Complex Frequency-shifted Perfectly Matched Layer Research Articles

Adaptive-Coefficient Finite Difference Frequency Domain Method for Solving Time-Fractional Cattaneo Equation with Absorbing Boundary Condition

The time-fractional Cattaneo (TFC) equation is a practical tool for simulating anomalous dynamics in physical diffusive processes. The existing numerical solutions to the TFC equation generally deal with the Dirichlet boundary conditions. In this paper, we incorporate the absorbing boundary condition as a complex-frequency-shifted (CFS) perfectly matched layer (PML) into the TFC equation. Then, we develop an adaptive-coefficient (AC) finite-difference frequency-domain (FDFD) method for solving the TFC with CFS PML. The corresponding analytical solution for homogeneous TFC equation with a point source is proposed for validation. The effectiveness of the developed AC FDFD method is verified by the numerical examples of four typical TFC models, including the different orders of time-fractional derivatives for both the homogeneous model and the layered model. The numerical examples show that the developed AC FDFD method is more accurate than the traditional second-order FDFD method for solving the TFC equation with the CFS PML absorbing boundary condition, while requiring similar computational costs.

A unified higher-order unsplit CFS-PML technique for solving second-order seismic equations using discontinuous Galerkin method

The absorbing boundary conditions play a key role in numerical simulations. The complex frequency-shifted perfectly matched layer (CFS-PML), one of the absorbing boundary conditions, has attracted widespread attention because it can efficiently attenuate near-grazing incident waves and evanescent waves. In practical application, the distinct boundary reflections still exist using the CFS-PML in a thin PML region. To further enhance the absorbing performance, we have developed a novel unified high-order unsplit CFS-PML technique. The high-order CFS-PML combined with the discontinuous Galerkin (DG) method is applied to solve the second-order seismic wave equation on an unstructured mesh. In this work, we establish a unified framework for high-order unsplit CFS-PML formulations. The classical CFS-PML is the first-order form of our proposed unsplit high-order approach. By using auxiliary variables and auxiliary ordinary differential equations (AODEs), a higher-order CFS-PML formulation can be reduced to a first-order CFS-PML formula. For an Nth-order AODE CFS-PML, it will lead to (2N-1) additional variables and (2N-1) auxiliary differential equations. Taking second- and third-order as examples, we have derived the high-order unsplit AODE CFS-PML expressions in detail. The DG method can effectively deal with complex geological structures, such as caves, cracks, faults, etc., which yields more accurate numerical solutions. We incorporate the high-order unsplit AODE CFS-PML technique into the DG method to approximate seismic wave equations. Their strengths are combined to calculate the propagating wavefield in more complex finite models and obtain better results. In addition, the stability conditions of the second- and third-order CFS-PMLs are provided. We perform the high-order unsplit CFS-PML implementation for seismic wave modeling in several models. The isotropic and anisotropic examples prove that the high-order unsplit CFS-PML enjoys better absorbing performance than the classical CFS-PML. The numerical result for the graben model certifies the feasibility and flexibility of our proposed high-order CFS-PML combined with the DG method.

3D forward modeling and response characteristics of low-resistivity overburden of the CFS-PML absorbing boundary for ground-well transient electromagnetic method

This study used the stable and convergent Dufort-Frankel method to differentially discretize the diffusion equation of the ground-well transient electromagnetic secondary field. The absorption boundary condition of complex frequency-shifted perfectly matched layer (CFS-PML) was used for truncation so that the low-frequency electromagnetic wave can be better absorbed at the model boundary. A typical three-dimensional (3D) homogeneous half-space model was established and a low-resistivity cube model was analyzed under the half-space condition. The response patterns and drivers of the low-resistivity cube model were discussed under the influence of a low-resistivity overburden. The absorption boundary conditions of CFS-PML significantly affected the low-frequency electromagnetic waves. For a low-resistivity cube around the borehole, its response curve exhibited a single-peak, and the extreme point of the curve corresponded to the center of the low-resistivity body. When the low-resistivity cube was directly below the borehole, the response curve showed three extreme values (two high and one low), with the low corresponding to the center of the low-resistivity body. The total field response of the low-resistivity overburden was stronger than that of the uniform half-space model due to the low-resistivity shielding effect of electromagnetic waves. When the receiving-transmitting distance gradually increased, the effect of the low-resistivity overburden was gradually weakened, and the response of the low-resistivity cube was strengthened. It was affected by the ratio of the overburden resistivity to the resistivity of the low-resistivity body.

