Perfectly Matched Layer Formulation Research Articles

An unsplit-field viscoelastic complex-frequency-shifted perfectly matched layer for analysis of transient waves in heterogeneous media based on an efficient voxel element method

Wave propagation analysis of heterogeneous unbounded media employing domain discretization methods necessitates truncation of the finite computational domain at its truncation boundaries. The Perfectly Matched Layer (PML) is a robust absorbing boundary condition, which is extensively employed to favorably absorb outgoing scattered waves from the interior computational domain, regardless of frequencies and directions of outgoing waves. Nonetheless, most available PMLs are developed for elastic domains, while their developments in domain discretization methods, which consider damping using the Rayleigh damping formulation, are insufficient. In this study, a viscoelastic PML formulation is developed that accommodates the Rayleigh-type damping. The PMLs are basically developed from a coordinate transformation concept using a complex stretching function. Suitable stretching functions can efficiently control the absorptive properties and the spectral character of the PML. Besides, the stretching functions play a crucial role in the long-time stability of transient wave simulations. The standard stretching function usually results in spurious reflections, especially when propagating waves incident to the truncation boundary at near-grazing incidence. On the other hand, the complex-frequency-shifted (CFS) stretching function has been shown to mitigate long-time instability. Therefore, the proposed viscoelastic PML in the time-domain and frequency-domain are derived using the CFS stretching function for the first time. Comprehensive formulations of the proposed viscoelastic CFS-PML are given, where the displacement field is represented as the sole unknown variable of the problem. Hence, the proposed CFS-PML can be incorporated into any domain discretization method. However, the proposed CFS-PML is further developed in the context of the voxel element method, where element-by-element iterative strategies are implemented to solve the governing equations without storing global matrices in the computer memory. Comprehensive numerical experiments are conducted to demonstrate the validity and efficiency of the proposed viscoelastic CFS-PML formulation in analyzing soil profiles with various damping ratios and properties. Moreover, long-time stability for problems involving waveguides and near-grazing wave incidence is studied, where no instabilities are detected.

A hybrid PML formulation for the 2D three-field dynamic poroelastic equations

Simulation of wave propagation in poroelastic half-spaces presents a common challenge in fields like geomechanics and biomechanics, requiring Absorbing Boundary Conditions (ABCs) at the semi-infinite space boundaries. Perfectly Matched Layers (PML) are a popular choice due to their excellent wave absorption properties. However, PML implementation can lead to problems with unknown stresses or strains, time convolutions, or PDE systems with Auxiliary Differential Equations (ADEs), which increases computational complexity and resource consumption.This article introduces a novel hybrid PML formulation for arbitrary poroelastic domains. Instead of using ADEs, this formulation utilizes time-history variables to reduce the number of unknowns and mathematical operations. The modification of the PDEs to account for the PML is limited to the PML domain only, resulting in smaller matrices while maintaining the governing equations in the interior domain and preserving the temporal structure of the problem. The hybrid approach introduces three scalar variables localized within the PML domain.The proposed formulation was tested in three numerical experiments in geophysics using realistic parameters for soft sites with free surfaces. The results were compared with numerical solutions from extended domains and simpler ABCs, such as paraxial approximation, demonstrating the accuracy, efficiency, and precision of the proposed method. The article also discusses the applicability of the method to complex media and its extension to the Multiaxial PML formulation.The codes used for the simulations are available for download from https://github.com/hmella/POROUS-HYBRID-PML.

A high-order perfectly matched layer scheme for second-order spectral-element time-domain elastic wave modelling

In this study, we develop a new high-order perfectly matched layer (PML) formulation for the 3D spectral-element time-domain (SETD) method to solve the second-order elastic wave equations in terms of displacements. The high-order PML consists of multiple well-known frequency-dependent complex stretching functions, leading to the complicated temporal convolution which requires large computational resources. To avoid the evaluation of convolution, we deal with the high-order stretching functions by introducing the recursive auxiliary first-order differential equations which can be easily solved in time domain. In addition, an equivalent concept of PML is adopted for absorbing waves, preserving the original form of wave equations in physical domain. Finally, we obtain a high-order, non-convolutional and unchanged-form PML formulation, which can be easily incorporated into the SETD codes designed for the physical domain, to make them applicable for the second-order wave equations in the unbounded media. Numerical experiments demonstrate the accuracy and stability of the high-order PML through the comparison with the reference solution. It is found that the high-order PML provides a much better absorption performance than the first-order PML, particularly for the grazing-incidence waves and surface waves in the large-scale domain. We also give a fluid-filled borehole model to illustrate that the high-order PML works well for absorbing the complex guided waves in the presence of fluid-solid interaction.

