Perfectly Matched Layer Implementation Research Articles

A novel algorithm of the perfectly matched layer based on the Runge–Kutta method of order 2 accuracy

AbstractThe perfectly matched layer (PML) has the ability to terminate the finite‐difference time‐domain (FDTD) lattice successfully. It is crucial to investigate the interaction between electromagnetic waves and the PML. Utilizing the Runge–Kutta method of order 2 accuracy, we derive a novel and efficient complex frequency‐shifted (CFS) PML implementation algorithm, which is named the RK‐PML. To validate the RK‐PML, we provide two numerical examples, which include rectangular waveguide and plasma cube truncated by the PML. By analyzing the time‐domain relative reflection errors, the frequency‐domain reflection coefficients, the computational time, and the memory footprint, the proposed RK‐PML outperforms slightly the convolutional PML (CPML).

Nearly perfectly matched layer implementation for time domain spectral element modelling of wave propagation in 3D heterogeneous and anisotropic porous media

Numerical modelling of wave propagation in the heterogeneous and anisotropic porous media is important for seismic exploration. When modelling time-domain poroelastic waves in an unbounded region, the implementation of perfectly matched layer (PML) is a common technique to absorb the outgoing waves. The PML in terms of first-order equations has been well proposed. However, the first-order PML needs fundamental and nontrivial reconstructions when applied to the second-order systems in displacements, such as finite element method (FEM) and spectral element method (SEM). It is well known SEM is an efficient algorithm for simulating seismic waves for its high accuracy and geometrical flexibility. However, at present, few literature studies the implementation of PML into time-domain SEM for simulating waves in the 3D unbounded heterogeneous porous media. In this work, combining with auxiliary ordinary differential equations (ODEs), we first systematically extend nearly PML (NPML) into time-domain SEM in terms of the second-order poroelastic system. The scheme preserves the original form of wave equations, leading to a simple and flexible incorporation into the existing algorithm. It avoids the temporary convolution through solving a set of first-order ODEs in time domain, resulting in high computational efficiency. Furthermore, to solve the coupling poroelastic/acoustic/elastic waves, we employ domain decomposition technique to handle the different regions independently, and apply the corresponding boundary conditions at the region interfaces to guarantee the coupling. Numerical examples demonstrate the accuracy and stability of the proposed method by comparing the numerical results with the analytical solutions and other independent numerical solutions. Finally, we apply the algorithm to solve wave propagation in a large-scale model involved a fluid-filled borehole embedded in heterogeneous porous formations.

An implementation of a finite difference time domain method for second-order nonlinear acoustic equations

The framework of the numerical scheme presented in this talk is that of the finite-difference time-domain (FDTD) method published by Sparrow and Raspet [J. Acoust. Soc. Am. 90, 1991]. The method is fourth-order in space and second-order in time when propagating in homogeneous media. Our implementation resides in a Cartesian coordinate system and is meant to be used in inhomogeneous media. In the neighborhood of inhomogeneities the FDTD method is reduced to second-order in space. Of interest to our applications is the simulation of infinite domains, which is achieved using the perfectly matched layer (PML). The particular PML implementation we consider is one that was designed to absorb linear acoustic waves neglecting thermoviscous effects. In this talk, we will give an overview of our FDTD implementation and show one- and two-dimensional examples to illustrate performance in heterogeneous media that include changes in material properties and scattering objects.

A new implementation of convolutional PML for second-order elastic wave equation

The perfectly matched layer (PML) boundary condition has been widely used as a very effective absorbing boundary condition for seismic wavefield simulations. Convolutional PML (CPML) achieved by using a complex frequency-shifted stretch function was the latest development to further improve PML’s absorption performance for near-grazing angle incident waves as well as for low-frequency incident waves. However, the mathematical theory of the PML method is derived from the first-order equation, all PML implementations of second-order equations are to introduce auxiliary equations or variables to rewrite original PML equations, which will complicate the implementation. In this article, we propose a simple and efficient CPML implementation method for the second-order elastic wave equation, which directly simulates the second-order CPML equation. The main advantage of this method is that there is no need to introduce auxiliary variables or auxiliary equations to convert the second-order PML equation from the complex coordinate space to the real axis. Compared with the conventional CPML method for the second-order elastic wave equation, it introduces only eight convolution variables. We demonstrate the validity and absorption performance through extensive numerical experiments.

