The Application of the Nonsplitting Perfectly Matched Layer in Numerical Modeling of Wave Propagation in Poroelastic Media

The nonsplitting perfectly matched layer （NPML） absorbing boundary condition （ABC） was first provided by Wang and Tang （2003） for the finite-difference simulation of elastic wave propagation in solids. In this paper, the method is developed to extend the NPML to simulating elastic wave propagation in poroelastic media. Biot＇s equations are discretized and approximated to a staggered-grid by applying a fourth-order accurate central difference in space and a second-order accurate central difference in time. A cylindrical twolayer seismic model and a borehole model are chosen to validate the effectiveness of the NPML. The results show that the numerical solutions agree well with the solutions of the discrete wavenumber （DW） method.

A study of perfectly matched layers for joint multicomponent reverse-time migration

Reverse-time migration in finite space requires effective boundary processing technology to eliminate the artificial truncation boundary effect in the migration result.On the basis of the elastic velocity-stress equations in vertical transversely isotropic media and the idea of the conventional split perfectly matched layer（PML）,the PML wave equations in reverse-time migration are derived in this paper and then the high order staggered grid discrete schemes are subsequently given.Aiming at the＂reflections＂from the boundary to the computational domain,as well as the effect of seismic event＇s abrupt changes at the two ends of the seismic array,the PML arrangement in reverse-time migration is given.The synthetic and real elastic,prestack,multi-component,reverse-time depth migration results demonstrate that this method has much better absorbing effects than other methods and the joint migration produces good imaging results.

Perfectly matched layer-absorbing boundary condition for finite-element time-domain modeling of elastic wave equations

The perfectly matched layer （PML） is a highly efficient absorbing boundary condition used for the numerical modeling of seismic wave equation. The article focuses on the application of this technique to finite-element time-domain numerical modeling of elastic wave equation. However, the finite-element time-domain scheme is based on the second- order wave equation in displacement formulation. Thus, the first-order PML in velocity-stress formulation cannot be directly applied to this scheme. In this article, we derive the finite- element matrix equations of second-order PML in displacement formulation, and accomplish the implementation of PML in finite-element time-domain modeling of elastic wave equation. The PML has an approximate zero reflection coefficients for bulk and surface waves in the finite-element modeling of P-SV and SH wave propagation in the 2D homogeneous elastic media. The numerical experiments using a two-layer model with irregular topography validate the efficiency of PML in the modeling of seismic wave propagation in geological models with complex structures and heterogeneous media.

A study of damping factors in perfectly matched layers for the numerical simulation of seismic waves

When simulating seismic wave propagation in free space, it is essential to introduce absorbing boundary conditions to eliminate reflections from artificially trtmcated boundaries. In this paper, a damping factor referred to as the Gaussian dmping factor is proposed. The Gaussian damping factor is based on the idea of perfectly matched layers （PMLs）. This work presents a detailed analysis of the theoretical foundations and advantages of the Gaussian damping factor. Additionally, numerical experiments for the simulation of seismic waves are presented based on two numerical models： a homogeneous model and a multi-layer model. The results show that the proposed factor works better. The Gaussian damping factor achieves a higher Signal-to-Noise Ratio （SNR） than previously used factors when using same number of PMLs, and requires less PMLs than other methods to achieve an identical SNR.

An Adaptive Uniaxial Perfectly Matched Layer Method for Time-Harmonic Scattering Problems

The uniaxial perfectly matched layer (PML) method uses rectangular domain to define the PML problem and thus provides greater flexibility and efficiency in deal- ing with problems involving anisotropic scatterers.In this paper an adaptive uniaxial PML technique for solving the time harmonic Helmholtz scattering problem is devel- oped.The PML parameters such as the thickness of the layer and the fictitious medium property are determined through sharp a posteriori error estimates.The adaptive finite element method based on a posteriori error estimate is proposed to solve the PML equa- tion which produces automatically a coarse mesh size away from the fixed domain and thus makes the total computational costs insensitive to the thickness of the PML absorb- ing layer.Numerical experiments are included to illustrate the competitive behavior of the proposed adaptive method.In particular,it is demonstrated that the PML layer can be chosen as close to one wave-length from the scatterer and still yields good accuracy and efficiency in approximating the far fields.

A Uniaxial Optimal Perfectly Matched Layer Method for Time-harmonic Scattering Problems

We develop a uniaxial optimal perfectly matched layer （opt PML） method for solving the time-harmonic scattering problems by choosing a particular absorbing function with unbounded integral in a rectangular domain. With this choice, the solution of the optimal PML problem not only converges exponentially to the solution of the original scatting problem, but also is insensitive to the thickness of the PML layer for sufficiently small parameter ε0. Numerical experiments are included to illustrate the competitive behavior of the proposed optimal method.

