Finite Difference Time Domain (FDTD) methods for solution ...ekashdan/page4/csss.pdf · Finite Difference Time Domain (FDTD) methods for solution of Maxwell’s equations Case Study

18th and 25th February, 2014 UCD - p. 1

Finite Difference Time Domain (FDTD)methods for solution of Maxwell’s equations

Case Study in Simulation Sciences

Dr. Eugene KashdanApplied and Computational Mathematics Group,

School of Mathematical Sciences, University College Dublin,

[email protected]

● Maxwell’s Equations

● Properties of materials

● Integral form

● Equations in scalar form

● Transverse mode

● Maxwell’s equations in 2D

FDTD methods

Divergence-free

Numerical stability


Maxwell’s Equations

The Maxwell equations in an isotropic medium are:



● Integral form


● Transverse mode


FDTD methods

Divergence-free

Numerical stability




∂ ~B

∂t+ ∇× ~E = 0, (Faraday’s Law)



● Integral form


● Transverse mode


FDTD methods

Divergence-free

Numerical stability




∂ ~B


∂ ~D

∂t−∇× ~H = − ~J, (Ampere’s law)



● Integral form


● Transverse mode


FDTD methods

Divergence-free

Numerical stability




∂ ~B


∂ ~D


coupled with Gauss’ law

∇ · ~B = 0 (magnetic field)

∇ · ~D = ρ (electric field),



● Integral form


● Transverse mode


FDTD methods

Divergence-free

Numerical stability




∂ ~B


∂ ~D


coupled with Gauss’ law

∇ · ~B = 0 (magnetic field)

∇ · ~D = ρ (electric field),

where ~J = σ ~E is electric current density, ρ is total electriccharge density, and the constitutive relations are given by~B = µ ~H and ~D = ε ~E.



● Integral form


● Transverse mode


FDTD methods

Divergence-free

Numerical stability


Properties of materials

µ – magnetic permeability.µ = µ0 · µr, where µ0 = 4π · 10−7 N

A2 is free space permeability.



● Integral form


● Transverse mode


FDTD methods

Divergence-free

Numerical stability





ε – dielectric permittivity.ε = ε0 · εr, ε0 = 1

c2µ0is free space permittivity in

[Fm

].



● Integral form


● Transverse mode


FDTD methods

Divergence-free

Numerical stability







[Fm

].

In general, µr and εr are frequency dependent. Materialswithout such dependence are called "the simple materials".



● Integral form


● Transverse mode


FDTD methods

Divergence-free

Numerical stability







[Fm

].


µr =1 for almost all simple materials except metals (perfectelectric conductors – PEC).



● Integral form


● Transverse mode


FDTD methods

Divergence-free

Numerical stability







[Fm

].



εr ≥ 1 is discontinuous at the interface between materials.



● Integral form


● Transverse mode


FDTD methods

Divergence-free

Numerical stability







[Fm

].



εr ≥ 1 is discontinuous at the interface between materials.

σ is electrical conductivity in[

Sm

], which represents conducting

properties of material.



● Integral form


● Transverse mode


FDTD methods

Divergence-free

Numerical stability


Integral form

∂

∂t

x

S

~B · dS = −∮

C

~E · dl (Faraday’s Law)



● Integral form


● Transverse mode


FDTD methods

Divergence-free

Numerical stability


Integral form

∂

∂t

x

S

~B · dS = −∮

C


∂

∂t

x

S

~D · dS =

∮

C

~H · dl −x

S

~J · dS (Ampere’s Law)



● Integral form


● Transverse mode


FDTD methods

Divergence-free

Numerical stability


Integral form

∂

∂t

x

S

~B · dS = −∮

C


∂

∂t

x

S

~D · dS =

∮

C

~H · dl −x

S

~J · dS (Ampere’s Law)

Gauss’ law:{

S

~D · dS =y

V

ρdV (electric field)

{

S

~B · dS = 0 (magnetic field)