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.

Computationally Efficient Locally One-Dimensional Algorithm for Open Region Ground Penetrating Radar Problem With Improved Absorption

By incorporating higher order formulation with perfectly matched layer (PML) implementation, unconditionally stable computationally efficient locally one-dimensional (LOD) algorithm with improved absorption is proposed for simulating ground penetrating radar and its open region problems in finite-difference time-domain (FDTD) algorithm. To take advantages of these methods, the proposed implementation shows characteristics of maintaining considerable accuracy, enhancing computational efficiency and improving the entire absorption. These above-mentioned advantages are illustrated through the ground penetrating radar related problem in open regions. It can be concluded from the calculation results that the proposal shows advantages of higher order formulation, PML implementation and LOD procedure which are efficient in remarkable accuracy, considerable efficiency and improved absorption with increment of CFLNs. Meanwhile, the simulation results also indicate that the proposal is an unconditionally stable procedure with the dissatisfaction of Courant-Friedrichs-Levy stability condition.

A Memory-Efficient Implementation of Perfectly Matched Layer With Smoothly Varying Coefficients in Discontinuous Galerkin Time-Domain Method

Wrapping a computation domain with a perfectly matched layer (PML) is one of the most effective methods of imitating/approximating the radiation boundary condition in Maxwell and wave equation solvers. Many PML implementations often use a smoothly-increasing attenuation coefficient to increase the absorption for a given layer thickness, and, at the same time, to reduce the numerical reflection from the interface between the computation domain and the PML. In discontinuous Galerkin time-domain (DGTD) methods, using a PML coefficient that varies within a mesh element requires a different mass matrix to be stored for every element and therefore significantly increases the memory footprint. In this work, this bottleneck is addressed by applying a weight-adjusted approximation to these mass matrices. The resulting DGTD scheme has the same advantages as the scheme that stores individual mass matrices, namely higher accuracy (due to reduced numerical reflection) and increased meshing flexibility (since the PML does not have to be defined layer by layer) but it requires significantly less memory.

Higher-Order Convolution PML (CPML) for FDTD Electromagnetic Modeling

A new simple formulation for incorporating a higher-order perfectly matched layer (PML) stretching function within a convolution PML (CPML) implementation in finite-difference time-domain (FDTD) electromagnetic modeling codes is developed. Obtaining in closed form the corresponding time domain impulse response of the inverse of a number of higher-order PML stretching functions enables the efficient and simple implementation of such higher-order PMLs using recursive convolution, in the same way as it was introduced initially for the complex frequency shifted (CFS) PML. This new higher-order CPML exhibits excellent performance that is comparable to the performance shown by other higher-order PML formulations whilst it retains the advantage of a relatively simpler implementation.

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.

A Complex Domain Mapping of the SCN for an Effective PML Implementation in TLM

An improved implementation of the perfectly matched layer (PML) is developed for the Transmission Line Modelling (TLM) method based on a mapping of the symmetrical condensed node (SCN) to an analytically extended geometric space. By mapping the TLM node, a medium – circuit equivalence is developed which maps transmission line parameters from real to complex domain. This consequently modifies the TLM scatter-connect process. The PML implementation is demonstrated for canonical cases where it is shown to have a comparable absorption performance and a significantly improved temporal stability relative to previously published TLM-PML formulations.

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.

DLSFEM–PML formulation for the steady-state response of a taut string on visco-elastic support under moving load

The numerical solution of the steady-state response of a uniform taut string on visco-elastic support under a concentrated transverse moving load is addressed. By recasting the governing second-order differential equation as a first-order system in convected coordinate, a local Discontinuous Least-Squares Finite Element Method (DLSFEM) formulation is developed within a complex-valued function space, to overcome numerical instabilities linked to high-velocity loads and handle far-field conditions through an effective Perfectly Matched Layer (PML) implementation. As an original advancement of the present DLSFEM–PML formulation, a coercivity theorem is proven for any first-order ordinary differential system and uniform error estimates are established for the finite element approximation for both$$L^2$$- and$$H^1$$-norms. Thus, the formulation newly joins a DLSFEM approach and a PML implementation, for solving the above-mentioned moving load problem. Numerical examples illustrate feasibility and accuracy of the method in reproducing the expected trends of solution and a priori error estimates.

