Conventional Perfectly Matched Layer Research Articles

An arbitrary Lagrangian–Eulerian method for analyzing finite-amplitude viscous acoustic waves radiated from vibrational solid boundaries: An implicit method

This paper is concerned with the development of an arbitrary Lagrangian–Eulerian (ALE) method for predicting nonlinear acoustic waves radiated from large-amplitude vibrational solid boundaries immersed in viscous fluids of infinite extent. The method is formulated based on a nonlinear acoustic model characterized by velocity potential and acoustic pressure. Unsplit implementations of three perfectly matched layer (PML) techniques are presented, namely conventional PML, convolution PML, and complex frequency-shifted PML (CFS-PML), to enable bounded-domain modeling of nonlinear acoustic waves propagating in an unbounded fluid medium. The long-time stability characteristics of the three distinct PML techniques are examined through theoretical analysis and numerical verification of their ability to absorb evanescent waves. Our results reveal that the CFS-PML technique exhibits superior long-time stability performance compared to the other two techniques. The CFS-PML is also compared to two different absorbing layer techniques: the Kosloff Absorbing Layer (KAL) technique and the Acoustic Black Hole (ABH) technique. The finite element method is adopted for spatial discretization of the finite-amplitude acoustic wave model, and an implicit predictor–corrector algorithm is proposed for time advancement. Several benchmark problems, including acoustic waves produced by a vibrational piston, and transversely oscillating and pulsating cylinders, are analyzed by the proposed method. It is found that the proposed method attains a second-order rate of convergence for spatial discretization, and the convergence rate in terms of the time step size is about first-order. The application of the method to acoustic levitation problems is also demonstrated, and the nonlinear phenomenon of acoustic waves in a single-axis acoustic levitator is discussed.

Application of the Reflectionless Discrete Perfectly Matched Layer for Acoustic Wave Simulation

The perfectly matched layer (PML) is one of the most popular absorbing boundary conditions for simulating seismic waves. In theory, the PML can absorb incident waves at any incident angle and any frequency in a medium. However, numerical reflections will be generated after the PML has been discretized. Therefore, how to reduce the reflections of discrete PML has been a research topic for more than 2 decades. In this paper, we adopt the reflectionless discrete PML (RD-PML) for seismic wave and implement the RD-PML based on the acoustic wave equation, and then compare its absorbing performance with that of the conventional discrete PML. Our numerical experiments show that the RD-PML has advantages over the conventional discrete PML. In homogeneous model, a thick enough RD-PML can effectively eliminate reflections. In heterogeneous model, a thin-layer RD-PML can obtain better absorbing performance even than the thick-layer conventional discrete PML. The absorbing performance of the RD-PML can be improved by using the periodic boundary without increasing the amount of computation and memory. RD-PML provides a new perspective to understand the discretization of PML, and may play an important role in promoting the development of PML technology.

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.

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

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

Machine-Learning-Based PML for the FDTD Method

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

Second-order PML: Optimal choice of [formula omitted]-order PML for truncating FDTD domains

The nth-order PMLs based on the unsplit-field formulations and the Z-transform methods are proposed to truncate the finite-difference time-domain (FDTD) domains, which will be validated through numerical simulations whether the higher-order PML with more than two poles, such as n=3, will hold better absorption performance as compared with the conventional, complex frequency shifted (CFS), second-order perfectly matched layer (PMLs). The advantages and disadvantages of different PMLs are demonstrated. It has shown that the higher-order PMLs have the advantages of both the conventional and the CFS PMLs in terms of absorption performance, since the conventional PML is ineffective at absorbing the evanescent waves and the CFS-PML is incapable of absorbing low-frequency propagating waves. It is clearly shown that the second-order PML is overall the optimal choice for truncating arbitrary FDTD domains, since it not only requires less computational time and memory, but holds almost the same absorption performance as compared with the third-order PML. Three numerical simulations have been carried out in three-dimensional (3D) problems to confirm the analysis.

Application of unsplit convolutional perfectly matched layer for scalar arbitrarily wide-angle wave equation

A classical split perfectly matched layer (PML) method has recently been applied to the scalar arbitrarily wide-angle wave equation (AWWE) in terms of displacement. However, the classical split PML obviously increases computational cost and cannot efficiently absorb waves propagating into the absorbing layer at grazing incidence. Our goal was to improve the computational efficiency of AWWE and to enhance the suppression of edge reflections by applying a convolutional PML (CPML). We reformulated the original AWWE as a first-order formulation and incorporated the CPML with a general complex frequency shifted stretching operator into the renewed formulation. A staggered-grid finite-difference (FD) method was adopted to discretize the first-order equation system. For wavefield depth continuation, the first-order AWWE with the CPML saved memory compared with the original second-order AWWE with the conventional split PML. With the help of numerical examples, we verified the correctness of the staggered-grid FD method and concluded that the CPML can efficiently absorb evanescent and propagating waves.

