Convolutional Perfectly Matched Layer Research Articles

Efficient Parallel FDTD Method Based on Non-Uniform Conformal Mesh

The finite-difference time-domain (FDTD) method is a versatile electromagnetic simulation technique, widely used for solving various broadband problems. However, when dealing with complex structures and large dimensions, especially when applying perfectly matched layer (PML) absorbing boundaries, tremendous computational burdens will occur. To reduce the computational time and memory, this paper presents a Message Passing Interface (MPI) parallel scheme based on non-uniform conformal FDTD, which is suitable for convolutional perfectly matched layer (CPML) absorbing boundaries, and adopts a domain decomposition approach, dividing the entire computational domain into several subdomains. More importantly, only one magnetic field exchange is required during the iterations, and the electric field update is divided into internal and external parts, facilitating the synchronous communication of magnetic fields between adjacent subdomains and internal electric field updates. Finally, unmanned helicopters, helical antennas, 100-period folded waveguides, and 16 × 16 phased array antennas are designed to verify the accuracy and efficiency of the algorithm. Moreover, we conducted parallel tests on a supercomputing platform, showing its satisfactory reduction in computational time and excellent parallel efficiency.

A time-domain SGFD-FK hybrid method for 2D teleseismic elastic wave modeling and inversion

AbstractFull waveform inversion (FWI) has proved to be a reliable tool for high-resolution imaging of lithospheric structures at various depths down to the upper mantle. However, when the size of the model is large, the computational burden is significative and applications are restricted to low frequencies. To tackle this issue, we developed a new 2D time-domain hybrid method to simulate high-frequency teleseismic body waves propagating through a local heterogeneous elastic Earth model: The frequency-wavenumber (FK) integration method is coupled with the staggered grid finite difference method (SGFD). The FK method is used to compute the wavefield due to obliquely incident plane P and SV waves in a 1D multilayered half space that excites a local heterogeneous region. Inside this region, the velocity-stress staggered grid FD method (SGFD) is used to accurately deal with wave propagation in heterogeneous media. Spurious waves that might be generated at the boundaries of the local region are avoided using convolutional perfectly matched layers (CPML). This new hybrid method inherits the low-memory requirements of the FK method and the accuracy, efficiency and easy implementation of the SGFD. The new hybrid method is benchmarked against the analytical FK method for some canonical models and shows good agreement with analytical solutions. Subsequently, our modeling tool is incorporated into a full waveform inversion algorithm adapted for teleseismic configurations to invert the incident P wave and its coda. The inversion is carried out using a gradient approach that is efficiently implemented via the adjoint-state method. The results suggest that our hybrid method and FWI algorithm represent a valuable tool for 2D forward and inverse regional applications using teleseismic data sets.

Numerical Simulation of Elastic Wave in Frequency Domain Based on Generalized Finite Difference Method With a Multiaxial Convolutional Perfectly Matched Layer Boundary

Numerical Simulation of Elastic Wave in Frequency Domain Based on Generalized Finite Difference Method With a Multiaxial Convolutional Perfectly Matched Layer Boundary

Scattering and Attenuation in 5G Electromagnetic Propagation (5 GHz and 25 GHz) in the Presence of Rainfall: A Numerical Study

Rainfall has always been a concern for wireless communications systems. As 5G technology relies on high-frequency bands, it is fundamental to model and simulate the interaction of such radio waves with rainfall, as the deployment of large-scale infrastructure for 5G is highly expensive. This research presents a reformulation of the Maxwell equations for a bi-dimensional space in a transverse electric propagation mode, for a linear, inhomogeneous, and isotropic propagation medium with its magnetic and electric properties dependent on time. This reformulation was solved using the Finite Differences in Time Domain (FDTD) method with the Convolutional Perfectly Matched Layer (CPML) boundary condition. Two main frequency propagation scenarios were studied: 5 GHz (corresponding to Wi-Fi in the 802.11n standard as well as to the lowest bands of 5G) and 25 GHz (corresponding to 5G), within a 10m×3m rectangular domain in air and with rain. The rainfall was simulated using a parallel Ziggurat algorithm. According to the findings, while 5 GHz waves experience scattering processes, 25 GHz waves experience substantial dispersion and attenuation throughout the domain in low- to moderate-intensity rain.

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).

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.

