Stretched Coordinate Perfectly Matched Layer Research Articles

A property-dependent Perfectly Matched Layer with a single additional layer for Maxwell’s equations in finite difference frequency domains

A Perfectly Matched Layer (PML) has become a widespread technique for preventing reflections from computational boundaries for electromagnetic wave propagation problems in frequency domains. Especially, a PML for the Helmholtz equation is well developed and applied to the finite difference method. In the currently used methods of PMLs, additional 4 to 6 layers surrounding a computational domain are required to be added in a PML region in order to terminate infinite domain calculations. Such a requirement not only demands a relatively large computer memory, but also increases execution time and complexity in implementations.The objectives of this paper are to show how to derive and implement a highly efficient PML on the termination of a computational grid in electromagnetic scattering simulations by one property-dependent PML (PPML) with just a single additional layer, based on the stretched-coordinate PML (SC-PML) and unsplit-field PML formulations. Its effectiveness is validated by examples with varying background media and applied frequencies in computational domains. These validate our proposed PPML to be a universal and accurate tool for solving frequency-domain Maxwell’s equations discretized by finite differences accurately in both lossless and lossy media under a wide range of applied frequencies. On one hand, PPML effectively reduces requirements of CPU time and computer memory by an additional single layer in the PPML region. On the other hand, the proposed technique simply calculates the property-dependent parameters of the PPML to obtain high computational accuracy over the PMLs which are used currently. In addition, with this novel implementation, the PPML can be easily specified in one or more coordinate systems in the same manner as the standard Finite-Difference-Frequency-Domain (FDFD) equations without any increase in their derivation work.

Efficient Implementations of SC-PML for Arbitrary Media Using DSP Techniques

Through the digital signal processing (DSP) techniques, two simple implementations of the stretched coordinate perfectly matched layer (SC-PML) are proposed in this letter. Specially, they are based on the bilinear Z-transform and matched Z-transform methods, respectively. In addition, incorporating the concept of generalized material independent PML into these formulations enable them to be completely independent of material properties. Thus, these new implementations can be used to terminate the finite-difference time-domain lattices composed of arbitrary media. As for resource utilization, the proposals can guarantee that only one auxiliary variable is needed for each field component in the whole PML regions. A brief comparison of the number of floating-point operations and constant coefficients successfully shows that they can achieve a bit higher efficiency compared with other published efficient SC-PML implementations. Finally, two numerical examples of electromagnetic waves radiating in vacuum and soil half-space are provided to validate these DSP-based algorithms.

A 3-D Stochastic FDTD Algorithm for Wave Propagation in Isotropic Cold Plasma Medium Based on Bilinear Transform

In this paper, a 3-D stochastic finite-difference time-domain algorithm is developed for simulating wave propagation in isotropic cold plasma medium by using the bilinear transform (BT) technique. The proposed algorithm has a simple formulation and is very easy to apply to isotropic dispersive material. It efficiently calculates the mean value and standard deviation of electromagnetic field components caused by the variability or uncertainty of the plasma's electron density and collision frequency. The stretched coordinate perfectly matched layer (SC-PML) with a complex-frequency-shifted factor is implemented as the absorbing boundary condition, and the BT technique is also incorporated into the SC-PML. Numerical examples are shown to demonstrate the accuracy and effectiveness of the proposed method.

Application of the Finite-Difference Frequency-Domain (FDFD) method on radiowave propagation in urban environments

In this work a Finite-Difference Frequency-Domain (FDFD) propagation method for complex urban environments is proposed. The formulation starts from the discretization of the Helmholtz equation for the magnetic field instead of the usual separate one order derivative Ampere's and Faraday's laws. The Stretched Coordinate Perfectly Matched Layer (SCPML) is used as an absorbing boundary condition. These procedures produce less field components to estimate and achieve high wave absorption at the computation domain boundaries. The main goal is the rigorous prediction of VHF/SHF signals in real urban scenarios through the evaluation of several propagation mechanisms: direct rays, diffraction, reflection and refraction effects. The method is validated through an analytic problem and preliminary results are generated by two case studies: a cellular system measurement campaign and an idealized urban scenario.

Simple PML implementation for finite‐difference time‐domain method

Via the auxiliary differential equation method, the authors propose a simple and efficient implementation of the stretched coordinate perfectly matched layer (SC-PML) in this Letter. The proposal can successfully halve the number of auxiliary variables associated with the PML while maintaining the capacity of the original SC-PML in terms of simple implementation. Thus, it can achieve a higher computational efficiency in the finite-difference time-domain simulations. The effectiveness of this new implementation is verified through an example carried out in a 3D Debye dispersive space.

