Perfectly Matched Layer Method Research Articles

Simplified Tunnel–Soil Model Based on Thin-Layer Method–Volume Method–Perfectly Matched Layer Method

In order to analyze the ground vibration responses induced by the dynamic loads in a tunnel, this paper proposes a new simplified tunnel–soil model. Specifically, based on the basic theory of the thin-layer method (TLM), the basic solution of three-dimensional layered foundation soil displacement was derived in the cylindrical coordinate system. The perfectly matched layer (PML) boundary condition was applied to the TLM. Subsequently, a tunnel–soil dynamic interaction analysis model was established using the volume method (VM) in conjunction with the TLM-PML method. The displacement frequency response function of the foundation soil around the tunnel foundation was derived. Finally, a ground vibration test under an impact load in a tunnel was carried out. The test and calculated results were compared. The comparison results show that the ground vibration acceleration response values within 25 m from the load are similar. Compared with the test results, the theoretical calculation results exhibit a decreasing trend in the range of 40–80 Hz between 25 and 60 m, with the maximum reduction being approximately one order of magnitude. In addition, the experimental comparison demonstrates that the model can be used to analyze the ground vibrations caused by underground loads.

The time‐harmonic electromagnetic wave scattering by a biperiodic elastic body

This paper concentrates on an interaction scattering problem between the time‐harmonic electromagnetic waves and an unbounded periodic elastic medium. The uniqueness results of the interaction problem are established for small frequencies or all frequencies except a discrete set in both the absorbing and nonabsorbing medium, and then the existence of solutions is derived by the classical Fredholm alternative. The perfectly matched layer (PML) method is proposed to truncate the unbounded scattering domain to a bounded computational domain. We prove the well‐posedness of the solution for the truncated PML problem, where a homogeneous boundary condition is imposed on the outer boundary of the PML. The exponential convergence of the PML method is established in terms of the thickness and parameters of the PML. The proof is based on the PML extension and the exponential decay properties of the modified fundamental solution.

Time domain modeling of elastic waves using a stress-based unsplit-field perfectly matched layer with enhanced numerical stability

A stress-based unsplit-field perfectly matched layer (PML) method with enhanced numerical stability is proposed to simulate elastic waves in heterogeneous unbounded domains. The proposed PML method attenuates the waves by forcing the satisfaction of mixed-field equations constructed using a complex coordinate stretching in the frequency domain. Unlike other unsplit-field PMLs, which introduce coordinate stretching before combining the constitutive and compatibility equations, the proposed method first combines the two equations and then stretches the resulting equation. The stretched equation is converted back to the time domain using the inverse Fourier transform to yield transient elastic wave equations in the PML-truncated domain. Semi-discrete equations of motion for displacement and stresses were obtained using standard Lagrange-family finite elements. In a series of numerical examples involving semi-infinite elastic media, the accuracy of the wave solutions calculated using the proposed PML method was evaluated using a displacement L2-norm, a normalized L2-error, and an energy norm. The examples further validated the wave absorption performance of the PML and evaluated the long-term stability of solutions utilizing the Ricker pulse excitation with various central frequencies. The numerical experiments indicate that the solutions obtained using the proposed PML are significantly more stable than those obtained using the existing strain-based unsplit-field PMLs. The proposed method is particularly useful for applications requiring explicit time integration of semi-discrete equations of motion because the mass matrix can be diagonalized. The PML method and its associated solution strategies can be applied to various engineering problems, such as soil-structure interaction analysis, monitoring of structures, and geophysical probing.

A New Perfectly Matched Layer Method for the Helmholtz Equation in Nonconvex Domains

A New Perfectly Matched Layer Method for the Helmholtz Equation in Nonconvex Domains

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.

Convergence of the uniaxial PML method for time-domain electromagnetic scattering problems

In this paper, we propose and study the uniaxial perfectly matched layer (PML) method for three-dimensional time-domain electromagnetic scattering problems, which has a great advantage over the spherical one in dealing with problems involving anisotropic scatterers. The truncated uniaxial PML problem is proved to be well-posed and stable, based on the Laplace transform technique and the energy method. Moreover, the L2-norm and L∞-norm error estimates in time are given between the solutions of the original scattering problem and the truncated PML problem, leading to the exponential convergence of the time-domain uniaxial PML method in terms of the thickness and absorbing parameters of the PML layer. The proof depends on the error analysis between the EtM operators for the original scattering problem and the truncated PML problem, which is different from our previous work (Wei et al. [SIAM J. Numer. Anal. 58 (2020) 1918–1940]).

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 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.

