Computing Electromagnetic Fields in Inhomogeneous Media Using Lattice Gas Automata. I. Introduction

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1 Comuting Electomagnetic Fields in Inhomogeneous Media Using Lattice Gas Automata M.Zhang, D. Cule, L. Shafai, G. Bidges and N.Simons Deatment of Electical and Comute Engineeing Univesity of Manitoba Winnieg, Canada R3T 5V6 VPRS, Diectoate of Antennas and Integated Electonics Communications Reseach Cente Ottawa, Canada KH 8S I. Intoduction Lattice Gas Automata (LGA can be consideed as an altenative to the conventional diffeential equation descition of oblems in electomagnetics. LGAs ae discete dynamical systems that ae based on a micoscoic model of the hysics being simulated. The basic constituents of an LGA ae discete cells. These cells ae inteconnected accoding to cetain symmetic equiements to fom an extemely lage egula lattice. The cells of an LGA ae extemely simle, equiing only a few bits to comletely descibe thei states. Even though they ae simle howeve, the collective behaviou of LGA micoscoic systems ae caable of exhibiting those behavious descibed by atial diffeential equations fo eal hysical systems. One tye of simle LGA, the HPP LGA, is constucted with only a few bits e cell and oeated on a ectangula lattice. We have demonstated [1] that it is caable of simulating two dimensional electomagnetic fields. Futhemoe, the inheent aallelism and simlicity of LGA algoithms make them ideally suited to imlementation in a aallel ocessing achitectue. In this ae we esent new HPP-tye mixtue LGA algoithms fo modelling wave oagation in inhomogeneous dense media. Change in sound seed of an LGA can be achieved by incooating est bits at a lattice site as well as moving o inteaction bits. We analytically show how this model can be alied to the simulation of electomagnetic fields in inhomogeneous media. In this ae ou analysis is based on the moe comlex hexagonal lattice with esults esented fo a ectangula (HPP lattice shown in Fig.1. With the small etubation assumtion, we develo and check the validity of a simle HPP model fo simulating wave oagation. The ability to model media with diffeent sound seeds is analogous to modelling diffeent dielectic constants in inhomogeneous media of electomagnetics. We theoetically give a geneal fomula fo the sound seed which enables a wide ange of dielectic constant to be modelled by secifying vaious inteaction models in an LGA. A vaiety of alications of this model fo oblems of wave inteaction with dielectic objects, fom a simle heteogeneous dielectic cylinde to comlex biological stuctues, ae eoted and comaed with taditional numeical methods. II. Lattice Gas Automata and Mixtue Model An LGA model without zeo-velocity est aticles can only yield a unifom sound seed. To enable the LGA lattice to model media with diffeent sound seeds (analogous to modelling diffeent dielectic constants in electomagnetics, cetain est aticles ae incooated within sites of the lattice. It will be seen in the analysis that thee ae only a few estictions (consevations of mass and momentum, and semidetail balance imosed on constucting such mixtue models, and thus thee ae many ways fo them to be emloyed. One can also secify that cetain egions of the lattice have diffeent est aticle numbes and masses. The enegy exchanges between moving and est aticles in the egions ae thus diffeent, and a lattice with inhomogeneous sound seeds can be ealized. 1