Perfectly matched layer implementation for ADI-FDTD in dispersive media

Alternating direction implicit finite difference time domain （ADI-FDTD） method is unconditionally stable and the maximum time step is not limited by the Courant stability condition, but rather by numerical error. Compared with the conventional FDTD method, the time step of ADI-FDTD can be enlarged arbitrarily and the CPU cost can be reduced. 2D perfectly matched layer （PML） absorbing boundary condition is proposed to truncate computation space for ADI-FDTD in dispersive media using recursive convolution（RC） method and the 2D PML formulations for dispersive media are derived. ADI-FDTD formulations for dispersive media can be obtained from the simplified PML formulations. The scattering of target in dispersive soil is simulated under sine wave and Gaussian pulse excitations and numerical results of ADI-FDTD with PML are compared with FDTD. Good agreement is observed. At the same time the CPU cost for ADI-FDTD is obviously reduced.

An improved convolution perfectly matched layer for elastic second-order wave equation

A convolution perfectly matched layer(CPML)can efficiently absorb boundary reflection in numerical simulation.However,the CPML is suitable for the first-order elastic wave equation and is difficult to apply directly to the second-order elastic wave equation.In view of this,based on the first-order CPML absorbing boundary condition,we propose a new CPML(NCPML)boundary which can be directly applied to the second-order wave equation.We first systematically extend the first-order CPML technique into second-order wave equations,neglecting the space-varying characteristics of the partial damping coefficient in the complex-frequency domain,avoiding the generation of convolution in the time domain.We then transform the technique back to the time domain through the inverse Fourier transform.Numerical simulation indicates that the space-varying characteristics of the attenuation factor have little influence on the absorption effect and increase the memory at the same time.A number of numerical examples show that the NCPML proposed in this study is effective in simulating elastic wave propagation,and this algorithm is more efficient and requires less memory allocation than the conventional PML absorbing boundary.

Application of perfectly matched layer to soil-foundation interaction analysis

Despite of the limitation in modeling infinite space, the finite element method(FEM) is one of the most used tools to numerically study the geotechnical problems regarding the capacity of simulating different geometries, conditions and material behaviors. A kind of absorbing layer named perfectly matched layer(PML) has been applied to modeling the radiation damping using FEM, which makes the dynamic analysis of soil-structure interaction more accurate. The PML is capable of absorbing incident waves under any angle and frequency, ensuring them to pass through the model boundaries without reflection.In this context, a new FEM program has been written and the PML formula has been implemented by rewriting the dynamic equation of motion and deriving new properties for the quadrilateral elements.The analysis of soil-foundation interaction by applying the PML is validated by the evaluation of impedance/compliance functions for different ground conditions. The results obtained from the PML model match the extended mesh results, even though the domain is small enough that other types of absorbing boundaries can reflect waves back to the foundation. The mechanism of the wave propagation in the region shows that the forced vibrations can be fully absorbed and damped by the boundaries surrounded by PMLs which is the role of radiation damping in FEM modeling.

Finite Element Analysis in Combination with Perfectly Matched Layer to the Numerical Modeling of Acoustic Devices in Piezoelectric Materials

The characterization of finite length Surface Acoustic Wave (SAW) and Bulk acoustic Wave (BAW) resonators is addressed here. The Finite Element Analysis (FEA) induces artificial wave reflections at the edges of the mesh. In fact, these ones do not contribute in practice to the corresponding experimental response. The Perfectly Matched Layer (PML) method, allows to suppress the boundary reflections. In this work, we first demonstrate the basis of PML adapted to FEA formalism. Next, the results of such a method are depicted allowing a discussion on the behavior of finite acoustic resonators.

A spherical higher-order finite-difference time-domain algorithm with perfectly matched layer

A higher-order finite-difference time-domain（HO-FDTD） in the spherical coordinate is presented in this paper. The stability and dispersion properties of the proposed scheme are investigated and an air-filled spherical resonator is modeled in order to demonstrate the advantage of this scheme over the finite-difference time-domain（FDTD） and the multiresolution time-domain（MRTD） schemes with respect to memory requirements and CPU time. Moreover, the Berenger＇s perfectly matched layer（PML） is derived for the spherical HO-FDTD grids, and the numerical results validate the efficiency of the PML.

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.

Comparison of perfectly matched layer and multi-transmitting formula artificial boundary condition based on hybrid finite element formulation

The theory of perfectly matched layer （PML） artificial boundary condition （ABC）, which is characterized by absorption any wave motions with arbitrary frequency and arbitrarily incident angle, is introduced. The construction process of PML boundary based on elastodynamic partial differential equation （PDE） system is developed. Combining with velocity-stress hybrid finite element formulation, the applicability of PML boundary is investigated and the numerical reflection of PML boundary is estimated. The reflectivity of PML and multi-transmitting formula （MTF） boundary is then compared based on body wave and surface wave simulations. The results show that although PML boundary yields some reflection, its absorption performance is superior to MTF boundary in the numerical simulations of near-fault wave propagation, especially in comer and large angle grazing incidence situations. The PML boundary does not arise any unstable phenomenon and the stability of PML boundary is better than MTF boundary in hybrid finite element method. For a specified problem and analysis tolerance, the computational efficiency of PML boundary is only a little lower than MTF boundary.

