IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 11, 2012
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Implementation of Absorbing Boundary Conditions Based on the Second-Order One-Way Wave Equation in the LOD- and the ADI-FDTD Methods José A. Pereda, Member, IEEE, Abdelaziz Serroukh, Ana Grande, and Angel Vegas, Member, IEEE
Abstract—This letter describes the implementation of second-order one-way wave-equation absorbing boundary conditions (ABCs) in two unconditionally stable finite-difference time-domain (FDTD) methods—namely the locally one-dimensional (LOD)- and the alternating-direction implicit (ADI)-FDTD methods. The Higdon second-order absorbing operator is discretized in the same way as when it is used with the conventional FDTD method. The resulting discrete expression is directly applied to the electric field in each time substep of the LOD- and the ADI-FDTD methods. Numerical examples are given to illustrate the validity of the proposed approach. Index Terms—Alternating-direction implicit finite-difference time-domain (ADI-FDTD) method, locally one-dimensional FDTD (LOD-FDTD) method, one-way wave-equation absorbing boundary condition.
I. INTRODUCTION
I
N THE past decade, there has been a growing interest in developing split-step finite-difference time-domain (FDTD) techniques such as the locally one-dimensional (LOD)- and the alternating-direction implicit (ADI)-FDTD methods [1]. These methods provide unconditional stability with an acceptable computational cost. When an FD method is applied to an open structure, absorbing boundary conditions (ABCs) are needed to truncate the domain of solution. Two main types of ABCs have been used in combination with unconditionally stable FDTD methods: ABCs based on the perfectly matched layer (PML) concept [2], [3], and those based on one-way wave equations [4]–[9]. Generally, the first approach provides smaller reflection errors, while the second one is simpler to implement and more efficient in terms of CPU time and memory requirements. The incorporation of ABCs based on the one-way wave equation into unconditionally stable FDTD methods is considerably more intriguing than in the case of the conventional FDTD method. Even for the simplest first-order ABC, several
formulations with different properties of accuracy and stability have been reported [4]–[7]. Up to now, attempts to incorporate high-order one-way wave-equation ABCs have encountered stability problems [8]. Moreover, it has been pointed out that this type of high-order ABC destroys the tridiagonal structure of the data matrix, which is a key property of the LOD- and ADI-FDTD methods [9]. An alternative high-order ABC based on linear interpolation has recently been introduced [9]. Unfortunately, this ABC is conditionally stable. This letter presents a successful implementation of secondorder one-way wave-equation ABC in the LOD- and the ADIFDTD methods. The considered ABC is based on the Higdon second-order absorbing operator [10], [11]. This operator is discretized by applying the same FD scheme as when it is used with the conventional FDTD method. The resulting discrete expression is applied to the electric field in each time substep of the LOD- and the ADI-FDTD methods. Numerical examples are given to illustrate the validity of the proposed approach.
