A thesis submitted to the University of Manchester for the degree of Doctor of Philosophy in the Faculty of Science and Engineering

Transcription

1 INTRODUCTION OF THE DEBYE MEDIA TO THE FILTERED FINITE-DIFFERENCE TIME-DOMAIN METHOD WITH COMPLEX-FREQUENCY-SHIFTED PERFECTLY MATCHED LAYER ABSORBING BOUNDARY CONDITIONS A thesis submitted to the University of Manchester for the degree of Doctor of Philosophy in the Faculty of Science and Engineering 07 By Zeyu Long School of Electrical and Electronic Engineering

2 Contents Abstract Declaration 3 Copyright 4 Acknowledgements 5 Introduction 6. Motivation Challenges Aim and Objectives Outline Frequency-Dependent Finite-Difference Time-Domain Method 9. Maxwell s equations Finite Difference Time Domain Method Finite Difference Approximation Yee Algorithm Yee Cell and Leapfrog Arrangement Expression in Three-Dimension Expression in Two-Dimension Expression in one-dimension Dispersive Materials Debye Model Formulations of the FD-FDTD method Recursive Convolution Scheme Auxiliary Differential Equations

3 .4 Perfectly Matched Layers Split PML Un-split PML Stretched-Coordinate PML The CFS Stretching Coefficient The CFS-PML Formulation Numerical Dispersion Dispersion Relation for the Waves Numerical Dispersion Courant-Friedrichs-Lewy Condition Numerical Phase Velocity One Dimensional Phase Velocity Two Dimensional Phase Velocity Numerical Dispersion for PML One dimensional Dispersion Two Dimensional Dispersion Further Investigation for the Dispersion of CFS-PML Investigation for One Dimensional Dispersion Manipulation of 3.78) Manipulation of 3.79) Investigation for Two Dimensional Dispersion PML in the Four Corners PML in the Four Sides Summary Spatial Filtering Approach Implicit FDTD Methods Idea of the Spatial Filtering Approach Revised D CFL Condition Revised D CFL Condition Revised Stability Condition for CFS-PML Revised Stability Condition for D CFS-PML Revised Stability Condition for D CFS-PML Spatial Filtering Discrete Sine and Cosine Transform

4 4.4. Low Pass Filter in the Spatial Frequency Domain Procedures Numerical Experiments of the Spatial Filtering Approach Experiments of the Filtered FDTD Method D Filtered FDTD Method Observed Signals Improvement of Computational Efficiency Experiments of the Filtered FD-FDTD Method with CFS-PML Relative Permittivity of Debye Media Scaling of the PML Conductivity D Filtered FD-FDTD Method with CFS-PML Observed Signals Improvement of Computational Efficiency Analysis of the Numerical Results Simulation Settings Analysis for the Stability PML Conductivity Comparison between Air and Fat media In Air In fat Improvement of the Computational Efficiency Analysis for the Accuracy Numerical Phase Velocity of the Filtering Approach Spectrum Loss of the Excitation signal due to the Filtering Procedure Impact to the PML Accuracy Assessment Guidance of Choosing the Filtering Wavenumber Practical Application Numerical Modeling of the Human Tissues Numerical Experiment of Deep Brain Stimulation Conclusion and Future Work Conclusion

5 7. Future Work A The 3D filtered FDTD Method 33 A. Revised 3D CFL Condition A. 3D Filtered FDTD Method A.. Observed Signals with Single Point Excitation A.. Observed Signals with Spherical Excitation A..3 Improvement of Computational Efficiency A..4 3D Filtered FD-FDTD Method with CFS-PML A..4. Observed Signals with Spherical Excitation A..4. Improvement of Computational Efficiency Bibliography 4 5

6 List of Tables. Notation of each electromagnetic field component in 3D Notation of each electromagnetic field component in TE mode Notation of each electromagnetic field component in TM mode The position of each electromagnetic field component in D The simulation parameters for the filtered FDTD method The four scenarios in the numerical experiments The parameters for human tissues in the one-pole Debye relaxation model [83] A. The four scenarios in the numerical experiments in the 3D filtered FDTD method A. The four scenarios in the numerical experiments of the 3D filtered FD-FDTD method with CFS-PML

7 List of Figures. Position of the electromagnetic field components in the 3D-Yee cell [7][8]. Electric field components are located in the middle of the edges and magnetic field components are located in the center of the faces In D TE mode, the electric field components are located in the edge of the cell and the magnetic field components are located in the center of the the cell Position of the electromagnetic field components in the D FDTD method The time domain susceptibility χ p t) in the one-pole Debye model for fat The three dimensional wavenumber [54] X A varying xω from zero to π for the cases of x = mm, c x = mm and x = 0.5 mm Y A varying xω from zero to π for the case of x = mm. t cfld c is the upper limit of the D CFL condition for t The difference between t pml and t cfld varying xω from zero c to π for the cases of x = mm, x = mm and x = 0.5 mm. t cfld is the upper limit of the D CFL condition The structure of the PML [35]. The interior is vacuum Y A varying xω from zero to π while x = mm. t cfld is the c upper limit of the D CFL condition The difference between t pml and t cfld varying xω from zero c to π for the cases of x = mm, x = mm and x = 0.5 mm. t cfld is the upper limit of the D CFL condition

8 3.8 X B varying xω from zero to π for the cases of x = mm, c x = mm and x = 0.5 mm Y B varying xω from zero to π for the case of x = mm. t cfld c is the upper limit of the D CFL condition for t The difference between t dpml and t cfld varying xω from zero c to π for the cases of x = mm, x = mm and x = 0.5 mm. t cfld is the upper limit of the D CFL condition for t K A varying ϕ from zero to π for the cases of k max x = π, k max x = π and k max x = π The value of K C, K D varying k max x from zero to π for the cases of x = mm, mm and 0.5 mm The value of K C, K E varying k max x from zero to π for the cases of x = mm, mm and 0.5 mm The value of K F and K G varying k max x from zero to π for the cases of x = mm, mm and 0.5 mm The value of K F, K H varying k max x from zero to π for the cases of x = mm, mm and 0.5 mm Transformation from the spatial domain to the spatial frequency domain. The FDTD space has N x N y cells Transformation from the spatial frequency domain to the spatial domain. The FDTD space has N x N y cells Flowchart of the conventional FDTD method and the filtered FDTD method Flowchart of the Filtered FD-FDTD method with CFS-PML The source excitation in the time domain The normalized spectrum of the source excitation. The maximum frequency of this source is f highest = 5.73 GHz and the minimum frequency of this source is 0.6 GHz The simulation environment. The computation domain is a D cavity filled with air and bounded by PEC. The source is excited in the centre of the E z field. The observation point is 0 cells away from the excitation point The electric field distribution of E z before filtering at t = 80 t cfl. 97 8

9 5.5 The spatial spectrum of the electric field distribution of E z before filtering at t = 80 t cfl The spatial spectrum of the electric field distribution of E z after filtering at t = 80 t cfl. The filtering wavenumber k max is 0π rad/m The electric field distribution of E z after filtering at t = 80 t cfl The observed signals at 90,00) in E z for four scenarios. The four scenarios are introduced in Table 5.. In Scenarios, 3 and 4, k max = 0π rad/m and CFLN= The maximum CFLN which gives stable computation with for each k max. The theoretical value is obtained from 4.6) The improvement of computational efficiency The simulation environment. The computational domain is a D cavity filled with Debye media. 3 CFS-PML layers are applied into this computational domain. The excitation point is 0 cells away from the centre of the whole domain, and the observation point is 0 cells away from the excitation point The observed signals at 80,00) in E z for four scenarios. All the fields are filled with fat. The four scenarios are introduced in Table 5.. In Scenarios, 3 and 4, k max = 0π rad/m and CFLN= The maximum CFLN which gives stable computation for each k max. The theoretical value is obtained from 4.6) The improvement of computational efficiency The maximum CFLN which gives stable computation for each k max with different PML layers in air. Theoretical values are obtained from 4.6). The shadow area is enlarged to display more details.the FDTD space is cells The maximum CFLN which gives stable computation for each k max. The FDTD space is filled with air. Theoretical values are obtained from 4.6). The shadow area is enlarged to display more details The maximum CFLN which gives stable computation for each k max. The FDTD space is filled with fat. Theoretical values are obtained from 4.6). The shadow area is enlarged to display more details

10 6.4 The improvement of computational efficiency for three FDTD sizes. 6.5 The numerical phase velocity errors for x = λ min, λ min, λ min and λ min. λ 5.4 min is 5.4 cm. x = λ min is the spatial size used in the 5.4 numerical experiments. P D is obtained from 3.4) The numerical velocity anisotropy error for x = λ min, λ min, λ min and λ min. λ 5.4 min is 5.4 cm. x = λ min is the spatial size used in 5.4 the numerical experiments. P Da is obtained from 3.4) The theoretical spectrum loss of the excitation E and the simulation error R. The FDTD size is cells and 3 CFS-PML layers were used. The computational domain is filled with air The impact of the PML boundary with the spatial filtering approach. The FDTD size is cells and 3 CFS-PML layers were used. The computational domain is filled with air. The signal is observed at 80,00) in E z The simulation error R for the FDTD size of in air. R is calculated by using 6.7) The simulation error R for the FDTD size of in air. R is calculated by using 6.7) The simulation error R for the FDTD size of in air. R is calculated by using 6.7) The simulation error R for the FDTD size of in fat. R is calculated by using 6.7) The simulation error R for the FDTD size of in fat. R is calculated by using 6.7) The simulation error R for the FDTD size of in fat. R is calculated by using 6.7) The cross section of human head on the xy-plane at z = 80 and the simulation environment. The size of the simulation field is cells and the thickness of the CFS-PML layers is 3 cells. The field is excited at 30,50) and observed at 30,80) The value of U, P D and E varying k max from zero to 50 π rad/m The signal observed at 30,80) Electrical field distribution at t = 00 t cfld Electrical field distribution at t = 00 t cfld Electrical field distribution at t = 500 t cfld

11 6. Electrical field distribution at t = 000 t cfld A. The observed signals at 90,00,00) in E z for four scenarios. The four scenarios are introduced in Table 5.. In scenarios, 3 and 4, k max = 0π rad/m and CFLN= A. The observed signals at 85,00,00) in E z for four scenarios with spherical excitation. The four scenarios are introduced in Table 5.. In scenarios, 3 and 4, k max = 0π rad/m and CFLN= A.3 The maximum CFLN which gives stable computation for each k max. The theoretical value is obtained from A.3) A.4 The improvement of computational efficiency A.5 The observed signals at 75,00,00) in E z for four scenarios. The computational domain is filled with fat. The four scenarios are introduced in Table 5.. In scenarios, 3 and 4, k max = 0π rad/m and CFLN= A.6 The maximum CFLN which gives stable computation for each k max. The theoretical value is obtained from A.3) A.7 The improvement of computational efficiency

12 Abstract The finite-difference time-domain FDTD) method is one of most widely used computational electromagnetics CEM) methods to solve the Maxwell s equations for modern engineering problems. In biomedical applications, like the microwave imaging for early disease detection and treatment, the human tissues are considered as lossy and dispersive materials. The most popular model to describe the material properties of human body is the Debye model. In order to simulate the computational domain as an open region for biomedical applications, the complex-frequency-shifted perfectly matched layers CFS- PML) are applied to absorb the outgoing waves. The CFS-PML is highly efficient at absorbing the evanescent or very low frequency waves. This thesis investigates the stability of the CFS-PML and presents some conditions to determine the parameters for the one dimensional and two dimensional CFS-PML. The advantages of the FDTD method are the simplicity of implementation and the capability for various applications. However the Courant-Friedrichs-Lewy CFL) condition limits the temporal size for stable FDTD computations. Due to the CFL condition, the computational efficiency of the FDTD method is constrained by the fine spatial-temporal sampling, especially in the simulations with the electrically small objects or dispersive materials. Instead of modifying the explicit time updating equations and the leapfrog integration of the conventional FDTD method, the spatial filtered FDTD method extends the CFL limit by filtering out the unstable components in the spatial frequency domain. This thesis implements filtered FDTD method with CFS-PML and one-pole Debye medium, then introduces a guidance to optimize the spatial filter for improving the computational speed with desired accuracy.

13 Declaration No portion of the work referred to in this thesis has been submitted in support of an application for another degree or qualification of this or any other university or other institute of learning. 3

14 Copyright The author of this thesis including any appendices and/or schedules to this thesis) owns certain copyright or related rights in it the Copyright ) and s/he has given The University of Manchester certain rights to use such Copyright, including for administrative purposes. Copies of this thesis, either in full or in extracts and whether in hard or electronic copy, may be made only in accordance with the Copyright, Designs and Patents Act 988 as amended) and regulations issued under it or, where appropriate, in accordance with licensing agreements which the University has from time to time. This page must form part of any such copies made. The ownership of certain Copyright, patents, designs, trade marks and other intellectual property the Intellectual Property ) and any reproductions of copyright works in the thesis, for example graphs and tables Reproductions), which may be described in this thesis, may not be owned by the author and may be owned by third parties. Such Intellectual Property and Reproductions cannot and must not be made available for use without the prior written permission of the owners) of the relevant Intellectual Property and/or Reproductions. Further information on the conditions under which disclosure, publication and commercialisation of this thesis, the Copyright and any Intellectual Property and/or Reproductions described in it may take place is available in the University IP Policy see policies/intellectual-property.pdf), in any relevant Thesis restriction declarations deposited in the University Library, The University Library s regulations see and in The University s policy on presentation of Theses. 4

15 Acknowledgements I m truly thankful to my supervisor, Dr. Fumie Costen, who guided my the research patiently and helpfully. I m grateful to RIHEN Integrated Cluster for the access to their high speed computers. 5

16 Chapter Introduction Since 940s, based on the evolution of the computer technology, the computational electromagnetics CEM) methods have been widely used to study the communication systems, bio-electromagnetics and other subjects that involve the electromagnetic waves []. The CEM methods can be classified into two categories, the frequency domain methods and time domain methods. The frequency domain methods include the finite element method FEM) [][3], the fast multipole method FMM) [4], the method of moments MoM) [5][6] and so on. As one of the time domain methods, the finite-difference time-domain FDTD) method is famous for its precise geometry structure and straightforward algorithm [7][8]. All these methods are developed for solving the electromagnetic problems in different circumstances.. Motivation In 966, Kane S. Yee published the original paper of the FDTD method [8]. Sampling the electromagnetic field components in both time and space, the Yee s algorithm discretizes the Maxwell s curl equations by using the central difference approximation, and solves these equations at each time step to update the electromagnetic field components. The solution of the FDTD method is second order accurate. As a time domain technique, the FDTD method is adaptable for various electromagnetic applications over a wide bandwidth. 6

17 . Challenges The major challenges of the FDTD method are to absorb the traveling waves at the boundary, to model the frequency dependent FD) materials, to minimize the numerical dispersion and to extend the numerical stability [9][0]. The studies of the boundary conditions and dispersive materials improve the capability of the FDTD method to deal with practical problems like the biomedical therapies or early disease detections. In this thesis, the dispersive materials are simulated by the one-pole Debye model and the outgoing waves are attenuated by complex-frequency-shifted perfectly matched layers CFS-PML) absorbing boundary conditions ABC) [][]. The numerical dispersion affects the accuracy of the FDTD method. To guarantee the accuracy, the spatial size should be shorter than one tenth of the minimum excited wavelength [7]. For stable computations, the upper limit of the temporal size is constrained by the Courant Friedrichs Lewy CFL) condition. As a result, the high temporal and spatial resolution reduce the computational efficiency and increase the memory cost for the FDTD method. To increase the computation speed, the filtered FDTD method extends the CFL limit conditionally by filtering out the unstable components in the high spatial frequencies [3]..3 Aim and Objectives The aim of this research is to accelerate the computation speed of the FDTD method for biomedical applications. Thus the first objective of this thesis is introducing the one-pole Debye media to the filtered FDTD method with CFS-PML ABC. The filtered FD-FDTD method with CFS-PML is studied based on two points, the stability and accuracy. The stability of the spatial filtering approach is governed by a filtering wavenumber based on the revised CFL condition, which is derived from the lossless materials. Therefore the second objective of this thesis is to derive the revised stability condition for the spatial filtering approach with CFS-PML. The choice of the filtering wavenumber depends on the accuracy. However, in the previous publications like [3] and [4], the filtering wavenumber is mainly decided by experiences. For determining the filtering wavenumber, the third objective of this thesis is to investigate the key aspects that affect the accuracy of the filtered FD-FDTD method with CFS-PML. 7

18 .4 Outline This thesis consists of 7 chapters. Chapter reviews the background of the FDTD method with one-pole Debye model and CFS-PML. The mathematical formulations for the conventional FDTD method, the auxiliary differential equationade) scheme for the Debye model and the convolutional PML with CFS stretching coefficient are presented. The numerical dispersion, stability and phase velocity of the convention FDTD method are shown in Chapter 3. However, it is a challenge to derive the numerical stability for the FDTD method with CFS-PML ABC, since the numerical wavenumber in the CFS-PML computation is a complex number. This thesis derives the numerical dispersion relation of the CFS-PML, then presents some conditions to determine the temporal size for the CFS-PML. Chapter 4 shows the basic idea of the spatial filtering approach and presents the revised CFL conditions for the filtered FDTD method. The revised stability of the filtered FDTD method with CFS-PML ABC is investigated based on the numerical dispersion relation of the CFS-PML. The last section of the chapter shows the procedures of the spatial filtering approach. Chapter 5 presents the numerical experiments of the spatial filtering approach. The selection of the field for filtering is discussed for D filtered FDTD method and D filtered FD-FDTD Method with CFS-PML. In order to find the optimized filtering wavenumber, Chapter 6 studies the numerical dispersion of the filtered FDTD method by assessing the numerical phase velocity error and introduces the major aspects that affect the stability and accuracy of the filtered FD-FDTD method with CFS-PML ABC. A practical example is demonstrated in Chapter 6 as well. Chapter 7 concludes the dissertation and presents the possible future research works. Appendix A implements the 3D spatial filtering approach. Spherical excitation is introduced for avoiding the high spatial frequency components which are generated from the single point excitation. 8