● Integral form


● Transverse mode


FDTD methods

Divergence-free

Numerical stability


Equations in scalar form

In Cartesian coordinates, in 3D, Maxwell’s equations are



● Integral form


● Transverse mode


FDTD methods

Divergence-free

Numerical stability




ε∂Ex

∂t=

∂Hz

∂y− ∂Hy

∂zµ

∂Hx

∂t= −∂Ez

∂y+

∂Ey

∂z

ε∂Ey

∂t=

∂Hx

∂z− ∂Hz

∂xµ

∂Hy

∂t= −∂Ex

∂z+

∂Ez

∂x

ε∂Ez

∂t=

∂Hy

∂x− ∂Hx

∂yµ

∂Hz

∂t= −∂Ey

∂x+

∂Ex

∂y



● Integral form


● Transverse mode


FDTD methods

Divergence-free

Numerical stability




ε∂Ex

∂t=

∂Hz

∂y− ∂Hy

∂zµ

∂Hx

∂t= −∂Ez

∂y+

∂Ey

∂z

ε∂Ey

∂t=

∂Hx

∂z− ∂Hz

∂xµ

∂Hy

∂t= −∂Ex

∂z+

∂Ez

∂x

ε∂Ez

∂t=

∂Hy

∂x− ∂Hx

∂yµ

∂Hz

∂t= −∂Ey

∂x+

∂Ex

∂y

We assume that the medium is loss-free (J = 0) and ε and µ arenot time-dependent)



● Integral form


● Transverse mode


FDTD methods

Divergence-free

Numerical stability


Transverse mode

■ Transverse electric (TE) modes: no electric field in thedirection of propagation.



● Integral form


● Transverse mode


FDTD methods

Divergence-free

Numerical stability


Transverse mode


■ Transverse magnetic (TM) modes: no magnetic field in thedirection of propagation.



● Integral form


● Transverse mode


FDTD methods

Divergence-free

Numerical stability


Transverse mode



■ Transverse electromagnetic (TEM) modes: neither electricnor magnetic field in the direction of propagation.



● Integral form


● Transverse mode


FDTD methods

Divergence-free

Numerical stability


Transverse mode




■ Hybrid modes: non-zero electric and magnetic fields in thedirection of propagation.



● Integral form


● Transverse mode


FDTD methods

Divergence-free

Numerical stability


Transverse mode





In rectangular waveguides, modes are marked as TEmn,where m is the number of half-wavelengths across the width ofthe waveguide and n – across the height of the waveguide.



● Integral form


● Transverse mode


FDTD methods

Divergence-free

Numerical stability


Transverse mode





In rectangular waveguides, modes are marked as TEmn,where m is the number of half-wavelengths across the width ofthe waveguide and n – across the height of the waveguide.

Example: a radio wave in a hollow metal waveguide must havezero tangential electric field amplitude at the walls of thewaveguide, so the transverse pattern of the electric field ofwaves is restricted to those that fit between the walls



● Integral form


● Transverse mode


FDTD methods

Divergence-free

Numerical stability


Maxwell’s equations in 2D

We choose z as transverse direction:



● Integral form


● Transverse mode


FDTD methods

Divergence-free

Numerical stability




■ TE mode

ε∂Ex

∂t=

∂Hz

∂y, ε

∂Ey

∂t= −∂Hz

∂x,

µ∂Hz

∂t= −∂Ey

∂x+

∂Ex

∂y.



● Integral form


● Transverse mode


FDTD methods

Divergence-free

Numerical stability




■ TE mode

ε∂Ex

∂t=

∂Hz

∂y, ε

∂Ey

∂t= −∂Hz

∂x,

µ∂Hz

∂t= −∂Ey

∂x+

∂Ex

∂y.

■ TM mode

µ∂Hx

∂t= −∂Ez

∂y, µ

∂Hy

∂t=

∂Ez

∂x,

ε∂Ez

∂t=

∂Hy

∂x− ∂Hx

∂y.



● Integral form


● Transverse mode


FDTD methods

● Yee Algorithm

● Spatial location in 3D

● Discretization – electric field

● Discretization – magnetic field

Divergence-free

Numerical stability


Yee Algorithm

"Classical" FDTD method (Yee, 1966) uses the second ordercentral difference scheme for integration in space and thesecond order Leapfrog scheme for integration in time. This is astaggered non-dissipative scheme both in space and in time.



