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IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS, VOL. 16, NO. 7, JULY 2006
Split-Field PML Implementations for the Unconditionally Stable LOD-FDTD Method Valtemir E. do Nascimento, Senior Member, IEEE, Ben-Hur V. Borges, and Fernando L. Teixeira
Abstract—We introduce and compare two split-field implementations of the perfectly matched layer (PML) for the unconditionally stable locally one-dimensional (LOD) finite-difference timedomain (FDTD) method. The LOD–FDTD formalism is expanded in terms of a symmetric source implementation. It is verified that the relative performance of both PML implementations is superior to the split PML performance in the alternating direction implicit (ADI) FDTD method. Index Terms—Alternating direction implicit (ADI) technique, finite-difference time-domain (FDTD) method, locally one-dimensional (LOD) technique, perfectly matched layer (PML).
I. INTRODUCTION HE finite-difference time-domain (FDTD) method [1] has been widely employed in the analysis and design of a large variety of devices, particularly in the microwave and mm-wave range. However, devices with disparate geometrical sizes that include small geometrical features pose a challenge for FDTD simulations due to the Courant stability criterion. This criterion sets a bound in the maximum time step size, and impacts the number of time steps necessary to complete a given simulation. This limitation of traditional FDTD was lifted with the introduction of FDTD approaches based on the alternating direction implicit (ADI) operator splitting technique [2]–[5]. This approach is unconditionally stable and only involves the solution of a tridiagonal matrix at each time step (in contrast to previous unconditionally stable schemes that require an iterative sparse solver at each time step). This allows for the use of time steps significantly larger than those imposed by the Courant criterion, now bounded only by the desired numerical accuracy. More recently, other operator splitting techniques to achieve unconditional stability have gained considerable attention. One such technique is based on locally one-dimensional (LOD) splitting [6]. A similar variant has been extended to solve Maxwell equations in time domain in [7], where it was denoted as a split step (SS) approach. The finite difference equations in this case were constructed based on a Strang splitting for the time update. Later, a simplified LOD approach for FDTD based on the Crank–Nicolson (CN) scheme to avoid numerical dissipation, was introduced in [8] and [9]. This approach is denoted here as LOD–FDTD.
In order to simulate open domain problems, it is necessary to include absorbing boundary conditions such as the perfectly matched layer (PML) [10], [11]. The PML has been previously extended to ADI–FDTD [12], [13]. This letter introduces and compares the performance of two split-field PML implementations for LOD–FDTD. For simplicity, the formalism is developed for the two-dimensional (2-D) transversal electric (TE) case, but it can be extended to three dimensions. II. LOD–FDTD EQUATIONS WITH PML Starting with TE Maxwell’s equations, and using a split-field PML formulation, one obtains
T
Manuscript received February 16, 2006; revised April 5, 2006. This work was supported in part by the Funda¸cäo de Amparo à Pesquisa do Estado de Säo Paulo (FAPESP), the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), the National Science Foundation (NSF), and the Air Force Office of Scientific Research (AFOSR). V. E. do Nascimento and B.-H. V. Borges are with the Electrical Engineering Department, São Carlos School of Engineering, University of São Paulo, São Carlos 13566-590, Brazil (e-mail:
[email protected]). F. L. Teixeira is with the Department of Electrical and Computer Engineering and ElectroScience Laboratory, The Ohio State University, Columbus, OH 43212 USA (e-mail:
[email protected]). Digital Object Identifier 10.1109/LMWC.2006.877132
(1) (2) (3) (4) is the magnetic current source. The first LOD–FDTD where step is obtained by discretizing (1) and (2), while the second step is obtained by discretizing (3) and (4). The finite difference equations for the first step at time instant (1/2) is given by
(5)
(6) where are PML coefficients that depend on the PML conductivity . In order to guarantee unconditional stability, the CN scheme is applied to the spatial derivatives of
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DO NASCIMENTO et al.: SPLIT-FIELD PML IMPLEMENTATIONS
the field components and in (1)–(2). The first step is then obtained by substituting (6) into (5) which results in the component: following implicit equation for the