2 Fig. shows two examle cells of ou mixtue models, one fo a ectangula (HPP lattice and the othe fo a hexagonal (FHP lattice []. Excet fo the moving aticles (4 moving aticles associating with a HPP cell, 6 moving aticles with a FHP cell, thee ae u to est aticles allowed at a lattice site. We denote these est aticles to be m 1, m,..., m b, esectively. To obtain a satial distibution of sound seeds on the lattice, is then defined to be a function of osition.a comletely abitay sound can then be secified by ceating mixtues of cells with diffeent est aticle numbes by using a weight aamete k (,..., whee k is the ceation o annihilation obability of est aticle m k. In Fig.3, one of many est aticle models ae shown, whee a stack ( 3 of aticles of vaious masses( m 1 4, m 8 and m 3 16 can be ceated. Exchange of enegy between moving and est aticles occus when a est aticle of mass 4 is ceated when fou unit mass moving aticles collide and whee thee is initially no mass 4 est aticle. Altenatively, if a mass 4 est aticle aleady exists at a site, and thee ae no initial moving aticles, fou moving aticles will be ceated afte the collision hase, and the est aticle will be annihilated. Even moe geneally, we could incooate stochastic ules which obabilistically allow est aticles to be ceated o annihilated l Fig.1: HPP lattice with ectangula cells. m b (n b o, n b m b (n b o, n b m k (n k 0, n k n 0 m (n 0, n n m 1 (n 10, n 1 03 n 01 n 04 n0 3 n 0 m k m (n 0, n (n 10, n 1 m 1 n 01 (n k 0, n k n 04 (a: A HPP cell. n 05 n 06 (b: A FHP cell. Fig.: LGA ule with obabilistically weighted est aticles: (a: HPP mixtue model, (b: FHP mixtue model Fig.3: LGA ule with thee weighted est aticles.

3 In the analysis of this model we will use two subscit binay vaiables n ik ( x, t to eesent the aticle states at a aticula site x and time ste t in the mixtue lattice. In this notation the fist subscit i eesents the velocity diection of aticles. Thus, the moving aticles in a aticula cell can be denoted as n i 0 ( x, t, whee i1,,3 and 4 fo the fou moving aticles of the HPP model, and i1,... 6 fo the six moving aticles of the FHP model, and i0 fo est aticles. The second subscit k eesents the mass index of the aticles. Fo moving aticles (unit mass k 0. Following the notation, the bits { n o k,... } eesent the est aticles with mass m k. The andom bit vaiables (... ae intoduced to stochastically descibe the esence o absence of a est aticles m k (... at a lattice site. These bits n k (... ae andomly samled with the aveage n k values of n k k (..., and satisfying the limitations k + 1 k and 1 k. III.Theoetical Analysis Consideing the exclusion of viscosity in the final analysis, we begin with the moe comlex FHP lattice and then late limit it to the Eule equation deived fom the lowest ode of the Chaman-Enskog exansions[3]. The FHP lattice is used since it enables a moe igoous analysis. The micoscoic dynamics can be exessed in tems of the bit vaiables fo the moving aticles as n io, ( x + c i, t + t n io, ( x, t + ω i,o [n (,] x t ( i 1,...,, (1 and fo the est aticles as n ok, ( x, t + t n ok, ( x, t + ω o,k [n ( x, t ] (,...,, ( whee eesents the velocity states in which the moving aticles at a site might exit, and whee 6 c i 4 fo the FHP and 4 fo HPP lattice. The collision oeato ω i k ( n descibes the change in bits due to collisions. The symbol ω ( n ( x, t indicates the deendency of the collision oeato on all bit vaiables n ik, at site x and timeste t. The macoscoic undestanding of the lattice dynamic system can be obtained by efoming an ensemble aveage by assuming Boltzmann molecula chaos assumtion. The Boltzmann equations in tems of the mean oulation of aticles can then be obtained as N io N i, k <n ik, >, ( x + c i, t + t N io, ( x, t + Ω i,o [N ( x, t ] ( i 1,...,, (3 N ok, ( x, t + t N ok, ( x, t + Ω o,k [N ( x, t ] (,...,, (4 whee ω ( n Ω( n Ω( N, and N indicates that the deendency of the collision oeato Ω ( N on all the mean oulation of aticles N ik,. To obtain a solutions fo the above equation and undestand the macoscoic behaviou, a Chaman-Enskog etubation exansion and multi-scale technique [] can be utilized. In ode to this we fist conside the solutions fo equations (3 and (4 at local equilibia with density and momentum slowing changing in sace and time. It can be oved [,4] if the collisions veify the semi-detailed balance and conseve mass and momentum, the mean oulation of aticle at equilibia N ik, ae descibed by the Femi-Diac distibution. By using the consevation of mass eq and the consevation of momentum, 3