The HIE-FDTD Method for Simulating Dispersion Media Represented by Drude, Debye, and Lorentz Models

The hybrid implicit-explicit finite-difference time-domain (HIE-FDTD) method is a weakly conditionally stable finite-difference time-domain (FDTD) method that has attracted much attention in recent years. However due to the dispersion media such as water, soil, plasma, biological tissue, optical materials, etc., the application of the HIE-FDTD method is still relatively limited. Therefore, in this paper, the HIE-FDTD method was extended to solve typical dispersion media by combining the Drude, Debye, and Lorentz models with hybrid implicit-explicit difference techniques. The advantage of the presented method is that it only needs to solve a set of equations, and then different dispersion media including water, soil, plasma, biological tissue, and optical materials can be analyzed. The convolutional perfectly matched layer (CPML) boundary condition was introduced to truncate the computational domain. Numerical examples were used to validate the absorbing performance of the CPML boundary and prove the accuracy and computational efficiency of the dispersion HIE-FDTD method proposed in this paper. The simulated results showed that the dispersion HIE-FDTD method could not only obtain accurate calculation results, but also had a much higher computational efficiency than the finite-difference time-domain (FDTD) method.

Higher Order CPML for Leapfrog Complying-Divergence Implicit FDTD Method and Its Numerical Properties

The leapfrog complying-divergence implicit finite-difference time-domain (CDI-FDTD) method is an implicit scheme that is unconditionally stable involving the simplest and most efficient leapfrog update procedures with minimum floating-point operations (flops). Moreover, the method possesses complying divergence satisfying Gauss’s law, while other methods, including the leapfrog alternating direction implicit (ADI)-FDTD counterpart, do not. These desirable features make the leapfrog CDI-FDTD useful for electromagnetic simulations when the unconditional stability is to be exploited. In this article, a higher order convolutional perfectly matched layer (CPML) for leapfrog CDI-FDTD is developed and implemented for simulating open space problems. To formulate the CPML for leapfrog CDI-FDTD, we have carefully considered the order of CPML update along with proper use of main and auxiliary fields to retain the unconditional stability and complying divergence. The proposed CPML for leapfrog CDI-FDTD remains stable beyond the Courant time step as ascertained through von Neumann analysis. By comparing the flops and computation time, the leapfrog CDI-FDTD is shown to be most efficient among common implicit FDTD methods with CPML. The formulation of numerical divergence for higher order CPML is also presented. Simulation results show that the proposed higher order CPML has many advantages of simplicity, complying divergence, and high absorption efficiency.

Decoupled acoustoelasticity equations and stable FD modeling for wave propagation in prestressed media

Seismic exploration of deep oil/gas reservoirs involves wave propagation in high-pressure media. Acoustoelasticity with the third-order elastic constants can account for nonlinear strain responses due to stresses of finite magnitude. Since current seismic exploration depends primarily on P waves, further development in prospecting technology to handle deep reflection data will benefit from the ability to decouple the elastic waves from high-pressure deep formations. We decouple the first-order velocity-stress acoustoelasticity equations for wave propagation under isotropic (confining) and anisotropic (uniaxial and pure shear) prestress conditions. A rotated staggered-grid finite-difference (RSG-FD) method is used to solve the decoupled acoustoelasticity equations with an unsplit convolutional perfectly matched layer (CPML) absorbing boundary. Numerical examples demonstrate the significant impact of prestressed conditions on seismic responses in both phase and amplitude. The stress-induced seismic anisotropy with orthotropic characteristics is strongly related to the orientation of prestresses. We compare the coupled and the decoupled wavefields to validate the proposed decoupling method. Investigation of the influence of prestresses on the acoustoelastic decoupling illustrates that the isotropic prestress does not affect the decoupling, but the anisotropic prestresses significantly affect the decoupling accuracy.

The Convolutional Perfectly Matched Layer for an Efficient 3-D WLP-FDTD Method

In this letter, the convolution perfectly matched layer (CPML) is modified and implemented for an efficient 3-D finite-difference time-domain method with weighted Laguerre polynomials (WLP-FDTD). The formulations are derived from the CPML equations for conventional WLP-FDTD. A factorization-splitting scheme is used to achieve the high efficiency. Numerical results are given to verify the absorbing performance and efficiency of the proposed CPML.