Double-pole unsplit complex-frequency-shifted multiaxial perfectly matched layer combined with strong-stability-preserved Runge-Kutta time discretization for seismic wave equation based on the discontinuous Galerkin method

The classical complex-frequency-shifted perfectly matched layer (CFS-PML) technique has attracted widespread attention for seismic wave simulations. However, few studies have addressed the double-pole variant of the CFS-PML scheme. The double-pole CFS-PML has a stronger capacity to absorb near-grazing incident waves and evanescent waves than the classical CFS-PML. Using the discontinuous Galerkin (DG) method, we derive a double-pole unsplit auxiliary ordinary differential equation CFS-multiaxial PML (AODE CFS-MPML) formulation, which combines a fourth-order strong-stability-preserved Runge-Kutta time discretization for wavefield simulation on an unstructured grid. The double-pole unsplit CFS-MPML formulations are obtained by introducing auxiliary memory variables and AODEs. The original stress-velocity equations and the double-pole unsplit AODE CFS-MPML equations are all first-order hyperbolic systems and suitable for the DG method. The attenuative variables are added directly to the original seismic wave equations without changing their formats. In contrast to the split perfectly matched layer (PML), we avoid reformulating PML equations in the nonattenuative modeling region. The original seismic wave equation is solved in the nonattenuative modeling domain, whereas the double-pole unsplit AODE CFS-MPML equation is implemented in the PML absorbing region. Three numerical examples validate the performance of the double-pole unsplit AODE CFS-MPML technique. The isotropic and anisotropic experiments demonstrate that our developed double-pole unsplit AODE CFS-MPML is more stable and obtains more accurate solutions than the classical CFS-PML. The second example indicates the flexibility of the combination of the DG method with the double-pole CFS-MPML on undulating topography. The final example displays the applicability and effectiveness of our method in a 3D situation.

Finite-difference modeling of Maxwell viscoelastic media developed from perfectly matched layer

In numerical simulation of wave propagation, both viscoelastic materials and perfectly matched layers (PMLs) attenuate waves. The wave equations for both the viscoelastic model and the PML contain convolution operators. However, convolution operator is intractable in finite-difference time-domain (FDTD) method. A great deal of progress has been made in using time stepping instead of convolution in FDTD. To incorporate PML into viscoelastic media, more memory variables need to be introduced, which increases the code complexity and computation costs. By modifying the nonsplitting PML formulation, I propose a viscoelastic model, which can be used as a viscoelastic material and/or a PML just by adjusting the parameters. The proposed viscoelastic model is essentially equivalent to a Maxwell model. Compared with existing PML methods, the proposed method requires less memory and its implementation in existing finite-difference codes is much easier. The attenuation and phase velocity of P- and S-waves are frequency independent in the viscoelastic model if the related quality factors (Q) are greater than 10. The numerical examples show that the method is stable for materials with high absorption (Q = 1), and for heterogeneous media with large contrast of acoustic impedance and large contrast of viscosity.

Hybrid asynchronous isogeometric Perfectly Matched Layer for transient elastodynamics

This paper presents an isogeometric formulation (IGA) of the Perfectly Matched Layer (PML) for two-dimensional time-domain elastodynamics in unbounded media. The displacement-based unsplit-field PML formulation is first discretized in space using B-spline basis functions of increasing polynomial order p, from 2 to 10, able to deal with the strong solution gradients in the thickness of the PML thanks to the high order continuity provided by the B-spline basis functions. In order to keep the high accuracy of IGA, implicit time integration is chosen in the PML subdomains, whereas the physical domain is integrated with an explicit time integration scheme. The domain decomposition is based on a dual formulation and Heterogeneous Asynchronous Time Integrator (HATI) approach. The coupling between non-conforming IGA-patches for the PML subdomains and the physical domain is carried out using a standard mortar method. It enables us to select for the IGA-PML subdomains both the appropriate space discretization and the appropriate time discretization with an implicit time integrator and a larger time step size. The hybrid multi-time step IGA-PML performance in terms of computation time, is assessed for two-dimensional elastodynamic problems in single and multi-layer media depending on the chosen discretization parameters in space and time.

