Perfectly Matched Layer Region Research Articles

A Novel Method for Divergence Clean in 3-D Conformal Particle-in-Cell Code CEMPIC

For realizing divergence clean in our self-developed 3-D conformal particle-in-cell code CEMPIC, a commonly modified hyperbolic formulation to solve Maxwell’s equations is given. Based on this scheme, charged particles can directly be conformally emitted and absorbed. The modified purely hyperbolic Maxwell scheme can also be applied to perfectly matched layer (PML) formulation by using the auxiliary-differential equation technique, which will allow particles being within the PML region. Finally, some numerical examples are used to test the algorithms.

A Perfectly Matched Layer Technique Applied to Lattice Spring Model in Seismic Wavefield Forward Modeling for Poisson’s Solids

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

High-Order Conformal Perfectly Matched Layer for the DGTD Method

Based on the discontinuous Galerkin time-domain (DGTD) method, a new conformal perfectly matched layer (PML) is introduced to truncate the 3-D open domain. Compared with the traditional (rectangular or cubic) PML, the conformal PML is a smooth convex shell, which significantly reduces the buffer space in the computational domain. In this article, we construct the conformal PML in an orthogonal curvilinear coordinate system. Conductivities are defined to absorb the outgoing waves, depending on the distance from the sampling point to the non-PML region and the principal curvature radii of the sampling point, which are calculated utilizing the Weingarten transformation. The analytical expression of 3-D conformal PML is derived with the complex coordinate stretching technique. Furthermore, to reduce the total degrees of freedom (DoFs) while maintaining accuracy, the hierarchical vector basis functions are chosen to discretize the conformal PML and physical region. Numerical results validate its good absorption performance.

Review and Recent Developments on the Perfectly Matched Layer (PML) Method for the Numerical Modeling and Simulation of Elastic Wave Propagation in Unbounded Domains

This review article revisits and outlines the perfectly matched layer (PML) method and its various formulations developed over the past 25 years for the numerical modeling and simulation of wave propagation in unbounded media. Based on the concept of complex coordinate stretching, an efficient mixed displacement-strain unsplit-field PML formulation for second-order (displacementbased) linear elastodynamic equations is then proposed for simulating the propagation and absorption of elastic waves in unbounded (infinite or semi-infinite) domains. Both time-harmonic (frequency-domain) and time-dependent (timedomain) PML formulations are derived for two-and three-dimensional linear elastodynamic problems. Through the introduction of only a few additional variables governed by low-order auxiliary differential equations, the resulting mixed timedomain PML formulation is second-order in time, thereby allowing the use of standard time integration schemes commonly employed in computational structural dynamics and thus facilitating the incorporation into existing displacementbased finite element codes. For computational efficiency, the proposed timedomain PML formulation is implemented using a hybrid approach that couples a mixed (displacement-strain) formulation for the PML region with a classical (displacement-based) formulation for the physical domain of interest, using a standard Galerkin finite element method (FEM) for spatial discretization and a Newmark time scheme coupled with a finite difference (Crank-Nicolson) time scheme for time sampling. Numerical experiments show the performances of the PML method in terms of accuracy, efficiency and stability for two-dimensional linear elastodynamic problems in single-and multi-layer isotropic homogeneous elastic media.

Perfectly matched discrete layers for modeling unbounded domains

The method of perfectly matched layers (PML) is based on the elegant idea of introducing attenuation while keeping impedance unaltered, thus resulting in effective absorption of waves without spurious reflections. Due to its simplicity and effectiveness, PML has been widely used over the past 25 years to model wave propagation in unbounded domains. However, an acknowledged challenge is that PML no longer preserves impedance once the domain and the PML region are discretized, leading to spurious reflection at the interface. This issue has been fixed with the help of so-called impedance preserving discretization, leading to the method of perfectly matched discrete layers (PMDL), developed more than a decade ago. The idea is based on the observation that midpoint integration completely eliminates the discretization error in impedance that arises from linear finite element discretization. Beyond eliminating reflections at the interface, impedance preserving discretization is shown to be crucial to achieving stability of absorbing boundary conditions in situations with differing signs of phase and group velocities, e.g., anisotropic media. This talk will be an exposition of the development of the PMDL by the author and his collaborators, staring with fundamental ideas impedance preserving discretization followed by more advanced ABCs for anisotropic and periodic media.