A Performance-Enhanced Absorbing Boundary Condition for FDTD Methods on Face-Centered Cubic Grids

Based on the auxiliary differential equation (ADE), a performance-enhanced convolution perfectly matched layer (CPML) for the face-centered cubic finite-difference time-domain (FCC-FDTD) method is developed in this letter. Unlike the traditional FDTD methods on Yee’s grids, for which the CPML technique is relatively straightforward to be implemented, poor performance, and even instability may occur in the FCC-FDTD method. To avoid these problems, the ADE technique is incorporated into the derivation, and the impact of node locations on absorbing boundary constitutive parameters is carefully considered. In addition, the derivation of the time-marching formulations is substantially simplified. The optimal ranges of constitutive parameters are found through a numerical sweep. Two numerical examples, the electromagnetic pulse propagation and absorbed power density of a brain model during mobile communication, are examined to validate the performance. The results suggest that the proposed ADE-CPML for the FCC-FDTD method shows good stability and accuracy.

Narrow-Bandpass One-Step Leapfrog Hybrid Implicit-Explicit Algorithm with Convolutional Boundary Condition for Its Applications in Sensors.

A large number of sensors work in the narrow bandpass circumstance. Meanwhile, some of them hold fine details merely along one and two dimensions. In order to efficiently simulate these sensors and devices, the one-step leapfrog hybrid implicit-explicit (HIE) algorithm with the complex envelope (CE) method and absorbing boundary condition is proposed in the narrow bandpass circumstance. To be more precise, absorbing boundary condition is implemented by the higher order convolutional perfectly matched layer (CPML) formulation to further enhance the absorption during the entire simulation. Numerical examples and their experiments are carried out to further illustrate the effectiveness of the proposed algorithm. The results show considerable agreement with the experiment and theory resolution. The relationship between the time step and mesh size can break the Courant–Friedrichs–Levy condition which indicates the physical size/selection mesh size. Such a condition indicates that the proposed algorithm behaviors are considerably accurate due to the rational choice in discretized mesh. It also shows decrement in simulation duration and memory consumption compared with the other algorithms. In addition, absorption performance can be improved by employing the proposed higher order CPML algorithm during the whole simulation.

Implementation of the convolutional perfectly matched layer (CPML) in the face-centered cubic (FCC) FDTD method

The implementation of convolutional perfectly matched layer (CPML) is presented and validated on the face-centered cubic (FCC) grids. The standard Cartesian (Yee) grid can be competently substituted by the FCC grid. Although the FCC grid has more lattice points when compared with the Yee grid, yet it shows better grid isotropy and reveals stability criterion which is much more relaxed. In this work, the iterative equations of CPML in the FCC grid are formulated, for using it as an absorbing boundary condition to truncate the computational domain. Numerical results are provided to prove the effectiveness of CPML.

Improved convolutional perfectly matched layer with auxiliary differential equation for the weakly conditionally stable finite‐difference time‐domain method

On basis of the auxiliary differential equation, an improved convolutional perfectly matched layer (CPML) scheme has been proposed and applied to the three-dimensional weakly conditionally stable (WCS) finite-difference time-domain (FDTD) method. With this method, the maximum error is reduced effectively, and the absorption effect is more obvious. The simulation results show the proposed improvement scheme with more calculation accuracy than initial CPML under various Courant-Friedrich-Levy number (CFLN) conditions while ensuring similar calculation time and memory cost. Moreover, when CFLN is increased to eight, the calculation time is <30% of the traditional method, which proved the proposed method with more calculation efficiency.

Compact System-Combined-Based FDTD Implementations With Approximate Crank–Nicolson Scheme and Electromagnetic Scattering

In this article, we develop the efficacious system-combined-based approximate Crank–Nicolson finite-difference time-domain (SC-ACN-FDTD) method which can be applied to electromagnetic scattering. After being processed by the matrix form with the discrete time, the electric and magnetic fields from Maxwell’s equations are both unified as the unique vector at the same time step. By means of the vector transformation, iterative procedures in the time domain of both electric and magnetic fields can be obtained in the Crank–Nicolson format. To model an open-domain problem, we apply convolutional perfectly matched layer (CPML) to truncate the finite computational region. To guarantee the numerical accuracy of the SC-ACN-FDTD method, we employ the empirical formula to forecast the maximum Courant–Friedrichs–Lewy (CFL) factor based on the spatial sampling density. Compared with the conventional finite-difference time-domain (FDTD) methods, the proposed SC-ACN-FDTD method can be well demonstrated and further validated by adopting electromagnetic scattering in the near and far fields. To further validate our results, the commercial software COMSOL is also adopted for the far field’s pattern in the single frequency.

