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21 MARCH 2012 IIT KHARAGPUR SUBHASH YADAV RF & MICROWAVE ENGG ROLL-NO-11EC63R10 Unconditionally Stable FDTD Methods

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Page 1: Unconditionally stable fdtd methods

21 MARCH 2012 IIT KHARAGPUR

SUBHASH YADAV

RF amp MICROWAVE ENGG

ROLL-NO-11EC63R10

Unconditionally Stable FDTD Methods

Outlineshelliphelliphelliphellip

1Some computational methods for maxwells equations

2First fdtd algorithm YEE 1966 discritizations

3Some problems and condition in conventional fdtd

4Unconditionally stable fdtd use to solve heat equation

21 march 2012 2

Outlineshellip

5Imclicit Cranknicolson ADI Methods

6-Von neumann stability 7Advantages

8Disadvantages

9Conclusion

10References

21 march 2012 3

Maxwell equations

0

0

1

1

H

E

H E

E H

t

t typermittivi electric

typermeabili magnetic

))()()((H zyxtHzyxtHzyxtH zyx

))()()((E zyxtEzyxtEzyxtE zyx

field magnetic

field electric

Computational Electromagnetics

Finite-differencetime-domain(FDTD)

Finite-differencefrequency-domain(FDFD)

Method of Moments(MoM)

Fast multipole method (FMM)

Finite element method (FEM)

Transmission line matrix (TLM)

Frequency

Computational Electromagnetics

Finite-differencetime-domain(FDTD)

Finite-differencefrequency-domain(FDFD)

Method of Moments(MoM)

Fast multipole method (FMM)

Finite element method (FEM)

Transmission line matrix (TLM)

FDTD Overview ndash Updating Equations

760

Three scalar equations can be obtained from one vector curl equation

EH

t

yx zx

y x zy

y xzz

HE H

t y z

E H H

t z xH HE

t x y

HE

t

yx zx

y xzy

yxzz

EH E

t z y

H EE

t x zEEH

t y x

Finite DifferenceFinite Difference

Taylorrsquos seriesTaylorrsquos series

II Finite DifferenceII Finite Difference

Taylorrsquos Taylorrsquos seriesseries

ErorEror

Finite Difference Time Domain Method

bull Divide the interval x into sub-intervals each of width h

bull Divide the interval t into sub-intervals each of width k

bull A grid of points is used forthe finite difference solution

bull Tij represents T(xi tj)bull Replace the derivates by

finite-difference formulas10

t

x

(i j) (i+1 j)

(i j+1)

Ey

Ex

Hz

The Yee Discretization (1966)

Staggered grid in space mdash every field component is stored on a different grid

(i j k) (i+1 j k)

(i j k+1)

(i+1 j+1 k)

(i+1 j+1 k+1)

Ez

Ex

Ey

HyHx

Hz

(i j) (i+1 j)

(i j+1)

Ey

Ex

Hz

The Yee Discretization (1966)

all derivatives become center differenceshellip

H

t

1

E

H zt i

1

2 j

1

2

1

Eyx

Exy

1

Ey (i 1 j 1

2) Ey (i j

1

2)

xEx (i

1

2 j 1) Ex (i

1

2 j)

y

+ O(∆x2) + O(∆y2)

FDTD Overview ndash Updating Equations

yx zx

HE H

t y z

1

05 0505 05

( ) ( )( )

( ) ( 1)( ) ( 1 )

n nx x

x

n nn ny yz z

E i j k E i j ki j k

t

H i j k H i j kH i j k H i j k

y z

FDTD Overview ndash Updating Equations

yx zx

EH E

t z y

05 05

05

( ) ( )( )

( 1) ( ) ( 1 ) ( )

n nx x

x

n n n ny y z z

H i j k H i j ki j k

t

E i j k E i j k E i j k E i j k

z y

FDTD Overview ndash Updating Equations

Express the future components in terms of the past components

05 05

1

05 05

( ) ( 1 )

( ) ( )( ) ( ) ( 1)

n nz z

n nx x n n

x y y

H i j k H i j k

ytE i j k E i j k

i j k H i j k H i j k

z

05

05 05

( 1) ( )

( ) ( )( ) ( 1 ) ( )

n ny y

n nx x n n

x z z

E i j k E i j k

t zH i j k H i j ki j k E i j k E i j k

y

Fundamentals of the FDTD methodFundamentals of the FDTD method

Accuracy and stabilityAccuracy and stability

AccuracyAccuracy 10

StabilityStability

222

max

z

1

y

1

x

1c

1tt

3c

tt max

2D2D

22 y

1

x

1c

1t

1D1Dc

t

Physically this condition means that the time Physically this condition means that the time step should be smaller than the time for the step should be smaller than the time for the wave to propagate from one cell to the wave to propagate from one cell to the neighbor oneneighbor one

Slides from 5-15 may be skipped because it was taught in the class

III Fundamentals of the FDTD methodIII Fundamentals of the FDTD method

Dispersion relationDispersion relation

ckkkk

22

z2

y2

x2

In free space (ideal)

In FDTD computation (numerical)

22

z

2

y

2

x

2

tsin

tc

1

2

zk~

sinz

1

2

yk~

siny

1

2

xk~

sinx

1

utcsinArctk

~2

k~vpnum

2

z

2

y

2

x

2

zk~

sinz

1

2

yk~

siny

1

2

xk~

sinx

1u

The numerical medium is dispersive the propagation of the wave varies with frequency and angle

III Fundamentals of the FDTD methodIII Fundamentals of the FDTD method

Dispersion relationDispersion relation

Limitations of FDTD method

1-Grid spacing should be ~λ102-According to Courantrsquos stability condition time step Δt becomes small when FDTDgrid spacing becomes small3-In 3-D simulation simulation time scales like N^4 and required memory size scales like N^34-Application is restricted to relatively small size

21 march 2012 20

Space Domain Discretization

bull Heat Conduction Equation

bull Central-Finite-Difference Approximation

pc

tyxg

y

tyxT

x

tyxT

t

tyxT

)()()()(

2

2

2

2

2

2

2

11

22

112

2

2

)(2

x

T

x

TTT

xOx

TTT

x

T

nx

nji

nji

nji

nji

nji

njin

ji

cc

21 march 2012 21

X

n

jiTn

j1-iT n

j1iT

n

1jiT

n

1jiT

(00) (i0)hellip

(0j)

x

y

(I0)

(0J)

hellip

hellip

hellip

Finite-Difference Formulation of the Heat Conduction on a Chip

bull Space Domainbull Time Domain

21 march 2012 22

Time domain discretization

bull Heat Conduction Equation

ndash Simple Explicit Method ndash Simple Implicit Methodndash Crank-Nicolson Method

gcy

T

x

T

t

TT

p

yxnn

1

)()( 2

2

2

21

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

23

Can we check if a numerical scheme

is stable without computation Von Neumann stability

analysisbull Analyze if (or for which conditions) a

numerical scheme is stable or unstablebull Makes a local analysis coefficients of PDE are

assumed to vary slowly (our example constant)

bull How will unavoidable errors (say rounding errors)evolve in time

John von Neumann1903-1957

24

Von Neumann stability analysis

A numerical scheme is unstable if

Ansatz Wave number k and amplification factor

21 march 2012 25

bull Accuracybull Stability Constraint

bull No matrix inversion but time steps are limited by space discretization

gcy

T

x

T

t

TT

p

ny

nx

nn

1

)()( 2

2

2

21

] [ 22 yxt

2

11122

yxt

Simple Explicit Method

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

21 march 2012 26

Accuracy

Unconditionally Stable

No limits on time step but involves with large scale matrix inversion

gcy

T

x

T

t

TT

p

ny

nx

nn

1

)()( 2

12

2

121

] [ 22 yxt

Simple Implicit Method

11

1

11

12

11

1

11

12

2

2nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

21 march 2012 27

Accuracy

Unconditionally stable

No limits on time step but involves with large scale matrix inversion

gcy

TT

x

TT

t

TT

p

ny

ny

nx

nx

nn

1

)(2)(2 2

212

2

2121

] [ 222 yxt

Crank-Nicolson Method

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

11

1

11

12

11

1

11

12

2

2nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

21 march 2012 28

n

m1616

ex m=4n=4

Total node number N = mn

Matrix size = NxN

Analysis of Crank-Nicolson Method

21 march 2012 29

Alternating Direction Implicit Method

Solves higher dimension problem by successive Lower dimension methods

Accuracy

Unconditionally stable

No limits on time step and no large scale matrix inversion

] [ 222 yxt

21 march2012 30

Step I x-direction implicit y-direction explicitStep II x-direction explicit y-direction implicit

n

Alternating Direction Implicit Method

bull Peaceman-Rachford Algorithmbull Douglas-Gunn Algorithm

21 march 2012 31

bull Step I

bull Step II

gc

tT

rrT

rr

p

ny

yx

xny

yx

x

)2

1)(2

1()2

1)(2

1( 22122

gc

tT

rT

r

p

ny

yn

xx

2)

21()

21( 22

12

gc

tT

rT

r

p

n

xxn

yy

2)

21()

21( 2

1212

Peaceman-Rachford Algorithm

21 march 2012 32

Douglas-Gunn Algorithm

bull Step I

bull Step II

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

12

1

22

gc

tTrTT

rTT

p

nyy

nnxxnn

22

122

1

)(2

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

2

121

2)(

2

21 march 2012 33

Illustration for ADI

X-direction implicit Y-direction implicit

i = 1 2 hellip m

1

njiT

Step I Step II

j = 1

n

2

hellip

1 2 hellip m

11

njiT

11

njiT

j = 1

n

2hellip

21

njiT

21

1

njiT

21

1

njiT

21 march 2012 34

Analysis of ADI Method

mm

Tridiagonal Matrix

Time complexity O(N)

2xnxm = 2nm =2N

2 steps n matrices tridaigonal matrix

X-direction implicit

21

njiT

i = 1 2 hellip mj = 1

n

2

hellip

21

1

njiT

21

1

njiT

ADVANTAGE over conditionally stable

Reduce simulation time

Good accuracy

Size of geometrical feature may be typical of wavelengh

Good geometrical flexibility to allow with corner or high curvature than fdtd

Advantages of the FDTD method over other methodsAdvantages of the FDTD method over other methods

bull It is conceptually simpleIt is conceptually simple

bull The algorithm does not require the formulation of integral equation and relatively complex The algorithm does not require the formulation of integral equation and relatively complex

scatters can be treated without inversion of large matrices scatters can be treated without inversion of large matrices

bull It is simple to implement for complicated inhomogeneous conducting or dielectric structures It is simple to implement for complicated inhomogeneous conducting or dielectric structures

because constitutive parameters can be assigned to each lattice pointbecause constitutive parameters can be assigned to each lattice point

bull Its computer memory requirement is not prohibitive for many complex structures of interest Its computer memory requirement is not prohibitive for many complex structures of interest

bull The algorithm make use of the memory in a simple sequential orderThe algorithm make use of the memory in a simple sequential order

bull It is much easier to obtain frequency domain data from time domain results than the converse It is much easier to obtain frequency domain data from time domain results than the converse

Thus it is more convenient to obtain frequency domain results via time domain when many Thus it is more convenient to obtain frequency domain results via time domain when many

frequencies are involvedfrequencies are involved

Include examples of unconditionally stable fdtd and prove the advantages

Disadvantages Disadvantages

bull Its implementation necessitates modeling object as well as its surroundings Thus the Its implementation necessitates modeling object as well as its surroundings Thus the

required program execution time may be excessiverequired program execution time may be excessive

bull Its accuracy is at least one order of magnitude worse than that of the method of Its accuracy is at least one order of magnitude worse than that of the method of

moments for example moments for example

bull Since the computational meshes are rectangular in shape they do not conform the Since the computational meshes are rectangular in shape they do not conform the

scatterers with curved surfaces as is the case of the cylindrical or spherical boundary scatterers with curved surfaces as is the case of the cylindrical or spherical boundary

Its computer memory requirement is not prohibitive for many complex structures of Its computer memory requirement is not prohibitive for many complex structures of

interest interest

bull As in all finite difference algorithms the field quantities are only known at grid nodes As in all finite difference algorithms the field quantities are only known at grid nodes

Applications of FDTD method

1048708Electromagnetic scattering amp antenna design1048708EMCEMI design1048708Simulation of wave propagation problem1048708Solving partial defferential equation 1048708Waveguide analysis

Stripline amp microstrip line analysishellipetc hellip

Referenceshelliphelliphellip1-Electromagnetic simulation using FDTD method

ldquoDennis M Sillevanrdquo IEEE press series on rf amp mivcrowave technology ldquoRoger DPollard amp Richard

Bootanrdquoseries

2Understanding the FDTD method ldquoJohn BSchneiderrdquo may 8 2011

3 3 D -ADI -Mehod-unconditionaly stable time domain algorithom for solving maxwell equations

ldquoIEEE Transaction on microwave theory amp techniques vol-48No10 october 2010 ldquoTakefumi

IEEE

4High order split step unconditionally stable FDTD method ampnumerical analysis ldquoIEEE

Transactions on antinna amp propagationvol 59 no9sept2011rdquo Yong-Dan Kong amp Qing xin chu

senior memberIEEE

5Genaral Finite DifferenceSchemes for Heat equations ldquoIndian J Pure Applemath

10(2)209-222Febuary 1979rdquoby PS Jain ampDN Holla department of methemetics

IITB 400076

  • Unconditionally Stable FDTD Methods
  • Outlineshelliphelliphelliphellip
  • Outlineshellip
  • Slide 4
  • Computational Electromagnetics
  • Slide 6
  • FDTD Overview ndash Updating Equations
  • Slide 8
  • Slide 9
  • Finite Difference Time Domain Method
  • The Yee Discretization (1966)
  • Slide 12
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Slide 17
  • Slide 18
  • Slide 19
  • Space Domain Discretization
  • Finite-Difference Formulation of the Heat Conduction on a Chip
  • Time domain discretization
  • Slide 23
  • Slide 24
  • Simple Explicit Method
  • Simple Implicit Method
  • Crank-Nicolson Method
  • Analysis of Crank-Nicolson Method
  • Alternating Direction Implicit Method
  • Slide 30
  • Peaceman-Rachford Algorithm
  • Douglas-Gunn Algorithm
  • Illustration for ADI
  • Analysis of ADI Method
  • Slide 35
  • Slide 36
  • Slide 37
  • Slide 38
  • Referenceshelliphelliphellip
  • Slide 40
Page 2: Unconditionally stable fdtd methods

Outlineshelliphelliphelliphellip

1Some computational methods for maxwells equations

2First fdtd algorithm YEE 1966 discritizations

3Some problems and condition in conventional fdtd

4Unconditionally stable fdtd use to solve heat equation

21 march 2012 2

Outlineshellip

5Imclicit Cranknicolson ADI Methods

6-Von neumann stability 7Advantages

8Disadvantages

9Conclusion

10References

21 march 2012 3

Maxwell equations

0

0

1

1

H

E

H E

E H

t

t typermittivi electric

typermeabili magnetic

))()()((H zyxtHzyxtHzyxtH zyx

))()()((E zyxtEzyxtEzyxtE zyx

field magnetic

field electric

Computational Electromagnetics

Finite-differencetime-domain(FDTD)

Finite-differencefrequency-domain(FDFD)

Method of Moments(MoM)

Fast multipole method (FMM)

Finite element method (FEM)

Transmission line matrix (TLM)

Frequency

Computational Electromagnetics

Finite-differencetime-domain(FDTD)

Finite-differencefrequency-domain(FDFD)

Method of Moments(MoM)

Fast multipole method (FMM)

Finite element method (FEM)

Transmission line matrix (TLM)

FDTD Overview ndash Updating Equations

760

Three scalar equations can be obtained from one vector curl equation

EH

t

yx zx

y x zy

y xzz

HE H

t y z

E H H

t z xH HE

t x y

HE

t

yx zx

y xzy

yxzz

EH E

t z y

H EE

t x zEEH

t y x

Finite DifferenceFinite Difference

Taylorrsquos seriesTaylorrsquos series

II Finite DifferenceII Finite Difference

Taylorrsquos Taylorrsquos seriesseries

ErorEror

Finite Difference Time Domain Method

bull Divide the interval x into sub-intervals each of width h

bull Divide the interval t into sub-intervals each of width k

bull A grid of points is used forthe finite difference solution

bull Tij represents T(xi tj)bull Replace the derivates by

finite-difference formulas10

t

x

(i j) (i+1 j)

(i j+1)

Ey

Ex

Hz

The Yee Discretization (1966)

Staggered grid in space mdash every field component is stored on a different grid

(i j k) (i+1 j k)

(i j k+1)

(i+1 j+1 k)

(i+1 j+1 k+1)

Ez

Ex

Ey

HyHx

Hz

(i j) (i+1 j)

(i j+1)

Ey

Ex

Hz

The Yee Discretization (1966)

all derivatives become center differenceshellip

H

t

1

E

H zt i

1

2 j

1

2

1

Eyx

Exy

1

Ey (i 1 j 1

2) Ey (i j

1

2)

xEx (i

1

2 j 1) Ex (i

1

2 j)

y

+ O(∆x2) + O(∆y2)

FDTD Overview ndash Updating Equations

yx zx

HE H

t y z

1

05 0505 05

( ) ( )( )

( ) ( 1)( ) ( 1 )

n nx x

x

n nn ny yz z

E i j k E i j ki j k

t

H i j k H i j kH i j k H i j k

y z

FDTD Overview ndash Updating Equations

yx zx

EH E

t z y

05 05

05

( ) ( )( )

( 1) ( ) ( 1 ) ( )

n nx x

x

n n n ny y z z

H i j k H i j ki j k

t

E i j k E i j k E i j k E i j k

z y

FDTD Overview ndash Updating Equations

Express the future components in terms of the past components

05 05

1

05 05

( ) ( 1 )

( ) ( )( ) ( ) ( 1)

n nz z

n nx x n n

x y y

H i j k H i j k

ytE i j k E i j k

i j k H i j k H i j k

z

05

05 05

( 1) ( )

( ) ( )( ) ( 1 ) ( )

n ny y

n nx x n n

x z z

E i j k E i j k

t zH i j k H i j ki j k E i j k E i j k

y

Fundamentals of the FDTD methodFundamentals of the FDTD method

Accuracy and stabilityAccuracy and stability

AccuracyAccuracy 10

StabilityStability

222

max

z

1

y

1

x

1c

1tt

3c

tt max

2D2D

22 y

1

x

1c

1t

1D1Dc

t

Physically this condition means that the time Physically this condition means that the time step should be smaller than the time for the step should be smaller than the time for the wave to propagate from one cell to the wave to propagate from one cell to the neighbor oneneighbor one

Slides from 5-15 may be skipped because it was taught in the class

III Fundamentals of the FDTD methodIII Fundamentals of the FDTD method

Dispersion relationDispersion relation

ckkkk

22

z2

y2

x2

In free space (ideal)

In FDTD computation (numerical)

22

z

2

y

2

x

2

tsin

tc

1

2

zk~

sinz

1

2

yk~

siny

1

2

xk~

sinx

1

utcsinArctk

~2

k~vpnum

2

z

2

y

2

x

2

zk~

sinz

1

2

yk~

siny

1

2

xk~

sinx

1u

The numerical medium is dispersive the propagation of the wave varies with frequency and angle

III Fundamentals of the FDTD methodIII Fundamentals of the FDTD method

Dispersion relationDispersion relation

Limitations of FDTD method

1-Grid spacing should be ~λ102-According to Courantrsquos stability condition time step Δt becomes small when FDTDgrid spacing becomes small3-In 3-D simulation simulation time scales like N^4 and required memory size scales like N^34-Application is restricted to relatively small size

21 march 2012 20

Space Domain Discretization

bull Heat Conduction Equation

bull Central-Finite-Difference Approximation

pc

tyxg

y

tyxT

x

tyxT

t

tyxT

)()()()(

2

2

2

2

2

2

2

11

22

112

2

2

)(2

x

T

x

TTT

xOx

TTT

x

T

nx

nji

nji

nji

nji

nji

njin

ji

cc

21 march 2012 21

X

n

jiTn

j1-iT n

j1iT

n

1jiT

n

1jiT

(00) (i0)hellip

(0j)

x

y

(I0)

(0J)

hellip

hellip

hellip

Finite-Difference Formulation of the Heat Conduction on a Chip

bull Space Domainbull Time Domain

21 march 2012 22

Time domain discretization

bull Heat Conduction Equation

ndash Simple Explicit Method ndash Simple Implicit Methodndash Crank-Nicolson Method

gcy

T

x

T

t

TT

p

yxnn

1

)()( 2

2

2

21

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

23

Can we check if a numerical scheme

is stable without computation Von Neumann stability

analysisbull Analyze if (or for which conditions) a

numerical scheme is stable or unstablebull Makes a local analysis coefficients of PDE are

assumed to vary slowly (our example constant)

bull How will unavoidable errors (say rounding errors)evolve in time

John von Neumann1903-1957

24

Von Neumann stability analysis

A numerical scheme is unstable if

Ansatz Wave number k and amplification factor

21 march 2012 25

bull Accuracybull Stability Constraint

bull No matrix inversion but time steps are limited by space discretization

gcy

T

x

T

t

TT

p

ny

nx

nn

1

)()( 2

2

2

21

] [ 22 yxt

2

11122

yxt

Simple Explicit Method

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

21 march 2012 26

Accuracy

Unconditionally Stable

No limits on time step but involves with large scale matrix inversion

gcy

T

x

T

t

TT

p

ny

nx

nn

1

)()( 2

12

2

121

] [ 22 yxt

Simple Implicit Method

11

1

11

12

11

1

11

12

2

2nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

21 march 2012 27

Accuracy

Unconditionally stable

No limits on time step but involves with large scale matrix inversion

gcy

TT

x

TT

t

TT

p

ny

ny

nx

nx

nn

1

)(2)(2 2

212

2

2121

] [ 222 yxt

Crank-Nicolson Method

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

11

1

11

12

11

1

11

12

2

2nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

21 march 2012 28

n

m1616

ex m=4n=4

Total node number N = mn

Matrix size = NxN

Analysis of Crank-Nicolson Method

21 march 2012 29

Alternating Direction Implicit Method

Solves higher dimension problem by successive Lower dimension methods

Accuracy

Unconditionally stable

No limits on time step and no large scale matrix inversion

] [ 222 yxt

21 march2012 30

Step I x-direction implicit y-direction explicitStep II x-direction explicit y-direction implicit

n

Alternating Direction Implicit Method

bull Peaceman-Rachford Algorithmbull Douglas-Gunn Algorithm

21 march 2012 31

bull Step I

bull Step II

gc

tT

rrT

rr

p

ny

yx

xny

yx

x

)2

1)(2

1()2

1)(2

1( 22122

gc

tT

rT

r

p

ny

yn

xx

2)

21()

21( 22

12

gc

tT

rT

r

p

n

xxn

yy

2)

21()

21( 2

1212

Peaceman-Rachford Algorithm

21 march 2012 32

Douglas-Gunn Algorithm

bull Step I

bull Step II

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

12

1

22

gc

tTrTT

rTT

p

nyy

nnxxnn

22

122

1

)(2

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

2

121

2)(

2

21 march 2012 33

Illustration for ADI

X-direction implicit Y-direction implicit

i = 1 2 hellip m

1

njiT

Step I Step II

j = 1

n

2

hellip

1 2 hellip m

11

njiT

11

njiT

j = 1

n

2hellip

21

njiT

21

1

njiT

21

1

njiT

21 march 2012 34

Analysis of ADI Method

mm

Tridiagonal Matrix

Time complexity O(N)

2xnxm = 2nm =2N

2 steps n matrices tridaigonal matrix

X-direction implicit

21

njiT

i = 1 2 hellip mj = 1

n

2

hellip

21

1

njiT

21

1

njiT

ADVANTAGE over conditionally stable

Reduce simulation time

Good accuracy

Size of geometrical feature may be typical of wavelengh

Good geometrical flexibility to allow with corner or high curvature than fdtd

Advantages of the FDTD method over other methodsAdvantages of the FDTD method over other methods

bull It is conceptually simpleIt is conceptually simple

bull The algorithm does not require the formulation of integral equation and relatively complex The algorithm does not require the formulation of integral equation and relatively complex

scatters can be treated without inversion of large matrices scatters can be treated without inversion of large matrices

bull It is simple to implement for complicated inhomogeneous conducting or dielectric structures It is simple to implement for complicated inhomogeneous conducting or dielectric structures

because constitutive parameters can be assigned to each lattice pointbecause constitutive parameters can be assigned to each lattice point

bull Its computer memory requirement is not prohibitive for many complex structures of interest Its computer memory requirement is not prohibitive for many complex structures of interest

bull The algorithm make use of the memory in a simple sequential orderThe algorithm make use of the memory in a simple sequential order

bull It is much easier to obtain frequency domain data from time domain results than the converse It is much easier to obtain frequency domain data from time domain results than the converse

Thus it is more convenient to obtain frequency domain results via time domain when many Thus it is more convenient to obtain frequency domain results via time domain when many

frequencies are involvedfrequencies are involved

Include examples of unconditionally stable fdtd and prove the advantages

Disadvantages Disadvantages

bull Its implementation necessitates modeling object as well as its surroundings Thus the Its implementation necessitates modeling object as well as its surroundings Thus the

required program execution time may be excessiverequired program execution time may be excessive

bull Its accuracy is at least one order of magnitude worse than that of the method of Its accuracy is at least one order of magnitude worse than that of the method of

moments for example moments for example

bull Since the computational meshes are rectangular in shape they do not conform the Since the computational meshes are rectangular in shape they do not conform the

scatterers with curved surfaces as is the case of the cylindrical or spherical boundary scatterers with curved surfaces as is the case of the cylindrical or spherical boundary

Its computer memory requirement is not prohibitive for many complex structures of Its computer memory requirement is not prohibitive for many complex structures of

interest interest

bull As in all finite difference algorithms the field quantities are only known at grid nodes As in all finite difference algorithms the field quantities are only known at grid nodes

Applications of FDTD method

1048708Electromagnetic scattering amp antenna design1048708EMCEMI design1048708Simulation of wave propagation problem1048708Solving partial defferential equation 1048708Waveguide analysis

Stripline amp microstrip line analysishellipetc hellip

Referenceshelliphelliphellip1-Electromagnetic simulation using FDTD method

ldquoDennis M Sillevanrdquo IEEE press series on rf amp mivcrowave technology ldquoRoger DPollard amp Richard

Bootanrdquoseries

2Understanding the FDTD method ldquoJohn BSchneiderrdquo may 8 2011

3 3 D -ADI -Mehod-unconditionaly stable time domain algorithom for solving maxwell equations

ldquoIEEE Transaction on microwave theory amp techniques vol-48No10 october 2010 ldquoTakefumi

IEEE

4High order split step unconditionally stable FDTD method ampnumerical analysis ldquoIEEE

Transactions on antinna amp propagationvol 59 no9sept2011rdquo Yong-Dan Kong amp Qing xin chu

senior memberIEEE

5Genaral Finite DifferenceSchemes for Heat equations ldquoIndian J Pure Applemath

10(2)209-222Febuary 1979rdquoby PS Jain ampDN Holla department of methemetics

IITB 400076

  • Unconditionally Stable FDTD Methods
  • Outlineshelliphelliphelliphellip
  • Outlineshellip
  • Slide 4
  • Computational Electromagnetics
  • Slide 6
  • FDTD Overview ndash Updating Equations
  • Slide 8
  • Slide 9
  • Finite Difference Time Domain Method
  • The Yee Discretization (1966)
  • Slide 12
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Slide 17
  • Slide 18
  • Slide 19
  • Space Domain Discretization
  • Finite-Difference Formulation of the Heat Conduction on a Chip
  • Time domain discretization
  • Slide 23
  • Slide 24
  • Simple Explicit Method
  • Simple Implicit Method
  • Crank-Nicolson Method
  • Analysis of Crank-Nicolson Method
  • Alternating Direction Implicit Method
  • Slide 30
  • Peaceman-Rachford Algorithm
  • Douglas-Gunn Algorithm
  • Illustration for ADI
  • Analysis of ADI Method
  • Slide 35
  • Slide 36
  • Slide 37
  • Slide 38
  • Referenceshelliphelliphellip
  • Slide 40
Page 3: Unconditionally stable fdtd methods

Outlineshellip

5Imclicit Cranknicolson ADI Methods

6-Von neumann stability 7Advantages

8Disadvantages

9Conclusion

10References

21 march 2012 3

Maxwell equations

0

0

1

1

H

E

H E

E H

t

t typermittivi electric

typermeabili magnetic

))()()((H zyxtHzyxtHzyxtH zyx

))()()((E zyxtEzyxtEzyxtE zyx

field magnetic

field electric

Computational Electromagnetics

Finite-differencetime-domain(FDTD)

Finite-differencefrequency-domain(FDFD)

Method of Moments(MoM)

Fast multipole method (FMM)

Finite element method (FEM)

Transmission line matrix (TLM)

Frequency

Computational Electromagnetics

Finite-differencetime-domain(FDTD)

Finite-differencefrequency-domain(FDFD)

Method of Moments(MoM)

Fast multipole method (FMM)

Finite element method (FEM)

Transmission line matrix (TLM)

FDTD Overview ndash Updating Equations

760

Three scalar equations can be obtained from one vector curl equation

EH

t

yx zx

y x zy

y xzz

HE H

t y z

E H H

t z xH HE

t x y

HE

t

yx zx

y xzy

yxzz

EH E

t z y

H EE

t x zEEH

t y x

Finite DifferenceFinite Difference

Taylorrsquos seriesTaylorrsquos series

II Finite DifferenceII Finite Difference

Taylorrsquos Taylorrsquos seriesseries

ErorEror

Finite Difference Time Domain Method

bull Divide the interval x into sub-intervals each of width h

bull Divide the interval t into sub-intervals each of width k

bull A grid of points is used forthe finite difference solution

bull Tij represents T(xi tj)bull Replace the derivates by

finite-difference formulas10

t

x

(i j) (i+1 j)

(i j+1)

Ey

Ex

Hz

The Yee Discretization (1966)

Staggered grid in space mdash every field component is stored on a different grid

(i j k) (i+1 j k)

(i j k+1)

(i+1 j+1 k)

(i+1 j+1 k+1)

Ez

Ex

Ey

HyHx

Hz

(i j) (i+1 j)

(i j+1)

Ey

Ex

Hz

The Yee Discretization (1966)

all derivatives become center differenceshellip

H

t

1

E

H zt i

1

2 j

1

2

1

Eyx

Exy

1

Ey (i 1 j 1

2) Ey (i j

1

2)

xEx (i

1

2 j 1) Ex (i

1

2 j)

y

+ O(∆x2) + O(∆y2)

FDTD Overview ndash Updating Equations

yx zx

HE H

t y z

1

05 0505 05

( ) ( )( )

( ) ( 1)( ) ( 1 )

n nx x

x

n nn ny yz z

E i j k E i j ki j k

t

H i j k H i j kH i j k H i j k

y z

FDTD Overview ndash Updating Equations

yx zx

EH E

t z y

05 05

05

( ) ( )( )

( 1) ( ) ( 1 ) ( )

n nx x

x

n n n ny y z z

H i j k H i j ki j k

t

E i j k E i j k E i j k E i j k

z y

FDTD Overview ndash Updating Equations

Express the future components in terms of the past components

05 05

1

05 05

( ) ( 1 )

( ) ( )( ) ( ) ( 1)

n nz z

n nx x n n

x y y

H i j k H i j k

ytE i j k E i j k

i j k H i j k H i j k

z

05

05 05

( 1) ( )

( ) ( )( ) ( 1 ) ( )

n ny y

n nx x n n

x z z

E i j k E i j k

t zH i j k H i j ki j k E i j k E i j k

y

Fundamentals of the FDTD methodFundamentals of the FDTD method

Accuracy and stabilityAccuracy and stability

AccuracyAccuracy 10

StabilityStability

222

max

z

1

y

1

x

1c

1tt

3c

tt max

2D2D

22 y

1

x

1c

1t

1D1Dc

t

Physically this condition means that the time Physically this condition means that the time step should be smaller than the time for the step should be smaller than the time for the wave to propagate from one cell to the wave to propagate from one cell to the neighbor oneneighbor one

Slides from 5-15 may be skipped because it was taught in the class

III Fundamentals of the FDTD methodIII Fundamentals of the FDTD method

Dispersion relationDispersion relation

ckkkk

22

z2

y2

x2

In free space (ideal)

In FDTD computation (numerical)

22

z

2

y

2

x

2

tsin

tc

1

2

zk~

sinz

1

2

yk~

siny

1

2

xk~

sinx

1

utcsinArctk

~2

k~vpnum

2

z

2

y

2

x

2

zk~

sinz

1

2

yk~

siny

1

2

xk~

sinx

1u

The numerical medium is dispersive the propagation of the wave varies with frequency and angle

III Fundamentals of the FDTD methodIII Fundamentals of the FDTD method

Dispersion relationDispersion relation

Limitations of FDTD method

1-Grid spacing should be ~λ102-According to Courantrsquos stability condition time step Δt becomes small when FDTDgrid spacing becomes small3-In 3-D simulation simulation time scales like N^4 and required memory size scales like N^34-Application is restricted to relatively small size

21 march 2012 20

Space Domain Discretization

bull Heat Conduction Equation

bull Central-Finite-Difference Approximation

pc

tyxg

y

tyxT

x

tyxT

t

tyxT

)()()()(

2

2

2

2

2

2

2

11

22

112

2

2

)(2

x

T

x

TTT

xOx

TTT

x

T

nx

nji

nji

nji

nji

nji

njin

ji

cc

21 march 2012 21

X

n

jiTn

j1-iT n

j1iT

n

1jiT

n

1jiT

(00) (i0)hellip

(0j)

x

y

(I0)

(0J)

hellip

hellip

hellip

Finite-Difference Formulation of the Heat Conduction on a Chip

bull Space Domainbull Time Domain

21 march 2012 22

Time domain discretization

bull Heat Conduction Equation

ndash Simple Explicit Method ndash Simple Implicit Methodndash Crank-Nicolson Method

gcy

T

x

T

t

TT

p

yxnn

1

)()( 2

2

2

21

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

23

Can we check if a numerical scheme

is stable without computation Von Neumann stability

analysisbull Analyze if (or for which conditions) a

numerical scheme is stable or unstablebull Makes a local analysis coefficients of PDE are

assumed to vary slowly (our example constant)

bull How will unavoidable errors (say rounding errors)evolve in time

John von Neumann1903-1957

24

Von Neumann stability analysis

A numerical scheme is unstable if

Ansatz Wave number k and amplification factor

21 march 2012 25

bull Accuracybull Stability Constraint

bull No matrix inversion but time steps are limited by space discretization

gcy

T

x

T

t

TT

p

ny

nx

nn

1

)()( 2

2

2

21

] [ 22 yxt

2

11122

yxt

Simple Explicit Method

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

21 march 2012 26

Accuracy

Unconditionally Stable

No limits on time step but involves with large scale matrix inversion

gcy

T

x

T

t

TT

p

ny

nx

nn

1

)()( 2

12

2

121

] [ 22 yxt

Simple Implicit Method

11

1

11

12

11

1

11

12

2

2nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

21 march 2012 27

Accuracy

Unconditionally stable

No limits on time step but involves with large scale matrix inversion

gcy

TT

x

TT

t

TT

p

ny

ny

nx

nx

nn

1

)(2)(2 2

212

2

2121

] [ 222 yxt

Crank-Nicolson Method

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

11

1

11

12

11

1

11

12

2

2nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

21 march 2012 28

n

m1616

ex m=4n=4

Total node number N = mn

Matrix size = NxN

Analysis of Crank-Nicolson Method

21 march 2012 29

Alternating Direction Implicit Method

Solves higher dimension problem by successive Lower dimension methods

Accuracy

Unconditionally stable

No limits on time step and no large scale matrix inversion

] [ 222 yxt

21 march2012 30

Step I x-direction implicit y-direction explicitStep II x-direction explicit y-direction implicit

n

Alternating Direction Implicit Method

bull Peaceman-Rachford Algorithmbull Douglas-Gunn Algorithm

21 march 2012 31

bull Step I

bull Step II

gc

tT

rrT

rr

p

ny

yx

xny

yx

x

)2

1)(2

1()2

1)(2

1( 22122

gc

tT

rT

r

p

ny

yn

xx

2)

21()

21( 22

12

gc

tT

rT

r

p

n

xxn

yy

2)

21()

21( 2

1212

Peaceman-Rachford Algorithm

21 march 2012 32

Douglas-Gunn Algorithm

bull Step I

bull Step II

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

12

1

22

gc

tTrTT

rTT

p

nyy

nnxxnn

22

122

1

)(2

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

2

121

2)(

2

21 march 2012 33

Illustration for ADI

X-direction implicit Y-direction implicit

i = 1 2 hellip m

1

njiT

Step I Step II

j = 1

n

2

hellip

1 2 hellip m

11

njiT

11

njiT

j = 1

n

2hellip

21

njiT

21

1

njiT

21

1

njiT

21 march 2012 34

Analysis of ADI Method

mm

Tridiagonal Matrix

Time complexity O(N)

2xnxm = 2nm =2N

2 steps n matrices tridaigonal matrix

X-direction implicit

21

njiT

i = 1 2 hellip mj = 1

n

2

hellip

21

1

njiT

21

1

njiT

ADVANTAGE over conditionally stable

Reduce simulation time

Good accuracy

Size of geometrical feature may be typical of wavelengh

Good geometrical flexibility to allow with corner or high curvature than fdtd

Advantages of the FDTD method over other methodsAdvantages of the FDTD method over other methods

bull It is conceptually simpleIt is conceptually simple

bull The algorithm does not require the formulation of integral equation and relatively complex The algorithm does not require the formulation of integral equation and relatively complex

scatters can be treated without inversion of large matrices scatters can be treated without inversion of large matrices

bull It is simple to implement for complicated inhomogeneous conducting or dielectric structures It is simple to implement for complicated inhomogeneous conducting or dielectric structures

because constitutive parameters can be assigned to each lattice pointbecause constitutive parameters can be assigned to each lattice point

bull Its computer memory requirement is not prohibitive for many complex structures of interest Its computer memory requirement is not prohibitive for many complex structures of interest

bull The algorithm make use of the memory in a simple sequential orderThe algorithm make use of the memory in a simple sequential order

bull It is much easier to obtain frequency domain data from time domain results than the converse It is much easier to obtain frequency domain data from time domain results than the converse