4 io, i 1 ( k 0 ρ N eq + m k N eq k (5 ρu N io, (6 k 1 ( i 0 ok, eq c i i 1 ( k 0 whee ρ and u ae the mass density and flow velocity e cell, esectively. The filling atio k is elated to the initial condition, eesenting that in this mixtue model sites on the lattice ae initialized with est aticle of mass m k. The andom bit obability of is elated to the local collision ules.the equilibium solution fo moving and est aticles can be obtained as N eq ρ ρ ( 1 d d c, (7 io, c ρ i u c m c 4 iγ c c iδ --- δ ρ m ( 1 d γδ + c c s δγδ u γ u δ (8 N eq ρ ( 1 d ( 1 d ok, d k k c 4 ρ m ( 1 d ( 1 d u m c k s In the above exessions, Geek index eesents the satial comonents of velocity of the aticles. At the equilibium the aveage moving aticle density d and the est aticle density d k with mass m k can be elated by d k d m k d m k ( 1 d m k +. c s is the sound seed of the lattice to be detemined latte. Now, to obtain the etubation solution to the Boltzmann equations (3 and (4 nea equilibia we exand aound the equilibia as a seies in owes of [,4] N ik, ε ( o ( 1 ( N ik, N ik, N i, k N ik, k k O ( ε 3, (9 whee ( o N ik, eq N ik,. Based on Chaman-Enskog exansion, a multi-scale technique is used by assuming ( n N ik, that the gadients of the and the elated time t ae vey small, thus satisfying ~ t O ( ε fo the fist ode deivative and ( n O ( ε n fo the nth ode deivative. Now we inset the exansion (7 in the Boltzmann equations (3 and (4 and use the above multi-scale technique. By identifying the tems at ode O( ε, the following equations can be witten fo the fist ode ( 1 N i, o ( 1 N ok, solutions of and, esectively, ( o t N io, + ( o t N ok, ( o c iβ β N i, o Λ ( 1 kl, N ol, l 1 Λ m ( 1 ij, N jo, j 1 Λ m ( 1 ik, N ok,, (10, ( , m, Λ Ω ik, io,,, Λ kl, Λ kj, ,. m Ω Λ io ij, N jo, N N eq N ok, N N eq It can be shown that by using the consevations of mass and momentum (5 and (6, the macoscoic equations fo mass and momentum can be witten as ρ + ( ρu 0 (1 t + + ( 1 Λ kj, N jo, j 1 Ω ok N ol, N N eq Ω ok N jo, N N eq t ( ρu + ( ρg ( ρuu P ( ρ, u (13 4