A perfectly matched layer for the absorption of electromagnetic waves using differential forms in three-dimensional domains

As a significant contribution, we present a novel formulation of the complex frequency-shifted perfectly matched layer (CFS-PML) for differential forms, introducing the innovative concept of a directional incidence matrix. This paper explores the application of differential forms in simulating wave propagation problems in three-dimensional domains. To confine the unbounded domain, we utilize a discretization technique based on the CFS-PML. Additionally, we provide an illustrative example that involves a system comprising a GPR (Ground Penetrating Radar) horn antenna, air, lossy soil, and a buried sphere.

Implementation of ADE-CFS-PML for the Single Field WCS-FDTD Method

In this letter, the auxiliary differential equation (ADE) form of the complex-frequency-shifted perfectly matched layer (CFS-PML) is implemented for the single field (SF) weakly conditionally stable (WCS) finite difference time domain (FDTD) method. The magnetic field terms in the auxiliary variables of ADE-CFS-PML are eliminated, conforming to the SF formulation. The reflection error and stability of the proposed method is verified with numerical examples. Results show that the proposed method is stable and features low reflection error when the maximum time step is chosen for the SF-WCS-FDTD method. The higher multithread computation efficiency of the proposed method is also demonstrated.

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.

Investigating the stability of an explicit ADE-FDTD scheme for modeling graphene: Avoiding erroneous conclusions

In recent years, an explicit auxiliary differential equation finite difference time domain (ADE-FDTD) scheme has been frequently used in graphene simulations. In this respect, a differencing scheme in which the magnetic field and the current density are collocated in time was developed and it has been recently stated that this explicit ADE-FDTD implementation is “conditionally stable with the maximum time-step size bounded by the common Courant–Friedrichs–Lewy (CFL) limit”. In this communication, the stability of this explicit ADE-FDTD scheme is revisited and it is shown that the conventional CFL constraint is not retained, and a time-step less than the CFL limit must be used in all domain grids to insure the stability in the whole system. Although this time-step stringent can be of little effect for some graphene applications, and in order to avoid drawing erroneous conclusions, it is shown that when the given ADE-FDTD scheme is used for simulating graphene structures that include noble and other plasmonic materials the derived time-step stringent can be of significant effect. To retain the CFL stability constraint, alternative formulation based on the Runge–Kutta (RK) time-differencing scheme is also presented in this communication. Contrary to the other existing CFL-stable ADE-FDTD approaches, the presented RK-FDTD method is equally applicable to both magnetized and unmagnetized graphene structures without incurring additional computational cost. Furthermore, the RK-FDTD scheme is extended for modeling the complex-frequency-shifted perfectly matched layer (CFS-PML) mesh truncating technique. Finally, the stability and accuracy of the explicit ADE/RK-FDTD schemes are demonstrated through numerical examples.

A High-Efficiency Spectral Element Method Based on CFS-PML for GPR Numerical Simulation and Reverse Time Migration

Improving the accuracy and efficiency of the numerical simulation of ground penetrating radar (GPR) becomes a pressing need with the rapidly increased amount of inversion data and the growing demand for migration imaging quality. In this article, we present a numerical spectral element time-domain (SETD) simulation procedure for GPR forward modeling and further apply it to the reverse time migration (RTM) with complex geoelectric models. This approach takes into account the flexibility of the finite element methods and the high precision of the spectral methods. Meanwhile, in this procedure, the complex frequency shifted perfectly matched layer (CFS-PML) is loaded to effectively suppress the echo at the truncated boundary, and the per-element GPU parallel framework used can achieve up to 5.7788 times the efficiency compared with the CPU calculation. The experiments on SETD spatial convergence and CFS-PML optimal parameter selection showed that, under the same degree of freedom, the SETD offered substantially better accuracy compared with the traditional FETD. The experiments on RTM of different profiles with different orders of SETD via a complex geoelectric model verify the universality of the algorithm. The results indicate that the RTM imaging effect has been significantly improved with the increase of SETD order. It fully proves the great potential of efficient and high-precision SETD simulation algorithm in the RTM imaging direction and shows certain guiding significance for underground target structure exploration.