Thus it is more convenient to obtain frequency domain results via time domain when many Thus it is more convenient to obtain frequency domain results via time domain when many

frequencies are involvedfrequencies are involved

Include examples of unconditionally stable fdtd and prove the advantages

Disadvantages Disadvantages

bull Its implementation necessitates modeling object as well as its surroundings Thus the Its implementation necessitates modeling object as well as its surroundings Thus the

required program execution time may be excessiverequired program execution time may be excessive

bull Its accuracy is at least one order of magnitude worse than that of the method of Its accuracy is at least one order of magnitude worse than that of the method of

moments for example moments for example

bull Since the computational meshes are rectangular in shape they do not conform the Since the computational meshes are rectangular in shape they do not conform the

scatterers with curved surfaces as is the case of the cylindrical or spherical boundary scatterers with curved surfaces as is the case of the cylindrical or spherical boundary

Its computer memory requirement is not prohibitive for many complex structures of Its computer memory requirement is not prohibitive for many complex structures of

interest interest

bull As in all finite difference algorithms the field quantities are only known at grid nodes As in all finite difference algorithms the field quantities are only known at grid nodes

Applications of FDTD method

1048708Electromagnetic scattering amp antenna design1048708EMCEMI design1048708Simulation of wave propagation problem1048708Solving partial defferential equation 1048708Waveguide analysis

Stripline amp microstrip line analysishellipetc hellip

Referenceshelliphelliphellip1-Electromagnetic simulation using FDTD method

ldquoDennis M Sillevanrdquo IEEE press series on rf amp mivcrowave technology ldquoRoger DPollard amp Richard

Bootanrdquoseries

2Understanding the FDTD method ldquoJohn BSchneiderrdquo may 8 2011

3 3 D -ADI -Mehod-unconditionaly stable time domain algorithom for solving maxwell equations

ldquoIEEE Transaction on microwave theory amp techniques vol-48No10 october 2010 ldquoTakefumi

IEEE

4High order split step unconditionally stable FDTD method ampnumerical analysis ldquoIEEE

Transactions on antinna amp propagationvol 59 no9sept2011rdquo Yong-Dan Kong amp Qing xin chu

senior memberIEEE

5Genaral Finite DifferenceSchemes for Heat equations ldquoIndian J Pure Applemath

10(2)209-222Febuary 1979rdquoby PS Jain ampDN Holla department of methemetics

IITB 400076

  • Unconditionally Stable FDTD Methods
  • Outlineshelliphelliphelliphellip
  • Outlineshellip
  • Slide 4
  • Computational Electromagnetics
  • Slide 6
  • FDTD Overview ndash Updating Equations
  • Slide 8
  • Slide 9
  • Finite Difference Time Domain Method
  • The Yee Discretization (1966)
  • Slide 12
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Slide 17
  • Slide 18
  • Slide 19
  • Space Domain Discretization
  • Finite-Difference Formulation of the Heat Conduction on a Chip
  • Time domain discretization
  • Slide 23
  • Slide 24
  • Simple Explicit Method
  • Simple Implicit Method
  • Crank-Nicolson Method
  • Analysis of Crank-Nicolson Method
  • Alternating Direction Implicit Method
  • Slide 30
  • Peaceman-Rachford Algorithm
  • Douglas-Gunn Algorithm
  • Illustration for ADI
  • Analysis of ADI Method
  • Slide 35
  • Slide 36
  • Slide 37
  • Slide 38
  • Referenceshelliphelliphellip
  • Slide 40
Page 4: Unconditionally stable fdtd methods

Maxwell equations

0

0

1

1

H

E

H E

E H

t

t typermittivi electric

typermeabili magnetic

))()()((H zyxtHzyxtHzyxtH zyx

))()()((E zyxtEzyxtEzyxtE zyx

field magnetic

field electric

Computational Electromagnetics

Finite-differencetime-domain(FDTD)

Finite-differencefrequency-domain(FDFD)

Method of Moments(MoM)

Fast multipole method (FMM)

Finite element method (FEM)

Transmission line matrix (TLM)

Frequency

Computational Electromagnetics

Finite-differencetime-domain(FDTD)

Finite-differencefrequency-domain(FDFD)

Method of Moments(MoM)

Fast multipole method (FMM)

Finite element method (FEM)

Transmission line matrix (TLM)

FDTD Overview ndash Updating Equations

760

Three scalar equations can be obtained from one vector curl equation

EH

t

yx zx

y x zy

y xzz

HE H

t y z

E H H

t z xH HE

t x y

HE

t

yx zx

y xzy

yxzz

EH E

t z y

H EE

t x zEEH

t y x

Finite DifferenceFinite Difference

Taylorrsquos seriesTaylorrsquos series

II Finite DifferenceII Finite Difference

Taylorrsquos Taylorrsquos seriesseries

ErorEror

Finite Difference Time Domain Method

bull Divide the interval x into sub-intervals each of width h

bull Divide the interval t into sub-intervals each of width k

bull A grid of points is used forthe finite difference solution

bull Tij represents T(xi tj)bull Replace the derivates by

finite-difference formulas10

t

x

(i j) (i+1 j)

(i j+1)

Ey

Ex

Hz

The Yee Discretization (1966)

Staggered grid in space mdash every field component is stored on a different grid

(i j k) (i+1 j k)

(i j k+1)

(i+1 j+1 k)

(i+1 j+1 k+1)

Ez

Ex

Ey

HyHx

Hz

(i j) (i+1 j)

(i j+1)

Ey

Ex

Hz

The Yee Discretization (1966)

all derivatives become center differenceshellip

H

t

1

E

H zt i

1

2 j

1

2

1

Eyx

Exy

1

Ey (i 1 j 1

2) Ey (i j

1

2)

xEx (i

1

2 j 1) Ex (i

1

2 j)

y

+ O(∆x2) + O(∆y2)

FDTD Overview ndash Updating Equations

yx zx

HE H

t y z

1

05 0505 05

( ) ( )( )

( ) ( 1)( ) ( 1 )

n nx x

x

n nn ny yz z

E i j k E i j ki j k

t

H i j k H i j kH i j k H i j k

y z

FDTD Overview ndash Updating Equations

yx zx

EH E

t z y

05 05

05

( ) ( )( )

( 1) ( ) ( 1 ) ( )

n nx x

x

n n n ny y z z

H i j k H i j ki j k

t

E i j k E i j k E i j k E i j k

z y

FDTD Overview ndash Updating Equations

Express the future components in terms of the past components

05 05

1

05 05

( ) ( 1 )

( ) ( )( ) ( ) ( 1)

n nz z

n nx x n n

x y y

H i j k H i j k

ytE i j k E i j k

i j k H i j k H i j k

z

05

05 05

( 1) ( )

( ) ( )( ) ( 1 ) ( )

n ny y

n nx x n n

x z z

E i j k E i j k

t zH i j k H i j ki j k E i j k E i j k

y

Fundamentals of the FDTD methodFundamentals of the FDTD method

Accuracy and stabilityAccuracy and stability

AccuracyAccuracy 10

StabilityStability

222

max

z

1

y

1

x

1c

1tt

3c

tt max

2D2D

22 y

1

x

1c

1t

1D1Dc

t

Physically this condition means that the time Physically this condition means that the time step should be smaller than the time for the step should be smaller than the time for the wave to propagate from one cell to the wave to propagate from one cell to the neighbor oneneighbor one

Slides from 5-15 may be skipped because it was taught in the class

III Fundamentals of the FDTD methodIII Fundamentals of the FDTD method

Dispersion relationDispersion relation

ckkkk

22

z2

y2

x2

In free space (ideal)

In FDTD computation (numerical)

22

z

2

y

2

x

2

tsin

tc

1

2

zk~

sinz

1

2

yk~

siny

1

2

xk~

sinx

1

utcsinArctk

~2

k~vpnum

2

z

2

y

2

x

2

zk~

sinz

1

2

yk~

siny

1

2

xk~

sinx

1u

The numerical medium is dispersive the propagation of the wave varies with frequency and angle

III Fundamentals of the FDTD methodIII Fundamentals of the FDTD method

Dispersion relationDispersion relation

Limitations of FDTD method

1-Grid spacing should be ~λ102-According to Courantrsquos stability condition time step Δt becomes small when FDTDgrid spacing becomes small3-In 3-D simulation simulation time scales like N^4 and required memory size scales like N^34-Application is restricted to relatively small size

21 march 2012 20

Space Domain Discretization

bull Heat Conduction Equation

bull Central-Finite-Difference Approximation

pc

tyxg

y

tyxT

x

tyxT

t

tyxT

)()()()(

2

2

2

2

2

2

2

11

22

112

2

2

)(2

x

T

x

TTT

xOx

TTT

x

T

nx

nji

nji

nji

nji

nji

njin

ji

cc

21 march 2012 21

X

n

jiTn

j1-iT n

j1iT

n

1jiT

n

1jiT

(00) (i0)hellip

(0j)

x

y

(I0)

(0J)

hellip

hellip

hellip

Finite-Difference Formulation of the Heat Conduction on a Chip

bull Space Domainbull Time Domain

21 march 2012 22

Time domain discretization

bull Heat Conduction Equation

ndash Simple Explicit Method ndash Simple Implicit Methodndash Crank-Nicolson Method

gcy

T

x

T

t

TT

p

yxnn

1

)()( 2

2

2

21

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

23

Can we check if a numerical scheme

is stable without computation Von Neumann stability

analysisbull Analyze if (or for which conditions) a

numerical scheme is stable or unstablebull Makes a local analysis coefficients of PDE are

assumed to vary slowly (our example constant)

bull How will unavoidable errors (say rounding errors)evolve in time

John von Neumann1903-1957

24

Von Neumann stability analysis

A numerical scheme is unstable if

Ansatz Wave number k and amplification factor

21 march 2012 25

bull Accuracybull Stability Constraint

bull No matrix inversion but time steps are limited by space discretization

gcy

T

x

T

t

TT

p

ny

nx

nn

1

)()( 2

2

2

21

] [ 22 yxt

2

11122

yxt

Simple Explicit Method

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

21 march 2012 26

Accuracy

Unconditionally Stable

No limits on time step but involves with large scale matrix inversion

gcy

T

x

T

t

TT

p

ny

nx

nn

1

)()( 2

12

2

121

] [ 22 yxt

Simple Implicit Method

11

1

11

12

11

1

11

12

2

2nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

21 march 2012 27

Accuracy

Unconditionally stable

No limits on time step but involves with large scale matrix inversion

gcy

TT

x

TT

t

TT

p

ny

ny

nx

nx

nn

1

)(2)(2 2

212

2

2121

] [ 222 yxt

Crank-Nicolson Method

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

11

1

11

12

11

1

11

12

2

2nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

21 march 2012 28

n

m1616

ex m=4n=4

Total node number N = mn

Matrix size = NxN

Analysis of Crank-Nicolson Method

21 march 2012 29

Alternating Direction Implicit Method

Solves higher dimension problem by successive Lower dimension methods

Accuracy

Unconditionally stable

No limits on time step and no large scale matrix inversion

] [ 222 yxt

21 march2012 30

Step I x-direction implicit y-direction explicitStep II x-direction explicit y-direction implicit

n

Alternating Direction Implicit Method

bull Peaceman-Rachford Algorithmbull Douglas-Gunn Algorithm

21 march 2012 31

bull Step I

bull Step II

gc

tT

rrT

rr

p

ny

yx

xny

yx

x

)2

1)(2

1()2

1)(2

1( 22122

gc

tT

rT

r

p

ny

yn

xx

2)

21()

21( 22

12

gc

tT

rT

r

p

n

xxn

yy

2)

21()

21( 2

1212

Peaceman-Rachford Algorithm

21 march 2012 32

Douglas-Gunn Algorithm

bull Step I

bull Step II

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

12

1

22

gc

tTrTT

rTT

p

nyy

nnxxnn

22

122

1

)(2

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

2

121

2)(

2

21 march 2012 33

Illustration for ADI

X-direction implicit Y-direction implicit

i = 1 2 hellip m

1

njiT

Step I Step II

j = 1

n

2

hellip

1 2 hellip m

11

njiT

11

njiT

j = 1

n

2hellip

21

njiT

21

1

njiT

21

1

njiT

21 march 2012 34

Analysis of ADI Method

mm

Tridiagonal Matrix

Time complexity O(N)

2xnxm = 2nm =2N

2 steps n matrices tridaigonal matrix

X-direction implicit

21

njiT

i = 1 2 hellip mj = 1

n

2

hellip

21

1

njiT

21

1

njiT

ADVANTAGE over conditionally stable

Reduce simulation time

Good accuracy

Size of geometrical feature may be typical of wavelengh

Good geometrical flexibility to allow with corner or high curvature than fdtd

Advantages of the FDTD method over other methodsAdvantages of the FDTD method over other methods

bull It is conceptually simpleIt is conceptually simple

bull The algorithm does not require the formulation of integral equation and relatively complex The algorithm does not require the formulation of integral equation and relatively complex

scatters can be treated without inversion of large matrices scatters can be treated without inversion of large matrices

bull It is simple to implement for complicated inhomogeneous conducting or dielectric structures It is simple to implement for complicated inhomogeneous conducting or dielectric structures

because constitutive parameters can be assigned to each lattice pointbecause constitutive parameters can be assigned to each lattice point

bull Its computer memory requirement is not prohibitive for many complex structures of interest Its computer memory requirement is not prohibitive for many complex structures of interest

bull The algorithm make use of the memory in a simple sequential orderThe algorithm make use of the memory in a simple sequential order

bull It is much easier to obtain frequency domain data from time domain results than the converse It is much easier to obtain frequency domain data from time domain results than the converse

Thus it is more convenient to obtain frequency domain results via time domain when many Thus it is more convenient to obtain frequency domain results via time domain when many

frequencies are involvedfrequencies are involved

Include examples of unconditionally stable fdtd and prove the advantages

Disadvantages Disadvantages

bull Its implementation necessitates modeling object as well as its surroundings Thus the Its implementation necessitates modeling object as well as its surroundings Thus the

required program execution time may be excessiverequired program execution time may be excessive

bull Its accuracy is at least one order of magnitude worse than that of the method of Its accuracy is at least one order of magnitude worse than that of the method of

moments for example moments for example

bull Since the computational meshes are rectangular in shape they do not conform the Since the computational meshes are rectangular in shape they do not conform the

scatterers with curved surfaces as is the case of the cylindrical or spherical boundary scatterers with curved surfaces as is the case of the cylindrical or spherical boundary

Its computer memory requirement is not prohibitive for many complex structures of Its computer memory requirement is not prohibitive for many complex structures of

interest interest

bull As in all finite difference algorithms the field quantities are only known at grid nodes As in all finite difference algorithms the field quantities are only known at grid nodes

Applications of FDTD method

1048708Electromagnetic scattering amp antenna design1048708EMCEMI design1048708Simulation of wave propagation problem1048708Solving partial defferential equation 1048708Waveguide analysis

Stripline amp microstrip line analysishellipetc hellip

Referenceshelliphelliphellip1-Electromagnetic simulation using FDTD method

ldquoDennis M Sillevanrdquo IEEE press series on rf amp mivcrowave technology ldquoRoger DPollard amp Richard

Bootanrdquoseries

2Understanding the FDTD method ldquoJohn BSchneiderrdquo may 8 2011

3 3 D -ADI -Mehod-unconditionaly stable time domain algorithom for solving maxwell equations

ldquoIEEE Transaction on microwave theory amp techniques vol-48No10 october 2010 ldquoTakefumi

IEEE

4High order split step unconditionally stable FDTD method ampnumerical analysis ldquoIEEE

Transactions on antinna amp propagationvol 59 no9sept2011rdquo Yong-Dan Kong amp Qing xin chu

senior memberIEEE

5Genaral Finite DifferenceSchemes for Heat equations ldquoIndian J Pure Applemath

10(2)209-222Febuary 1979rdquoby PS Jain ampDN Holla department of methemetics

IITB 400076

  • Unconditionally Stable FDTD Methods
  • Outlineshelliphelliphelliphellip
  • Outlineshellip
  • Slide 4
  • Computational Electromagnetics
  • Slide 6
  • FDTD Overview ndash Updating Equations
  • Slide 8
  • Slide 9
  • Finite Difference Time Domain Method
  • The Yee Discretization (1966)
  • Slide 12
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Slide 17
  • Slide 18
  • Slide 19
  • Space Domain Discretization
  • Finite-Difference Formulation of the Heat Conduction on a Chip
  • Time domain discretization
  • Slide 23
  • Slide 24
  • Simple Explicit Method
  • Simple Implicit Method
  • Crank-Nicolson Method
  • Analysis of Crank-Nicolson Method
  • Alternating Direction Implicit Method
  • Slide 30
  • Peaceman-Rachford Algorithm
  • Douglas-Gunn Algorithm
  • Illustration for ADI
  • Analysis of ADI Method
  • Slide 35
  • Slide 36
  • Slide 37
  • Slide 38
  • Referenceshelliphelliphellip
  • Slide 40
Page 5: Unconditionally stable fdtd methods

Computational Electromagnetics

Finite-differencetime-domain(FDTD)

Finite-differencefrequency-domain(FDFD)

Method of Moments(MoM)

Fast multipole method (FMM)

Finite element method (FEM)

Transmission line matrix (TLM)

Frequency

Computational Electromagnetics

Finite-differencetime-domain(FDTD)

Finite-differencefrequency-domain(FDFD)

Method of Moments(MoM)

Fast multipole method (FMM)

Finite element method (FEM)

Transmission line matrix (TLM)

FDTD Overview ndash Updating Equations

760

Three scalar equations can be obtained from one vector curl equation

EH

t

yx zx

y x zy

y xzz

HE H

t y z

E H H

t z xH HE

t x y

HE

t

yx zx

y xzy

yxzz

EH E

t z y

H EE

t x zEEH

t y x

Finite DifferenceFinite Difference

Taylorrsquos seriesTaylorrsquos series

II Finite DifferenceII Finite Difference

Taylorrsquos Taylorrsquos seriesseries

ErorEror

Finite Difference Time Domain Method

bull Divide the interval x into sub-intervals each of width h

bull Divide the interval t into sub-intervals each of width k

bull A grid of points is used forthe finite difference solution

bull Tij represents T(xi tj)bull Replace the derivates by

finite-difference formulas10

t

x

(i j) (i+1 j)

(i j+1)

Ey

Ex

Hz

The Yee Discretization (1966)

Staggered grid in space mdash every field component is stored on a different grid

(i j k) (i+1 j k)

(i j k+1)

(i+1 j+1 k)

(i+1 j+1 k+1)

Ez

Ex

Ey

HyHx

Hz

(i j) (i+1 j)

(i j+1)

Ey

Ex

Hz

The Yee Discretization (1966)

all derivatives become center differenceshellip

H

t

1

E

H zt i

1

2 j

1

2

1

Eyx

Exy

1

Ey (i 1 j 1

2) Ey (i j

1

2)

xEx (i

1

2 j 1) Ex (i

1

2 j)

y

+ O(∆x2) + O(∆y2)

FDTD Overview ndash Updating Equations

yx zx

HE H

t y z

1

05 0505 05

( ) ( )( )

( ) ( 1)( ) ( 1 )

n nx x

x

n nn ny yz z

E i j k E i j ki j k

t

H i j k H i j kH i j k H i j k

y z

FDTD Overview ndash Updating Equations

yx zx

EH E

t z y

05 05

05

( ) ( )( )

( 1) ( ) ( 1 ) ( )

n nx x

x

n n n ny y z z

H i j k H i j ki j k

t

E i j k E i j k E i j k E i j k

z y

FDTD Overview ndash Updating Equations

Express the future components in terms of the past components

05 05

1

05 05

( ) ( 1 )

( ) ( )( ) ( ) ( 1)

n nz z

n nx x n n

x y y

H i j k H i j k

ytE i j k E i j k

i j k H i j k H i j k

z

05

05 05

( 1) ( )

( ) ( )( ) ( 1 ) ( )

n ny y

n nx x n n

x z z

E i j k E i j k

t zH i j k H i j ki j k E i j k E i j k

y

Fundamentals of the FDTD methodFundamentals of the FDTD method

Accuracy and stabilityAccuracy and stability

AccuracyAccuracy 10

StabilityStability

222

max

z

1

y

1

x

1c

1tt

3c

tt max

2D2D

22 y

1

x

1c

1t

1D1Dc

t

Physically this condition means that the time Physically this condition means that the time step should be smaller than the time for the step should be smaller than the time for the wave to propagate from one cell to the wave to propagate from one cell to the neighbor oneneighbor one

Slides from 5-15 may be skipped because it was taught in the class

III Fundamentals of the FDTD methodIII Fundamentals of the FDTD method

Dispersion relationDispersion relation

ckkkk

22

z2

y2

x2

In free space (ideal)

In FDTD computation (numerical)

22

z

2

y

2

x

2

tsin

tc

1

2

zk~

sinz

1

2

yk~

siny

1

2

xk~

sinx

1

utcsinArctk

~2

k~vpnum

2

z

2

y

2

x

2

zk~

sinz

1

2

yk~

siny

1

2

xk~

sinx

1u

The numerical medium is dispersive the propagation of the wave varies with frequency and angle

III Fundamentals of the FDTD methodIII Fundamentals of the FDTD method

Dispersion relationDispersion relation

Limitations of FDTD method

1-Grid spacing should be ~λ102-According to Courantrsquos stability condition time step Δt becomes small when FDTDgrid spacing becomes small3-In 3-D simulation simulation time scales like N^4 and required memory size scales like N^34-Application is restricted to relatively small size

21 march 2012 20

Space Domain Discretization

bull Heat Conduction Equation

bull Central-Finite-Difference Approximation

pc

tyxg

y

tyxT

x

tyxT

t

tyxT

)()()()(

2

2

2

2

2

2

2

11

22

112

2

2

)(2

x

T

x

TTT

xOx

TTT

x

T

nx

nji

nji

nji

nji

nji

njin

ji

cc

21 march 2012 21

X

n

jiTn

j1-iT n

j1iT

n

1jiT

n

1jiT

(00) (i0)hellip

(0j)

x

y

(I0)

(0J)

hellip

hellip

hellip

Finite-Difference Formulation of the Heat Conduction on a Chip

bull Space Domainbull Time Domain

21 march 2012 22

Time domain discretization

bull Heat Conduction Equation

ndash Simple Explicit Method ndash Simple Implicit Methodndash Crank-Nicolson Method

gcy

T

x

T

t

TT

p

yxnn

1

)()( 2

2

2

21

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

23

Can we check if a numerical scheme

is stable without computation Von Neumann stability

analysisbull Analyze if (or for which conditions) a

numerical scheme is stable or unstablebull Makes a local analysis coefficients of PDE are

assumed to vary slowly (our example constant)

bull How will unavoidable errors (say rounding errors)evolve in time

John von Neumann1903-1957

24

Von Neumann stability analysis

A numerical scheme is unstable if

Ansatz Wave number k and amplification factor

21 march 2012 25

bull Accuracybull Stability Constraint

bull No matrix inversion but time steps are limited by space discretization

gcy

T

x

T

t

TT

p

ny

nx

nn

1

)()( 2

2

2

21

] [ 22 yxt

2

11122

yxt

Simple Explicit Method

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

21 march 2012 26

Accuracy

Unconditionally Stable

No limits on time step but involves with large scale matrix inversion

gcy

T

x

T

t

TT

p

ny

nx

nn

1

)()( 2

12

2

121

] [ 22 yxt

Simple Implicit Method

11

1

11

12

11

1

11

12

2

2nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

21 march 2012 27

Accuracy

Unconditionally stable

No limits on time step but involves with large scale matrix inversion

gcy

TT

x

TT

t

TT

p

ny

ny

nx

nx

nn

1

)(2)(2 2

212

2

2121

] [ 222 yxt

Crank-Nicolson Method

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

11

1

11

12

11

1

11

12

2

2nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

21 march 2012 28

n

m1616

ex m=4n=4

Total node number N = mn

Matrix size = NxN

Analysis of Crank-Nicolson Method

21 march 2012 29

Alternating Direction Implicit Method

Solves higher dimension problem by successive Lower dimension methods

Accuracy

Unconditionally stable

No limits on time step and no large scale matrix inversion

] [ 222 yxt

21 march2012 30

Step I x-direction implicit y-direction explicitStep II x-direction explicit y-direction implicit

n

Alternating Direction Implicit Method

bull Peaceman-Rachford Algorithmbull Douglas-Gunn Algorithm

21 march 2012 31

bull Step I

bull Step II

gc

tT

rrT

rr

p

ny

yx

xny

yx

x

)2

1)(2

1()2

1)(2

1( 22122

gc

tT

rT

r

p

ny

yn

xx

2)

21()

21( 22

12

gc

tT

rT

r

p

n

xxn

yy

2)

21()

21( 2

1212

Peaceman-Rachford Algorithm

21 march 2012 32

Douglas-Gunn Algorithm

bull Step I

bull Step II

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

12

1

22

gc

tTrTT

rTT

p

nyy

nnxxnn

22

122

1

)(2

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

2

121

2)(

2

21 march 2012 33

Illustration for ADI

X-direction implicit Y-direction implicit

i = 1 2 hellip m

1

njiT

Step I Step II

j = 1

n

2

hellip

1 2 hellip m

11

njiT

11

njiT

j = 1

n

2hellip

21

njiT

21

1

njiT

21

1

njiT

21 march 2012 34

Analysis of ADI Method

mm

Tridiagonal Matrix

Time complexity O(N)

2xnxm = 2nm =2N

2 steps n matrices tridaigonal matrix

X-direction implicit

21

njiT

i = 1 2 hellip mj = 1

n

2

hellip

21

1

njiT

21

1

njiT

ADVANTAGE over conditionally stable

Reduce simulation time

Good accuracy

Size of geometrical feature may be typical of wavelengh

Good geometrical flexibility to allow with corner or high curvature than fdtd

Advantages of the FDTD method over other methodsAdvantages of the FDTD method over other methods

bull It is conceptually simpleIt is conceptually simple

bull The algorithm does not require the formulation of integral equation and relatively complex The algorithm does not require the formulation of integral equation and relatively complex

scatters can be treated without inversion of large matrices scatters can be treated without inversion of large matrices

bull It is simple to implement for complicated inhomogeneous conducting or dielectric structures It is simple to implement for complicated inhomogeneous conducting or dielectric structures

because constitutive parameters can be assigned to each lattice pointbecause constitutive parameters can be assigned to each lattice point

bull Its computer memory requirement is not prohibitive for many complex structures of interest Its computer memory requirement is not prohibitive for many complex structures of interest

bull The algorithm make use of the memory in a simple sequential orderThe algorithm make use of the memory in a simple sequential order

bull It is much easier to obtain frequency domain data from time domain results than the converse It is much easier to obtain frequency domain data from time domain results than the converse

Thus it is more convenient to obtain frequency domain results via time domain when many Thus it is more convenient to obtain frequency domain results via time domain when many

frequencies are involvedfrequencies are involved

Include examples of unconditionally stable fdtd and prove the advantages

Disadvantages Disadvantages

bull Its implementation necessitates modeling object as well as its surroundings Thus the Its implementation necessitates modeling object as well as its surroundings Thus the

required program execution time may be excessiverequired program execution time may be excessive

bull Its accuracy is at least one order of magnitude worse than that of the method of Its accuracy is at least one order of magnitude worse than that of the method of

moments for example moments for example

bull Since the computational meshes are rectangular in shape they do not conform the Since the computational meshes are rectangular in shape they do not conform the

scatterers with curved surfaces as is the case of the cylindrical or spherical boundary scatterers with curved surfaces as is the case of the cylindrical or spherical boundary

Its computer memory requirement is not prohibitive for many complex structures of Its computer memory requirement is not prohibitive for many complex structures of

interest interest

bull As in all finite difference algorithms the field quantities are only known at grid nodes As in all finite difference algorithms the field quantities are only known at grid nodes

Applications of FDTD method

1048708Electromagnetic scattering amp antenna design1048708EMCEMI design1048708Simulation of wave propagation problem1048708Solving partial defferential equation 1048708Waveguide analysis

Stripline amp microstrip line analysishellipetc hellip

Referenceshelliphelliphellip1-Electromagnetic simulation using FDTD method

ldquoDennis M Sillevanrdquo IEEE press series on rf amp mivcrowave technology ldquoRoger DPollard amp Richard

Bootanrdquoseries

2Understanding the FDTD method ldquoJohn BSchneiderrdquo may 8 2011

3 3 D -ADI -Mehod-unconditionaly stable time domain algorithom for solving maxwell equations

ldquoIEEE Transaction on microwave theory amp techniques vol-48No10 october 2010 ldquoTakefumi

IEEE

4High order split step unconditionally stable FDTD method ampnumerical analysis ldquoIEEE

Transactions on antinna amp propagationvol 59 no9sept2011rdquo Yong-Dan Kong amp Qing xin chu

senior memberIEEE

5Genaral Finite DifferenceSchemes for Heat equations ldquoIndian J Pure Applemath

10(2)209-222Febuary 1979rdquoby PS Jain ampDN Holla department of methemetics

IITB 400076

  • Unconditionally Stable FDTD Methods
  • Outlineshelliphelliphelliphellip
  • Outlineshellip
  • Slide 4
  • Computational Electromagnetics
  • Slide 6
  • FDTD Overview ndash Updating Equations
  • Slide 8
  • Slide 9
  • Finite Difference Time Domain Method
  • The Yee Discretization (1966)
  • Slide 12
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Slide 17
  • Slide 18
  • Slide 19
  • Space Domain Discretization
  • Finite-Difference Formulation of the Heat Conduction on a Chip
  • Time domain discretization
  • Slide 23
  • Slide 24
  • Simple Explicit Method
  • Simple Implicit Method
  • Crank-Nicolson Method
  • Analysis of Crank-Nicolson Method
  • Alternating Direction Implicit Method
  • Slide 30
  • Peaceman-Rachford Algorithm
  • Douglas-Gunn Algorithm
  • Illustration for ADI
  • Analysis of ADI Method
  • Slide 35
  • Slide 36
  • Slide 37
  • Slide 38
  • Referenceshelliphelliphellip
  • Slide 40
Page 6: Unconditionally stable fdtd methods

Frequency

Computational Electromagnetics

Finite-differencetime-domain(FDTD)

Finite-differencefrequency-domain(FDFD)

Method of Moments(MoM)

Fast multipole method (FMM)

Finite element method (FEM)

Transmission line matrix (TLM)

FDTD Overview ndash Updating Equations

760

Three scalar equations can be obtained from one vector curl equation

EH

t

yx zx

y x zy

y xzz

HE H

t y z

E H H

t z xH HE

t x y

HE

t

yx zx

y xzy

yxzz

EH E

t z y

H EE

t x zEEH

t y x

Finite DifferenceFinite Difference

Taylorrsquos seriesTaylorrsquos series

II Finite DifferenceII Finite Difference

Taylorrsquos Taylorrsquos seriesseries

ErorEror

Finite Difference Time Domain Method

bull Divide the interval x into sub-intervals each of width h

bull Divide the interval t into sub-intervals each of width k

bull A grid of points is used forthe finite difference solution

bull Tij represents T(xi tj)bull Replace the derivates by

finite-difference formulas10

t

x

(i j) (i+1 j)

(i j+1)

Ey

Ex

Hz

The Yee Discretization (1966)

Staggered grid in space mdash every field component is stored on a different grid

(i j k) (i+1 j k)

(i j k+1)

(i+1 j+1 k)

(i+1 j+1 k+1)

Ez

Ex

Ey

HyHx

Hz

(i j) (i+1 j)

(i j+1)

Ey

Ex

Hz

The Yee Discretization (1966)

all derivatives become center differenceshellip

H

t

1

E

H zt i

1

2 j

1

2

1

Eyx

Exy

1

Ey (i 1 j 1

2) Ey (i j

1

2)

xEx (i

1

2 j 1) Ex (i

1

2 j)

y

+ O(∆x2) + O(∆y2)

FDTD Overview ndash Updating Equations

yx zx

HE H

t y z

1

05 0505 05

( ) ( )( )

( ) ( 1)( ) ( 1 )

n nx x

x

n nn ny yz z

E i j k E i j ki j k

t

H i j k H i j kH i j k H i j k

y z

FDTD Overview ndash Updating Equations

yx zx

EH E

t z y

05 05

05

( ) ( )( )

( 1) ( ) ( 1 ) ( )

n nx x

x

n n n ny y z z

H i j k H i j ki j k

t

E i j k E i j k E i j k E i j k

z y

FDTD Overview ndash Updating Equations

Express the future components in terms of the past components

05 05

1

05 05

( ) ( 1 )

( ) ( )( ) ( ) ( 1)

n nz z

n nx x n n

x y y

H i j k H i j k

ytE i j k E i j k

i j k H i j k H i j k

z

05

05 05

( 1) ( )

( ) ( )( ) ( 1 ) ( )

n ny y

n nx x n n

x z z

E i j k E i j k

t zH i j k H i j ki j k E i j k E i j k

y

Fundamentals of the FDTD methodFundamentals of the FDTD method

Accuracy and stabilityAccuracy and stability

AccuracyAccuracy 10

StabilityStability

222

max

z

1

y

1

x

1c

1tt

3c

tt max

2D2D

22 y

1

x

1c

1t

1D1Dc

t

Physically this condition means that the time Physically this condition means that the time step should be smaller than the time for the step should be smaller than the time for the wave to propagate from one cell to the wave to propagate from one cell to the neighbor oneneighbor one

Slides from 5-15 may be skipped because it was taught in the class

III Fundamentals of the FDTD methodIII Fundamentals of the FDTD method

Dispersion relationDispersion relation

ckkkk

22

z2

y2

x2

In free space (ideal)

In FDTD computation (numerical)

22

z

2

y

2

x

2

tsin

tc

1

2

zk~

sinz

1

2

yk~

siny

1

2

xk~

sinx

1

utcsinArctk

~2

k~vpnum

2

z

2

y

2

x

2

zk~

sinz

1

2

yk~

siny

1

2

xk~

sinx

1u

The numerical medium is dispersive the propagation of the wave varies with frequency and angle

III Fundamentals of the FDTD methodIII Fundamentals of the FDTD method

Dispersion relationDispersion relation

Limitations of FDTD method

1-Grid spacing should be ~λ102-According to Courantrsquos stability condition time step Δt becomes small when FDTDgrid spacing becomes small3-In 3-D simulation simulation time scales like N^4 and required memory size scales like N^34-Application is restricted to relatively small size

21 march 2012 20

Space Domain Discretization

bull Heat Conduction Equation

bull Central-Finite-Difference Approximation

pc

tyxg

y

tyxT

x

tyxT

t

tyxT

)()()()(

2

2

2

2

2

2

2

11

22

112

2

2

)(2

x

T

x

TTT

xOx

TTT

x

T

nx

nji

nji

nji

nji

nji

njin

ji

cc

21 march 2012 21

X

n

jiTn

j1-iT n

j1iT

n

1jiT

n

1jiT

(00) (i0)hellip

(0j)

x

y

(I0)

(0J)

hellip

hellip

hellip

Finite-Difference Formulation of the Heat Conduction on a Chip

bull Space Domainbull Time Domain

21 march 2012 22

Time domain discretization

bull Heat Conduction Equation

ndash Simple Explicit Method ndash Simple Implicit Methodndash Crank-Nicolson Method

gcy

T

x

T

t

TT

p

yxnn

1

)()( 2

2

2

21

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

23

Can we check if a numerical scheme

is stable without computation Von Neumann stability

analysisbull Analyze if (or for which conditions) a

numerical scheme is stable or unstablebull Makes a local analysis coefficients of PDE are

assumed to vary slowly (our example constant)

bull How will unavoidable errors (say rounding errors)evolve in time

John von Neumann1903-1957

24

Von Neumann stability analysis

A numerical scheme is unstable if

Ansatz Wave number k and amplification factor

21 march 2012 25

bull Accuracybull Stability Constraint

bull No matrix inversion but time steps are limited by space discretization

gcy

T

x

T

t

TT

p

ny

nx

nn

1

)()( 2

2

2

21

] [ 22 yxt

2

11122

yxt

Simple Explicit Method

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

21 march 2012 26

Accuracy

Unconditionally Stable

No limits on time step but involves with large scale matrix inversion

gcy

T

x

T

t

TT

p

ny

nx

nn

1

)()( 2

12

2

121

] [ 22 yxt

Simple Implicit Method

11

1

11

12

11

1

11

12

2

2nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

21 march 2012 27

Accuracy

Unconditionally stable

No limits on time step but involves with large scale matrix inversion

gcy

TT

x

TT

t

TT

p

ny

ny

nx

nx

nn

1

)(2)(2 2

212

2

2121

] [ 222 yxt

Crank-Nicolson Method

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

11

1

11

12

11

1

11

12

2

2nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

21 march 2012 28

n

m1616

ex m=4n=4

Total node number N = mn

Matrix size = NxN

Analysis of Crank-Nicolson Method

21 march 2012 29

Alternating Direction Implicit Method

Solves higher dimension problem by successive Lower dimension methods

Accuracy

Unconditionally stable

No limits on time step and no large scale matrix inversion

] [ 222 yxt

21 march2012 30

Step I x-direction implicit y-direction explicitStep II x-direction explicit y-direction implicit

n

Alternating Direction Implicit Method

bull Peaceman-Rachford Algorithmbull Douglas-Gunn Algorithm

21 march 2012 31

bull Step I

bull Step II

gc

tT

rrT

rr

p

ny

yx

xny

yx

x

)2

1)(2

1()2

1)(2

1( 22122

gc

tT

rT

r

p

ny

yn

xx

2)

21()

21( 22

12

gc

tT

rT

r

p

n

xxn

yy

2)

21()