5 ( ρ, u ρ m c ρ ρ 1 ρ dc s c m 1 d u c c s (14 Equations (1-(15 show that the fist ode etubation on the lattice obey the Eule equation. To detemine the linea wave behaviou of the lattice and associated sound seed c s, we conside a case in which a small etubation( ρ, u is sueosed onto an equilibium state with density ρ o and zeo flow u o 0. We can wite ρ ρ o + ρ and u u, whee ρ o is the unifom backgound density, ρ and u ae, esectively, weak density and flow etubation with the ode of O ( ε. Fo this situation, the consevation of mass and mass at ode O ( ε can be exessed as t ρ + ρ o u 0, (16 t u + c s ρ o ρ 0, (17 whee the aamete, the sound seed, is calculated fom (17as c P s c b ρ m (18 ρ ρ c d ( 1 d d ( 1 d + m kk d k ( 1 d k u 0 u 0 Equations(16 and (17 can be combined to eliminate u and lead to the linea wave equation in tems of ρ as t ρ - c s ρ 0 (19 Hee we can note that the egime of undamed sound wave involves only the lowe symmetic equiments[,4], and thus the simle ectangula lattice of HPP is valid fo modelling linea wave oagation. An analogy now can be made between the above two-dimensional wave equation (19 and twodimensional TM o TE electomagnetic fields. Fo the TM z case, the macoscoic etubative density, can be equated to the electic field E z, and the x- and y- comonents of the etubative flow velocity, u ( u x, u y, ρ can be equated to magnetic field comonents, H y and H x, esectively. Similaly, fo the case of TE z, ρ can be equated to the magnetic field H z, and u ( u x, u y can then be equated to electic field comonents E y and E x esectively. In additional to this analogy, the mixtue HPP enables us to contol the sound seed (dielectic constant in a vey flexible way. IV.Numeical Results Seveal esults ae then esented as examles of the mixtue HPP LGA fo modelling inhomogeneous dielectic media in two-dimensional electomagnetic oblems. Fig.4 shows the time-domain scatteing intensity inside a dielectic cylindical shell with ε 5 and of inne adius a80 l and oute adius b100 l. The shell was ceated using a single est aticle m 1 4. Fig.5 demonstates a gaussian lane oagating though a dielectic cylindical shell with ρ m d, ρ m k k d k c s ε 85 (15. To obtain a elative dielectic constant 5

6 ε 85, a mixtue model with u to thee est aticles with masses m 1 4, m 8, and m 3 16 was constucted. As an examle of wave inteaction with a comlex biological stuctue, the scatteing field fom a human body coss-section model (dielectic values only was simulated and shown in Fig.6. In this body coss-section model moe than eight tissues with diffeent dielectic constants anging fom 4 to 85 wee modelled. V. Conclusion We have esented and analysed a new LGA mixtue model fo inhomogeneous media. As indicated by the sound seed fomula (18, only with a few estictions on such things as the consevation laws and the semi-detailed balance condition, thee ae a geat numbe of the collision ules which ae qualified to secify the lattice to enable a wide ange of dielectic constant to be modelled. This has been confimed by a vaiety of simulation exeiments[5]. VI. Refeences [1] N.R.Simons, G. Bidges,B.Podaima and A.Sebak, Cellula Automata as Envionment fo Simulating Electomagnetic Phenomena, IEEE Micowave and Guided Wave lettes, vol.4,.47-49,1994. [] U.Fisch, D.d Humiees, B.Hasslache, P.Lallemand, Y.Pomeau and J.P.Rivet, Lattice Hydodynamics in two and thee dimensions, Comlex Syst. vol.1, , [3] S.Chaman and T. Cowling, The Mathematical Theoy of Non-Unifom Gases, 3d.ed., Cambidge Univesity, Cambidge, England,1970. [4] D.Rothman and S.Zaleski, Lattice-Gas models fo Phase Seaation: Inteface, Phase Tansitions and Multihase Flow, Rev. of Moden Phys., vol.66, No.4, ,1994. [5] G. Bidges,N.R.Simons, D.Cule, M. Zhang and M.Cuhaci, Alication of the Lattice Gas Automata Technique to modelling Wave Inteaction with Biological Media, Poceedings of the IEE10th Intenational Confeence Antennas and Poagation, vol., , Ail, [6] P.B. Johns, A Symmetic Condensed Node fo the TLM Method, IEEE Tans. Micowave Theoy and Techniques, vol.35, No.4, , Ail Fig.4: Time-domain field intensities fo the electic field intensity inside a dielectic cylindical shell with ε 5. Comaison is made to the esults obtained using the TLM method[6]. 6

Fig.6: Image of the instantaneous field intensity fo hamonic

7 Fig.5: Snashot of the micoscoic field intensity fo a gausian lane wave oagating though a dielectic cylindical shell with ε 85. Fig.6: Image of the instantaneous field intensity fo hamonic lane wave incidence on the the coss section of human toso at 975 MHz. 7

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