EEE 431 Lecture 7

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EEE 431Computational Methods in

Electrodynamics

Lecture 7By

Dr. Rasime [email protected]

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FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD)

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FDTD References [1]M.N.O. Sadiku, Numerical Techniques in Electromagnetics, CRC

Press, 2001, pp. 159-192 [2] A. Taflove, Computational Elertodynamics: the Finite-Difference Time-Domain Method, Artech, 1995 [3] A. Taflove, S.C. Hagness, same as above, 2nd ed., Artech, 2000 [4] K. Kunz and R. Luebbers, Finite-Difference Time-Domain Method for Electromagnetics, CRC Press, 1993 [5]Kane S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,”

IEEE Trans. Antennas Propagat., vol. AP-14, No. 3, pp. 302-307, May,1966.

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References

Introduction to the Finite Time Domain (FDTD) Technique, S.Connor

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The FDTD method, proposed by Yee, 1966, is another numerical method, used widely for the solution of EM problems. It is used to solve open-region scattering, radiation, diffusion, microwave circuit modeling, biomedical etc. problems.

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One of the most important concerns of the FDTD method is the requirement of the artificial mesh truncation (boundary) conditions. These conditions are used to truncate the solution domain and they are known as absorbing boundary conditions (ABCs), as they theoretically absorb fields. Imperfect ABCs create reflections and the accuracy of the FDTD method depends on the accuracy of the ABCs.

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FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) The following advantages make the FDTD

method popular: Its a direct solution of Maxwell’s equations, no

integral equations are required and no matrix inversions are necessary.

Its implementation is easy and it is conceptually simple.

It can be applied to the three-dimensional, arbitrary geometries.

It can be applied to materials with any conductivity.

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The FDTD method has also the following disadvantages:

When the FDTD method is applied, the object and its surroundings must be defined.

Since computational meshes are rectangular in shape it is difficult to apply the method to the curved scaterers.

It has low order of accuracy and stability unless fine mesh is used

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In this method the coupled Maxwell’s curl equations in the differential form are discretized, approximating the derivatives with centered difference approximations in both time and space domains.

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The six scalar components of electric and magnetic fields are obtained in a time-stepped manner.

The space domain includes the object and it is terminated by Absorbing Boundary Conditions (ABCs).

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Basic Finite-Difference Time-Domain Algorithm

Differential Forms of Maxwell’s Equations

. 0

DXH J

tB

. v

BXE

tD

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In linear, isotropic and homogeneous materials and are related to and with the following constitutive relations:

D B E H

D E

B H

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Also is related to as,

in a conducting medium.

J E

J E

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Substituting the constitutive relations into the Maxwell’s equation’s, we can write six scalar equations in the Cartesian coordinate system.

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Electric Field Intensity:

1

1

1

yx zx

y x zy

y xzz

HE HE

t y z

E H HE

t z x

H HEE

t x y

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Magnetic field intensity:

1

1

1

yx z

y xz

yxz

EH E

t z y

H EE

t x z

EEH

t y x

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Yee Algorithm: Yee algorithm solves for both electric

and magnetic fields in time and space using the coupled Maxwell’s curl equations.

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the Yee algorithm centers its and field components in three-dimensional space so that every component is surrounded by four circulating components, and every component is surrounded by four circulating components.

E H

EH

H

E

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Unit cell of the Yee space lattice.

x

y

z

E x E x

E x

E y

E y

E y

E z

E z

H x

H z

H y

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The computational domain is divided into a number of rectangular unit cells

According to Yee algorithm and field components are separated by in time.

E H

2

t

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t= 0

t= 0.5 t

t= t

x= 0 x= x x= 2x x= 3x

E E E E

EEEE

H H H

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Finite Differences ( Discretization ) A space point in a uniform

rectangular lattice is denoted as

( , , ) ( , , )i j k i x j y k z

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Here , and are the lattice space increments in the x, y, and z coordinate directions respectively and i, j, and k are integers. If any scalar function of space and time evaluated at a discrete point in the grid and at a discrete point in time is denoted by u, then

x y z

, ,( , , , ) ni j ku i x j y k z n t u

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Using central, finite difference approximation in space, i.e w.r.t.x:

1 1, , , , 22 2, , ,

n n

i j k i j ku uu

i x j y k z n t O xx x

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Using central, finite difference approximation in time:

1 1

2 22, , , ,, , ,

n n

i j k i j ku uui x j y k z n t O t

t t

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Ex

, , , ,

, 1, , , , , 1 , ,

1

1 2 1 2 1 2 1 2

, ,1

, ,i j k i j k

i j k i j k i j k i j k

n nx x

n n n nz z y y

i j k tE E

i j k

t tH H H H

y y

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Ey

, , , ,

, , 1 , , 1, , , ,

1

1 2 1 2 1 2 1 2

, ,1

, ,i j k i j k


n ny y

n n n nx x z z

i j k tE E

i j k

t tH H H H

z x

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Ez

, , , ,

1, , , , , 1, , ,

1

1 2 1 2 1 2 1 2

, ,1

, ,i j k i j k


n nz z

n n n ny y x x

i j k tE E

i j k

t tH H H H

x y

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Hx

, , , ,

( , , 1) ( , , ) ( , 1, ), ,

1 2 1 2

i j k i j k

i j k i j k i j ki j k

n nx x

n n n ny y z z

H H

t tE E E E

z y

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Hy

, , , ,

( 1, , ) ( , , ) ( , , 1), ,

1 2 1 2

i j k i j k


n ny y

n n n nz z x x

H H

t tE E E E

x z

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Hz

, , , ,

( , 1, ) ( , , ) ( 1, , ), ,

1 2 1 2

i j k i j k


n nz z

n n n nx x y y

H H

t tE E E E

y x

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EEE 431 Lecture 7

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