DIAZ-FITZGERALD TIME DOMAIN (D-FTD) METHOD APPLIED TODIELECTRIC AND LOSSY MATERIALS

Franco De Flaviis*, Massimo Noro**, Nicolaos G. Alexopoulos*

* University of California at Los Angeles, Dep. of Electrical Engineering, Los Angeles, CA 90024** University of California at Los Angeles, Dep.of Chemistry, Los Angeles, CA 90024

Abstract - We present a new approach to the discretizedsolution of Maxwell's Equations for a real medium. Theproblem is equivalent to the solution of the equations ofmotion of a mechanical system of spheres properlyconnected. This model is an extension of the modelproposed by G. F. Fitzgerald in 1885. The method isdifferent from the usual FDTD implementation in that itprovides a condensed node representation for the electricfield, and it also uses different leap-frog scheme topropagate the field. In addition it provides an essentialpedagogical tool for further understanding of the solutionof Maxwell's equations in the time domain. As a validationof the technique the scattered field from an infinitely longcylinder is calculated and is compared to the exact solutionfor the two cases with and without losses, excellentagreement is obtained.

1.Introduction

The discretization of Maxwell's Equations can be carriedout via the introduction of the vector potential. If space isdiscretized, the time evolution of electric and magnetic fieldin each mesh position can be correctly described by a leap-frogging scheme between the electric field and the vectorpotential, in distinction to the FDTD where the leap-frogging involves the electric and magnetic fields. TheHamiltonian of this system corresponds to that of an arrayof spheres with fixed axes all connected with rubberspheres. A similar mechanical model was first proposed byFitzgerald[1] in 1885, and recently implemented as anumerical technique by Rodolfo Diaz[2]. Two of theprincipal strengths of the D-FTD are the fact it avoidsconvolution integrals in the description of dispersivemedia, and the fact that the electric field is localized ateach node of the computational grid.

2. Mathematical model

For the general case we report here the coupled equationsbetween Electric Field and Vector Potential i.e.:

∂E∂t

= − 1εµ

∇ 2A

E = − ∂A∂t

− ∇ φ

(1.a)

(1.b)

For simplicity we present in this summary the derivationof the method for the analog 2-D mechanical model;

extension to 3-D is straightforward. Consider an orderedarray (for clarity only one line is shown) of spheresconnected to each other with small rubber spheres asdescribed in Fig.1.

y

z

x

Fig.1 Rigid spheres connected by rubber spheres

Each sphere is connected to the four neighbor spheres byrubber spheres which are assumed not to slip on the surfaceof the spheres. The rotational motion of the sphere inposition (i,j) is therefore coupled to the motion of thespheres (i-1,j), (i+1,j), (i,j-1) and (i,j+1) as shown on thex-y cut plane in Fig.2.

θi,j θi+1,jθi-1,j

θi,j+1

θi,j-1

F1y

F2y

F1x

F2x

Fig.2 Cut x-y plane of the model of Fig.1

Let us assume that the (i,j) sphere with moment of inertiaI(i,j) is free to rotate around the z-axis with angularvelocity ω = ∂θ(i, j) ∂t , where θ(i, j) is the rotationangle with respect to a common reference axis on the x-y

plane. Let the radius of the sphere be a and the distancefrom two adjacent spheres 2a, then a difference in rotation

angle between two adjacent spheres ∆θ generates atension T = ka∆θ on the rubber sphere connecting them.This is manifested as tangential force (F1x,F2x,F1y,F2y)on the rigid sphere. Combining the tensions due to thefour neighbors together with Newton's second lawT = Iα where α = ∂ω ∂t , and discretizing in time [3]we obtain:

α i, jn+1 =

ki, ja2

Ii, j

4θi, jn − θi+1, j

n − θi−1, jn − θi, j +1

n − θi, j −1n{ }(2)

ωi, jn+1 = ωi, j

n + α i, jn δt (3)

θi, jn+1 = θi, j

n + ωi, jn+1δt (4)

where the superscript n indicates the time-step. Note thatin this model both the moment of inertia Ii,j and therubber sphere force constant ki,j depend on the particularsphere. A simple analogy can be made at this point torecover Fitzgerald's Aether model. If we identify thedielectric constant with some kind of "electric mass" of thesphere , and the magnetic permeability with some kind ofsphere elasticity , we can write

I = εk = µ −1

a = (∆s)−1

(5)

so that the quantity ka2 I becomes 1 ∆s2εµ and we

can substitute ∂E

∂t, E, A for α ,ω,θ to complete the

analogy. In a dielectric material with no losses we simplyincrease the moment of inertia of the pulleys to take intoaccount the actual dielectric constant of the mediumε ≠ ε0 , while a lossy material can be easily modeled by

immersing the spheres in a viscous fluid bath, andtherefore by adding a viscous damping term γ to thedifferential equation of the motion (3) which becomes.