21( 2

1212

Peaceman-Rachford Algorithm

21 march 2012 32

Douglas-Gunn Algorithm

bull Step I

bull Step II

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

12

1

22

gc

tTrTT

rTT

p

nyy

nnxxnn

22

122

1

)(2

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

2

121

2)(

2

21 march 2012 33

Illustration for ADI

X-direction implicit Y-direction implicit

i = 1 2 hellip m

1

njiT

Step I Step II

j = 1

n

2

hellip

1 2 hellip m

11

njiT

11

njiT

j = 1

n

2hellip

21

njiT

21

1

njiT

21

1

njiT

21 march 2012 34

Analysis of ADI Method

mm

Tridiagonal Matrix

Time complexity O(N)

2xnxm = 2nm =2N

2 steps n matrices tridaigonal matrix

X-direction implicit

21

njiT

i = 1 2 hellip mj = 1

n

2

hellip

21

1

njiT

21

1

njiT

ADVANTAGE over conditionally stable

Reduce simulation time

Good accuracy

Size of geometrical feature may be typical of wavelengh

Good geometrical flexibility to allow with corner or high curvature than fdtd

Advantages of the FDTD method over other methodsAdvantages of the FDTD method over other methods

bull It is conceptually simpleIt is conceptually simple

bull The algorithm does not require the formulation of integral equation and relatively complex The algorithm does not require the formulation of integral equation and relatively complex

scatters can be treated without inversion of large matrices scatters can be treated without inversion of large matrices

bull It is simple to implement for complicated inhomogeneous conducting or dielectric structures It is simple to implement for complicated inhomogeneous conducting or dielectric structures

because constitutive parameters can be assigned to each lattice pointbecause constitutive parameters can be assigned to each lattice point

bull Its computer memory requirement is not prohibitive for many complex structures of interest Its computer memory requirement is not prohibitive for many complex structures of interest

bull The algorithm make use of the memory in a simple sequential orderThe algorithm make use of the memory in a simple sequential order

bull It is much easier to obtain frequency domain data from time domain results than the converse It is much easier to obtain frequency domain data from time domain results than the converse

Thus it is more convenient to obtain frequency domain results via time domain when many Thus it is more convenient to obtain frequency domain results via time domain when many

frequencies are involvedfrequencies are involved

Include examples of unconditionally stable fdtd and prove the advantages

Disadvantages Disadvantages

bull Its implementation necessitates modeling object as well as its surroundings Thus the Its implementation necessitates modeling object as well as its surroundings Thus the

required program execution time may be excessiverequired program execution time may be excessive

bull Its accuracy is at least one order of magnitude worse than that of the method of Its accuracy is at least one order of magnitude worse than that of the method of

moments for example moments for example

bull Since the computational meshes are rectangular in shape they do not conform the Since the computational meshes are rectangular in shape they do not conform the

scatterers with curved surfaces as is the case of the cylindrical or spherical boundary scatterers with curved surfaces as is the case of the cylindrical or spherical boundary

Its computer memory requirement is not prohibitive for many complex structures of Its computer memory requirement is not prohibitive for many complex structures of

interest interest

bull As in all finite difference algorithms the field quantities are only known at grid nodes As in all finite difference algorithms the field quantities are only known at grid nodes

Applications of FDTD method

1048708Electromagnetic scattering amp antenna design1048708EMCEMI design1048708Simulation of wave propagation problem1048708Solving partial defferential equation 1048708Waveguide analysis

Stripline amp microstrip line analysishellipetc hellip

Referenceshelliphelliphellip1-Electromagnetic simulation using FDTD method

ldquoDennis M Sillevanrdquo IEEE press series on rf amp mivcrowave technology ldquoRoger DPollard amp Richard

Bootanrdquoseries

2Understanding the FDTD method ldquoJohn BSchneiderrdquo may 8 2011

3 3 D -ADI -Mehod-unconditionaly stable time domain algorithom for solving maxwell equations

ldquoIEEE Transaction on microwave theory amp techniques vol-48No10 october 2010 ldquoTakefumi

IEEE

4High order split step unconditionally stable FDTD method ampnumerical analysis ldquoIEEE

Transactions on antinna amp propagationvol 59 no9sept2011rdquo Yong-Dan Kong amp Qing xin chu

senior memberIEEE

5Genaral Finite DifferenceSchemes for Heat equations ldquoIndian J Pure Applemath

10(2)209-222Febuary 1979rdquoby PS Jain ampDN Holla department of methemetics

IITB 400076

  • Unconditionally Stable FDTD Methods
  • Outlineshelliphelliphelliphellip
  • Outlineshellip
  • Slide 4
  • Computational Electromagnetics
  • Slide 6
  • FDTD Overview ndash Updating Equations
  • Slide 8
  • Slide 9
  • Finite Difference Time Domain Method
  • The Yee Discretization (1966)
  • Slide 12
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Slide 17
  • Slide 18
  • Slide 19
  • Space Domain Discretization
  • Finite-Difference Formulation of the Heat Conduction on a Chip
  • Time domain discretization
  • Slide 23
  • Slide 24
  • Simple Explicit Method
  • Simple Implicit Method
  • Crank-Nicolson Method
  • Analysis of Crank-Nicolson Method
  • Alternating Direction Implicit Method
  • Slide 30
  • Peaceman-Rachford Algorithm
  • Douglas-Gunn Algorithm
  • Illustration for ADI
  • Analysis of ADI Method
  • Slide 35
  • Slide 36
  • Slide 37
  • Slide 38
  • Referenceshelliphelliphellip
  • Slide 40
Page 7: Unconditionally stable fdtd methods

FDTD Overview ndash Updating Equations

760

Three scalar equations can be obtained from one vector curl equation

EH

t

yx zx

y x zy

y xzz

HE H

t y z

E H H

t z xH HE

t x y

HE

t

yx zx

y xzy

yxzz

EH E

t z y

H EE

t x zEEH

t y x

Finite DifferenceFinite Difference

Taylorrsquos seriesTaylorrsquos series

II Finite DifferenceII Finite Difference

Taylorrsquos Taylorrsquos seriesseries

ErorEror

Finite Difference Time Domain Method

bull Divide the interval x into sub-intervals each of width h

bull Divide the interval t into sub-intervals each of width k

bull A grid of points is used forthe finite difference solution

bull Tij represents T(xi tj)bull Replace the derivates by

finite-difference formulas10

t

x

(i j) (i+1 j)

(i j+1)

Ey

Ex

Hz

The Yee Discretization (1966)

Staggered grid in space mdash every field component is stored on a different grid

(i j k) (i+1 j k)

(i j k+1)

(i+1 j+1 k)

(i+1 j+1 k+1)

Ez

Ex

Ey

HyHx

Hz

(i j) (i+1 j)

(i j+1)

Ey

Ex

Hz

The Yee Discretization (1966)

all derivatives become center differenceshellip

H

t

1

E

H zt i

1

2 j

1

2

1

Eyx

Exy

1

Ey (i 1 j 1

2) Ey (i j

1

2)

xEx (i

1

2 j 1) Ex (i

1

2 j)

y

+ O(∆x2) + O(∆y2)

FDTD Overview ndash Updating Equations

yx zx

HE H

t y z

1

05 0505 05

( ) ( )( )

( ) ( 1)( ) ( 1 )

n nx x

x

n nn ny yz z

E i j k E i j ki j k

t

H i j k H i j kH i j k H i j k

y z

FDTD Overview ndash Updating Equations

yx zx

EH E

t z y

05 05

05

( ) ( )( )

( 1) ( ) ( 1 ) ( )

n nx x

x

n n n ny y z z

H i j k H i j ki j k

t

E i j k E i j k E i j k E i j k

z y

FDTD Overview ndash Updating Equations

Express the future components in terms of the past components

05 05

1

05 05

( ) ( 1 )

( ) ( )( ) ( ) ( 1)

n nz z

n nx x n n

x y y

H i j k H i j k

ytE i j k E i j k

i j k H i j k H i j k

z

05

05 05

( 1) ( )

( ) ( )( ) ( 1 ) ( )

n ny y

n nx x n n

x z z

E i j k E i j k

t zH i j k H i j ki j k E i j k E i j k

y

Fundamentals of the FDTD methodFundamentals of the FDTD method

Accuracy and stabilityAccuracy and stability

AccuracyAccuracy 10

StabilityStability

222

max

z

1

y

1

x

1c

1tt

3c

tt max

2D2D

22 y

1

x

1c

1t

1D1Dc

t

Physically this condition means that the time Physically this condition means that the time step should be smaller than the time for the step should be smaller than the time for the wave to propagate from one cell to the wave to propagate from one cell to the neighbor oneneighbor one

Slides from 5-15 may be skipped because it was taught in the class

III Fundamentals of the FDTD methodIII Fundamentals of the FDTD method

Dispersion relationDispersion relation

ckkkk

22

z2

y2

x2

In free space (ideal)

In FDTD computation (numerical)

22

z

2

y

2

x

2

tsin

tc

1

2

zk~

sinz

1

2

yk~

siny

1

2

xk~

sinx

1

utcsinArctk

~2

k~vpnum

2

z

2

y

2

x

2

zk~

sinz

1

2

yk~

siny

1

2

xk~

sinx

1u

The numerical medium is dispersive the propagation of the wave varies with frequency and angle

III Fundamentals of the FDTD methodIII Fundamentals of the FDTD method

Dispersion relationDispersion relation

Limitations of FDTD method

1-Grid spacing should be ~λ102-According to Courantrsquos stability condition time step Δt becomes small when FDTDgrid spacing becomes small3-In 3-D simulation simulation time scales like N^4 and required memory size scales like N^34-Application is restricted to relatively small size

21 march 2012 20

Space Domain Discretization

bull Heat Conduction Equation

bull Central-Finite-Difference Approximation

pc

tyxg

y

tyxT

x

tyxT

t

tyxT

)()()()(

2

2

2

2

2

2

2

11

22

112

2

2

)(2

x

T

x

TTT

xOx

TTT

x

T

nx

nji

nji

nji

nji

nji

njin

ji

cc

21 march 2012 21

X

n

jiTn

j1-iT n

j1iT

n

1jiT

n

1jiT

(00) (i0)hellip

(0j)

x

y

(I0)

(0J)

hellip

hellip

hellip

Finite-Difference Formulation of the Heat Conduction on a Chip

bull Space Domainbull Time Domain

21 march 2012 22

Time domain discretization

bull Heat Conduction Equation

ndash Simple Explicit Method ndash Simple Implicit Methodndash Crank-Nicolson Method

gcy

T

x

T

t

TT

p

yxnn

1

)()( 2

2

2

21

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

23

Can we check if a numerical scheme

is stable without computation Von Neumann stability

analysisbull Analyze if (or for which conditions) a

numerical scheme is stable or unstablebull Makes a local analysis coefficients of PDE are

assumed to vary slowly (our example constant)

bull How will unavoidable errors (say rounding errors)evolve in time

John von Neumann1903-1957

24

Von Neumann stability analysis

A numerical scheme is unstable if

Ansatz Wave number k and amplification factor

21 march 2012 25

bull Accuracybull Stability Constraint

bull No matrix inversion but time steps are limited by space discretization

gcy

T

x

T

t

TT

p

ny

nx

nn

1

)()( 2

2

2

21

] [ 22 yxt

2

11122

yxt

Simple Explicit Method

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

21 march 2012 26

Accuracy

Unconditionally Stable

No limits on time step but involves with large scale matrix inversion

gcy

T

x

T

t

TT

p

ny

nx

nn

1

)()( 2

12

2

121

] [ 22 yxt

Simple Implicit Method

11

1

11

12

11

1

11

12

2

2nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

21 march 2012 27

Accuracy

Unconditionally stable

No limits on time step but involves with large scale matrix inversion

gcy

TT

x

TT

t

TT

p

ny

ny

nx

nx

nn

1

)(2)(2 2

212

2

2121

] [ 222 yxt

Crank-Nicolson Method

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

11

1

11

12

11

1

11

12

2

2nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

21 march 2012 28

n

m1616

ex m=4n=4

Total node number N = mn

Matrix size = NxN

Analysis of Crank-Nicolson Method

21 march 2012 29

Alternating Direction Implicit Method

Solves higher dimension problem by successive Lower dimension methods

Accuracy

Unconditionally stable

No limits on time step and no large scale matrix inversion

] [ 222 yxt

21 march2012 30

Step I x-direction implicit y-direction explicitStep II x-direction explicit y-direction implicit

n

Alternating Direction Implicit Method

bull Peaceman-Rachford Algorithmbull Douglas-Gunn Algorithm

21 march 2012 31

bull Step I

bull Step II

gc

tT

rrT

rr

p

ny

yx

xny

yx

x

)2

1)(2

1()2

1)(2

1( 22122

gc

tT

rT

r

p

ny

yn

xx

2)

21()

21( 22

12

gc

tT

rT

r

p

n

xxn

yy

2)

21()

21( 2

1212

Peaceman-Rachford Algorithm

21 march 2012 32

Douglas-Gunn Algorithm

bull Step I

bull Step II

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

12

1

22

gc

tTrTT

rTT

p

nyy

nnxxnn

22

122

1

)(2

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

2

121

2)(

2

21 march 2012 33

Illustration for ADI

X-direction implicit Y-direction implicit

i = 1 2 hellip m

1

njiT

Step I Step II

j = 1

n

2

hellip

1 2 hellip m

11

njiT

11

njiT

j = 1

n

2hellip

21

njiT

21

1

njiT

21

1

njiT

21 march 2012 34

Analysis of ADI Method

mm

Tridiagonal Matrix

Time complexity O(N)

2xnxm = 2nm =2N

2 steps n matrices tridaigonal matrix

X-direction implicit

21

njiT

i = 1 2 hellip mj = 1

n

2

hellip

21

1

njiT

21

1

njiT

ADVANTAGE over conditionally stable

Reduce simulation time

Good accuracy

Size of geometrical feature may be typical of wavelengh

Good geometrical flexibility to allow with corner or high curvature than fdtd

Advantages of the FDTD method over other methodsAdvantages of the FDTD method over other methods

bull It is conceptually simpleIt is conceptually simple

bull The algorithm does not require the formulation of integral equation and relatively complex The algorithm does not require the formulation of integral equation and relatively complex

scatters can be treated without inversion of large matrices scatters can be treated without inversion of large matrices

bull It is simple to implement for complicated inhomogeneous conducting or dielectric structures It is simple to implement for complicated inhomogeneous conducting or dielectric structures

because constitutive parameters can be assigned to each lattice pointbecause constitutive parameters can be assigned to each lattice point

bull Its computer memory requirement is not prohibitive for many complex structures of interest Its computer memory requirement is not prohibitive for many complex structures of interest

bull The algorithm make use of the memory in a simple sequential orderThe algorithm make use of the memory in a simple sequential order

bull It is much easier to obtain frequency domain data from time domain results than the converse It is much easier to obtain frequency domain data from time domain results than the converse

Thus it is more convenient to obtain frequency domain results via time domain when many Thus it is more convenient to obtain frequency domain results via time domain when many

frequencies are involvedfrequencies are involved

Include examples of unconditionally stable fdtd and prove the advantages

Disadvantages Disadvantages

bull Its implementation necessitates modeling object as well as its surroundings Thus the Its implementation necessitates modeling object as well as its surroundings Thus the

required program execution time may be excessiverequired program execution time may be excessive

bull Its accuracy is at least one order of magnitude worse than that of the method of Its accuracy is at least one order of magnitude worse than that of the method of

moments for example moments for example

bull Since the computational meshes are rectangular in shape they do not conform the Since the computational meshes are rectangular in shape they do not conform the

scatterers with curved surfaces as is the case of the cylindrical or spherical boundary scatterers with curved surfaces as is the case of the cylindrical or spherical boundary

Its computer memory requirement is not prohibitive for many complex structures of Its computer memory requirement is not prohibitive for many complex structures of

interest interest

bull As in all finite difference algorithms the field quantities are only known at grid nodes As in all finite difference algorithms the field quantities are only known at grid nodes

Applications of FDTD method

1048708Electromagnetic scattering amp antenna design1048708EMCEMI design1048708Simulation of wave propagation problem1048708Solving partial defferential equation 1048708Waveguide analysis

Stripline amp microstrip line analysishellipetc hellip

Referenceshelliphelliphellip1-Electromagnetic simulation using FDTD method

ldquoDennis M Sillevanrdquo IEEE press series on rf amp mivcrowave technology ldquoRoger DPollard amp Richard

Bootanrdquoseries

2Understanding the FDTD method ldquoJohn BSchneiderrdquo may 8 2011

3 3 D -ADI -Mehod-unconditionaly stable time domain algorithom for solving maxwell equations

ldquoIEEE Transaction on microwave theory amp techniques vol-48No10 october 2010 ldquoTakefumi

IEEE

4High order split step unconditionally stable FDTD method ampnumerical analysis ldquoIEEE

Transactions on antinna amp propagationvol 59 no9sept2011rdquo Yong-Dan Kong amp Qing xin chu

senior memberIEEE

5Genaral Finite DifferenceSchemes for Heat equations ldquoIndian J Pure Applemath

10(2)209-222Febuary 1979rdquoby PS Jain ampDN Holla department of methemetics

IITB 400076

  • Unconditionally Stable FDTD Methods
  • Outlineshelliphelliphelliphellip
  • Outlineshellip
  • Slide 4
  • Computational Electromagnetics
  • Slide 6
  • FDTD Overview ndash Updating Equations
  • Slide 8
  • Slide 9
  • Finite Difference Time Domain Method
  • The Yee Discretization (1966)
  • Slide 12
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Slide 17
  • Slide 18
  • Slide 19
  • Space Domain Discretization
  • Finite-Difference Formulation of the Heat Conduction on a Chip
  • Time domain discretization
  • Slide 23
  • Slide 24
  • Simple Explicit Method
  • Simple Implicit Method
  • Crank-Nicolson Method
  • Analysis of Crank-Nicolson Method
  • Alternating Direction Implicit Method
  • Slide 30
  • Peaceman-Rachford Algorithm
  • Douglas-Gunn Algorithm
  • Illustration for ADI
  • Analysis of ADI Method
  • Slide 35
  • Slide 36
  • Slide 37
  • Slide 38
  • Referenceshelliphelliphellip
  • Slide 40
Page 8: Unconditionally stable fdtd methods

Finite DifferenceFinite Difference

Taylorrsquos seriesTaylorrsquos series

II Finite DifferenceII Finite Difference

Taylorrsquos Taylorrsquos seriesseries

ErorEror

Finite Difference Time Domain Method

bull Divide the interval x into sub-intervals each of width h

bull Divide the interval t into sub-intervals each of width k

bull A grid of points is used forthe finite difference solution

bull Tij represents T(xi tj)bull Replace the derivates by

finite-difference formulas10

t

x

(i j) (i+1 j)

(i j+1)

Ey

Ex

Hz

The Yee Discretization (1966)

Staggered grid in space mdash every field component is stored on a different grid

(i j k) (i+1 j k)

(i j k+1)

(i+1 j+1 k)

(i+1 j+1 k+1)

Ez

Ex

Ey

HyHx

Hz

(i j) (i+1 j)

(i j+1)

Ey

Ex

Hz

The Yee Discretization (1966)

all derivatives become center differenceshellip

H

t

1

E

H zt i

1

2 j

1

2

1

Eyx

Exy

1

Ey (i 1 j 1

2) Ey (i j

1

2)

xEx (i

1

2 j 1) Ex (i

1

2 j)

y

+ O(∆x2) + O(∆y2)

FDTD Overview ndash Updating Equations

yx zx

HE H

t y z

1

05 0505 05

( ) ( )( )

( ) ( 1)( ) ( 1 )

n nx x

x

n nn ny yz z

E i j k E i j ki j k

t

H i j k H i j kH i j k H i j k

y z

FDTD Overview ndash Updating Equations

yx zx

EH E

t z y

05 05

05

( ) ( )( )

( 1) ( ) ( 1 ) ( )

n nx x

x

n n n ny y z z

H i j k H i j ki j k

t

E i j k E i j k E i j k E i j k

z y

FDTD Overview ndash Updating Equations

Express the future components in terms of the past components

05 05

1

05 05

( ) ( 1 )

( ) ( )( ) ( ) ( 1)

n nz z

n nx x n n

x y y

H i j k H i j k

ytE i j k E i j k

i j k H i j k H i j k

z

05

05 05

( 1) ( )

( ) ( )( ) ( 1 ) ( )

n ny y

n nx x n n

x z z

E i j k E i j k

t zH i j k H i j ki j k E i j k E i j k

y

Fundamentals of the FDTD methodFundamentals of the FDTD method

Accuracy and stabilityAccuracy and stability

AccuracyAccuracy 10

StabilityStability

222

max

z

1

y

1

x

1c

1tt

3c

tt max

2D2D

22 y

1

x

1c

1t

1D1Dc

t

Physically this condition means that the time Physically this condition means that the time step should be smaller than the time for the step should be smaller than the time for the wave to propagate from one cell to the wave to propagate from one cell to the neighbor oneneighbor one

Slides from 5-15 may be skipped because it was taught in the class

III Fundamentals of the FDTD methodIII Fundamentals of the FDTD method

Dispersion relationDispersion relation

ckkkk

22

z2

y2

x2

In free space (ideal)

In FDTD computation (numerical)

22

z

2

y

2

x

2

tsin

tc

1

2

zk~

sinz

1

2

yk~

siny

1

2

xk~

sinx

1

utcsinArctk

~2

k~vpnum

2

z

2

y

2

x

2

zk~

sinz

1

2

yk~

siny

1

2

xk~

sinx

1u

The numerical medium is dispersive the propagation of the wave varies with frequency and angle

III Fundamentals of the FDTD methodIII Fundamentals of the FDTD method

Dispersion relationDispersion relation

Limitations of FDTD method

1-Grid spacing should be ~λ102-According to Courantrsquos stability condition time step Δt becomes small when FDTDgrid spacing becomes small3-In 3-D simulation simulation time scales like N^4 and required memory size scales like N^34-Application is restricted to relatively small size

21 march 2012 20

Space Domain Discretization

bull Heat Conduction Equation

bull Central-Finite-Difference Approximation

pc

tyxg

y

tyxT

x

tyxT

t

tyxT

)()()()(

2

2

2

2

2

2

2

11

22

112

2

2

)(2

x

T

x

TTT

xOx

TTT

x

T

nx

nji

nji

nji

nji

nji

njin

ji

cc

21 march 2012 21

X

n

jiTn

j1-iT n

j1iT

n

1jiT

n

1jiT

(00) (i0)hellip

(0j)

x

y

(I0)

(0J)

hellip

hellip

hellip

Finite-Difference Formulation of the Heat Conduction on a Chip

bull Space Domainbull Time Domain

21 march 2012 22

Time domain discretization

bull Heat Conduction Equation

ndash Simple Explicit Method ndash Simple Implicit Methodndash Crank-Nicolson Method

gcy

T

x

T

t

TT

p

yxnn

1

)()( 2

2

2

21

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

23

Can we check if a numerical scheme

is stable without computation Von Neumann stability

analysisbull Analyze if (or for which conditions) a

numerical scheme is stable or unstablebull Makes a local analysis coefficients of PDE are

assumed to vary slowly (our example constant)

bull How will unavoidable errors (say rounding errors)evolve in time

John von Neumann1903-1957

24

Von Neumann stability analysis

A numerical scheme is unstable if

Ansatz Wave number k and amplification factor

21 march 2012 25

bull Accuracybull Stability Constraint

bull No matrix inversion but time steps are limited by space discretization

gcy

T

x

T

t

TT

p

ny

nx

nn

1

)()( 2

2

2

21

] [ 22 yxt

2

11122

yxt

Simple Explicit Method

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

21 march 2012 26

Accuracy

Unconditionally Stable

No limits on time step but involves with large scale matrix inversion

gcy

T

x

T

t

TT

p

ny

nx

nn

1

)()( 2

12

2

121

] [ 22 yxt

Simple Implicit Method

11

1

11

12

11

1

11

12

2

2nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

21 march 2012 27

Accuracy

Unconditionally stable

No limits on time step but involves with large scale matrix inversion

gcy

TT

x

TT

t

TT

p

ny

ny

nx

nx

nn

1

)(2)(2 2

212

2

2121

] [ 222 yxt

Crank-Nicolson Method

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

11

1

11

12

11

1

11

12

2

2nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

21 march 2012 28

n

m1616

ex m=4n=4

Total node number N = mn

Matrix size = NxN

Analysis of Crank-Nicolson Method

21 march 2012 29

Alternating Direction Implicit Method

Solves higher dimension problem by successive Lower dimension methods

Accuracy

Unconditionally stable

No limits on time step and no large scale matrix inversion

] [ 222 yxt

21 march2012 30

Step I x-direction implicit y-direction explicitStep II x-direction explicit y-direction implicit

n

Alternating Direction Implicit Method

bull Peaceman-Rachford Algorithmbull Douglas-Gunn Algorithm

21 march 2012 31

bull Step I

bull Step II

gc

tT

rrT

rr

p

ny

yx

xny

yx

x

)2

1)(2

1()2

1)(2

1( 22122

gc

tT

rT

r

p

ny

yn

xx

2)

21()

21( 22

12

gc

tT

rT

r

p

n

xxn

yy

2)

21()

21( 2

1212

Peaceman-Rachford Algorithm

21 march 2012 32

Douglas-Gunn Algorithm

bull Step I

bull Step II

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

12

1

22

gc

tTrTT

rTT

p

nyy

nnxxnn

22

122

1

)(2

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

2

121

2)(

2

21 march 2012 33

Illustration for ADI

X-direction implicit Y-direction implicit

i = 1 2 hellip m

1

njiT

Step I Step II

j = 1

n

2

hellip

1 2 hellip m

11

njiT

11

njiT

j = 1

n

2hellip

21

njiT

21

1

njiT

21

1

njiT

21 march 2012 34

Analysis of ADI Method

mm

Tridiagonal Matrix

Time complexity O(N)

2xnxm = 2nm =2N

2 steps n matrices tridaigonal matrix

X-direction implicit

21

njiT

i = 1 2 hellip mj = 1

n

2

hellip

21

1

njiT

21

1

njiT

ADVANTAGE over conditionally stable

Reduce simulation time

Good accuracy

Size of geometrical feature may be typical of wavelengh

Good geometrical flexibility to allow with corner or high curvature than fdtd

Advantages of the FDTD method over other methodsAdvantages of the FDTD method over other methods

bull It is conceptually simpleIt is conceptually simple

bull The algorithm does not require the formulation of integral equation and relatively complex The algorithm does not require the formulation of integral equation and relatively complex

scatters can be treated without inversion of large matrices scatters can be treated without inversion of large matrices

bull It is simple to implement for complicated inhomogeneous conducting or dielectric structures It is simple to implement for complicated inhomogeneous conducting or dielectric structures

because constitutive parameters can be assigned to each lattice pointbecause constitutive parameters can be assigned to each lattice point

bull Its computer memory requirement is not prohibitive for many complex structures of interest Its computer memory requirement is not prohibitive for many complex structures of interest

bull The algorithm make use of the memory in a simple sequential orderThe algorithm make use of the memory in a simple sequential order

bull It is much easier to obtain frequency domain data from time domain results than the converse It is much easier to obtain frequency domain data from time domain results than the converse

Thus it is more convenient to obtain frequency domain results via time domain when many Thus it is more convenient to obtain frequency domain results via time domain when many

frequencies are involvedfrequencies are involved

Include examples of unconditionally stable fdtd and prove the advantages

Disadvantages Disadvantages

bull Its implementation necessitates modeling object as well as its surroundings Thus the Its implementation necessitates modeling object as well as its surroundings Thus the

required program execution time may be excessiverequired program execution time may be excessive

bull Its accuracy is at least one order of magnitude worse than that of the method of Its accuracy is at least one order of magnitude worse than that of the method of

moments for example moments for example

bull Since the computational meshes are rectangular in shape they do not conform the Since the computational meshes are rectangular in shape they do not conform the

scatterers with curved surfaces as is the case of the cylindrical or spherical boundary scatterers with curved surfaces as is the case of the cylindrical or spherical boundary

Its computer memory requirement is not prohibitive for many complex structures of Its computer memory requirement is not prohibitive for many complex structures of

interest interest

bull As in all finite difference algorithms the field quantities are only known at grid nodes As in all finite difference algorithms the field quantities are only known at grid nodes

Applications of FDTD method

1048708Electromagnetic scattering amp antenna design1048708EMCEMI design1048708Simulation of wave propagation problem1048708Solving partial defferential equation 1048708Waveguide analysis

Stripline amp microstrip line analysishellipetc hellip

Referenceshelliphelliphellip1-Electromagnetic simulation using FDTD method

ldquoDennis M Sillevanrdquo IEEE press series on rf amp mivcrowave technology ldquoRoger DPollard amp Richard

Bootanrdquoseries

2Understanding the FDTD method ldquoJohn BSchneiderrdquo may 8 2011

3 3 D -ADI -Mehod-unconditionaly stable time domain algorithom for solving maxwell equations

ldquoIEEE Transaction on microwave theory amp techniques vol-48No10 october 2010 ldquoTakefumi

IEEE

4High order split step unconditionally stable FDTD method ampnumerical analysis ldquoIEEE

Transactions on antinna amp propagationvol 59 no9sept2011rdquo Yong-Dan Kong amp Qing xin chu

senior memberIEEE

5Genaral Finite DifferenceSchemes for Heat equations ldquoIndian J Pure Applemath

10(2)209-222Febuary 1979rdquoby PS Jain ampDN Holla department of methemetics

IITB 400076

  • Unconditionally Stable FDTD Methods
  • Outlineshelliphelliphelliphellip
  • Outlineshellip
  • Slide 4
  • Computational Electromagnetics
  • Slide 6
  • FDTD Overview ndash Updating Equations
  • Slide 8
  • Slide 9
  • Finite Difference Time Domain Method
  • The Yee Discretization (1966)
  • Slide 12
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Slide 17
  • Slide 18
  • Slide 19
  • Space Domain Discretization
  • Finite-Difference Formulation of the Heat Conduction on a Chip
  • Time domain discretization
  • Slide 23
  • Slide 24
  • Simple Explicit Method
  • Simple Implicit Method
  • Crank-Nicolson Method
  • Analysis of Crank-Nicolson Method
  • Alternating Direction Implicit Method
  • Slide 30
  • Peaceman-Rachford Algorithm
  • Douglas-Gunn Algorithm
  • Illustration for ADI
  • Analysis of ADI Method
  • Slide 35
  • Slide 36
  • Slide 37
  • Slide 38
  • Referenceshelliphelliphellip
  • Slide 40
Page 9: Unconditionally stable fdtd methods

II Finite DifferenceII Finite Difference

Taylorrsquos Taylorrsquos seriesseries

ErorEror

Finite Difference Time Domain Method

bull Divide the interval x into sub-intervals each of width h

bull Divide the interval t into sub-intervals each of width k

bull A grid of points is used forthe finite difference solution

bull Tij represents T(xi tj)bull Replace the derivates by

finite-difference formulas10

t

x

(i j) (i+1 j)

(i j+1)

Ey

Ex

Hz

The Yee Discretization (1966)

Staggered grid in space mdash every field component is stored on a different grid

(i j k) (i+1 j k)

(i j k+1)

(i+1 j+1 k)

(i+1 j+1 k+1)

Ez

Ex

Ey

HyHx

Hz

(i j) (i+1 j)

(i j+1)

Ey

Ex

Hz

The Yee Discretization (1966)

all derivatives become center differenceshellip

H

t

1

E

H zt i

1

2 j

1

2

1

Eyx

Exy

1

Ey (i 1 j 1

2) Ey (i j

1

2)

xEx (i

1

2 j 1) Ex (i

1

2 j)

y

+ O(∆x2) + O(∆y2)

FDTD Overview ndash Updating Equations

yx zx

HE H

t y z

1

05 0505 05

( ) ( )( )

( ) ( 1)( ) ( 1 )

n nx x

x

n nn ny yz z

E i j k E i j ki j k

t

H i j k H i j kH i j k H i j k

y z

FDTD Overview ndash Updating Equations

yx zx

EH E

t z y

05 05

05

( ) ( )( )

( 1) ( ) ( 1 ) ( )

n nx x

x

n n n ny y z z

H i j k H i j ki j k

t

E i j k E i j k E i j k E i j k

z y

FDTD Overview ndash Updating Equations

Express the future components in terms of the past components

05 05

1

05 05

( ) ( 1 )

( ) ( )( ) ( ) ( 1)

n nz z

n nx x n n

x y y

H i j k H i j k

ytE i j k E i j k

i j k H i j k H i j k

z

05

05 05

( 1) ( )

( ) ( )( ) ( 1 ) ( )

n ny y

n nx x n n

x z z

E i j k E i j k

t zH i j k H i j ki j k E i j k E i j k

y

Fundamentals of the FDTD methodFundamentals of the FDTD method

Accuracy and stabilityAccuracy and stability

AccuracyAccuracy 10

StabilityStability

222

max

z

1

y

1

x

1c

1tt

3c

tt max

2D2D

22 y

1

x

1c

1t

1D1Dc

t

Physically this condition means that the time Physically this condition means that the time step should be smaller than the time for the step should be smaller than the time for the wave to propagate from one cell to the wave to propagate from one cell to the neighbor oneneighbor one

Slides from 5-15 may be skipped because it was taught in the class

III Fundamentals of the FDTD methodIII Fundamentals of the FDTD method

Dispersion relationDispersion relation

ckkkk

22

z2

y2

x2

In free space (ideal)

In FDTD computation (numerical)

22

z

2

y

2

x

2

tsin

tc

1

2

zk~

sinz

1

2

yk~

siny

1

2

xk~

sinx

1

utcsinArctk

~2

k~vpnum

2

z

2

y

2

x

2

zk~

sinz

1

2

yk~

siny

1

2

xk~

sinx

1u

The numerical medium is dispersive the propagation of the wave varies with frequency and angle

III Fundamentals of the FDTD methodIII Fundamentals of the FDTD method

Dispersion relationDispersion relation

Limitations of FDTD method

1-Grid spacing should be ~λ102-According to Courantrsquos stability condition time step Δt becomes small when FDTDgrid spacing becomes small3-In 3-D simulation simulation time scales like N^4 and required memory size scales like N^34-Application is restricted to relatively small size

21 march 2012 20

Space Domain Discretization

bull Heat Conduction Equation

bull Central-Finite-Difference Approximation

pc

tyxg

y

tyxT

x

tyxT

t

tyxT

)()()()(

2

2

2

2

2

2

2

11

22

112

2

2

)(2

x

T

x

TTT

xOx

TTT

x

T

nx

nji

nji

nji

nji

nji

njin

ji

cc

21 march 2012 21

X

n

jiTn

j1-iT n

j1iT

n

1jiT

n

1jiT

(00) (i0)hellip

(0j)

x

y

(I0)

(0J)

hellip

hellip

hellip

Finite-Difference Formulation of the Heat Conduction on a Chip

bull Space Domainbull Time Domain

21 march 2012 22

Time domain discretization

bull Heat Conduction Equation

ndash Simple Explicit Method ndash Simple Implicit Methodndash Crank-Nicolson Method

gcy

T

x

T

t

TT

p

yxnn

1

)()( 2

2

2

21

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

23

Can we check if a numerical scheme

is stable without computation Von Neumann stability

analysisbull Analyze if (or for which conditions) a

numerical scheme is stable or unstablebull Makes a local analysis coefficients of PDE are

assumed to vary slowly (our example constant)

bull How will unavoidable errors (say rounding errors)evolve in time

John von Neumann1903-1957

24

Von Neumann stability analysis

A numerical scheme is unstable if

Ansatz Wave number k and amplification factor

21 march 2012 25

bull Accuracybull Stability Constraint

bull No matrix inversion but time steps are limited by space discretization

gcy

T

x

T

t

TT

p

ny

nx

nn

1

)()( 2

2

2

21

] [ 22 yxt

2

11122

yxt

Simple Explicit Method

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

21 march 2012 26

Accuracy

Unconditionally Stable

No limits on time step but involves with large scale matrix inversion

gcy

T

x

T

t

TT

p

ny

nx

nn

1

)()( 2

12

2

121

] [ 22 yxt

Simple Implicit Method

11

1

11

12

11

1

11

12

2

2nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

21 march 2012 27

Accuracy

Unconditionally stable

No limits on time step but involves with large scale matrix inversion

gcy

TT

x

TT

t

TT

p

ny

ny

nx

nx

nn

1

)(2)(2 2

212

2

2121

] [ 222 yxt

Crank-Nicolson Method

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

11

1

11

12

11

1

11

12

2

2nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

21 march 2012 28

n

m1616

ex m=4n=4

Total node number N = mn

Matrix size = NxN

Analysis of Crank-Nicolson Method

21 march 2012 29

Alternating Direction Implicit Method

Solves higher dimension problem by successive Lower dimension methods

Accuracy

Unconditionally stable

No limits on time step and no large scale matrix inversion

] [ 222 yxt

21 march2012 30

Step I x-direction implicit y-direction explicitStep II x-direction explicit y-direction implicit

n

Alternating Direction Implicit Method

bull Peaceman-Rachford Algorithmbull Douglas-Gunn Algorithm

21 march 2012 31

bull Step I

bull Step II

gc

tT

rrT

rr

p

ny

yx

xny

yx

x

)2

1)(2

1()2

1)(2

1( 22122

gc

tT

rT

r

p

ny

yn

xx

2)

21()

21( 22

12

gc

tT

rT

r

p

n

xxn

yy

2)

21()

21( 2

1212

Peaceman-Rachford Algorithm

21 march 2012 32

Douglas-Gunn Algorithm

bull Step I

bull Step II

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

12

1

22

gc

tTrTT

rTT

p

nyy

nnxxnn

22

122

1

)(2

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

2

121

2)(

2

21 march 2012 33

Illustration for ADI

X-direction implicit Y-direction implicit

i = 1 2 hellip m

1

njiT

Step I Step II

j = 1

n

2

hellip

1 2 hellip m

11

njiT

11

njiT

j = 1

n

2hellip

21

njiT

21

1

njiT

21

1

njiT

21 march 2012 34

Analysis of ADI Method

mm

Tridiagonal Matrix

Time complexity O(N)

