological model, in which 3D domains may be prohibitively large, and for a near-surface cross-borehole survey configuration, a case in which full-waveform inversion may typically be used.

Parameter Design of Conformal PML Based on 2D Monostatic RCS Optimization

In this study, 2D finite element (FE) solving process with the conformal perfectly matched layer (PML) is elucidated to perform the electromagnetic scattering computation. With the 2D monostatic RCS as the optimization objective, a sensitivity analysis of the basic design parameters of conformal PML (e.g., layer thickness, loss factor, extension order and layer number) is conducted to identify the major parameters of conformal PML that exerts more significant influence on 2D RCS. Lastly, the major design parameters of conformal PML are optimized by the simulated annealing algorithm (SA). As revealed from the numerical examples, the parameter design and optimization method of conformal PML based on SA is capable of enhancing the absorption effect exerted by the conformal PML and decreasing the error of the RCS calculation. It is anticipated that the parameter design method of conformal PML based on RCS optimization can be applied to the cognate absorbing boundary and 3D electromagnetic computation.

Constitutive Parameter Optimization Method of Obliquely Incident Reflectivity for Conformal PML

The conformal perfectly matched layer (PML), i.e., an efficient absorbing boundary condition, is commonly employed to address the open-field scattering problem of electromagnetic wave. To develope a conformal PML exhibiting a significant absorption effect and small reflection error, the present study proposes the constitutive parameter optimization method of obliquely incident reflectivity in terms of the conformal PML. First, the recurrence formula of obliquely incident reflectivity is desired. Subsequently, by the sensitivity analysis of constitutive parameters, the major optimal design variables are determined for the conformal PML. Lastly, with the reflectivity of the conformal PML as the optimization target, this study adopts the genetic algorithm (GA), simulated annealing algorithm (SA) and particle swarm optimization algorithm (PSO) to optimize the constitutive parameters of the conformal PML. As revealed from the results, the optimization method is capable of significantly reducing the reflection error and applying to the parameter design of conformal PML.

An Optimized CPML Formulation for High Order FVTD Schemes for CED

Developing an optimal perfectly matched layer (PML) formulation is crucial for efficient CED calculations. This becomes imperative for higher order CED schemes. If a PML causes spurious back-reflection of waves into the computational domain, those waves will be evolved by the higher order scheme as if they are physical. We present a PML that is conformant in its collocation and discretization with second, third and fourth order finite volume time-domain (FVTD) schemes that preserve global divergence. We present optimal PML parameters for second, third and fourth order FVTD schemes based on a careful numerically-motivated, optimization. At each order of accuracy we have to repeat the optimization study in order to get the best performance. We find that with increasing order of accuracy we can achieve greater suppression of spuriously reflected waves from the PML layer, especially at late times. Taking the finite difference time-domain (FDTD) method as a baseline, our schemes show as much as three orders of magnitude improvement in the suppression of late time reflection of waves from the PML layer. The schemes rely on several newly-invented reconstruction strategies and a very novel Implicit Taylor ADER (Arbitrary accuracy DERivatives) predictor step. Riemann solvers provide the corrector step.

Enhanced conformal perfectly matched layers for Bernstein–Bézier finite element modelling of short wave scattering

The aim of this paper is to accurately solve short wave scattering problems governed by the Helmholtz equation using the Bernstein–Bézier Finite Element method (BBFEM), combined with a conformal perfectly matched layer (PML). Enhanced PMLs, where curved geometries are represented by means of the blending map method of Gordon and Hall, are numerically investigated. In particular, the performance of radial and elliptical shaped PMLs, with a parabolic absorption function, are assessed and compared in terms of accuracy against second order Bayliss–Gunzberger–Turkel (BGT2) based local absorbing boundary conditions. Numerical results dealing with problems of Hankel source radiation and wave scattering by a rigid cylinder show that the radial shaped PML, with the h and p versions of BBFEM, enables the recovery of the predicted algebraic and exponential convergence rates of a high order finite element method (FEM). Furthermore, radial shaped BGT2 and PML have a similar performance, as long as the wave is not sufficiently well resolved. But, BGT2 performs poorly as the wave resolution increases. Additionally, the effect of harmonics of higher modes on accuracy is examined. The study reveals that the PML outperforms BGT2 for almost all propagating modes. However, a similar performance is achieved with both methods either with higher modes or a low wave resolution. Results from a multiple scattering benchmark problem provide evidence of the good performance of the proposed PMLs and the benefit of elliptical shaped PMLs in reducing significantly the size of the computational domain, without altering accuracy. The choice of the PML parameters ensuring optimal performance is also discussed.