Convolutional Implementation and Analysis of the CFS-PML ABC for the FDTD Method Based on Wave Equation

In this letter, the convolutional implementation of the complex frequency shifted perfectly matched layer (CFS-PML) absorbing boundary condition (ABC) is proposed for the finite-difference time-domain (FDTD) method of solving wave equation. The FDTD solution time-marching formulations of the developed wave-equation convolutional CFS-PML (CPML) ABC are derived. Several numerical experiments are used to verify the capacity of the proposed algorithm. Solution results confirm that the proposed CPML ABC can obtain excellent absorption for both traveling waves and evanescent waves.

Z-Transform-Based FDTD Implementations of Biaxial Anisotropy for Radar Target Scattering Problems

In this article, an efficient Z-transform-based finite-difference time-domain (Z-FDTD) is developed to implement and analyze electromagnetic scatterings in the 3D biaxial anisotropy. In terms of the Z-transform technique, we first discuss the conversion relationship between time- or frequency-domain derivative operators and the corresponding Z-domain operator, then build up the Z-transform-based iteration from the electric flux D converted to the electric field E based on dielectric tensor ε (and from the magnetic flux B converted to the magnetic field H in line with permeability tensor μ) by combining the constitutive formulations about the biaxial anisotropy. As a result, the iterative process about the Z-FDTD implementation can be smoothly carried out by means of combining with the Maxwell’s equations. To our knowledge, it is inevitably necessary for the absorbing boundary condition (ABC) to be considered in the electromagnetic scattering; hence, we utilize the unsplit-field complex-frequency-shifted perfectly matched layer (CFS-PML) to truncate the Z-FDTD’s physical region, and then capture time- and frequency-domain radiation with the electric dipole. In the 3D simulations, we select two different biaxial anisotropic models to validate the proposed formulations by using the popular commercial software COMSOL. Moreover, it is certain that those results are effective and available for electromagnetic scattering problems under the oblique incidence executed by the Z-FDTD method.

An Unsplit CFS-PML Scheme for the Second-Order Wave Equation With Its Application in Fractional Viscoacoustic Simulation

The unsplit complex frequency-shifted perfectly matched layer (CFS-PML) has been widely used in the first-order wave equation in velocity and stress while rarely formulated for the wave equation recast as a second-order system in displacement. Among different variants of PML, the unsplit CFS-PML for the second-order wave equation enjoys better absorbing performance and numerical stability, compared to the traditional PML, due to the presence of the general form of CFS stretching factors, as well as higher computational efficiency over the split PMLs since it avoids wave equation order reduction and splitting the state variables into multiple directional components. This study aims to develop an unsplit CFS-PML scheme for the second-order wave equation and devote specific attention to fractional viscoacoustic simulation where fractional time derivatives are involved and hard to be reformulated into a first-order form. In the complex space, PML is typically regarded as an analytical continuation of the real coordinates; thus, we define an explicit coordinate stretching operator acting on the Laplacian operator. This stretching operator consists of several convolution terms; each of them can be efficiently resolved by a recursive convolution updating strategy. Viscoacoustic simulations on homogeneous Pierre Shale, Marmousi model, and 3-D SEG/EAGE overthrust model verify the feasibility and absorbing the performance of our proposed scheme.

Implicit Approximate Crank–Nicolson Theory for Anisotropic Ferrite Structure Simulation with Enhanced Absorption

AbstractTo simulate ferrite structures with anisotropic characteristics efficiently, an unconditionally stable implementation is proposed by incorporating the approximate Crank–Nicolson (CN) theory and a complex‐frequency‐shifted perfectly matched layer (CFS‐PML) formulation in the finite‐difference time‐domain lattice. More specifically, the proposed implementation combines the CN approximate factorization splitting procedure with bilinear transformation in the z‐domain within the CFS‐PML regions. To simulate the unique anisotropic characteristic of the ferrite structure, the auxiliary differential equation (ADE) method, which is modified at the integer time step, is introduced during the formulation. Numerical examples are presented to demonstrate the effectiveness and efficiency theoretically and experimentally. According to the results, it can be concluded that the proposed scheme takes full advantage of the approximate CN theory, CFS‐PML formulation, and modified ADE method in terms of the considerable computational accuracy, outstanding absorbing performance, and remarkable efficiency. Moreover, it can be demonstrated that the proposed algorithm can maintain its stability with the dissatisfaction of the Courant–Friedrichs–Lewy condition.

Perfectly Matched Layer for Accurate FDTD for Anisotropic Magnetized Plasma

In this work, we propose a stable perfectly matched layer (PML) for accurate finite-difference time-domain (FDTD) methods for analyzing electromagnetic wave propagation in the anisotropic magnetized plasma region. Toward this purpose, we apply the complex frequency-shifted PML systematically to the E-J collocated FDTD method. In specific, auxiliary PML variables are included in the matrix calculation involved in the final update equations of the E-J collocated FDTD method. Numerical examples are used to validate the proposed PML-FDTD for anisotropic magnetized plasma.