19 Chapter Frequency-Dependent Finite-Difference Time-Domain Method As a robust and accurate technique for solving the Maxwell s equations in time domain, the finite-difference time-domain FDTD) method is a widely used method to simulate the wave propagation of the various applications in the contemporary engineering. However, the dielectric parameters are specified as a constant in the conventional FDTD method, which ignores the dispersion property of the frequency-dependant FD) materials. To overcome this drawback, the FD-FDTD method utilizes the material models like Debye model, Cole-Cole and other models to generate the dispersive materials for numerical computing [7]. This thesis uses the Debye model, which is simple in implementation and accurate in modeling the bio-tissues, for bio-electromagnetic applications. This chapter reviews the Maxwell s equations and the FD-FDTD method with one-pole Debye model and complex-frequency-shifted perfectly matched layers CFS-PML).. Maxwell s equations The Maxwell s equations, which can be written in either integral form or differential form, interpret the interaction of the electricity and magnetism mathematically. The integral form presents the integral of the electromagnetic fields over a closed surface or volume), which is sufficient for analytic calculations. Equivalent to the integral form in mathematics, the differential form describes amplitude of 9

20 the electromagnetic field at each point in both space and time. Thus, to display the wave propagation in the electromagnetic fields, the differential form of the Maxwell s equations is a better choice than the integral form. In the regions without electromagnetic currents, the differential form of the time-varying Maxwell s equations with absorbing materials are composed of the Faraday s Law of the Ampere s Law of the Gauss s Law for the electric field of B t = E J m,.) D t = H J e,.) D = ρ.3) and the Gauss s Law for the magnetic field of B = 0.4) where E is the electric field vector, H is the magnetic field vector, D is the electric flux density vector, B is the magnetic flux density vector, J m is the equivalent magnetic current density, J e is the electric current density and ρ is the electric charge density. The Maxwell s equations introduce two curl equations.) and.)) and two divergence equations.3) and.4)). The Faraday s Law illustrates the physical phenomenon that a time-dependent magnetic field can create an electric field, and the Ampere s Law expresses the reverse phenomenon of the Faraday s Law. The Gauss s Law for the electric field describes the relation between the electric charge density and the electric flux density. The Gauss s Law for the magnetic field proves that magnetic monopoles do not exist. Therefore the total magnetic flux through a closed surface is zero. The constitutive relations of the field-independent, direction-independent and frequency-independent materials are defined as B = µh = µ 0 µ r H,.5) 0

21 D = ɛe = ɛ 0 ɛ r E,.6) J e = σe.7) and J m = σ H.8) where ɛ is the electric permittivity, ɛ 0 is the electric permittivity in free space, ɛ r is the relative electric permittivity, µ is the magnetic permeability, µ 0 is the magnetic permeability in free space, µ r is the relative magnetic permeability, σ is the electric conductivity and σ is the equivalent magnetic conductivity. yields and Substituting the constitutive relations from.5) to.8) into.) and.) E t = ɛ H σ ɛ E.9) H t = µ E σ µ H.0) which are the Maxwell s curl equations for linear, isotropic and non-dispersive materials. Applying.9) and.0) into the rectangular coordinate system, we obtain E x t = ɛ Hz y H ) y z σe x,.) E y t = ɛ Hx z H ) z x σe y,.) E z t = ɛ Hy x H ) x y σe z,.3)

22 H x t = µ Ey z E ) z y σ H x,.4) and H y t H z t = µ = µ Ez x E ) x z σ H y,.5) Ex y E ) y x σ H z..6) where E x and H x are electromagnetic field components in the x-direction, E y and H y are electromagnetic field components in the y-direction and E z and H z are electromagnetic field components in the z-direction.. Finite Difference Time Domain Method.. Finite Difference Approximation The basic idea of the FDTD method is to discretize the Maxwell s curl equations by using finite difference approximations. To estimate the field value at grid point x 0, the finite difference approximation are derived from the forward Taylor expansion of f x 0 + x ), t = f x 0, t) + f x 0, t) x + f ) x 0, t)! and the backward Taylor expansion of f... + f m) x 0, t) m! x 0 x ), t = f x 0, t) f x 0, t) x + f ) x 0, t)!... + ) m f m) x 0, t) m! ) x + ) m x ) ) x ) m x ) where m is the order of polynomial, x is the discretized space size and fx, t) is the one-dimensional wave solution that fx, t) = e jkx ωt) k is the wavenumber

23 and ω is the angular frequency). Subtracting.8) from.7) yields f x 0 + x ), t f x 0 x ), t = f x 0, t) x + f 3) x 0, t) 3 ) 3 x ) Dividing both sides of.9) by x and then rearranging it, we derive the expression of central difference approximation as fx 0, t) x = f x 0 + x, t) f x 0 x, t) x f 3) x 0, t) x... 4 = f x 0 + x, t) f x 0 x, t) O x ).0) x where O x ) is the discretization error or truncation error) which represents all the terms of order higher than two. The central difference approximation is explicit for updating the field value at next time step. Thus, by removing O x ), Yee applied the central difference approximation with second order accurate to discretize the Maxwell s curl equations. as Replacing the spatial interval x in.7) by x,.7) can be manipulated fx 0, t) x = f x 0 + x, t) f x 0, t) x O x).) where O x) is the discretization error which represents the terms of order higher than one..) is the expression of forward difference approximation. Similarly, from manipulation of.8), the backward difference approximation is obtained as fx 0, t) x = f x 0, t) f x 0 x, t) x O x)..) The forward and backward difference approximations are first order accurate, if we remove the discretization error O x) of.) and.). Equations.0),.) and.) are the difference equations for the first derivative of function f x) at grid x 0. To estimate the function f x) at t 0, we can use the semi-implicit approximation central difference) of f x, t 0 ) = f x, t 0 + t 3 ) ) + f x, t0 t..3)

24 Ey Z Ex Hz Ex Ey Ez Ez Hx Ez Hy Ex Ey Y X Figure.: Position of the electromagnetic field components in the 3D-Yee cell [7][8]. Electric field components are located in the middle of the edges and magnetic field components are located in the center of the faces... Yee Algorithm The finite difference scheme establishes the basis of the FDTD method to obtain the approximate solutions of the Maxwell s curl equations at each point and time step. Reported by Yee [8], the FDTD method discretizes the electromagnetic fields by Yee cells and applies the leapfrog arrangement to update the fields information at each time step.... Yee Cell and Leapfrog Arrangement As demonstrated in Figure., each electric field component of E is surrounded by four magnetic field components of H, and each magnetic field component of H is surrounded by four electric field components of E. According to their half 4

25 Electromagnetic Spatial Notation Time Step Component X-axis Y-axis Z-axis E x i + j k n E y i j + k n E z i j k + n H x i j + k + n + H y i + j k + n + H z i + j + k n + Table.: Notation of each electromagnetic field component in 3D. space interval noted in Table.), the electric field components are placed in the middle of the cell edges and the magnetic field components are placed in the center of the cell faces. The Yee cell provides a solid geometry structure to obtain the solutions of the time-dependent Maxwell s curl equations. This spatial arrangement of the electromagnetic components satisfies the physical properties of Faraday s law and Ampere s law [7]. Sampled alternatively in both time and space, the value of the electromagnetic field components are approximated by a set of difference equations. For example, the electric field component in time step n + is updated by its information in the previous time step and the magnetic field component in time step n+. This time marching procedure is called leapfrog arrangement.... Expression in Three-Dimension Following the sampling information in Figure. and Table., the three dimension FDTD method is derived by applying the finite difference equations into the scale equations from.) to.6). For example [8][7], at time step t = n + ) t, the update equation for the electric field component E x at 5

26 i +, j, k) is obtained by discretizing.) to Ex n+ i+,j,k) En xi+,j,k) = t ɛi+,j,k) H n+ z i+,j+,k) Hn+ z i+,j,k) y H n+ y i+,j,k+ ) H n+ y i+,j,k ) z σ i +, j, k) E n+ x i +, j, k)..4) Based on the semi-implicit approximation.3), the difference form of the electric field component E n+ x i +, j, k) at time step n + is expressed as E n+ x i + ), j, k = En+ x i +, j, k) + E n x i +, j, k)..5) Substituting.5) into equation.4) yields Ex n+ i+,j,k) En xi+,j,k) = t ɛi+,j,k) H n+ z i+,j+,k)n+ n+ H z i+,j,k) y H n+ y i+,j,k+ ) H n+ y i+,j,k ) z σ i +, j, k) Ex n+ i+,j,k)+en xi+,j,k)..6) Rearranging.6) by collecting Ex n+ i +, j, k) to the left side and the rest to the right side, we obtain Ex i n+ + ), j, k = t σi+,j,k) ɛi+,j,k) + t σi+,j,k) ɛi+,j,k) t ɛi+,j,k) + t σi+,j,k) ɛi+,j,k) ) ) E n x. i +, j, k) + H n+ z i+,j+,k ) H n+ z i+,j,k ) y H n+ y i+,j,k+ ) H n+ z y i+,j,k )..7) Equation.7) is the standard form of the Yee algorithm to update the electric component E x at t = n + ) t. Based on.) and.3), the discretized 6

27 form of E y at t = n + ) t is expressed as Ey i, n+ j + ), k = t σi,j+,k) ɛi,j+,k) + t σi,j+,k) ɛi,j+,k) t ɛi,j+,k) + t σi,j+,k) ɛi,j+,k) and E z at t = n + ) t is expressed as Ez n+ i, j, k + ) = ) ) E n y. i, j +, k) + H n+ x i,j+,k+ ) H n+ x i,j+,k ) z H n+ z i+,j+,k ) H n+ t σi,j,k+ ) ɛi,j,k+ ) + t σi,j,k+ ) ɛi,j,k+ ) t ɛi,j,k+ ) + t σi,j,k+ ) ɛi,j,k+ ) ) ) x E n z. z i,j+,k ) i, j, k + ) + H n+ y i+,j,k+ ) H n+ y i,j,k+ ) x H n+ x i,j+,k+ ) H n+ y x i,j,k+ )..8).9) Discretizing the equations.4),.5) and.6), the update equations for 7

28 the magnetic field components at t = n + ) t are obtained as H n+ x i, j +, k + ) = t σ i,j+,k+ ) µi,j+,k+ ) + t σ i,j+,k+ ) µi,j+,k+ ) t µi,j+,k+ ) + t σ i,j+,k+ ) µi,j+,k+ ) ) ) H n x Ez n i,j+,k+ ) Ez n i,j,k+ ) y Ey n i,j+,k+ ) Ey n i,j+,k ) z. i, j +, k + ),.30) H n+ y i +, j, k + ) = t σ i+,j,k+ ) µi+,j,k+ ) + t σ i+,j,k+ ) µi+,j,k+ ) t µi+,j,k+ ) + t σ i+,j,k+ ) µi+,j,k+ ) ) ) H n y Exi+ n,j,k+ ) Exi+ n,j,k ) z Ez n i+,j,k+ ) Ez n i,j,k+ ) x. i +, j, k + ).3) and H n+ z i +, j +, k) = t σ i+,j+,k) µi+,j+,k) + t σ i+,j+,k) µi+,j+,k) t µi+,j+,k) + t σ i+,j+,k) µi+,j+,k) ) ) H n z Ey n i+,j+,k ) Ey n i,j+,k ) x Exi+ n,j+,k ) Exi+ n,j,k ) y. i +, j +, k)..3) 8

29 The finite-difference equations from.7) to.3) present the time marching algorithm of the electromagnetic field components in a Cartesian coordinate system. With the magnetic field components at t = n ) t and the electric field components at t = n t, we can update the magnetic field to t = n + ) t. Then with the electric field components at t = n t and the updated value of the magnetic field components at t = n + ) t, the next time step of the electric field components is obtained by equations.7),.8) and.9)....3 Expression in Two-Dimension To derive the equations for the two-dimensional FDTD method, we can remove the electric or magnetic vector components in one of the directions in the threedimensional expression. Assuming the electric field components in z-direction are removed E z = 0), the equations from.) to.6) can be reduced to the transverse electric TE) mode only involves E x, E y and H z ) of E x t = ɛ H z y σe x),.33) E y t = ɛ H z x σe y).34) and H z t = µ E x y E y x σ H z )..35) When the z-direction components for the magnetic field H z = 0) is removed, the transverse magnetic TM) mode only involves H x, H y and E z ) is written as H x t = µ E z y σ H x ),.36) H y t = µ E z x σ H y ).37) and E z t = ɛ H x x H x y σe z)..38) 9

30 Ex Ey Hz Ey Ex y z x Figure.: In D TE mode, the electric field components are located in the edge of the cell and the magnetic field components are located in the center of the the cell. Figure. shows the two dimensional Yee cell for the TE mode and Table. is the notation of the electromagnetic field components of TE mode. Therefore the discretized form of the TE mode are expressed as Ex i n+ + ), j = + t σi+,j) ɛi+,j) + t σi+,j) Ex i n + ), j ɛi+,j) t ɛi+,j) + t σi+,j) H n+ z i +, j + ) / y ɛi+,j) H n+ z i +, j ) / y,.39) 30

31 Electromagnetic Spatial Notation Time Step Component X-axis Y-axis E x i + j n E y i j + n H z i + j + n + Table.: Notation of each electromagnetic field component in TE mode. Spatial Notation Electromagnetic Component X-axis Y-axis Time Step H x i j + n + H y i + j n + E z i j n Table.3: Notation of each electromagnetic field component in TM mode. Ey n+ i, j + ) = t σi,j+ ) ɛi,j+ ) E + t σi,j+ y n ) ɛi,j+ ) t ɛi,j+ ) H n+ z + t σi,j+ ) ɛi,j+ ) i, j + ) H n+ z i +, j + ) / x i, j + ) / x.40) and H n+ z i +, j + ) = t σ i+,j+ ) µi+,j+ ) H n + t σ i+,j+ z i + ), j + ) µi+,j+ ) Ey n i +, j + t ) / x µi+,j+ ) Ey n i, j + ) / x + t σ i+,j+ ) E n µi+,j+ x i + ), j + ) / y. +Ex n i +, j) / y.4) 3

32 Derived from.36),.37) and.38), with the notations in Table.3, the finite difference expression for TM mode are obtained as H n+ x i, j + ) = t σ i,j+ ) µi,j+ ) H n σ + t i,j+ x i, j + ) ) µi,j+,k+ ) t ) µi,j+ ) Ez n i, j + ) / y, + t σ i,j+ ) Ez n i, j) / y µi,j+ ).4) H n+ y i +, j ) = + t σ i+,j) µi+,j) + t σ i+,j) H n y i + ), j µi+,j) t µi+,j) Ez n i +, j) / y + t σ i+,j) µi+,j) Ez n i, j) / y ).43) and E n+ z i, j) = + t σi,j) ) ɛi,j) E + t σi,j) z n i, j) ɛi,j) ) t ɛi,j) + t σi,j) ɛi,j) H n+ y H n+ y H n+ x +H n+ x i +, j) / x i, j) / x i, j + ) / y i, j ) / y..44)...4 Expression in one-dimension In the one-dimensional Cartesian coordinate system, let us assume the electromagnetic wave is only propagating along the z-direction. The Maxwell s curl 3

33 Hy Hy Hy Hy Z Ex Ex Ex Ex Figure.3: Position of the electromagnetic field components in the D FDTD method. equations from.) to.6) are reduced to E x t = ɛ H y z + σe x).45) and H y t = µ E x z + σ H y )..46) This D mode only involves the field components E x and H y. Following the spatial notation in Table.4, the one-dimensional equations for the FDTD method are derived as E n+ x k) = t σk) ) ɛk) E + t σk) x n k) ɛk) ) t ɛk) H n+ + t σk) y k + ) / z H n+ y k ) ) / z ɛk).47) and H n+ y k + ) = t σ k+ ) µk+ ) H n + t σ k+ ) y µk+ ) t µk+ ) + t σ k+ ) µk+ ) k + ) Ex n k + ) / z Ez n k) / z).48) 33

34 Electromagnetic Component Spatial Notation Time Step E x k n H y k + n + Table.4: The position of each electromagnetic field component in D..3 Dispersive Materials The electric permittivity of the dispersive material is complex and frequencydependent. Thus, propagating in the dispersive medium, the electromagnetic waves with different frequencies have different phase velocities [5]. Regarding the media parameters as constants, the conventional FDTD method ignores the dielectric properties of the dispersive materials. Describing the material characteristics over a wide band of frequencies, many models like Debye model and Cole-Cole model, are developed for biological simulations. These two models can be adapted to different frequency ranges. The Debye model can accurately generate the material properties of the biological tissues for the frequencies which are higher than GHz [6]. However, the Debye model is not sufficient to describe the media characteristics for the low frequencies, especially for frequencies below 0 MHz [6]. For such frequency range, the Cole-Cole model is a suggested alternative. This thesis uses the Debye model to simulate the dispersive materials for its accurate performance in the high frequency range. To simplify the implementation, the one-pole Debye relaxation model with auxiliary differential equations is adopted [][7][8]..3. Debye Model The constitutive relation for the electric flux density D with dielectric materials is D = ɛ 0 ɛ E + P.49) where ɛ is the relative permittivity at infinite frequency and P is the electric polarization caused by dispersive materials which increase the total flux density [9][0]. For the linear material dispersion, the polarization P is P = ɛ 0 χ p E.50) 34

35 χ p t) t 0 3 s) Figure.4: The time domain susceptibility χ p t) in the one-pole Debye model for fat. where χ p is the electric susceptibility. χ p is a function that can be expressed in either time domain or frequency domain. The dielectric dispersion is defined by the recursive convolution of χ p t) in time domain. Therefore, in the frequency domain, the frequency dependence of χ p ω) must have a causal inverse Fourier transform from the time domain. Note that, the linear dispersion represents the linearity of the polarization P to the electric field E, not the susceptibility χ p. Substituting.7),.49) and.50) into Ampere s Law.) yields H = ɛ 0 ɛ + ɛ 0 χ p ) E t + σe = ɛ 0 ɛ + ɛ 0 χ p ) E t + σ jω σ = ɛ 0 ɛ + χ p + jωɛ 0 E t ) E t..5) Equation.5) is adoptable for all kinds of the material models. The imaginary part of ɛ + χ p + σ jωɛ 0 ) is the loss factor in the medium, and the real part determines how much energy can be stored into the medium from an external electric field. 35