II. FORMULATION The Higdon second-order ABC for the boundary can be expressed as [10] (1)
Each first-order operator provides perfect absorption for any linear combination of plane waves impinging on the boundary with phase velocity where is the incidence angle and the speed of light in free space. Approximating according to the so-called box scheme in [10] (i.e., by using central differences and averages), the following discrete expression is obtained: (2)
Manuscript received June 04, 2012; revised July 27, 2012; accepted August 02, 2012. Date of publication August 08, 2012; date of current version August 30, 2012. This work was supported by the Spanish MICINN under Projects TEC2010-21496-C03-01 and CONSOLIDER CSD2008-00066. J. A. Pereda, A. Serroukh, and A. Vegas are with the Departamento de Ingeniería de Comunicaciones (DICOM), Universidad de Cantabria, 39005 Santander, Spain (e-mail:
[email protected]). A. Grande is with the Departamento de Electricidad y Electrónica, Universidad de Valladolid, 47071 Valladolid, Spain. Color versions of one or more of the figures in this letter are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LAWP.2012.2212411
where
and
are shift operators defined by
and the coefficient
1536-1225/$31.00 © 2012 IEEE
is given by
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IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 11, 2012
Cascading two discrete first-order operators of the form given in (2), the following second-order ABC is obtained:
(3) , and . where This expression has been widely used with the conventional FDTD method. Herein, we will describe how it can be incorporated within LOD- and ADI-FDTD codes. To this end, we will assume that the time-dependent Maxwell curl equations for waves are solved in a rectangular region of the -plane. This space region is discretized by using cells, and is solved at grid nodes ( ; ). For both methods, in the first substep, is updated implicitly in the -direction, and in the second substep, is updated implicitly in the -direction. For the LOD-FDTD method, the ABC at given by (3) is applied in each substep as follows. A. Substep 1 To update (3) is applied to
at the boundary
, (4)
resulting in the following expression: (5) involves only past values of . where the term Equation (5) can be incorporated directly into the tridiagonal matrix equation used to solve for along the -direction. However, the augmented data matrix is no longer tridiagonal because of the term in (5). This problem can be easily overcome by using the LOD-FDTD equation at node to eliminate in (5). B. Substep 2 To update to
at the boundary
, (3) is applied
1.b) The values at the absorbing boundaries , (i.e., for and , ) are updated explicitly. 1.c) is updated explicitly according to the standard LOD-FDTD method Substep 2: 2.a) The values in the interior and at the absorbing boundaries , (i.e., for and ) are updated implicitly in the -direction. This calculation involves the solution of systems of linear equations. Each system has equations, including the ABCs at and . 2.b) The values at the absorbing boundaries , (i.e., for , , and ) are updated explicitly. 2.c) is updated explicitly according to the standard LOD-FDTD method The implementation of (3) into the ADI-FDTD algorithm follows the same path as the one introduced previously for the LOD-FDTD method. In fact, (4) and (6) can be incorporated directly into an ADI-FDTD code without any change. III. NUMERICAL VALIDATION To illustrate the validity of the formulation proposed above, we have considered two examples: a line electric current radiating in free-space and the matched termination of a rectangular waveguide port. In the first example, we consider a free-space domain with 80 80 cells. This test domain is truncated by ABCs. The phase velocities in (1) used for this example are . A line electric current directed along the -direction is applied at the center of the domain ( ). The time dependence of the source is a modulated Gaussian pulse with central frequency GHz and effective lateral bandwidth GHz. The minimum effective wavelength in the problem is then cm. The cell size is mm. The reflection error at a point is defined, in the time domain, as
: (6)
explicitly, as in the conThis expression is solved for ventional FDTD method. The resulting LOD-FDTD algorithm incorporating secondorder one-way ABCs in all four boundaries can be summarized as follows: Substep 1: 1.a) The values in the interior and at the absorbing boundaries , (i.e., for and ) are updated implicitly in the -direction. This task involves the solution of systems of linear equations (for ). Each system has equations (the standard LOD-FDTD equations for the interior and the ABC equations for and ).
where is the field computed using the test domain and is the reference field, free of reflection errors, computed by using a larger domain. The reflection error was computed at two grid points: and Fig. 1 shows the reflection error computed at points A and B as a function of time for the LOD- and ADI-FDTD methods. These results have been obtained by using a stability factor , where is the time used in the simulation and is the maximum time-step allowed by the conventional FDTD method (i.e., ). As expected, the error at point A, which is predominately due to normally incident waves, is less than the error at point B.
PEREDA et al.: IMPLEMENTATION OF ABSORBING BOUNDARY CONDITIONS
Fig. 1. Reflection error as a function of time.
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Fig. 3. Reflection coefficient as a function of the frequency for a WR75 waveguide port terminated by first- or second-order ABCs.
IV. CONCLUSION Second-order one-way wave-equation ABCs have been successfully incorporated into the LOD- and the ADI-FDTD methods. The proposed ABC is based on the Higdon operator, which is discretized in the same way as when it is used with the conventional FDTD method. A key feature of the proposed approach is that, in the first substep, the ABC involves electric field quantities defined at half-integer time-steps only and, in the second substep, involves electric field quantities defined exclusively at integer time-steps. Fig. 2. Maximum reflection error at points A and B for different values of the stability factor.