A simple implementation of PML for second-order elastic wave equations

When modeling time-domain elastic wave propagation in an unbound space, the standard perfectly matched layer (PML) is straightforward for the first-order partial differential equations (PDEs); by contrast, the PML requires tremendous re-constructions of the governing equations in the second-order PDE form, which is however preferable, because of much less memory and time consumption. Therefore, it is imperative to explore a simple implementation of PML for the second-order system. In this work, we first systematically extend the first-order Nearly PML (NPML) technique into second-order systems, implemented by the spectral element and finite difference time-domain algorithms. It merits the following advantages: the simplicity in implementation, by keeping the second-order PDE-based governing equations exactly the same; the efficiency in computation, by introducing a set of auxiliary ordinary differential equations (ODEs). Mathematically, this PML technique effectively hybridizes the second-order PDEs and first-order ODEs, and locally attenuates outgoing waves, thus efficiently avoid either spatial or temporal global convolutions. Numerical experiments demonstrate that the NPML for the second-order PDE has an excellent absorbing performance for elastic, anelastic and anisotropic media in terms of the absorption accuracy, implementation complexity, and computation efficiency.

An Alternative PML Implementation for Arbitrary Media

An efficient implementation of the complex frequency-shifted perfectly matched layer (PML) is proposed to terminate the finite-difference time domain (FDTD) lattices. The proposal is based on the matrix exponential algorithm originally used for modeling frequency dispersive media. As the equations of describing $E (D)$ fields, $H (B)$ fields, and auxiliary variables associated with PML are formulated as a compact first-order differential matrix form, no convolution operations, variable substitutions, or equations rearrangements are needed in its mathematical derivations. Via the concept of generalized material-independent PML, this new implementation can also be used to truncate arbitrary media. Furthermore, since its iteration forms are similar to those of the convolutional PML (CPML) that has been widely used in FDTD simulations, the proposal can be directly applied to existing codes with only minor changes. Meanwhile, it can achieve better absorption effect than CPML. Through a numerical example in terms of the printed F antenna and corresponding analyses of performance and efficiency, we find that the proposal works well as absorbing boundary condition.

The Auxiliary Differential Equations Perfectly Matched Layers Based on the Hybrid SETD and PSTD Algorithms for Acoustic Waves

The perfectly matched layer (PML) absorbing boundary condition has been proven to absorb body waves and surface waves very efficiently at non-grazing incidence. However, the traditional PML would generate large spurious reflections at grazing incidence, for example, when the sources are located near the truncating boundary and the receivers are at a large offset. In this paper, a new PML implementation is presented for the boundary truncation in three-dimensional spectral element time domain (SETD) for solving acoustic wave equations. This method utilizes pseudospectral time-domain (PSTD) method to solve first-order auxiliary differential equations (ADEs), which is more straightforward than that in the classical FEM framework.

Anisotropic Locally Conformal Perfectly Matched Layer for Higher Order Curvilinear Finite-Element Modeling

A perfectly matched layer (PML) method is proposed for electrically large curvilinear meshes based on a higher order finite-element modeling paradigm and the concept of transformation electromagnetics. The method maps the non-Maxwellian formulation of the locally conformal PML to a purely Maxwellian implementation using continuously varying anisotropic and inhomogeneous material parameters. An approach to the implementation of a conformal PML for higher order meshes is also presented, based on a method of normal projection for PML mesh generation around an already existing convex volume mesh of a dielectric scatterer, with automatically generated constitutive material parameters. Once the initial mesh is generated, a PML optimization method based on gradient descent is implemented to most accurately match the PML material parameters to the geometrical interface. The numerical results show that the implementation of a conformal PML in the higher order finite-element modeling paradigm dramatically reduces the reflection error when compared to traditional PMLs with piecewise constant material parameters. The ability of the new PML to accurately and efficiently model scatterers with a large variation in geometrical shape and those with complex material compositions is demonstrated in examples of a dielectric almond and a continuously inhomogeneous and anisotropic transformation-optics cloaking structure, respectively.