2xnxm = 2nm =2N

2 steps n matrices tridaigonal matrix

X-direction implicit

21

njiT

i = 1 2 hellip mj = 1

n

2

hellip

21

1

njiT

21

1

njiT

ADVANTAGE over conditionally stable

Reduce simulation time

Good accuracy

Size of geometrical feature may be typical of wavelengh

Good geometrical flexibility to allow with corner or high curvature than fdtd

Advantages of the FDTD method over other methodsAdvantages of the FDTD method over other methods

bull It is conceptually simpleIt is conceptually simple

bull The algorithm does not require the formulation of integral equation and relatively complex The algorithm does not require the formulation of integral equation and relatively complex

scatters can be treated without inversion of large matrices scatters can be treated without inversion of large matrices

bull It is simple to implement for complicated inhomogeneous conducting or dielectric structures It is simple to implement for complicated inhomogeneous conducting or dielectric structures

because constitutive parameters can be assigned to each lattice pointbecause constitutive parameters can be assigned to each lattice point

bull Its computer memory requirement is not prohibitive for many complex structures of interest Its computer memory requirement is not prohibitive for many complex structures of interest

bull The algorithm make use of the memory in a simple sequential orderThe algorithm make use of the memory in a simple sequential order

bull It is much easier to obtain frequency domain data from time domain results than the converse It is much easier to obtain frequency domain data from time domain results than the converse

Thus it is more convenient to obtain frequency domain results via time domain when many Thus it is more convenient to obtain frequency domain results via time domain when many

frequencies are involvedfrequencies are involved

Include examples of unconditionally stable fdtd and prove the advantages

Disadvantages Disadvantages

bull Its implementation necessitates modeling object as well as its surroundings Thus the Its implementation necessitates modeling object as well as its surroundings Thus the

required program execution time may be excessiverequired program execution time may be excessive

bull Its accuracy is at least one order of magnitude worse than that of the method of Its accuracy is at least one order of magnitude worse than that of the method of

moments for example moments for example

bull Since the computational meshes are rectangular in shape they do not conform the Since the computational meshes are rectangular in shape they do not conform the

scatterers with curved surfaces as is the case of the cylindrical or spherical boundary scatterers with curved surfaces as is the case of the cylindrical or spherical boundary

Its computer memory requirement is not prohibitive for many complex structures of Its computer memory requirement is not prohibitive for many complex structures of

interest interest

bull As in all finite difference algorithms the field quantities are only known at grid nodes As in all finite difference algorithms the field quantities are only known at grid nodes

Applications of FDTD method

1048708Electromagnetic scattering amp antenna design1048708EMCEMI design1048708Simulation of wave propagation problem1048708Solving partial defferential equation 1048708Waveguide analysis

Stripline amp microstrip line analysishellipetc hellip

Referenceshelliphelliphellip1-Electromagnetic simulation using FDTD method

ldquoDennis M Sillevanrdquo IEEE press series on rf amp mivcrowave technology ldquoRoger DPollard amp Richard

Bootanrdquoseries

2Understanding the FDTD method ldquoJohn BSchneiderrdquo may 8 2011

3 3 D -ADI -Mehod-unconditionaly stable time domain algorithom for solving maxwell equations

ldquoIEEE Transaction on microwave theory amp techniques vol-48No10 october 2010 ldquoTakefumi

IEEE

4High order split step unconditionally stable FDTD method ampnumerical analysis ldquoIEEE

Transactions on antinna amp propagationvol 59 no9sept2011rdquo Yong-Dan Kong amp Qing xin chu

senior memberIEEE

5Genaral Finite DifferenceSchemes for Heat equations ldquoIndian J Pure Applemath

10(2)209-222Febuary 1979rdquoby PS Jain ampDN Holla department of methemetics

IITB 400076

  • Unconditionally Stable FDTD Methods
  • Outlineshelliphelliphelliphellip
  • Outlineshellip
  • Slide 4
  • Computational Electromagnetics
  • Slide 6
  • FDTD Overview ndash Updating Equations
  • Slide 8
  • Slide 9
  • Finite Difference Time Domain Method
  • The Yee Discretization (1966)
  • Slide 12
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Slide 17
  • Slide 18
  • Slide 19
  • Space Domain Discretization
  • Finite-Difference Formulation of the Heat Conduction on a Chip
  • Time domain discretization
  • Slide 23
  • Slide 24
  • Simple Explicit Method
  • Simple Implicit Method
  • Crank-Nicolson Method
  • Analysis of Crank-Nicolson Method
  • Alternating Direction Implicit Method
  • Slide 30
  • Peaceman-Rachford Algorithm
  • Douglas-Gunn Algorithm
  • Illustration for ADI
  • Analysis of ADI Method
  • Slide 35
  • Slide 36
  • Slide 37
  • Slide 38
  • Referenceshelliphelliphellip
  • Slide 40
Page 10: Unconditionally stable fdtd methods

Finite Difference Time Domain Method

bull Divide the interval x into sub-intervals each of width h

bull Divide the interval t into sub-intervals each of width k

bull A grid of points is used forthe finite difference solution

bull Tij represents T(xi tj)bull Replace the derivates by

finite-difference formulas10

t

x

(i j) (i+1 j)

(i j+1)

Ey

Ex

Hz

The Yee Discretization (1966)

Staggered grid in space mdash every field component is stored on a different grid

(i j k) (i+1 j k)

(i j k+1)

(i+1 j+1 k)

(i+1 j+1 k+1)

Ez

Ex

Ey

HyHx

Hz

(i j) (i+1 j)

(i j+1)

Ey

Ex

Hz

The Yee Discretization (1966)

all derivatives become center differenceshellip

H

t

1

E

H zt i

1

2 j

1

2

1

Eyx

Exy

1

Ey (i 1 j 1

2) Ey (i j

1

2)

xEx (i

1

2 j 1) Ex (i

1

2 j)

y

+ O(∆x2) + O(∆y2)

FDTD Overview ndash Updating Equations

yx zx

HE H

t y z

1

05 0505 05

( ) ( )( )

( ) ( 1)( ) ( 1 )

n nx x

x

n nn ny yz z

E i j k E i j ki j k

t

H i j k H i j kH i j k H i j k

y z

FDTD Overview ndash Updating Equations

yx zx

EH E

t z y

05 05

05

( ) ( )( )

( 1) ( ) ( 1 ) ( )

n nx x

x

n n n ny y z z

H i j k H i j ki j k

t

E i j k E i j k E i j k E i j k

z y

FDTD Overview ndash Updating Equations

Express the future components in terms of the past components

05 05

1

05 05

( ) ( 1 )

( ) ( )( ) ( ) ( 1)

n nz z

n nx x n n

x y y

H i j k H i j k

ytE i j k E i j k

i j k H i j k H i j k

z

05

05 05

( 1) ( )

( ) ( )( ) ( 1 ) ( )

n ny y

n nx x n n

x z z

E i j k E i j k

t zH i j k H i j ki j k E i j k E i j k

y

Fundamentals of the FDTD methodFundamentals of the FDTD method

Accuracy and stabilityAccuracy and stability

AccuracyAccuracy 10

StabilityStability

222

max

z

1

y

1

x

1c

1tt

3c

tt max

2D2D

22 y

1

x

1c

1t

1D1Dc

t

Physically this condition means that the time Physically this condition means that the time step should be smaller than the time for the step should be smaller than the time for the wave to propagate from one cell to the wave to propagate from one cell to the neighbor oneneighbor one

Slides from 5-15 may be skipped because it was taught in the class

III Fundamentals of the FDTD methodIII Fundamentals of the FDTD method

Dispersion relationDispersion relation

ckkkk

22

z2

y2

x2

In free space (ideal)

In FDTD computation (numerical)

22

z

2

y

2

x

2

tsin

tc

1

2

zk~

sinz

1

2

yk~

siny

1

2

xk~

sinx

1

utcsinArctk

~2

k~vpnum

2

z

2

y

2

x

2

zk~

sinz

1

2

yk~

siny

1

2

xk~

sinx

1u

The numerical medium is dispersive the propagation of the wave varies with frequency and angle

III Fundamentals of the FDTD methodIII Fundamentals of the FDTD method

Dispersion relationDispersion relation

Limitations of FDTD method

1-Grid spacing should be ~λ102-According to Courantrsquos stability condition time step Δt becomes small when FDTDgrid spacing becomes small3-In 3-D simulation simulation time scales like N^4 and required memory size scales like N^34-Application is restricted to relatively small size

21 march 2012 20

Space Domain Discretization

bull Heat Conduction Equation

bull Central-Finite-Difference Approximation

pc

tyxg

y

tyxT

x

tyxT

t

tyxT

)()()()(

2

2

2

2

2

2

2

11

22

112

2

2

)(2

x

T

x

TTT

xOx

TTT

x

T

nx

nji

nji

nji

nji

nji

njin

ji

cc

21 march 2012 21

X

n

jiTn

j1-iT n

j1iT

n

1jiT

n

1jiT

(00) (i0)hellip

(0j)

x

y

(I0)

(0J)

hellip

hellip

hellip

Finite-Difference Formulation of the Heat Conduction on a Chip

bull Space Domainbull Time Domain

21 march 2012 22

Time domain discretization

bull Heat Conduction Equation

ndash Simple Explicit Method ndash Simple Implicit Methodndash Crank-Nicolson Method

gcy

T

x

T

t

TT

p

yxnn

1

)()( 2

2

2

21

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

23

Can we check if a numerical scheme

is stable without computation Von Neumann stability

analysisbull Analyze if (or for which conditions) a

numerical scheme is stable or unstablebull Makes a local analysis coefficients of PDE are

assumed to vary slowly (our example constant)

bull How will unavoidable errors (say rounding errors)evolve in time

John von Neumann1903-1957

24

Von Neumann stability analysis

A numerical scheme is unstable if

Ansatz Wave number k and amplification factor

21 march 2012 25

bull Accuracybull Stability Constraint

bull No matrix inversion but time steps are limited by space discretization

gcy

T

x

T

t

TT

p

ny

nx

nn

1

)()( 2

2

2

21

] [ 22 yxt

2

11122

yxt

Simple Explicit Method

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

21 march 2012 26

Accuracy

Unconditionally Stable

No limits on time step but involves with large scale matrix inversion

gcy

T

x

T

t

TT

p

ny

nx

nn

1

)()( 2

12

2

121

] [ 22 yxt

Simple Implicit Method

11

1

11

12

11

1

11

12

2

2nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

21 march 2012 27

Accuracy

Unconditionally stable

No limits on time step but involves with large scale matrix inversion

gcy

TT

x

TT

t

TT

p

ny

ny

nx

nx

nn

1

)(2)(2 2

212

2

2121

] [ 222 yxt

Crank-Nicolson Method

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

11

1

11

12

11

1

11

12

2

2nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

21 march 2012 28

n

m1616

ex m=4n=4

Total node number N = mn

Matrix size = NxN

Analysis of Crank-Nicolson Method

21 march 2012 29

Alternating Direction Implicit Method

Solves higher dimension problem by successive Lower dimension methods

Accuracy

Unconditionally stable

No limits on time step and no large scale matrix inversion

] [ 222 yxt

21 march2012 30

Step I x-direction implicit y-direction explicitStep II x-direction explicit y-direction implicit

n

Alternating Direction Implicit Method

bull Peaceman-Rachford Algorithmbull Douglas-Gunn Algorithm

21 march 2012 31

bull Step I

bull Step II

gc

tT

rrT

rr

p

ny

yx

xny

yx

x

)2

1)(2

1()2

1)(2

1( 22122

gc

tT

rT

r

p

ny

yn

xx

2)

21()

21( 22

12

gc

tT

rT

r

p

n

xxn

yy

2)

21()

21( 2

1212

Peaceman-Rachford Algorithm

21 march 2012 32

Douglas-Gunn Algorithm

bull Step I

bull Step II

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

12

1

22

gc

tTrTT

rTT

p

nyy

nnxxnn

22

122

1

)(2

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

2

121

2)(

2

21 march 2012 33

Illustration for ADI

X-direction implicit Y-direction implicit

i = 1 2 hellip m

1

njiT

Step I Step II

j = 1

n

2

hellip

1 2 hellip m

11

njiT

11

njiT

j = 1

n

2hellip

21

njiT

21

1

njiT

21

1

njiT

21 march 2012 34

Analysis of ADI Method

mm

Tridiagonal Matrix

Time complexity O(N)

2xnxm = 2nm =2N

2 steps n matrices tridaigonal matrix

X-direction implicit

21

njiT

i = 1 2 hellip mj = 1

n

2

hellip

21

1

njiT

21

1

njiT

ADVANTAGE over conditionally stable

Reduce simulation time

Good accuracy

Size of geometrical feature may be typical of wavelengh

Good geometrical flexibility to allow with corner or high curvature than fdtd

Advantages of the FDTD method over other methodsAdvantages of the FDTD method over other methods

bull It is conceptually simpleIt is conceptually simple

bull The algorithm does not require the formulation of integral equation and relatively complex The algorithm does not require the formulation of integral equation and relatively complex

scatters can be treated without inversion of large matrices scatters can be treated without inversion of large matrices

bull It is simple to implement for complicated inhomogeneous conducting or dielectric structures It is simple to implement for complicated inhomogeneous conducting or dielectric structures

because constitutive parameters can be assigned to each lattice pointbecause constitutive parameters can be assigned to each lattice point

bull Its computer memory requirement is not prohibitive for many complex structures of interest Its computer memory requirement is not prohibitive for many complex structures of interest

bull The algorithm make use of the memory in a simple sequential orderThe algorithm make use of the memory in a simple sequential order

bull It is much easier to obtain frequency domain data from time domain results than the converse It is much easier to obtain frequency domain data from time domain results than the converse

Thus it is more convenient to obtain frequency domain results via time domain when many Thus it is more convenient to obtain frequency domain results via time domain when many

frequencies are involvedfrequencies are involved

Include examples of unconditionally stable fdtd and prove the advantages

Disadvantages Disadvantages

bull Its implementation necessitates modeling object as well as its surroundings Thus the Its implementation necessitates modeling object as well as its surroundings Thus the

required program execution time may be excessiverequired program execution time may be excessive

bull Its accuracy is at least one order of magnitude worse than that of the method of Its accuracy is at least one order of magnitude worse than that of the method of

moments for example moments for example

bull Since the computational meshes are rectangular in shape they do not conform the Since the computational meshes are rectangular in shape they do not conform the

scatterers with curved surfaces as is the case of the cylindrical or spherical boundary scatterers with curved surfaces as is the case of the cylindrical or spherical boundary

Its computer memory requirement is not prohibitive for many complex structures of Its computer memory requirement is not prohibitive for many complex structures of

interest interest

bull As in all finite difference algorithms the field quantities are only known at grid nodes As in all finite difference algorithms the field quantities are only known at grid nodes

Applications of FDTD method

1048708Electromagnetic scattering amp antenna design1048708EMCEMI design1048708Simulation of wave propagation problem1048708Solving partial defferential equation 1048708Waveguide analysis

Stripline amp microstrip line analysishellipetc hellip

Referenceshelliphelliphellip1-Electromagnetic simulation using FDTD method

ldquoDennis M Sillevanrdquo IEEE press series on rf amp mivcrowave technology ldquoRoger DPollard amp Richard

Bootanrdquoseries

2Understanding the FDTD method ldquoJohn BSchneiderrdquo may 8 2011

3 3 D -ADI -Mehod-unconditionaly stable time domain algorithom for solving maxwell equations

ldquoIEEE Transaction on microwave theory amp techniques vol-48No10 october 2010 ldquoTakefumi

IEEE

4High order split step unconditionally stable FDTD method ampnumerical analysis ldquoIEEE

Transactions on antinna amp propagationvol 59 no9sept2011rdquo Yong-Dan Kong amp Qing xin chu

senior memberIEEE

5Genaral Finite DifferenceSchemes for Heat equations ldquoIndian J Pure Applemath

10(2)209-222Febuary 1979rdquoby PS Jain ampDN Holla department of methemetics

IITB 400076

  • Unconditionally Stable FDTD Methods
  • Outlineshelliphelliphelliphellip
  • Outlineshellip
  • Slide 4
  • Computational Electromagnetics
  • Slide 6
  • FDTD Overview ndash Updating Equations
  • Slide 8
  • Slide 9
  • Finite Difference Time Domain Method
  • The Yee Discretization (1966)
  • Slide 12
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Slide 17
  • Slide 18
  • Slide 19
  • Space Domain Discretization
  • Finite-Difference Formulation of the Heat Conduction on a Chip
  • Time domain discretization
  • Slide 23
  • Slide 24
  • Simple Explicit Method
  • Simple Implicit Method
  • Crank-Nicolson Method
  • Analysis of Crank-Nicolson Method
  • Alternating Direction Implicit Method
  • Slide 30
  • Peaceman-Rachford Algorithm
  • Douglas-Gunn Algorithm
  • Illustration for ADI
  • Analysis of ADI Method
  • Slide 35
  • Slide 36
  • Slide 37
  • Slide 38
  • Referenceshelliphelliphellip
  • Slide 40
Page 11: Unconditionally stable fdtd methods

(i j) (i+1 j)

(i j+1)

Ey

Ex

Hz

The Yee Discretization (1966)

Staggered grid in space mdash every field component is stored on a different grid

(i j k) (i+1 j k)

(i j k+1)

(i+1 j+1 k)

(i+1 j+1 k+1)

Ez

Ex

Ey

HyHx

Hz

(i j) (i+1 j)

(i j+1)

Ey

Ex

Hz

The Yee Discretization (1966)

all derivatives become center differenceshellip

H

t

1

E

H zt i

1

2 j

1

2

1

Eyx

Exy

1

Ey (i 1 j 1

2) Ey (i j

1

2)

xEx (i

1

2 j 1) Ex (i

1

2 j)

y

+ O(∆x2) + O(∆y2)

FDTD Overview ndash Updating Equations

yx zx

HE H

t y z

1

05 0505 05

( ) ( )( )

( ) ( 1)( ) ( 1 )

n nx x

x

n nn ny yz z

E i j k E i j ki j k

t

H i j k H i j kH i j k H i j k

y z

FDTD Overview ndash Updating Equations

yx zx

EH E

t z y

05 05

05

( ) ( )( )

( 1) ( ) ( 1 ) ( )

n nx x

x

n n n ny y z z

H i j k H i j ki j k

t

E i j k E i j k E i j k E i j k

z y

FDTD Overview ndash Updating Equations

Express the future components in terms of the past components

05 05

1

05 05

( ) ( 1 )

( ) ( )( ) ( ) ( 1)

n nz z

n nx x n n

x y y

H i j k H i j k

ytE i j k E i j k

i j k H i j k H i j k

z

05

05 05

( 1) ( )

( ) ( )( ) ( 1 ) ( )

n ny y

n nx x n n

x z z

E i j k E i j k

t zH i j k H i j ki j k E i j k E i j k

y

Fundamentals of the FDTD methodFundamentals of the FDTD method

Accuracy and stabilityAccuracy and stability

AccuracyAccuracy 10

StabilityStability

222

max

z

1

y

1

x

1c

1tt

3c

tt max

2D2D

22 y

1

x

1c

1t

1D1Dc

t

Physically this condition means that the time Physically this condition means that the time step should be smaller than the time for the step should be smaller than the time for the wave to propagate from one cell to the wave to propagate from one cell to the neighbor oneneighbor one

Slides from 5-15 may be skipped because it was taught in the class

III Fundamentals of the FDTD methodIII Fundamentals of the FDTD method

Dispersion relationDispersion relation

ckkkk

22

z2

y2

x2

In free space (ideal)

In FDTD computation (numerical)

22

z

2

y

2

x

2

tsin

tc

1

2

zk~

sinz

1

2

yk~

siny

1

2

xk~

sinx

1

utcsinArctk

~2

k~vpnum

2

z

2

y

2

x

2

zk~

sinz

1

2

yk~

siny

1

2

xk~

sinx

1u

The numerical medium is dispersive the propagation of the wave varies with frequency and angle

III Fundamentals of the FDTD methodIII Fundamentals of the FDTD method

Dispersion relationDispersion relation

Limitations of FDTD method

1-Grid spacing should be ~λ102-According to Courantrsquos stability condition time step Δt becomes small when FDTDgrid spacing becomes small3-In 3-D simulation simulation time scales like N^4 and required memory size scales like N^34-Application is restricted to relatively small size

21 march 2012 20

Space Domain Discretization

bull Heat Conduction Equation

bull Central-Finite-Difference Approximation

pc

tyxg

y

tyxT

x

tyxT

t

tyxT

)()()()(

2

2

2

2

2

2

2

11

22

112

2

2

)(2

x

T

x

TTT

xOx

TTT

x

T

nx

nji

nji

nji

nji

nji

njin

ji

cc

21 march 2012 21

X

n

jiTn

j1-iT n

j1iT

n

1jiT

n

1jiT

(00) (i0)hellip

(0j)

x

y

(I0)

(0J)

hellip

hellip

hellip

Finite-Difference Formulation of the Heat Conduction on a Chip

bull Space Domainbull Time Domain

21 march 2012 22

Time domain discretization

bull Heat Conduction Equation

ndash Simple Explicit Method ndash Simple Implicit Methodndash Crank-Nicolson Method

gcy

T

x

T

t

TT

p

yxnn

1

)()( 2

2

2

21

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

23

Can we check if a numerical scheme

is stable without computation Von Neumann stability

analysisbull Analyze if (or for which conditions) a

numerical scheme is stable or unstablebull Makes a local analysis coefficients of PDE are

assumed to vary slowly (our example constant)

bull How will unavoidable errors (say rounding errors)evolve in time

John von Neumann1903-1957

24

Von Neumann stability analysis

A numerical scheme is unstable if

Ansatz Wave number k and amplification factor

21 march 2012 25

bull Accuracybull Stability Constraint

bull No matrix inversion but time steps are limited by space discretization

gcy

T

x

T

t

TT

p

ny

nx

nn

1

)()( 2

2

2

21

] [ 22 yxt

2

11122

yxt

Simple Explicit Method

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

21 march 2012 26

Accuracy

Unconditionally Stable

No limits on time step but involves with large scale matrix inversion

gcy

T

x

T

t

TT

p

ny

nx

nn

1

)()( 2

12

2

121

] [ 22 yxt

Simple Implicit Method

11

1

11

12

11

1

11

12

2

2nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

21 march 2012 27

Accuracy

Unconditionally stable

No limits on time step but involves with large scale matrix inversion

gcy

TT

x

TT

t

TT

p

ny

ny

nx

nx

nn

1

)(2)(2 2

212

2

2121

] [ 222 yxt

Crank-Nicolson Method

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

11

1

11

12

11

1

11

12

2

2nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

21 march 2012 28

n

m1616

ex m=4n=4

Total node number N = mn

Matrix size = NxN

Analysis of Crank-Nicolson Method

21 march 2012 29

Alternating Direction Implicit Method

Solves higher dimension problem by successive Lower dimension methods

Accuracy

Unconditionally stable

No limits on time step and no large scale matrix inversion

] [ 222 yxt

21 march2012 30

Step I x-direction implicit y-direction explicitStep II x-direction explicit y-direction implicit

n

Alternating Direction Implicit Method

bull Peaceman-Rachford Algorithmbull Douglas-Gunn Algorithm

21 march 2012 31

bull Step I

bull Step II

gc

tT

rrT

rr

p

ny

yx

xny

yx

x

)2

1)(2

1()2

1)(2

1( 22122

gc

tT

rT

r

p

ny

yn

xx

2)

21()

21( 22

12

gc

tT

rT

r

p

n

xxn

yy

2)

21()

21( 2

1212

Peaceman-Rachford Algorithm

21 march 2012 32

Douglas-Gunn Algorithm

bull Step I

bull Step II

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

12

1

22

gc

tTrTT

rTT

p

nyy

nnxxnn

22

122

1

)(2

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

2

121

2)(

2

21 march 2012 33

Illustration for ADI

X-direction implicit Y-direction implicit

i = 1 2 hellip m

1

njiT

Step I Step II

j = 1

n

2

hellip

1 2 hellip m

11

njiT

11

njiT

j = 1

n

2hellip

21

njiT

21

1

njiT

21

1

njiT

21 march 2012 34

Analysis of ADI Method

mm

Tridiagonal Matrix

Time complexity O(N)

2xnxm = 2nm =2N

2 steps n matrices tridaigonal matrix

X-direction implicit

21

njiT

i = 1 2 hellip mj = 1

n

2

hellip

21

1

njiT

21

1

njiT

ADVANTAGE over conditionally stable

Reduce simulation time

Good accuracy

Size of geometrical feature may be typical of wavelengh

Good geometrical flexibility to allow with corner or high curvature than fdtd

Advantages of the FDTD method over other methodsAdvantages of the FDTD method over other methods

bull It is conceptually simpleIt is conceptually simple

bull The algorithm does not require the formulation of integral equation and relatively complex The algorithm does not require the formulation of integral equation and relatively complex

scatters can be treated without inversion of large matrices scatters can be treated without inversion of large matrices

bull It is simple to implement for complicated inhomogeneous conducting or dielectric structures It is simple to implement for complicated inhomogeneous conducting or dielectric structures

because constitutive parameters can be assigned to each lattice pointbecause constitutive parameters can be assigned to each lattice point

bull Its computer memory requirement is not prohibitive for many complex structures of interest Its computer memory requirement is not prohibitive for many complex structures of interest

bull The algorithm make use of the memory in a simple sequential orderThe algorithm make use of the memory in a simple sequential order

bull It is much easier to obtain frequency domain data from time domain results than the converse It is much easier to obtain frequency domain data from time domain results than the converse

Thus it is more convenient to obtain frequency domain results via time domain when many Thus it is more convenient to obtain frequency domain results via time domain when many

frequencies are involvedfrequencies are involved

Include examples of unconditionally stable fdtd and prove the advantages

Disadvantages Disadvantages

bull Its implementation necessitates modeling object as well as its surroundings Thus the Its implementation necessitates modeling object as well as its surroundings Thus the

required program execution time may be excessiverequired program execution time may be excessive

bull Its accuracy is at least one order of magnitude worse than that of the method of Its accuracy is at least one order of magnitude worse than that of the method of

moments for example moments for example

bull Since the computational meshes are rectangular in shape they do not conform the Since the computational meshes are rectangular in shape they do not conform the

scatterers with curved surfaces as is the case of the cylindrical or spherical boundary scatterers with curved surfaces as is the case of the cylindrical or spherical boundary

Its computer memory requirement is not prohibitive for many complex structures of Its computer memory requirement is not prohibitive for many complex structures of

interest interest

bull As in all finite difference algorithms the field quantities are only known at grid nodes As in all finite difference algorithms the field quantities are only known at grid nodes

Applications of FDTD method

1048708Electromagnetic scattering amp antenna design1048708EMCEMI design1048708Simulation of wave propagation problem1048708Solving partial defferential equation 1048708Waveguide analysis

Stripline amp microstrip line analysishellipetc hellip

Referenceshelliphelliphellip1-Electromagnetic simulation using FDTD method

ldquoDennis M Sillevanrdquo IEEE press series on rf amp mivcrowave technology ldquoRoger DPollard amp Richard

Bootanrdquoseries

2Understanding the FDTD method ldquoJohn BSchneiderrdquo may 8 2011

3 3 D -ADI -Mehod-unconditionaly stable time domain algorithom for solving maxwell equations

ldquoIEEE Transaction on microwave theory amp techniques vol-48No10 october 2010 ldquoTakefumi

IEEE

4High order split step unconditionally stable FDTD method ampnumerical analysis ldquoIEEE

Transactions on antinna amp propagationvol 59 no9sept2011rdquo Yong-Dan Kong amp Qing xin chu

senior memberIEEE

5Genaral Finite DifferenceSchemes for Heat equations ldquoIndian J Pure Applemath

10(2)209-222Febuary 1979rdquoby PS Jain ampDN Holla department of methemetics

IITB 400076

  • Unconditionally Stable FDTD Methods
  • Outlineshelliphelliphelliphellip
  • Outlineshellip
  • Slide 4
  • Computational Electromagnetics
  • Slide 6
  • FDTD Overview ndash Updating Equations
  • Slide 8
  • Slide 9
  • Finite Difference Time Domain Method
  • The Yee Discretization (1966)
  • Slide 12
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Slide 17
  • Slide 18
  • Slide 19
  • Space Domain Discretization
  • Finite-Difference Formulation of the Heat Conduction on a Chip
  • Time domain discretization
  • Slide 23
  • Slide 24
  • Simple Explicit Method
  • Simple Implicit Method
  • Crank-Nicolson Method
  • Analysis of Crank-Nicolson Method
  • Alternating Direction Implicit Method
  • Slide 30
  • Peaceman-Rachford Algorithm
  • Douglas-Gunn Algorithm
  • Illustration for ADI
  • Analysis of ADI Method
  • Slide 35
  • Slide 36
  • Slide 37
  • Slide 38
  • Referenceshelliphelliphellip
  • Slide 40
Page 12: Unconditionally stable fdtd methods

(i j) (i+1 j)

(i j+1)

Ey

Ex

Hz

The Yee Discretization (1966)

all derivatives become center differenceshellip

H

t

1

E

H zt i

1

2 j

1

2

1

Eyx

Exy

1

Ey (i 1 j 1

2) Ey (i j

1

2)

xEx (i

1

2 j 1) Ex (i

1

2 j)

y

+ O(∆x2) + O(∆y2)

FDTD Overview ndash Updating Equations

yx zx

HE H

t y z

1

05 0505 05

( ) ( )( )

( ) ( 1)( ) ( 1 )

n nx x

x

n nn ny yz z

E i j k E i j ki j k

t

H i j k H i j kH i j k H i j k

y z

FDTD Overview ndash Updating Equations

yx zx

EH E

t z y

05 05

05

( ) ( )( )

( 1) ( ) ( 1 ) ( )

n nx x

x

n n n ny y z z

H i j k H i j ki j k

t

E i j k E i j k E i j k E i j k

z y

FDTD Overview ndash Updating Equations

Express the future components in terms of the past components

05 05

1

05 05

( ) ( 1 )

( ) ( )( ) ( ) ( 1)

n nz z

n nx x n n

x y y

H i j k H i j k

ytE i j k E i j k

i j k H i j k H i j k

z

05

05 05

( 1) ( )

( ) ( )( ) ( 1 ) ( )

n ny y

n nx x n n

x z z

E i j k E i j k

t zH i j k H i j ki j k E i j k E i j k

y

Fundamentals of the FDTD methodFundamentals of the FDTD method

Accuracy and stabilityAccuracy and stability

AccuracyAccuracy 10

StabilityStability

222

max

z

1

y

1

x

1c

1tt

3c

tt max

2D2D

22 y

1

x

1c

1t

1D1Dc

t

Physically this condition means that the time Physically this condition means that the time step should be smaller than the time for the step should be smaller than the time for the wave to propagate from one cell to the wave to propagate from one cell to the neighbor oneneighbor one

Slides from 5-15 may be skipped because it was taught in the class

III Fundamentals of the FDTD methodIII Fundamentals of the FDTD method

Dispersion relationDispersion relation

ckkkk

22

z2

y2

x2

In free space (ideal)

In FDTD computation (numerical)

22

z

2

y

2

x

2

tsin

tc

1

2

zk~

sinz

1

2

yk~

siny

1

2

xk~

sinx

1

utcsinArctk

~2

k~vpnum

2

z

2

y

2

x

2

zk~

sinz

1

2

yk~

siny

1

2

xk~

sinx

1u

The numerical medium is dispersive the propagation of the wave varies with frequency and angle

III Fundamentals of the FDTD methodIII Fundamentals of the FDTD method

Dispersion relationDispersion relation

Limitations of FDTD method

1-Grid spacing should be ~λ102-According to Courantrsquos stability condition time step Δt becomes small when FDTDgrid spacing becomes small3-In 3-D simulation simulation time scales like N^4 and required memory size scales like N^34-Application is restricted to relatively small size

21 march 2012 20

Space Domain Discretization

bull Heat Conduction Equation

bull Central-Finite-Difference Approximation

pc

tyxg

y

tyxT

x

tyxT

t

tyxT

)()()()(

2

2

2

2

2

2

2

11

22

112

2

2

)(2

x

T

x

TTT

xOx

TTT

x

T

nx

nji

nji

nji

nji

nji

njin

ji

cc

21 march 2012 21

X

n

jiTn

j1-iT n

j1iT

n

1jiT

n

1jiT

(00) (i0)hellip

(0j)

x

y

(I0)

(0J)

hellip

hellip

hellip

Finite-Difference Formulation of the Heat Conduction on a Chip

bull Space Domainbull Time Domain

21 march 2012 22

Time domain discretization

bull Heat Conduction Equation

ndash Simple Explicit Method ndash Simple Implicit Methodndash Crank-Nicolson Method

gcy

T

x

T

t

TT

p

yxnn

1

)()( 2

2

2

21

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

23

Can we check if a numerical scheme

is stable without computation Von Neumann stability

analysisbull Analyze if (or for which conditions) a

numerical scheme is stable or unstablebull Makes a local analysis coefficients of PDE are

assumed to vary slowly (our example constant)

bull How will unavoidable errors (say rounding errors)evolve in time

John von Neumann1903-1957

24

Von Neumann stability analysis

A numerical scheme is unstable if

Ansatz Wave number k and amplification factor

21 march 2012 25

bull Accuracybull Stability Constraint

bull No matrix inversion but time steps are limited by space discretization

gcy

T

x

T

t

TT

p

ny

nx

nn

1

)()( 2

2

2

21

] [ 22 yxt

2

11122

yxt

Simple Explicit Method

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

21 march 2012 26

Accuracy

Unconditionally Stable

No limits on time step but involves with large scale matrix inversion

gcy

T

x

T

t

TT

p

ny

nx

nn

1

)()( 2

12

2

121

] [ 22 yxt

Simple Implicit Method

11

1

11

12

11

1

11

12

2

2nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

21 march 2012 27

Accuracy

Unconditionally stable

No limits on time step but involves with large scale matrix inversion

gcy

TT

x

TT

t

TT

p

ny

ny

nx

nx

nn

1

)(2)(2 2

212

2

2121

] [ 222 yxt

Crank-Nicolson Method

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

11

1

11

12

11

1

11

12

2

2nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

21 march 2012 28

n

m1616

ex m=4n=4

Total node number N = mn

Matrix size = NxN

Analysis of Crank-Nicolson Method

21 march 2012 29

Alternating Direction Implicit Method

Solves higher dimension problem by successive Lower dimension methods

Accuracy

Unconditionally stable

No limits on time step and no large scale matrix inversion

] [ 222 yxt

21 march2012 30

Step I x-direction implicit y-direction explicitStep II x-direction explicit y-direction implicit

n

Alternating Direction Implicit Method

bull Peaceman-Rachford Algorithmbull Douglas-Gunn Algorithm

21 march 2012 31

bull Step I

bull Step II

gc

tT

rrT

rr

p

ny

yx

xny

yx

x

)2

1)(2

1()2

1)(2

1( 22122

gc

tT

rT

r

p

ny

yn

xx

2)

21()

21( 22

12

gc

tT

rT

r

p

n

xxn

yy

2)

21()

21( 2

1212

Peaceman-Rachford Algorithm

21 march 2012 32

Douglas-Gunn Algorithm

bull Step I

bull Step II

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

12

1

22

gc

tTrTT

rTT

p

nyy

nnxxnn

22

122

1

)(2

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

2

121

2)(

2

21 march 2012 33

Illustration for ADI

X-direction implicit Y-direction implicit

i = 1 2 hellip m

1

njiT

Step I Step II

j = 1

n

2

hellip

1 2 hellip m

11

njiT

11

njiT

j = 1

n

2hellip

21

njiT

21

1

njiT

21

1

njiT

21 march 2012 34

Analysis of ADI Method

mm

Tridiagonal Matrix

Time complexity O(N)

2xnxm = 2nm =2N

2 steps n matrices tridaigonal matrix

X-direction implicit

21

njiT

i = 1 2 hellip mj = 1

n

2

hellip

21

1

njiT

21

1

njiT

ADVANTAGE over conditionally stable

Reduce simulation time

Good accuracy

Size of geometrical feature may be typical of wavelengh

Good geometrical flexibility to allow with corner or high curvature than fdtd

Advantages of the FDTD method over other methodsAdvantages of the FDTD method over other methods

bull It is conceptually simpleIt is conceptually simple

bull The algorithm does not require the formulation of integral equation and relatively complex The algorithm does not require the formulation of integral equation and relatively complex

scatters can be treated without inversion of large matrices scatters can be treated without inversion of large matrices

bull It is simple to implement for complicated inhomogeneous conducting or dielectric structures It is simple to implement for complicated inhomogeneous conducting or dielectric structures

because constitutive parameters can be assigned to each lattice pointbecause constitutive parameters can be assigned to each lattice point

bull Its computer memory requirement is not prohibitive for many complex structures of interest Its computer memory requirement is not prohibitive for many complex structures of interest

bull The algorithm make use of the memory in a simple sequential orderThe algorithm make use of the memory in a simple sequential order

bull It is much easier to obtain frequency domain data from time domain results than the converse It is much easier to obtain frequency domain data from time domain results than the converse

Thus it is more convenient to obtain frequency domain results via time domain when many Thus it is more convenient to obtain frequency domain results via time domain when many

frequencies are involvedfrequencies are involved

Include examples of unconditionally stable fdtd and prove the advantages

Disadvantages Disadvantages

bull Its implementation necessitates modeling object as well as its surroundings Thus the Its implementation necessitates modeling object as well as its surroundings Thus the

required program execution time may be excessiverequired program execution time may be excessive

bull Its accuracy is at least one order of magnitude worse than that of the method of Its accuracy is at least one order of magnitude worse than that of the method of

moments for example moments for example

bull Since the computational meshes are rectangular in shape they do not conform the Since the computational meshes are rectangular in shape they do not conform the

scatterers with curved surfaces as is the case of the cylindrical or spherical boundary scatterers with curved surfaces as is the case of the cylindrical or spherical boundary

Its computer memory requirement is not prohibitive for many complex structures of Its computer memory requirement is not prohibitive for many complex structures of

interest interest

bull As in all finite difference algorithms the field quantities are only known at grid nodes As in all finite difference algorithms the field quantities are only known at grid nodes

Applications of FDTD method

1048708Electromagnetic scattering amp antenna design1048708EMCEMI design1048708Simulation of wave propagation problem1048708Solving partial defferential equation 1048708Waveguide analysis

Stripline amp microstrip line analysishellipetc hellip

Referenceshelliphelliphellip1-Electromagnetic simulation using FDTD method

ldquoDennis M Sillevanrdquo IEEE press series on rf amp mivcrowave technology ldquoRoger DPollard amp Richard