Computationally efficient complex envelope approximate Crank–Nicolson scheme and its open region problem for anisotropic gyrotropic plasma

By incorporating a complex envelope (CE) method, higher order formulation, and approximate Crank–Nicolson (CN) procedure, unconditionally stable complex frequency shifted perfectly matched layer (CFS-PML) implementation is proposed for anisotropic gyrotropic plasma bandpass simulation in open region problems. More precisely, the CE based higher order CN approximate-factorization-splitting (AFS) PML implementation is introduced to terminate the unbounded finite-difference time-domain (FDTD) lattice. The proposed implementation can not only improve computational efficiency but also enhance the absorption at boundaries during the whole bandpass simulation. Numerical examples which include plasma slab models and ridge waveguide structures are introduced to further demonstrate accuracy, absorption, and efficiency. It can be concluded in the results that our proposal can achieve considerable performance in modeling bandpass signals, improving computational efficiency, enhancing absorption, and maintaining remarkable accuracy. Meanwhile, it can be observed that the proposed implementation is stable when the time step surpasses far beyond the Courant–Friedrichs–Lewy condition.

An interpolating scaling functions method with low-storage five-stage fourth-order explicit Runge-Kutta schemes for 3D ground penetrating radar simulation

Accurate and efficient 3D simulation is extremely important for both grasping the features of radar wave propagation in realistic models and the high-resolution inversion. In this paper, we propose a new numerical algorithm called the low-storage five-stage fourth-order explicit Runge-Kutta interpolation scaling function method (LSERK-ISFM) and apply it into the 3D simulation of ground penetrating radar (GPR). Firstly, we derive the auxiliary differential equation (ADE) of the 3D Maxwell's equations based on the complex-frequency shifted perfectly matched layer (CFS-PML), discretize the spatial derivative of the Maxwell equation by ISFM, and integrate the temporal dimension through the LSERK scheme. Compared with the traditional finite-difference time-domain (FDTD) method, the LSERK-ISFM algorithm can effectively improve the precision of space-time dispersion of the Maxwell's equations. Specifically, by the homogeneous vacuum medium model, we compare the forward modeling results of the different number of grids per wavelength, the comparison results prove that a smaller number of grids per wavelength can be selected in the LSERK-ISFM scheme, which can greatly contribute to reducing the consumption of memory. Then we build a simple 3D model to properly compare the proposed algorithm with the coarse-mesh FDTD and the fine-mesh FDTD, the results show that, with the same mesh generation, the LSERK-ISFM algorithm outperforms the conventional FDTD algorithms in terms of both accuracy and efficiency. Finally, we test the algorithm on a 3D complex model by a series of 3D forward profiles and the corresponding slices, both simulated reflections and diffractions clearly depict the shape and location of the anomalous bodies, which shows great potential of application in the field of 3D exploration and interpretation for GPR community.

Higher‐order perfectly matched layer for the implicit CNDG‐FDTD algorithm

AbstractOn the basis of complex frequency‐shifted perfectly matched layer (CFS‐PML) formulation, an implementation of the higher‐order PML is proposed to terminate unbounded finite‐difference time domain (FDTD) computational domain. By incorporating the Crank‐Nicolson Douglas‐Gunn algorithm and the bilinear transform method, the proposed scheme can not only maintain the unconditional stability of the CN‐FDTD algorithm in terms of reducing computational time but also take advantage of the higher‐order PML in terms of improving absorbing performance. Numerical examples are provided to demonstrate the performance of the proposal in the homogenous free space and half‐space soil vacuum problems, respectively. It is demonstrated that the proposed unconditionally stable higher‐order CFS‐PML can not only efficiently absorb low‐frequency propagation waves, low‐frequency evanescent waves, and late‐time reflections but also overcome Courant‐Friedrich‐Levy limit.