One-Step Crank-Nicolson Direct-Splitting Algorithm with Enhanced Absorption to Evaluate Low-Pressure Discharge for Satellite Sensors in Outer Space.

Low-pressure discharge causes air ionization resulting in performance degeneration or failure for the satellite sensors in outer space. Here, a one-step Crank-Nicolson Direct-Splitting (CNDS) algorithm is proposed to evaluate the electrical behavior of satellite sensors under the low-pressure discharge circumstance. To be more specific, the CNDS algorithm is proposed in the Lorentz medium, which can accurately analyze the ionized air and generated plasma. Higher order perfectly matched layer (PML) is modified in the Lorentz medium to efficiently terminate the unbounded lattice. It can be concluded that the proposed algorithm shows entire considerable performance in the low-pressure discharge evaluation. The proposed PML formulation behaviors enhanced absorbing performance compared with the existing algorithm. Through the experiments, it can be observed that the low-pressure discharge phenomenon causes performance variation, which shows a significant influence on the satellite sensors. Meanwhile, results show considerable agreement between the simulation and experiment results which indicates the effectiveness of the algorithm.

Time domain numerical modelling of guided wave excitation in fluid-filled pipes

Acoustic waves have been widely used to inspect pipeline defects, including leakage, blockage and corrosion etc. The time-domain excitation of guided waves in the fluid/solid coupled pipeline system has rarely been studied theoretically. This requires the incorporation of a source term in the coupled system. This article introduces a finite element based numerical model to study the excitation of guided waves in the coupled system with a sound source either in the fluid or on the pipe wall. A compatible pair of time-domain Perfectly Matched Layers (PMLs) have been proposed to absorb pipe end reflections in the elastic wall and in the fluid respectively. These fluid/solid PML numerical formulations are coupled and the implementation of the coupling is introduced. The numerical model is validated against an enlarged model without PML, and excellent agreement has been achieved. The numerical model shows that the dominant excited wave mode is a fluid type wave mode when the sound source is a fluid source or a radial elastic line source. However, the dominant excited wave mode is a structure type wave mode when the sound source is an axial elastic line source. • A compatible pair of fluid and elastic PML formulations have been proposed. • A mixed displacement-strain un-split formulation is proposed in the elastic domain. • A mixed velocity potential-velocity un-split formulation is proposed in the fluid domain. • Fluid or radial elastic line source excites a fluid dominant wave mode in the low frequency range. • Axial elastic line source excites a structure dominant wave mode in the low frequency range.

Isogeometric analysis of acoustic scattering with perfectly matched layers (IGAPML)

The perfectly matched layer (PML) formulation is a prominent way of handling radiation problems in unbounded domain and has gained interest due to its simple implementation in finite element codes. However, its simplicity can be advanced further using the isogeometric framework. This work presents a spline based PML formulation which avoids additional coordinate transformation as the formulation is based on the same space in which the numerical solution is sought. The procedure can be automated for any convex artificial boundary. This removes restrictions on the domain construction using PML and can therefore reduce computational cost and improve mesh quality. The usage of spline basis functions with higher continuity also improves the accuracy of the PML-approximation and the numerical solution.

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 isogeometric formulation of locally-conformal perfectly matched layer for acoustic scattering problems

This paper presents an isogeometric formulation of the locally-conformal perfectly matched layer (PML) for time-harmonic acoustic scattering problems. The new formulation is a generalization of the conventional locally-conformal PML, in which the NURBS patch supporting the PML domain is transformed from real space to complex space. This is achieved by complex coordinate stretching, based on a stretching vector field indicating the directions in which incident sound waves are absorbed. The performance of the isogeometric PML formulation is discussed through several acoustic scattering problems, spanning from one to three dimensions. It is found that the proposed method presents superior computational accuracy, high geometric adaptivity, and good robustness against challenging geometric features. The geometry-preserving ability inherent in the isogeometric framework could bring extra benefits by eliminating geometric errors that are unavoidable in the conventional PML. Meanwhile, these properties are not sensitive to the location of the sound source or the depth of the PML domain.