● Integral form


● Transverse mode


FDTD methods

● Yee Algorithm




Divergence-free

Numerical stability


Yee Algorithm


A 1D space-time chart of the Yee algorithm



● Integral form


● Transverse mode


FDTD methods

● Yee Algorithm




Divergence-free

Numerical stability


Yee Algorithm



E∆t∆x−E

0∆x

∆t=

H0.5∆t1.5∆x−H

0.5∆t0.5∆x

∆x



● Integral form


● Transverse mode


FDTD methods

● Yee Algorithm




Divergence-free

Numerical stability


Yee Algorithm



E∆t∆x−E

0∆x

∆t=

H0.5∆t1.5∆x−H

0.5∆t0.5∆x

∆x



● Integral form


● Transverse mode


FDTD methods

● Yee Algorithm




Divergence-free

Numerical stability


Spatial location in 3D

In Cartesian coordinates and three dimensions we have thefollowing spatial distribution of the components:



● Integral form


● Transverse mode


FDTD methods

● Yee Algorithm




Divergence-free

Numerical stability


Spatial location in 3D

In Cartesian coordinates and three dimensions we have thefollowing spatial distribution of the components:



● Integral form


● Transverse mode


FDTD methods

● Yee Algorithm




Divergence-free

Numerical stability


Discretization – electric field

Et+∆tx,(i+ 1

2,j,k)

= Etx,(i+ 1

2,j,k) +

∆t

εi+ 12,j,k

[(Ht+∆t

2

z,(i+ 12,j+ 1

2,k)

− Ht+∆t

2

z,(i+ 12,j− 1

2,k)

∆y

)

(Ht+∆t

2

y,(i+ 12,j,k 1

2)− H

t+∆t2

y,(i+ 12,j,k− 1

2)

∆z

)]

Et+∆ty,(i,j+ 1

2,k)

= Ety,(i,j+ 1

2,k) +

∆t

εi,j+ 12,k

[(Ht+∆t

2

x,(i,j+ 12,k+ 1

2)− H

t+∆t2

x,(i,j+ 12,k− 1

2)

∆z

)

(Ht+∆t

2

z,(i+ 12,j+ 1

2,k)

− Ht+∆t

2

z,(i− 12,j+ 1

2,k)

∆x

)

Et+∆tz,(i,j,k+ 1

2)= Et

z,(i,j,k+ 12) +

∆t

εi,j,k+ 12

[(Ht+∆t

2

y,(i+ 12,j,k+ 1

2)− H

t+∆t2

y,(i− 12,j,k+ 1

2)

∆x

)

(Ht+∆t

2

x,(i,j+ 12,k+ 1

2)− H

t+∆t2

x,(i,j− 12,k+ 1

2)

∆y

)]



● Integral form


● Transverse mode


FDTD methods

● Yee Algorithm




Divergence-free

Numerical stability


Discretization – magnetic field

Ht+ 3∆t

2

x,(i,j+ 12,k+ 1

2)= H

t+∆t2

x,(i,j+ 12,k+ 1

2)+

∆t

µi,j+ 12,k+ 1

2

×

[(Et+∆ty,(i,j+ 1

2,k+1)

− Et+∆ty,(i,j+ 1

2,k)

∆z

)

−(Et+∆t

z,(i,j,k+ 12)− Et+∆t

z,(i,j+1,k+ 12)

∆y

)]

Ht+ 3∆t

2

y,(i+ 12,j,k+ 1

2)= H

t+∆t2

y,(i+ 12,j,k+ 1

2)+

∆t

µi+ 12,j,k+ 1

2

×

[(Et+∆tz,(i+1,j,k+ 1

2)− Et+∆t

z,(i,j,k+ 12)

∆x

)

−(Et+∆t

x,(i+ 12,j,k+1)

− Et+∆tx,(i+ 1

2,j,k)

∆z

)]

Ht+ 3∆t

2

z,(i+ 12,j+ 1

2,k)

= Ht+∆t

2

z,(i+ 12,j+ 1

2,k)

+∆t

µi+ 12,j+ 1

2,k

×

[(Et+∆tx,(i+ 1

2,j+1,k)

− Et+∆tx,(i+ 1

2,j,k)

∆y

)

−(Et+∆t

y,(i+1,j+ 12,k)

− Et+∆ty,(i,j+ 1

2,k)

∆x

)]



● Integral form


● Transverse mode


FDTD methods

Divergence-free

● No charges, no current:

∂ρ∂t

+ ∇ · J = 0

● Applying Yee algorithm

● Concept of order of accuracy

Numerical stability


No charges, no current: ∂ρ∂t

+ ∇ · J = 0

{

Yee cell

~D · dS =ε0

∂t

(

Ex,(i,j+ 12,k+ 1

2) − Ex,(i−1,j+ 1

2,k+ 1

2)

)

∆y∆z

︸︷︷︸

Term 1

+

ε0

∂t

(

Ey,(i− 12,j+1,k+ 1

2) − Ey,(i− 1

2,j,k+ 1

2)

)

∆x∆z

︸︷︷︸

Term 2

+

ε0

∂t

(

Ez,(i− 12,j+ 1

2,k+1) − Ez,(i− 1

2,j+ 1

2,k)

)

∆x∆y

︸︷︷︸

Term 3



● Integral form


● Transverse mode


FDTD methods

Divergence-free


∂ρ∂t

+ ∇ · J = 0



Numerical stability


Applying Yee algorithm

T1 =

(Hz,(i,j+1,k+ 1

2) − Hz,(i,j,k+ 1

2)

∆y−

Hy,(i,j+ 12,j,k+1) − Hy,(i,j+ 1

2,j,k)

∆z

)

−(

Hz,(i−1,j+1,k+ 12) − Hz,(i−1,j,k+ 1

2)

∆y

−Hy,(i−1,j+ 1

2,j,k+1) − Hy,(i−1,j+ 1

2,j,k)

∆z

)

. . .



● Integral form


● Transverse mode


FDTD methods

Divergence-free


∂ρ∂t

+ ∇ · J = 0



Numerical stability


Applying Yee algorithm

T1 =

(Hz,(i,j+1,k+ 1

2) − Hz,(i,j,k+ 1

2)

∆y−

Hy,(i,j+ 12,j,k+1) − Hy,(i,j+ 1

2,j,k)

∆z

)

−(

Hz,(i−1,j+1,k+ 12) − Hz,(i−1,j,k+ 1

2)

∆y

−Hy,(i−1,j+ 1

2,j,k+1) − Hy,(i−1,j+ 1

2,j,k)

∆z

)

. . .After substitution:

{

Yee cell

~D · dS = (T1)∆y∆z + (T2)∆x∆z + (T3)∆x∆y = 0



● Integral form


● Transverse mode


FDTD methods

Divergence-free


∂ρ∂t

+ ∇ · J = 0



Numerical stability


Concept of order of accuracy

According to the Lax-Richtmyer Equivalence Theorem , if a finitedifference scheme has a truncation error of order (p, q) and thescheme is stable, then the difference between the analyticsolution and the numerical solution in appropriate norm is oforder (∆t)p + hq for all finite time.



● Integral form


● Transverse mode


FDTD methods

Divergence-free


∂ρ∂t

+ ∇ · J = 0



Numerical stability




∂u

∂x(i∆x, j∆y, k∆z, n∆t) =

uni+1/2,j,k − un

i−1/2,j,k

∆x+ O[(∆x)2] ,

∂u

∂t(i∆x, j∆y, k∆z, n∆t) =

un+1/2i,j,k − u

n−1/2i,j,k

∆t+ O[(∆t)2]



● Integral form


● Transverse mode


FDTD methods

Divergence-free


∂ρ∂t

+ ∇ · J = 0



Numerical stability




∂u

∂x(i∆x, j∆y, k∆z, n∆t) =

uni+1/2,j,k − un

i−1/2,j,k

∆x+ O[(∆x)2] ,

∂u

∂t(i∆x, j∆y, k∆z, n∆t) =

un+1/2i,j,k − u

n−1/2i,j,k

∆t+ O[(∆t)2]

However, there is no equivalent of the Lax-Richtmyer Theore m thatextends these results to approximation of equations with va riablecoefficients and includes boundary conditions and forcing f unctions ∗.