Fig. 2 compares the maximum relative error at points A and B as a function for both the LOD- and the ADI-FDTD methods. The results obtained by the conventional FDTD method for the case have also been included as a reference. It can be seen that the curves shown are nearly flat, so that the error is nearly independent of the time-step. In the second example, we consider a WR75 guide propagating the mode in the frequency band [10–15] GHz. The width of this waveguide is mm. The cell size is mm and mm, where is the wavelength in the waveguide at 15 GHz. For this application, the phase velocities in (1) are taken to be and , with being the waveguide phase velocity at 12.5 GHz. Fig. 3 shows the reflection coefficient of the ABC as a function of the frequency for . It can be seen that the reflection coefficient is better than 50 dB in the whole frequency band, which is enough for most applications. For the sake of comparison, the results obtained with the Higdon first-order operator with are also included. It can be seen that the second-order ABC provides an improvement of about 28 dB over the first-order one. The first-order ABC has been discretized according to [5] for the ADI-FDTD method and according to the scheme labeled ”novel consistent scheme” in [7] for the LOD-FDTD method.
REFERENCES [1] J. Shibayama, M. Muraki, R. Takahashi, J. Yamauchi, and H. Nakano, “Performance evaluation of several implicit FDTD methods for optical waveguide analyses,” J. Lightw. Technol., vol. 24, no. 6, pp. 2465–2472, Jun. 2006. [2] G. Liu and S. D. Gedney, “Perfectly matched layer media for an unconditionally stable three-dimensional ADI-FDTD method,” IEEE Microw. Guided Wave Lett., vol. 10, no. 7, pp. 261–263, Jul. 2000. [3] V. E. do Nascimento, B.-H. V. Borges, and F. L. Teixeira, “Split-field PML implementations for the unconditionally stable LOD-FDTD method,” IEEE Microw. Wireless Compon. Lett., vol. 16, no. 7, pp. 398–400, Jul. 2006. [4] J.-N. Hwang and F.-C. Chen, “Stability analysis of the Mur’s absorbing boundary condition in the alternating direction implicit finite-difference method,” Microw. Antennas Propag., vol. 1, no. 3, pp. 597–601, Jun. 2007. [5] T. Stefanski and T. D. Drysdale, “Improved implementation of the Mur first-order absorbing boundary condition in the ADI-FDTD method,” Microw. Opt. Technol. Lett., vol. 50, p. 1757, Jul. 2008. [6] W. C. Tay and E. L. Tan, “Mur absorbing boundary condition for efficient fundamental 3-D LOD-FDTD,” IEEE Microw. Wireless Compon. Lett., vol. 20, no. 2, pp. 61–63, Feb. 2010. [7] F. Liang, G. Wang, H. Lin, and B.-Z. Wang, “Mur absorbing boundary condition for three-step 3-D LOD-FDTD method,” IEEE Microw. Wireless Compon. Lett., vol. 20, no. 11, pp. 589–591, Nov. 2010. [8] M. H. Kermani, X. Wu, and O. M. Ramahi, “Instable ADI-FDTD openregion simulation,” in Proc. IEEE Antennas Propag. Soc. Symp., 2004, vol. 1, pp. 595–598. [9] J. Zhou and J. Zhao, “Efficient high-order absorbing boundary condition for the ADI-FDTD method,” IEEE Microw. Wireless Compon. Lett., vol. 19, no. 1, pp. 6–8, Jan. 2009. [10] R. L. Higdon, “Numerical absorbing boundary conditions for the wave equation,” Math. Comput., vol. 49, no. 179, pp. 65–90, Jul. 1987. [11] L. A. Vielva, J. A. Pereda, A. Prieto, and A. Vegas, “FDTD multimode characterization of waveguide devices using absorbing boundary conditions for propagating and evanescent modes,” IEEE Microw. Guided Wave Lett., vol. 4, no. 6, pp. 160–162, Jun. 1994.