Isogeometric locally-conformal perfectly matched layer for time-harmonic acoustics

This paper presents an isogeometric formulation of the perfectly matched layer (PML) for time-harmonic acoustic simulations. The new formulation is a generalization of the conventional locally-conformal PML, in which the NURBS patch supporting the PML domain is transformed from real space to complex space. This is achieved by complex coordinate stretching, based on a stretching vector field indicating the directions in which incident sound waves are absorbed. To solve the difficulty that stretching vectors cannot be defined for control points not lying on the patch, an interpolation method is proposed in which the control-point stretching vectors are calculated by interpolating those at a set of integration points. An automatically matched layer (AML) is also formulated in the isogeometric context. Due to its form-invariant nature, the Helmholtz equation which governs time-harmonic acoustics is kept unchanged in the complex space, except that the Laplacian operator and the sound pressure should be transformed to their complex forms. The performance of the isogeometric PML formulation is discussed through several numerical examples, spanning one to three dimensions and involving both sound radiation and sound scattering problems. It is found that the proposed method presents superior computational accuracy, high geometric adaptivity, and good robustness against challenging geometric features. The geometry-preserving ability inherent in the isogeometric framework brings extra benefits by eliminating geometric errors that are unavoidable in the conventional PML. Meanwhile, these properties are not sensitive to the location of the sound source or the depth of the PML domain. The implementation of the isogeometric PML only requires the replacement of real control-point coordinates by complex ones, making it possible to incorporate the proposed method to various existing numerical routines.

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.

Non-convolutional second-order complex-frequency-shifted perfectly matched layers for transient elastic wave propagation

The numerical simulation of wave propagation in heterogeneous unbounded media using domain discretization techniques requires truncation of the physical domain: at the truncation boundary, Perfectly Matched Layers (PMLs) – buffers wherein wave attenuation is imposed – are often used to mimic outgoing wave motion and prevent waves from re-entering the interior computational domain. The PML’s wave-dissipative properties derive from a coordinate mapping concept, where the physical coordinate is mapped onto a frequency-dependent complex coordinate in the PML via a complex stretching function. The choice of the stretching function controls not only the spectral character and the absorptive properties of the PML, but also, more critically, the long-time stability when the computational domain-PML ensemble is used for transient wave simulations.The standard PML stretching function can lead to error growth, particularly when propagating waves impinge at grazing incidence on the truncation boundary. By contrast, a modification to the standard stretching function that has led to the Complex-Frequency-Shifted CFS-PML has been shown to alleviate the temporal instability. However, whereas PML formulations using the standard stretching function can lead to second-order in time semi-discrete forms, affording multifaceted benefits, all CFS-PML formulations to date require the evaluation of convolutions. In this paper, we discuss a new CFS-PML formulation that avoids the evaluation of convolutions, while preserving the second-order temporal character of elastic waves. It is shown that, upon spatial discretization, the CFS-PML can be completely described by a triad of stiffness, damping, and mass matrices, which can be readily incorporated into existing finite element codes originally designed for interior problems, to endow them with wave simulation capabilities on unbounded domains. Numerical experiments in the time-domain demonstrate the efficacy of the proposed approach; we also report long-time stability for problems involving waveguides and grazing wave incidence.

Three-dimensional hybrid asynchronous perfectly matched layer for wave propagation in heterogeneous semi-infinite media

This paper presents an efficient hybrid asynchronous three-dimensional (3D) perfectly matched layer (PML) for modeling unbounded domains. The proposed unsplit explicit or implicit 3D PML formulation is implemented in the framework of a heterogeneous asynchronous time integrator. It is fully versatile in terms of time integrators and time step sizes according to partitions while conserving classical finite element formulations in the elastic domain without complex-valued stretched coordinates. Examples of a semi-infinite bar, Lamb’s test, and a soil–structure interaction problem with PML-truncated semi-infinite heterogeneous media are investigated to illustrate the efficiency of the proposed PML in terms of accuracy and CPU time.