36 For the one-pole Debye medium, the time domain form of the electric susceptibility is expressed as χ p t) = ɛ s ɛ e τ t D ut).5) τ D where ɛ s is the static relative permittivity, τ D is the characteristic relaxation time and ut) is a function given by ut) = {, t > 0 0, t 0.53) Figure.4 displays the susceptibility χ p t) for the electric field with fat, which is generated by ɛ = 3.998, ɛ s = and τ D = ps. χ p t) rises to its maximum value just after t = 0 and decreases to zero exponentially. The value of χ p t) is zero before t = 0, thus the dispersive material at the FDTD grid is not excited before the wave arrives. To ensure the causality, the susceptibility χ p ω) in the frequency domain is derived as χ p ω) = ɛ s ɛ + jωτ D..54) Substituting.54) into.5), Ampere s Law for one-pole Debye medium can be rewritten as H = ɛ 0 ɛ + ɛ s ɛ + σ ) E + jωτ D jωɛ 0 t..55).3. Formulations of the FD-FDTD method The key point of the biomedical simulations is to generate the lossy biological medium for the microwave imaging, bio-electromagnetic hazard and other applications. Since the dispersive materials involve frequency-dependent dielectrics, the FD media can not be applied to the conventional FDTD method directly. There are two major techniques, the recursive convolution approach and the auxiliary differential equation ADE) approach, that can implement the material models like one-pole Debye model) to the time marching equation of the standard Yee algorithm. This thesis chooses the ADE scheme for its simplicity in implementation. 36

37 .3.. Recursive Convolution Scheme In 990 [0], Luebbers first derived the formulations of the FD-FDTD by using the recursive convolution scheme for Debye media. With the convolutional form of the polarization P, the flux density D is written as [7][8][0] D = ɛ 0 ɛ E + P = ɛ 0 ɛ E + n t τ=0 Et τ)χ p τ)dτ..56) Substituting.56) into Ampere s Law.), we can derive the update equations for the electric field E with dispersive materials, which are presented in [0][]. Although the recursive convolution approach can guarantee a second order accuracy for the FD-FDTD method, this approach requires a lot of floatingpoint operations and is only suitable for linear media..3.. Auxiliary Differential Equations A simple implementation for the FD-FDTD method was reported in 995 [][3], which is called the ADE scheme. This ADE scheme requires less floating-point operations than the recursive convolution scheme. However, the memory usage of ADE is nearly twice as the recursive convolution method. Two years later [4], Okoniewski improved the ADE scheme for Debye model, which demands the same or less memory than the recursive convolution method. According to Ampere s Law for one-pole Debye model in.55), the ADE scheme assumes that the electric flux density D is composed of the polarization P, the electric field component E and the electric current density J e, which is defined as.57) can be rewritten as D = ɛ 0 ɛ + ɛ s ɛ + σ ) E..57) + jωτ D jωɛ 0 j ω τ D D + jωd = j ω τ D ɛ ɛ 0 E + jω στ D + ɛ s ɛ 0 ) E + σe..58) Then transforming.58) into the time domain, we obtain τ D D) + D t t = τ D ɛ ɛ 0 E) + στ D+ɛ s ɛ 0 )E) +σe..59) t t 37

38 Thus the discretization form of.59) is expressed as E n+ =S E n S E n +S 3 D n+ S 4 D n +S 5 D n.60) where S = 4τ Dɛ ɛ 0 + στ D + ɛ s ɛ 0 ) t σ t τ D ɛ ɛ 0 + στ D + ɛ s ɛ 0 ) t + σ t, S = τ D ɛ ɛ 0 τ D ɛ ɛ 0 + στ D + ɛ s ɛ 0 ) t + σ t, S 3 = τ D + t τ D ɛ ɛ 0 + στ D + ɛ s ɛ 0 ) t + σ t, and S 4 = S 5 = 4τ D + t τ D ɛ ɛ 0 + στ D + ɛ s ɛ 0 ) t + σ t τ D τ D ɛ ɛ 0 + στ D + ɛ s ɛ 0 ) t + σ t. Equation.60) is the ADE of the FD-FDTD method for updating E. The equations for updating H are the same as.30),.3) and.3) for the FD- FDTD method. The time update equations for D are presented in [5]. To renew the electric field components at time step t = n + ) t, we need the information of the electric field and the electric flux density at t = n t and t = n ) t, and the electric flux density at t = n + ) t. Therefore, to maintain the explicit computation, the time marching algorithm for the FD-FDTD method should follow the order of H n = D n = E n = H n+. The implementation of ADE requires two additional vector variables to store the field information of D and E in the time step which is two steps earlier than the next time step. 38

39 .4 Perfectly Matched Layers The perfect electrical conductor PEC) boundary reflects all the outgoing waves for the conventional FDTD method, which bounds the electromagnetic components in a limited region. Simulating an infinite space in a truncated computational domain, the absorbing boundary condition ABC) is developed to eliminate the wave reflections from the boundary [6][7]. The Lindman boundary condition [8], Engquist-Majda boundary condition [9], Mur boundary condition [30], Bayliss boundary condition [3] and Liao boundary condition [3] are the most famous boundary conditions in 970s and 980s. All these ABCs adjust the value of the solutions at the edge of the grid by modifying the wave equations, and eliminate the reflections perfectly in the one dimensional scenario. In the two or three dimensional cases, since the reflections are generated at the boundary with different angles, these ABCs can only minimize the reflections but not absorb them completely [33]. Another remarkable drawback of these ABCs is that, the numerical calculation may become unstable with inhomogeneous medium [7]. As reported by Bérenger in 994 [34], instead of modifying the wave equations, the perfectly matched layer PML) ABC attenuates the traveling waves by inserting several PML layers, which are the layers of lossy materials or PML materials). The impedance of the PML is perfectly matched between the PML and non-pml region, thus there are no spurious reflections at the interior-pml interface. Surrounded by the PML ABC, the computational domain can be terminated by the PEC boundary condition. The PML can absorb not only the outgoing wave from the interior-pml interface to the PEC boundary, but also the returning waves from the PEC boundary. To improve the absorbing ability of PML, we can simply increase the thickness of layers. The PML is efficient in absorbing the incident waves with different propagation angles and frequencies, and has a great capability for lossy, dispersive, inhomogeneous materials. The implementations of PML are classified into two types, the split PML and un-split PML [35]. The split PML separates each field into two split-fields to solve the Maxwell s equations in Cartesian coordinates [34][36]. The un-split PML defines a special stretching coefficient to combine the split field components, which is a straightforward method to absorb the traveling waves. The un-split PML approaches, including the stretched-coordinate PML SC-PML) [37], the uniaxial PML UPML) [38][39][40] and the convolutional PML CPML) [][], keep the same propagation characteristics and reflection properties as the split 39

40 PML [4]. This thesis selects the CPML with complex frequency shifted CFS) stretching coefficient often called CFS-PML) to absorb the outgoing waves [4]. The CFS- PML is highly sufficient at absorbing the evanescent waves or very low frequency components [43][44]. For the metric coefficients of the CFS-PML formulation are irrelevant to the material characteristics, the CFS-PML formulation is applicable for all kinds of material models [7]..4. Split PML The Bérenger s split PML is developed by separating the electromagnetic field components into two subcomponents. For example, E x = E xy + E xz, E y = E yx + E yz and E z = E zx + E zy. Therefore.) can be split to ɛ E xy t + σ y E xy = H zx + H zy ) y.6) and ɛ E xz t + σ z E xz = H yx + H yz ) z.6) where σ y and σ z are the artificial PML conductivities. The remaining 0 split PML equations are derived from.) to.6) [35]. Since the traveling wave is attenuated rapidly in the PML, it is usual to implement the PML by the exponential time difference rather than the central difference [45][46]. The solution of the non-homogeneous differential equation.6) is composed of two parts [47][48]. The first one is E xygen t) = β e σ ɛ t.63) where β is not equal to zero..63) is the general solution of equation E xy t + σ y ɛ E xy = 0.64) which is the homogeneous differential equation that corresponds to.6). Based on.63), the electric field component E n+ xy gen can be written as E n+ xy gen = e σ ɛ t E n xy gen..65) 40

41 The second part of the solution of.6) is E xypar t) = σ y H zx + H zy ) y + β e σ ɛ t.66) which is a particular solution of.6) and β is a coefficient that needs to be determined by the initial conditions. In the FDTD method, at t = 0, the electric field components are initialized as zero E xypar t = 0) = 0). Thus.66) can be manipulated as 0 = σ y H zx + H zy ) y + β β = H zx + H zy )..67) σ y y Substituting the β of.67) into.66), at the next step t = t), we obtain E xypar t) = e σ ɛ t) H zx + H zy )..68) σ y y With the general solution of.65) and the particular solution of.68), the total increment of E xy from t = n t to t = n + ) t is obtained as E n+ xy = e σ ɛ t E n xy gen + e σ ɛ t) H zx + H zy )..69) σ y y Substituting the notations in Table. into.69), the exponential discretization of.6) is expressed as ),j,k) t ɛi+,j,k) e σyi+ Exy i+ n+ ), j, k = e σ yi+,j,k ) y n+ H zx i+ H n+ zx σyi+,j,k) t ɛi+,j,k) Exy n i +, j, k) +., j+, k) +H n+ zy i+, j+, k) ) i+, j, k) H n+ zy i+, j, k)..70) 4

42 Equation.70) is the update equation for the E xy components in PML. The rest time marching equations can be derived by applying the exponential discretization to the split Maxwell s equations, which are presented in [49]..4. Un-split PML There are three major Un-split PMLs, the SC-PML, UPML and CFS-PML. Adopting the same absorbing medium as Bérenger s PML, the SC-PML is a more effective implementation than the split PML. Based on the Maxwellian formulation, the UMPL applies an anisotropic medium to the FDTD method, which has the same absorbing performance as the split PML. Compared to the SC-PML and UPML, the CFS-PML has a better absorbing performance the reflection is smaller), a stronger capability efficient to absorb the evanescent waves or very low frequency components) and a simpler computational complexity less memory requirement) [4][50][5]. Thus we implement the CFS-PML to absorb the outgoing waves of the FD-FDTD method. Since the CPML is developed from the SC-PML, let us start from the SC-PML..4.. Stretched-Coordinate PML In [34], Bérenger pointed out that the PML can be realized with un-split electromagnetic fields, which is a straightforward implementation. Based on his idea, Chew and Weedon introduced a complex stretching coefficient to the split PML and derived a set of un-split equations. The implementation of Chew s un-split PML is based on the stretched coordinate method [37]. Substituting the plane wave solutions of E xy = E 0xy e jωt jkxx jkyy jkzz and E xz = E 0xz e jωt jkxx jkyy jkzz.7) into.6) and.6) yields jωɛ + σ y jωɛ )E xy = H zx + H zy ) y jωɛs y ω)e xy = H z y jωɛe xy = s y ω) H z y.7) 4

43 and jωɛ + σ z jωɛ )E xz = H yx + H yz ) z jωɛs z ω)e xz = H y z jωɛe xz = H y s z ω) z.73) where k x, k y and k z are the three-dimensional wavenumbers, and s y ω) = + σy jωɛ and s z ω) = + σz are the stretching coefficients of the split PML. The normalized jωɛ stretching coefficient is defined as s I ω) = + σ I jωɛ and s Iω) = + σ I jωɛ.74) where I represents the directions. The combination of.7) and.73) yields obtain jωɛe x = H z s y ω) y H y s z ω) z..75) Transforming.75) to the time domain with a stretched coordinate [37], we ɛ E x t = s y t) H z y s zt) H y z.76) where signifies the convolution product and s y t) and s z t) are the inverse Laplace transform of /s y ω) and /s z ω)..76) is an un-split Maxwell s equation for the E x component. The equations for the remaining electromagnetic field components are expressed as [37] ɛ E y t = s z t) H x z s xt) H z x,.77) ɛ E z t = s x t) H y x s yt) H x y,.78) ɛ H x t = s yt) E z y + s zt) E y z,.79) 43

44 ɛ H y t = s zt) E x z + s xt) E z x.80) and ɛ H z t = s xt) E y x + s yt) E x y..8) Based on the equations from.76) to.8), Chew implemented the FD- FDTD method in [37]. Equations from.76) to.8) are the un-split Maxwell s curl equations for both SC-PML and CFS-PML. The only difference between these two PML approaches is their stretching coefficient..4.. The CFS Stretching Coefficient With the stretching coefficient in.74), the SC-PML is incapable of absorbing the very low frequency or evanescent waves [5][53]. To overcome this limitation, Kuzuoglu and Mittra inserted a degree of freedom α x into.74) to make the PML medium causal [4]. Thus the CFS stretching coefficients are defined as and s I ω) = κ I + σ I α I + jωɛ 0.8) σ I s Iω) = κ I + αi + jωµ 0.83) where κ I amplifies the attenuation of the incident wave, and α I is the key parameter that can enable or disable the absorbing ability of the CFS-PML [7]. Note that, α I is greater than zero, and κ I is equal or greater than one. If ω α I /ɛ 0, the CFS-PML absorbs the traveling waves as the standard PML. If ω α I /ɛ 0, the imaginary part of the CFS stretching coefficient can be ignored and no wave attenuation happens in the CFS-PML The CFS-PML Formulation The time domain equation of the convolutional CFS-PML is derived from the stretched coordinate equations from.76) to.8). Applying the inverse Laplace 44

45 transform to the CFS stretching coefficient in.8), we obtain s I t) = δt) κ I σ I ɛ 0 κ e ) σ I ɛ 0 κ + I α I tut).84) I where δt) is the Dirac function [35]..84) into.76) yields For non-dispersive media, substituting ɛ 0 E x t = H z κ y y H y κ z z + ζ yt) H z y ζ zt) H y z.85) where ζ I t) = σ I ɛ 0 κ e ) σ I ɛ 0 κ + I α I tut)..86) I The discretization of.85) presents the explicit CFS-PML update equation for E x of E n+ x = E n x + t ɛ 0 [ Hz n+/ κ y y Hy n+/ κ z z + ψ n+/ hz ψ n+/ hy ].87) where ψ hy and ψ hz are the discrete form of ζ i t) convolved with the magnetic field components. They are realized by using the recursive convolution as ψ n+/ hz = p y ψ n / hz ψ n+/ hy = p z ψ n / hy Hz n+/ + q y y Hy n+/ + q z z.88).89) where p I = e t ) σ I ɛ 0 κ + I α I.90) q I = σ I κ I σ I + κ I α I ) p I )..9) Following the equations from.85) to.87), we can derive the update equations for E y and E z from.77) and.78). Based on the same procedure, the update equations for H x, H y and H z can be derived from.79),.80) and.8). In [5], Gedney introduced a set of ADE equations to implement the CFS-PML, which is equivalent to the implementation in this section. 45

46 Chapter 3 Numerical Dispersion The computational domain of the FDTD method is discretized by Yee cells in space and sampled by t in time. As a nature of the numerical discretization, the numerical dispersion means that the phase velocity of the traveling wave in the FDTD grid is different from that in the physical world. The coarse spatial sampling causes delay and distortion, which can be accumulated and lead to incorrect simulation results, for the signals in the computational domain. Derived from the numerical dispersion relation, the Courant-Friedrich-Lewy CFL) condition defines the upper bound of t with respect to the spatial step size x, y and z. For any t greater than the CFL limit, the numerical waves are traveling faster than the speed of light in the FDTD domain. Therefore the solutions of the Maxwell s curl equations are increased spuriously and the computation becomes unstable. Sections from 3. to 3.4 review the numerical dispersion, stability and phase velocity for the FDTD method. Section 3.5 presents the numerical dispersion relation for the CFS-PML and Section 3.6 derives the stability conditions of the CFS-PML. 3. Dispersion Relation for the Waves The dispersion relation is a function that relates the frequency and wavenumber, which introduces the interrelationship for the electromagnetic waves propagating in the temporal and spatial domain. The three-dimensional wave equation for homogeneous lossless medium is f x,y,z,t) x + f x,y,z,t) y + f x,y,z,t) z + ω c f x,y,z,t) = 0 3.) 46

47 k z k θ k y ϕ k x Figure 3.: The three dimensional wavenumber [54]. where c = ɛ0 µ 0 is the speed of light and f x,y,z,t) is the wave solution of f x,y,z,t) = f 0 e jkxx+kyy+kzz ωt). 3.) Substituting 3.) into 3.) yields k x k y k z + ω c )f x,y,z,t) = 0 k x + k y + k z = ω c. 3.3) Equation 3.3) is three dimensional dispersion relation for the electromagnetic waves. As shown in Figure 3., the three dimensional wavenumber is defined as k x = k sin θ cos ϕ, k y = k sin θ sin ϕ and k z = k cos θ k is the amplitude, θ and ϕ are the propagation angles [54].), 3.3) can be simplified to k = ω c. 3.4) In 3.4), it is clear that the wavenumber is proportional to the frequency. 47

48 3. Numerical Dispersion For an idealized space that has no electric and magnetic conductivities σ = 0 and σ = 0), the time recursive equations from.7) to.3) are manipulated as Ex i n+ + ), j, k = Ex n + t ɛ i +, j, k ) H n+ z i+,j+,k ) H n+ z i+,j,k ) y H n+ y i+,j,k+ ) H n+ y i+,j,k ) z,3.5) Ey i, n+ j + ), k Ez n+ i, j, k + ) = Ey n + t ɛ i, j +, k ) = Ez n + t ɛ H n+ x i,j+,k+ ) H n+ x i,j+,k ) z H n+ z i+,j+,k ) H n+ i, j, k + ) x z i,j+,k ) H n+ y i+,j,k+ ) H n+ y i,j,k+ ) x H n+ x i,j+,k+ ) H n+ y x i,j,k+ ),3.6),3.7) H n+ x i, j +, k + ) = H n x t µ i, j +, k + ) Ez n i,j+,k+ ) E z n i,j,k+ ) y Ey n i,j+,k+ ) Ey n i,j+,k ) z, 3.8) 48