A property-dependent Perfectly Matched Layer with a single additional layer for Maxwell’s equations in finite difference frequency domains

A Perfectly Matched Layer (PML) has become a widespread technique for preventing reflections from computational boundaries for electromagnetic wave propagation problems in frequency domains. Especially, a PML for the Helmholtz equation is well developed and applied to the finite difference method. In the currently used methods of PMLs, additional 4 to 6 layers surrounding a computational domain are required to be added in a PML region in order to terminate infinite domain calculations. Such a requirement not only demands a relatively large computer memory, but also increases execution time and complexity in implementations.The objectives of this paper are to show how to derive and implement a highly efficient PML on the termination of a computational grid in electromagnetic scattering simulations by one property-dependent PML (PPML) with just a single additional layer, based on the stretched-coordinate PML (SC-PML) and unsplit-field PML formulations. Its effectiveness is validated by examples with varying background media and applied frequencies in computational domains. These validate our proposed PPML to be a universal and accurate tool for solving frequency-domain Maxwell’s equations discretized by finite differences accurately in both lossless and lossy media under a wide range of applied frequencies. On one hand, PPML effectively reduces requirements of CPU time and computer memory by an additional single layer in the PPML region. On the other hand, the proposed technique simply calculates the property-dependent parameters of the PPML to obtain high computational accuracy over the PMLs which are used currently. In addition, with this novel implementation, the PPML can be easily specified in one or more coordinate systems in the same manner as the standard Finite-Difference-Frequency-Domain (FDFD) equations without any increase in their derivation work.

Applying PML technique with additional differential equation method in non-uniform mixed-element DGTD method

In discontinuous Galerkin time domain method (DGTD), the computational region is divided into multiple subdomains instead of using a single mesh for the multiscale structure in finite-difference time-domain (FDTD) method. Accordingly, electrically coarse structures are modeled by hexahedral elements with higher interpolation degrees, while electrically fine structures with complex shapes are spatial discretized by tetrahedral elements with lower interpolation degrees. A nonconforming mixed-element DGTD method is implemented to simulate transient electromagnetic response of general 3-D targets in this paper. Perfectly matched layer (PML) boundary condition is utilized to simulate the propagation of electromagnetic waves in infinite space. PML is formulated by auxiliary differential equation with transformed field variables E and H. The numerical flux is employed to communicate fields between adjacent subdomains including physical region and PML region. The accuracy and efficiency of the proposed method is demonstrated by several numerical experiments.

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.

Study on the Stability of Perfectly Matched Layer Implementations Using a Hybrid Implicit–Explicit Update Scheme

The perfectly matched layer (PML) technique is a general-purpose absorbing boundary condition for finite methods which is usually applied within explicit time-domain schemes. There are a number of slightly different implementations of the extended update equations, and many of them have been reported in the literature to show some long-term instabilities, typically only weakly depending on the time step size. In this contribution, we present a hybrid implicit–explicit algorithm that introduces two additional parameters to control the stability. The implicit part of the algorithm is used only locally within the PML region, and for a simple setup, stability is reached for a certain range of the additional parameters.

PML Implementation in a Nonconforming Mixed-Element DGTD Method for Periodic Structure Analysis

A nonconforming mixed-element (tetrahedral/hexahedral) discontinuous Galerkin time-domain (DGTD) method with a perfectly matched layer (PML) absorber is proposed to analyze the electromagnetic scattering from doubly periodic structures. An efficient auxiliary differential equation (ADE)-based PML implementation is presented with transformed field variables introduced by the time-domain periodic boundary conditions (PBCs). The proposed PML implementation performs well in absorbing the waves with high-order Floquet modes. Additionally, a mixed-order DGTD method is introduced to improve the proposed PML implementation’s long-term stability and reduce the total computational cost. Based on the mixed tetrahedral and hexahedral grids, the nonconforming DGTD method can provide accurate field distribution near complex scattering geometries while simultaneously reducing the degrees of freedom (DoF) in the rest of the computational domain, including the open space and PML regions. Finally, electromagnetic simulations are presented to demonstrate the applicability, accuracy, and efficiency of the proposed method.

Analytical Derivation of Ellipsoid Conformal Perfectly Matched Layers for Electromagnetic Simulations

The 3-D ellipsoid conformal perfectly matched layer (PML) is proposed and realized by the discontinuous Galerkin time-domain (DGTD) method. The interface between free space and the PML region is an ellipsoid in this scheme. The analytical formulas are derived from differential geometry. The numerical results show that compared with planar-PML, the ellipsoid-PML (EPML) avoids the corner problem and provides an efficient absorbing boundary condition for the DGTD simulation. Meanwhile, this new EPML scheme has higher accuracy than that of the old EPML scheme that we proposed.

