832 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 3, MARCH 2010 Development of the CPML for Three-Dimensional Unconditionally Stable LOD-FDTD Method Iftikhar Ahmed, Member, IEEE, Eng Huat Khoo, and Erping Li, Fellow, IEEE Abstract—A convolutional perfectly matched layer (CPML) is developed for three-dimensional unconditionally stable, locally one dimensional (LOD)-finite-difference time-domain (FDTD) method. The formulation of the LOD-FDTD CPML is derived and numerical results are demonstrated at different positions for different Courant Friedrich Levy numbers in the simulation domain. The method is validated numerically with FDTD-CPML. Index Terms—Alternating direction implicit-finite-difference time-domain (ADI-FDTD), convolutional perfectly matched layer (CPML), Courant Friedrich Levy (CFL) limit, finite-difference time-domain (FDTD), locally one dimensional (LOD)-FDTD, perfectly matched layer (PML). I. INTRODUCTION I T IS KNOWN that Courant Friedrich Levy (CFL) limit re- duces computational efficiency of the finite-difference time- domain (FDTD) method [1] for the simulation of structures where fine mesh is needed. The CFL limit was removed with the development of unconditionally stable alternating direction implicit (ADI)-FDTD method [2], [3]. Nevertheless, recently another unconditionally stable technique, known as the locally one-dimensional (LOD)-FDTD has been introduced [4] for two dimensional structures. Later on, it has been extended to the three-dimensional LOD-FDTD method [5] and compared with the ADI-FDTD method. It is found that the LOD-FDTD method is efficient than the ADI-FDTD method due to the requirement of less number of arithmetic operations [5]. The FDTD and ADI-FDTD methods have been applied to open structures, where to truncate the open boundaries numerous absorbing boundary conditions (ABCs) have been developed [1]. Among absorbing boundary conditions perfectly matched layer (PML) [6], uniaxial perfectly matched layer (UPML) [7] and CPML [8] are most commonly used boundary conditions. The PML and the UPML approaches show ab- sorption errors at low frequencies and evanescent waves. In addition, for these both approaches formulation modification is needed if the material of a structure changes. On the other hand, the CPML is completely independent of the host medium and there is no need of modifications in formulation, when apply to lossy, dispersive, anisotropic, nonlinear, and inhomogeneous media. The use of CPML provides significant saving in memory Manuscript received June 09, 2009; revised August 26, 2009. First published December 31, 2009; current version published March 03, 2010. The authors are with the Department of Computational Electromagnetics and Photonics, Institute of High Performance Computing, Singapore 138632, Sin- gapore (e-mail: iahmed; eplee@ihpc.a-star.edu.sg). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2039334 [8]. In addition, the CPML permits an easy implementation of the complex frequency shifted (CFS) stretching factor that allows the reflection of the low frequency evanescent waves to be significantly reduced [9]. Due to generality and simplicity of the CPML, it is implemented with FDTD method on graph- ical processing unit (GPU) [10] to accelerate the simulation speed. For three dimensional ADI-FDTD and two-dimensional LOD-FDTD methods CPML is developed in [11], [12]. Due to significance of the CPML absorbing boundary condition, in this paper it is developed for the three dimensional LOD-FDTD method. It is abbreviated as LOD-CPML. The formulation and numerical results are discussed in the following sections. II. FORMULATION OF LOD-CPML Maxwell’s equations for an isotropic and lossless media are For the three dimensional LOD-FDTD method, these equations are written as follows. Step 1 (1a) (1b) (1c) (1d) Step2 (2a) (2b) (2c) (2d) 0018-926X/$26.00 © 2010 IEEE