49 and H n+ y H n+ z i +, j, k + ) i +, j +, k ) = H n y t µ = H n z t µ i +, j, k + ) Exi+ n,j,k+ ) Exi+ n,j,k ) z Ez n i+,j,k+ ) E z n i,j,k+ ) x i +, j +, k ) Ey n i+,j+,k ) Ey n i,j+,k ) x Exi+ n,j+,k ) Exi+ n,j,k ) y 3.9).3.0) The discretized form of the wave solutions for the three dimensional electromagnetic wave are expressed as E n x i, j, k) = E x0 e ji x k x+j y k y+k z k z n tω), 3.) E n y i, j, k) = E y0 e ji x k x+j y k y+k z k z n tω), 3.) E n z i, j, k) = E z0 e ji x k x+j y k y+k z k z n tω), 3.3) H n x i, j, k) = H x0 e ji x k x+j y k y+k z k z n tω), 3.4) H n y i, j, k) = H y0 e ji x k x+j y k y+k z k z n tω) 3.5) and H n z i, j, k) = H z0 e ji x k x+j y k y+k z k z n tω) 3.6) where k x = k sin θ cos ϕ, k y = k sin θ sin ϕ and k z = k cos θ are the components of the numerical wavevector in x, y and z directions. Note that, k is the amplitude of numerical wavenumber. Substituting the wave solutions 3.), 3.5) and 3.6) into the discretized 49

50 Yee equation 3.5) yields e j tω e j x k x Ex n i, j, k) = e j x k x Ex n i, j, k) + t ɛ e j tω e j x kx H n z i,j,k) y e j tω e j x kx H n y i,j,k) z ) e j y k y e j y k y ) e j z k z e j z k z. 3.7) 3.7) can be simplified to e j tω e j tω ) E n x i, j, k) = t ɛ Hz n i,j,k) y Hy n i,j,k) z ) e j y k y e j y k y ) e j z k z e j z k z. 3.8) Applying Euler s formulae sin x = ejx e jx ) into 3.8), we obtain j sin tω )En x i, j, k) = t sin z k z ɛ )Hn y i, j, k) z ) sin y k y ) Hn z i, j, k). y 3.9) Following the steps from equation 3.7) to 3.9), the discretized Yee equations from 3.6) to 3.0) can be rewritten as ) sin tω )En y i, j, k) = t sin x k x ) Hn z i, j, k) sin z k z ɛ x )Hn x i, j, k), z sin tω )En z i, j, k) = t ɛ sin tω )Hn x i, j, k) = t µ sin y k y ) Hn x i, j, k) y sin y k y ) En z i, j, k) y ) sin x k x ) Hn y i, j, k), x ) sin x k x ) En y i, j, k), z 3.0) 3.) 3.) 50

51 sin tω )Hn y i, j, k) = t sin z k z µ )En x i, j, k) z and sin tω )Hn z i, j, k) = t µ sin x k x ) En y i, j, k) x ) sin x k x ) En z i, j, k) x 3.3) ) sin y k y ) En x i, j, k). y 3.4) Based on the above six equations from 3.9) to 3.4), we obtain the three dimensional numerical dispersion relation of [7] ɛµ t sin tω ) = sin x kx x ) + sin y ky ) y + sin z kz ) z. 3.5) As discussed in Section 3., the dispersion relation 3.3) for the waves is linear. However, for the numerical dispersion 3.5), the relationship between the frequency and wavenumber is mainly governed by t, x, y and z. According to l Hopital s rule, we can derive that lim t 0 sin tω/) t = ω, lim sin x k x/) = k x, x 0 x sin y k lim y/) = k y sin z k and lim z/) = k z y 0 y z 0 z. Thus if and only if t, x, y and z approach zero, the numerical dispersion equals the wave dispersion. Based on the two dimensional FDTD equations from.4) to.44), the two dimensional dispersion equation is derived as ɛµ t sin tω ) = sin x kx x ) + sin y ky ) y 3.6) where k x = k cos ϕ and k y = k sin ϕ are the D numerical wavenumbers. The D numerical wavenumbers are a special case of the 3D numerical wavenumbers with cos θ = 0. where The one dimensional dispersion equation is derived from.47) and.48), ɛµ t sin tω ) = sin x kx ) x 3.7) where k x = k is the D numerical wavenumber. The D numerical wavenumber 5

52 is a special case of the 3D numerical wavenumbers with cos θ = 0 and sin ϕ = Courant-Friedrichs-Lewy Condition If the angular frequency is a complex value of ω = ω real +jω imag, the wave solution 3.) may either attenuate to zero ω imag is positive) or rise to infinity ω imag is negative) with respect to the time t. Thus the amplitude of the wave depends on the value of ω imag, which may cause instable wave behaviors. In order to ensure the stability, the angular frequency should be a real number and the term sin tω ) should be less than one. Then the numerical dispersion relation 3.5) can be written as ) t sin kx x εµ x + sin ky y y ) + sin kz z z ). 3.8) 3.8) can be written as t ) sin kx x x εµ ) + sin ky y y ) + sin. 3.9) kz z z Since the maximum value of the sinusoidal functions in 3.9) is one, thus the general form of 3.9) is [55][56] t εµ ) ) ). 3.30) + + x y z If x = y = z, 3.30) can be simplified to t x εµ 3. Inequality 3.30) is the Courant-Friedrichs-Lewy condition for the FDTD method, which defines the upper limit of t for stable computation. Based on 3.6), the CFL condition 5

53 for the two-dimensional case is t εµ ) ). 3.3) + x y If x = y, 3.3) can be simplified to t x εµ. Based on 3.7), the one-dimensional CFL condition is t εµ ) = x εµ. 3.3) x In simulations, the wavenumber may become a complex value, if we set t greater than the upper limit of the CFL condition [7][57]. As a result, the phase velocity of the numerical wave may be faster than the speed of light and the amplitude of the wave in the FDTD grid increases exponentially [57]. 3.4 Numerical Phase Velocity Section 3.3 reviewed the CFL condition of the FDTD method to set the temporal size for stable computation. However, we can not set the Yee cell with any spatial size, even if t satisfies the CFL condition. The insufficient spatial sampling generates numerical dispersion, which causes incorrect numerical phase velocities. In the numerical calculation, the shortest wavelength of the excitation is λ min = c/f highest, where f highest is the highest frequency of the excitation. Thus, under the Nyquist sampling limit, the spatial size of each Yee cell should smaller than x = λ min /. However the Nyquist limit only guarantees the minimum accuracy for the FDTD computation and leads to large distortion for the numerical wave in different propagation angles. To improve the accuracy, more than two samples per minimum-wavelength is required. This section shows how to obtain the numerical phase velocity and measure the phase velocity error. 53

54 3.4. One Dimensional Phase Velocity Based on the numerical dispersion relation in 3.7), the numerical wavenumber k x can be derived as k x = ɛµ x x arcsin sin ω t ) t ). 3.33) Thus the D numerical phase velocity is defined as c D = ω kx = ω x )). 3.34) x ω t arcsin c t sin We can analyze the numerical dispersion by comparing the numerical phase velocity with the physical phase velocity. Thus the D numerical phase velocity error is given by P D = c c D c. 3.35) 3.4. Two Dimensional Phase Velocity In the two dimensional case, the electromagnetic wave involves a propagation angle ϕ. For the square Yee cell x = y), the dispersion relation 3.6) in free space is rewritten as ) x sin tω c t ) = sin x k cos ϕ ) + sin x k sin ϕ ) 3.36) If the numerical wave is propagating on the angles of ϕ = 0, ϕ = π/, ϕ = π and ϕ = 3π/, equation 3.36) can be reduced to x c t sin tω ) = sin x k ). 3.37) Then the numerical phase velocity for the wave propagating along the major 54

55 grid axes is expressed as c D = ω k = ω x )). 3.38) x ω t arcsin c t sin If the numerical wave is propagating on the angles of ϕ = π/4, ϕ = 3π/4, ϕ = 5π/4 and ϕ = 7π/4, equation 3.36) can be reduced to x sin tω c t ) = sin x k ). 3.39) Thus the numerical phase velocity for the wave propagating along the grid diagonals is expressed as c D = ω kx = ω x )). 3.40) x ω t arcsin c t sin Based on 3.38) and 3.40), the D numerical phase velocity error is given by P D = c min[ c D, c D ] c, 3.4) and the anisotropy error for the D FDTD method is P Da = c D c D min[ c D, c D ]. 3.4) If t satisfies the CFL condition, the numerical phase velocity is always slower than the physical phase velocity. Thus P D shows the loss of the wave propagation speed in numerical computations and P Da indicates the distortion of the traveling waves in the electromagnetic fields. The Nyquist limit provides the maximum spatial size for each Yee cells. To improve the accuracy, we can minimize x to adequately sample the spatial field. Based on the numerical phase velocity error and anisotropy error, the sampling rate of at least ten cells per λ min is often accepted [7]. 55

56 3.5 Numerical Dispersion for PML Equation 3.5) is the dispersion relation for the electromagnetic field with lossless materials, which is derived from the explicit central difference equations. In the lossy PML region, the amplitude of the signal is reduced rapidly. Therefore, in the PML, the central difference cannot sufficiently approximate the wave attenuation, especially in the first layers. Thus the PML introduces the exponential difference to discretize time steps of the electromagnetic fields One dimensional Dispersion Derived from.9) and.0), the one dimensional Maxwell s curl equations with PML conductivity are written as ɛ E y t + σ x E y = H z x 3.43) and µ H z t + σ xh z = E y x 3.44) where σ x and σx is perfectly matched as σ x = σ x. Applying the exponential ɛ µ difference for the partial differentials of time and central difference for the partial differentials of space to 3.43) and 3.44), the discretized PML equations with non-dispersive materials ɛ = ɛ 0 and µ = µ 0 ) are written as and where y i) = a x Ey n H n+ z i) b x E n+ H n+ z ) i + n+ H z x ) i i + ) = a xh n z i + ) E b y n i + ) Ey n i) x x 3.45) 3.46) a I = e σ I t ɛ 0, a I = e σ I t µ 0, 3.47) 56

57 b I = a I and b I = a I. 3.48) σ I The discretized wave solution for the x-directed electromagnetic waves are σ I E n y i) = E y0 e jωn t j k xi x 3.49) and Hz n i) = H z0 e jωn t j k xi x. 3.50) According to the PML s matching condition σ x µ 0 = σxɛ 0 ), we can derive a = a and bɛ 0 = b µ 0 [35]. Then substitution of 3.49) and 3.50) into 3.45) and 3.46) yields e jω t E n y i) = a x E n y i) b x e j ω t H n z i) x ) e j k x x e j k x x 3.5) and e j ω t e j k x x Hz n i)=a x e j ω t e j k E x x Hz n i) b y n i) ) x e j k x x.3.5) x Reorganizing 3.5) by collecting E n y to the left hand side and the rest terms to the right hand side, then substituting this E n y into 3.5), the numerical dispersion relation for the one-dimensional PML is derived as ) jω t e a x e jω t = b xb j k x x x e j k x x e x = a x) c ɛ 0 σ x x j k x x e j k x x e.3.53) If the matched conductivities σ x and σ x equal zero, then we can obtain that a x = a x =, b x = t ɛ 0 and b x = t µ 0 L Hopital s rule). Therefore, in the nonconductivities case, the dispersion relation 3.53) with the exponential discretization is equivalent to the one with central finite discretization in 3.7), although their time marching equations are different. 57

58 3.5. Two Dimensional Dispersion The two dimensional dispersion can be derived from either the TE mode or TM mode. For example, the TM mode equations for the PML with non-dispersive materials are expressed as ɛ 0 E zx t + σ x E zx = H y x, 3.54) ɛ 0 E zy t + σ y E zy = H x y, 3.55) µ 0 H x t + σ yh x = E zx + E zy ) y 3.56) and µ 0 H y t + σxh y = E zx + E zy ). 3.57) x Applying the exponential and central difference approximation, the discretization of 3.54), 3.55), 3.56) and 3.57) yields zx i, j) = a x Ezxi, n H n+ y j) + b x E n+ ) i + n+ H y x ) i, 3.58) zy i, j) = a y Ezyi, n H n+ x j) b y E n+ ) i + n+ H x y ) i, 3.59) and H n+ x i, j + ) = a yh n x i, j + ) b y [ E n y zx i, j + ) + Ezyi, n j + ) Ezxi, n j) Ezyi, n j) ] 3.60) H n+ y i +, j) = a xh n y i +, j) + b [ x E n x zx i +, j) + Ezyi n +, j) Ezxi, n j) Ezyi, n j) ]. 3.6) 58

59 Applying the discrete wave solutions of E n zx i, j) = E zx0 e jωn t j k xi x j k yj y, 3.6) E n zy i, j) = E zy0 e jωn t j k xi x j k yj y, 3.63) H n x i, j) = H x0 e jωn t j k xi x j k yj y 3.64) and H n y i, j) = H y0 e jωn t j k xi x j k yj y 3.65) into 3.58), 3.59), 3.60) and 3.6), we obtain e j e jω t Ezxi, n j) = a x Ezxi, n ω t Hy n i) ) j) + b x e j k x x e j k x x, 3.66) x e jω t E n zyi, j) = a y E n zyi, j) b y e j ω t H n x i) y ) e j k y y e j k y y, 3.67) e j ω t e j k y y Hx n i, j) = e j ω t e j k y y a yhx n i, j) b y E n zx i, j) + Ezyi, n j) ) ) e j k y y y 3.68) and e j ω t e j k x x Hy n i, j) = e j ω t e j k x x a xhy n i, j) + b x E n zx i, j) + Ezyi, n j) ) ) e j k x x. 3.69) x Then substitution of 3.68) and 3.69) into 3.67) and 3.66) yields e j ω t a x e j ω t ) E n zx i, j) = a x) c ɛ ) 0 e j k x x σx x e j k x x E n zxi, j) + Ezyi, n j) ) 3.70) 59

60 and e j ω t a y e j ω t ) E n zy i, j) = a y) c ɛ ) 0 e j k y y σy y e j k y y E n zxi, j) + Ezyi, n j) ). 3.7) Reorganizing 3.70) by collecting E n zx to the left side and the rest terms to the right side, then substituting this E n zx into 3.7), the numerical dispersion relation for two-dimensional PML is derived as a x ) c ɛ ) 0 e j k x x σx x e j k ) x x e j ω t a y e j ω t + a y) c ɛ ) 0 e j k y y σy y e j k ) y y e j ω t a x e j ω t ) ) = e j ω t a x e j ω t e j ω t a y e j ω t. 3.7) 3.6 Further Investigation for the Dispersion of CFS-PML It is a challenge to derive the stability condition for the PML, since the complex stretching coefficient of the PML generates a complex wavenumber. Thus it is usual to evaluate the stability of PML by numerical experiments. Based on the dispersion relations in Section 3.5, this section presents some conditions to determine the temporal size of the D and D CFS-PML Investigation for One Dimensional Dispersion The CFS-PML has an additional matched condition, which is α x µ 0 = α xɛ 0 [35][43]. If κ x = κ x, the one dimensional wavenumber in the CFS-PML is expressed as [58] k x = ω c sx s x = ω c κ x + σ x α x + jωɛ 0 ). 3.73) 60

61 e where Substituting 3.73) into 3.53), we obtain ) jω t a x e jω t = a x) c ɛ 0 e j xω c A e xω ) c B e j xω c A e xω c B σx x 3.74) A = κ x + σ xα x αx + ω ɛ 0 and B = ωσ xɛ 0 αx + ω ɛ. 3.75) 0 where Application of Euler s formulae e jx = cos x + j sin x) to 3.74) yields a x ) c ɛ 0 σ x x a x ) cos ω t ) + j + a x) sin ω t ) ) = Ψ Ψ ) cos xω c A) + jψ + Ψ ) sin xω ) c A) 3.76) Ψ = e xω c B. 3.77) Equation 3.76) can be separated into a real part equation of a x ) c ɛ 0 σ x x + a x) cos ω t ) ) a x = Ψ + Ψ ) cos xω ) A) ) c 3.78) and an imaginary part equation of a x) sinω t) = a x) c ɛ 0 σ x x Manipulation of 3.78) Ψ Ψ ) sin xω A). 3.79) c Since 0 cos ω t ), the real part equation 3.78) can be written as + a x ) a x) c ɛ 0 σ x x Ψ + Ψ ) cos xω ) A) ) c a x ). 3.80) 6

62 In the absence of PML conductivityσ x = 0), Equation 3.80) is always satisfied since a x equals to one. However the term a x ) does not equal zero in the CFS-PML, and 3.80) can be divided into two conditions. The first one is and the second one is ax 0 + a x X A c ɛ 0 Ψ + Ψ ) cos xω ) A) σx x c ) c ɛ 0 σx x Ψ + Ψ ) cos xω A) c 3.8) ) + Y A. 3.8) 0 X A x = 0.5mm x = mm x = mm π 5 π 5 3π 5 4π 5 π xω c Figure 3.: X A varying xω from zero to π for the cases of x = mm, x = c mm and x = 0.5 mm. Let us define the left hand side of 3.8) as X A, the right hand side of 3.8) as Y A and the maximum temporal size for stable computation in the D FDTD method as t cfld. Since σ x should be a positive number and κ x should be equal to or greater than one [35][5][59], let us set σ x = 0.00, κ x = and α x = to calculate the value of X A. Figure 3. displays the variations of X A with xω for the cases c of x = mm, x = mm and x = 0.5 mm. According to the Nyquist 6