Bootanrdquoseries

2Understanding the FDTD method ldquoJohn BSchneiderrdquo may 8 2011

3 3 D -ADI -Mehod-unconditionaly stable time domain algorithom for solving maxwell equations

ldquoIEEE Transaction on microwave theory amp techniques vol-48No10 october 2010 ldquoTakefumi

IEEE

4High order split step unconditionally stable FDTD method ampnumerical analysis ldquoIEEE

Transactions on antinna amp propagationvol 59 no9sept2011rdquo Yong-Dan Kong amp Qing xin chu

senior memberIEEE

5Genaral Finite DifferenceSchemes for Heat equations ldquoIndian J Pure Applemath

10(2)209-222Febuary 1979rdquoby PS Jain ampDN Holla department of methemetics

IITB 400076

  • Unconditionally Stable FDTD Methods
  • Outlineshelliphelliphelliphellip
  • Outlineshellip
  • Slide 4
  • Computational Electromagnetics
  • Slide 6
  • FDTD Overview ndash Updating Equations
  • Slide 8
  • Slide 9
  • Finite Difference Time Domain Method
  • The Yee Discretization (1966)
  • Slide 12
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Slide 17
  • Slide 18
  • Slide 19
  • Space Domain Discretization
  • Finite-Difference Formulation of the Heat Conduction on a Chip
  • Time domain discretization
  • Slide 23
  • Slide 24
  • Simple Explicit Method
  • Simple Implicit Method
  • Crank-Nicolson Method
  • Analysis of Crank-Nicolson Method
  • Alternating Direction Implicit Method
  • Slide 30
  • Peaceman-Rachford Algorithm
  • Douglas-Gunn Algorithm
  • Illustration for ADI
  • Analysis of ADI Method
  • Slide 35
  • Slide 36
  • Slide 37
  • Slide 38
  • Referenceshelliphelliphellip
  • Slide 40
Page 13: Unconditionally stable fdtd methods

FDTD Overview ndash Updating Equations

yx zx

HE H

t y z

1

05 0505 05

( ) ( )( )

( ) ( 1)( ) ( 1 )

n nx x

x

n nn ny yz z

E i j k E i j ki j k

t

H i j k H i j kH i j k H i j k

y z

FDTD Overview ndash Updating Equations

yx zx

EH E

t z y

05 05

05

( ) ( )( )

( 1) ( ) ( 1 ) ( )

n nx x

x

n n n ny y z z

H i j k H i j ki j k

t

E i j k E i j k E i j k E i j k

z y

FDTD Overview ndash Updating Equations

Express the future components in terms of the past components

05 05

1

05 05

( ) ( 1 )

( ) ( )( ) ( ) ( 1)

n nz z

n nx x n n

x y y

H i j k H i j k

ytE i j k E i j k

i j k H i j k H i j k

z

05

05 05

( 1) ( )

( ) ( )( ) ( 1 ) ( )

n ny y

n nx x n n

x z z

E i j k E i j k

t zH i j k H i j ki j k E i j k E i j k

y

Fundamentals of the FDTD methodFundamentals of the FDTD method

Accuracy and stabilityAccuracy and stability

AccuracyAccuracy 10

StabilityStability

222

max

z

1

y

1

x

1c

1tt

3c

tt max

2D2D

22 y

1

x

1c

1t

1D1Dc

t

Physically this condition means that the time Physically this condition means that the time step should be smaller than the time for the step should be smaller than the time for the wave to propagate from one cell to the wave to propagate from one cell to the neighbor oneneighbor one

Slides from 5-15 may be skipped because it was taught in the class

III Fundamentals of the FDTD methodIII Fundamentals of the FDTD method

Dispersion relationDispersion relation

ckkkk

22

z2

y2

x2

In free space (ideal)

In FDTD computation (numerical)

22

z

2

y

2

x

2

tsin

tc

1

2

zk~

sinz

1

2

yk~

siny

1

2

xk~

sinx

1

utcsinArctk

~2

k~vpnum

2

z

2

y

2

x

2

zk~

sinz

1

2

yk~

siny

1

2

xk~

sinx

1u

The numerical medium is dispersive the propagation of the wave varies with frequency and angle

III Fundamentals of the FDTD methodIII Fundamentals of the FDTD method

Dispersion relationDispersion relation

Limitations of FDTD method

1-Grid spacing should be ~λ102-According to Courantrsquos stability condition time step Δt becomes small when FDTDgrid spacing becomes small3-In 3-D simulation simulation time scales like N^4 and required memory size scales like N^34-Application is restricted to relatively small size

21 march 2012 20

Space Domain Discretization

bull Heat Conduction Equation

bull Central-Finite-Difference Approximation

pc

tyxg

y

tyxT

x

tyxT

t

tyxT

)()()()(

2

2

2

2

2

2

2

11

22

112

2

2

)(2

x

T

x

TTT

xOx

TTT

x

T

nx

nji

nji

nji

nji

nji

njin

ji

cc

21 march 2012 21

X

n

jiTn

j1-iT n

j1iT

n

1jiT

n

1jiT

(00) (i0)hellip

(0j)

x

y

(I0)

(0J)

hellip

hellip

hellip

Finite-Difference Formulation of the Heat Conduction on a Chip

bull Space Domainbull Time Domain

21 march 2012 22

Time domain discretization

bull Heat Conduction Equation

ndash Simple Explicit Method ndash Simple Implicit Methodndash Crank-Nicolson Method

gcy

T

x

T

t

TT

p

yxnn

1

)()( 2

2

2

21

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

23

Can we check if a numerical scheme

is stable without computation Von Neumann stability

analysisbull Analyze if (or for which conditions) a

numerical scheme is stable or unstablebull Makes a local analysis coefficients of PDE are

assumed to vary slowly (our example constant)

bull How will unavoidable errors (say rounding errors)evolve in time

John von Neumann1903-1957

24

Von Neumann stability analysis

A numerical scheme is unstable if

Ansatz Wave number k and amplification factor

21 march 2012 25

bull Accuracybull Stability Constraint

bull No matrix inversion but time steps are limited by space discretization

gcy

T

x

T

t

TT

p

ny

nx

nn

1

)()( 2

2

2

21

] [ 22 yxt

2

11122

yxt

Simple Explicit Method

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

21 march 2012 26

Accuracy

Unconditionally Stable

No limits on time step but involves with large scale matrix inversion

gcy

T

x

T

t

TT

p

ny

nx

nn

1

)()( 2

12

2

121

] [ 22 yxt

Simple Implicit Method

11

1

11

12

11

1

11

12

2

2nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

21 march 2012 27

Accuracy

Unconditionally stable

No limits on time step but involves with large scale matrix inversion

gcy

TT

x

TT

t

TT

p

ny

ny

nx

nx

nn

1

)(2)(2 2

212

2

2121

] [ 222 yxt

Crank-Nicolson Method

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

11

1

11

12

11

1

11

12

2

2nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

21 march 2012 28

n

m1616

ex m=4n=4

Total node number N = mn

Matrix size = NxN

Analysis of Crank-Nicolson Method

21 march 2012 29

Alternating Direction Implicit Method

Solves higher dimension problem by successive Lower dimension methods

Accuracy

Unconditionally stable

No limits on time step and no large scale matrix inversion

] [ 222 yxt

21 march2012 30

Step I x-direction implicit y-direction explicitStep II x-direction explicit y-direction implicit

n

Alternating Direction Implicit Method

bull Peaceman-Rachford Algorithmbull Douglas-Gunn Algorithm

21 march 2012 31

bull Step I

bull Step II

gc

tT

rrT

rr

p

ny

yx

xny

yx

x

)2

1)(2

1()2

1)(2

1( 22122

gc

tT

rT

r

p

ny

yn

xx

2)

21()

21( 22

12

gc

tT

rT

r

p

n

xxn

yy

2)

21()

21( 2

1212

Peaceman-Rachford Algorithm

21 march 2012 32

Douglas-Gunn Algorithm

bull Step I

bull Step II

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

12

1

22

gc

tTrTT

rTT

p

nyy

nnxxnn

22

122

1

)(2

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

2

121

2)(

2

21 march 2012 33

Illustration for ADI

X-direction implicit Y-direction implicit

i = 1 2 hellip m

1

njiT

Step I Step II

j = 1

n

2

hellip

1 2 hellip m

11

njiT

11

njiT

j = 1

n

2hellip

21

njiT

21

1

njiT

21

1

njiT

21 march 2012 34

Analysis of ADI Method

mm

Tridiagonal Matrix

Time complexity O(N)

2xnxm = 2nm =2N

2 steps n matrices tridaigonal matrix

X-direction implicit

21

njiT

i = 1 2 hellip mj = 1

n

2

hellip

21

1

njiT

21

1

njiT

ADVANTAGE over conditionally stable

Reduce simulation time

Good accuracy

Size of geometrical feature may be typical of wavelengh

Good geometrical flexibility to allow with corner or high curvature than fdtd

Advantages of the FDTD method over other methodsAdvantages of the FDTD method over other methods

bull It is conceptually simpleIt is conceptually simple

bull The algorithm does not require the formulation of integral equation and relatively complex The algorithm does not require the formulation of integral equation and relatively complex

scatters can be treated without inversion of large matrices scatters can be treated without inversion of large matrices

bull It is simple to implement for complicated inhomogeneous conducting or dielectric structures It is simple to implement for complicated inhomogeneous conducting or dielectric structures

because constitutive parameters can be assigned to each lattice pointbecause constitutive parameters can be assigned to each lattice point

bull Its computer memory requirement is not prohibitive for many complex structures of interest Its computer memory requirement is not prohibitive for many complex structures of interest

bull The algorithm make use of the memory in a simple sequential orderThe algorithm make use of the memory in a simple sequential order

bull It is much easier to obtain frequency domain data from time domain results than the converse It is much easier to obtain frequency domain data from time domain results than the converse

Thus it is more convenient to obtain frequency domain results via time domain when many Thus it is more convenient to obtain frequency domain results via time domain when many

frequencies are involvedfrequencies are involved

Include examples of unconditionally stable fdtd and prove the advantages

Disadvantages Disadvantages

bull Its implementation necessitates modeling object as well as its surroundings Thus the Its implementation necessitates modeling object as well as its surroundings Thus the

required program execution time may be excessiverequired program execution time may be excessive

bull Its accuracy is at least one order of magnitude worse than that of the method of Its accuracy is at least one order of magnitude worse than that of the method of

moments for example moments for example

bull Since the computational meshes are rectangular in shape they do not conform the Since the computational meshes are rectangular in shape they do not conform the

scatterers with curved surfaces as is the case of the cylindrical or spherical boundary scatterers with curved surfaces as is the case of the cylindrical or spherical boundary

Its computer memory requirement is not prohibitive for many complex structures of Its computer memory requirement is not prohibitive for many complex structures of

interest interest

bull As in all finite difference algorithms the field quantities are only known at grid nodes As in all finite difference algorithms the field quantities are only known at grid nodes

Applications of FDTD method

1048708Electromagnetic scattering amp antenna design1048708EMCEMI design1048708Simulation of wave propagation problem1048708Solving partial defferential equation 1048708Waveguide analysis

Stripline amp microstrip line analysishellipetc hellip

Referenceshelliphelliphellip1-Electromagnetic simulation using FDTD method

ldquoDennis M Sillevanrdquo IEEE press series on rf amp mivcrowave technology ldquoRoger DPollard amp Richard

Bootanrdquoseries

2Understanding the FDTD method ldquoJohn BSchneiderrdquo may 8 2011

3 3 D -ADI -Mehod-unconditionaly stable time domain algorithom for solving maxwell equations

ldquoIEEE Transaction on microwave theory amp techniques vol-48No10 october 2010 ldquoTakefumi

IEEE

4High order split step unconditionally stable FDTD method ampnumerical analysis ldquoIEEE

Transactions on antinna amp propagationvol 59 no9sept2011rdquo Yong-Dan Kong amp Qing xin chu

senior memberIEEE

5Genaral Finite DifferenceSchemes for Heat equations ldquoIndian J Pure Applemath

10(2)209-222Febuary 1979rdquoby PS Jain ampDN Holla department of methemetics

IITB 400076

  • Unconditionally Stable FDTD Methods
  • Outlineshelliphelliphelliphellip
  • Outlineshellip
  • Slide 4
  • Computational Electromagnetics
  • Slide 6
  • FDTD Overview ndash Updating Equations
  • Slide 8
  • Slide 9
  • Finite Difference Time Domain Method
  • The Yee Discretization (1966)
  • Slide 12
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Slide 17
  • Slide 18
  • Slide 19
  • Space Domain Discretization
  • Finite-Difference Formulation of the Heat Conduction on a Chip
  • Time domain discretization
  • Slide 23
  • Slide 24
  • Simple Explicit Method
  • Simple Implicit Method
  • Crank-Nicolson Method
  • Analysis of Crank-Nicolson Method
  • Alternating Direction Implicit Method
  • Slide 30
  • Peaceman-Rachford Algorithm
  • Douglas-Gunn Algorithm
  • Illustration for ADI
  • Analysis of ADI Method
  • Slide 35
  • Slide 36
  • Slide 37
  • Slide 38
  • Referenceshelliphelliphellip
  • Slide 40
Page 14: Unconditionally stable fdtd methods

FDTD Overview ndash Updating Equations

yx zx

EH E

t z y

05 05

05

( ) ( )( )

( 1) ( ) ( 1 ) ( )

n nx x

x

n n n ny y z z

H i j k H i j ki j k

t

E i j k E i j k E i j k E i j k

z y

FDTD Overview ndash Updating Equations

Express the future components in terms of the past components

05 05

1

05 05

( ) ( 1 )

( ) ( )( ) ( ) ( 1)

n nz z

n nx x n n

x y y

H i j k H i j k

ytE i j k E i j k

i j k H i j k H i j k

z

05

05 05

( 1) ( )

( ) ( )( ) ( 1 ) ( )

n ny y

n nx x n n

x z z

E i j k E i j k

t zH i j k H i j ki j k E i j k E i j k

y

Fundamentals of the FDTD methodFundamentals of the FDTD method

Accuracy and stabilityAccuracy and stability

AccuracyAccuracy 10

StabilityStability

222

max

z

1

y

1

x

1c

1tt

3c

tt max

2D2D

22 y

1

x

1c

1t

1D1Dc

t

Physically this condition means that the time Physically this condition means that the time step should be smaller than the time for the step should be smaller than the time for the wave to propagate from one cell to the wave to propagate from one cell to the neighbor oneneighbor one

Slides from 5-15 may be skipped because it was taught in the class

III Fundamentals of the FDTD methodIII Fundamentals of the FDTD method

Dispersion relationDispersion relation

ckkkk

22

z2

y2

x2

In free space (ideal)

In FDTD computation (numerical)

22

z

2

y

2

x

2

tsin

tc

1

2

zk~

sinz

1

2

yk~

siny

1

2

xk~

sinx

1

utcsinArctk

~2

k~vpnum

2

z

2

y

2

x

2

zk~

sinz

1

2

yk~

siny

1

2

xk~

sinx

1u

The numerical medium is dispersive the propagation of the wave varies with frequency and angle

III Fundamentals of the FDTD methodIII Fundamentals of the FDTD method

Dispersion relationDispersion relation

Limitations of FDTD method

1-Grid spacing should be ~λ102-According to Courantrsquos stability condition time step Δt becomes small when FDTDgrid spacing becomes small3-In 3-D simulation simulation time scales like N^4 and required memory size scales like N^34-Application is restricted to relatively small size

21 march 2012 20

Space Domain Discretization

bull Heat Conduction Equation

bull Central-Finite-Difference Approximation

pc

tyxg

y

tyxT

x

tyxT

t

tyxT

)()()()(

2

2

2

2

2

2

2

11

22

112

2

2

)(2

x

T

x

TTT

xOx

TTT

x

T

nx

nji

nji

nji

nji

nji

njin

ji

cc

21 march 2012 21

X

n

jiTn

j1-iT n

j1iT

n

1jiT

n

1jiT

(00) (i0)hellip

(0j)

x

y

(I0)

(0J)

hellip

hellip

hellip

Finite-Difference Formulation of the Heat Conduction on a Chip

bull Space Domainbull Time Domain

21 march 2012 22

Time domain discretization

bull Heat Conduction Equation

ndash Simple Explicit Method ndash Simple Implicit Methodndash Crank-Nicolson Method

gcy

T

x

T

t

TT

p

yxnn

1

)()( 2

2

2

21

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

23

Can we check if a numerical scheme

is stable without computation Von Neumann stability

analysisbull Analyze if (or for which conditions) a

numerical scheme is stable or unstablebull Makes a local analysis coefficients of PDE are

assumed to vary slowly (our example constant)

bull How will unavoidable errors (say rounding errors)evolve in time

John von Neumann1903-1957

24

Von Neumann stability analysis

A numerical scheme is unstable if

Ansatz Wave number k and amplification factor

21 march 2012 25

bull Accuracybull Stability Constraint

bull No matrix inversion but time steps are limited by space discretization

gcy

T

x

T

t

TT

p

ny

nx

nn

1

)()( 2

2

2

21

] [ 22 yxt

2

11122

yxt

Simple Explicit Method

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

21 march 2012 26

Accuracy

Unconditionally Stable

No limits on time step but involves with large scale matrix inversion

gcy

T

x

T

t

TT

p

ny

nx

nn

1

)()( 2

12

2

121

] [ 22 yxt

Simple Implicit Method

11

1

11

12

11

1

11

12

2

2nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

21 march 2012 27

Accuracy

Unconditionally stable

No limits on time step but involves with large scale matrix inversion

gcy

TT

x

TT

t

TT

p

ny

ny

nx

nx

nn

1

)(2)(2 2

212

2

2121

] [ 222 yxt

Crank-Nicolson Method

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

11

1

11

12

11

1

11

12

2

2nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

21 march 2012 28

n

m1616

ex m=4n=4

Total node number N = mn

Matrix size = NxN

Analysis of Crank-Nicolson Method

21 march 2012 29

Alternating Direction Implicit Method

Solves higher dimension problem by successive Lower dimension methods

Accuracy

Unconditionally stable

No limits on time step and no large scale matrix inversion

] [ 222 yxt

21 march2012 30

Step I x-direction implicit y-direction explicitStep II x-direction explicit y-direction implicit

n

Alternating Direction Implicit Method

bull Peaceman-Rachford Algorithmbull Douglas-Gunn Algorithm

21 march 2012 31

bull Step I

bull Step II

gc

tT

rrT

rr

p

ny

yx

xny

yx

x

)2

1)(2

1()2

1)(2

1( 22122

gc

tT

rT

r

p

ny

yn

xx

2)

21()

21( 22

12

gc

tT

rT

r

p

n

xxn

yy

2)

21()

21( 2

1212

Peaceman-Rachford Algorithm

21 march 2012 32

Douglas-Gunn Algorithm

bull Step I

bull Step II

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

12

1

22

gc

tTrTT

rTT

p

nyy

nnxxnn

22

122

1

)(2

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

2

121

2)(

2

21 march 2012 33

Illustration for ADI

X-direction implicit Y-direction implicit

i = 1 2 hellip m

1

njiT

Step I Step II

j = 1

n

2

hellip

1 2 hellip m

11

njiT

11

njiT

j = 1

n

2hellip

21

njiT

21

1

njiT

21

1

njiT

21 march 2012 34

Analysis of ADI Method

mm

Tridiagonal Matrix

Time complexity O(N)

2xnxm = 2nm =2N

2 steps n matrices tridaigonal matrix

X-direction implicit

21

njiT

i = 1 2 hellip mj = 1

n

2

hellip

21

1

njiT

21

1

njiT

ADVANTAGE over conditionally stable

Reduce simulation time

Good accuracy

Size of geometrical feature may be typical of wavelengh

Good geometrical flexibility to allow with corner or high curvature than fdtd

Advantages of the FDTD method over other methodsAdvantages of the FDTD method over other methods

bull It is conceptually simpleIt is conceptually simple

bull The algorithm does not require the formulation of integral equation and relatively complex The algorithm does not require the formulation of integral equation and relatively complex

scatters can be treated without inversion of large matrices scatters can be treated without inversion of large matrices

bull It is simple to implement for complicated inhomogeneous conducting or dielectric structures It is simple to implement for complicated inhomogeneous conducting or dielectric structures

because constitutive parameters can be assigned to each lattice pointbecause constitutive parameters can be assigned to each lattice point

bull Its computer memory requirement is not prohibitive for many complex structures of interest Its computer memory requirement is not prohibitive for many complex structures of interest

bull The algorithm make use of the memory in a simple sequential orderThe algorithm make use of the memory in a simple sequential order

bull It is much easier to obtain frequency domain data from time domain results than the converse It is much easier to obtain frequency domain data from time domain results than the converse

Thus it is more convenient to obtain frequency domain results via time domain when many Thus it is more convenient to obtain frequency domain results via time domain when many

frequencies are involvedfrequencies are involved

Include examples of unconditionally stable fdtd and prove the advantages

Disadvantages Disadvantages

bull Its implementation necessitates modeling object as well as its surroundings Thus the Its implementation necessitates modeling object as well as its surroundings Thus the

required program execution time may be excessiverequired program execution time may be excessive

bull Its accuracy is at least one order of magnitude worse than that of the method of Its accuracy is at least one order of magnitude worse than that of the method of

moments for example moments for example

bull Since the computational meshes are rectangular in shape they do not conform the Since the computational meshes are rectangular in shape they do not conform the

scatterers with curved surfaces as is the case of the cylindrical or spherical boundary scatterers with curved surfaces as is the case of the cylindrical or spherical boundary

Its computer memory requirement is not prohibitive for many complex structures of Its computer memory requirement is not prohibitive for many complex structures of

interest interest

bull As in all finite difference algorithms the field quantities are only known at grid nodes As in all finite difference algorithms the field quantities are only known at grid nodes

Applications of FDTD method

1048708Electromagnetic scattering amp antenna design1048708EMCEMI design1048708Simulation of wave propagation problem1048708Solving partial defferential equation 1048708Waveguide analysis

Stripline amp microstrip line analysishellipetc hellip

Referenceshelliphelliphellip1-Electromagnetic simulation using FDTD method

ldquoDennis M Sillevanrdquo IEEE press series on rf amp mivcrowave technology ldquoRoger DPollard amp Richard

Bootanrdquoseries

2Understanding the FDTD method ldquoJohn BSchneiderrdquo may 8 2011

3 3 D -ADI -Mehod-unconditionaly stable time domain algorithom for solving maxwell equations

ldquoIEEE Transaction on microwave theory amp techniques vol-48No10 october 2010 ldquoTakefumi

IEEE

4High order split step unconditionally stable FDTD method ampnumerical analysis ldquoIEEE

Transactions on antinna amp propagationvol 59 no9sept2011rdquo Yong-Dan Kong amp Qing xin chu

senior memberIEEE

5Genaral Finite DifferenceSchemes for Heat equations ldquoIndian J Pure Applemath

10(2)209-222Febuary 1979rdquoby PS Jain ampDN Holla department of methemetics

IITB 400076

  • Unconditionally Stable FDTD Methods
  • Outlineshelliphelliphelliphellip
  • Outlineshellip
  • Slide 4
  • Computational Electromagnetics
  • Slide 6
  • FDTD Overview ndash Updating Equations
  • Slide 8
  • Slide 9
  • Finite Difference Time Domain Method
  • The Yee Discretization (1966)
  • Slide 12
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Slide 17
  • Slide 18
  • Slide 19
  • Space Domain Discretization
  • Finite-Difference Formulation of the Heat Conduction on a Chip
  • Time domain discretization
  • Slide 23
  • Slide 24
  • Simple Explicit Method
  • Simple Implicit Method
  • Crank-Nicolson Method
  • Analysis of Crank-Nicolson Method
  • Alternating Direction Implicit Method
  • Slide 30
  • Peaceman-Rachford Algorithm
  • Douglas-Gunn Algorithm
  • Illustration for ADI
  • Analysis of ADI Method
  • Slide 35
  • Slide 36
  • Slide 37
  • Slide 38
  • Referenceshelliphelliphellip
  • Slide 40
Page 15: Unconditionally stable fdtd methods

FDTD Overview ndash Updating Equations

Express the future components in terms of the past components

05 05

1

05 05

( ) ( 1 )

( ) ( )( ) ( ) ( 1)

n nz z

n nx x n n

x y y

H i j k H i j k

ytE i j k E i j k

i j k H i j k H i j k

z

05

05 05

( 1) ( )

( ) ( )( ) ( 1 ) ( )

n ny y

n nx x n n

x z z

E i j k E i j k

t zH i j k H i j ki j k E i j k E i j k

y

Fundamentals of the FDTD methodFundamentals of the FDTD method

Accuracy and stabilityAccuracy and stability

AccuracyAccuracy 10

StabilityStability

222

max

z

1

y

1

x

1c

1tt

3c

tt max

2D2D

22 y

1

x

1c

1t

1D1Dc

t

Physically this condition means that the time Physically this condition means that the time step should be smaller than the time for the step should be smaller than the time for the wave to propagate from one cell to the wave to propagate from one cell to the neighbor oneneighbor one

Slides from 5-15 may be skipped because it was taught in the class

III Fundamentals of the FDTD methodIII Fundamentals of the FDTD method

Dispersion relationDispersion relation

ckkkk

22

z2

y2

x2

In free space (ideal)

In FDTD computation (numerical)

22

z

2

y

2

x

2

tsin

tc

1

2

zk~

sinz

1

2

yk~

siny

1

2

xk~

sinx

1

utcsinArctk

~2

k~vpnum

2

z

2

y

2

x

2

zk~

sinz

1

2

yk~

siny

1

2

xk~

sinx

1u

The numerical medium is dispersive the propagation of the wave varies with frequency and angle

III Fundamentals of the FDTD methodIII Fundamentals of the FDTD method

Dispersion relationDispersion relation

Limitations of FDTD method

1-Grid spacing should be ~λ102-According to Courantrsquos stability condition time step Δt becomes small when FDTDgrid spacing becomes small3-In 3-D simulation simulation time scales like N^4 and required memory size scales like N^34-Application is restricted to relatively small size

21 march 2012 20

Space Domain Discretization

bull Heat Conduction Equation

bull Central-Finite-Difference Approximation

pc

tyxg

y

tyxT

x

tyxT

t

tyxT

)()()()(

2

2

2

2

2

2

2

11

22

112

2

2

)(2

x

T

x

TTT

xOx

TTT

x

T

nx

nji

nji

nji

nji

nji

njin

ji

cc

21 march 2012 21

X

n

jiTn

j1-iT n

j1iT

n

1jiT

n

1jiT

(00) (i0)hellip

(0j)

x

y

(I0)

(0J)

hellip

hellip

hellip

Finite-Difference Formulation of the Heat Conduction on a Chip

bull Space Domainbull Time Domain

21 march 2012 22

Time domain discretization

bull Heat Conduction Equation

ndash Simple Explicit Method ndash Simple Implicit Methodndash Crank-Nicolson Method

gcy

T

x

T

t

TT

p

yxnn

1

)()( 2

2

2

21

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

23

Can we check if a numerical scheme

is stable without computation Von Neumann stability

analysisbull Analyze if (or for which conditions) a

numerical scheme is stable or unstablebull Makes a local analysis coefficients of PDE are

assumed to vary slowly (our example constant)

bull How will unavoidable errors (say rounding errors)evolve in time

John von Neumann1903-1957

24

Von Neumann stability analysis

A numerical scheme is unstable if

Ansatz Wave number k and amplification factor

21 march 2012 25

bull Accuracybull Stability Constraint

bull No matrix inversion but time steps are limited by space discretization

gcy

T

x

T

t

TT

p

ny

nx

nn

1

)()( 2

2

2

21

] [ 22 yxt

2

11122

yxt

Simple Explicit Method

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

21 march 2012 26

Accuracy

Unconditionally Stable

No limits on time step but involves with large scale matrix inversion

gcy

T

x

T

t

TT

p

ny

nx

nn

1

)()( 2

12

2

121

] [ 22 yxt

Simple Implicit Method

11

1

11

12

11

1

11

12

2

2nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

21 march 2012 27

Accuracy

Unconditionally stable

No limits on time step but involves with large scale matrix inversion

gcy

TT

x

TT

t

TT

p

ny

ny

nx

nx

nn

1

)(2)(2 2

212

2

2121

] [ 222 yxt

Crank-Nicolson Method

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

11

1

11

12

11

1

11

12

2

2nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

21 march 2012 28

n

m1616

ex m=4n=4

Total node number N = mn

Matrix size = NxN

Analysis of Crank-Nicolson Method

21 march 2012 29

Alternating Direction Implicit Method

Solves higher dimension problem by successive Lower dimension methods

Accuracy

Unconditionally stable

No limits on time step and no large scale matrix inversion

] [ 222 yxt

21 march2012 30

Step I x-direction implicit y-direction explicitStep II x-direction explicit y-direction implicit

n

Alternating Direction Implicit Method

bull Peaceman-Rachford Algorithmbull Douglas-Gunn Algorithm

21 march 2012 31

bull Step I

bull Step II

gc

tT

rrT

rr

p

ny

yx

xny

yx

x

)2

1)(2

1()2

1)(2

1( 22122

gc

tT

rT

r

p

ny

yn

xx

2)

21()

21( 22

12

gc

tT

rT

r

p

n

xxn

yy

2)

21()

21( 2

1212

Peaceman-Rachford Algorithm

21 march 2012 32

Douglas-Gunn Algorithm

bull Step I

bull Step II

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

12

1

22

gc

tTrTT

rTT

p

nyy

nnxxnn

22

122

1

)(2

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

2

121

2)(

2

21 march 2012 33

Illustration for ADI

X-direction implicit Y-direction implicit

i = 1 2 hellip m

1

njiT

Step I Step II

j = 1

n

2

hellip

1 2 hellip m

11

njiT

11

njiT

j = 1

n

2hellip

21

njiT

21

1

njiT

21

1

njiT

21 march 2012 34

Analysis of ADI Method

mm

Tridiagonal Matrix

Time complexity O(N)

2xnxm = 2nm =2N

2 steps n matrices tridaigonal matrix

X-direction implicit

21

njiT

i = 1 2 hellip mj = 1

n

2

hellip

21

1

njiT

21

1

njiT

ADVANTAGE over conditionally stable

Reduce simulation time

Good accuracy

Size of geometrical feature may be typical of wavelengh

Good geometrical flexibility to allow with corner or high curvature than fdtd

Advantages of the FDTD method over other methodsAdvantages of the FDTD method over other methods

bull It is conceptually simpleIt is conceptually simple

bull The algorithm does not require the formulation of integral equation and relatively complex The algorithm does not require the formulation of integral equation and relatively complex

scatters can be treated without inversion of large matrices scatters can be treated without inversion of large matrices

bull It is simple to implement for complicated inhomogeneous conducting or dielectric structures It is simple to implement for complicated inhomogeneous conducting or dielectric structures

because constitutive parameters can be assigned to each lattice pointbecause constitutive parameters can be assigned to each lattice point

bull Its computer memory requirement is not prohibitive for many complex structures of interest Its computer memory requirement is not prohibitive for many complex structures of interest

bull The algorithm make use of the memory in a simple sequential orderThe algorithm make use of the memory in a simple sequential order

bull It is much easier to obtain frequency domain data from time domain results than the converse It is much easier to obtain frequency domain data from time domain results than the converse

Thus it is more convenient to obtain frequency domain results via time domain when many Thus it is more convenient to obtain frequency domain results via time domain when many

frequencies are involvedfrequencies are involved

Include examples of unconditionally stable fdtd and prove the advantages

Disadvantages Disadvantages

bull Its implementation necessitates modeling object as well as its surroundings Thus the Its implementation necessitates modeling object as well as its surroundings Thus the

required program execution time may be excessiverequired program execution time may be excessive

bull Its accuracy is at least one order of magnitude worse than that of the method of Its accuracy is at least one order of magnitude worse than that of the method of

moments for example moments for example

bull Since the computational meshes are rectangular in shape they do not conform the Since the computational meshes are rectangular in shape they do not conform the

scatterers with curved surfaces as is the case of the cylindrical or spherical boundary scatterers with curved surfaces as is the case of the cylindrical or spherical boundary

Its computer memory requirement is not prohibitive for many complex structures of Its computer memory requirement is not prohibitive for many complex structures of

interest interest

bull As in all finite difference algorithms the field quantities are only known at grid nodes As in all finite difference algorithms the field quantities are only known at grid nodes

Applications of FDTD method

1048708Electromagnetic scattering amp antenna design1048708EMCEMI design1048708Simulation of wave propagation problem1048708Solving partial defferential equation 1048708Waveguide analysis

Stripline amp microstrip line analysishellipetc hellip

Referenceshelliphelliphellip1-Electromagnetic simulation using FDTD method

ldquoDennis M Sillevanrdquo IEEE press series on rf amp mivcrowave technology ldquoRoger DPollard amp Richard

Bootanrdquoseries

2Understanding the FDTD method ldquoJohn BSchneiderrdquo may 8 2011

3 3 D -ADI -Mehod-unconditionaly stable time domain algorithom for solving maxwell equations

ldquoIEEE Transaction on microwave theory amp techniques vol-48No10 october 2010 ldquoTakefumi

IEEE

4High order split step unconditionally stable FDTD method ampnumerical analysis ldquoIEEE

Transactions on antinna amp propagationvol 59 no9sept2011rdquo Yong-Dan Kong amp Qing xin chu

senior memberIEEE

5Genaral Finite DifferenceSchemes for Heat equations ldquoIndian J Pure Applemath

10(2)209-222Febuary 1979rdquoby PS Jain ampDN Holla department of methemetics

IITB 400076

  • Unconditionally Stable FDTD Methods
  • Outlineshelliphelliphelliphellip
  • Outlineshellip
  • Slide 4
  • Computational Electromagnetics
  • Slide 6
  • FDTD Overview ndash Updating Equations
  • Slide 8
  • Slide 9
  • Finite Difference Time Domain Method
  • The Yee Discretization (1966)
  • Slide 12
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Slide 17
  • Slide 18
  • Slide 19
  • Space Domain Discretization
  • Finite-Difference Formulation of the Heat Conduction on a Chip
  • Time domain discretization
  • Slide 23
  • Slide 24
  • Simple Explicit Method
  • Simple Implicit Method
  • Crank-Nicolson Method
  • Analysis of Crank-Nicolson Method
  • Alternating Direction Implicit Method
  • Slide 30
  • Peaceman-Rachford Algorithm
  • Douglas-Gunn Algorithm
  • Illustration for ADI
  • Analysis of ADI Method
  • Slide 35
  • Slide 36
  • Slide 37
  • Slide 38
  • Referenceshelliphelliphellip
  • Slide 40
Page 16: Unconditionally stable fdtd methods

Fundamentals of the FDTD methodFundamentals of the FDTD method

Accuracy and stabilityAccuracy and stability

AccuracyAccuracy 10

StabilityStability

222

max

z

1

y

1

x

1c

1tt

3c

tt max

2D2D

22 y

1

x

1c

1t

1D1Dc

t

Physically this condition means that the time Physically this condition means that the time step should be smaller than the time for the step should be smaller than the time for the wave to propagate from one cell to the wave to propagate from one cell to the neighbor oneneighbor one

Slides from 5-15 may be skipped because it was taught in the class

III Fundamentals of the FDTD methodIII Fundamentals of the FDTD method

Dispersion relationDispersion relation

ckkkk

22

z2

y2

x2

In free space (ideal)

In FDTD computation (numerical)

22

z

2

y

2

x

2

tsin

tc

1

2

zk~

sinz

1

2

yk~

siny

1

2

xk~

sinx

1

utcsinArctk

~2

k~vpnum

2

z

2

y

2

x

2

zk~

sinz

1

2

yk~

siny

1

2

xk~

sinx

1u

The numerical medium is dispersive the propagation of the wave varies with frequency and angle

III Fundamentals of the FDTD methodIII Fundamentals of the FDTD method

Dispersion relationDispersion relation

Limitations of FDTD method

1-Grid spacing should be ~λ102-According to Courantrsquos stability condition time step Δt becomes small when FDTDgrid spacing becomes small3-In 3-D simulation simulation time scales like N^4 and required memory size scales like N^34-Application is restricted to relatively small size

21 march 2012 20

Space Domain Discretization

bull Heat Conduction Equation

bull Central-Finite-Difference Approximation

pc

tyxg

y

tyxT

x

tyxT

t

tyxT

)()()()(

2

2

2

2

2

2

2

11

22

112

2

2

)(2

x

T

x

TTT

xOx

TTT

x

T

nx

nji

nji

nji

nji

nji

njin

ji

cc

21 march 2012 21

X

n

jiTn

j1-iT n

j1iT

n

1jiT

n

1jiT

(00) (i0)hellip

(0j)

x

y

(I0)

(0J)

hellip

hellip

hellip

Finite-Difference Formulation of the Heat Conduction on a Chip

bull Space Domainbull Time Domain

21 march 2012 22

Time domain discretization

bull Heat Conduction Equation

ndash Simple Explicit Method ndash Simple Implicit Methodndash Crank-Nicolson Method

gcy

T

x

T

t

TT

p

yxnn

1

)()( 2

2

2

21

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

23

Can we check if a numerical scheme

is stable without computation Von Neumann stability

analysisbull Analyze if (or for which conditions) a

numerical scheme is stable or unstablebull Makes a local analysis coefficients of PDE are

assumed to vary slowly (our example constant)

bull How will unavoidable errors (say rounding errors)evolve in time

John von Neumann1903-1957

24

Von Neumann stability analysis

A numerical scheme is unstable if

Ansatz Wave number k and amplification factor

21 march 2012 25

bull Accuracybull Stability Constraint

bull No matrix inversion but time steps are limited by space discretization

gcy

T

x

T

t

TT

p

ny

nx

nn

1

)()( 2

2

2

21

] [ 22 yxt

2

11122

yxt

Simple Explicit Method

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

21 march 2012 26

Accuracy

Unconditionally Stable

No limits on time step but involves with large scale matrix inversion

gcy

T

x

T

t

TT

p

ny

nx

nn

1

)()( 2

12

2

121

] [ 22 yxt

Simple Implicit Method

11

1

11

12

11

1

11

12

2

2nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

21 march 2012 27

Accuracy

Unconditionally stable

No limits on time step but involves with large scale matrix inversion

gcy

TT

x

TT

t

TT

p

ny

ny

nx

nx

nn

1

)(2)(2 2

212

2

2121

] [ 222 yxt

Crank-Nicolson Method

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

11

1

11

12

11

1

11

12

2

2nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

21 march 2012 28

n

m1616

ex m=4n=4

Total node number N = mn

Matrix size = NxN

Analysis of Crank-Nicolson Method

21 march 2012 29

Alternating Direction Implicit Method

Solves higher dimension problem by successive Lower dimension methods

Accuracy

Unconditionally stable

No limits on time step and no large scale matrix inversion

] [ 222 yxt

21 march2012 30

Step I x-direction implicit y-direction explicitStep II x-direction explicit y-direction implicit

n

Alternating Direction Implicit Method

bull Peaceman-Rachford Algorithmbull Douglas-Gunn Algorithm

21 march 2012 31

bull Step I

bull Step II

gc

tT

rrT

rr

p

ny

yx

xny

yx

x

)2

1)(2

1()2

1)(2

1( 22122

gc

tT

rT

r

p

ny

yn

xx

2)