Error analysis of PML-FEM approximations for the Helmholtz equation in waveguides

In this paper, we study finite element approximate solutions to the Helmholtz equation in waveguides by using a perfectly matched layer (PML). The PML is defined in terms of a piecewise linear coordinate stretching function with two parameters for absorbing propagating and evanescent components respectively, and truncated with a Neumann condition on an artificial boundary rather than a Dirichlet condition for cutoff modes that waveguides may allow. In the finite element analysis for the PML problem, we have to deal with two difficulties arising from the lack of full regularity of PML solutions and the anisotropic nature of the PML problem with, in particular, large PML damping parameters. Anisotropic finite element meshes in the PML regions depending on the damping parameters are used to handle anisotropy of the PML problem. As a main goal, we establish quasi-optimal a priori error estimates, that does not depend on anisotropy of the PML problem (when no cutoff mode is involved), including the exponentially convergent PML error with respect to the width and the strength of PML. The numerical experiments that confirm the convergence analysis will be presented.

On perfectly matched layers for discontinuous Petrov–Galerkin methods

In this article, several discontinuous Petrov–Galerkin (DPG) methods with perfectly matched layers (PMLs) are derived along with their quasi-optimal graph test norms. Ultimately, two different complex coordinate stretching strategies are considered in these derivations. Unlike with classical formulations used by Bubnov–Galerkin methods, with so-called ultraweak variational formulations, these two strategies in fact deliver different formulations in the PML region. One of the strategies, which is argued to be more physically natural, is employed for numerically solving two- and three-dimensional time-harmonic acoustic, elastic, and electromagnetic wave propagation problems, defined in unbounded domains. Through these numerical experiments, efficacy of the new DPG methods with PMLs is verified.

The use of generalized finite difference method in perfectly matched layer analysis

• GFDM is extended to general form, which includes complex and real situations as special cases. • GFDM is implemented in perfectly matched layer (PML) analysis. • GFDM assessed through the use of compliance analysis of a rigid strip footing. • The comparisons with other methods show that GFDM can be used effectively and reliably in PML analysis. This study deals with the use of Generalized Finite Difference Method (GFDM) in Perfectly Matched Layer (PML) analysis. There are two options for performing PML analysis. First option is to express PML equations in terms of real coordinates of the points in actual (real) PML region; the second is to use governing equations (expressed in terms of complex stretching coordinates) as they are in complex PML region. The first option is implemented in this study; the implementation of the second option is under way and will be reported in another study. For the integration of PML equations, the use of GFDM is proposed. Finally, the suggested procedure is assessed computationally by considering the compliance functions of surface and embedded rigid strip foundations. GFDM with PML results are compared to those obtained by using Finite Element Method (FEM) with PML and Boundary Element Method (BEM). Excellent matches in results showed the reliability of the proposed procedure in PML analysis.

A Memory-Saving Realization of the Perfectly Matched Layer in the Precise-Integration Time-Domain Method

The memory demand of the precise-integration time-domain (PITD) method is enlarged quite a lot by the implementation of the perfectly matched layers following the conventional way, which is enclosing the field components of the perfectly matched layer (PML) regions and the main region into one field component vector. In this letter, a region-splitting (RS) technique, for the memory-saving realization of the PML in the PITD method, is proposed. In the RS technique, the field components in the main region and each PML region are calculated separately. The accurate mutual information transmissions between any two regions are guaranteed by advancing the sampling time points of the connecting field components by half-time-step. Several numerical experiments are performed to verify the effectiveness and the efficiency of the proposed technique.

A Novel Coupling Algorithm for Perfectly Matched Layer With Wave Equation-Based Discontinuous Galerkin Time-Domain Method

The second-order wave equation-based discontinuous Galerkin time-domain (DGTD) methods typically employ the first-order absorbing boundary condition for modeling open problems. To improve the modeling accuracy, this paper proposes a novel coupling algorithm of the well-posed perfectly matched layer (PML) for wave equation-based DGTD methods. Based on the domain decomposition technique, the proposed coupling algorithm divides the computational domain into two regions, that is, the physical and PML regions, whose meshes can be nonconformal with each other. Instead of introducing time convolution terms, the new coupling scheme is implemented through employing different DGTD frameworks for the two regions. Specifically, the physical region employs the wave equation-based DGTD framework, while the PML region employs the first-order Maxwell’s curl equations-based DGTD framework. To facilitate modeling of electrically small problems, the implicit Newmark-beta time integration is used for the physical region. To conveniently couple with the physical region, the implicit Crank–Nicolson algorithm is used for the PML region. Numerical results are shown to examine the accuracy and efficiency of the proposed coupling algorithm for modeling electrically small problems.