63 Y A t = t cfld t = 0.8 t cfld t = 0.5 t cfld 0 π 5 π 5 3π 5 4π 5 π xω c Figure 3.3: Y A varying xω from zero to π for the case of x = mm. t cfld c is the upper limit of the D CFL condition for t. sampling theorem, the greatest xω c in numerical simulation is π. Thus we plot Figure 3. in the range of 0 xω c π. In Figure 3., with the parameters of σ x = 0.00, κ x = and α x = 0.006, X A is smaller than one, which satisfies the condition in 3.8). Therefore this set of parameters is used to generate the stretching coefficient of CFS-PML in later calculations and simulations. The maximum value of X A is obtained when cos xω c A) =. Therefore 3.8) can be further manipulated to c ɛ 0 σ x x Ψ + Ψ ). 3.83) Since a x is a function of t, we plot Y A with several values of t for the case of x = mm in Figure 3.3. According to the results in Figure 3.3, Y A is greater than zero if t is equal to or smaller than t cfld. In further calculations, the results of using x = mm and x = 0.5 mm are identical to the results in Figure 3.3, which proves that the change of x does not affect the value of Y A. The lower bound of Y A takes place at xω A = π. Then condition 3.8) can be c 63

64 further manipulated as ) ax c ɛ ) 0 e πb + a x σx x A + e πb A. 3.84) Substituting 3.47) into 3.84) yields e σ x t ɛ 0 + e σ x t ɛ 0 c ɛ ) 0 e πb σx x A + e πb A e σ x t ɛ 0 + e σ x t ɛ 0 + e σ x t ɛ 0 cɛ ) 0 e πb A + e πb A σ x x cɛ ) 0 e πb A + e πb A + σ x x cɛ ) 0 e πb A + e πb A σ x x cɛ ) 0 e πb A + e πb A σ x x + e σ x t ɛ 0 e σ cɛ 0 x t ɛ 0 σ x x cɛ 0 σ x x cɛ 0 t ɛ 0 σ ln x x σ cɛ 0 x σ x x Since t is a real number, the term cɛ 0 σ x x greater than one. e πb A e πb A + e πb A ) + e πb A + e πb A ) e πb A + e πb e πb A + e πb A ) +. A ) 3.85) + e πb A ) in 3.85) should be Manipulation of 3.79) In numerical calculation, sinω t) is a positive number from zero to one for 0 ω t π. Thus 3.79) can be written as 0 a x) c ɛ 0 σ x x Ψ Ψ ) sin xω A) a c x). 3.86) 64

65 t pml / t cfld ) x = mm x = mm x = 0.5mm 0 0 π 5 π 5 3π 5 4π 5 xω c Figure 3.4: The difference between t pml and t cfld varying xω from zero to c π for the cases of x = mm, x = mm and x = 0.5 mm. t cfld is the upper limit of the D CFL condition. π In the CFS-PML, since Ψ > Ψ and 0 < a x <, 3.86) can be simplified to ) ax c ɛ 0 + a x σx x Ψ Ψ ) sin xω A). 3.87) c Substituting 3.47) into 3.87) and applying the mathematical manipulation done in 3.85) to 3.87), the upper limit of t is obtained as t ɛ c ɛ 0 Ψ Ψ ) sin xω A) + 0 σ ln x x c t σ c ɛ pml. 3.88) x 0 Ψ σ Ψ ) sin xω A) x x c c ɛ 0 Note that, σx x Ψ Ψ ) sin xω A) should be greater than one in 3.88). c Equation 3.88) defines the upper limit of t for stable computation with CFS-PML ABC. Let us define the right hand side of 3.88) as t pml. In order to use any t under the CFL condition for CFS-PML, t pml should be equal to or greater than t cfld for any xω c from zero to π. Otherwise the computation may 65

66 suffer from the late-time instability. t pml can be further manipulated as t pml = ɛ 0 ln + σ x c ɛ 0 σ x x Ψ Ψ ) sin xω c A) 3.89) which indicates that, t pml decreases with increasing Ψ Ψ ) sin xω A). However, in the range of 0 xω π, the maximum value of Ψ Ψ ) takes place c c at xω = π, while sin xω c c minimum value of t pml is in the range of A) takes its maximum value at xω π A < xω c A = π. Thus the c < π. With the parameters of CFS-PML which are used to produce Figure 3., Figure 3.4 compares t pml with t cfld, and shows that t pml is greater than t cfld wavenumber. for any complex 3.6. Investigation for Two Dimensional Dispersion PMLσ x, σ x, σ y, σ y) Region Top PML0, 0, σ y, σ y) PMLσ x, σ x, σ y, σ y) PEC Region Left PMLσ x, σ x, 0, 0) Interior Region PML interface Region Right PMLσ x, σ x, 0, 0) y ϕ x PMLσ x, σ x, σ y, σ y) Region Down PML0, 0, σ y, σ y) PMLσ x, σ x, σ y, σ y) Figure 3.5: The structure of the PML [35]. The interior is vacuum. For the traveling waves to pass through the interior-pml interface without any reflections, the conductivities in the PML should be perfectly matched to the 66

67 conductivities in the interior region. By using only one pair of the conductivities σ x and σx) in the x-direction of the PML region while keeping the other conductivities σ y and σy) as the same as those in the interior field, and only using σ y and σy in the y-direction while σ x and σx are the same as the interior conductivities, the PML eliminates the interface reflection for any incidence angles and frequencies. Figure 3.5 shows that, the interior region of vacuum is surrounded by the CFS-PML and the total computational domain is bounded by PEC. The whole PML region is composed of four corners and four sides. The CFS-PML conductivities σ x and σx are set to zero in Region Top and Region Down, while σ y and σy are perfectly matched as σ y µ = σyɛ. For CFS-PML in Region Left and Region Right, the conductivities are set to σ x,σx,0,0). Note that, the four corners are overlapped by two adjacent CFS-PMLs, which have two pairs of the PML conductivities PML in the Four Corners For the CFS-PML in the corners of the FDTD space, let us assume that σ x = σ y, σ x = σ y and x = y. Then 3.7) is simplified to ) jω t e a x e jω t = a x) c ɛ j k x x 0 e j k x x e σx x + a x) c ɛ 0 σ x x j k y x e j k y x e where k x = ω c κ x + ) σx α x+jωɛ 0 cos ϕ and k y = ω κ c y + ) σy α y+jωɛ 0 sin ϕ. 3.90) The case of ϕ = 0, π/, π and 3π/ With α x = α y and κ x = κ y, if the wave is propagating along the major grid axes ϕ = 0, π/, π and 3π/), 3.90) can be reduced to ) a x ) c ɛ 0 σ e j k x x x x e j k ) x x = e j ω t a x e j ω t 3.9) 67

68 0.8 Y A 0.6 t = t cfld t = 0.8 t cfld t = 0.5 t cfld 0.4 π 0 π 3π 4π xω c Figure 3.6: Y A varying xω from zero to π while x = mm. t cfld is the c upper limit of the D CFL condition. π which is identical to the D PML dispersion equation in 3.53). Thus equations 3.83), 3.85) and 3.88) are the conditions for choosing the parameters of 3.9). In order to guarantee the stability in D computation, 3.85) and 3.88) have to be greater than t cfld, which is the maximum t under D CFL limit. With the parameters which are used to produce Figure 3., Y A is displayed in Figure 3.6 and the comparison of t pml and t cfld is plotted in Figure 3.7. Since the results of x = mm and x = 0.5 mm are identical to x = mm, here we only show the variation of Y A for the case of x = mm in Figure 3.6. In Figure 3.6 and Figure 3.7, both 3.8) and 3.88) are satisfied for any complex wavenumber under the D CFL condition with the above parameters. The case of ϕ = π/4, 3π/4, 5π/4 and 7π/4 With α x = α y and κ x = κ y, for the propagation angles of ϕ = π/4, 3π/4, 5π/4 and 7π/4, the dispersion equation 68

69 t pml / t cfld ) x = mm x = mm x = 0.5mm 0 0 π 5 π 5 3π 5 4π 5 xω c Figure 3.7: The difference between t pml and t cfld varying xω from zero to c π for the cases of x = mm, x = mm and x = 0.5 mm. t cfld is the upper limit of the D CFL condition. π 3.90) can be written as ) jω t e a x e jω t = a x) j k c ɛ x x 0 e j k x x e σx x. 3.9) yields Substituting Euler s formulae and the complex wavenumber 3.73) into 3.9) a x ) c ɛ 0 σ x x where a x ) cos ω t ) + j + a x) sin ω t ) ) = Ψ Ψ ) cos xω c A) + jψ + Ψ ) sin xω ) c A) 3.93) xω Ψ = e c B. 3.94) 69

70 3.93) consists of two parts. The real part is a x ) c ɛ 0 σ x x and the imaginary part is + a x) cos ω t ) ) a x Ψ + Ψ ) cos xω ) A) ) = ) c a x) sinω t) = a x) c ɛ 0 Ψ σx x Ψ ) sin xω c A). 3.96) Since 0 cos ω t ), the real part 3.95) is written as 0 a x) c ɛ 0 σ x x Ψ + Ψ ) cos xω ) A) ) + + a x ) + a x) c 3.97) which can be split to two inequalities. The first one is and the second one is X B c ɛ 0 Ψ σx x + Ψ ) cos xω ) A) 3.98) c ) ax c ɛ 0 0 Ψ + a x σx x + Ψ ) cos xω ) c A) + Y B. 3.99) Since 0 sinω t), Ψ > Ψ and 0 < a x <, the imaginary part 3.96) is written as c ɛ 0 σ x x Ψ Ψ ) sin xω c A) + a x a x. 3.00) Substituting 3.47) into 3.00), the condition for t is derived as t ɛ c ɛ 0 Ψ 0 σ ln x x Ψ ) sin xω c A) + σ c ɛ x 0 Ψ σx x Ψ ) sin xω c A) t dpml 3.0) where c ɛ 0 Ψ σx x Ψ ) sin xω c A) should greater than one. Let us define the left hand side of 3.98) as X B, the right hand side of 3.99) 70

71 X B x = 0.5mm x = mm x = mm Figure 3.8: X B varying xω c mm and x = 0.5 mm. π 5 π 5 xω c 3π 5 from zero to π for the cases of x = mm, x = 4π 5 π as Y B and the right hand side of 3.0) as t dpml. The minimal Y B is obtained at cos xω c xω A) =. Then substituting 3.47) and c = π A cɛ0 t ɛ 0 ln σ x x σ x e πb A + e πb cɛ0 πb e A + e πb σ x x into 3.99) yields A ) + 3.0) A ) where cɛ πb 0 σ x x e A + e πb A ) should be greater than one. The conditions 3.98), 3.0) and 3.0) are stricter than 3.83), 3.85) and 3.88). Thus, for the CFS-PMLs in the four corners of the FDTD space, we should use conditions 3.98), 3.0) and 3.0) to choose the parameters. With the CFS stretching coefficient which is used to produce Figure 3., Figure 3.8, 3.9 and 3.0 plot the the value of X B, Y B and t dpml over xω. The results in c Figure 3.8 all fulfill condition 3.98), which indicates that the parameters of the CFS-PML can be used in both D and D calculations. The variation of Y B for the case of x = mm is displayed in Figure 3.6. The results of Y B in the cases 7

72 Y B 0.5 t = t cfld t = 0.8 t cfld t = 0.5 t cfld 0 0 π π 5 π 5 xω c Figure 3.9: Y B varying xω from zero to π for the case of x = mm. t cfld c is the upper limit of the D CFL condition for t. 3π 5 4π 5 of x = mm and x = 0.5 mm are identical to x = mm. Figures 3.9 and 3.0 show that, conditions 3.99) and 3.0) are satisfied for any complex wavenumber with the above parameters PML in the Four Sides As shown in Figure 3.5, the conductivity is set to σ x, σ x, 0, 0) for the CFS-PML in Region Left and Region Right, and 0, 0, σ y, σ y) in Region Top and Region Down. If σ x = σ y, κ x = κ y, α x = α y and x = y, the dispersion relations in the left and right sides are identical to the top and bottom. For the CFS-PML in Region Left and Region Right, when we assume that σ x = σ y and x = y, the numerical dispersion 3.7) is simplified to a x ) c ɛ ) 0 e j k x x σx x e j k ) x x e j ω t e j ω t + c t x = ) e j k y x e j k ) y x e j ω t a x e j ω t e j ω t a x e j ω t ) e j ω t e j ω t ) 3.03) 7

73 t dpml / t cfld ) x = mm x = mm x = 0.5mm 0 0 π 5 π 5 xω c Figure 3.0: The difference between t dpml and t cfld varying xω from zero c to π for the cases of x = mm, x = mm and x = 0.5 mm. t cfld is the upper limit of the D CFL condition for t. 3π 5 4π 5 π where k x = ω c ) κ x + σx α x+jωɛ 0 cos ϕ and k y = ωκ c y sin ϕ. The case of ϕ = 0 and π If the wave is propagating along the x-axis ϕ = 0 and π), the wavenumber in the y-direction is zero k y = 0). In this case, 3.03) is reduced to 3.9). Therefore the conditions 3.83), 3.85) and 3.88) can be used to choose the parameters for the propagation angles of ϕ = 0 and π in Region Left and Region Right. The case of ϕ = π/ and 3π/ If the wave is propagating in the y-direction ϕ = π/ and 3π/), we obtain that k x = 0 and k y = ω c κ y. Then 3.03) is reduced to c t e j k y x x e j k ) y x) = e j ω t e j ω t c t x sin ω c κ y x) = sin ω t). 3.04) Since 0 sin ω t), we can derive that 0 c t sin ω κ x c y x). 73

74 The maximal value of sin ω κ c y x) is one, thus t x, which is the same as c the CFL condition. In summary, in the four sides of the PML and the propagation angles of ϕ = 0, π/, π and 3π/, the computation is stable with the t, which is chosen under the CFL condition and the conditions 3.83), 3.85) and 3.88). The case of ϕ = π/4, 3π/4, 5π/4 and 7π/4 Let us assume that κ x = κ y, for the wave is propagating along the diagonals of the grid ϕ = π/4, 3π/4, 5π/4 and 7π/4) in Region Left and Region Right, the wavenumbers are manipulated as k ) x = ω c κ x + σx α x+jωɛ 0 and k y = ω c κ x. Then 3.03) is reduced to a x ) c ɛ 0 σ e j xω x x A c e xω B xω j c e A c e xω B) ) c e j ω t e j ω t + c t x e j κxω x κxω j c e x) ) c e j ω t a x e j ω t ) ) = e j ω t a x e j ω t e j ω t e j ω t. Substituting Euler s formulae into 3.05) yields + c t x where 3.05) a x ) c ɛ 0 F σx x A + jf B ) sin ω t ) sin κ yω x c ) + a x) sin ω t ) )) a x + j a x) sinω t) = + a x) sin ω t ) )) a x + j a x) sinω t) sin ω t ) 3.06) F A = Ψ + Ψ ) cos xω c A) 3.07) and F B = Ψ Ψ ) sin xω c A). 3.08) 74

75 3.06) can be separated into two parts. The real part is a x ) c ɛ 0 σ x x c t x sin ω t )F A + c t x a x) c ɛ 0 σ x x ) ) sin κ yω x c ) a x ) + a x) sin ω t = a x ) + a x) sin ω t ) ) sin ω t ) sin ω t )F A + c t x sin κ yω x c ) a x) sin κ yω x c ) + a x) sin ω t ) = a x) sin ω t ) + a x) sin 4 ω t ) and the imaginary part is a x ) c ɛ 0 σ x x sin ω t )F B + c t x sin κ yω x c ) a x) sinω t) a x) sinω t) sin ω t ) = ) 3.09) + Let us manipulate the real part equation 3.09) as + a x) sin 4 ω t ) ) a x ) c ɛ 0 F σx x A a x ) c t x sin κ yω x c ) + a x) sin ω t ) where + c t x sin κ yω x c ) a x) = 0. Since + a x) is a positive value, 3.) can be written as 3.) sin 4 ω t ) + Z A sin ω t ) + Z A = 0 3.) Z A = a x) c ɛ 0 + a x)σx x F A a x) + a x) c t x sin κ yω x c ) 3.3) 75

76 and Z A = c t x sin κ yω x c ) a x) + a x). 3.4) If Z A 4Z A < 0, the left hand side of 3.) is greater than zero for any real ω, which is instable for any t. For Z A 4Z A 0, let us write 3.) as sin ω t ) + Z A ) ZA 4Z A sin ω t ) + Z A + which can be separated into two equations of and sin ω t ) + Z A ZA 4Z A sin ω t ) + Z A + ZA 4Z A Since 0 sin ω t ), 3.6) and 3.7) can be written as ) ZA 4Z A = 0 = 0 3.6) = 0 3.7) 3.5) 0 Z A + Z A 4Z A 3.8) and 0 Z A Z A 4Z A 3.9) When Z A 0, Z A + Z A 4Z A ) and Z A Z A 4Z A ) are equal to or smaller than zero due to Z A 0. Z A + Z A 4Z A ) equals zero when Z A = 0. Z A Z A 4Z A ) equals zero when Z A = Z A = 0. However, Z A equals zero only at ω x = 0. Therefore, for Z c A 0, 3.5) is not satisfied for any real ω x, which may cause numerical instability. c When Z A < 0, Z A Z A 4Z A ) and Z A + Z A 4Z A ) is alway equal to or greater than zero due to Z A 0. Thus 3.8) and 3.9) 76

77 can be simplified to and Z A 4Z A + Z A 3.0) Z A 4Z A + Z A. 3.) 3.5) is valid when either 3.0) or 3.) is satisfied. Consequently, the parameters in 3.) can be chosen based on the following conditions Z A 4Z A 0, Z A < 0 and Z A 4Z A + Z A and ZA 4Z A + Z A. In the range of 0 ω t π, sin ω t ) equals zero only at ω t = 0. When ) = 0, the left hand side of 3.09) equal zero since ω t = 0. Thus the sin ω t imaginary part equation 3.09) is valid for any t if sin ω t ) = Summary After reviewing the conception of numerical dispersion and stability of the FDTD method, this chapter presented the dispersion relation and stability condition of the CFS-PML in Sections 3.5 and 3.6. Section 3.5 introduced the numerical dispersion relation for the CFS-PML by discretizing Maxwell s curl equations with exponential difference. Based on the derived dispersion relation in Section 3.5, Section 3.6 derived the stability conditions of the CFS-PML in both D and D scenarios. In the D computation, the parameters of the CFS-PML should satisfy condition in 3.8) and the temporal increment should satisfy conditions in 3.85) and 3.88). In the D case, 3.98) is the condition for setting parameters of the CFS-PML and 3.0) and 3.0) are the conditions for determining the time discretization of the CFS-PML. 77