21()

21( 22

12

gc

tT

rT

r

p

n

xxn

yy

2)

21()

21( 2

1212

Peaceman-Rachford Algorithm

21 march 2012 32

Douglas-Gunn Algorithm

bull Step I

bull Step II

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

12

1

22

gc

tTrTT

rTT

p

nyy

nnxxnn

22

122

1

)(2

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

2

121

2)(

2

21 march 2012 33

Illustration for ADI

X-direction implicit Y-direction implicit

i = 1 2 hellip m

1

njiT

Step I Step II

j = 1

n

2

hellip

1 2 hellip m

11

njiT

11

njiT

j = 1

n

2hellip

21

njiT

21

1

njiT

21

1

njiT

21 march 2012 34

Analysis of ADI Method

mm

Tridiagonal Matrix

Time complexity O(N)

2xnxm = 2nm =2N

2 steps n matrices tridaigonal matrix

X-direction implicit

21

njiT

i = 1 2 hellip mj = 1

n

2

hellip

21

1

njiT

21

1

njiT

ADVANTAGE over conditionally stable

Reduce simulation time

Good accuracy

Size of geometrical feature may be typical of wavelengh

Good geometrical flexibility to allow with corner or high curvature than fdtd

Advantages of the FDTD method over other methodsAdvantages of the FDTD method over other methods

bull It is conceptually simpleIt is conceptually simple

bull The algorithm does not require the formulation of integral equation and relatively complex The algorithm does not require the formulation of integral equation and relatively complex

scatters can be treated without inversion of large matrices scatters can be treated without inversion of large matrices

bull It is simple to implement for complicated inhomogeneous conducting or dielectric structures It is simple to implement for complicated inhomogeneous conducting or dielectric structures

because constitutive parameters can be assigned to each lattice pointbecause constitutive parameters can be assigned to each lattice point

bull Its computer memory requirement is not prohibitive for many complex structures of interest Its computer memory requirement is not prohibitive for many complex structures of interest

bull The algorithm make use of the memory in a simple sequential orderThe algorithm make use of the memory in a simple sequential order

bull It is much easier to obtain frequency domain data from time domain results than the converse It is much easier to obtain frequency domain data from time domain results than the converse

Thus it is more convenient to obtain frequency domain results via time domain when many Thus it is more convenient to obtain frequency domain results via time domain when many

frequencies are involvedfrequencies are involved

Include examples of unconditionally stable fdtd and prove the advantages

Disadvantages Disadvantages

bull Its implementation necessitates modeling object as well as its surroundings Thus the Its implementation necessitates modeling object as well as its surroundings Thus the

required program execution time may be excessiverequired program execution time may be excessive

bull Its accuracy is at least one order of magnitude worse than that of the method of Its accuracy is at least one order of magnitude worse than that of the method of

moments for example moments for example

bull Since the computational meshes are rectangular in shape they do not conform the Since the computational meshes are rectangular in shape they do not conform the

scatterers with curved surfaces as is the case of the cylindrical or spherical boundary scatterers with curved surfaces as is the case of the cylindrical or spherical boundary

Its computer memory requirement is not prohibitive for many complex structures of Its computer memory requirement is not prohibitive for many complex structures of

interest interest

bull As in all finite difference algorithms the field quantities are only known at grid nodes As in all finite difference algorithms the field quantities are only known at grid nodes

Applications of FDTD method

1048708Electromagnetic scattering amp antenna design1048708EMCEMI design1048708Simulation of wave propagation problem1048708Solving partial defferential equation 1048708Waveguide analysis

Stripline amp microstrip line analysishellipetc hellip

Referenceshelliphelliphellip1-Electromagnetic simulation using FDTD method

ldquoDennis M Sillevanrdquo IEEE press series on rf amp mivcrowave technology ldquoRoger DPollard amp Richard

Bootanrdquoseries

2Understanding the FDTD method ldquoJohn BSchneiderrdquo may 8 2011

3 3 D -ADI -Mehod-unconditionaly stable time domain algorithom for solving maxwell equations

ldquoIEEE Transaction on microwave theory amp techniques vol-48No10 october 2010 ldquoTakefumi

IEEE

4High order split step unconditionally stable FDTD method ampnumerical analysis ldquoIEEE

Transactions on antinna amp propagationvol 59 no9sept2011rdquo Yong-Dan Kong amp Qing xin chu

senior memberIEEE

5Genaral Finite DifferenceSchemes for Heat equations ldquoIndian J Pure Applemath

10(2)209-222Febuary 1979rdquoby PS Jain ampDN Holla department of methemetics

IITB 400076

  • Unconditionally Stable FDTD Methods
  • Outlineshelliphelliphelliphellip
  • Outlineshellip
  • Slide 4
  • Computational Electromagnetics
  • Slide 6
  • FDTD Overview ndash Updating Equations
  • Slide 8
  • Slide 9
  • Finite Difference Time Domain Method
  • The Yee Discretization (1966)
  • Slide 12
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Slide 17
  • Slide 18
  • Slide 19
  • Space Domain Discretization
  • Finite-Difference Formulation of the Heat Conduction on a Chip
  • Time domain discretization
  • Slide 23
  • Slide 24
  • Simple Explicit Method
  • Simple Implicit Method
  • Crank-Nicolson Method
  • Analysis of Crank-Nicolson Method
  • Alternating Direction Implicit Method
  • Slide 30
  • Peaceman-Rachford Algorithm
  • Douglas-Gunn Algorithm
  • Illustration for ADI
  • Analysis of ADI Method
  • Slide 35
  • Slide 36
  • Slide 37
  • Slide 38
  • Referenceshelliphelliphellip
  • Slide 40
Page 17: Unconditionally stable fdtd methods

III Fundamentals of the FDTD methodIII Fundamentals of the FDTD method

Dispersion relationDispersion relation

ckkkk

22

z2

y2

x2

In free space (ideal)

In FDTD computation (numerical)

22

z

2

y

2

x

2

tsin

tc

1

2

zk~

sinz

1

2

yk~

siny

1

2

xk~

sinx

1

utcsinArctk

~2

k~vpnum

2

z

2

y

2

x

2

zk~

sinz

1

2

yk~

siny

1

2

xk~

sinx

1u

The numerical medium is dispersive the propagation of the wave varies with frequency and angle

III Fundamentals of the FDTD methodIII Fundamentals of the FDTD method

Dispersion relationDispersion relation

Limitations of FDTD method

1-Grid spacing should be ~λ102-According to Courantrsquos stability condition time step Δt becomes small when FDTDgrid spacing becomes small3-In 3-D simulation simulation time scales like N^4 and required memory size scales like N^34-Application is restricted to relatively small size

21 march 2012 20

Space Domain Discretization

bull Heat Conduction Equation

bull Central-Finite-Difference Approximation

pc

tyxg

y

tyxT

x

tyxT

t

tyxT

)()()()(

2

2

2

2

2

2

2

11

22

112

2

2

)(2

x

T

x

TTT

xOx

TTT

x

T

nx

nji

nji

nji

nji

nji

njin

ji

cc

21 march 2012 21

X

n

jiTn

j1-iT n

j1iT

n

1jiT

n

1jiT

(00) (i0)hellip

(0j)

x

y

(I0)

(0J)

hellip

hellip

hellip

Finite-Difference Formulation of the Heat Conduction on a Chip

bull Space Domainbull Time Domain

21 march 2012 22

Time domain discretization

bull Heat Conduction Equation

ndash Simple Explicit Method ndash Simple Implicit Methodndash Crank-Nicolson Method

gcy

T

x

T

t

TT

p

yxnn

1

)()( 2

2

2

21

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

23

Can we check if a numerical scheme

is stable without computation Von Neumann stability

analysisbull Analyze if (or for which conditions) a

numerical scheme is stable or unstablebull Makes a local analysis coefficients of PDE are

assumed to vary slowly (our example constant)

bull How will unavoidable errors (say rounding errors)evolve in time

John von Neumann1903-1957

24

Von Neumann stability analysis

A numerical scheme is unstable if

Ansatz Wave number k and amplification factor

21 march 2012 25

bull Accuracybull Stability Constraint

bull No matrix inversion but time steps are limited by space discretization

gcy

T

x

T

t

TT

p

ny

nx

nn

1

)()( 2

2

2

21

] [ 22 yxt

2

11122

yxt

Simple Explicit Method

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

21 march 2012 26

Accuracy

Unconditionally Stable

No limits on time step but involves with large scale matrix inversion

gcy

T

x

T

t

TT

p

ny

nx

nn

1

)()( 2

12

2

121

] [ 22 yxt

Simple Implicit Method

11

1

11

12

11

1

11

12

2

2nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

21 march 2012 27

Accuracy

Unconditionally stable

No limits on time step but involves with large scale matrix inversion

gcy

TT

x

TT

t

TT

p

ny

ny

nx

nx

nn

1

)(2)(2 2

212

2

2121

] [ 222 yxt

Crank-Nicolson Method

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

11

1

11

12

11

1

11

12

2

2nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

21 march 2012 28

n

m1616

ex m=4n=4

Total node number N = mn

Matrix size = NxN

Analysis of Crank-Nicolson Method

21 march 2012 29

Alternating Direction Implicit Method

Solves higher dimension problem by successive Lower dimension methods

Accuracy

Unconditionally stable

No limits on time step and no large scale matrix inversion

] [ 222 yxt

21 march2012 30

Step I x-direction implicit y-direction explicitStep II x-direction explicit y-direction implicit

n

Alternating Direction Implicit Method

bull Peaceman-Rachford Algorithmbull Douglas-Gunn Algorithm

21 march 2012 31

bull Step I

bull Step II

gc

tT

rrT

rr

p

ny

yx

xny

yx

x

)2

1)(2

1()2

1)(2

1( 22122

gc

tT

rT

r

p

ny

yn

xx

2)

21()

21( 22

12

gc

tT

rT

r

p

n

xxn

yy

2)

21()

21( 2

1212

Peaceman-Rachford Algorithm

21 march 2012 32

Douglas-Gunn Algorithm

bull Step I

bull Step II

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

12

1

22

gc

tTrTT

rTT

p

nyy

nnxxnn

22

122

1

)(2

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

2

121

2)(

2

21 march 2012 33

Illustration for ADI

X-direction implicit Y-direction implicit

i = 1 2 hellip m

1

njiT

Step I Step II

j = 1

n

2

hellip

1 2 hellip m

11

njiT

11

njiT

j = 1

n

2hellip

21

njiT

21

1

njiT

21

1

njiT

21 march 2012 34

Analysis of ADI Method

mm

Tridiagonal Matrix

Time complexity O(N)

2xnxm = 2nm =2N

2 steps n matrices tridaigonal matrix

X-direction implicit

21

njiT

i = 1 2 hellip mj = 1

n

2

hellip

21

1

njiT

21

1

njiT

ADVANTAGE over conditionally stable

Reduce simulation time

Good accuracy

Size of geometrical feature may be typical of wavelengh

Good geometrical flexibility to allow with corner or high curvature than fdtd

Advantages of the FDTD method over other methodsAdvantages of the FDTD method over other methods

bull It is conceptually simpleIt is conceptually simple

bull The algorithm does not require the formulation of integral equation and relatively complex The algorithm does not require the formulation of integral equation and relatively complex

scatters can be treated without inversion of large matrices scatters can be treated without inversion of large matrices

bull It is simple to implement for complicated inhomogeneous conducting or dielectric structures It is simple to implement for complicated inhomogeneous conducting or dielectric structures

because constitutive parameters can be assigned to each lattice pointbecause constitutive parameters can be assigned to each lattice point

bull Its computer memory requirement is not prohibitive for many complex structures of interest Its computer memory requirement is not prohibitive for many complex structures of interest

bull The algorithm make use of the memory in a simple sequential orderThe algorithm make use of the memory in a simple sequential order

bull It is much easier to obtain frequency domain data from time domain results than the converse It is much easier to obtain frequency domain data from time domain results than the converse

Thus it is more convenient to obtain frequency domain results via time domain when many Thus it is more convenient to obtain frequency domain results via time domain when many

frequencies are involvedfrequencies are involved

Include examples of unconditionally stable fdtd and prove the advantages

Disadvantages Disadvantages

bull Its implementation necessitates modeling object as well as its surroundings Thus the Its implementation necessitates modeling object as well as its surroundings Thus the

required program execution time may be excessiverequired program execution time may be excessive

bull Its accuracy is at least one order of magnitude worse than that of the method of Its accuracy is at least one order of magnitude worse than that of the method of

moments for example moments for example

bull Since the computational meshes are rectangular in shape they do not conform the Since the computational meshes are rectangular in shape they do not conform the

scatterers with curved surfaces as is the case of the cylindrical or spherical boundary scatterers with curved surfaces as is the case of the cylindrical or spherical boundary

Its computer memory requirement is not prohibitive for many complex structures of Its computer memory requirement is not prohibitive for many complex structures of

interest interest

bull As in all finite difference algorithms the field quantities are only known at grid nodes As in all finite difference algorithms the field quantities are only known at grid nodes

Applications of FDTD method

1048708Electromagnetic scattering amp antenna design1048708EMCEMI design1048708Simulation of wave propagation problem1048708Solving partial defferential equation 1048708Waveguide analysis

Stripline amp microstrip line analysishellipetc hellip

Referenceshelliphelliphellip1-Electromagnetic simulation using FDTD method

ldquoDennis M Sillevanrdquo IEEE press series on rf amp mivcrowave technology ldquoRoger DPollard amp Richard

Bootanrdquoseries

2Understanding the FDTD method ldquoJohn BSchneiderrdquo may 8 2011

3 3 D -ADI -Mehod-unconditionaly stable time domain algorithom for solving maxwell equations

ldquoIEEE Transaction on microwave theory amp techniques vol-48No10 october 2010 ldquoTakefumi

IEEE

4High order split step unconditionally stable FDTD method ampnumerical analysis ldquoIEEE

Transactions on antinna amp propagationvol 59 no9sept2011rdquo Yong-Dan Kong amp Qing xin chu

senior memberIEEE

5Genaral Finite DifferenceSchemes for Heat equations ldquoIndian J Pure Applemath

10(2)209-222Febuary 1979rdquoby PS Jain ampDN Holla department of methemetics

IITB 400076

  • Unconditionally Stable FDTD Methods
  • Outlineshelliphelliphelliphellip
  • Outlineshellip
  • Slide 4
  • Computational Electromagnetics
  • Slide 6
  • FDTD Overview ndash Updating Equations
  • Slide 8
  • Slide 9
  • Finite Difference Time Domain Method
  • The Yee Discretization (1966)
  • Slide 12
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Slide 17
  • Slide 18
  • Slide 19
  • Space Domain Discretization
  • Finite-Difference Formulation of the Heat Conduction on a Chip
  • Time domain discretization
  • Slide 23
  • Slide 24
  • Simple Explicit Method
  • Simple Implicit Method
  • Crank-Nicolson Method
  • Analysis of Crank-Nicolson Method
  • Alternating Direction Implicit Method
  • Slide 30
  • Peaceman-Rachford Algorithm
  • Douglas-Gunn Algorithm
  • Illustration for ADI
  • Analysis of ADI Method
  • Slide 35
  • Slide 36
  • Slide 37
  • Slide 38
  • Referenceshelliphelliphellip
  • Slide 40
Page 18: Unconditionally stable fdtd methods

III Fundamentals of the FDTD methodIII Fundamentals of the FDTD method

Dispersion relationDispersion relation

Limitations of FDTD method

1-Grid spacing should be ~λ102-According to Courantrsquos stability condition time step Δt becomes small when FDTDgrid spacing becomes small3-In 3-D simulation simulation time scales like N^4 and required memory size scales like N^34-Application is restricted to relatively small size

21 march 2012 20

Space Domain Discretization

bull Heat Conduction Equation

bull Central-Finite-Difference Approximation

pc

tyxg

y

tyxT

x

tyxT

t

tyxT

)()()()(

2

2

2

2

2

2

2

11

22

112

2

2

)(2

x

T

x

TTT

xOx

TTT

x

T

nx

nji

nji

nji

nji

nji

njin

ji

cc

21 march 2012 21

X

n

jiTn

j1-iT n

j1iT

n

1jiT

n

1jiT

(00) (i0)hellip

(0j)

x

y

(I0)

(0J)

hellip

hellip

hellip

Finite-Difference Formulation of the Heat Conduction on a Chip

bull Space Domainbull Time Domain

21 march 2012 22

Time domain discretization

bull Heat Conduction Equation

ndash Simple Explicit Method ndash Simple Implicit Methodndash Crank-Nicolson Method

gcy

T

x

T

t

TT

p

yxnn

1

)()( 2

2

2

21

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

23

Can we check if a numerical scheme

is stable without computation Von Neumann stability

analysisbull Analyze if (or for which conditions) a

numerical scheme is stable or unstablebull Makes a local analysis coefficients of PDE are

assumed to vary slowly (our example constant)

bull How will unavoidable errors (say rounding errors)evolve in time

John von Neumann1903-1957

24

Von Neumann stability analysis

A numerical scheme is unstable if

Ansatz Wave number k and amplification factor

21 march 2012 25

bull Accuracybull Stability Constraint

bull No matrix inversion but time steps are limited by space discretization

gcy

T

x

T

t

TT

p

ny

nx

nn

1

)()( 2

2

2

21

] [ 22 yxt

2

11122

yxt

Simple Explicit Method

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

21 march 2012 26

Accuracy

Unconditionally Stable

No limits on time step but involves with large scale matrix inversion

gcy

T

x

T

t

TT

p

ny

nx

nn

1

)()( 2

12

2

121

] [ 22 yxt

Simple Implicit Method

11

1

11

12

11

1

11

12

2

2nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

21 march 2012 27

Accuracy

Unconditionally stable

No limits on time step but involves with large scale matrix inversion

gcy

TT

x

TT

t

TT

p

ny

ny

nx

nx

nn

1

)(2)(2 2

212

2

2121

] [ 222 yxt

Crank-Nicolson Method

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

11

1

11

12

11

1

11

12

2

2nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

21 march 2012 28

n

m1616

ex m=4n=4

Total node number N = mn

Matrix size = NxN

Analysis of Crank-Nicolson Method

21 march 2012 29

Alternating Direction Implicit Method

Solves higher dimension problem by successive Lower dimension methods

Accuracy

Unconditionally stable

No limits on time step and no large scale matrix inversion

] [ 222 yxt

21 march2012 30

Step I x-direction implicit y-direction explicitStep II x-direction explicit y-direction implicit

n

Alternating Direction Implicit Method

bull Peaceman-Rachford Algorithmbull Douglas-Gunn Algorithm

21 march 2012 31

bull Step I

bull Step II

gc

tT

rrT

rr

p

ny

yx

xny

yx

x

)2

1)(2

1()2

1)(2

1( 22122

gc

tT

rT

r

p

ny

yn

xx

2)

21()

21( 22

12

gc

tT

rT

r

p

n

xxn

yy

2)

21()

21( 2

1212

Peaceman-Rachford Algorithm

21 march 2012 32

Douglas-Gunn Algorithm

bull Step I

bull Step II

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

12

1

22

gc

tTrTT

rTT

p

nyy

nnxxnn

22

122

1

)(2

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

2

121

2)(

2

21 march 2012 33

Illustration for ADI

X-direction implicit Y-direction implicit

i = 1 2 hellip m

1

njiT

Step I Step II

j = 1

n

2

hellip

1 2 hellip m

11

njiT

11

njiT

j = 1

n

2hellip

21

njiT

21

1

njiT

21

1

njiT

21 march 2012 34

Analysis of ADI Method

mm

Tridiagonal Matrix

Time complexity O(N)

2xnxm = 2nm =2N

2 steps n matrices tridaigonal matrix

X-direction implicit

21

njiT

i = 1 2 hellip mj = 1

n

2

hellip

21

1

njiT

21

1

njiT

ADVANTAGE over conditionally stable

Reduce simulation time

Good accuracy

Size of geometrical feature may be typical of wavelengh

Good geometrical flexibility to allow with corner or high curvature than fdtd

Advantages of the FDTD method over other methodsAdvantages of the FDTD method over other methods

bull It is conceptually simpleIt is conceptually simple

bull The algorithm does not require the formulation of integral equation and relatively complex The algorithm does not require the formulation of integral equation and relatively complex

scatters can be treated without inversion of large matrices scatters can be treated without inversion of large matrices

bull It is simple to implement for complicated inhomogeneous conducting or dielectric structures It is simple to implement for complicated inhomogeneous conducting or dielectric structures

because constitutive parameters can be assigned to each lattice pointbecause constitutive parameters can be assigned to each lattice point

bull Its computer memory requirement is not prohibitive for many complex structures of interest Its computer memory requirement is not prohibitive for many complex structures of interest

bull The algorithm make use of the memory in a simple sequential orderThe algorithm make use of the memory in a simple sequential order

bull It is much easier to obtain frequency domain data from time domain results than the converse It is much easier to obtain frequency domain data from time domain results than the converse

Thus it is more convenient to obtain frequency domain results via time domain when many Thus it is more convenient to obtain frequency domain results via time domain when many

frequencies are involvedfrequencies are involved

Include examples of unconditionally stable fdtd and prove the advantages

Disadvantages Disadvantages

bull Its implementation necessitates modeling object as well as its surroundings Thus the Its implementation necessitates modeling object as well as its surroundings Thus the

required program execution time may be excessiverequired program execution time may be excessive

bull Its accuracy is at least one order of magnitude worse than that of the method of Its accuracy is at least one order of magnitude worse than that of the method of

moments for example moments for example

bull Since the computational meshes are rectangular in shape they do not conform the Since the computational meshes are rectangular in shape they do not conform the

scatterers with curved surfaces as is the case of the cylindrical or spherical boundary scatterers with curved surfaces as is the case of the cylindrical or spherical boundary

Its computer memory requirement is not prohibitive for many complex structures of Its computer memory requirement is not prohibitive for many complex structures of

interest interest

bull As in all finite difference algorithms the field quantities are only known at grid nodes As in all finite difference algorithms the field quantities are only known at grid nodes

Applications of FDTD method

1048708Electromagnetic scattering amp antenna design1048708EMCEMI design1048708Simulation of wave propagation problem1048708Solving partial defferential equation 1048708Waveguide analysis

Stripline amp microstrip line analysishellipetc hellip

Referenceshelliphelliphellip1-Electromagnetic simulation using FDTD method

ldquoDennis M Sillevanrdquo IEEE press series on rf amp mivcrowave technology ldquoRoger DPollard amp Richard

Bootanrdquoseries

2Understanding the FDTD method ldquoJohn BSchneiderrdquo may 8 2011

3 3 D -ADI -Mehod-unconditionaly stable time domain algorithom for solving maxwell equations

ldquoIEEE Transaction on microwave theory amp techniques vol-48No10 october 2010 ldquoTakefumi

IEEE

4High order split step unconditionally stable FDTD method ampnumerical analysis ldquoIEEE

Transactions on antinna amp propagationvol 59 no9sept2011rdquo Yong-Dan Kong amp Qing xin chu

senior memberIEEE

5Genaral Finite DifferenceSchemes for Heat equations ldquoIndian J Pure Applemath

10(2)209-222Febuary 1979rdquoby PS Jain ampDN Holla department of methemetics

IITB 400076

  • Unconditionally Stable FDTD Methods
  • Outlineshelliphelliphelliphellip
  • Outlineshellip
  • Slide 4
  • Computational Electromagnetics
  • Slide 6
  • FDTD Overview ndash Updating Equations
  • Slide 8
  • Slide 9
  • Finite Difference Time Domain Method
  • The Yee Discretization (1966)
  • Slide 12
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Slide 17
  • Slide 18
  • Slide 19
  • Space Domain Discretization
  • Finite-Difference Formulation of the Heat Conduction on a Chip
  • Time domain discretization
  • Slide 23
  • Slide 24
  • Simple Explicit Method
  • Simple Implicit Method
  • Crank-Nicolson Method
  • Analysis of Crank-Nicolson Method
  • Alternating Direction Implicit Method
  • Slide 30
  • Peaceman-Rachford Algorithm
  • Douglas-Gunn Algorithm
  • Illustration for ADI
  • Analysis of ADI Method
  • Slide 35
  • Slide 36
  • Slide 37
  • Slide 38
  • Referenceshelliphelliphellip
  • Slide 40
Page 19: Unconditionally stable fdtd methods

Limitations of FDTD method

1-Grid spacing should be ~λ102-According to Courantrsquos stability condition time step Δt becomes small when FDTDgrid spacing becomes small3-In 3-D simulation simulation time scales like N^4 and required memory size scales like N^34-Application is restricted to relatively small size

21 march 2012 20

Space Domain Discretization

bull Heat Conduction Equation

bull Central-Finite-Difference Approximation

pc

tyxg

y

tyxT

x

tyxT

t

tyxT

)()()()(

2

2

2

2

2

2

2

11

22

112

2

2

)(2

x

T

x

TTT

xOx

TTT

x

T

nx

nji

nji

nji

nji

nji

njin

ji

cc

21 march 2012 21

X

n

jiTn

j1-iT n

j1iT

n

1jiT

n

1jiT

(00) (i0)hellip

(0j)

x

y

(I0)

(0J)

hellip

hellip

hellip

Finite-Difference Formulation of the Heat Conduction on a Chip

bull Space Domainbull Time Domain

21 march 2012 22

Time domain discretization

bull Heat Conduction Equation

ndash Simple Explicit Method ndash Simple Implicit Methodndash Crank-Nicolson Method

gcy

T

x

T

t

TT

p

yxnn

1

)()( 2

2

2

21

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

23

Can we check if a numerical scheme

is stable without computation Von Neumann stability

analysisbull Analyze if (or for which conditions) a

numerical scheme is stable or unstablebull Makes a local analysis coefficients of PDE are

assumed to vary slowly (our example constant)

bull How will unavoidable errors (say rounding errors)evolve in time

John von Neumann1903-1957

24

Von Neumann stability analysis

A numerical scheme is unstable if

Ansatz Wave number k and amplification factor

21 march 2012 25

bull Accuracybull Stability Constraint

bull No matrix inversion but time steps are limited by space discretization

gcy

T

x

T

t

TT

p

ny

nx

nn

1

)()( 2

2

2

21

] [ 22 yxt

2

11122

yxt

Simple Explicit Method

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

21 march 2012 26

Accuracy

Unconditionally Stable

No limits on time step but involves with large scale matrix inversion

gcy

T

x

T

t

TT

p

ny

nx

nn

1

)()( 2

12

2

121

] [ 22 yxt

Simple Implicit Method

11

1

11

12

11

1

11

12

2

2nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

21 march 2012 27

Accuracy

Unconditionally stable

No limits on time step but involves with large scale matrix inversion

gcy

TT

x

TT

t

TT

p

ny

ny

nx

nx

nn

1

)(2)(2 2

212

2

2121

] [ 222 yxt

Crank-Nicolson Method

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

11

1

11

12

11

1

11

12

2

2nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

21 march 2012 28

n

m1616

ex m=4n=4

Total node number N = mn

Matrix size = NxN

Analysis of Crank-Nicolson Method

21 march 2012 29

Alternating Direction Implicit Method

Solves higher dimension problem by successive Lower dimension methods

Accuracy

Unconditionally stable

No limits on time step and no large scale matrix inversion

] [ 222 yxt

21 march2012 30

Step I x-direction implicit y-direction explicitStep II x-direction explicit y-direction implicit

n

Alternating Direction Implicit Method

bull Peaceman-Rachford Algorithmbull Douglas-Gunn Algorithm

21 march 2012 31

bull Step I

bull Step II

gc

tT

rrT

rr

p

ny

yx

xny

yx

x

)2

1)(2

1()2

1)(2

1( 22122

gc

tT

rT

r

p

ny

yn

xx

2)

21()

21( 22

12

gc

tT

rT

r

p

n

xxn

yy

2)

21()

21( 2

1212

Peaceman-Rachford Algorithm

21 march 2012 32

Douglas-Gunn Algorithm

bull Step I

bull Step II

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

12

1

22

gc

tTrTT

rTT

p

nyy

nnxxnn

22

122

1

)(2

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

2

121

2)(

2

21 march 2012 33

Illustration for ADI

X-direction implicit Y-direction implicit

i = 1 2 hellip m

1

njiT

Step I Step II

j = 1

n

2

hellip

1 2 hellip m

11

njiT

11

njiT

j = 1

n

2hellip

21

njiT

21

1

njiT

21

1

njiT

21 march 2012 34

Analysis of ADI Method

mm

Tridiagonal Matrix

Time complexity O(N)

2xnxm = 2nm =2N

2 steps n matrices tridaigonal matrix

X-direction implicit

21

njiT

i = 1 2 hellip mj = 1

n

2

hellip

21

1

njiT

21

1

njiT

ADVANTAGE over conditionally stable

Reduce simulation time

Good accuracy

Size of geometrical feature may be typical of wavelengh

Good geometrical flexibility to allow with corner or high curvature than fdtd

Advantages of the FDTD method over other methodsAdvantages of the FDTD method over other methods

bull It is conceptually simpleIt is conceptually simple

bull The algorithm does not require the formulation of integral equation and relatively complex The algorithm does not require the formulation of integral equation and relatively complex

scatters can be treated without inversion of large matrices scatters can be treated without inversion of large matrices

bull It is simple to implement for complicated inhomogeneous conducting or dielectric structures It is simple to implement for complicated inhomogeneous conducting or dielectric structures

because constitutive parameters can be assigned to each lattice pointbecause constitutive parameters can be assigned to each lattice point

bull Its computer memory requirement is not prohibitive for many complex structures of interest Its computer memory requirement is not prohibitive for many complex structures of interest

bull The algorithm make use of the memory in a simple sequential orderThe algorithm make use of the memory in a simple sequential order

bull It is much easier to obtain frequency domain data from time domain results than the converse It is much easier to obtain frequency domain data from time domain results than the converse

Thus it is more convenient to obtain frequency domain results via time domain when many Thus it is more convenient to obtain frequency domain results via time domain when many

frequencies are involvedfrequencies are involved

Include examples of unconditionally stable fdtd and prove the advantages

Disadvantages Disadvantages

bull Its implementation necessitates modeling object as well as its surroundings Thus the Its implementation necessitates modeling object as well as its surroundings Thus the

required program execution time may be excessiverequired program execution time may be excessive

bull Its accuracy is at least one order of magnitude worse than that of the method of Its accuracy is at least one order of magnitude worse than that of the method of

moments for example moments for example

bull Since the computational meshes are rectangular in shape they do not conform the Since the computational meshes are rectangular in shape they do not conform the

scatterers with curved surfaces as is the case of the cylindrical or spherical boundary scatterers with curved surfaces as is the case of the cylindrical or spherical boundary

Its computer memory requirement is not prohibitive for many complex structures of Its computer memory requirement is not prohibitive for many complex structures of

interest interest

bull As in all finite difference algorithms the field quantities are only known at grid nodes As in all finite difference algorithms the field quantities are only known at grid nodes

Applications of FDTD method

1048708Electromagnetic scattering amp antenna design1048708EMCEMI design1048708Simulation of wave propagation problem1048708Solving partial defferential equation 1048708Waveguide analysis

Stripline amp microstrip line analysishellipetc hellip

Referenceshelliphelliphellip1-Electromagnetic simulation using FDTD method

ldquoDennis M Sillevanrdquo IEEE press series on rf amp mivcrowave technology ldquoRoger DPollard amp Richard

Bootanrdquoseries

2Understanding the FDTD method ldquoJohn BSchneiderrdquo may 8 2011

3 3 D -ADI -Mehod-unconditionaly stable time domain algorithom for solving maxwell equations

ldquoIEEE Transaction on microwave theory amp techniques vol-48No10 october 2010 ldquoTakefumi

IEEE

4High order split step unconditionally stable FDTD method ampnumerical analysis ldquoIEEE

Transactions on antinna amp propagationvol 59 no9sept2011rdquo Yong-Dan Kong amp Qing xin chu

senior memberIEEE

5Genaral Finite DifferenceSchemes for Heat equations ldquoIndian J Pure Applemath

10(2)209-222Febuary 1979rdquoby PS Jain ampDN Holla department of methemetics

IITB 400076

  • Unconditionally Stable FDTD Methods
  • Outlineshelliphelliphelliphellip
  • Outlineshellip
  • Slide 4
  • Computational Electromagnetics
  • Slide 6
  • FDTD Overview ndash Updating Equations
  • Slide 8
  • Slide 9
  • Finite Difference Time Domain Method
  • The Yee Discretization (1966)
  • Slide 12
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Slide 17
  • Slide 18
  • Slide 19
  • Space Domain Discretization
  • Finite-Difference Formulation of the Heat Conduction on a Chip
  • Time domain discretization
  • Slide 23
  • Slide 24
  • Simple Explicit Method
  • Simple Implicit Method
  • Crank-Nicolson Method
  • Analysis of Crank-Nicolson Method
  • Alternating Direction Implicit Method
  • Slide 30
  • Peaceman-Rachford Algorithm
  • Douglas-Gunn Algorithm
  • Illustration for ADI
  • Analysis of ADI Method
  • Slide 35
  • Slide 36
  • Slide 37
  • Slide 38
  • Referenceshelliphelliphellip
  • Slide 40
Page 20: Unconditionally stable fdtd methods

21 march 2012 20

Space Domain Discretization

bull Heat Conduction Equation

bull Central-Finite-Difference Approximation

pc

tyxg

y

tyxT

x

tyxT

t

tyxT

)()()()(

2

2

2

2

2

2

2

11

22

112

2

2

)(2

x

T

x

TTT

xOx

TTT

x

T

nx

nji

nji

nji

nji

nji

njin

ji

cc

21 march 2012 21

X

n

jiTn

j1-iT n

j1iT

n

1jiT

n

1jiT

(00) (i0)hellip

(0j)

x

y

(I0)

(0J)

hellip

hellip

hellip

Finite-Difference Formulation of the Heat Conduction on a Chip

bull Space Domainbull Time Domain

21 march 2012 22

Time domain discretization

bull Heat Conduction Equation

ndash Simple Explicit Method ndash Simple Implicit Methodndash Crank-Nicolson Method

gcy

T

x

T

t

TT

p

yxnn

1

)()( 2

2

2

21

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

23

Can we check if a numerical scheme

is stable without computation Von Neumann stability

analysisbull Analyze if (or for which conditions) a

numerical scheme is stable or unstablebull Makes a local analysis coefficients of PDE are

assumed to vary slowly (our example constant)

bull How will unavoidable errors (say rounding errors)evolve in time

John von Neumann1903-1957

24

Von Neumann stability analysis

A numerical scheme is unstable if

Ansatz Wave number k and amplification factor

21 march 2012 25

bull Accuracybull Stability Constraint

bull No matrix inversion but time steps are limited by space discretization

gcy

T

x

T

t

TT

p

ny

nx

nn

1

)()( 2

2

2

21

] [ 22 yxt

2

11122

yxt

Simple Explicit Method

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

21 march 2012 26

Accuracy

Unconditionally Stable

No limits on time step but involves with large scale matrix inversion

gcy

T

x

T

t

TT

p

ny

nx

nn

1

)()( 2

12

2

121

] [ 22 yxt

Simple Implicit Method

11

1

11

12

11

1

11

12

2

2nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

21 march 2012 27

Accuracy

Unconditionally stable

No limits on time step but involves with large scale matrix inversion

gcy

TT

x

TT

t

TT

p

ny

ny

nx

nx

nn

1

)(2)(2 2

212

2

2121

] [ 222 yxt

Crank-Nicolson Method

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

11

1

11

12

11

1

11

12

2

2nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

21 march 2012 28

n

m1616

ex m=4n=4

Total node number N = mn

Matrix size = NxN

Analysis of Crank-Nicolson Method

21 march 2012 29

Alternating Direction Implicit Method

Solves higher dimension problem by successive Lower dimension methods

Accuracy

Unconditionally stable

No limits on time step and no large scale matrix inversion

] [ 222 yxt

21 march2012 30

Step I x-direction implicit y-direction explicitStep II x-direction explicit y-direction implicit

n

Alternating Direction Implicit Method

bull Peaceman-Rachford Algorithmbull Douglas-Gunn Algorithm

21 march 2012 31

bull Step I

bull Step II

gc

tT

rrT

rr

p

ny

yx

xny

yx

x

)2

1)(2

1()2

1)(2

1( 22122

gc

tT

rT

r

p

ny

yn

xx

2)

21()

21( 22

12

gc

tT

rT

r

p

n

xxn

yy

2)

21()

21( 2

1212

Peaceman-Rachford Algorithm

21 march 2012 32

Douglas-Gunn Algorithm

bull Step I

bull Step II

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

12

1

22

gc

tTrTT

rTT

p

nyy

nnxxnn

22

122

1

)(2

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

2

121

2)(

2

21 march 2012 33

Illustration for ADI

X-direction implicit Y-direction implicit

i = 1 2 hellip m

1

njiT

Step I Step II

j = 1

n

2

hellip

1 2 hellip m

11

njiT

11

njiT

j = 1

n

2hellip

21

njiT

21

1

njiT

21

1

njiT

21 march 2012 34

Analysis of ADI Method

mm

Tridiagonal Matrix

Time complexity O(N)