The Auxiliary Differential Equations Perfectly Matched Layers Based on the Hybrid SETD and PSTD Algorithms for Acoustic Waves

The perfectly matched layer (PML) absorbing boundary condition has been proven to absorb body waves and surface waves very efficiently at non-grazing incidence. However, the traditional PML would generate large spurious reflections at grazing incidence, for example, when the sources are located near the truncating boundary and the receivers are at a large offset. In this paper, a new PML implementation is presented for the boundary truncation in three-dimensional spectral element time domain (SETD) for solving acoustic wave equations. This method utilizes pseudospectral time-domain (PSTD) method to solve first-order auxiliary differential equations (ADEs), which is more straightforward than that in the classical FEM framework.

PML Implementation for Efficient 3-D Laguerre-Based FDTD Method Using a New Perturbation Term

In this letter, the three-dimentional Berenger's perfectly matched layer (PML) implementation is proposed for the efficient Laguerre-based finite-difference time-domain method using a new perturbation term. By using this new term and the factorization-splitting scheme, the huge matrix equation of Berenger's PML implementation is split into six tridiagonal matrices and solved efficiently. In the numerical validation, the absorbing performance of the proposed PML implementation is studied as a function of the constitutive parameters and is compared to the existing absorbing boundary conditions (ABCs). Numerical results show that the proposed PML implementation has a better absorbing performance.

Electromagnetic simulation of VLSI circuits by the modified ADI‐FDTD method

ABSTRACTThe alternating direction implicit (ADI) finite‐difference time‐domain (FDTD) method can be used to simulate very large scale integration (VLSI) circuits efficiently as the time step is not restricted by the Courant–Friedrich–Levy stability condition. When the Berenger's split‐field perfectly matched layer (PML) absorbing boundary condition is used for the ADI‐FDTD method for open region simulation, the PML implementation will make this scheme unstable. In this article, the modified PML conductivity profiles are proposed to improve the stability of this scheme. Numerical simulations of the VLSI interconnect and RF inductor in time domain and frequency domain will be demonstrated to show the efficiency and accuracy of this method. © 2014 Wiley Periodicals, Inc. Microwave Opt Technol Lett 56:2530–2534, 2014

Complex frequency-shifted multi-axial perfectly matched layer for elastic wave modelling on curvilinear grids

SUMMARY Perfectly matched layer (PML) is an efficient absorbing technique for numerical wave simulations.Sinceitappeared,variousimprovementshavebeenmade.Thecomplexfrequency-shifted PML (CFS-PML) improves the absorbing performance for near-grazing incident waves and evanescent waves. The auxiliary differential equation (ADE) formulation of the PML provides a convenient unsplit-field PML implementation that can be directly used with high order time marching schemes. The multi-axial PML (MPML) stabilizes the PML on anisotropic media. However, these improvements were generally developed for Cartesian grids. In this paper, we extend the ADE CFS-PML to general curvilinear (non-orthogonal) grids for elastic wave modelling. Unlike the common implementations to absorb the waves in the computational space, we apply the damping along the perpendicular direction of the PML layer in the local Cartesian coordinates. Further, we relate the perpendicular and parallel components of the gradient operator in the local Cartesian coordinates to the derivatives in the curvilinear coordinates, to avoid mapping the wavefield to the local Cartesian coordinates. It is thus easy to be incorporated with numerical schemes on curvilinear grids. We derive the PML equations for the interior region and for the free surface separately because the free surface boundary condition modifies the elastic wave equations. We show that the elastic wave modelling on curvilinear grids exhibits anisotropic effects in the computational space, which may lead to unstable simulations. To stabilize the simulation, we adapt the MPML strategy to also absorb the wavefield along the two parallel directions of the PML. We illustrate the stability of this ADE CFS-MPML for finite-difference elastic wave simulations on curvilinear grids by two numerical experiments.