Frequency-Domain Finite-Difference Elastic Wave Modeling in the Presence of Surface Topography

Seismic numerical modeling in the presence of surface topography has become a valuable tool to characterize seismic wave propagation in basin or mountain areas. Regarding advantages of frequency-domain seismic wavefield simulations (e.g., easy implementation of multiple sources and straightforward extension of adding attenuation factors), we propose a frequency-domain finite-difference seismic wavefield simulation in 2D elastic media with an irregular free surface. In the frequency domain, we first transform second-order elastic wave equations and first-order free surface boundary conditions from the Cartesian coordinate system to the curvilinear coordinate system. Then we apply complex frequency-shifted perfectly matched layer (CFS-PML) absorbing boundary condition to second-order elastic wave equations in the curvilinear coordinate. To better couple free surface boundary conditions and CFS-PML absorbing boundary condition, we also apply the complex coordinate stretching method used in CFS-PML to free surface boundary conditions in the curvilinear coordinate. In the first numerical test, the comparison of the seismograms calculated by our algorithm with an analytical solution indicates that our algorithm can accurately simulate seismic wavefield in the frequency domain. Finally, we choose three more elastic models with different types of surface topographies to further characterize seismic wave propagation.

Direct-Splitting-Based CN-FDTD for Modeling 2D Material Nanostructure Problems

Incorporating a truncation of the complex-frequency-shifted perfectly matched layer (CFS-PML), the direct-splitting- based Crank-Nicolson finite-difference time-domain (CNDS-FDTD) is developed and applied to the infrared two-dimensional layered material (2DLM) black phosphorous (BP) metasurface implementations on the all-dielectric nanostructure. To improve extremely low efficiencies in solving infrared terahertz (THz) problems with the few-atomic-layer thickness of 2DLMs, the CFS-CNDS-FDTD is proposed in demand due to the fact that it possesses capabilities of implicit FDTD method and unsplit-field CFS-PML truncation, respectively, in completely conquering the Courant-Friedrich-Levy condition (CFL) limit and holding good performance. The temporal incremental in the CFS-CNDS-FDTD can reach 1000 times larger than that in the regular FDTD for infrared nanoscale problems centered at the 2.5 THz and then keep accurate. Three-dimensional (3D) numerical cases have been carried out to corroborate the proposed method. The CFS-CNDS- FDTD can not only achieve high accuracies and then saves several dozen times of CPU time as compared to the regular FDTD, but also pave the way for designing all-dielectric nanostructures with other 2DLM metasurfaces.

PML and CFS-PML boundary conditions for a mesh-free finite difference solution of the elastic wave equation

Mesh-free finite difference (FD) methods can improve the geometric flexibility of modeling without the need for lattice mapping or complex meshing process. Radial-basis-function-generated FD is among the most commonly used mesh-free FD methods and can accurately simulate seismic wave propagation in the non-rectangular computational domain. In this paper, we propose a perfectly matched layer (PML) boundary condition for a mesh-free FD solution of the elastic wave equation, which can be applied to the boundaries of the non-rectangular velocity model. The performance of the PML is, however, severely reduced for near-grazing incident waves and low-frequency waves. We thus also propose the complex-frequency-shifted PML (CFS-PML) boundary condition for a mesh-free FD solution of the elastic wave equation. For two PML boundary conditions, we derive unsplit time-domain expressions by constructing auxiliary differential equations, both of which require less memory and are easy for programming. Numerical experiments demonstrate that these two PML boundary conditions effectively eliminate artificial boundary reflections in mesh-free FD simulations. When compared with the PML boundary condition, the CFS-PML boundary condition results in better absorption for near-grazing incident waves and evanescent waves.

Implementation of the Crank‐Nicolson Douglas‐Gunn finite‐difference time domain with complex frequency‐shifted perfectly matched layer for modeling unbounded isotropic dispersive media in two dimensions

AbstractTo model open‐domain problems with Drude, Lorentz, and Debye media, the complex frequency‐shifted perfectly matched layer (CFS‐PML) is adopted to truncate the Crank‐Nicolson Douglas‐Gunn finite‐difference time‐domain (CNDG‐FDTD) region. The auxiliary differential equation (ADE) and the bilinear Z‐transform methods are incorporated separately into the implementations of CNDG‐CFS‐PML formulations, while the ADE, piecewise linear recursive convolution (RC), and trapezoidal RC methods are utilized to model dispersive media. The proposed formulations can not only circumvent the stability condition, but also have the advantages of the CFS‐PML in attenuating the evanescent waves and reducing the late‐time reflection. Three numerical examples have been carried out to validate these formulations. The simulation results show that the proposed CNDG‐CFS‐PML algorithm is efficient in absorbing performance and saving more computational time compared with the conventional FDTD method, which leads to extensive applicability and acts as a very good prospect.