Perfectly Matched Layers for time-harmonic transverse electric wave propagation in cylindrical and toroidal gyrotropic media

In this paper we implement the stretched-coordinate Perfectly Matched Layer (PML) technique in [28] to emulate full power absorption outside the simulation domain for time-harmonic electromagnetic wave propagation in presence of gyrotropic dielectric tensor and curved geometry relevant for magnetized plasma devices. We recall the PML formulation as an artificial inhomogeneous lossy medium, following the stretching into the complex plane of a general system of three orthogonal curvilinear coordinates. We apply the general method in cylindrical and toroidal geometries. We then assess this technique in a simple case combining gyrotropy and coordinate curvature. Our test problem analytically quantifies the reflection of Transverse Electric (TE) cylindrical eigenmodes in a gyrotropic medium by a radial PML in cylindrical geometry. The obtained reflection coefficient involves wave, PML and geometric parameters at the PML location. The new coefficient generalizes the one obtained earlier with Cartesian coordinates, and becomes equivalent when the effects of the local cylindrical curvature at the PML (stretched) location can be neglected. These curvature effects are outlined and the limitations they impose on the properties of the PML are quantified as a function of the relevant parameters. Peculiarities related to the gyrotropy are also highlighted. Finite element calculations of the test problem in two-dimensional cylindrical geometry are exploited to verify these properties numerically. Indications are finally given on how to choose the PML parameters in order to obtain a minimal wave reflection at given numerical cost, taking into account errors associated with the numerical scheme.

An Absorbing Boundary Condition Based on Perfectly Matched Layer Technique Combined with Discontinuous Galerkin Boltzmann Method for Low Mach Number Flow Noise

For flow noise simulations, the nonreflecting boundary condition (NRBC) is significant to confine the computational domain to a small domain. Lattice Boltzmann method (LBM) has advantages for noise because of its low dissipation, but is limited to the uniform grid. In this paper, an absorbing boundary condition (ABC) based on perfectly matched layer (PML) technique is introduced to LBM. Then PML stability is analyzed and a new strategy is developed to achieve robustness. Invoking the decoupling time integration, the underlying equation for streaming is solved with the nodal discontinuous Galerkin method. Benchmark acoustic problems were used to demonstrate the PML absorption. Moreover, PML parameters, long time behavior and inhomogeneous pseudo mean flow are discussed. The methodology appears to work very well and would be hoped for practical flow noise computation.

Performance Analysis of Perfectly Matched Layers Applied to Spherical FDTD Grids

A numerical analysis of the performance of the spherical perfectly matched layer (PML) applied to the finite-difference time-domain (FDTD) algorithm implemented in spherical staggered grids is presented in this communication. The spherical PML is derived through a complex transformation (analytical continuation) of the radial coordinate. We evaluate the performance of the spherical PML by simulating fields produced by electric dipoles within the spherical FDTD terminated by a spherical PML and computing the reflection error. The influence of several PML parameters on the reflection error is examined.

An Absorbing Boundary Condition Based on Perfectly Matched Layer Technique Combined with Discontinuous Galerkin Boltzmann Method for Low Mach Number Flow Noise

For flow noise simulations, the nonreflecting boundary condition (NRBC) is significant to confine the computational domain to a small domain. Lattice Boltzmann method (LBM) has advantages for noise because of its low dissipation, but is limited to the uniform grid. In this paper, an absorbing boundary condition (ABC) based on perfectly matched layer (PML) technique is introduced to LBM. Then PML stability is analyzed and a new strategy is developed to achieve robustness. Invoking the decoupling time integration, the underlying equation for streaming is solved with the nodal discontinuous Galerkin method. Benchmark acoustic problems were used to demonstrate the PML absorption. Moreover, PML parameters, long time behavior and inhomogeneous pseudo mean flow are discussed. The methodology appears to work very well and would be hoped for practical flow noise computation.

An adaptive finite element PML method for the elastic wave scattering problem in periodic structures

An adaptive finite element method is presented for the elastic scattering of a time-harmonic plane wave by a periodic surface. First, the unbounded physical domain is truncated into a bounded computational domain by introducing the perfectly matched layer (PML) technique. The well-posedness and exponential convergence of the solution are established for the truncated PML problem by developing an equivalent transparent boundary condition. Second, an a posteriori error estimate is deduced for the discrete problem and is used to determine the finite elements for refinements and to determine the PML parameters. Numerical experiments are included to demonstrate the competitive behavior of the proposed adaptive method.

PML solution of longitudinal wave propagation in heterogeneous media

This paper describes the development of a model for unbounded heterogeneous domains with radiation damping produced by an unphysical wave absorbing layer. The Perfectly Matched Layer (PML) approach is used along with a displacement-based finite element. The heterogeneous model is validated using the closed-form solution of a benchmark problem: a free rod with two-part modulus subjected to a specified time history. Both elastically supported and unsupported semi-infinite rods with different degrees of inhomogeneity and loading are considered. Numerical results illustrate the effects of inhomogeneity on the response and are compared with those for equivalent homogeneous domains. The effects of characteristic features of the inhomogeneous problem, presence of local maxima and cut-off frequency are determined. A degenerate case of a homogeneous semi-infinite rod on elastic foundations is produced by tending the magnitude of the foundation stiffness to zero. The response of the latter is compared with that of a free rod. The importance of proper selection of the PML parameters to highly accurate and efficient results is demonstrated by example problems.