78 Chapter 4 Spatial Filtering Approach The numerical dispersion is a non-physical characteristic of the FDTD method that constrains the spatial and temporal size to guarantee the accuracy and stability. For the simulations with electrically small objects or dispersive materials, the spatio-temporal resolution is required to be high, which costs massive memory and long execution time. To improve the computational speed, the implicit schemes are developed for extending the CFL limit unconditionally. However these schemes generate undesirable errors when the temporal interval is increased. As an alternative to these implicit schemes, the spatial filtering approach removes the spurious frequency components in the electromagnetic fields to gain large temporal interval [3][4][60]. This chapter describes the principle of the spatial filtering approach and shows the procedures of this approach. 4. Implicit FDTD Methods The conventional explicit FDTD method, which is computationally simple and numerically accurate, updates the electromagnetic fields with the information at the previous time steps. However, due to the CFL condition, the maximum temporal discretization of the explicit FDTD method depends on the spatial discretization. The implicit schemes, such as the alternating direction implicit FDTD ADI-FDTD) method [6][6][63], Crank-Nicolson FDTD CN-FDTD) method [64] and locally one dimensional FDTD LOD-FDTD) method [65], are developed to relax or overcome the CFL condition. These implicit schemes reformulate the conventional Yee algorithm and update the electric and magnetic fields simultaneously at each time steps. In the implicit FDTD methods, the 78

79 maximum temporal discretization is independent of the spatial discretization. To exceed the CFL limit unconditionally, the Crank-Nicolson CN) scheme is applied to Maxwell s equations in order to reformulate the conventional FDTD method [66][67]. The irreducible heavy matrix computation restrains the further application of the CN-FDTD method, although the CN-FDTD method presents the potential of giving high accuracy computation and small numerical dispersion error [68]. The ADI-FDTD, Douglas-Gunn FDTD and split-step FDTD methods are developed to reduce the computational complexity of the CN-FDTD method [64][69][70]. However these methods generate additional numerical errors. The numerical approximation of the ADI-FDTD method is second order accurate in both time and space. The accuracy of the solution is only first order accurate in the first sub-step, but rises to second order accurate following the second sub-step [64]. If t is less than the upper limit of the CFL condition, the numerical dispersion error of the ADI-FDTD method is similar to the conventional FDTD method. However, if t exceeds the CFL limit, the numerical dispersion of the ADI-FDTD method increases while t enlarges [7][7]. Therefore, although the ADI-FDTD method is unconditionally stable, the choice of t is dependent on the desired accuracy. Apart from the high dispersion error with large t, the accuracy of the ADI- FDTD method is also affected by its explicit source excitation [73][74]. Mathematically, the ADI-FDTD method is a simplification of the CN-FDTD method, and can be regarded as an O t ) perturbation of the CN-FDTD method [7]. Reported by Jun Shibayama [65], the LOD-FDTD method only has a splitting error of O t) by comparing to the CN-FDTD method [75]. The three dimensional LOD-FDTD method discretizes Maxwell s equation with Crank-Nicolson approximation and splits a single time step to three sub-steps [76]. Each sub-step solves electromagnetic field components in one direction. The recursive equations of each direction are independent of the other directions, which can be solved by a linear tridiagonal system. Thus, comparing to the ADI-FDTD method, the LOD- FDTD method is simpler in implementation and more efficient in computation [76]. To summarize, the implicit FDTD methods are unconditionally stable, but increase the computational complexity of the conventional FDTD and sacrifice the accuracy to exceed the CFL condition. 79

80 4. Idea of the Spatial Filtering Approach In high accuracy computations, the fine spatial sampling provides a wide bandwidth in the spatial frequency domain. However the excited signal only engages a small part of the bandwidth. The remaining part of the bandwidth contains no signal information, but increases the numerical dispersion which restrains the temporal size for stable computations. In contrast with the implicit schemes, instead of reformulating the time marching equations of the conventional FDTD method, the spatial filtering approach extends the CFL limit by removing the unstable harmonics in the high spatial frequency range [3][4]. This approach is conditionally stable since the bandwidth with signals still carries spurious frequency components that cause numerical dispersion. 4.. Revised D CFL Condition For any real ω, the sinusoidal term sin tω ) of 3.7) is always smaller than one. Thus 3.7) can be written as ) c t kx x sin x. 4.) Based on 4.), the condition for t is expressed as t x c sin kx x ). 4.) To guarantee the stability, t should be smaller than the minimum value of the right hand side of 4.). Since the maximum value of sin k x x) is one for any real wavenumber, the upper limit of t for stable computation is x, which is c the D CFL limit. For the purpose of decreasing the numerical dispersion, x should be shorter than one-tenth of λ min. If x = λ min, the highest wavenumber of the excited signal is k highest = π λ min = π 5 x 0. However, according to the Nyquist sampling theorem, the maximum spatial frequency in the FDTD computation is x, thus the greatest wavenumber that the FDTD method can accommodate is k 0 = π = π. Therefore the wavenumbers from π to π do not carry x x 5 x x source excitations but generate undesired dispersion errors. By filtering out these spurious frequency components, we can extend the CFL condition conditionally. 80

81 The revised CFL condition is defined as t x c sin kmax x ) 4.3) where k max is the filtering wavenumber and the spatial frequency components in the range of k max < k π are filtered out [3]. The upper limit of 4.3) x is greater than) the upper limit of the D CFL condition x ) with a factor of c / sin kmax x. 4.. Revised D CFL Condition To satisfy any real ω sin tω ) ), the D numerical dispersion 3.6) can be written as t ) k cos ϕ x c sin + x sin k sin ϕ y y ). 4.4) If x = y, by filtering out the wavenumbers higher than k max in 4.4), the revised D CFL condition for t is obtained as t x c sin kmax cos ϕ x ) + sin kmax sin ϕ x ). 4.5) Let us define / sin k max x cos ϕ ) + sin k max x sin ϕ ) as K A ϕ). According to the Nyquist sampling theorem, k max x can take any value from zero to π. Figures 4. plots the variation of K A ϕ) for the cases of k max x = π, π and π 4, which shows that the minimum value of K A occurs at ϕ = π and 3π. In the range 4 4 of 0 ϕ π, the first derivative of K A ϕ) with respect to ϕ equals zero at ϕ = 0, π, π, 3π and π [77]. Substituting the above five angles into the second derivative 4 4 of K A ϕ) with respect to ϕ, K A ϕ) is positive at ϕ = π and 3π. Therefore the 4 4 minimum value of K A ϕ) takes place at ϕ = π and 3π, which are the angles at 4 4 8

82 K A ϕ) k max x = π k max x = π k max x = π 4 π 5 π 5 3π 5 4π 5 π ϕ Figure 4.: K A varying ϕ from zero to π for the cases of k max x = π, k max x = π and k max x = π 4. the diagonals of a D cell. Substituting ϕ = π 4 and 3π 4 into 4.5) yields t x c sin kmax x ) 4.6) which is the revised D CFL condition for any propagation angle [4]. The upper limit of 4.6) is greater than the upper ) limit of the conventional D CFL condition x c ) with a factor of / sin kmax x. 4.3 Revised Stability Condition for CFS-PML 4.3. Revised Stability Condition for D CFS-PML 3.85) and 3.88) are the two conditions that define the upper bound of t for stable computation in the CFS-PML. The revised stability condition of 3.85) is derived from 3.8). First, let us write 3.8) as ax + a x ) c ɛ 0 Ψ + Ψ ) cos xω ) A) σx x c. 4.7) 8

83 If Ψ + Ψ ) cos xω A) is equal to or greater than two, the left hand side c of 4.7) is always smaller than zero, which is stable for any t. If Ψ + Ψ ) cos xω A) is smaller than two, 4.7) can be written as c ) ax + a x Substituting 3.47) into 4.8) yields e σ x t ɛ 0 + e σ x t ɛ 0 c ɛ 0 σ x x Ψ + Ψ ) cos xω A) ). 4.8) c c ɛ 0 σ x x Ψ + Ψ ) cos xω A) ). 4.9) c Since 0 < e σ x t ɛ 0 <, 4.0) can be simplified to e σ x t ɛ 0 + e σ x t ɛ 0 cɛ 0 σ x x Ψ + Ψ ) cos xω A) c ). 4.0) With the similar derivation in 3.85), 4.0) can be manipulated as t ɛ 0 ln σ x cɛ 0 σ x x cɛ 0 σ x x Ψ + Ψ ) cos xω A) ) + c Ψ + Ψ ) cos xω A) ) 4.) c where cɛ 0 σ x x Ψ + Ψ ) cos xω A) ) should be greater than one. Based on c 3.73) and 3.75), the D numerical wavenumber is written as k x = ω A + jb). 4.) c In the numerical calculations, 0 ω c k 0 = the wavenumbers from k max to k 0, 4.) can be written as π. Thus, when we filter out x ) cɛ t ɛ 0 0 σ x x e x k maxb + e x k maxb ) cos x k max A) + ln σ x ) cɛ 0 σ x x 4.3) e x k maxb + e x k maxb ) cos x k max A) 83

84 ts) x = mm, K D x = mm, K C x = mm, K D x = mm, K C x = 0.5 mm, K D x = 0.5 mm, K C π 5 π 5 3π 5 4π 5 π k max x Figure 4.: The value of K C, K D varying k max x from zero to π for the cases of x = mm, mm and 0.5 mm. which is the first revised stability condition for the D CFS-PML. Another revised stability condition is obtained from 3.88). With the procedures which have been done to derive 4.3), the revised stability condition of 3.88) is derived as t ɛ c ɛ 0 e x k maxb e x k maxb ) sin x k 0 σ ln x max A) + x. 4.4) σ c ɛ x 0 e σ x k maxb e x k maxb ) sin x k x max A) x 4.3) and 4.4) are the two revised stability conditions for the filtered FDTD method with CFS-PML in D. Let us define the upper limit of t in 4.3) as K C, in 4.3) as K D and in 4.4) as K E. With the same σ x, κ x and α x used in Figure 3., Figures 4. and 4.3 show the variation of K C, K D and K E with respect to k max x. For the same x, the curve of K D is identical to K C in Figure 4.. In Figure 4.3, for the case of x = mm, K E is always greater than K C with the same filtering wavenumber k max. However K E is smaller than K C in the range of π < k max x < 3π when x = mm. In the case of x = 0.5 mm, K 5 E is smaller than K C in the range of π < k 5 max x < π. Thus, we should set t based on 3 both 4.3) and 4.4). 84

85 ts) ts) ts) k max x K E K C π π 3π 4π π a) x = mm k max x K E K C π π 3π 4π π b) x = mm k max x K E K C π π 3π 4π π c) x = 0.5 mm Figure 4.3: The value of K C, K E varying k max x from zero to π for the cases of x = mm, mm and 0.5 mm. 85

86 ts) x = mm, K G x = mm, K F x = mm, K G x = mm, K F x = 0.5 mm, K G x = 0.5 mm, K F π 5 π 5 3π 5 4π 5 π k max x Figure 4.4: The value of K F and K G varying k max x from zero to π for the cases of x = mm, mm and 0.5 mm Revised Stability Condition for D CFS-PML For the CFS-PML in the four corners of the D FDTD space, the stability conditions are given by 3.0) and 3.0). The revised stability of 3.0) is derived from 3.99). If Ψ + Ψ ) cos xω c A) is equal to or greater than two, the right hand side of 3.99) is always greater than zero, which is stable for any t. If Ψ + Ψ ) cos xω A) is smaller than two, 3.99) can be written as c ) ax + a x ). 4.5) c ɛ 0 Ψ σx x + Ψ ) cos xω c A) Substituting 3.47) into 4.5), the condition for t is derived as t ɛ 0 ln σ x cɛ0 σ x x cɛ0 σ x x Ψ + Ψ ) cos xω Ψ + Ψ ) cos xω c A) ) ) c A) + 4.6) 86

87 ts) K H K F ts) π π 3π 4π π k max x a) x = mm π π 3π 4π π k max x b) x = mm K H K F 0 9 K H ts) K F 0 0 π π 3π 4π π k max x c) x = 0.5 mm Figure 4.5: The value of K F, K H varying k max x from zero to π for the cases of x = mm, mm and 0.5 mm. 87

88 where ) cɛ 0 σ x x Ψ + Ψ ) cos xω c A) should be greater than one. Filtering out the wavenumbers higher than k max, 4.6) is written as t ɛ 0 ln σ x cɛ0 σ x x cɛ0 σ x x e x kmaxb e x kmaxb ) + e x kmaxb ) cos x k max A) ) + e x kmaxb ) cos x k max A) + 4.7) and 3.0) is written as t ɛ 0 ln σ x c ɛ 0 e x kmaxb σx x c ɛ 0 e x kmaxb σx x e x kmaxb ) sin x k max e x kmaxb ) sin x k max A) ) A) Let us define the upper limit of t in 4.6) as K F, in 4.7) as K G and in 4.8) as K H. Figures 4.4 and 4.5 plot the value of K F, K G and K H with respect to k max x. For the same x, the curve of K G is identical to K F in Figure 4.4. In Figure 4.5, K H is always greater than K F with the same filtering wavenumber k max when x = mm, K H is smaller than K F in the range of 7π < k 0 max x < 4π 5 when x = mm and K H is smaller than K F in the range of 3π < k 5 max x < 9π 0 when x = 0.5 mm. 4.4 Spatial Filtering 4.4. Discrete Sine and Cosine Transform The filtered FDTD method extends the CFL limit conditionally by filtering out the high frequency components in the spatial domain. To obtain the spatial frequency components, many techniques, such as the discrete Fourier transform DFT) and the discrete sine/cosine transform DST/DCT), can be applied to transform the electromagnetic field from the spatial domain to the spatial frequency domain [78][79]. The DFT is widely used for spectral analysis due to the accurate representation of a signal in the frequency domain. This thesis selects the DST/DCT for implementing the filtered FDTD method. There are two advantages of the DST/DCT over the DFT. The first one is that the computational complexity of the DST/DCT is simpler than the DFT. The DST/DCT computes 88

89 only the real part of the input signals while DFT computes both the real and imaginary part. The second one is that the DST/DCT is odd/even symmetric at the boundaries of the input signals. The DFT repeats the input signal periodically, which may create spurious high frequencies from the boundary discontinuity. However, assuming the input signal to be odd/even symmetric, the boundaries of the DST/DCT computation tend to be continuous. For example, if the input signal is an array of ABCDE, the DCT computation treats it as a periodical array of ABCDEEDCBA and the DST computation treats it as ABCDE0-E)-D)-C)-B)-A). In the spatial filtering approach, both DST or DCT can be applied to transform the electromagnetic fields from the spatial domain into the spatial frequency domain without creating spurious high frequencies from the boundary discontinuity. DCT is adopted in this thesis for implementing the spatial filtering procedure Low Pass Filter in the Spatial Frequency Domain Instead of modifying the conventional Yee s equations, the filtered FDTD method introduces a low pass filter to remove the unstable spatial frequency components. In the spatial frequency domain, the two dimensional low-pass filter is defined as F k x, k, y ) = 0, k x + k y k max k x + k y > k. 4.9) max Note that, the ideal low pass filters of 4.9) is the only filter that can be applied in the filtered FDTD method. The numerical dispersion rises in the coming time steps, if the unstable spatial components are not removed completely. Thus, other filters, like the Butterworth filter or Chebyshev filter, may cause late time instability Procedures To maintain the stable computation with t greater than the CFL limit, the spatial filtering procedure should be applied at each time steps. Figure 4.8 shows the flowchart of the conventional FDTD method and the filtered FDTD method. Figure 4.9 shows the flowchart for the filtered FD-FDTD method with CFS-PML. Since the filtering approach does not reformulate the conventional Yee equations, 89

90 N y y D FDTD space y Spatial frequency domain DCT = y y N y y N y y x x N x x N x x N x x x Figure 4.6: Transformation from the spatial domain to the spatial frequency domain. The FDTD space has N x N y cells. y Spatial frequency domain N y y D FDTD space IDCT = k max N y y N y y y y N x x N x x k max x x x N x x Figure 4.7: Transformation from the spatial frequency domain to the spatial domain. The FDTD space has N x N y cells. 90

91 the filtered FDTD method remains second order accuracy for the numerical approximation. As shown in Figure 4.9, at t = n + ) t, H x, H y and H z are updated by the Yee algorithm with the filtered electromagnetic fields. The outgoing waves in the magnetic field are absorbed by the CFS-PML. However, since t is greater than the CFL limit, unstable harmonics are generated in the magnetic field, especially in the high spatial frequencies. As shown in Figure 4.6, the magnetic field is transformed to spatial frequency domain by using DCT. With the low pass filter in 4.9), Figure 4.7 filters out the wavenumbers that are higher than k max and applies the inverse DCT to obtain the filtered magnetic field. Then, at t = n + ) t, the electric field is updated by the Yee algorithm with the filtered magnetic field at t = n + ) t. Theoretically, we should apply the filtering procedure to the electric field as well. However this procedure can be excluded by reducing the value of k max or using the t which is smaller than the upper limit of the revised CFL condition [4]. 9

92 The Conventional FDTD Method Start t = 0 t = t + t Update the magnetic field components H x, H y and H z by Yee algorithm t = t + t Source excitation E z = source hard source) or E z = E z + source soft source) Update the electric field components E x, E y and E z by Yee algorithm t t max No End simulation Yes The Filtered FDTD Method with CFS-PML Start t = 0 t = t + t Update the magnetic field components H x, H y and H z by Yee algorithm Apply the CFS-PML into the magnetic fields Transform the magnetic field components to the spatial frequency form of H kx, H ky and H kz by DCT Obtain the filtered magnetic field components H kmaxx, H kmaxy and H kmaxz by filtering out the high spatial frequency components Transform the filtered magnetic field components to the spatial domain form of H xfiltered, H yfiltered and H zfiltered by IDCT t = t + t Source excitation E z = source hard source) or E z = E z + source soft source) Update the electric field components E x, E y and E z with H xfiltered, H yfiltered and H zfiltered by Yee algorithm Apply the CFS-PML into the electric fields Repeat the spatial filtering procedure for the electric field components to obtain E xfiltered, E yfiltered and E zfiltered t t max Yes No End simulation Figure 4.8: Flowchart of the conventional FDTD method and the filtered FDTD method. 9

