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

탄성파의 수치 모델링은 유한한 경계에서 발생하는 인공적인 반사파의 제거를 위한 경계조건을 필요로 한다. 이 연구에서는 주파수영역의 탄성파 수치 모델링에 CFS-PML (Complex Frequency Shifted-Perfectly Matched Layer) 경계조건을 적용하였다. 수치모델링 검증을 위해 Lamb's Problem의 해석해와 수치모델링 결과를 비교한 결과 일치하였다. 모형 내의 운동에너지, 최대크기오차, 그리고 스펙트럼오차를 통하여 CFS-PML경계조건이 기존의 흡수경계조건들 보다 유한경계에서 발생한 인공적인 반사파를 효과적으로 제거할 수 있음을 확인하였다. CFS-PML경계조건의 변수 <TEX>${\kappa}_{max}$</TEX>와 <TEX>${\alpha}_{max}$</TEX>의 최적값은 운동에너지를 이용하여 산정할 수 있었다. 또한, 주파수에 따른 함수로 정의된 <TEX>${\alpha}_{max}$</TEX>를 변수로 갖는 변형된 CFS-PML경계조건을 제안하여 기존 PML경계조건, CFS-PML경계조건, 그리고 변형된 CFS-PML경계조건의 성능을 운동에너지, 최대크기오차, 스펙트럼 오차로 비교하였다. 기존 PML경계조건에서 나타난 스쳐가는 입사각에 대한 반사파 문제가 CFS-PML경계조건, 그리고 변형된 CFS-PML경계조건에서는 개선되었다. Absorbing boundary conditions are used to mitigate undesired reflections that can arise at the model's truncation boundaries. We apply a complex frequency shifted perfectly matched layer (CFS-PML) to elastic wave modeling in the frequency domain. Modeling results show that the performance of our implementation is superior to other absorbing boundaries. We consider the coefficients of CFS-PML to be optimal when the kinetic energy becomes to the minimum, and propose the modified CFS-PML that has the CFS-PML coefficient <TEX>${\alpha}_{max}$</TEX> defined as a function of frequency. Results with CFS-PML and modified CFS-PML are significantly improved compared with those of the classical PML technique suffering from large spurious reflections at grazing incidence.

In the simulation of airborne transient electromagnetic method (ATEM) in time-domain, the truncated boundary reflection can bring a big error to the results. The complex frequency shifted perfectly matched layer (CFS-PML) absorbing boundary condition has been proved to have a better absorption of low frequency incident wave and can reduce the late reflection greatly. In this paper, we apply the CFS-PML to three-dimensional numerical simulation of ATEM in time-domain to achieve a high precision .The expression of divergence equation in CFS-PML is confirmed and its explicit iteration format based on the finite difference method and the recursive convolution technique is deduced. Finally, we use the uniformity half space model and the anomalous model to test the validity of this method. Results show that the CFS-PML can reduce the average relative error to 2.87% and increase the accuracy of the anomaly recognition.

Weighted Laguerre finite-difference time-domain (Laguerre-FDTD) method is an efficient and unconditionally stable time-domain numerical method. The complex frequency-shifted perfectly matched layer (CFS-PML) is the main absorbing boundary condition in the Laguerre-FDTD method. The implementations of CFS-PML in the Laguerre-FDTD method, which have been reported, include the use of auxiliary variables and auxiliary differential equations. These implementations make CFS-PML sensitive to its parameters and timescale parameter s of weighted Laguerre polynomials and result in potential instability and poor absorption issues in numerical simulations. In this paper, we proposed an efficient and stable implementation of CFS-PML based on recursive convolution in the Laguerre-FDTD method. The accuracy of the proposed implementation is theoretically validated. Its numerical dispersion is theoretically derived for choosing the key parameters of CFS-PML in the Laguerre-FDTD scheme. The numerical dispersion demonstrates that the timescale factor s needs to match the simulated signal frequency and the CFS-PML parameter $\alpha $ is chosen as $0.5s\varepsilon _{0}$ to obtain the minimum dispersive error. Numerical examples validate the theoretical predictions and verify that the proposed implementation of CFS-PML is robust and retains the advantages of CFS-PML in classical FDTD method, such as effectively absorbing evanescent as well as guided waves. It requires nonsplits of electromagnetic field components, no auxiliary variables, and no modifications when applying it to inhomogeneous, lossy, and dispersive media. The CFS-PML formulas can directly be converted into computer codes of the Laguerre-FDTD method.

A new numerical discretization of the complex frequency shifted‐perfectly matched layer (CFS‐PML) is presented for truncation of open domains. This discretization is based on the theory of discrete differential forms applied to electromagnetism, which guarantees both the simplification of the algorithms and a more elegant formulation of expression. The curl Maxwell's equations are solved in time domain in terms of electric e magnetic fields. The Ampere‐Maxwell law with, the constitutive relations included through the Hodge operator, is discretized in space by applying the Galerkin method. Conversely, the Faraday law is represented by topological relations via incidence matrix. The time domain discretization uses the leap‐frog scheme with a recursive convolution inside PML region. Two examples are shown, the first one to validate the CFS‐PML procedure and the second one to solve a typical ground penetrating radar scenario which include a lossy media. © 2012 Wiley Periodicals, Inc. Microwave Opt Technol Lett 55:351–357, 2012; View this article online at wileyonlinelibrary.com. DOI 10.1002/mop.27298

With the great growth continuing in computers' power that is associated with a big growth in the numerical algorithms for solving radiation problems, the finite difference time domain (FDTD) method has taken its chance to demonstrate its power in simplifying the sophisticated radiation problems. In this research, the FDTD method is used, with suitable absorbing boundary conditions (ABC) and feeding model, to solve the dielectric resonator antenna (DRA) problems both in two dimensions (2D) and three dimensions (3D) using different shapes of the antennas and different types of the feeding signals including transient signals. The resultant electric field strengths and radiation patterns are obtained for these cases.