Efficient PML implementation based on the unconditionally stable CN-FDTD algorithm for anisotropic magnetized plasma

Based on the bilinear transform (BT) method and the Crank-Nicolson-approximate-decoupling (CNAD) algorithm, an efficient unconditionally stable implementation of the stretched coordinate perfectly matched layer (SC-PML) is proposed to truncate anisotropic magnetized plasma. The dispersive magnetized plasma can be calculated by the modified auxiliary differential equation (ADE) method. The proposed SC-PML formulations not only have the advantage of the conventional PML in terms of the absorbing performances but also take the advantage of unconditional stability of the original CN algorithm. In this paper, numerical examples are provided in the two-dimensional domain with anisotropic magnetized plasma to validate the effectiveness of the algorithm. The results show that the proposed SC-PML is an unconditionally stable scheme for the time step which surpasses the Courant–Friedrichs–Lewy (CFL) condition and it has considerable absorbing performance.

Z ‐transform for unconditional stable Crank‐Nicolson FDTD implementation of SC‐PML for dispersive Debye media

A novel and unsplit-field implementation of the stretched coordinate perfectly matched layer (SC-PML) is proposed for truncating the one-dimensional (1D) frequency dispersive first-order Debye medium. These formulations are based on the unconditional stable Crank-Nicolson scheme and the Z-transform method. Numerical results validate the effectiveness and performance of this new PML.

Implementation of Perfectly Matched Layer (PML) for the WCS-FDTD Method Using DSP Techniques

An efficient implementation of perfectly matched layer (PML), based on the stretched coordinate PML (SC-PML) and digital signal processing (DSP) techniques, is proposed to truncate the weakly conditional stable finite-difference time-domain (WCS-FDTD) lattices. The updating PML equations are further simplified with the introduction of auxiliary quantities. Compared to the convolutional PML (CPML), the proposed PML achieves better absorbing performance at the same computational expenditure. With its implementation, energy leakage from small cross-slots of a rectangular waveguide coupler is predicted accurately, which agrees very well with that of the other methods.

Efficient DSP-Higher-Order PML Formulations for the Metal Plate Buried in Dispersive Soil Half-Space Problem

An efficient and unsplit-field implementation of the higher order perfectly matched layer (PML) based on the stretched coordinate PML (SC-PML) formulations and digital signal processing (DSP) techniques is proposed to truncate the finite-difference time-domain (FDTD) lattices. The higher order PML has the advantages of both the conventional PML and the complex frequency-shifted PML (CFS-PML) in terms of absorbing performances. A 3-D FDTD simulation of the metal plate buried in dispersive soil half-space problem has been carried out to validate these formulations. It is shown that the proposed PML formulations with the higher order scheme are efficient in terms of attenuating both the low-frequency propagating waves and evanescent waves and reducing late-time reflections.

Novel and efficient FDTD implementation of higher-order perfectly matched layer based on ADE method

Based on the stretched coordinate perfectly matched layer (SC-PML) formulations and the auxiliary differential equation (ADE) method, an efficient and unsplit-field implementation of the higher-order PML scheme with more than one pole is proposed to truncate the finite-difference time-domain (FDTD) lattices. The higher-order PML has the advantages of both the conventional PML and the complex frequency shifted PML (CFS-PML) in terms of absorbing performances. The proposed algorithm is validated through two numerical tests carried out in three dimensional and two dimensional domains. It is shown in the numerical simulations that the proposed PML formulations with the higher-order scheme are efficient in terms of attenuating both the low-frequency propagating waves and evanescent waves and reducing late-time reflections, and also hold much better absorbing performances than the conventional SC-PML and the convolutional PML (CPML) with the CFS scheme.

Choice of the perfectly matched layer boundary condition for frequency-domain Maxwell’s equations solvers

We show that the performance of frequency-domain solvers of Maxwell’s equations is greatly affected by the kind of the perfectly matched layer (PML) used. In particular, we demonstrate that using the stretched-coordinate PML (SC-PML) results in significantly faster convergence speed than using the uniaxial PML (UPML). Such a difference in convergence behavior is explained by an analysis of the condition number of the coefficient matrices. Additionally, we develop a diagonal preconditioning scheme that significantly improves solver performance when UPML is used.

An absorbing boundary condition for general dispersive medium and general FDTD spatial scheme

Based on the stretched coordinate perfectly matched layer (SC-PML) formulation and the auxiliary differential equation method, an absorbing boundary condition for general dispersive medium is presented, and applied to both the standard finite difference time domain (FDTD) method and the high-order FDTD method. The proposed D-H formulations are completely independent of the material properties of the FDTD computational domain. Thus they can be directly applied to the simulations involving arbitrary dielectrics. Numerical results show that compared with the CPML, the proposed method is versatile, has an improved absorbing performance and low computational complexity, and can substantially reduce the computational time.