2xnxm = 2nm =2N

2 steps n matrices tridaigonal matrix

X-direction implicit

21

njiT

i = 1 2 hellip mj = 1

n

2

hellip

21

1

njiT

21

1

njiT

ADVANTAGE over conditionally stable

Reduce simulation time

Good accuracy

Size of geometrical feature may be typical of wavelengh

Good geometrical flexibility to allow with corner or high curvature than fdtd

Advantages of the FDTD method over other methodsAdvantages of the FDTD method over other methods

bull It is conceptually simpleIt is conceptually simple

bull The algorithm does not require the formulation of integral equation and relatively complex The algorithm does not require the formulation of integral equation and relatively complex

scatters can be treated without inversion of large matrices scatters can be treated without inversion of large matrices

bull It is simple to implement for complicated inhomogeneous conducting or dielectric structures It is simple to implement for complicated inhomogeneous conducting or dielectric structures

because constitutive parameters can be assigned to each lattice pointbecause constitutive parameters can be assigned to each lattice point

bull Its computer memory requirement is not prohibitive for many complex structures of interest Its computer memory requirement is not prohibitive for many complex structures of interest

bull The algorithm make use of the memory in a simple sequential orderThe algorithm make use of the memory in a simple sequential order

bull It is much easier to obtain frequency domain data from time domain results than the converse It is much easier to obtain frequency domain data from time domain results than the converse

Thus it is more convenient to obtain frequency domain results via time domain when many Thus it is more convenient to obtain frequency domain results via time domain when many

frequencies are involvedfrequencies are involved

Include examples of unconditionally stable fdtd and prove the advantages

Disadvantages Disadvantages

bull Its implementation necessitates modeling object as well as its surroundings Thus the Its implementation necessitates modeling object as well as its surroundings Thus the

required program execution time may be excessiverequired program execution time may be excessive

bull Its accuracy is at least one order of magnitude worse than that of the method of Its accuracy is at least one order of magnitude worse than that of the method of

moments for example moments for example

bull Since the computational meshes are rectangular in shape they do not conform the Since the computational meshes are rectangular in shape they do not conform the

scatterers with curved surfaces as is the case of the cylindrical or spherical boundary scatterers with curved surfaces as is the case of the cylindrical or spherical boundary

Its computer memory requirement is not prohibitive for many complex structures of Its computer memory requirement is not prohibitive for many complex structures of

interest interest

bull As in all finite difference algorithms the field quantities are only known at grid nodes As in all finite difference algorithms the field quantities are only known at grid nodes

Applications of FDTD method

1048708Electromagnetic scattering amp antenna design1048708EMCEMI design1048708Simulation of wave propagation problem1048708Solving partial defferential equation 1048708Waveguide analysis

Stripline amp microstrip line analysishellipetc hellip

Referenceshelliphelliphellip1-Electromagnetic simulation using FDTD method

ldquoDennis M Sillevanrdquo IEEE press series on rf amp mivcrowave technology ldquoRoger DPollard amp Richard

Bootanrdquoseries

2Understanding the FDTD method ldquoJohn BSchneiderrdquo may 8 2011

3 3 D -ADI -Mehod-unconditionaly stable time domain algorithom for solving maxwell equations

ldquoIEEE Transaction on microwave theory amp techniques vol-48No10 october 2010 ldquoTakefumi

IEEE

4High order split step unconditionally stable FDTD method ampnumerical analysis ldquoIEEE

Transactions on antinna amp propagationvol 59 no9sept2011rdquo Yong-Dan Kong amp Qing xin chu

senior memberIEEE

5Genaral Finite DifferenceSchemes for Heat equations ldquoIndian J Pure Applemath

10(2)209-222Febuary 1979rdquoby PS Jain ampDN Holla department of methemetics

IITB 400076

  • Unconditionally Stable FDTD Methods
  • Outlineshelliphelliphelliphellip
  • Outlineshellip
  • Slide 4
  • Computational Electromagnetics
  • Slide 6
  • FDTD Overview ndash Updating Equations
  • Slide 8
  • Slide 9
  • Finite Difference Time Domain Method
  • The Yee Discretization (1966)
  • Slide 12
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Slide 17
  • Slide 18
  • Slide 19
  • Space Domain Discretization
  • Finite-Difference Formulation of the Heat Conduction on a Chip
  • Time domain discretization
  • Slide 23
  • Slide 24
  • Simple Explicit Method
  • Simple Implicit Method
  • Crank-Nicolson Method
  • Analysis of Crank-Nicolson Method
  • Alternating Direction Implicit Method
  • Slide 30
  • Peaceman-Rachford Algorithm
  • Douglas-Gunn Algorithm
  • Illustration for ADI
  • Analysis of ADI Method
  • Slide 35
  • Slide 36
  • Slide 37
  • Slide 38
  • Referenceshelliphelliphellip
  • Slide 40
Page 21: Unconditionally stable fdtd methods

21 march 2012 21

X

n

jiTn

j1-iT n

j1iT

n

1jiT

n

1jiT

(00) (i0)hellip

(0j)

x

y

(I0)

(0J)

hellip

hellip

hellip

Finite-Difference Formulation of the Heat Conduction on a Chip

bull Space Domainbull Time Domain

21 march 2012 22

Time domain discretization

bull Heat Conduction Equation

ndash Simple Explicit Method ndash Simple Implicit Methodndash Crank-Nicolson Method

gcy

T

x

T

t

TT

p

yxnn

1

)()( 2

2

2

21

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

23

Can we check if a numerical scheme

is stable without computation Von Neumann stability

analysisbull Analyze if (or for which conditions) a

numerical scheme is stable or unstablebull Makes a local analysis coefficients of PDE are

assumed to vary slowly (our example constant)

bull How will unavoidable errors (say rounding errors)evolve in time

John von Neumann1903-1957

24

Von Neumann stability analysis

A numerical scheme is unstable if

Ansatz Wave number k and amplification factor

21 march 2012 25

bull Accuracybull Stability Constraint

bull No matrix inversion but time steps are limited by space discretization

gcy

T

x

T

t

TT

p

ny

nx

nn

1

)()( 2

2

2

21

] [ 22 yxt

2

11122

yxt

Simple Explicit Method

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

21 march 2012 26

Accuracy

Unconditionally Stable

No limits on time step but involves with large scale matrix inversion

gcy

T

x

T

t

TT

p

ny

nx

nn

1

)()( 2

12

2

121

] [ 22 yxt

Simple Implicit Method

11

1

11

12

11

1

11

12

2

2nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

21 march 2012 27

Accuracy

Unconditionally stable

No limits on time step but involves with large scale matrix inversion

gcy

TT

x

TT

t

TT

p

ny

ny

nx

nx

nn

1

)(2)(2 2

212

2

2121

] [ 222 yxt

Crank-Nicolson Method

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

11

1

11

12

11

1

11

12

2

2nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

21 march 2012 28

n

m1616

ex m=4n=4

Total node number N = mn

Matrix size = NxN

Analysis of Crank-Nicolson Method

21 march 2012 29

Alternating Direction Implicit Method

Solves higher dimension problem by successive Lower dimension methods

Accuracy

Unconditionally stable

No limits on time step and no large scale matrix inversion

] [ 222 yxt

21 march2012 30

Step I x-direction implicit y-direction explicitStep II x-direction explicit y-direction implicit

n

Alternating Direction Implicit Method

bull Peaceman-Rachford Algorithmbull Douglas-Gunn Algorithm

21 march 2012 31

bull Step I

bull Step II

gc

tT

rrT

rr

p

ny

yx

xny

yx

x

)2

1)(2

1()2

1)(2

1( 22122

gc

tT

rT

r

p

ny

yn

xx

2)

21()

21( 22

12

gc

tT

rT

r

p

n

xxn

yy

2)

21()

21( 2

1212

Peaceman-Rachford Algorithm

21 march 2012 32

Douglas-Gunn Algorithm

bull Step I

bull Step II

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

12

1

22

gc

tTrTT

rTT

p

nyy

nnxxnn

22

122

1

)(2

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

2

121

2)(

2

21 march 2012 33

Illustration for ADI

X-direction implicit Y-direction implicit

i = 1 2 hellip m

1

njiT

Step I Step II

j = 1

n

2

hellip

1 2 hellip m

11

njiT

11

njiT

j = 1

n

2hellip

21

njiT

21

1

njiT

21

1

njiT

21 march 2012 34

Analysis of ADI Method

mm

Tridiagonal Matrix

Time complexity O(N)

2xnxm = 2nm =2N

2 steps n matrices tridaigonal matrix

X-direction implicit

21

njiT

i = 1 2 hellip mj = 1

n

2

hellip

21

1

njiT

21

1

njiT

ADVANTAGE over conditionally stable

Reduce simulation time

Good accuracy

Size of geometrical feature may be typical of wavelengh

Good geometrical flexibility to allow with corner or high curvature than fdtd

Advantages of the FDTD method over other methodsAdvantages of the FDTD method over other methods

bull It is conceptually simpleIt is conceptually simple

bull The algorithm does not require the formulation of integral equation and relatively complex The algorithm does not require the formulation of integral equation and relatively complex

scatters can be treated without inversion of large matrices scatters can be treated without inversion of large matrices

bull It is simple to implement for complicated inhomogeneous conducting or dielectric structures It is simple to implement for complicated inhomogeneous conducting or dielectric structures

because constitutive parameters can be assigned to each lattice pointbecause constitutive parameters can be assigned to each lattice point

bull Its computer memory requirement is not prohibitive for many complex structures of interest Its computer memory requirement is not prohibitive for many complex structures of interest

bull The algorithm make use of the memory in a simple sequential orderThe algorithm make use of the memory in a simple sequential order

bull It is much easier to obtain frequency domain data from time domain results than the converse It is much easier to obtain frequency domain data from time domain results than the converse

Thus it is more convenient to obtain frequency domain results via time domain when many Thus it is more convenient to obtain frequency domain results via time domain when many

frequencies are involvedfrequencies are involved

Include examples of unconditionally stable fdtd and prove the advantages

Disadvantages Disadvantages

bull Its implementation necessitates modeling object as well as its surroundings Thus the Its implementation necessitates modeling object as well as its surroundings Thus the

required program execution time may be excessiverequired program execution time may be excessive

bull Its accuracy is at least one order of magnitude worse than that of the method of Its accuracy is at least one order of magnitude worse than that of the method of

moments for example moments for example

bull Since the computational meshes are rectangular in shape they do not conform the Since the computational meshes are rectangular in shape they do not conform the

scatterers with curved surfaces as is the case of the cylindrical or spherical boundary scatterers with curved surfaces as is the case of the cylindrical or spherical boundary

Its computer memory requirement is not prohibitive for many complex structures of Its computer memory requirement is not prohibitive for many complex structures of

interest interest

bull As in all finite difference algorithms the field quantities are only known at grid nodes As in all finite difference algorithms the field quantities are only known at grid nodes

Applications of FDTD method

1048708Electromagnetic scattering amp antenna design1048708EMCEMI design1048708Simulation of wave propagation problem1048708Solving partial defferential equation 1048708Waveguide analysis

Stripline amp microstrip line analysishellipetc hellip

Referenceshelliphelliphellip1-Electromagnetic simulation using FDTD method

ldquoDennis M Sillevanrdquo IEEE press series on rf amp mivcrowave technology ldquoRoger DPollard amp Richard

Bootanrdquoseries

2Understanding the FDTD method ldquoJohn BSchneiderrdquo may 8 2011

3 3 D -ADI -Mehod-unconditionaly stable time domain algorithom for solving maxwell equations

ldquoIEEE Transaction on microwave theory amp techniques vol-48No10 october 2010 ldquoTakefumi

IEEE

4High order split step unconditionally stable FDTD method ampnumerical analysis ldquoIEEE

Transactions on antinna amp propagationvol 59 no9sept2011rdquo Yong-Dan Kong amp Qing xin chu

senior memberIEEE

5Genaral Finite DifferenceSchemes for Heat equations ldquoIndian J Pure Applemath

10(2)209-222Febuary 1979rdquoby PS Jain ampDN Holla department of methemetics

IITB 400076

  • Unconditionally Stable FDTD Methods
  • Outlineshelliphelliphelliphellip
  • Outlineshellip
  • Slide 4
  • Computational Electromagnetics
  • Slide 6
  • FDTD Overview ndash Updating Equations
  • Slide 8
  • Slide 9
  • Finite Difference Time Domain Method
  • The Yee Discretization (1966)
  • Slide 12
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Slide 17
  • Slide 18
  • Slide 19
  • Space Domain Discretization
  • Finite-Difference Formulation of the Heat Conduction on a Chip
  • Time domain discretization
  • Slide 23
  • Slide 24
  • Simple Explicit Method
  • Simple Implicit Method
  • Crank-Nicolson Method
  • Analysis of Crank-Nicolson Method
  • Alternating Direction Implicit Method
  • Slide 30
  • Peaceman-Rachford Algorithm
  • Douglas-Gunn Algorithm
  • Illustration for ADI
  • Analysis of ADI Method
  • Slide 35
  • Slide 36
  • Slide 37
  • Slide 38
  • Referenceshelliphelliphellip
  • Slide 40
Page 22: Unconditionally stable fdtd methods

21 march 2012 22

Time domain discretization

bull Heat Conduction Equation

ndash Simple Explicit Method ndash Simple Implicit Methodndash Crank-Nicolson Method

gcy

T

x

T

t

TT

p

yxnn

1

)()( 2

2

2

21

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

23

Can we check if a numerical scheme

is stable without computation Von Neumann stability

analysisbull Analyze if (or for which conditions) a

numerical scheme is stable or unstablebull Makes a local analysis coefficients of PDE are

assumed to vary slowly (our example constant)

bull How will unavoidable errors (say rounding errors)evolve in time

John von Neumann1903-1957

24

Von Neumann stability analysis

A numerical scheme is unstable if

Ansatz Wave number k and amplification factor

21 march 2012 25

bull Accuracybull Stability Constraint

bull No matrix inversion but time steps are limited by space discretization

gcy

T

x

T

t

TT

p

ny

nx

nn

1

)()( 2

2

2

21

] [ 22 yxt

2

11122

yxt

Simple Explicit Method

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

21 march 2012 26

Accuracy

Unconditionally Stable

No limits on time step but involves with large scale matrix inversion

gcy

T

x

T

t

TT

p

ny

nx

nn

1

)()( 2

12

2

121

] [ 22 yxt

Simple Implicit Method

11

1

11

12

11

1

11

12

2

2nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

21 march 2012 27

Accuracy

Unconditionally stable

No limits on time step but involves with large scale matrix inversion

gcy

TT

x

TT

t

TT

p

ny

ny

nx

nx

nn

1

)(2)(2 2

212

2

2121

] [ 222 yxt

Crank-Nicolson Method

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

11

1

11

12

11

1

11

12

2

2nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

21 march 2012 28

n

m1616

ex m=4n=4

Total node number N = mn

Matrix size = NxN

Analysis of Crank-Nicolson Method

21 march 2012 29

Alternating Direction Implicit Method

Solves higher dimension problem by successive Lower dimension methods

Accuracy

Unconditionally stable

No limits on time step and no large scale matrix inversion

] [ 222 yxt

21 march2012 30

Step I x-direction implicit y-direction explicitStep II x-direction explicit y-direction implicit

n

Alternating Direction Implicit Method

bull Peaceman-Rachford Algorithmbull Douglas-Gunn Algorithm

21 march 2012 31

bull Step I

bull Step II

gc

tT

rrT

rr

p

ny

yx

xny

yx

x

)2

1)(2

1()2

1)(2

1( 22122

gc

tT

rT

r

p

ny

yn

xx

2)

21()

21( 22

12

gc

tT

rT

r

p

n

xxn

yy

2)

21()

21( 2

1212

Peaceman-Rachford Algorithm

21 march 2012 32

Douglas-Gunn Algorithm

bull Step I

bull Step II

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

12

1

22

gc

tTrTT

rTT

p

nyy

nnxxnn

22

122

1

)(2

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

2

121

2)(

2

21 march 2012 33

Illustration for ADI

X-direction implicit Y-direction implicit

i = 1 2 hellip m

1

njiT

Step I Step II

j = 1

n

2

hellip

1 2 hellip m

11

njiT

11

njiT

j = 1

n

2hellip

21

njiT

21

1

njiT

21

1

njiT

21 march 2012 34

Analysis of ADI Method

mm

Tridiagonal Matrix

Time complexity O(N)

2xnxm = 2nm =2N

2 steps n matrices tridaigonal matrix

X-direction implicit

21

njiT

i = 1 2 hellip mj = 1

n

2

hellip

21

1

njiT

21

1

njiT

ADVANTAGE over conditionally stable

Reduce simulation time

Good accuracy

Size of geometrical feature may be typical of wavelengh

Good geometrical flexibility to allow with corner or high curvature than fdtd

Advantages of the FDTD method over other methodsAdvantages of the FDTD method over other methods

bull It is conceptually simpleIt is conceptually simple

bull The algorithm does not require the formulation of integral equation and relatively complex The algorithm does not require the formulation of integral equation and relatively complex

scatters can be treated without inversion of large matrices scatters can be treated without inversion of large matrices

bull It is simple to implement for complicated inhomogeneous conducting or dielectric structures It is simple to implement for complicated inhomogeneous conducting or dielectric structures

because constitutive parameters can be assigned to each lattice pointbecause constitutive parameters can be assigned to each lattice point

bull Its computer memory requirement is not prohibitive for many complex structures of interest Its computer memory requirement is not prohibitive for many complex structures of interest

bull The algorithm make use of the memory in a simple sequential orderThe algorithm make use of the memory in a simple sequential order

bull It is much easier to obtain frequency domain data from time domain results than the converse It is much easier to obtain frequency domain data from time domain results than the converse

Thus it is more convenient to obtain frequency domain results via time domain when many Thus it is more convenient to obtain frequency domain results via time domain when many

frequencies are involvedfrequencies are involved

Include examples of unconditionally stable fdtd and prove the advantages

Disadvantages Disadvantages

bull Its implementation necessitates modeling object as well as its surroundings Thus the Its implementation necessitates modeling object as well as its surroundings Thus the

required program execution time may be excessiverequired program execution time may be excessive

bull Its accuracy is at least one order of magnitude worse than that of the method of Its accuracy is at least one order of magnitude worse than that of the method of

moments for example moments for example

bull Since the computational meshes are rectangular in shape they do not conform the Since the computational meshes are rectangular in shape they do not conform the

scatterers with curved surfaces as is the case of the cylindrical or spherical boundary scatterers with curved surfaces as is the case of the cylindrical or spherical boundary

Its computer memory requirement is not prohibitive for many complex structures of Its computer memory requirement is not prohibitive for many complex structures of

interest interest

bull As in all finite difference algorithms the field quantities are only known at grid nodes As in all finite difference algorithms the field quantities are only known at grid nodes

Applications of FDTD method

1048708Electromagnetic scattering amp antenna design1048708EMCEMI design1048708Simulation of wave propagation problem1048708Solving partial defferential equation 1048708Waveguide analysis

Stripline amp microstrip line analysishellipetc hellip

Referenceshelliphelliphellip1-Electromagnetic simulation using FDTD method

ldquoDennis M Sillevanrdquo IEEE press series on rf amp mivcrowave technology ldquoRoger DPollard amp Richard

Bootanrdquoseries

2Understanding the FDTD method ldquoJohn BSchneiderrdquo may 8 2011

3 3 D -ADI -Mehod-unconditionaly stable time domain algorithom for solving maxwell equations

ldquoIEEE Transaction on microwave theory amp techniques vol-48No10 october 2010 ldquoTakefumi

IEEE

4High order split step unconditionally stable FDTD method ampnumerical analysis ldquoIEEE

Transactions on antinna amp propagationvol 59 no9sept2011rdquo Yong-Dan Kong amp Qing xin chu

senior memberIEEE

5Genaral Finite DifferenceSchemes for Heat equations ldquoIndian J Pure Applemath

10(2)209-222Febuary 1979rdquoby PS Jain ampDN Holla department of methemetics

IITB 400076

  • Unconditionally Stable FDTD Methods
  • Outlineshelliphelliphelliphellip
  • Outlineshellip
  • Slide 4
  • Computational Electromagnetics
  • Slide 6
  • FDTD Overview ndash Updating Equations
  • Slide 8
  • Slide 9
  • Finite Difference Time Domain Method
  • The Yee Discretization (1966)
  • Slide 12
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Slide 17
  • Slide 18
  • Slide 19
  • Space Domain Discretization
  • Finite-Difference Formulation of the Heat Conduction on a Chip
  • Time domain discretization
  • Slide 23
  • Slide 24
  • Simple Explicit Method
  • Simple Implicit Method
  • Crank-Nicolson Method
  • Analysis of Crank-Nicolson Method
  • Alternating Direction Implicit Method
  • Slide 30
  • Peaceman-Rachford Algorithm
  • Douglas-Gunn Algorithm
  • Illustration for ADI
  • Analysis of ADI Method
  • Slide 35
  • Slide 36
  • Slide 37
  • Slide 38
  • Referenceshelliphelliphellip
  • Slide 40
Page 23: Unconditionally stable fdtd methods

23

Can we check if a numerical scheme

is stable without computation Von Neumann stability

analysisbull Analyze if (or for which conditions) a

numerical scheme is stable or unstablebull Makes a local analysis coefficients of PDE are

assumed to vary slowly (our example constant)

bull How will unavoidable errors (say rounding errors)evolve in time

John von Neumann1903-1957

24

Von Neumann stability analysis

A numerical scheme is unstable if

Ansatz Wave number k and amplification factor

21 march 2012 25

bull Accuracybull Stability Constraint

bull No matrix inversion but time steps are limited by space discretization

gcy

T

x

T

t

TT

p

ny

nx

nn

1

)()( 2

2

2

21

] [ 22 yxt

2

11122

yxt

Simple Explicit Method

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

21 march 2012 26

Accuracy

Unconditionally Stable

No limits on time step but involves with large scale matrix inversion

gcy

T

x

T

t

TT

p

ny

nx

nn

1

)()( 2

12

2

121

] [ 22 yxt

Simple Implicit Method

11

1

11

12

11

1

11

12

2

2nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

21 march 2012 27

Accuracy

Unconditionally stable

No limits on time step but involves with large scale matrix inversion

gcy

TT

x

TT

t

TT

p

ny

ny

nx

nx

nn

1

)(2)(2 2

212

2

2121

] [ 222 yxt

Crank-Nicolson Method

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

11

1

11

12

11

1

11

12

2

2nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

21 march 2012 28

n

m1616

ex m=4n=4

Total node number N = mn

Matrix size = NxN

Analysis of Crank-Nicolson Method

21 march 2012 29

Alternating Direction Implicit Method

Solves higher dimension problem by successive Lower dimension methods

Accuracy

Unconditionally stable

No limits on time step and no large scale matrix inversion

] [ 222 yxt

21 march2012 30

Step I x-direction implicit y-direction explicitStep II x-direction explicit y-direction implicit

n

Alternating Direction Implicit Method

bull Peaceman-Rachford Algorithmbull Douglas-Gunn Algorithm

21 march 2012 31

bull Step I

bull Step II

gc

tT

rrT

rr

p

ny

yx

xny

yx

x

)2

1)(2

1()2

1)(2

1( 22122

gc

tT

rT

r

p

ny

yn

xx

2)

21()

21( 22

12

gc

tT

rT

r

p

n

xxn

yy

2)

21()

21( 2

1212

Peaceman-Rachford Algorithm

21 march 2012 32

Douglas-Gunn Algorithm

bull Step I

bull Step II

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

12

1

22

gc

tTrTT

rTT

p

nyy

nnxxnn

22

122

1

)(2

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

2

121

2)(

2

21 march 2012 33

Illustration for ADI

X-direction implicit Y-direction implicit

i = 1 2 hellip m

1

njiT

Step I Step II

j = 1

n

2

hellip

1 2 hellip m

11

njiT

11

njiT

j = 1

n

2hellip

21

njiT

21

1

njiT

21

1

njiT

21 march 2012 34

Analysis of ADI Method

mm

Tridiagonal Matrix

Time complexity O(N)

2xnxm = 2nm =2N

2 steps n matrices tridaigonal matrix

X-direction implicit

21

njiT

i = 1 2 hellip mj = 1

n

2

hellip

21

1

njiT

21

1

njiT

ADVANTAGE over conditionally stable

Reduce simulation time

Good accuracy

Size of geometrical feature may be typical of wavelengh

Good geometrical flexibility to allow with corner or high curvature than fdtd

Advantages of the FDTD method over other methodsAdvantages of the FDTD method over other methods

bull It is conceptually simpleIt is conceptually simple

bull The algorithm does not require the formulation of integral equation and relatively complex The algorithm does not require the formulation of integral equation and relatively complex

scatters can be treated without inversion of large matrices scatters can be treated without inversion of large matrices

bull It is simple to implement for complicated inhomogeneous conducting or dielectric structures It is simple to implement for complicated inhomogeneous conducting or dielectric structures

because constitutive parameters can be assigned to each lattice pointbecause constitutive parameters can be assigned to each lattice point

bull Its computer memory requirement is not prohibitive for many complex structures of interest Its computer memory requirement is not prohibitive for many complex structures of interest

bull The algorithm make use of the memory in a simple sequential orderThe algorithm make use of the memory in a simple sequential order

bull It is much easier to obtain frequency domain data from time domain results than the converse It is much easier to obtain frequency domain data from time domain results than the converse

Thus it is more convenient to obtain frequency domain results via time domain when many Thus it is more convenient to obtain frequency domain results via time domain when many

frequencies are involvedfrequencies are involved

Include examples of unconditionally stable fdtd and prove the advantages

Disadvantages Disadvantages

bull Its implementation necessitates modeling object as well as its surroundings Thus the Its implementation necessitates modeling object as well as its surroundings Thus the

required program execution time may be excessiverequired program execution time may be excessive

bull Its accuracy is at least one order of magnitude worse than that of the method of Its accuracy is at least one order of magnitude worse than that of the method of

moments for example moments for example

bull Since the computational meshes are rectangular in shape they do not conform the Since the computational meshes are rectangular in shape they do not conform the

scatterers with curved surfaces as is the case of the cylindrical or spherical boundary scatterers with curved surfaces as is the case of the cylindrical or spherical boundary

Its computer memory requirement is not prohibitive for many complex structures of Its computer memory requirement is not prohibitive for many complex structures of

interest interest

bull As in all finite difference algorithms the field quantities are only known at grid nodes As in all finite difference algorithms the field quantities are only known at grid nodes

Applications of FDTD method

1048708Electromagnetic scattering amp antenna design1048708EMCEMI design1048708Simulation of wave propagation problem1048708Solving partial defferential equation 1048708Waveguide analysis

Stripline amp microstrip line analysishellipetc hellip

Referenceshelliphelliphellip1-Electromagnetic simulation using FDTD method

ldquoDennis M Sillevanrdquo IEEE press series on rf amp mivcrowave technology ldquoRoger DPollard amp Richard

Bootanrdquoseries

2Understanding the FDTD method ldquoJohn BSchneiderrdquo may 8 2011

3 3 D -ADI -Mehod-unconditionaly stable time domain algorithom for solving maxwell equations

ldquoIEEE Transaction on microwave theory amp techniques vol-48No10 october 2010 ldquoTakefumi

IEEE

4High order split step unconditionally stable FDTD method ampnumerical analysis ldquoIEEE

Transactions on antinna amp propagationvol 59 no9sept2011rdquo Yong-Dan Kong amp Qing xin chu

senior memberIEEE

5Genaral Finite DifferenceSchemes for Heat equations ldquoIndian J Pure Applemath

10(2)209-222Febuary 1979rdquoby PS Jain ampDN Holla department of methemetics

IITB 400076

  • Unconditionally Stable FDTD Methods
  • Outlineshelliphelliphelliphellip
  • Outlineshellip
  • Slide 4
  • Computational Electromagnetics
  • Slide 6
  • FDTD Overview ndash Updating Equations
  • Slide 8
  • Slide 9
  • Finite Difference Time Domain Method
  • The Yee Discretization (1966)
  • Slide 12
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Slide 17
  • Slide 18
  • Slide 19
  • Space Domain Discretization
  • Finite-Difference Formulation of the Heat Conduction on a Chip
  • Time domain discretization
  • Slide 23
  • Slide 24
  • Simple Explicit Method
  • Simple Implicit Method
  • Crank-Nicolson Method
  • Analysis of Crank-Nicolson Method
  • Alternating Direction Implicit Method
  • Slide 30
  • Peaceman-Rachford Algorithm
  • Douglas-Gunn Algorithm
  • Illustration for ADI
  • Analysis of ADI Method
  • Slide 35
  • Slide 36
  • Slide 37
  • Slide 38
  • Referenceshelliphelliphellip
  • Slide 40
Page 24: Unconditionally stable fdtd methods

24

Von Neumann stability analysis

A numerical scheme is unstable if

Ansatz Wave number k and amplification factor

21 march 2012 25

bull Accuracybull Stability Constraint

bull No matrix inversion but time steps are limited by space discretization

gcy

T

x

T

t

TT

p

ny

nx

nn

1

)()( 2

2

2

21

] [ 22 yxt

2

11122

yxt

Simple Explicit Method

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

21 march 2012 26

Accuracy

Unconditionally Stable

No limits on time step but involves with large scale matrix inversion

gcy

T

x

T

t

TT

p

ny

nx

nn

1

)()( 2

12

2

121

] [ 22 yxt

Simple Implicit Method

11

1

11

12

11

1

11

12

2

2nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

21 march 2012 27

Accuracy

Unconditionally stable

No limits on time step but involves with large scale matrix inversion

gcy

TT

x

TT

t

TT

p

ny

ny

nx

nx

nn

1

)(2)(2 2

212

2

2121

] [ 222 yxt

Crank-Nicolson Method

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

11

1

11

12

11

1

11

12

2

2nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

21 march 2012 28

n

m1616

ex m=4n=4

Total node number N = mn

Matrix size = NxN

Analysis of Crank-Nicolson Method

21 march 2012 29

Alternating Direction Implicit Method

Solves higher dimension problem by successive Lower dimension methods

Accuracy

Unconditionally stable

No limits on time step and no large scale matrix inversion

] [ 222 yxt

21 march2012 30

Step I x-direction implicit y-direction explicitStep II x-direction explicit y-direction implicit

n

Alternating Direction Implicit Method

bull Peaceman-Rachford Algorithmbull Douglas-Gunn Algorithm

21 march 2012 31

bull Step I

bull Step II

gc

tT

rrT

rr

p

ny

yx

xny

yx

x

)2

1)(2

1()2

1)(2

1( 22122

gc

tT

rT

r

p

ny

yn

xx

2)

21()

21( 22

12

gc

tT

rT

r

p

n

xxn

yy

2)

21()

21( 2

1212

Peaceman-Rachford Algorithm

21 march 2012 32

Douglas-Gunn Algorithm

bull Step I

bull Step II

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

12

1

22

gc

tTrTT

rTT

p

nyy

nnxxnn

22

122

1

)(2

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

2

121

2)(

2

21 march 2012 33

Illustration for ADI

X-direction implicit Y-direction implicit

i = 1 2 hellip m

1

njiT

Step I Step II

j = 1

n

2

hellip

1 2 hellip m

11

njiT

11

njiT

j = 1

n

2hellip

21

njiT

21

1

njiT

21

1

njiT

21 march 2012 34

Analysis of ADI Method

mm

Tridiagonal Matrix

Time complexity O(N)

2xnxm = 2nm =2N

2 steps n matrices tridaigonal matrix

X-direction implicit

21

njiT

i = 1 2 hellip mj = 1

n

2

hellip

21

1

njiT

21

1

njiT

ADVANTAGE over conditionally stable

Reduce simulation time

Good accuracy

Size of geometrical feature may be typical of wavelengh

Good geometrical flexibility to allow with corner or high curvature than fdtd

Advantages of the FDTD method over other methodsAdvantages of the FDTD method over other methods

bull It is conceptually simpleIt is conceptually simple

bull The algorithm does not require the formulation of integral equation and relatively complex The algorithm does not require the formulation of integral equation and relatively complex

scatters can be treated without inversion of large matrices scatters can be treated without inversion of large matrices

bull It is simple to implement for complicated inhomogeneous conducting or dielectric structures It is simple to implement for complicated inhomogeneous conducting or dielectric structures

because constitutive parameters can be assigned to each lattice pointbecause constitutive parameters can be assigned to each lattice point

bull Its computer memory requirement is not prohibitive for many complex structures of interest Its computer memory requirement is not prohibitive for many complex structures of interest

bull The algorithm make use of the memory in a simple sequential orderThe algorithm make use of the memory in a simple sequential order

bull It is much easier to obtain frequency domain data from time domain results than the converse It is much easier to obtain frequency domain data from time domain results than the converse

Thus it is more convenient to obtain frequency domain results via time domain when many Thus it is more convenient to obtain frequency domain results via time domain when many

frequencies are involvedfrequencies are involved

Include examples of unconditionally stable fdtd and prove the advantages

Disadvantages Disadvantages

bull Its implementation necessitates modeling object as well as its surroundings Thus the Its implementation necessitates modeling object as well as its surroundings Thus the

required program execution time may be excessiverequired program execution time may be excessive

bull Its accuracy is at least one order of magnitude worse than that of the method of Its accuracy is at least one order of magnitude worse than that of the method of

moments for example moments for example

bull Since the computational meshes are rectangular in shape they do not conform the Since the computational meshes are rectangular in shape they do not conform the

scatterers with curved surfaces as is the case of the cylindrical or spherical boundary scatterers with curved surfaces as is the case of the cylindrical or spherical boundary

Its computer memory requirement is not prohibitive for many complex structures of Its computer memory requirement is not prohibitive for many complex structures of

interest interest

bull As in all finite difference algorithms the field quantities are only known at grid nodes As in all finite difference algorithms the field quantities are only known at grid nodes

Applications of FDTD method

1048708Electromagnetic scattering amp antenna design1048708EMCEMI design1048708Simulation of wave propagation problem1048708Solving partial defferential equation 1048708Waveguide analysis

Stripline amp microstrip line analysishellipetc hellip

Referenceshelliphelliphellip1-Electromagnetic simulation using FDTD method

ldquoDennis M Sillevanrdquo IEEE press series on rf amp mivcrowave technology ldquoRoger DPollard amp Richard

Bootanrdquoseries

2Understanding the FDTD method ldquoJohn BSchneiderrdquo may 8 2011

3 3 D -ADI -Mehod-unconditionaly stable time domain algorithom for solving maxwell equations

ldquoIEEE Transaction on microwave theory amp techniques vol-48No10 october 2010 ldquoTakefumi

IEEE

4High order split step unconditionally stable FDTD method ampnumerical analysis ldquoIEEE

Transactions on antinna amp propagationvol 59 no9sept2011rdquo Yong-Dan Kong amp Qing xin chu

senior memberIEEE

5Genaral Finite DifferenceSchemes for Heat equations ldquoIndian J Pure Applemath

10(2)209-222Febuary 1979rdquoby PS Jain ampDN Holla department of methemetics

IITB 400076

  • Unconditionally Stable FDTD Methods
  • Outlineshelliphelliphelliphellip
  • Outlineshellip
  • Slide 4
  • Computational Electromagnetics
  • Slide 6
  • FDTD Overview ndash Updating Equations
  • Slide 8
  • Slide 9
  • Finite Difference Time Domain Method
  • The Yee Discretization (1966)
  • Slide 12
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Slide 17
  • Slide 18
  • Slide 19
  • Space Domain Discretization
  • Finite-Difference Formulation of the Heat Conduction on a Chip
  • Time domain discretization
  • Slide 23
  • Slide 24
  • Simple Explicit Method
  • Simple Implicit Method
  • Crank-Nicolson Method
  • Analysis of Crank-Nicolson Method
  • Alternating Direction Implicit Method
  • Slide 30
  • Peaceman-Rachford Algorithm
  • Douglas-Gunn Algorithm
  • Illustration for ADI
  • Analysis of ADI Method
  • Slide 35
  • Slide 36
  • Slide 37
  • Slide 38
  • Referenceshelliphelliphellip
  • Slide 40
Page 25: Unconditionally stable fdtd methods

21 march 2012 25

bull Accuracybull Stability Constraint

bull No matrix inversion but time steps are limited by space discretization

gcy

T

x

T

t

TT

p

ny

nx

nn

1

)()( 2

2

2

21

] [ 22 yxt

2

11122

yxt

Simple Explicit Method

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

21 march 2012 26

Accuracy

Unconditionally Stable

No limits on time step but involves with large scale matrix inversion

gcy

T

x

T

t

TT

p

ny

nx

nn

1

)()( 2

12

2

121

] [ 22 yxt

Simple Implicit Method

11

1

11

12

11

1

11

12

2

2nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

21 march 2012 27

Accuracy

Unconditionally stable

No limits on time step but involves with large scale matrix inversion

gcy

TT

x

TT

t

TT

p

ny

ny

nx

nx

nn

1

)(2)(2 2

212

2

2121

] [ 222 yxt

Crank-Nicolson Method

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

11

1

11

12

11

1

11

12

2

2nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

21 march 2012 28

n

m1616

ex m=4n=4

Total node number N = mn

Matrix size = NxN

Analysis of Crank-Nicolson Method

21 march 2012 29

Alternating Direction Implicit Method

Solves higher dimension problem by successive Lower dimension methods

Accuracy

Unconditionally stable

No limits on time step and no large scale matrix inversion

] [ 222 yxt

21 march2012 30

Step I x-direction implicit y-direction explicitStep II x-direction explicit y-direction implicit

n

Alternating Direction Implicit Method

bull Peaceman-Rachford Algorithmbull Douglas-Gunn Algorithm

21 march 2012 31

bull Step I

bull Step II

gc

tT

rrT

rr

p

ny

yx

xny

yx

x

)2

1)(2

1()2

1)(2

1( 22122

gc

tT

rT

r

p

ny

yn

xx

2)

21()

21( 22

12

gc

tT

rT

r

p

n

xxn

yy

2)

21()

