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.

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.

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.

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.

DISCRETE SINGULAR CONVOLUTION METHOD WITH PERFECTLY MATCHED ABSORBING LAYERS FOR THE WAVE SCATTERING BY PERIODIC STRUCTURES

A new computational algorithm is introduced for solving scattering problem in periodic structure. The PML technique is used to deal with the difficulty on truncating the unbounded domain while the DSC algorithm is utilized for the spatial discretization. The present study reveals that the method is efficient for solving the problem.

Unconditionally stable Crank-Nicolson algorithm with enhanced absorption for rotationally symmetric multi-scale problems in anisotropic magnetized plasma

Large calculation error can be formed by directly employing the conventional Yee’s grid to curve surfaces.In order to alleviate such condition,unconditionally stable CrankNicolson Douglas-Gunn(CNDG)algorithm with is proposed for rotationally symmetric multi-scale problems in anisotropic magnetized plasma.Within the CNDG algorithm,an alternative scheme for the simulation of anisotropic plasma is proposed in body-of-revolution domains.Convolutional perfectly matched layer(CPML)formulation is proposed to efficiently solve the open region problems.Numerical example is carried out for the illustration of effectiveness including the efficiency,resources,and absorption.Through the results,it can be concluded that the proposed scheme shows considerable performance during the simulation.

Investigation of composite electromagnetic scattering from ship-like target on the randomly rough sea surface using FDTD method

Composite electromagnetic scattering from a two-dimensional （2D） ship-like target on a one-dimensional sea surface is investigated by using the finite-difference time-domain （FDTD） method. A uniaxial perfectly matched layer is adopted for truncation of FDTD lattices.The FDTD updated equations can be used for the total computation domain by choosing the uniaxial parameters properly. To validate the proposed numerical technique,a 2D infinitely long cylinder over the sea surface is taken into account first.The variation of angular distribution of the scattering changing with incident angle is calculated. The results show good agreement with the conventional moment method. Finally,the influence of the incident angle,the polarization,and the size of the ship-like target on the composite scattering coefficient is discussed in detail.

A PML Method for Electromagnetic Scattering from Two-dimensional Overfilled Cavities

In this paper, we consider electromagnetic scattering problems for two-dimensional overfilled cavities. A half ringy absorbing perfectly matched layer （PML） is introduced to enclose the cavity, and the PML formulations for both TM and TE polarizations are presented. Existence, uniqueness and convergence of the PML solutions are considered. Numerical experiments demonstrate that the PML method is efficient and accurate for solving cavity scattering problems.

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.

ANALYSIS OF FDTD TO UPML FOR MAXWELL EQUATIONS IN POLAR COORDINATES

An FDTD system associated with uniaxial perfectly matched layer（UPML） for an electromagnetic scattering problem in two-dimensional space in polar coordinates is considered.Particularly the FDTD system of an initial-boundary value problems of the transverse magnetic（TM） mode to Maxwell＇s equations is obtained by Yee＇s algorithm,and the open domain of the scattering problem is truncated by a circle with a UPML.Besides,an artificial boundary condition is imposed on the outer boundary of the UPML.Afterwards,stability of the FDTD system on the truncated domain is established through energy estimates by the Gronwall inequality.Numerical experiments are designed to approve the theoretical analysis.

Exponential time differencing based efficient SC-PML for RCS simulation

To efficiently simulate and calculate the radar cross section(RCS) related electromagnetic problems by employing the finite-difference time-domain(FDTD) algorithm, an efficient stretched coordinate perfectly matched layer(ESC-PML) based upon the exponential time differencing(ETD) method is proposed.The proposed implementation can not only reduce the number of auxiliary variables in the SC-PML regions but also maintain the ability of the original SC-PML in terms of the absorbing performance. Compared with the other existed algorithms, the ETDFDTD method shows the least memory consumption resulting in the computational efficiency. The effectiveness and efficiency of the proposed ESC-PML scheme is verified through the RCS relevant problems including the perfect E conductor(PEC) sphere model and the patch antenna model. The results indicate that the proposed scheme has the advantages of the ETD-FDTD method and ESC-PML scheme in terms of high computational efficiency and considerable computational accuracy.

A FINITE ELEMENT METHOD WITH PERFECTLY MATCHED ABSORBING LAYERS FOR THE WAVE SCATTERING BY A PERIODIC CHIRAL STRUCTURE

Consider the diffraction of a time-harmonic wave incident upon a periodic chiral structure. The diffraction problem may be simplified to a two-dimensional one. In this paper, the diffraction problem is solved by a finite element method with perfectly matched absorbing layers （PMLs）. We use the PML technique to truncate the unbounded domain to a bounded one which attenuates the outgoing waves in the PML region. Our computational experiments indicate that the proposed method with complicated chiral grating structures. is efficient, which is capable of dealing

Approximate CN scheme and its open region problems for metamaterial rotational symmetric simulation

In order to simulate metamaterial rotational symmetric open region problems,unconditionally stable perfectly match layer(PML)implementation is proposed in the body of revolution(BOR)finite-difference time-domain(FDTD)lattice.More precisely,the proposed algorithm is implemented by the Crank-Nicolson(CN)Douglas-Gunn(DG)procedure for BOR metamaterial simulation.The constitutive relationship of metamaterial can be expressed by the Drude model and calculated by the piecewise linear recursive convolution(PLRC)approach.The effectiveness including absorption,efficiency,and accuracy is demonstrated through the numerical example.It can be concluded that the proposed implementation is to take the advantages of the CNDG-PML procedure,PLRC approach,and BORFDTD algorithm in terms of considerable accuracy,enhanced absorption and remarkable efficiency.Meanwhile,it can be demonstrated that the proposed scheme can maintain its unconditional stability when the time step exceeds the CourantFriedrichs-Levy(CFL)condition.

Unsplit-field higher-order nearly PML for arbitrary media in EM simulation

An unsplit-field higher order nearly perfectly matched layer(NPML)based on the auxiliary differential equation approach is introduced in three-dimensional finite-difference timedomain lattices.The proposed scheme has the advantage of both the NPML scheme and the higher order concept in terms of the improved absorbing performance and considerable computational efficiency.By incorporating with the generalized material independent concept,the proposed implementation is indepen dent of the material’s type.Thus,it has the advantages of terminating arbitrary media without changing the updated equations in the PML regions.Its effectiveness and efficiency is further demonstrated through numerical examples.

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.