Hybrid perfectly-matched-layers for transient simulation of scalar elastic waves

This paper presents a new formulation for forward scalar wave simulations in semi-infinite media. Perfectly-Matched-Layers (PMLs) are used as a wave absorbing boundary layer to surround a finite computational domain truncated from the semi-infinite domain. In this work, a hybrid formulation was developed for the simulation of scalar wave motion in two-dimensional PML-truncated domains. In this formulation, displacements and stresses are considered as unknowns in the PML domain, while only displacements are considered to be unknowns in the interior domain. This formulation reduces computational cost compared to fully-mixed formulations. To obtain governing wave equations in the PML region, complex coordinate stretching transformation was introduced to equilibrium, constitutive, and compatibility equations in the frequency domain. Then, equations were converted back to the time-domain using the inverse Fourier transform. The resulting equations are mixed (contain both displacements and stresses), and are coupled with the displacement-only equation in the regular domain. The Newmark method was used for the time integration of the semi-discrete equations.

A modified and stable version of a perfectly matched layer technique for the 3-d second order wave equation in time domain with an application to aeroacoustics

We consider the second order wave equation in an unbounded domain and propose an advanced perfectly matched layer (PML) technique for its efficient and reliable simulation. In doing so, we concentrate on the time domain case and use the finite-element (FE) method for the space discretization. Our un-split-PML formulation requires four auxiliary variables within the PML region in three space dimensions. For a reduced version (rPML), we present a long time stability proof based on an energy analysis. The numerical case studies and an application example demonstrate the good performance and long time stability of our formulation for treating open domain problems.

Dispersive Waves and PML Performance for Large-Stencil Differencing in the Parabolic Equation

The discrete reflection/absorption performance of perfectly matched layer (PML) boundary conditions is a function of the field discretization scheme throughout the computational domain and the form of the complex coordinate extension that characterizes the PML region. Most previous work analyzing PML reflection performance has assumed use of low-order (second- or fourth-), small-stencil (three- or five-point) spatial discretizations whose dispersion is poor outside a narrow low-wavenumber band. Large-stencil differencing schemes, regardless of order, are easily created with physically realistic, wide-band dispersive characteristics, but the effect their adoption has on PML performance has not been analyzed previously. In the context of a semi-discrete approximation to the parabolic/paraxial equation, this work derives the analytic form of the discrete PML reflection coefficient for an arbitrary-order/arbitrary-stencil interior differencing scheme coupled to an arbitrary-order/arbitrary-stencil PML region. This result is then used to show that the absorption performance of the PML regions is largely insensitive to changes in interior differencing. As a consequence, the benefits of superior dispersion characteristics provided by use of large-stencil differencing in the interior physical domain can be had without incurring discrete reflection penalties at the PML.

An effective perfectly matched layer design for acoustic fourth-order frequency-domain finite-difference scheme

SUMMARY In finite-difference (FD) acoustic forward modelling, the parameter settings for the perfectly matched layer (PML) are case-dependent. There is no explicit PML formula that can be applied for most acoustic models without tuning, especially, for the fourth-order FD scheme. In this paper, we propose an explicit PML formula for the acoustic frequency-domain FD with second-order and fourth-order accuracies, respectively. The fourth-order FD scheme uses a special treatment for the boundary. The number of points in the PML is fixed to be 10 and 15 for the second-order and fourth-order FD schemes, respectively. The maximum artificial attenuation parameter associated with the PML formula is automatically calculated based on the FD grid size and the value of the compressional velocity of the boundary cells of the interior domain. Numerical tests confirm that this empirical formula achieves the desired accuracy for 2-D and 3-D media for grid sizes varying from 1 to 200 m. For the fourth-order FD scheme, the proposed PML formula works effectively up to 25 points per wavelength for both 2-D and 3-D media. Beyond that, the error of the PML discretization becomes larger than the discretization error in the interior domain. For such cases and to keep a fourth-order accuracy, a larger number of points in the PML (thicker PML region) needs to be employed.