Exponential convergence of Cartesian PML method for Maxwell’s equations in a two-layer medium

The perfectly matched layer (PML) method is extensively studied for scattering problems in homogeneous background media. However, rigorous studies on the PML method in layered media are very rare in the literature, particularly, for three-dimensional electromagnetic scattering problems. Cartesian PML method is favorable in numerical solutions since it is apt to deal with anisotropic scatterers and to construct finite element meshes. Its theories are more difficult than circular PML method due to anisotropic wave-absorbing materials. This paper presents a systematic study on the Cartesian PML method for three-dimensional electromagnetic scattering problem in a two-layer medium. We prove the well-posedness of the PML truncated problem and that the PML solution converges exponentially to the exact solution as either the material parameter or the thickness of PML increases. To the best of the authors’ knowledge, this is the first theoretical work on Cartesian PML method for Maxwell’s equations in layered media.

Convergence Analysis of the PML Method for Time-Domain Electromagnetic Scattering Problems

In this paper, a perfectly matched layer (PML) method is proposed to solve the time-domain electromagnetic scattering problems in three dimensions effectively. The PML problem is defined in a spherical layer and derived by using the Laplace transform and real coordinate stretching in the frequency domain. The well-posedness and the stability estimate of the PML problem are first proved based on the Laplace transform and the energy method. The exponential convergence of the PML method is then established in terms of the thickness of the layer and the PML absorbing parameter. As far as we know, this is the first convergence result for the time-domain PML method for the three-dimensional Maxwell equations. Our proof is mainly based on the stability estimates of solutions of the truncated PML problem and the exponential decay estimates of the stretched dyadic Green's function for the Maxwell equations in the free space.

Developing Perfectly Matched Layer method to solve Heat Equation numerically

Perfectly Matched Layer (PML) techniques, although having been studied extensively to study systems with nonrestrictive boundaries in many physical fields, are not readily adaptable to the study of thermodynamic systems governed by the heat equation. Using the explicit finite difference method, we can easily describe systems with perfectly absorbing or reflecting boundaries. These, however, are highly idealized physical states, so we have begun extending our abilities to simulating more realistic physical situations by defining arbitrary spatial differential operators, which govern how heat at the boundaries of the system of interest propagates. As a first attempt, we transformed the linear spatial domain to a trigonometric domain to combat reflection from boundaries, but this is quite crude. We then approached the problem using Fourier Transforms in order to define the problem in a heat distribution's frequency space, damp excessive heat past the boundaries, and transform back to position space. Initial results using the finite difference method verify the current computer simulation's ability to solve problems with ideal circumstances, and the development of a PML method which accurately simulates nonrestrictive boundaries and can be easily translated into a computer algorithm is in progress.

Generalized stiffness reduction method to remove the artificial edge-effects for seismic wave modelling in elastic anisotropic media

SUMMARYIn seismic wave modelling, the boundary reflections caused by the computational grid edges should be reduced to produce accurate simulation results. The perfectly matched layer (PML) method is one of the popular techniques to suppress such artificial reflections, because it can be easily applied to the first-order wave equation in many numerical methods. However, one issue of the PML method is that the stability condition might be violated in complex elastic anisotropic media. In these cases, the PML method will not attenuate the boundary reflections but rather introduce strong artefacts in the simulation results. To tackle this problem, we propose a generalized stiffness reduction method (GSRM) as a substitute for the PML method. We first derive the stability conditions of the PML method and analyse the suitable conditions for their application to time- and frequency-domain seismic wave modelling. Then, we develop a simple and effective numerical implementation of the GSRM to attenuate the boundary reflections and apply it to seismic wave modelling in elastic anisotropic media. We give some numerical experiments to demonstrate the feasibility and advantages of the GSRM compared to the PML method. Numerical examples show the GSRM is conceptually simpler, more computationally efficient and more straightforward in terms of numerical implementation than the PML method for seismic modelling using either first- or second-order time- and frequency-domain wave equations.

A Time-Domain Formulation of Elastic Waves in Heterogeneous Unbounded Domains

A time-domain formulation and its numerical implementation of three-dimensional elastic waves in unbounded domains are discussed. To simulate elastic waves in unbounded domains, it is necessary to truncate the originally infinite extent and remove artificial reflections at truncated boundaries. In this work, perfectly matched layer (PML) is used to surround the truncated regular domain and to absorb outgoing waves beyond the truncated boundaries. To derive governing equations for elastic waves in the PML-truncated domain, complex coordinate stretching is applied to equilibrium, constitutive, and compatibility equations in the frequency domain. Then the stretched equations were converted to the time domain by applying the inverse Fourier transform to yield a system of mixed displacement-stress equations. Introducing a mixed finite element method leads to mixed semi-discrete equations of motion which can be integrated in time for displacements and stresses. Numerical experiments show that reflections from truncated boundaries are significantly reduced by using the developed PML method. For the reduction of computational cost in higher dimensions, an MPI-based parallel computation has been performed on assembly and time integration routines of the finite element wave simulation, which shows good scalability.