ωi, jn +1 = α i, j

n +1δt + ωi, jn (1 + γδt) (6)

In a similar way it is possible to use mechanical analogsfor single or multiple Debye relaxation dielectricmaterials[2]. The formulation of 1-D model and 3-D modelis straightforward by respectively removing two spheres inthe x or y direction, for the 1-D case or adding two spherein the z direction for the 3-D case.

3.Results

In this section, we present the computer results for theinternal electric field of a uniform, circular dielectric

cylindrical scatterer. The cylinder is assumed to be infinitein the z-direction. The incident radiation is assumed to be aTM plane wave constituted by a Gaussian pulse in shape.The pulse width chosen in this work is 40 time stepcorresponding to a width of 20 ps. Because there is novariation of either scatterer geometry or incident field in thez direction, this problem may be treated as the twodimensional scattering of the incident wave. A twodimensional grid of 400 by 300 mesh points is used. Thecylinder axis is positioned at (200,125). Second orderabsorbing boundary condition is used to truncate the grid[4]. The grid coordinates internal to the cylinder with radiuso f 0 . 0 6 m , d e t e r m i n e d b y

(i − 200)2 + ( j −125)2 ≤ 202 , are assigned thedielectric parameters . All the grid points outside this gridare assigned the parameters of free space. The plane wavesource is activated at j=100. The program is time-steppedfor a long enough time so that the plane wave has thetime to scatter from the cylinder and the scattered field hasthe time to reach the observation points. DFT algorithm isused to extract the information of the field distribution atthe frequency of 1.5 GHz. In the first simulation for thecylinder scatterer we use the following parameters: εd=2-j0,

∆s=3mm, δt = 0.5psec . The computer results for thisrun are detailed in Fig.3

0

0.5

1

1.5

2

-0.06 -0.04 -0.02 0 0.02 0.04 0.06

Exact

Computed

|E|

x (m)

Fig. 3 Comparison between computed and Exact solutionof the inner scattered Electric field for a lossless cylinderalong his axis.

which graphs the DFT of the electric field along his axis,after 3500 time steps. The exact solution, in the frequencydomain, is calculated using the summed series technique asin Jones [5]. The computer solution locates the positionsof all the peaks and nulls of the electric field with error lessthen 0.3%. For the second example the cylindrical scattererhas the same parameters except that it is lossy with adielectric constant εd=2-j0.43 The result of the simulationis reported in Fig.4.

0

0.5

1

1.5

2

-0.06 -0.04 -0.02 0 0.02 0.04 0.06

Exact

Computed

|E|

x (m)

Fig. 4 Comparison between computed and Exact solutionof the inner scattered Electric field for a lossy cylinder,along a cut at i=100

excellent agreement is obtained between theory andcalculation.

4.Conclusions

The correspondence of Maxwell's equation with amechanical system is of great importance since it provideseasy and straightforward understanding for the modeling oflossy as well as single and multiple relaxation materials.This new model also provides a condensed noderepresentation of the electric field which results in a moreprecise implementation of boundary condition. It alsoprovides a new way of thinking to address and tackleconcepts for new absorbing boundary conditions. At thisstage we can assert that the ordinary FDTD technique hasexplored all possibilities and that state-of-the-art [6]refinements cannot improve what has already beenachieved. This is why a completely new approach has beenfound and is explored in this paper. Our proposed newmethod in the time domain, the D-FTD, provides universalmodeling capabilities since it is based on naturalelectromechanical analogs. It allows for the development ofa highly accurate method based on physical principles, itcircumvents convolution integrals in the characterization ofhighly dispersive materials and it easily allows for locallyvariable mesh size.

5. Acknowledgments

The authors are indebted to Dr. R. Diaz for many usefuldiscussions and criticism of this work. We also extend ourappreciation to Professor Engquist for his comments onthe absorbing boundary condition we used.

References

[1] G.F.Fitzgerald in a letter to Oliver Lodge, 3 Mar.1894, as quoted in B.J.Hount, The Maxwellians.Cornell University Press, Ithaca, 1991.

[2] R. E. Diaz "A Discret Fitzgerald Time Domain Methodfor Computational Electromagnetics" InternationalConference on Electromagnetics for advancedapplications (ICEAA) 1994, Torino, Italy.

[3] Leon Lapidus, George F. Pinder, Numerical solution ofpartial differential equation in science andengineering . John Wiley & Sons, Inc. 1982.

[4] R. Clayton and B. Engquist "Absorbing boundaryconditions for acoustic and elastic wave equations"Bulletin of Seismological Society of America. Vol.67, No.6, pp.1529-1540. December 1977.

[5] D.S. Jones, The teory of Electromagnetism. NewYork: Macmillan, 1964.

[6] C. E. Reuter, R. M. Joseph, E. T. Thiele, D. S. Katz,A. Taflove "Ultrawideband Absorbing BoundaryConditions for Termination of Waveguiding Structuresin FD-TD Simulations", IEEE Microwave and GuidedWave Lett., vol.4, pp.344-346, Oct. 1994.