Unconditionally Stable Finite Difference Time Domain Methods for Frequency Dependent Media

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Abstract

The efficiency of the conventional, explicit finite difference time domain (FDTD)method is constrained by the upper limit on the temporal discretization, imposedby the Courant–Friedrich–Lewy (CFL) stability condition. Therefore, there is agrowing interest in overcoming this limitation by employing unconditionally stableFDTD methods for which time-step and space-step can be independentlychosen. Unconditionally stable Crank Nicolson method has not been widely usedin time domain electromagnetics despite its high accuracy and low anisotropy.There has been no work on the Crank Nicolson FDTD (CN–FDTD) method forfrequency dependent medium.In this thesis a new three-dimensional frequency dependent CN–FDTD (FD–CN–FDTD) method is proposed. Frequency dependency of single–pole Debyematerials is incorporated into the CN–FDTD method by means of an auxiliarydifferential formulation. In order to provide a convenient and straightforwardalgorithm, Mur’s first-order absorbing boundary conditions are used in the FD–CN–FDTD method. Numerical tests validate and confirm that the FD–CN–FDTD method is unconditionally stable beyond the CFL limit.The proposed method yields a sparse system of linear equations which canbe solved by direct or iterative methods, but numerical experiments demonstratethat for large problems of practical importance iterative solvers are to be used.The FD–CN–FDTD sparse matrix is diagonally dominant when the time-stepis near the CFL limit but the diagonal dominance of the matrix deteriorateswith the increase of the time-step, making the solution time longer. Selectionof the matrix solver to handle the FD–CN–FDTD sparse system is crucial tofully harness the advantages of using larger time-step, because the computationalcosts associated with the solver must be kept as low as possible. Two best–knowniterative solvers, Bi-Conjugate Gradient Stabilised (BiCGStab) and GeneralisedMinimal Residual (GMRES), are extensively studied in terms of the number ofiteration requirements for convergence, CPU time and memory requirements.BiCGStab outperforms GMRES in every aspect. Many of these findings do notmatch with the existing literature on frequency–independent CN–FDTD methodand the possible reasons for this are pointed out.The proposed method is coded in Fortran and major implementation techniquesof the serial code as well as its parallel implementation in Open Multi-Processing (OpenMP) are presented. As an application, a simulation model ofthe human body is developed in the FD–CN–FDTD method and numerical simulationof the electromagnetic wave propagation inside the human head is shown.Finally, this thesis presents a new method modifying the frequency dependentalternating direction implicit FDTD (FD–ADI–FDTD) method. Although theADI–FDTD method provides a computationally affordable approximation of theCN–FDTD method, it exhibits a loss of accuracy with respect to the CN-FDTDmethod which may become severe for some practical applications. The modifiedFD–ADI–FDTD method can improve the accuracy of the normal FD–ADI–FDTDmethod without significantly increasing the computational costs.

Keyword(s)

Alternating direction implicit (ADI)-FDTD method; Bioelectromagnetics; Crank Nicolson method; Finite difference time domain (FDTD) method; Frequency dependent materials; Sparse matrix solvers

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