● Integral form


● Transverse mode


FDTD methods

Divergence-free

Numerical stability

● TM mode

● Time eigenvalue problem

● Condition of temporal stability

● Space eigenvalue problem

● Condition of spatial stability

● Enforcing stability of Yee

algorithm


TM mode

Hn+1/2x,(i,j) − H

n−1/2x,(i,j)

∆t= − 1

µ

(Enz,(i,j+ 1

2)− En

z,(i,j− 12)

∆y

)

Hn+1/2y,(i,j) − H

n−1/2y,(i,j)

∆t=

1

µ

(Enz,(i+ 1

2,j)

− Enz,(i− 1

2,j)

∆x

)

En+1z,(i,j) − En

z,(i,j)

∆t=

1

ε

[(Hn+1/2

y,(i+ 12,j)

− Hn+1/2

y,(i− 12)

∆x

)

−(H

n+1/2

x,(i,j+ 12)− H

n+1/2

x,(i,j− 12)

∆y

)]



● Integral form


● Transverse mode


FDTD methods

Divergence-free

Numerical stability

● TM mode






algorithm


Time eigenvalue problem

Introduce the generic vector component

Vn+1/2(i,j) − V

n−1/2(i,j)

∆t= ΛV n

(i,j)



● Integral form


● Transverse mode


FDTD methods

Divergence-free

Numerical stability

● TM mode






algorithm




Vn+1/2(i,j) − V

n−1/2(i,j)

∆t= ΛV n

(i,j)

and define a solution growth factor

qi,j =V

n+1/2(i,j)

V n(i,j)

=V n

(i,j)

Vn−1/2(i,j)

for all n.



● Integral form


● Transverse mode


FDTD methods

Divergence-free

Numerical stability

● TM mode






algorithm




Vn+1/2(i,j) − V

n−1/2(i,j)

∆t= ΛV n

(i,j)

and define a solution growth factor

qi,j =V

n+1/2(i,j)

V n(i,j)

=V n

(i,j)

Vn−1/2(i,j)

for all n.

The goal: |qi,j | ≤ 1 to avoid uncontrolled growth and blow-upfor all points (i, j).



● Integral form


● Transverse mode


FDTD methods

Divergence-free

Numerical stability

● TM mode






algorithm


Condition of temporal stability

After substitution and excluding V n(i,j):

q2i,j − Λ∆tqi,j − 1 = 0



● Integral form


● Transverse mode


FDTD methods

Divergence-free

Numerical stability

● TM mode






algorithm




q2i,j − Λ∆tqi,j − 1 = 0

The solution:

qi,j =Λ∆t

2+

√(

Λ∆t

2

)2

+ 1 = α +√

α1 + 1



● Integral form


● Transverse mode


FDTD methods

Divergence-free

Numerical stability

● TM mode






algorithm




q2i,j − Λ∆tqi,j − 1 = 0

The solution:

qi,j =Λ∆t

2+

√(

Λ∆t

2

)2

+ 1 = α +√

α1 + 1

|qi,j | = 1 always if Re[α] = 0 and −1 ≤ Im[α] ≤ 1; henceα = i · Im[α] and Λ = i · Im[Λ].



● Integral form


● Transverse mode


FDTD methods

Divergence-free

Numerical stability

● TM mode






algorithm




q2i,j − Λ∆tqi,j − 1 = 0

The solution:

qi,j =Λ∆t

2+

√(

Λ∆t

2

)2

+ 1 = α +√

α1 + 1

|qi,j | = 1 always if Re[α] = 0 and −1 ≤ Im[α] ≤ 1; henceα = i · Im[α] and Λ = i · Im[Λ].

In this case

−1 ≤ Im[Λ]∆t

2≤ 1 ⇒ − 2

∆t≤ Im[Λ] ≤ 2

∆t



● Integral form


● Transverse mode


FDTD methods

Divergence-free

Numerical stability

● TM mode






algorithm


Space eigenvalue problem

We isolate space differentiation operation:

− 1

µ

(Ez,(i,j+ 1

2) − Ez,(i,j− 1

2)

∆y

)

= ΛHx,(i,j)

1

µ

(Ez,(i+ 1

2,j) − Ez,(i− 1

2,j)

∆x

)

= ΛHy,(i,j)

1

ε

[(Hy,(i+ 1

2,j) − Hy,(i− 1

2)

∆x

)

−(

Hx,(i,j+ 12) − Hx,(i,j− 1

2)

∆y

)]

= ΛEz,(i,j)



● Integral form


● Transverse mode


FDTD methods

Divergence-free

Numerical stability

● TM mode






algorithm


Space eigenvalue problem

We isolate space differentiation operation:

− 1

µ

(Ez,(i,j+ 1

2) − Ez,(i,j− 1

2)

∆y

)

= ΛHx,(i,j)

1

µ

(Ez,(i+ 1

2,j) − Ez,(i− 1

2,j)

∆x

)

= ΛHy,(i,j)

1

ε

[(Hy,(i+ 1

2,j) − Hy,(i− 1

2)

∆x

)

−(

Hx,(i,j+ 12) − Hx,(i,j− 1

2)

∆y

)]

= ΛEz,(i,j)

and look for solution in form of the plane wave with kx and ky arethe components of wave-vector in x and y directions:

Ez,(I,J) = Ez0exp[i(kxI∆x + kyJ∆y)]

Ex,(I,J) = Hx0exp[i(kxI∆x + kyJ∆y)]

Ey,(I,J) = Hy0exp[i(kxI∆x + kyJ∆y)]



● Integral form


● Transverse mode


FDTD methods

Divergence-free

Numerical stability

● TM mode






algorithm


Condition of spatial stability

After substitution and applying Euler’s Identity:

Hx0=

2iEz0

λµ∆ysin

(ky∆y

2

)

, Hy0=

2iEz0

λµ∆xsin

(kx∆x

2

)

Ez0=

2i

Λε

[2iHy0

∆xsin

(kx∆x

2

)

− 2iHx0

∆ysin

(ky∆y

2

)]



● Integral form


● Transverse mode


FDTD methods

Divergence-free

Numerical stability

● TM mode






algorithm




Hx0=

2iEz0

λµ∆ysin

(ky∆y

2

)

, Hy0=

2iEz0

λµ∆xsin

(kx∆x

2

)

Ez0=

2i

Λε

[2iHy0

∆xsin

(kx∆x

2

)

− 2iHx0

∆ysin

(ky∆y

2

)]

After substituting Hx0and Hy0

and eliminating Ez0we obtain:

Λ2 = − 4

µε

[1

(∆x)2sin2

(kx∆x

2

)

+1

(∆y)2sin2

(ky∆y

2

)]



● Integral form


● Transverse mode


FDTD methods

Divergence-free

Numerical stability

● TM mode






algorithm




Hx0=

2iEz0

λµ∆ysin

(ky∆y

2

)

, Hy0=

2iEz0

λµ∆xsin

(kx∆x

2

)

Ez0=

2i

Λε

[2iHy0

∆xsin

(kx∆x

2

)

− 2iHx0

∆ysin

(ky∆y

2

)]

After substituting Hx0and Hy0

and eliminating Ez0we obtain:

Λ2 = − 4

µε

[1

(∆x)2sin2

(kx∆x

2

)

+1

(∆y)2sin2

(ky∆y

2

)]

Using the fact that Re[Λ] = 0 and the bounds of sin(.) we write:

−2c

√

1

(∆x)2+

1

(∆y)2≤ Im[Λ] ≤ 2c

√

1

(∆x)2+

1

(∆y)2,

where c = 1√

εµ is a speed of light.



● Integral form


● Transverse mode


FDTD methods

Divergence-free

Numerical stability

● TM mode






algorithm


Enforcing stability of Yee algorithm

2c

√

1

(∆x)2+

1

(∆y)2≤ 2

∆t⇒ ∆t ≤ 1

c√

1(∆x)2 + 1

(∆y)2



● Integral form


● Transverse mode


FDTD methods

Divergence-free

Numerical stability

● TM mode






algorithm



2c

√

1

(∆x)2+

1

(∆y)2≤ 2

∆t⇒ ∆t ≤ 1

c√

1(∆x)2 + 1

(∆y)2

When ∆x = ∆y = ∆:

∆t =∆

c√

2



● Integral form


● Transverse mode


FDTD methods

Divergence-free

Numerical stability

● TM mode






algorithm



2c

√

1

(∆x)2+

1

(∆y)2≤ 2

∆t⇒ ∆t ≤ 1

c√

1(∆x)2 + 1

(∆y)2

When ∆x = ∆y = ∆:

∆t =∆

c√

2

In practical applications, we always intend to keep ∆t as large as possible tominimize the computational time, but its value should be always less (at leastslightly) than the upper bound to avoid instability caused by the “numericaljunk” accumulation.

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