93 The Filtered FD-FDTD Method with CFS-PML Start t = 0 t = t + t Update the magnetic field components H x, H y and H z by Yee algorithm Apply the CFS-PML into the magnetic fields Transform the magnetic field components to the spatial frequency form of H kx, H ky and H kz by DCT t = t + t Source excitation D z = source hard source) or D z = D z + source soft source) Update the electric flux density field components D x, D y and D z with H xfiltered, H yfiltered and H zfiltered Obtain the filtered magnetic field components H kmaxx, H kmaxy and H kmaxz by filtering out the high spatial frequency components Transform the filtered magnetic field components to the spatial domain form of H xfiltered, H yfiltered and H zfiltered by IDCT Update the electric field components E x, E y and E z by ADE scheme Apply the CFS-PML into the electric fields Repeat the spatial filtering procedure for the electric field components to obtain E xfiltered, E yfiltered and E zfiltered t t max Yes No End simulation Figure 4.9: Flowchart of the Filtered FD-FDTD method with CFS-PML. 93

94 Chapter 5 Numerical Experiments of the Spatial Filtering Approach The FDTD method becomes a widely used tool for solving the electromagnetic problems in modern engineerings. Programmers can easily write the FDTD s explicit equations using such computer languages as C, C++, JAVA, MATLAB and FORTAN. As a scientific programming language, FORTRAN is more efficient than many other languages for its concise array operations. Therefore all the numerical tests in this thesis are simulated by FORTRAN in the Linux workstation. By filtering out the spurious frequency components in the electromagnetic field, the spatial filtering procedure extends the CFL limit conditionally. The key point of realizing the spatial filtering approach is to select the electromagnetic field for filtering. This chapter presents the way to select fields for filtering based on the numerical experiments in the filtered FDTD method and the filtered FD- FDTD method with CFS-PML ABC. 5. Experiments of the Filtered FDTD Method Table 5. shows the basic parameters for the filtered FDTD method with lossless materials. Under this condition, the electromagnetic field is excited by the Gaussian derivative pulse of E source t) = 00 t t 0 e t t 0) τ 5.) πτ 94

95 Parameters Speed of light c m/s) Vacuum magnetic permeability µ 0 H/m) 4π 0 7 Relative magnetic permeability µ r H/m) Magnetic permeability µ = µ 0.µ r H/m) 4π 0 7 Vacuum electrical permittivity ε 0 = c µ F/m) 36π 0 9 Relative electrical permittivity ε r F/m) Electrical permittivity ε = ε 0.ε r F/m) 36π 0 9 Source type soft Table 5.: The simulation parameters for the filtered FDTD method. where t 0 = 0.33 ns and τ = 60 ps [80][8]. Figure 5. shows the source excitation in time and Figure 5. shows the spectrum of the excitation. According to the definition in [8], the lowest frequency of interest is 0.6 GHz and the highest frequency of interest is f highest = 5.73 GHz. This section shows the numerical results of the filtered FDTD method. 5.. D Filtered FDTD Method The two dimensional filtered FDTD method is implemented in the TM mode, which consists of H x, H y and E z. The computation space is discretized by Yee cells and each cell is a mm mm square. The temporal interval of the filtered FDTD method is defined as t = CFLN t cfl, where CFLN is the CFL number and t cfl is the maximum temporal interval under the CFL condition. Since x = y = mm, the maximum wavenumber that our simulation can accommodate is k 0 = 000π rad/m and t cfl is.36 ps. Figure 5.3 shows the simulation setting where the source is excited at 00,00) in the E z field and the observation point is at 90,00). Based on the filtering procedures shown in Section 4.4.3, we filter out the spatial frequency components whose wavenumbers are higher than 00π rad/m in both the electric field and magnetic field. With the revised CFL condition 4.6), for k max = 0π rad/m and x = mm, the upper limit of t is 4.3 t cfl. 95

96 4 Esourcet) t 0 0 s) Figure 5.: The source excitation in the time domain. Normalized Spectrum 0.5 e 0.6 GHz f highest =5.73 GHz Frequency GHz) Figure 5.: The normalized spectrum of the source excitation. The maximum frequency of this source is f highest = 5.73 GHz and the minimum frequency of this source is 0.6 GHz. 96

97 PEC excitation point observation point y x Figure 5.3: The simulation environment. The computation domain is a D cavity filled with air and bounded by PEC. The source is excited in the centre of the E z field. The observation point is 0 cells away from the excitation point. Figure 5.4: The electric field distribution of E z before filtering at t = 80 t cfl. 97

98 Figure 5.5: The spatial spectrum of the electric field distribution of E z before filtering at t = 80 t cfl. Figure 5.6: The spatial spectrum of the electric field distribution of E z filtering at t = 80 t cfl. The filtering wavenumber k max is 0π rad/m. after 98

99 Figure 5.7: The electric field distribution of E z after filtering at t = 80 t cfl. Hence we set CFLN to 4 in this test. Figure 5.4 shows the electric field distribution of E z before filtering at t = 80 t cfl. As t is four times greater than the CFL limit, the unstable components are generated in the computational domain. These components may grow up and cause late time instability. Figure 5.5 shows the spatial spectrum of E z. Many spurious frequency components are generated in the high spatial frequencies. Figure 5.6 displays the spatial spectrum after filtering out the spatial frequency components whose wavenumbers are higher than 0π rad/m. Figure 5.7 displays the filtered E z field in the spatial domain. By using the spatial filtering procedure, the unstable components in Figure 5.7 is less than those in Figure Observed Signals By filtering out the high spatial frequency components whose wavenumbers are higher than 0π rad/m in both the electric field and magnetic field, the computation is stable within 0 8 time steps with CFLN=4. However, as suggested in Section 4.4.3, the filtering procedure can be applied only in the electric field or magnetic field. To determine the field for filtering, we compare the observed signals in the filtered FDTD method and the conventional FDTD method. Figure 99

100 0. Electric Field Ez V/m) 0-0. Scenario -0.4 Scenario Scenario 3 Scenario t 0 0 s) Figure 5.8: The observed signals at 90,00) in E z for four scenarios. The four scenarios are introduced in Table 5.. In Scenarios, 3 and 4, k max = 0π rad/m and CFLN=4. Scenario Description The conventional FDTD method with CFLN=. The filtered FDTD method that applies the low pass filter in both E and H. DCT technique is used and source is excited in E field. 3 The filtered FDTD method that applies the low pass filter only in E. DCT technique is used and source is excited in E field. 4 The filtered FDTD method that applies the low pass filter only in H. DCT technique is used and source is excited in E field. Table 5.: The four scenarios in the numerical experiments. 00

101 The theoretical value Filtering E and H Filtering only E CFLN π 00π 300π 400π k max rad/m) 500π Figure 5.9: The maximum CFLN which gives stable computation with for each k max. The theoretical value is obtained from 4.6). 5.8 shows the signals observed at 90,00) in E z for four scenarios which are introduced in Table 5.. All these scenarios excite the source at electric field. Scenario updates the electromagnetic fields by using the conventional FDTD method. Transforming the electromagnetic fields between spatial and spatial frequency domain via DCT/IDCT technique, the spatial filtering procedure is applied in Scenarios, 3 and 4 to gain larger time step, which is set to CFLN=4. As a reference, the observed signal in scenario is obtained from the conventional FDTD method with CFLN=. The signals observed in are obtained from the filtered FDTD method with different filtered fields. As shown in Figure 5.8, the variation among the observed signals in scenario, and 3 is negligible, however the observed signal in scenario 4 is different from the reference in scenario. Thus, in the TM mode, to guarantee the accuracy, the electric field should be filtered. Furthermore, in our D test for the TE mode with the excitation in either the electric field or magnetic field, the magnetic field has to be filtered because only filtering the unstable components in the electric field causes incorrect results Improvement of Computational Efficiency As discussed in Section 5.., for the TM mode, the filtered FDTD method can be implemented by removing the spurious frequency components in both the electric and magnetic fields or only the electric field. The results of the stability test for 0

102 5 4 Filtering only E Filtering E and H 3 W CFLN Figure 5.0: The improvement of computational efficiency. the TM mode are plotted in Figure 5.9. The y-axis of Figure 5.9 represents the maximum CFLN that gives stable computation at each k max. The results of the numerical experiments are stable with the maximum CFLN obtained from 4.6). At each k max, the maximum CFLN that gives stable computation with two filtered fields both E and H ) is slightly greater than one filtered field only E). The improvement of the computational efficiency is defined as W = The run time of the conventional FDTD The run time of the filtered FDTD 5.) where the run time of the conventional FDTD is measured with CFLN= for 0000 time steps and the run time of the filtered FDTD is measured for 0000 CFLN time steps. Figure 5.0 shows that, in the D filtered FDTD method, the computational speed of applying the filtering procedure in two fields is faster than the conventional FDTD method when CFLN is greater than 9. However, for the filtered FDTD method with one filtered field, the computational speed is faster than the conventional FDTD method when CFLN is greater than 3. Thus, although implementing the filtering procedure to both E and H fields slightly increases the maximum CFLN for stable computation, its computational speed is about three times slower than that of the case for applying the low-pass filter to only the electric field. 0

103 5. Experiments of the Filtered FD-FDTD Method with CFS-PML 5.. Relative Permittivity of Debye Media This section presents the numerical experiments of the filtered FD-FDTD method with CFS-PML. Based on the definition in.57), the relative permittivity in the one-pole Debye model is expressed as ɛ r = ɛ + ɛ s ɛ + jωτ D + σ jωɛ ) Thus the revised CFL condition of the filtered FDTD method is obtained by replacing c with v = ɛµ = c ɛr µ r 5.4) where v is the speed of wave propagation in the dispersive medium, for equations 4.3), 4.6) and A.3). 5.. Scaling of the PML Conductivity If the PML conductivity σ x in.8) is regarded as a lossy constant, a spurious reflection is produced at the PML interface [35]. To improve this phenomenon, the PML conductivity σ x has to be growing from an infinitesimal value at the interior-pml interface to the relatively high value at the bound of PML layers. Practically, scaled by a geometrical progression, the PML conductivity σ x in the x-direction is defined as σ x d) = σ 0 g / x ) d 5.5) where g is the scaling factor that determines the increment from one FDTD cell to the next, d is the distance from the interior-pml interface and σ 0 is the initial value for σ x at the interface [35]. σ 0 is given as σ 0 = ɛ 0c ln g) ln R 0 ) xg N ) 5.6) 03

104 PML PEC excitation point PML centre PML PML interface observation point PML Figure 5.: The simulation environment. The computational domain is a D cavity filled with Debye media. 3 CFS-PML layers are applied into this computational domain. The excitation point is 0 cells away from the centre of the whole domain, and the observation point is 0 cells away from the excitation point. where N is the number of total PML layers, R 0 is the reflection coefficient [35]. In this thesis, the scaled PML conductivity is produced by g =., N = 3 and R 0 = D Filtered FD-FDTD Method with CFS-PML Figure 5. shows the simulation environment of the D filtered FD-FDTD method with CFS-PML in the TM mode. The FDTD space of cells is truncated by the PEC boundary. Each cell is a mm mm square. 3 CFS-PML layers are applied to absorb the outgoing waves. The whole computational domain is filled with fat and excited at 90,00) with the source in 5.). To extend the CFL limit, the spatial filter can be carried out either before or after applying the boundary conditions. However, in the numerical tests, the accuracy of implementing the filtering procedure after applying the boundary conditions is better than before applying the boundary conditions. 04

105 Electric Field Ez 0 9 V/m) Scenario Scenario Scenario 3 Scenario t 0 0 s) Figure 5.: The observed signals at 80,00) in E z for four scenarios. All the fields are filled with fat. The four scenarios are introduced in Table 5.. In Scenarios, 3 and 4, k max = 0π rad/m and CFLN= Observed Signals The one-pole Debye model is implemented by the ADE scheme, which consists of H, D and E. In our tests, the filtered FD-FDTD method cannot extend the CFL limit by applying the filtering procedure only in the D field. Figure 5. shows the signal observed at 80,00) for four scenarios. The input signal of these four scenarios are excited in the electric field. Scenario is the case of conventional FD-FDTD method with CFS-PML. Scenarios, 3 and 4 are the cases of filtered FD-FDTD method with CFS-PML by using DCT/IDCT technique to transform the field components between spatial domain and spatial frequency domain. Scenario applies the filtering procedure in both E and H fields, Scenario 3 applies the filtering procedure only in E field and Scenario 4 applies the filtering procedure only in H field. The observation in scenario is obtained by setting CFLN=. CFLN is set to 4 in Scenarios, 3 and 4. For the filtered FD-FDTD method, except for the scenario 4, the results of the rest scenarios are identical to reference in scenario. Thus, in the TM mode, the filtering procedure can be applied in only E or both E and H. In the further numerical experiments for the TE mode, the magnetic field should be filtered because only filtering the electric field gives the incorrect results. 05

106 CFLN The theoretical value Filtering E and H Filtering only E π 00π 300π k max rad/m) 400π 500π Figure 5.3: The maximum CFLN which gives stable computation for each k max. The theoretical value is obtained from 4.6). 9 8 Filtering only E Filtering E and H W CFLN Figure 5.4: The improvement of computational efficiency. 06

107 5..5 Improvement of Computational Efficiency Figure 5.3 compares the maximum CFLN that gives stable computation in the numerical experiments with the theoretical value in 4.6). The maximum CFLN for the filtered FDTD method with two filtered fields E and H ) is greater than the theoretical value when k max is smaller than 00π rad/m. For the case with one filtered field only E), the maximum CFLN of the numerical experiments is greater than the theoretical value when k max is smaller than 0π rad/m. From k max = 330π rad/m to 000π rad/m, the maximum stable CFLN of these two kinds of filtering implementations are the same as the theoretical value. Figure 5.4 shows the improvement of the computational efficiency in the filtered FDTD method. With the same CFLN, the computational efficiency for filtering both H and E decreases about three times in comparison to the scenario that only applies the spatial filtering procedure to E. Since the maximum stable CFLN in the numerical experiments with one filtered field is similar to that with two filtered field, we can filter only the electric field in the D TM mode. 07

108 Chapter 6 Analysis of the Numerical Results For the spatial filtering approach, the upper limit of t is obtained by the revised CFL condition with a given k max. In [3][4], the lower bound of k max is defined π as to minimize the numerical dispersion. However the numerical dispersion of 5 x the filtered FDTD method depends on both x and the frequency of excitation. In addition, other aspects such as the spectrum of the excitation and the absorbing ability of the PML are affected by the filtering procedure when k max is small. To find the optimized k max, the chapter investigates major factors that affect the stability and accuracy of the filtered FD-FDTD method with CFS-PML ABC. 6. Simulation Settings In the D TM mode, removing the high spatial frequency components only in the electric field, the filtered FD-FDTD with CFS-PML method was tested with the FDTD sizes of cells, cells and cells. The spatial size of the Yee cells is x = y = mm. Thus k 0 = 000π rad/m and t cfld =.36 ps. As the same simulation environment in Figure 5., the FDTD space is terminated with the CFS-PML and excited ten cells away from the centre with the source in 5.). The entire computational domain is filled with either air or fat. Based on these two cases, the performance of the spatial filtering approach with dispersive material is analyzed. 08

109 CFLN PML layers 8 PML layers 4 PML layers The Theoretical Value π 3π 5π 5π k max 0 rad/m) 50π Figure 6.: The maximum CFLN which gives stable computation for each k max with different PML layers in air. Theoretical values are obtained from 4.6). The shadow area is enlarged to display more details.the FDTD space is cells. 6. Analysis for the Stability Both the dispersive material and the conductivity of CFS-PML influence the upper limit of t for stable computations in the spatial filtering approach. In the conventional FDTD method, with the same x, the maximum t for stable computation is identical for any sizes of the FDTD space. However, in the filtered FDTD method, the upper limit of t is impacted by the FDTD sizes with the same x. Thus this section analyzes the stability of the filtered FD-FDTD method with CFS-PML based on the sizes of the FDTD space. 6.. PML Conductivity As shown in 5.5) and 5.6), the scaled σ x d) is obtained by three variables of g, R 0 and N. With g =. and R 0 = 0 4, Figure 6. displays the maximum CFLN that gives stable computation for N = 4, 8 and 3 in the FDTD space of cells. The maximum stable CFLN of the numerical experiments are equal to or greater than the theoretical values which are obtained from 4.6). The 09

110 CFLN PML layers 8 PML layers 4 PML layers The Theoretical Value π 3.5π 5.5π 0 0 5π 50π k max 0 rad/m) Figure 6.: The maximum CFLN which gives stable computation for each k max. The FDTD space is filled with air. Theoretical values are obtained from 4.6). The shadow area is enlarged to display more details. upper limit of the CFLN is decreased slightly by reducing the layers of the CFS- PML. In the further tests for the reflection coefficient R 0 and the scaling factor g, the maximum stable CFLN decreases by increasing g and R 0. In the numerical experiments with the FDTD space of cells and cells, the influence of g, R 0 and N on the stability are the same as those of cells. 6.. Comparison between Air and Fat media 6... In Air Figure 6. shows the results of the stability test with 3 CFS-PML layers in air. The upper limit of the CFLN for stable computation decreases when the size of the FDTD space increases. In the range of k max < 500π rad/m, the stability performance for the FDTD space of cells is better than the cases of and cells, and the case of cells is better than cells. The maximum stable CFLN in the FDTD space of cells almost coincides with the theoretical curve. 0