General ADE formulations of SC-PML for modelling multi-term dispersive FDTD applications

General and efficient auxiliary differential equation (ADE) formulations of the stretched-coordinate perfectly matched layer (SC-PML) absorbing boundary conditions are presented for truncating open region multi-term dispersive finite-difference time-domain (FDTD) applications. In the proposed formulations, appropriate digital-filtering techniques are used for efficient implementations of the constitutive relations of the dispersive materials in the discrete-time domain. The validity of the new formulations has been demonstrated by computing wave propagation in a two-term Lorentz dispersive domain.

Simple and Efficient PML Algorithm for Truncating Nonlinear FDTD Domains Based on the Z-Transform Method

Based on the Z-transform method, a simple, efficient and unsplit-field implementation of the Stretched Coordinate Perfectly Matched Layer (SC-PML) is proposed for truncating the nonlinear dispersive Finite Difference Time Domain (FDTD) lattices. In addition, the nonlinear FDTD formulations using the Z-transform method are reformulated with the advantage of a simple derivative process. The proposed algorithm is validated through two numerical examples carried out in one dimensional and two dimensional domains which include Lorentz dispersion as well as Kerr and Raman nonlinearities.

Efficient parallel PML algorithms for truncating finite difference time domain simulations

In this paper, the parallel implementation of the stretched coordinate perfectly matched layer (SC-PML) and the wave equation PML (WE-PML) formulations is presented for truncating three-dimensional (3-D) finite difference time domain (FDTD) grids. In the proposed parallel algorithms, the FDTD computational domain is divided into contiguous non-overlapping subdomains using two-dimensional topology and the interprocessor communications between the neighboring subdomains are carried out by using the message passing interface (MPI) system. The performance of the proposed parallel algorithms has been studied by using a point source radiating in 3-D domains and performed on a network of PCs interconnected with Ethernet. It has been observed that the WE-PML parallel algorithm is approximately 2.3 faster than the SC-PML parallel algorithm.

An Efficient Implementation of the Stretched Coordinate Perfectly Matched Layer

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> An unsplit-field and efficient algorithm to implement the stretched coordinate perfectly matched layer (SC–PML) is proposed for truncating the finite-difference time-domain lattices. The main advantage of the proposed algorithm is that only one auxiliary variable is required in all corners and some edges of the PML regions by means of direct discretizing the second-order differential equation and a memory-minimized algorithm. Two numerical tests have been provided to validate the proposed algorithm. </para>

Efficient implementation of the stretched co-ordinate perfectly matched layer based on the Z-transform method

A novel, efficient and unsplit-field algorithm for implementing the stretched co-ordinate perfectly matched layer (SC-PML) based on the Z-transform method is discussed in detail for truncating the finite-difference time-domain lattices. The advantage of the proposed algorithm is that fewer auxiliary variables are required in the PML regions and thus the memory requirement is less than that of other implementations of the SC-PML. Four numerical tests have been carried out to validate the proposed algorithm. It is demonstrated in the numerical tests that the proposed algorithm provides good absorbing performance, requires less computational time and is stable.

Efficient implementation of stretched co-ordinate perfectly matched layer using DSP techniques

An efficient and unsplit-field algorithm for implementing the stretched co-ordinate perfectly matched layer (SC-PML) based on digital signal processing (DSP) techniques is presented for truncating the finite-difference time-domain (FDTD) lattices. The advantage of the proposed algorithm is that only one additional auxiliary variable is required in all corners and at all edges of the PML regions by means of a new memory-minimised algorithm for a high-order difference equation and thus memory requirement is saved significantly compared with that in published papers about the SC-PML. These formulations are independent of the material properties of the FDTD computational domain.

Perfectly matched layer implementation using bilinear transform for microwave device applications

This paper presents an extensive study on the perfectly matched layer (PML) implementation using bilinear transform in the finite-difference time-domain (FDTD) simulation. The bilinear transform is used to implement both the stretched coordinate PML (SC-PML) and the uniaxial PML (UPML) with the complex frequency-shifted (CFS) equations. It is shown that with the CFS factor, the implemented SC-PML and UPML attain significantly lower relative reflection error over wide frequency range with both superior in performance to the split-field PML. The FDTD algorithm incorporating these PMLs is applied to analyze wide-band responses of some complex microwave devices, including RF microelectromechanical systems switch and coplanar waveguide crossover junction.