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The PML coefficients depend on the conductivity . It is noteworthy mentioning that the resulting LOD–FDTD update yields only two equations per step, while the ADI–FDTD yields four equations per step. Therefore, a decrease in the number of FLOPs for the LOD–FDTD is expected when compared to the ADI–FDTD. The next section shows how to obtain the PML coefficients for two different implementations. III. SPLIT PML COEFFICIENTS IN LOD–FDTD A. First Scheme In this case, a central differencing approximation in time is applied to the conduction terms on the left hand side of (1)–(4), similar as done in the ADI–FDTD case [12]. After this expansion, the following coefficients are obtained for the first step: e 21t (2"0 0 xh1t) h ey = (2"0 0 xe 1t) ; ye = = ; x [(2"0 + xe 1t)1x] (2"0 + x 1t) (2"0 + xh1t) 2 h h h x =21t=[(20 + 0 x 1t)1x] and x =1t=(20 + 02 xh 1t):
The coefficients for the second step are similarly obtained as
(2"0 0 ye 1t) ; e = (2"0 0 yh1t) 21t h = ; x y (2"0 + ye 1t) [(2"0 + ye 1t)1y] (2"0 + yh1t) h 2 h h y =21t=[(20 + 0 y 1t)1y] and y =1t=(20 + 02 yh 1t)
ex =
where
and
The second step can be similarly obtained at time instant 1 leading to the following implicit equation for the component:
where represents the time step size, and and are the grid cell sizes in and directions, respectively. B. Second Scheme In this case, the forward approximation is applied to the conduction terms on the left hand side of (1)–(4), akin to the ADI–FDTD implementation in [13]. The following coefficients are obtained for the first step:
and Similarly, the coefficients for the second step are
and
where
IV. NUMERICAL RESULTS The performance of the two PML implementations is studied for different values of Courant–Friedrich–Levy number (CFLN), defined as the ratio between the time step size adopted for LOD–FDTD to the maximum one allowed in conventional FDTD. A point source is assumed in free space. A differentiated Gaussian pulse is utilized as excitation and applied to the component, having both central frequency and half bandwidth equal to 3.175 GHz. The computational domain, including the PML region, has 170 170 cells, with 0.33 mm. This corresponds to approximately 200 grid points per wavelength (ppw) at 4.5 GHz. The PML has eight cells along and directions with a conductivity profile as in [13] and maximum conductivity as in [12]. The excitation is located
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IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS, VOL. 16, NO. 7, JULY 2006
It should be mentioned that, for same mesh resolution and CFLN, LOD–FDTD yields a smaller computational cost compared to ADI–FDTD [8], [9]. For example, a reduction about 50% can be obtained for a CFLN 4 and a 600 600 mesh resolution. However, the LOD exhibits an extra truncation error term (not present in ADI) due to the noncommutativity of the operators resulting from the particular split in the curl operators in LOD. A detailed study of PML–LOD–FDTD, incorporating issues such as truncation error and overall computational cost, is beyond the scope of this letter and will be provided elsewhere. V. CONCLUSION Fig. 1. Reflection error from the PML–LOD–FDTD method, first scheme.
In this letter, we have presented and compared two split-field PML implementations for the LOD–FDTD. The first one is based on a central differencing approximation of the resulting finite difference update, while the second one is based on forward differencing. In both implementations, the CN scheme was employed to guarantee unconditional stability. It was shown that the performance of both PML implementations in LOD–FDTD yielded superior results to the traditional split PML applied to ADI–FDTD using a similar set of parameters. Typical reflection errors of about 75 dB were obtained for an eight-cell PML, which are considerably lower than the inherent dispersion error at the discretization scales considered. REFERENCES
Fig. 2. Reflection error from the PML–LOD–FDTD method, second scheme.
at the center of the computational domain, and the observation point is located 41 cells away from the source. The performance of the PML is measured by the reflection error, given as
log where is a reference value calculated with a computational domain sufficiently large so that any reflections from the boundary are isolated. This calculation is carried out for different CFLN. Simulation results for the first scheme are shown in Fig. 1. The reflection values for the PML–LOD–FDTD are around 80 dB for CFLN 6. In contrast, the PML reflection for the PML–ADI–FDTD with same parameters and CFLN 6 is about 35 dB [12]. The simulation results for the second scheme are shown in Fig. 2. These results indicate that PML reflection levels for the LOD–FDTD are very satisfactory even for large CFLN values. The reflection error obtained using an eight-cell PML in the LOD–FDTD is around 75 dB for CFLN 6. This is considerably lower than the inherent numerical dispersion error of LOD–FDTD at the discretization scale and CFLN considered. These results also show that the reflection error for the split PML–LOD–FDTD, based either on the first or second scheme, is less sensitive to CFLN variations than the PML approach for ADI in [12]. This is an intrinsic characteristic of the LOD approach, because LOD considers the electric and magnetic PML conductivities along a single direction per step, with no overlapping of directions.
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