Essay: Perfectly matched layer (PML), weighted Laguerre polynomials (WLP), finite-difference-time-domain (FDTD)

Abstract: An unsplit-field and stretched coordinate (SC) based perfectly matched layer (PML) is presented for JE Collocated finite-difference-time-domain method with weighted Laguerre polynomials (JEC-WLP-FDTD) in nonmagnetic plasma medium. Through adding a perturbation, the huge sparse matrix equation is solved with a factorization-splitting scheme. This leads to much less CPU time and memory storage than in the conventional implementation. With the complex-frequency-shifted (CFS) factor, the SC-PML which is used to truncate plasma media performs better than the traditional PML especially for attenuating low frequency and evanescent waves. To verify the accuracy and efficiency of the proposed method, two numerical examples are given.
Index Terms: perfectly matched layer (PML), weighted Laguerre polynomials (WLP), finite-difference-time-domain (FDTD), complex-frequency-shifted (CFS).
I. INTRODUCTION
T
HE finite-difference-time-domain (FDTD) method has been widely used to simulate the wave propagation in dispersion media due to its easy implementation. However, the well-known Courant-Friedrichs-Levy (CFL) stability condition constrains the application of the conventional FDTD method for the simulation of structure with fine geometries [1]. To overcome these limitations, unconditionally stable FDTD method has been developed, such as alternating direction implicit (ADI) FDTD [2’3], locally one dimensional (LOD) 7FDTD [4], WLP-FDTD [5] method. Among them, the WLP-FDTD method not only removes the CFL stability restriction, but also avoids the numerical dispersion error of the ADI-FDTD with the increase of the time-step [6]. Recently , The WLP-FDTD method has been employed to simulate electromagnetic wave propagation in a variety of dispersive Media by ADE technique [7]. The original WLP-FDTD method generates a huge sparse matrix equation. Directly solving this matrix equation is numerically expensive.
The perfectly matched layer, introduced by Berenger, has been widely used for truncating FDTD computational domains[8]. The original formulation is based on splitting field. From then on, different unsplit-field PML implementations have been presented for FDTD method, such as uniaxial PML (UPML) [9] and SC-PML [10, 11]. Among the various implementations of the PML, the SC-PML has the advantage of simple implementation in the corners and edges of the PML regions and is independent of background media. Recently, a split-field PML [12, 13] based on Berenger’s original formulation was employed within the WLP-FDTD formulation and so as the UPML [14]. More recently, a novel nearly PML implementation for WLP-FDTD is proposed in [15] for general dispersive material. Later, we proposed an effective SC-PML implementation for JEC-WLP-FDTD method in nonmagnetic plasma media.
In this paper, we present the factorization splitting JEC-WLP-FDTD algorithm in nonmagnetic plasma media. The PML is implemented by forming Maxwell’s equations in the stretched- coordinate (SC) system, and it can be easily combined with a complex-frequency-shifted (CFS) factor which was proved to be more efficient for low frequency and evanescent wave absorption [16, 17]. The proposed PML avoids field splitting and is easy to be implemented for dispersion media. Then the SC-PML is used to truncate the plasma lattices. Numerical results show the effectiveness of the proposed WLP-FDTD algorithm as well as the CFS-PML.
II. FORMULATION
Using the stretched coordinate PML formulation and considering the kinetic equations for cold electron plasma, the field equations for a TEMz wave propagation in two-dimensional nonmagnetic plasma media can be written as
(1)
(2)
(3)
(4)
(5)
where is the polarization current density, is the density of the electron, is the collision frequency, m is the electron mass, e is the electron charge, is the coordinate-stretching variable defined as
(6)
with the CFS factor, is modified to
(7)
Introducing the following auxiliary variables
(8)
These variables can be written into time domain by replacing in (8) with a differential operator , given by
(9)
With reference to Chung [5], the field components and the electron velocity components can be transferred into the Laguerre polynomial domain as
(10)
(11)
(12)
(13)
(14)
where , , s > 0 is a time scaling factor, is the order of WLP. and are the difference operators along and directions, respectively. The coefficients depend on PML parameters, we have
(15)
The coefficients are related to plasma parameters, given by
(16)
, , and are the lower order sums of the fields and auxiliary variables in Laguerre polynomial domain, given by
(17)
In (17), the auxiliary variables are calculated by
(18)
Substituting (13) into (10), (14) into (11), we have
(19)
(20)
Up to now, we have completed the formulations of the JEC-WLP-FDTD scheme with the SC-PML. Traditional JEC-WLP-FDTD algorithm solves the field components directly leading to solution of large sparse matrix equation, which is computational intensive in terms of both time and memory. To overcome this problem, we add a perturbation term, then use factorization splitting’ this new efficient JEC-WLP-FDTD implementation transfers the sparse matrix equation into two equations with tri-diagonal matrixes, which can be solved efficiently using a chasing algorithm.