This paper considers speed-up of a calculation of an absorbing boundary condition (ABC) of the finite-difference time-domain (FDTD) method based on a time-domain surface integral equation. The most popular ABC of the FDTD method, the perfectly matched layer (PML), is still required to be modified for its accuracy in cases of near field and large incident angle to ABC. The ABC based on the time-domain surface integral equation is free from such confinements of the ABC location and incident angle. On the other hand, it is known that extremely heavy computation cost is required for the time-domain surface integral. In this work, it is attempted to reduce the computation cost of the surface integral equation to use spherical harmonics expression for harmonic Green's function in the surface integral equation.

A modal absorbing boundary condition (ABC) based on the discrete Green's function (DGF) is introduced and applied for termination of waveguides simulated by means of the finite-difference time-domain (FDTD) method. The differences between the developed approach and implementations already demonstrated in the literature are presented. By applying DGF, a consistent theoretical approach to modal ABC in the FDTD method is obtained. Developed ABC is applicable to propagating and evanescent modes yielding the reflection error below -250 dB. Because the perfect matching may be computationally expensive, results of investigations of the tradeoff between computational efficiency and accuracy of the developed method are presented.

The absorbing boundary conditions (ABCs) are essential to terminate the computational space when the finite-difference time-domain (FDTD) method is applied to analyze electromagnetic (EM) problems. With the ABCs, the fields on the truncated boundaries are evaluated by the interior fields. In this paper, the relationship between the fields on the terminated boundaries and the interior fields is expressed as the transfer functions in the Z-domain. The proper transfer functions are determined from the radiation condition or the transmission condition. Simplifying these transfer functions into rational functions, we obtain different schemes of the ABCs. In this paper, both the transfer functions and coefficients of the final ABCs are derived and expressed as recurrence formulas for the convenience of programming. This method has the property of simplicity and flexibility. The ABCs obtained by this method show good absorbing performance and numerical stability in the practical applications.

The importance of dielectric losses in planar microstrip structures is evaluated with the finite-difference time-domain (FDTD) method. This analysis was previously not possible in many FDTD simulators due to a lack of absorbing boundary conditions (ABCs), which appropriately terminate air/dielectric interfaces for which the dielectric is lossy. The newly proposed lossy two-time derivative Lorentzian material (L2TDLM) model ABC allows for these terminations and is presented and implemented here for three-dimensional FDTD simulations. The effect of dielectric losses on several well-known planar microstrip structures is evaluated. It is shown that the inclusion of these losses in FDTD simulations, which is facilitated by the L2TDLM ABC, is, in fact, important to predict the performance of resonant structure on lossy dielectric substrates.

Both in the transmission line matrix (TLM) and the finite difference time domain (FDTD) methods, absorbing boundary conditions (ABCs) are used to truncate the computational domain for the simulation of open boundary problems. A comparison of the performance of ABCs applied to TLM and FDTD is presented. The results indicate a significant improvement in the performance of an ABC when applied to TLM compared to the performance of the same ABC when applied to FDTD. An explanation for this improvement in the performance is given.

The finite-difference time-domain (FDTD) method in conjunction with the perfectly matched layers (PML) absorbing boundary condition (ABC) has been a popular choice for the simulation of ground penetrating radars (GPRs) due to the following reasons: FDTD is suitable for modeling inhomogeneities. FDTD is also suitable for the computation of time-dependent radar signals. PML is a very effective ABC that renders the FDTD method quite efficient. In this paper, in addition to the conventional usage of PML to truncate the FDTD computational domain, we report on how the PML can also be used to simulate an absorbing material inside the FDTD computational domain.

In this paper, based on quadratic interpolation in time domain, an improved first-order absorbing boundary condition (ABC) for the alternating-direction-implicit finite-difference time-domain (ADI-FDTD) method is developed. The ABC is derived in two half-time steps iteration form which is very suitable for the ADI-FDTD method. The reflection of the ABC caused by both the truncated error and the phase velocity error is analyzed in detail. The phase velocity estimation and nonuniform cell are adopted to improve the performance of the ABC. Numerical results show that efficient improvement on the absorbing performance is achieved based on these methods.

Generalized Finite-Difference Time-Domain method with absorbing boundary conditions for solving the nonlinear Schrödinger equation on a GPU

The Perfectly Matched Layer (PML) concept, introduced by Berenger with the aim of synthesizing an absorbing boundary condition (ABC) for the Finite Difference Time Domain (FDTD) method, was recently modified and extended to Finite Element Frequency Domain (FEFD) applications. The modified equations, which neither require the splitting of the field components of interest nor involve negative conductivity parameters, are employed, in this paper, in conjunction with analytic ABC's to obtain a boundary condition (BC) for scattering problems. This BC exhibits an improved performance over a PML medium terminated by a perfect conductor, or an ABC termination alone. >

Absorbing boundary conditions for time harmonic wave propagation in discretized domains