Numerical modeling of three-dimensional open elastic waveguides combining semi-analytical finite element and perfectly matched layer methods

Among the numerous techniques of non-destructive evaluation, elastic guided waves are of particular interest to evaluate defects inside industrial and civil elongated structures owing to their ability to propagate over long distances. However for guiding structures buried in large solid media, waves can be strongly attenuated along the guide axis due to the energy radiation into the surrounding medium, usually considered as unbounded. Hence, searching the less attenuated modes becomes necessary in order to maximize the inspection distance. In the numerical modeling of embedded waveguides, the main difficulty is to account for the unbounded section. This paper presents a numerical approach combining a semi-analytical finite element method and a perfectly matched layer (PML) technique to compute the so-called trapped and leaky modes in three-dimensional embedded elastic waveguides of arbitrary cross-section. Two kinds of PML, namely the Cartesian PML and the radial PML, are considered. In order to understand the various spectral objects obtained by the method, the PML parameters effects upon the eigenvalue spectrum are highlighted through analytical studies and numerical experiments. Then, dispersion curves are computed for test cases taken from the literature in order to validate the approach.

An Accurate Derivation of Spatial Green's Function for Finite Dielectric Structures Using Characteristic Green's Function-Perfectly Matched Layer Method

A closed-form spatial Green's function for a truncated dielectric slab is derived by using a combination of the characteristic Green's function (CGF) and perfectly matched layer (PML) method. The original structure is terminated by PML that is backed by perfect electric conductor (PEC) in semi-infinite layer at the top and/or bottom. The eigenmodes of the closed structure by PML construct the continuous spectrum contribution of the original structure efficiently. Derived Green's function is exact for separable finite structure while it is approximate for nonseparable one especially in the corners. Generalized scattering matrix (GSM) of truncating surface which contains possible conversions between all modes is computed with mode-matching method. It is demonstrated that by importing GSM in CGF formulation, more exact results for nonseparable structures can be obtained. The source and observation points dependence is analytically available in derived expression and allows very efficient computation and storage of the spatial Green's function. Very close to the source, where the large number of modes must be considered, Shank's transform is used to preserve the efficiency of the method. The main advantage of the method lies in its accuracy and low dependency of the result to the PML parameters. Excellent agreements with the rigorous method of moments (MoM) are shown in several examples.

Stability of finite difference numerical simulations of acoustic logging-while-drilling with different perfectly matched layer schemes

In acoustic logging-while-drilling (ALWD) finite difference in time domain (FDTD) simulations, large drill collar occupies, most of the fluid-filled borehole and divides the borehole fluid into two thin fluid columns (radius ∼27 mm). Fine grids and large computational models are required to model the thin fluid region between the tool and the formation. As a result, small time step and more iterations are needed, which increases the cumulative numerical error. Furthermore, due to high impedance contrast between the drill collar and fluid in the borehole (the difference is >30 times), the stability and efficiency of the perfectly matched layer (PML) scheme is critical to simulate complicated wave modes accurately. In this paper, we compared four different PML implementations in a staggered grid finite difference in time domain (FDTD) in the ALWD simulation, including field-splitting PML (SPML), multiaxial PML(MPML), non-splitting PML (NPML), and complex frequency-shifted PML (CFS-PML). The comparison indicated that NPML and CFS-PML can absorb the guided wave reflection from the computational boundaries more efficiently than SPML and M-PML. For large simulation time, SPML, M-PML, and NPML are numerically unstable. However, the stability of M-PML can be improved further to some extent. Based on the analysis, we proposed that the CFS-PML method is used in FDTD to eliminate the numerical instability and to improve the efficiency of absorption in the PML layers for LWD modeling. The optimal values of CFS-PML parameters in the LWD simulation were investigated based on thousands of 3D simulations. For typical LWD cases, the best maximum value of the quadratic damping profile was obtained using one d 0. The optimal parameter space for the maximum value of the linear frequency-shifted factor (α 0) and the scaling factor (β 0) depended on the thickness of the PML layer. For typical formations, if the PML thickness is 10 grid points, the global error can be reduced to <1% using the optimal PML parameters, and the error will decrease as the PML thickness increases.

An adaptive anisotropic perfectly matched layer method for 3-D time harmonic electromagnetic scattering problems

We develop an anisotropic perfectly matched layer (PML) method for solving the time harmonic electromagnetic scattering problems in which the PML coordinate stretching is performed only in one direction outside a cuboid domain. The PML parameters such as the thickness of the layer and the absorbing medium property are determined through sharp a posteriori error estimates. Combined with the adaptive finite element method, the proposed adaptive anisotropic PML method provides a complete numerical strategy to solve the scattering problem in the framework of FEM which produces automatically a coarse mesh size away from the fixed domain and thus makes the total computational costs insensitive to the choice of the thickness of the PML layer. Numerical experiments are included to illustrate the competitive behavior of the proposed adaptive method.