An Exponential Time Differencing Algorithm for the FDTD-PML Analysis of Nonlinear Photonic Bandgap Structures

An exponential time differencing (ETD) algorithm is presented for the analysis of photonic crystal structures comprising the third-order nonlinear materials. Compared with the well known Z-transform method and auxiliary differential equation method for incorporating the nonlinearity, the proposed ETD algorithm has the least memory requirement and shows the same accuracy. The stretched coordinates perfectly matched layer (PML) with the complex frequency shifted variable implemented using the ETD algorithm is applied to terminate nonlinear photonic bandgap structures. Compared with the conventional PML and the convolutional PML, the proposed implementation can improve the absorbing performance significantly with similar memory requirement.

Efficient DSP-Higher-Order PML Formulations for the Metal Plate Buried in Dispersive Soil Half-Space Problem

An efficient and unsplit-field implementation of the higher order perfectly matched layer (PML) based on the stretched coordinate PML (SC-PML) formulations and digital signal processing (DSP) techniques is proposed to truncate the finite-difference time-domain (FDTD) lattices. The higher order PML has the advantages of both the conventional PML and the complex frequency-shifted PML (CFS-PML) in terms of absorbing performances. A 3-D FDTD simulation of the metal plate buried in dispersive soil half-space problem has been carried out to validate these formulations. It is shown that the proposed PML formulations with the higher order scheme are efficient in terms of attenuating both the low-frequency propagating waves and evanescent waves and reducing late-time reflections.

An unsplit complex-frequency-shifted PML based on matchedZ-transform for FDTD modelling of seismic wave equations

The perfectly matched layer (PML) is an efficient absorbing boundary condition for mitigating undesired reflections in seismic wave modelling. However, the performance of the conventional split-field PML (S-PML) is severely reduced at grazing incidence. Very low frequency waves and evanescent waves can also cause spurious reflections on the PML interface. Such problems could be circumvented by using the complex-frequency-shifted (CFS) stretching tensor in the PML. However, it is difficult to adopt the CFS stretching tensor in the S-PML. A matched Z-transform PML (MZT-PML) technique applied in electromagnetic modelling results in a complete unsplit-field form. In this paper, we show that the MZT-PML can easily implement the CFS stretching tensor and be adopted for elastic wave finite-difference time-domain modelling. The numerical tests in an elongated model illustrate that the complex-frequency-shifted matched Z-transform perfectly matched layer (CFS MZT-PML) can substantially improve performance by eliminating the spurious reflections at grazing incidence. The numerical stability of the CFS MZT-PML is verified by long time computation of the total energy. Also, we conduct a numerical test in a two-layer heterogeneous model showing that the CFS MZT-PML can also be used to simulate efficiently the wave propagation in more complex structures.

Implementation of perfectly matched layers in an arbitrary geometrical boundary for elastic wave modelling

SUMMARY The perfectly matched layer (PML) absorbing boundary condition is incorporated into an irregular-grid elastic-wave modelling scheme, thus resulting in an irregular-grid PML method. We develop the irregular-grid PML method using the local coordinate system based PML splitting equations and integral formulation of the PML equations. The irregular-grid PML method is implemented under a discretization of triangular grid cells, which has the ability to absorb incident waves in arbitrary directions. This allows the PML absorbing layer to be imposed along arbitrary geometrical boundaries. As a result, the computational domain can be constructed with smaller nodes, for instance, to represent the 2-D half-space by a semicircle rather than a rectangle. By using a smooth artificial boundary, the irregular-grid PML method can also avoid the special treatments to the corners, which lead to complex computer implementations in the conventional PML method. We implement the irregular-grid PML method in both 2-D elastic isotropic and anisotropic media. The numerical simulations of a VTI lamb’s problem, wave propagation in an isotropic elastic medium with curved surface and in a TTI medium demonstrate the good behaviour of the irregular-grid PML method.