Complex Envelope Approximate Crank-Nicolson Method and Its Open Boundary Implementation for Bandpass Problem

To efficiently simulate the bandpass electromagnetic problems by employing the finite-difference time-domain (FDTD) algorithm, unconditionally stable implementation is proposed based upon the complex envelope method and the Crank-Nicolson Direct-Splitting (CNDS) algorithm. To further absorb outgoing waves and reduce wave reflections in open region problems, absorbing boundary condition with improved absorption is proposed by incorporating the proposed scheme with the higher order perfectly matched layer (PML) formulation. The proposed scheme takes advantage of the complex envelope method, the CNDS algorithm and the higher order PML formulation in terms of simulating bandpass signals and enhancing absorption and improving computational efficiency. Numerical examples are carried out to further demonstrate the effectiveness and efficiency. Through the resultants, it can be illustrated that the proposed scheme can not only obtain considerable accuracy, favorable absorbing performance, outstanding efficiency in the simulation of bandpass open region problems but also maintain the stability of the algorithm when the time step surpasses far beyond the Courant-Friedrich-Levy condition.

Perfectly Matched Layer Formulation of the INBC-FDTD Algorithm for Electromagnetic Analysis of Thin Film Materials

The impedance network boundary condition (INBC)-based finite-difference time-domain (FDTD) method has been widely used for electromagnetic analysis of highly conductive thin film materials. In the INBC-FDTD formulation, the electromagnetic field variations inside the thin film material are taken into account mathematically and thus extremely small FDTD grids are not necessary for the FDTD modeling of the material. Therefore, computational efficiency of the INBC-FDTD formulation is significantly better than other FDTD formulations. Albeit with this great advantage, the INBC-FDTD formulation cannot be fully employed for thin film materials because the corresponding perfectly matched layer (PML) formulation has not been reported in literature. In this work, we propose a PML formulation suitable for the INBC-FDTD algorithm. Numerical examples illustrate that the proposed PML-INBC-FDTD formulation can yield good absorption performance and also it can improve computational efficiency while maintaining numerical accuracy.

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.

Performance Enhanced Crank-Nicolson Boundary Conditions for EM Problems

Based upon the approximate Crank–Nicolson (CN) algorithms and the higher order (HO) concept, unconditionally stable perfectly matched layer (PML) implementations are proposed for electromagnetic problems in the finite-difference time-domain (FDTD) lattice. The approximate CN algorithms include the CN direct-splitting (CNDS) and CN approximate-factorization-splitting (CNAFS). The proposed schemes take advantage of the approximate CN algorithms in terms of eliminating the Courant–Friedrichs–Levy stability condition and maintaining the computational accuracy. By introducing the HO concept to the PML formulation, the absorbing performance can be enhanced during the simulation. Furthermore, to cope with the existence of half-time-step problems in simulating the anisotropic magnetized plasma, a modified ADE method is proposed at the integer time step. Thus, the modified ADE method can be directly employed in the approximate CN algorithms. Numerical examples and experiments are carried out to demonstrate the effectiveness, efficiency, and performance. Through the results, it can be observed that the computational efficiency and absorbing performance can be significantly improved. Meanwhile, it can be concluded that the CNAFS-PML and CNDS-PML algorithms show higher accuracy and considerable efficiency, respectively, when the time step surpasses far beyond the CFL condition.

Perfectly matched layers for Schrödinger-type equations with nontrivial energy–momentum dispersion

The construction of perfectly-matched-layer (PML) boundary conditions is extended to wave equations with non-trivial energy–momentum dispersion relation. Complex coordinate extension is based upon the relative sign between the surface-normal component of group velocity and the k-vector. This ensures selective damping of propagating and counter-propagating modes at the simulation boundaries. Moreover, the technical complexity of the PML method can be inferred from the energy–momentum dispersion relation of the underlying wave equation, respectively, the finite-difference scheme employed. The PML transform is derived for selected Schrödinger-type (i.e. first-order-in-time) kinetic transport equations. Spectral analysis demonstrates proper damping behaviour of out-propagating modes. Results are presented for the Schrödinger equation for particles and holes, the s- and p-wave Bogoliubov–de Gennes equation, the Dirac equation, and the Dohnal k⋅p model. Where available, a comparison to previous PML formulations is made.