Trapezoid-Grid Finite-Difference Time-Domain Method for 3D Seismic Wavefield Modeling Using CPML Absorbing Boundary Condition

The large computational memory requirement is an important issue in 3D large-scale wave modeling, especially for GPU calculation. Based on the observation that wave propagation velocity tends to gradually increase with depth, we propose a 3D trapezoid-grid finite-difference time-domain (FDTD) method to achieve the reduction of memory usage without a significant increase of computational time or a decrease of modeling accuracy. It adopts the size-increasing trapezoid-grid mesh to fit the increasing trend of seismic wave velocity in depth, which can significantly reduce the oversampling in the high-velocity region. The trapezoid coordinate transformation is used to alleviate the difficulty of processing ununiform grids. We derive the 3D acoustic equation in the new trapezoid coordinate system and adopt the corresponding trapezoid-grid convolutional perfectly matched layer (CPML) absorbing boundary condition to eliminate the artificial boundary reflection. Stability analysis is given to generate stable modeling results. Numerical tests on the 3D homogenous model verify the effectiveness of our method and the trapezoid-grid CPML absorbing boundary condition, while numerical tests on the SEG/EAGE overthrust model indicate that for comparable computational time and accuracy, our method can achieve about 50% reduction on memory usage compared with those on the uniform-grid FDTD method.

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.

Universal Vector–Scalar Potential Framework for Inhomogeneous Electromagnetic System and Its Application in Semiclassical Quantum Electromagnetics

In this work, numerical solution to a general electromagnetic (EM) system is studied using a formalism based on the formulas for the E–B–A– <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\phi $ </tex-math></inline-formula> formulas with different gauge conditions. The finite-difference time-domain (FDTD) method is employed to discretize these formulas. In addition, the convolutional perfectly matched layer (CPML) technique is successfully applied to absorb outgoing scattered waves described by the proposed formulas. The gauge invariance of EM fields in inhomogeneous environment is demonstrated by numerical examples. Moreover, the proposed EM framework integrated with the Schrödinger equation is introduced to investigate the mesoscopic phenomenon for light–matter interaction, which is useful to design laser pulses for controlling discrete quantum states. The work offers a simple and general numerical EM framework, which is essential to bridge the classical EM and quantum mechanical systems.

A 3-D fourth-order one-step leapfrog HIE-FDTD method with CPML

ABSTRACT A 3-D fourth-order one-step leapfrog HIE-FDTD method with convolutional perfectly matched layer (CPML) has been proposed in this paper. When the time step size satisfies with the time stability condition, the maximum reflection error of the proposed method is below −79 dB, which demonstrates good absorbing performance of the CPML method. To verify the accuracy, we give the numerical dispersions analysis of the proposed method with the traditional FDTD method and the traditional HIE-FDTD method. The results demonstrate that when the time step size of the new method is the same with the other methods, the proposed method has the smallest dispersion error. Besides, numerical examples demonstrate that the proposed method spend much less CPU time than the traditional FDTD method and the traditional HIE-FDTD method. Last but not least, simulations for thin slots present that the proposed method is very useful to solve problems with coupled field.

FPGA Acceleration of 3D FDTD for Multi- Antennas Microwave Imaging Using HLS

Microwave Imaging (MI) for biomedical applications has attracted attention due to its harmless radiation compared to X-ray or MRI. One of the commonly used computing methods in MI is Finite Difference Time Domain (FDTD), which is executed several times in iterative loops, hence resulting in a high execution time. Although several hardware accelerators for FDTD have been recently introduced, they are not specifically designed for MI applications. In particular, only simple absorbing boundary conditions have been investigated, and the impact of dispersive materials on FDTD has not been considered. In this paper, we propose a multi-FPGA accelerator for 3D FDTD that is integrated in an MI algorithm, with Convolutional Perfectly Matched Layer (CPML) boundary conditions and an exact model for dispersive materials. By using High Level Synthesis (HLS), we obtain an optimized hardware accelerator that uses an efficient blocking method to reduce the data transfer time between external and local memories. We propose two alternative architectures that trade off performance and resource usage. In addition, our code, being developed at a high level, can also be run on GPUs whenever necessary. The results show that our multi-FPGA accelerator is superior to three similar GPU-based designs in terms of execution time and power consumption.