Conformal Perfectly Matched Layer for the Mixed Finite Element Time-Domain Method

We introduce a conformal perfectly matched layer (PML) for the finite-element time-domain (FETD) solution of transient Maxwell equations in open domains. The conformal PML is implemented in a mixed FETD setting based on a direct discretization of the first-order coupled Maxwell curl equations (as opposed to the second-order vector wave equation) that employs edge elements (Whitney 1-form) to expand the electric field and face elements (Whitney 2-form) to expand the magnetic field. We show that the conformal PML can be easily incorporated into the mixed FETD algorithm by utilizing PML constitutive tensors whose discretization is naturally decoupled from that of Maxwell curl equations (spatial derivatives). Compared to the conventional (rectangular) PML, a conformal PML allows for a considerable reduction on the amount of buffer space in the computational domain around the scatterer(s).

A nonsplit complex frequency-shifted PML based on recursive integration for FDTD modeling of elastic waves

In finite-difference time-domain (FDTD) modeling of elastic waves, absorbing boundary conditions are used to mitigate undesired reflections that can arise at the model’s truncation boundaries. The perfectly matched layer (PML) is generally considered to be the best available absorbing boundary condition. An important but rarely addressed limitation of current PML implementations is that their performance is severely reduced when waves are incident on the PML interface at near-grazing angles. In addition, very low frequency waves as well as evanescent waves could cause spurious reflections at the PML interface. In electromagnetic modeling, similar problems are circumvented by using a complex frequency-shifted stretching function in the PML formulas. However, in elastic-wave modeling using the conventional PML formulation — based on splitting the velocityand stressfields — it is difficult to adopt a complex frequency--shifted stretching function. We present an alternative implemen-tation of a PML that is based on recursive integration and does not require splitting of the velocity and stress fields. Modeling re-sults show that the performance of our implementation using a standard stretching function is identical to that of the convention-al split-field PML. Then we show that the new PML can be modi-fied easily to include the complex frequency-shifted stretching function. Results of models with an elongated domain show that this modification can substantially improve the performance of the PML boundary condition. An efficient implementation of the new PML requires less memory than the conventional split-field PML, and, therefore, is a very attractive alternative to the con-ventional PML. By adopting the complex frequency-shifted stretching function, the PML can accommodate a wide variety of model problems, and hence it is more generic.

Efficient Absorbing Layers for the Guided-mode Analysis Using the Imaginary-Distance BPM

The imaginary-distance beam-propagation method (ID-BPM) with the conventional perfectly matched layer (PML) often yields unstable results for the guided-mode analysis of an optical waveguide. To address this issue, two efficient absorbing layers are discussed on the basis of suitable modifications to the stretching coordinate variables of the conventional PML. Theoretical descriptions of these layers are presented using the plane wave solutions. The performance of these layers is investigated for several two- and three-dimensional problems. It is found for the analysis of a light-guiding metal line near the cutoff condition that the calculation time is reduced to one sixth of that using a zero boundary.

An efficient implementation of the PML for truncating FDTD domains

AbstractAn efficient implementation of the perfectly matched layer (PML) is presented for truncating lossy and lossless finite‐difference time‐domain (FDTD) computational domains. The implementation is based on the stretched coordinate PML formulations and on the derivation of the wave equation in the PML region. Significant savings in computational time and memory storage requirements are achieved. In addition, the FDTD implementation of the new formulations is simpler than the conventional PML formulations. Two dimensional numerical tests have been carried out to validate the proposed formulations. © 2002 Wiley Periodicals, Inc. Microwave Opt Technol Lett 36: 55–60, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.10669

High-performance absorbing boundary conditions for photonic crystal waveguide simulations

A high-performance absorbing boundary condition is newly developed for the reduction of spurious reflections in photonic crystal (PC) waveguide simulations. The PC waveguide is terminated with a perfectly matched layer (PML) in which the original PC structure remains as is. This PC-based PML works well, compared to a conventional PML, which acts as a homogeneous absorbing medium, simulating a semi-infinite free space and to a distributed-Bragg-reflector waveguide, which was recently developed to reduce reflections from PC waveguide ends, improving a wavenumber matching condition for PC waveguide modes.

A new anisotropic perfectly matched layer medium for mesh truncation in finite difference time domain analysis

In this paper an unsplit anisotropic perfectly matched layer (PML) medium, previously utilized in the context of finite element analysis, is implemented in the finite difference time domain (FDTD) algorithm. The FDTD anisotropic PML is easy to implement in the existing FDTD codes, and is well suited for truncating inhomogeneous and layered media without special treatment required in the conventional PML approach. A further advantage of the present approach is improved performance at lower frequencies. The applications of the unsplit anisotropic PML/FDTD method are illustrated by considering the problems of a plane wave propagation and an open microstrip line.