Rewriting (19), (20) and (12) into a matrix form
(21)
where
(22)
(23)
As the next step, let
, and moving the second term of the right hand side of (21) to the left, (21) can be written as
(24)
where I is a identity matrix, and are two 3 3 matrixes with two none zero elements each, given by , and .
Adding a perturbation term , (24) can be split into two equations’as follows:
(25)
where .
After some manipulations, we have
(26)
Expanding (26) leads to
(27)
(28)
(29)
(30)
(31)
Substituting (30) into (28), (30) and (29) into(27), (30) into (31), we get
(32)
(33)
(34)
(13), (14), (32),(33) and (34) can be discretized using Yee’s central difference scheme. After discretization, the left hand side coefficients of (32) and (33) become tri-diagonal matrixes because they have two differential operators each. The right hand side terms of (32) and (33) are known, which can be solved efficiently using a chasing algorithm. After computing in sequence, and can be updated from (13), (14) and (34), respectively. With the solution of (32), (33), (34), (13) and (14), the time domain fields can be re-constructed according to the method in [].
III. NUMERICAL STUDY
To verify the accuracy and efficiency of proposed JEC-WLP-FDTD method as well as the CFS-PML. Two numerical examples are given. First, we calculate a plane wave traveling in an unmagnetized plasma with the proposed JEC-WLP-FDTD method. The simulation model is shown in Fig.1. The computational domain is discretized into 100 50 lattices along the and direction, respectively. The grid size is defined as . Each boundary of the computational domain are terminated with 10 grids PML. The PML parameters are scaled following []:
(35)
(36)
(37)
where , represents the interface between FDTD and PML grids. is the thickness of the PML. is a constant number. The plasma occupied 30~70 grid along direction, with , . The other grids are free space.
The excitation source located at grid 20 is defined as a differential Gaussian pulse given by
(38)
where , . We choose the order of the WLP , and the time scale factor . The JEC-WLP-FDTD takes a time step of , such that the CFL number is 5. The total time duration is . Two observation points at grid 30 and 70 is used to calculate the plasma reflection coefficient and transmission coefficient, respectively. The frequency results is obtained by discrete time Fourier transform (DTFT).
Fig. 1 Simulation model for plane wave traveling in plasma
Fig. 2 Plasma reflection and transmission coefficients
TABLE I
THE COMPUTATIONAL TIME AND MEMORY FOR THE TWO JEC-WLP-FDTD METHOD
Method
CPU time (s) Memory(MB)
Conventional
Proposed
Fig. 3 Simulation model for point source radiation in plasma
Fig. 4 Transient magnetic fields of z component at the observation point A
(a)
(b)
Fig. 5 Maximum relative error in dB as a function of PML parameters at A. (a) PML without CFS, (b) CFS-PML
”
”
As shown in Fig. 2, excellent agreement between the proposed method, the conventional JEC-WLP-FDTD method and the analytical solution is obtained for reflection coefficient and transmission coefficient. The memory requirements and computational time of the two JEC-WLP-FDTD methods are compared in Table I. As expected, the proposed JEC-WLP-FDTD occupies much less memory and time.
As a second example, we simulate a magnetic current source radiated in plasma , the computation domain, including 10 layers PML, is divided into 50??50 cells with a uniform size of , .The simulation model is shown in Fig. 3. A magnetic current source in shape of differential Gaussian pulse given by
(39)
is excited at in the center of the simulation domain with , , . The order of WLP is 250, and the time scale factor is set to be . Fig.4 shows the transient component at the observation point. We compare the result of the proposed method with the conventional FDTD method. The good agreement between the two methods verifies the proposed solution.
We also calculate the reflection error from PML using
(39)
where is the magnetic field of the observation point and is the reference solution from an extended model where no reflection is captured during the simulation period.
In order to have a fair comparison, we first do parametric studies for both cases. Fig. 5(a) and (b) plot the peak reflection errors as function of PML parameters. Then we select the best parameter sets for each, and compare the relative reflection error in time domain, as shown in Fig. 4. The results show that the SC-PML with CFS achieves more than 5 dB improvement, as compared to the case without CFS.
IV. CONCLUSION
We presented an unsplit-field and stretched coordinate based perfectly matched layer for WLP-FDTD. Numerical results show the effectiveness of the proposed PML algorithm. In a similar manner, the formulation can be extended to two dimensions and other types of dispersive media.