Numerical study of exterior acoustic problems using a novel finite element-least square point interpolation method with perfectly matched layer

In recent years the finite element-least square point interpolation method (FE-LSPIM) is proposed by previous researchers to solve mechanical problems. Then, FE-LSPIM is also presented by the authors in acoustic analysis. This work introduces this idea into the two-dimensional (2D) exterior acoustic problems including the perfectly matched layer (PML) method and proposes a FE-LSPIM/PML model for exterior acoustic problems. In present work, the computational region, truncated by PML, is discretized into quadrilateral element meshes, and the shape functions of quadrilateral element are utilized for partition of unity (PU) and the least-square point interpolation method (LSPIM) are utilized for local approximation. The proposed model inherits the compatibility attributes of finite element method (FEM) and LSPIM, and will greatly depress numerical dispersion error; meanwhile, the present model can deal with the exterior domain problems by combining with the PML method. Numerical example shows that, the FE-LSPIM/PML achieves better absorbing effect in PML, and higher accuracy in the computational region as compared to the corresponding FEM/PML and the Element-Free Galerkin method (EFGM)/PML. Therefore, the FE-LSPIM/PML can be well applied to solve the exterior acoustic problems.

The Split-Field PML Absorbing Boundary Condition for the Unconditionally Stable Node-Based LOD-RPIM Method

To numerically model open-domain structures and radiation problems, an absorbing boundary condition (ABC) has to be applied to emulate infinite open space. In this letter, the split-field perfectly matched layer (PML) ABC is developed and presented for the unconditionally stable node-based locally one-dimensional radial point interpolation meshless method. Numerical experiments are conducted to demonstrate the performance of the proposed PML ABC; they show that the proposed PML method has a good absorbing capability.

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.

The PML method to solve a class of potential scattering problem with tapered wave incidence

This paper concerns about the two-dimensional direct scattering problem for Schrödinger equation with a class of decaying potential under the tapered wave incidence. We focus on the numerical solution of the problem by using the perfectly matched layer (PML) method. Based on the idea of complex coordinate stretching, the boundary value problem for the PML equation is formulated. Then the variational equation to the boundary value problem is proved to exist a unique solution except possibly for a set of values of wavenumber. The numerical experiments illustrate the influence of the thickness of PML layer, the absorption parameter, the wavenumber, the control bandwidth and even the incidence angle of the tapered wave on the effectiveness of this method.

An Adaptive Finite Element PML Method for the Acoustic-Elastic Interaction in Three Dimensions

AbstractConsider the scattering of a time-harmonic acoustic incident wave by a bounded, penetrable, and isotropic elastic solid, which is immersed in a homogeneous compressible air or fluid. The paper concerns the numerical solution for such an acoustic-elastic interaction problem in three dimensions. An exact transparent boundary condition (TBC) is developed to reduce the problem equivalently into a boundary value problem in a bounded domain. The perfectly matched layer (PML) technique is adopted to truncate the unbounded physical domain into a bounded computational domain. The well-posedness and exponential convergence of the solution are established for the truncated PML problem by using a PML equivalent TBC. An a posteriori error estimate based adaptive finite element method is developed to solve the scattering problem. Numerical experiments are included to demonstrate the competitive behavior of the proposed method.

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

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

Frequency domain finite-element and spectral-element acoustic wave modeling using absorbing boundaries and perfectly matched layer

Wave propagation modeling as a vital tool in seismology can be done via several different numerical methods among them are finite-difference, finite-element, and spectral-element methods (FDM, FEM and SEM). Some advanced applications in seismic exploration benefit the frequency domain modeling. Regarding flexibility in complex geological models and dealing with the free surface boundary condition, we studied the frequency domain acoustic wave equation using FEM and SEM. The results demonstrated that the frequency domain FEM and SEM have a good accuracy and numerical efficiency with the second order interpolation polynomials. Furthermore, we developed the second order Clayton and Engquist absorbing boundary condition (CE-ABC2) and compared it with the perfectly matched layer (PML) for the frequency domain FEM and SEM. In spite of PML method, CE-ABC2 does not add any additional computational cost to the modeling except assembling boundary matrices. As a result, considering CE-ABC2 is more efficient than PML for the frequency domain acoustic wave propagation modeling especially when computational cost is high and high-level absorbing performance is unnecessary.