111 CFLN PML layers 8 PML layers 4 PML layers The Theoretical Value π 0π 7.5π 5π k max 0 rad/m) 50π Figure 6.3: The maximum CFLN which gives stable computation for each k max. The FDTD space is filled with fat. Theoretical values are obtained from 4.6). The shadow area is enlarged to display more details In fat Figure 6.3 displays the results of the stability test with 3 CFS-PML layers in fat. At the highest excited frequency of 5.73 GHz, the relative permittivity of fat increases 5.53 times in comparison to the air. Due to the increment of the relative permittivity, based on 4.6) and 5.4), the theoretical value for the maximum stable CFLN in fat is.35 times greater than that in air. As shown in Figure 6.3, the maximum stable CFLN in the numerical experiments is smaller than the theoretical value when k max is large. In the case of cells, the maximum stable CFLN is greater than the theoretical value when k max is less than 00π rad/m. However the maximum stable CFLN in the FDTD space of cells is always smaller than the theoretical value by one or two. Thus, with the dispersive materials, the capability of the spatial filtering approach to extend the CFL limit is decreased Improvement of the Computational Efficiency Figure 6.4 shows the improvement of the computational efficiency W for three FDTD sizes. In the two dimensional computation, the filtered FD-FDTD method

112 W cells cells cells CFLN Figure 6.4: The improvement of computational efficiency for three FDTD sizes. costs less execution time than the conventional FD-FDTD method when CFLN is equal to or greater than two. 6.3 Analysis for the Accuracy 6.3. Numerical Phase Velocity of the Filtering Approach As mentioned in Section 3.4, the variation between the numerical propagation velocity and the physical propagation velocity is the key point to evaluate the numerical dispersion. Substituting the upper limit of t in 4.6) into 3.38), for the wave propagating along the major grid axes ϕ = 0, π/, π and 3π/), the numerical phase velocity of the spatial filtering approach is expressed as c Dfilter = ω x arcsin ). 6.) kmax x sin sin ω x c ) sin kmax x

113 P D x = λ min 0 = 5.3 mm x = λ min 0 =.6 mm x = λ min 40 =.3 mm x = λ min 5.4 = mm π 0.4π 0.6π 0.8π π k max x Figure 6.5: The numerical phase velocity errors for x = λ min, λ min, λ min and λ min. λ 5.4 min is 5.4 cm. x = λ min is the spatial size used in the numerical 5.4 experiments. P D is obtained from 3.4). Substituting the upper limit of t in 4.6) into 3.40) yields c Dfilter = ω x ) kmax x arcsin sin sin ω x c ) sin kmax x 6.) which is the numerical phase velocity of the propagation angles of ϕ = π/4, 3π/4, 5π/4 and 7π/4 in the spatial filtering approach. Figure 6.5 shows the numerical phase velocity error of the spatial filtering approach. At each value of k max x, the numerical phase velocity error P D reduces when x decreases. Figure 6.6 shows the anisotropy error of the spatial filtering approach. The numerical velocity anisotropy error P Da decreasing x. reduces with The highest excited frequency of the source in 5.) is 5.73 GHz, thus the minimum wavelength of the excitation is 5.4 cm. In the numerical experiments, x is mm, which equals λ min 5.4. As shown in Figure 6.5, for the case of x = λ min 5.4, when k max is smaller than 0π rad/m, P D is greater than one. At k max = 000π 3

114 0 0 3 P D a x = λ min 0 = 5.3 mm x = λ min 0 =.6 mm x = λ min 40 =.3 mm x = λ min 5.4 = mm 0 0.π 0.4π 0.6π 0.8π π k max x Figure 6.6: The numerical velocity anisotropy error for x = λ min, λ min, λ min and λ min. λ 5.4 min is 5.4 cm. x = λ min is the spatial size used in the numerical 5.4 experiments. P Da is obtained from 3.4). rad/m, the numerical phase velocity error of the filtered FDTD method is identical to the numerical phase velocity error of the conventional FDTD method. The numerical velocity anisotropy error of our tests is shown in Figure 6.6 with x = 5.3 mm,.6 mm,.3 mm and mm. In comparison to the numerical velocity anisotropy error of the conventional FDTD method, the numerical velocity anisotropy error of the filtered FDTD method is increased by times when x = mm Spectrum Loss of the Excitation signal due to the Filtering Procedure As reported in [5], the numerical dispersion reduces while t approaches to the CFL limit. Thus t is set to x c for the D FDTD computation in the case of x = y. The maximum frequency that the numerical computation can afford is f 0 = = c t x = π c k 0. In order to relax the CFL condition, the field components in the wavenumbers from k max to k 0 are removed in the filtered FDTD method. Therefore the frequency components from c k π max to f 0 are affected by the spatial filtering procedure while the electromagnetic waves are 4

115 propagating through the FDTD space. For the excitation with wide bandwidth, if part of the frequencies of the excitation are greater than c k π max, the accuracy of the calculation is affected by both the dispersion error and spectrum loss. The spectrum loss E of the excitation signal is expressed as T s nofilter l) s filtered l) E = l= 6.3) T s nonfilter l) l= where T is the number of maximum time steps, s nofilter l) = T T/ m= jπlm Sm) exp T ), 6.4) s filtered l) = T T f m= jπlm Sm) exp T ). 6.5) In 6.4) and 6.5), Sm) is the spectrum of the excitation and T f = f max / f, where f = /T t) and f max = c k max / π). Note that, 6.3) is the deviation between the filtered signal and the non-filtered signal in time domain. s nofilter l) and s filtered l) are derived from Sm) by using the inverse discrete Fourier transform. Substitution of 6.4) and 6.5) into 6.3) yields T l= E = T l= T/ m=t f + T/ m= Sm) exp ) jπlm T Sm) exp ) 6.6) jπlm T The simulation error R between the conventional FD-FDTD method and the filtered FD-FDTD method is calculated by F filtered i t, k max, CF LN) F nofilter i t) i R = 6.7) F nofilter i t) i 5

116 00 80 E R E and R %) π 90π k max rad/m) 35π 80π Figure 6.7: The theoretical spectrum loss of the excitation E and the simulation error R. The FDTD size is cells and 3 CFS-PML layers were used. The computational domain is filled with air. where F filtered and F nofilter represent the observed signals in time domain for the filtered and conventional FDTD method, respectively. Figure 6.7 compares the spectrum loss of excitation with the calculated error of simulation results. In Figure 6.7, R is greater than 00% in the range of k max < 0π rad/m. This is caused by the numerical phase velocity error as shown in Figure 6.5. However, the error of simulation is about 0% higher than E from k max = 0π rad/m to 0π rad/m. This difference is caused by the PML boundary. Figure 6.8 shows the signal observed at 80,00) in E z. With k max = 80π rad/m, the outgoing wave is not completely absorbed by the PML boundary conditions Impact to the PML The conductivity σ x d) is increased geometrically from the interior-pml interface to the PEC boundary, which absorbs the incident wave rapidly after the wave traveled into the CFS-PML. The strong attenuation of the traveling wave in the CFS-PML causes high spatial frequency components. If these high spatial frequency components are filtered out, the absorbing ability of the PML boundary condition is degraded. Figure 6.8 shows that the absorbing ability of the PML 6

117 Electric Field Ez 0 9 V/m) Conventional FDTD Filtered FD-FDTD k max = 80π Filtered FD-FDTD k max = 30π Time 0 9 s) Figure 6.8: The impact of the PML boundary with the spatial filtering approach. The FDTD size is cells and 3 CFS-PML layers were used. The computational domain is filled with air. The signal is observed at 80,00) in E z. boundary condition is reduced by decreasing the value of k max. The rapid wave attenuation in the CFS-PML causes high spatial frequency components, which requires a high k max to protect the absorbing ability of the CFS-PML. As σ x d) controls the absorbing ability of the CFS-PML, a well designed σ x d) can reduce the impact of the spatial filtering approach to the CFS-PML boundary condition. In our simulations, with the σ x d) of g =. and R 0 = 0 4, the filtered FDTD method yields the best accuracy. As displayed in Figure 6.7, the absorbing ability of the CFS-PML is affected by the spatial filtering procedure when k max is smaller than 80π rad/m. To match the value of R, E need to be shifted along the positive direction of the x-axis by approximately 75π rad/m Accuracy Assessment The simulation error R of the numerical experiments are calculated by 6.7). We only present R which is smaller than 50%. Figures from 6.9 to 6.4 show R with different k max and CFLN in either air or fat. Although the low pass filter is applied only in the electric field, all simulation results show that we can increase 7

118 Figure 6.9: The simulation error R for the FDTD size of in air. R is calculated by using 6.7). Figure 6.0: The simulation error R for the FDTD size of in air. R is calculated by using 6.7). Figure 6.: The simulation error R for the FDTD size of in air. R is calculated by using 6.7). 8

119 Figure 6.: The simulation error R for the FDTD size of in fat. R is calculated by using 6.7). Figure 6.3: The simulation error R for the FDTD size of in fat. R is calculated by using 6.7). Figure 6.4: The simulation error R for the FDTD size of in fat. R is calculated by using 6.7). 9

120 the CFLN up to 3 with R smaller than 0%. Obviously, at the same filtering wavenumber k max, R increases when CFLN increases. Comparing to the simulation errors in air, with the same FDTD size, we need a larger k max to gain the same R for the case of fat. For example, in the FDTD space of cells with air, R is smaller than 0% when k max is less than 85π rad/m. However, when the computational domain is filled with fat, R is smaller than 0% when k max is less than 5π rad/m. In the filtering approach, R decreases by increasing the size of the FDTD space. For the experiments with air, to keep the error less than 0%, the maximum stable CFLN we can use is 0 at k max = 00π rad/m for the case of cells, the maximum stable CFLN is at k max = 90π rad/m for the case of cells and the maximum stable CFLN is 3 at k max = 80π rad/m for the case of cells. The accuracy is clearly improved by enlarging the FDTD size for the case of air. However, in the computational domain filled with fat, the accuracy improvement by enlarging the FDTD size is negligible. For example, to keep the error smaller than 0%, the highest CFLN for the all three FDTD sizes are 4 at k max = 50π rad/m. 6.4 Guidance of Choosing the Filtering Wavenumber Based on the analysis in Section 6. and Section 6.3, we can run simulations as follows:. Choose the excitation and the size of the FDTD space.. Set the parameters of CFS-PML. These parameters should satisfy condition 3.98). 3. Calculate the numerical velocity error P D with respect to k max by substituting 6.) and 6.) into 3.4). 4. Based on 6.6), calculate the spectrum loss E with respect to k max. 5. Add up the results in step 3 and 4 and let U k max ) = P D k max ) + E k max ). U k max ) is the expected simulation error of the filtered FDTD method. 6. Select the k max with a desirable accuracy based on U k max ). 0

121 7. To reduce the impact of the spatial filtering procedure on PML, an additional value should be added to the k max which is selected in step 6 i.e. with the σ x d) in our test, the additional wavenumber is 75π rad/m.). 8. Find the CFLN by the revised CFL conditions. 9. Choose the field for filtering based on the results in Chapter 5 and apply the spatial filtering procedure at each time step. 6.5 Practical Application Following the guidance in Section 6.4, this section shows a realistic test of deep brain stimulation DBS), which is a biomedical treatment for neurological disorders. The DBS is applicable to treat the symptoms of Parkinson s disease and essential tremor. The surgical treatment of DBS has some potential risks such as infection, seizure, electrode fracture, skin irritation, intracranial hemorrhage and so on. In order to reduce the risks of surgical treatment, the non-invasive treatment, which stimulates the targeted tissue by exciting the electromagnetic waves outside the human body, is adopted. To simulate non-invasive treatment, the electromagnetic wave propagation in human body is studied by the proposed filtered FD-FDTD method with CFS-PML Numerical Modeling of the Human Tissues The digital human phantom DHP) used in this project was provided by RIKEN Saitama,Japan) under no-disclosure agreement between RIKEN and the University of Manchester. The usage was approved by RIKEN ethical committee. To obtain the DHP data for human body, the bio-research infrastructure construction team of RIKEN scanned the human body of a male by using the magnetic resonance imaging MRI) technique with the spatial resolution of mm. The 3D DHP data is composed of voxels and 53 kinds of tissues [84]. Each voxel is a uniform cell mm 3 ) and can be assembled to represent the human tissues or organs. Table 6. shows the parameters for human tissues in the Debye relaxation model [83]. The data is provided by the U.S. Air Force and the parameters are fitted by Fumie s group [85][86][87]. In 996, the U.S. Air Force reported their experimental data of the dielectric parameters of human tissues in [87]. Based on their data, the dielectric parameters for Debye model are fitted

122 Code Tissue σs/m) ɛ s ɛ τ D ps) cerebral cortex white matter cerebellum midbrain eyeball optic nerve cornea crystalline lens pituitary gland thalamus pineal gland tongue cerebrospinal fluid right suprarenal body urinary bladder large intestine duodenum esophagus gall bladder heart right kidney liver right lung pancreas small intestine spleen stomach thyroid gland trachea bone fat muscle skin diaphragm seminal vesicle erectile tissue spinal cord testicle epiglottis salivary gland middle ear inner ear pharynx Table 6.: The parameters for human tissues in the one-pole Debye relaxation model [83].

123 x y CFS-PML CFS-PML Excitation Point Cross Section of Human Head Observation Point CFS-PML The cross section of the human head on the xy-plane at z = 80. CFS-PML The simulation environment Figure 6.5: The cross section of human head on the xy-plane at z = 80 and the simulation environment. The size of the simulation field is cells and the thickness of the CFS-PML layers is 3 cells. The field is excited at 30,50) and observed at 30,80). by using Newton s method and least square fitting method [86]. Each human tissue is assigned with a unique number which is presented in the first column of Table 6.. For the bio-medical applications, the DHP data can be mapped to the corresponding human body tissues according to this unique number Numerical Experiment of Deep Brain Stimulation The DHP data in this experiment are plotted by a number of portable graymap PGM) files, which are written in ASCII code. Each PGM file represents one slice of the human body on the xy-plane. Thus the size of each PGM file is and the total number of these PGM files is 68. The left hand side of Figure 6.5 shows the cross section of human head on the xy-plane at z = 80 and the right hand side of Figure 6.5 shows the simulation environment. The FDTD 3

124 P D, E and U %) π P D E U 00π 50π 00π 50π k max rad/m) Figure 6.6: The value of U, P D and E varying k max from zero to 50 π rad/m. space is discretized by Yee cells and filled with dispersive materials based on Figure 6.5 and Table 6.. x and y are set to mm due to the mm spatial resolution of the DHP data. The electric field is excited at 30,50) and observed at 30,80) With x = mm and the highest excited frequency of 5.73 GHz, the numerical phase velocity error P D k max ) is derived by substituting 6.) and 6.) into 3.4). With T f = f max = π f c k max and the spectrum of the excitation, the spectrum f loss E k max ) of the excitation is obtained. Then we can derive the expected error of this simulation by using U k max ) = P D k max ) + E k max ), which is displayed in Figure 6.6. If the expected simulation error is 6.3%, k max is 00π rad/m according to the value of U k max ). To guarantee the absorbing ability of the CFS-PML, k max = 75π rad/m is used in the simulation at k max = 75π, U k max ) = 0.0). Substituting k max = 75π rad/m and x = mm into the revised CFL condition 4.6), the maximum t for stable computation is t = 5.8 t cfld. As shown in Figure 6.3, the stability of the filtered FDTD method can be decreased by the dispersive media. As a results, the CFLN in this application is set to 4. Similarly, if the expected simulation is 0.63%, k max should be set to 300π and CFLN=. Figure 6.8, Figure 6.9, Figure 6.0 and Figure 6. compare the electric field distribution in the conventional FDTD method with that in the filtered 4

125 .5 Electric Field V/m) Filtered FDTD, CFLN=4 Filtered FDTD, CFLN= Conventional FDTD, CFLN= 3 t 0 9 s) Figure 6.7: The signal observed at 30,80) FDTD method at t = 00 t cfld, 00 t cfld, 500 t cfld and 000 t cfld, respectively. The conventional FDTD method is computed with CFLN=. For the filtered FDTD method, k max = 75π rad/m is adopted for the simulation with CFLN=4, k max = 300π rad/m is adopted for the simulation with CFLN= and k max = 000π rad/m is adopted for the simulation with CFLN=. Figure 6.7 shows the signals observed at 30,80). The observed signal in the conventional FDTD method with CFLN= is used to compare with the signals observed in the filtered FDTD method with CFLN= and 4. Based on results in Figure 6.7, the simulation error R in the filtered FDTD method with CFLN= is 5.6% and that with CFLN=4 is 8.5%. At CFLN=4, the simulation error is higher than the expected simulation error 6.3%). 5

126 Max V/m) Conventional FDTD CFLN= Max = V/m Min = 0 V/m Filtered FDTD CFLN= Max = V/m Min = 0 V/m Filtered FDTD CFLN= Max = 0.04 V/m Min = 0 V/m Filtered FDTD CFLN=4 Max = V/m Min = 0 V/m Min V/m) Figure 6.8: Electrical field distribution at t = 00 t cfld. 6

127 Max V/m) Conventional FDTD CFLN= Max =.7403 V/m Min = 0 V/m Filtered FDTD CFLN= Max =.7403 V/m Min = 0 V/m Filtered FDTD CFLN= Max =.706 V/m Min = 0 V/m Filtered FDTD CFLN=4 Max =.6533 V/m Min = 0 V/m Min V/m) Figure 6.9: Electrical field distribution at t = 00 t cfld. 7

128 Max V/m) Conventional FDTD CFLN= Max =.0439 V/m Min = 0 V/m Filtered FDTD CFLN= Max =.0439 V/m Min = 0 V/m Filtered FDTD CFLN= Max =.30 V/m Min = 0 V/m Filtered FDTD CFLN=4 Max =.097 V/m Min = 0 V/m Min V/m) Figure 6.0: Electrical field distribution at t = 500 t cfld. 8

129 Max V/m) Conventional FDTD CFLN= Max = 0.80 V/m Min = 0 V/m Filtered FDTD CFLN= Max = 0.80 V/m Min = 0 V/m Filtered FDTD CFLN= Max = V/m Min = 0 V/m Filtered FDTD CFLN=4 Max = 0.70 V/m Min = 0 V/m Min V/m) Figure 6.: Electrical field distribution at t = 000 t cfld. 9