Stability of Perfectly Matched Layers for Time Fractional Schrödinger Equation

It is an important issue to numerically solve the time fractional Schrödinger equation on unbounded domains, which models the dynamics of optical solitons propagating via optical fibers. The perfectly matched layer approach is applied to truncate the unbounded physical domain, and obtain an initial boundary value problem on a bounded computational domain, which can be efficiently solved by the finite difference method. The stability of the reduced initial boundary value problem is rigorously analyzed. Some numerical results are presented to illustrate the accuracy and feasibility of the perfectly matched layer approach. According to these examples, the absorption parameters and the width of the absorption layer will affect the absorption effect. The larger the absorption width, the better the absorption effect. There is an optimal absorption parameter, the absorption effect is the best.

Layer-oriented Element Integration Algorithm of Conformal Perfectly Matched Layer

As an efficient artificial truncating boundary condition, conformal perfectly matched layer (CPML) is a kind of multilayer anisotropic absorbing media. To reduce computing effort of CPML, this article proposes a layer-oriented element integration algorithm. In this algorithm, the relative dielectric constant and permeability are considered as constants for each the very thin monolayer of CPML, and the element integration of multilayer along the normal direction is substituted by the element integration of m...

A FINITE ELEMENT METHOD WITH RECTANGULAR PERFECTLY MATCHED LAYERS FOR THE SCATTERING FROM CAVITIES

We develop a finite element method with rectangular perfectly matched layers （PMLs） for the wave scattering from two-dimensional cavities. The unbounded computational domain is truncated to a bounded one by using of a rectangular perfectly matched layer at the open aperture. The PML parameters such as the thickness of the layer and the fictitious medium property are determined through sharp a posteriori error estimates. Numerical experiments are carried out to illustrate the competitive behavior of the proposed method.

A Well-Posed and Discretely Stable Perfectly Matched Layer for Elastic Wave Equations in Second Order Formulation

We present a well-posed and discretely stable perfectly matched layer for the anisotropic(and isotropic)elastic wave equations without first re-writing the governing equations as a first order system.The new model is derived by the complex coordinate stretching technique.Using standard perturbation methods we show that complex frequency shift together with a chosen real scaling factor ensures the decay of eigen-modes for all relevant frequencies.To buttress the stability properties and the robustness of the proposed model,numerical experiments are presented for anisotropic elastic wave equations.The model is approximated with a stable node-centered finite difference scheme that is second order accurate both in time and space.

An hp Adaptive Uniaxial Perfectly Matched Layer Method for Helmholtz Scattering Problems

We propose an adaptive strategy for solving high frequency Helmholtz scattering problems.The method is based on the uniaxial PML method to truncate the scattering problem which is defined in the unbounded domain into the bounded domain.The parameters in the uniaxial PML method are determined by sharp a posteriori error estimates developed by Chen and Wu[8].An hp-adaptive finite element strategy is proposed to solve the uniaxial PML equation.Numerical experiments are included which indicate the desirable exponential decay property of the error.

An Optimized Perfectly Matched Layer for the Schrödinger Equation

We derive a perfectly matched layer(PML)for the Schrödinger equation using a modal ansatz.We derive approximate error formulas for the modeling error from the outer boundary of the PML and the error from the discretization in the layer and show how to choose layer parameters so that these errors are matched and optimal performance of the PML is obtained.Numerical computations in 1D and 2D demonstrate that the optimized PML works efficiently at a prescribed accuracy for the zero potential case,with a layer of width less than a third of the de Broglie wavelength corresponding to the dominating frequency.

Numerical Solution of a One-Dimensional Nonlocal Helmholtz Equation by Perfectly Matched Layers

We consider the computation of a nonlocal Helmholtz equation by using perfectly matched layer(PML).We first derive the nonlocal PML equation by extending PML modifications from the local operator to the nonlocal operator of integral form.After that,we give stability estimates of some weighted-average values of the nonlocal Helmholtz solution and prove that(i)the weighted-average value of the nonlocal PML solution decays exponentially in PML layers in one case;(ii)in the other case,the weighted-average value of the nonlocal Helmholtz solution itself decays exponentially outside some domain.Particularly for a typical kernel functionγ_(1)(s)=1/2 e^(−|s|),we obtain the Green’s function of the nonlocal Helmholtz equation,and use the Green’s function to further prove that(i)the nonlocal PML solution decays exponentially in PML layers in one case;(ii)in the other case,the nonlocal Helmholtz solution itself decays exponentially outside some domain.Based on our theoretical analysis,the truncated nonlocal problems are discussed and an asymptotic compatibility scheme is also introduced to solve the resulting truncated problems.Finally,numerical examples are provided to verify the effectiveness and validation of our nonlocal PML strategy and theoretical findings.