21( 2

1212

Peaceman-Rachford Algorithm

21 march 2012 32

Douglas-Gunn Algorithm

bull Step I

bull Step II

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

12

1

22

gc

tTrTT

rTT

p

nyy

nnxxnn

22

122

1

)(2

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

2

121

2)(

2

21 march 2012 33

Illustration for ADI

X-direction implicit Y-direction implicit

i = 1 2 hellip m

1

njiT

Step I Step II

j = 1

n

2

hellip

1 2 hellip m

11

njiT

11

njiT

j = 1

n

2hellip

21

njiT

21

1

njiT

21

1

njiT

21 march 2012 34

Analysis of ADI Method

mm

Tridiagonal Matrix

Time complexity O(N)

2xnxm = 2nm =2N

2 steps n matrices tridaigonal matrix

X-direction implicit

21

njiT

i = 1 2 hellip mj = 1

n

2

hellip

21

1

njiT

21

1

njiT

ADVANTAGE over conditionally stable

Reduce simulation time

Good accuracy

Size of geometrical feature may be typical of wavelengh

Good geometrical flexibility to allow with corner or high curvature than fdtd

Advantages of the FDTD method over other methodsAdvantages of the FDTD method over other methods

bull It is conceptually simpleIt is conceptually simple

bull The algorithm does not require the formulation of integral equation and relatively complex The algorithm does not require the formulation of integral equation and relatively complex

scatters can be treated without inversion of large matrices scatters can be treated without inversion of large matrices

bull It is simple to implement for complicated inhomogeneous conducting or dielectric structures It is simple to implement for complicated inhomogeneous conducting or dielectric structures

because constitutive parameters can be assigned to each lattice pointbecause constitutive parameters can be assigned to each lattice point

bull Its computer memory requirement is not prohibitive for many complex structures of interest Its computer memory requirement is not prohibitive for many complex structures of interest

bull The algorithm make use of the memory in a simple sequential orderThe algorithm make use of the memory in a simple sequential order

bull It is much easier to obtain frequency domain data from time domain results than the converse It is much easier to obtain frequency domain data from time domain results than the converse

Thus it is more convenient to obtain frequency domain results via time domain when many Thus it is more convenient to obtain frequency domain results via time domain when many

frequencies are involvedfrequencies are involved

Include examples of unconditionally stable fdtd and prove the advantages

Disadvantages Disadvantages

bull Its implementation necessitates modeling object as well as its surroundings Thus the Its implementation necessitates modeling object as well as its surroundings Thus the

required program execution time may be excessiverequired program execution time may be excessive

bull Its accuracy is at least one order of magnitude worse than that of the method of Its accuracy is at least one order of magnitude worse than that of the method of

moments for example moments for example

bull Since the computational meshes are rectangular in shape they do not conform the Since the computational meshes are rectangular in shape they do not conform the

scatterers with curved surfaces as is the case of the cylindrical or spherical boundary scatterers with curved surfaces as is the case of the cylindrical or spherical boundary

Its computer memory requirement is not prohibitive for many complex structures of Its computer memory requirement is not prohibitive for many complex structures of

interest interest

bull As in all finite difference algorithms the field quantities are only known at grid nodes As in all finite difference algorithms the field quantities are only known at grid nodes

Applications of FDTD method

1048708Electromagnetic scattering amp antenna design1048708EMCEMI design1048708Simulation of wave propagation problem1048708Solving partial defferential equation 1048708Waveguide analysis

Stripline amp microstrip line analysishellipetc hellip

Referenceshelliphelliphellip1-Electromagnetic simulation using FDTD method

ldquoDennis M Sillevanrdquo IEEE press series on rf amp mivcrowave technology ldquoRoger DPollard amp Richard

Bootanrdquoseries

2Understanding the FDTD method ldquoJohn BSchneiderrdquo may 8 2011

3 3 D -ADI -Mehod-unconditionaly stable time domain algorithom for solving maxwell equations

ldquoIEEE Transaction on microwave theory amp techniques vol-48No10 october 2010 ldquoTakefumi

IEEE

4High order split step unconditionally stable FDTD method ampnumerical analysis ldquoIEEE

Transactions on antinna amp propagationvol 59 no9sept2011rdquo Yong-Dan Kong amp Qing xin chu

senior memberIEEE

5Genaral Finite DifferenceSchemes for Heat equations ldquoIndian J Pure Applemath

10(2)209-222Febuary 1979rdquoby PS Jain ampDN Holla department of methemetics

IITB 400076

  • Unconditionally Stable FDTD Methods
  • Outlineshelliphelliphelliphellip
  • Outlineshellip
  • Slide 4
  • Computational Electromagnetics
  • Slide 6
  • FDTD Overview ndash Updating Equations
  • Slide 8
  • Slide 9
  • Finite Difference Time Domain Method
  • The Yee Discretization (1966)
  • Slide 12
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Slide 17
  • Slide 18
  • Slide 19
  • Space Domain Discretization
  • Finite-Difference Formulation of the Heat Conduction on a Chip
  • Time domain discretization
  • Slide 23
  • Slide 24
  • Simple Explicit Method
  • Simple Implicit Method
  • Crank-Nicolson Method
  • Analysis of Crank-Nicolson Method
  • Alternating Direction Implicit Method
  • Slide 30
  • Peaceman-Rachford Algorithm
  • Douglas-Gunn Algorithm
  • Illustration for ADI
  • Analysis of ADI Method
  • Slide 35
  • Slide 36
  • Slide 37
  • Slide 38
  • Referenceshelliphelliphellip
  • Slide 40
Page 26: Unconditionally stable fdtd methods

21 march 2012 26

Accuracy

Unconditionally Stable

No limits on time step but involves with large scale matrix inversion

gcy

T

x

T

t

TT

p

ny

nx

nn

1

)()( 2

12

2

121

] [ 22 yxt

Simple Implicit Method

11

1

11

12

11

1

11

12

2

2nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

21 march 2012 27

Accuracy

Unconditionally stable

No limits on time step but involves with large scale matrix inversion

gcy

TT

x

TT

t

TT

p

ny

ny

nx

nx

nn

1

)(2)(2 2

212

2

2121

] [ 222 yxt

Crank-Nicolson Method

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

11

1

11

12

11

1

11

12

2

2nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

21 march 2012 28

n

m1616

ex m=4n=4

Total node number N = mn

Matrix size = NxN

Analysis of Crank-Nicolson Method

21 march 2012 29

Alternating Direction Implicit Method

Solves higher dimension problem by successive Lower dimension methods

Accuracy

Unconditionally stable

No limits on time step and no large scale matrix inversion

] [ 222 yxt

21 march2012 30

Step I x-direction implicit y-direction explicitStep II x-direction explicit y-direction implicit

n

Alternating Direction Implicit Method

bull Peaceman-Rachford Algorithmbull Douglas-Gunn Algorithm

21 march 2012 31

bull Step I

bull Step II

gc

tT

rrT

rr

p

ny

yx

xny

yx

x

)2

1)(2

1()2

1)(2

1( 22122

gc

tT

rT

r

p

ny

yn

xx

2)

21()

21( 22

12

gc

tT

rT

r

p

n

xxn

yy

2)

21()

21( 2

1212

Peaceman-Rachford Algorithm

21 march 2012 32

Douglas-Gunn Algorithm

bull Step I

bull Step II

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

12

1

22

gc

tTrTT

rTT

p

nyy

nnxxnn

22

122

1

)(2

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

2

121

2)(

2

21 march 2012 33

Illustration for ADI

X-direction implicit Y-direction implicit

i = 1 2 hellip m

1

njiT

Step I Step II

j = 1

n

2

hellip

1 2 hellip m

11

njiT

11

njiT

j = 1

n

2hellip

21

njiT

21

1

njiT

21

1

njiT

21 march 2012 34

Analysis of ADI Method

mm

Tridiagonal Matrix

Time complexity O(N)

2xnxm = 2nm =2N

2 steps n matrices tridaigonal matrix

X-direction implicit

21

njiT

i = 1 2 hellip mj = 1

n

2

hellip

21

1

njiT

21

1

njiT

ADVANTAGE over conditionally stable

Reduce simulation time

Good accuracy

Size of geometrical feature may be typical of wavelengh

Good geometrical flexibility to allow with corner or high curvature than fdtd

Advantages of the FDTD method over other methodsAdvantages of the FDTD method over other methods

bull It is conceptually simpleIt is conceptually simple

bull The algorithm does not require the formulation of integral equation and relatively complex The algorithm does not require the formulation of integral equation and relatively complex

scatters can be treated without inversion of large matrices scatters can be treated without inversion of large matrices

bull It is simple to implement for complicated inhomogeneous conducting or dielectric structures It is simple to implement for complicated inhomogeneous conducting or dielectric structures

because constitutive parameters can be assigned to each lattice pointbecause constitutive parameters can be assigned to each lattice point

bull Its computer memory requirement is not prohibitive for many complex structures of interest Its computer memory requirement is not prohibitive for many complex structures of interest

bull The algorithm make use of the memory in a simple sequential orderThe algorithm make use of the memory in a simple sequential order

bull It is much easier to obtain frequency domain data from time domain results than the converse It is much easier to obtain frequency domain data from time domain results than the converse

Thus it is more convenient to obtain frequency domain results via time domain when many Thus it is more convenient to obtain frequency domain results via time domain when many

frequencies are involvedfrequencies are involved

Include examples of unconditionally stable fdtd and prove the advantages

Disadvantages Disadvantages

bull Its implementation necessitates modeling object as well as its surroundings Thus the Its implementation necessitates modeling object as well as its surroundings Thus the

required program execution time may be excessiverequired program execution time may be excessive

bull Its accuracy is at least one order of magnitude worse than that of the method of Its accuracy is at least one order of magnitude worse than that of the method of

moments for example moments for example

bull Since the computational meshes are rectangular in shape they do not conform the Since the computational meshes are rectangular in shape they do not conform the

scatterers with curved surfaces as is the case of the cylindrical or spherical boundary scatterers with curved surfaces as is the case of the cylindrical or spherical boundary

Its computer memory requirement is not prohibitive for many complex structures of Its computer memory requirement is not prohibitive for many complex structures of

interest interest

bull As in all finite difference algorithms the field quantities are only known at grid nodes As in all finite difference algorithms the field quantities are only known at grid nodes

Applications of FDTD method

1048708Electromagnetic scattering amp antenna design1048708EMCEMI design1048708Simulation of wave propagation problem1048708Solving partial defferential equation 1048708Waveguide analysis

Stripline amp microstrip line analysishellipetc hellip

Referenceshelliphelliphellip1-Electromagnetic simulation using FDTD method

ldquoDennis M Sillevanrdquo IEEE press series on rf amp mivcrowave technology ldquoRoger DPollard amp Richard

Bootanrdquoseries

2Understanding the FDTD method ldquoJohn BSchneiderrdquo may 8 2011

3 3 D -ADI -Mehod-unconditionaly stable time domain algorithom for solving maxwell equations

ldquoIEEE Transaction on microwave theory amp techniques vol-48No10 october 2010 ldquoTakefumi

IEEE

4High order split step unconditionally stable FDTD method ampnumerical analysis ldquoIEEE

Transactions on antinna amp propagationvol 59 no9sept2011rdquo Yong-Dan Kong amp Qing xin chu

senior memberIEEE

5Genaral Finite DifferenceSchemes for Heat equations ldquoIndian J Pure Applemath

10(2)209-222Febuary 1979rdquoby PS Jain ampDN Holla department of methemetics

IITB 400076

  • Unconditionally Stable FDTD Methods
  • Outlineshelliphelliphelliphellip
  • Outlineshellip
  • Slide 4
  • Computational Electromagnetics
  • Slide 6
  • FDTD Overview ndash Updating Equations
  • Slide 8
  • Slide 9
  • Finite Difference Time Domain Method
  • The Yee Discretization (1966)
  • Slide 12
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Slide 17
  • Slide 18
  • Slide 19
  • Space Domain Discretization
  • Finite-Difference Formulation of the Heat Conduction on a Chip
  • Time domain discretization
  • Slide 23
  • Slide 24
  • Simple Explicit Method
  • Simple Implicit Method
  • Crank-Nicolson Method
  • Analysis of Crank-Nicolson Method
  • Alternating Direction Implicit Method
  • Slide 30
  • Peaceman-Rachford Algorithm
  • Douglas-Gunn Algorithm
  • Illustration for ADI
  • Analysis of ADI Method
  • Slide 35
  • Slide 36
  • Slide 37
  • Slide 38
  • Referenceshelliphelliphellip
  • Slide 40
Page 27: Unconditionally stable fdtd methods

21 march 2012 27

Accuracy

Unconditionally stable

No limits on time step but involves with large scale matrix inversion

gcy

TT

x

TT

t

TT

p

ny

ny

nx

nx

nn

1

)(2)(2 2

212

2

2121

] [ 222 yxt

Crank-Nicolson Method

nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

112

112

2

2

11

1

11

12

11

1

11

12

2

2nji

nji

nji

ny

nji

nji

nji

nx

TTTT

TTTT

21 march 2012 28

n

m1616

ex m=4n=4

Total node number N = mn

Matrix size = NxN

Analysis of Crank-Nicolson Method

21 march 2012 29

Alternating Direction Implicit Method

Solves higher dimension problem by successive Lower dimension methods

Accuracy

Unconditionally stable

No limits on time step and no large scale matrix inversion

] [ 222 yxt

21 march2012 30

Step I x-direction implicit y-direction explicitStep II x-direction explicit y-direction implicit

n

Alternating Direction Implicit Method

bull Peaceman-Rachford Algorithmbull Douglas-Gunn Algorithm

21 march 2012 31

bull Step I

bull Step II

gc

tT

rrT

rr

p

ny

yx

xny

yx

x

)2

1)(2

1()2

1)(2

1( 22122

gc

tT

rT

r

p

ny

yn

xx

2)

21()

21( 22

12

gc

tT

rT

r

p

n

xxn

yy

2)

21()

21( 2

1212

Peaceman-Rachford Algorithm

21 march 2012 32

Douglas-Gunn Algorithm

bull Step I

bull Step II

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

12

1

22

gc

tTrTT

rTT

p

nyy

nnxxnn

22

122

1

)(2

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

2

121

2)(

2

21 march 2012 33

Illustration for ADI

X-direction implicit Y-direction implicit

i = 1 2 hellip m

1

njiT

Step I Step II

j = 1

n

2

hellip

1 2 hellip m

11

njiT

11

njiT

j = 1

n

2hellip

21

njiT

21

1

njiT

21

1

njiT

21 march 2012 34

Analysis of ADI Method

mm

Tridiagonal Matrix

Time complexity O(N)

2xnxm = 2nm =2N

2 steps n matrices tridaigonal matrix

X-direction implicit

21

njiT

i = 1 2 hellip mj = 1

n

2

hellip

21

1

njiT

21

1

njiT

ADVANTAGE over conditionally stable

Reduce simulation time

Good accuracy

Size of geometrical feature may be typical of wavelengh

Good geometrical flexibility to allow with corner or high curvature than fdtd

Advantages of the FDTD method over other methodsAdvantages of the FDTD method over other methods

bull It is conceptually simpleIt is conceptually simple

bull The algorithm does not require the formulation of integral equation and relatively complex The algorithm does not require the formulation of integral equation and relatively complex

scatters can be treated without inversion of large matrices scatters can be treated without inversion of large matrices

bull It is simple to implement for complicated inhomogeneous conducting or dielectric structures It is simple to implement for complicated inhomogeneous conducting or dielectric structures

because constitutive parameters can be assigned to each lattice pointbecause constitutive parameters can be assigned to each lattice point

bull Its computer memory requirement is not prohibitive for many complex structures of interest Its computer memory requirement is not prohibitive for many complex structures of interest

bull The algorithm make use of the memory in a simple sequential orderThe algorithm make use of the memory in a simple sequential order

bull It is much easier to obtain frequency domain data from time domain results than the converse It is much easier to obtain frequency domain data from time domain results than the converse

Thus it is more convenient to obtain frequency domain results via time domain when many Thus it is more convenient to obtain frequency domain results via time domain when many

frequencies are involvedfrequencies are involved

Include examples of unconditionally stable fdtd and prove the advantages

Disadvantages Disadvantages

bull Its implementation necessitates modeling object as well as its surroundings Thus the Its implementation necessitates modeling object as well as its surroundings Thus the

required program execution time may be excessiverequired program execution time may be excessive

bull Its accuracy is at least one order of magnitude worse than that of the method of Its accuracy is at least one order of magnitude worse than that of the method of

moments for example moments for example

bull Since the computational meshes are rectangular in shape they do not conform the Since the computational meshes are rectangular in shape they do not conform the

scatterers with curved surfaces as is the case of the cylindrical or spherical boundary scatterers with curved surfaces as is the case of the cylindrical or spherical boundary

Its computer memory requirement is not prohibitive for many complex structures of Its computer memory requirement is not prohibitive for many complex structures of

interest interest

bull As in all finite difference algorithms the field quantities are only known at grid nodes As in all finite difference algorithms the field quantities are only known at grid nodes

Applications of FDTD method

1048708Electromagnetic scattering amp antenna design1048708EMCEMI design1048708Simulation of wave propagation problem1048708Solving partial defferential equation 1048708Waveguide analysis

Stripline amp microstrip line analysishellipetc hellip

Referenceshelliphelliphellip1-Electromagnetic simulation using FDTD method

ldquoDennis M Sillevanrdquo IEEE press series on rf amp mivcrowave technology ldquoRoger DPollard amp Richard

Bootanrdquoseries

2Understanding the FDTD method ldquoJohn BSchneiderrdquo may 8 2011

3 3 D -ADI -Mehod-unconditionaly stable time domain algorithom for solving maxwell equations

ldquoIEEE Transaction on microwave theory amp techniques vol-48No10 october 2010 ldquoTakefumi

IEEE

4High order split step unconditionally stable FDTD method ampnumerical analysis ldquoIEEE

Transactions on antinna amp propagationvol 59 no9sept2011rdquo Yong-Dan Kong amp Qing xin chu

senior memberIEEE

5Genaral Finite DifferenceSchemes for Heat equations ldquoIndian J Pure Applemath

10(2)209-222Febuary 1979rdquoby PS Jain ampDN Holla department of methemetics

IITB 400076

  • Unconditionally Stable FDTD Methods
  • Outlineshelliphelliphelliphellip
  • Outlineshellip
  • Slide 4
  • Computational Electromagnetics
  • Slide 6
  • FDTD Overview ndash Updating Equations
  • Slide 8
  • Slide 9
  • Finite Difference Time Domain Method
  • The Yee Discretization (1966)
  • Slide 12
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Slide 17
  • Slide 18
  • Slide 19
  • Space Domain Discretization
  • Finite-Difference Formulation of the Heat Conduction on a Chip
  • Time domain discretization
  • Slide 23
  • Slide 24
  • Simple Explicit Method
  • Simple Implicit Method
  • Crank-Nicolson Method
  • Analysis of Crank-Nicolson Method
  • Alternating Direction Implicit Method
  • Slide 30
  • Peaceman-Rachford Algorithm
  • Douglas-Gunn Algorithm
  • Illustration for ADI
  • Analysis of ADI Method
  • Slide 35
  • Slide 36
  • Slide 37
  • Slide 38
  • Referenceshelliphelliphellip
  • Slide 40
Page 28: Unconditionally stable fdtd methods

21 march 2012 28

n

m1616

ex m=4n=4

Total node number N = mn

Matrix size = NxN

Analysis of Crank-Nicolson Method

21 march 2012 29

Alternating Direction Implicit Method

Solves higher dimension problem by successive Lower dimension methods

Accuracy

Unconditionally stable

No limits on time step and no large scale matrix inversion

] [ 222 yxt

21 march2012 30

Step I x-direction implicit y-direction explicitStep II x-direction explicit y-direction implicit

n

Alternating Direction Implicit Method

bull Peaceman-Rachford Algorithmbull Douglas-Gunn Algorithm

21 march 2012 31

bull Step I

bull Step II

gc

tT

rrT

rr

p

ny

yx

xny

yx

x

)2

1)(2

1()2

1)(2

1( 22122

gc

tT

rT

r

p

ny

yn

xx

2)

21()

21( 22

12

gc

tT

rT

r

p

n

xxn

yy

2)

21()

21( 2

1212

Peaceman-Rachford Algorithm

21 march 2012 32

Douglas-Gunn Algorithm

bull Step I

bull Step II

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

12

1

22

gc

tTrTT

rTT

p

nyy

nnxxnn

22

122

1

)(2

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

2

121

2)(

2

21 march 2012 33

Illustration for ADI

X-direction implicit Y-direction implicit

i = 1 2 hellip m

1

njiT

Step I Step II

j = 1

n

2

hellip

1 2 hellip m

11

njiT

11

njiT

j = 1

n

2hellip

21

njiT

21

1

njiT

21

1

njiT

21 march 2012 34

Analysis of ADI Method

mm

Tridiagonal Matrix

Time complexity O(N)

2xnxm = 2nm =2N

2 steps n matrices tridaigonal matrix

X-direction implicit

21

njiT

i = 1 2 hellip mj = 1

n

2

hellip

21

1

njiT

21

1

njiT

ADVANTAGE over conditionally stable

Reduce simulation time

Good accuracy

Size of geometrical feature may be typical of wavelengh

Good geometrical flexibility to allow with corner or high curvature than fdtd

Advantages of the FDTD method over other methodsAdvantages of the FDTD method over other methods

bull It is conceptually simpleIt is conceptually simple

bull The algorithm does not require the formulation of integral equation and relatively complex The algorithm does not require the formulation of integral equation and relatively complex

scatters can be treated without inversion of large matrices scatters can be treated without inversion of large matrices

bull It is simple to implement for complicated inhomogeneous conducting or dielectric structures It is simple to implement for complicated inhomogeneous conducting or dielectric structures

because constitutive parameters can be assigned to each lattice pointbecause constitutive parameters can be assigned to each lattice point

bull Its computer memory requirement is not prohibitive for many complex structures of interest Its computer memory requirement is not prohibitive for many complex structures of interest

bull The algorithm make use of the memory in a simple sequential orderThe algorithm make use of the memory in a simple sequential order

bull It is much easier to obtain frequency domain data from time domain results than the converse It is much easier to obtain frequency domain data from time domain results than the converse

Thus it is more convenient to obtain frequency domain results via time domain when many Thus it is more convenient to obtain frequency domain results via time domain when many

frequencies are involvedfrequencies are involved

Include examples of unconditionally stable fdtd and prove the advantages

Disadvantages Disadvantages

bull Its implementation necessitates modeling object as well as its surroundings Thus the Its implementation necessitates modeling object as well as its surroundings Thus the

required program execution time may be excessiverequired program execution time may be excessive

bull Its accuracy is at least one order of magnitude worse than that of the method of Its accuracy is at least one order of magnitude worse than that of the method of

moments for example moments for example

bull Since the computational meshes are rectangular in shape they do not conform the Since the computational meshes are rectangular in shape they do not conform the

scatterers with curved surfaces as is the case of the cylindrical or spherical boundary scatterers with curved surfaces as is the case of the cylindrical or spherical boundary

Its computer memory requirement is not prohibitive for many complex structures of Its computer memory requirement is not prohibitive for many complex structures of

interest interest

bull As in all finite difference algorithms the field quantities are only known at grid nodes As in all finite difference algorithms the field quantities are only known at grid nodes

Applications of FDTD method

1048708Electromagnetic scattering amp antenna design1048708EMCEMI design1048708Simulation of wave propagation problem1048708Solving partial defferential equation 1048708Waveguide analysis

Stripline amp microstrip line analysishellipetc hellip

Referenceshelliphelliphellip1-Electromagnetic simulation using FDTD method

ldquoDennis M Sillevanrdquo IEEE press series on rf amp mivcrowave technology ldquoRoger DPollard amp Richard

Bootanrdquoseries

2Understanding the FDTD method ldquoJohn BSchneiderrdquo may 8 2011

3 3 D -ADI -Mehod-unconditionaly stable time domain algorithom for solving maxwell equations

ldquoIEEE Transaction on microwave theory amp techniques vol-48No10 october 2010 ldquoTakefumi

IEEE

4High order split step unconditionally stable FDTD method ampnumerical analysis ldquoIEEE

Transactions on antinna amp propagationvol 59 no9sept2011rdquo Yong-Dan Kong amp Qing xin chu

senior memberIEEE

5Genaral Finite DifferenceSchemes for Heat equations ldquoIndian J Pure Applemath

10(2)209-222Febuary 1979rdquoby PS Jain ampDN Holla department of methemetics

IITB 400076

  • Unconditionally Stable FDTD Methods
  • Outlineshelliphelliphelliphellip
  • Outlineshellip
  • Slide 4
  • Computational Electromagnetics
  • Slide 6
  • FDTD Overview ndash Updating Equations
  • Slide 8
  • Slide 9
  • Finite Difference Time Domain Method
  • The Yee Discretization (1966)
  • Slide 12
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Slide 17
  • Slide 18
  • Slide 19
  • Space Domain Discretization
  • Finite-Difference Formulation of the Heat Conduction on a Chip
  • Time domain discretization
  • Slide 23
  • Slide 24
  • Simple Explicit Method
  • Simple Implicit Method
  • Crank-Nicolson Method
  • Analysis of Crank-Nicolson Method
  • Alternating Direction Implicit Method
  • Slide 30
  • Peaceman-Rachford Algorithm
  • Douglas-Gunn Algorithm
  • Illustration for ADI
  • Analysis of ADI Method
  • Slide 35
  • Slide 36
  • Slide 37
  • Slide 38
  • Referenceshelliphelliphellip
  • Slide 40
Page 29: Unconditionally stable fdtd methods

21 march 2012 29

Alternating Direction Implicit Method

Solves higher dimension problem by successive Lower dimension methods

Accuracy

Unconditionally stable

No limits on time step and no large scale matrix inversion

] [ 222 yxt

21 march2012 30

Step I x-direction implicit y-direction explicitStep II x-direction explicit y-direction implicit

n

Alternating Direction Implicit Method

bull Peaceman-Rachford Algorithmbull Douglas-Gunn Algorithm

21 march 2012 31

bull Step I

bull Step II

gc

tT

rrT

rr

p

ny

yx

xny

yx

x

)2

1)(2

1()2

1)(2

1( 22122

gc

tT

rT

r

p

ny

yn

xx

2)

21()

21( 22

12

gc

tT

rT

r

p

n

xxn

yy

2)

21()

21( 2

1212

Peaceman-Rachford Algorithm

21 march 2012 32

Douglas-Gunn Algorithm

bull Step I

bull Step II

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

12

1

22

gc

tTrTT

rTT

p

nyy

nnxxnn

22

122

1

)(2

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

2

121

2)(

2

21 march 2012 33

Illustration for ADI

X-direction implicit Y-direction implicit

i = 1 2 hellip m

1

njiT

Step I Step II

j = 1

n

2

hellip

1 2 hellip m

11

njiT

11

njiT

j = 1

n

2hellip

21

njiT

21

1

njiT

21

1

njiT

21 march 2012 34

Analysis of ADI Method

mm

Tridiagonal Matrix

Time complexity O(N)

2xnxm = 2nm =2N

2 steps n matrices tridaigonal matrix

X-direction implicit

21

njiT

i = 1 2 hellip mj = 1

n

2

hellip

21

1

njiT

21

1

njiT

ADVANTAGE over conditionally stable

Reduce simulation time

Good accuracy

Size of geometrical feature may be typical of wavelengh

Good geometrical flexibility to allow with corner or high curvature than fdtd

Advantages of the FDTD method over other methodsAdvantages of the FDTD method over other methods

bull It is conceptually simpleIt is conceptually simple

bull The algorithm does not require the formulation of integral equation and relatively complex The algorithm does not require the formulation of integral equation and relatively complex

scatters can be treated without inversion of large matrices scatters can be treated without inversion of large matrices

bull It is simple to implement for complicated inhomogeneous conducting or dielectric structures It is simple to implement for complicated inhomogeneous conducting or dielectric structures

because constitutive parameters can be assigned to each lattice pointbecause constitutive parameters can be assigned to each lattice point

bull Its computer memory requirement is not prohibitive for many complex structures of interest Its computer memory requirement is not prohibitive for many complex structures of interest

bull The algorithm make use of the memory in a simple sequential orderThe algorithm make use of the memory in a simple sequential order

bull It is much easier to obtain frequency domain data from time domain results than the converse It is much easier to obtain frequency domain data from time domain results than the converse

Thus it is more convenient to obtain frequency domain results via time domain when many Thus it is more convenient to obtain frequency domain results via time domain when many

frequencies are involvedfrequencies are involved

Include examples of unconditionally stable fdtd and prove the advantages

Disadvantages Disadvantages

bull Its implementation necessitates modeling object as well as its surroundings Thus the Its implementation necessitates modeling object as well as its surroundings Thus the

required program execution time may be excessiverequired program execution time may be excessive

bull Its accuracy is at least one order of magnitude worse than that of the method of Its accuracy is at least one order of magnitude worse than that of the method of

moments for example moments for example

bull Since the computational meshes are rectangular in shape they do not conform the Since the computational meshes are rectangular in shape they do not conform the

scatterers with curved surfaces as is the case of the cylindrical or spherical boundary scatterers with curved surfaces as is the case of the cylindrical or spherical boundary

Its computer memory requirement is not prohibitive for many complex structures of Its computer memory requirement is not prohibitive for many complex structures of

interest interest

bull As in all finite difference algorithms the field quantities are only known at grid nodes As in all finite difference algorithms the field quantities are only known at grid nodes

Applications of FDTD method

1048708Electromagnetic scattering amp antenna design1048708EMCEMI design1048708Simulation of wave propagation problem1048708Solving partial defferential equation 1048708Waveguide analysis

Stripline amp microstrip line analysishellipetc hellip

Referenceshelliphelliphellip1-Electromagnetic simulation using FDTD method

ldquoDennis M Sillevanrdquo IEEE press series on rf amp mivcrowave technology ldquoRoger DPollard amp Richard

Bootanrdquoseries

2Understanding the FDTD method ldquoJohn BSchneiderrdquo may 8 2011

3 3 D -ADI -Mehod-unconditionaly stable time domain algorithom for solving maxwell equations

ldquoIEEE Transaction on microwave theory amp techniques vol-48No10 october 2010 ldquoTakefumi

IEEE

4High order split step unconditionally stable FDTD method ampnumerical analysis ldquoIEEE

Transactions on antinna amp propagationvol 59 no9sept2011rdquo Yong-Dan Kong amp Qing xin chu

senior memberIEEE

5Genaral Finite DifferenceSchemes for Heat equations ldquoIndian J Pure Applemath

10(2)209-222Febuary 1979rdquoby PS Jain ampDN Holla department of methemetics

IITB 400076

  • Unconditionally Stable FDTD Methods
  • Outlineshelliphelliphelliphellip
  • Outlineshellip
  • Slide 4
  • Computational Electromagnetics
  • Slide 6
  • FDTD Overview ndash Updating Equations
  • Slide 8
  • Slide 9
  • Finite Difference Time Domain Method
  • The Yee Discretization (1966)
  • Slide 12
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Slide 17
  • Slide 18
  • Slide 19
  • Space Domain Discretization
  • Finite-Difference Formulation of the Heat Conduction on a Chip
  • Time domain discretization
  • Slide 23
  • Slide 24
  • Simple Explicit Method
  • Simple Implicit Method
  • Crank-Nicolson Method
  • Analysis of Crank-Nicolson Method
  • Alternating Direction Implicit Method
  • Slide 30
  • Peaceman-Rachford Algorithm
  • Douglas-Gunn Algorithm
  • Illustration for ADI
  • Analysis of ADI Method
  • Slide 35
  • Slide 36
  • Slide 37
  • Slide 38
  • Referenceshelliphelliphellip
  • Slide 40
Page 30: Unconditionally stable fdtd methods

21 march2012 30

Step I x-direction implicit y-direction explicitStep II x-direction explicit y-direction implicit

n

Alternating Direction Implicit Method

bull Peaceman-Rachford Algorithmbull Douglas-Gunn Algorithm

21 march 2012 31

bull Step I

bull Step II

gc

tT

rrT

rr

p

ny

yx

xny

yx

x

)2

1)(2

1()2

1)(2

1( 22122

gc

tT

rT

r

p

ny

yn

xx

2)

21()

21( 22

12

gc

tT

rT

r

p

n

xxn

yy

2)

21()

21( 2

1212

Peaceman-Rachford Algorithm

21 march 2012 32

Douglas-Gunn Algorithm

bull Step I

bull Step II

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

12

1

22

gc

tTrTT

rTT

p

nyy

nnxxnn

22

122

1

)(2

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

2

121

2)(

2

21 march 2012 33

Illustration for ADI

X-direction implicit Y-direction implicit

i = 1 2 hellip m

1

njiT

Step I Step II

j = 1

n

2

hellip

1 2 hellip m

11

njiT

11

njiT

j = 1

n

2hellip

21

njiT

21

1

njiT

21

1

njiT

21 march 2012 34

Analysis of ADI Method

mm

Tridiagonal Matrix

Time complexity O(N)

2xnxm = 2nm =2N

2 steps n matrices tridaigonal matrix

X-direction implicit

21

njiT

i = 1 2 hellip mj = 1

n

2

hellip

21

1

njiT

21

1

njiT

ADVANTAGE over conditionally stable

Reduce simulation time

Good accuracy

Size of geometrical feature may be typical of wavelengh

Good geometrical flexibility to allow with corner or high curvature than fdtd

Advantages of the FDTD method over other methodsAdvantages of the FDTD method over other methods

bull It is conceptually simpleIt is conceptually simple

bull The algorithm does not require the formulation of integral equation and relatively complex The algorithm does not require the formulation of integral equation and relatively complex

scatters can be treated without inversion of large matrices scatters can be treated without inversion of large matrices

bull It is simple to implement for complicated inhomogeneous conducting or dielectric structures It is simple to implement for complicated inhomogeneous conducting or dielectric structures

because constitutive parameters can be assigned to each lattice pointbecause constitutive parameters can be assigned to each lattice point

bull Its computer memory requirement is not prohibitive for many complex structures of interest Its computer memory requirement is not prohibitive for many complex structures of interest

bull The algorithm make use of the memory in a simple sequential orderThe algorithm make use of the memory in a simple sequential order

bull It is much easier to obtain frequency domain data from time domain results than the converse It is much easier to obtain frequency domain data from time domain results than the converse

Thus it is more convenient to obtain frequency domain results via time domain when many Thus it is more convenient to obtain frequency domain results via time domain when many

frequencies are involvedfrequencies are involved

Include examples of unconditionally stable fdtd and prove the advantages

Disadvantages Disadvantages

bull Its implementation necessitates modeling object as well as its surroundings Thus the Its implementation necessitates modeling object as well as its surroundings Thus the

required program execution time may be excessiverequired program execution time may be excessive

bull Its accuracy is at least one order of magnitude worse than that of the method of Its accuracy is at least one order of magnitude worse than that of the method of

moments for example moments for example

bull Since the computational meshes are rectangular in shape they do not conform the Since the computational meshes are rectangular in shape they do not conform the

scatterers with curved surfaces as is the case of the cylindrical or spherical boundary scatterers with curved surfaces as is the case of the cylindrical or spherical boundary

Its computer memory requirement is not prohibitive for many complex structures of Its computer memory requirement is not prohibitive for many complex structures of

interest interest

bull As in all finite difference algorithms the field quantities are only known at grid nodes As in all finite difference algorithms the field quantities are only known at grid nodes

Applications of FDTD method

1048708Electromagnetic scattering amp antenna design1048708EMCEMI design1048708Simulation of wave propagation problem1048708Solving partial defferential equation 1048708Waveguide analysis

Stripline amp microstrip line analysishellipetc hellip

Referenceshelliphelliphellip1-Electromagnetic simulation using FDTD method

ldquoDennis M Sillevanrdquo IEEE press series on rf amp mivcrowave technology ldquoRoger DPollard amp Richard

Bootanrdquoseries

2Understanding the FDTD method ldquoJohn BSchneiderrdquo may 8 2011

3 3 D -ADI -Mehod-unconditionaly stable time domain algorithom for solving maxwell equations

ldquoIEEE Transaction on microwave theory amp techniques vol-48No10 october 2010 ldquoTakefumi

IEEE

4High order split step unconditionally stable FDTD method ampnumerical analysis ldquoIEEE

Transactions on antinna amp propagationvol 59 no9sept2011rdquo Yong-Dan Kong amp Qing xin chu

senior memberIEEE

5Genaral Finite DifferenceSchemes for Heat equations ldquoIndian J Pure Applemath

10(2)209-222Febuary 1979rdquoby PS Jain ampDN Holla department of methemetics

IITB 400076

  • Unconditionally Stable FDTD Methods
  • Outlineshelliphelliphelliphellip
  • Outlineshellip
  • Slide 4
  • Computational Electromagnetics
  • Slide 6
  • FDTD Overview ndash Updating Equations
  • Slide 8
  • Slide 9
  • Finite Difference Time Domain Method
  • The Yee Discretization (1966)
  • Slide 12
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Slide 17
  • Slide 18
  • Slide 19
  • Space Domain Discretization
  • Finite-Difference Formulation of the Heat Conduction on a Chip
  • Time domain discretization
  • Slide 23
  • Slide 24
  • Simple Explicit Method
  • Simple Implicit Method
  • Crank-Nicolson Method
  • Analysis of Crank-Nicolson Method
  • Alternating Direction Implicit Method
  • Slide 30
  • Peaceman-Rachford Algorithm
  • Douglas-Gunn Algorithm
  • Illustration for ADI
  • Analysis of ADI Method
  • Slide 35
  • Slide 36
  • Slide 37
  • Slide 38
  • Referenceshelliphelliphellip
  • Slide 40
Page 31: Unconditionally stable fdtd methods

21 march 2012 31

bull Step I

bull Step II

gc

tT

rrT

rr

p

ny

yx

xny

yx

x

)2

1)(2

1()2

1)(2

1( 22122

gc

tT

rT

r

p

ny

yn

xx

2)

21()

21( 22

12

gc

tT

rT

r

p

n

xxn

yy

2)

21()

21( 2

1212

Peaceman-Rachford Algorithm

21 march 2012 32

Douglas-Gunn Algorithm

bull Step I

bull Step II

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

12

1

22

gc

tTrTT

rTT

p

nyy

nnxxnn

22

122

1

)(2

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

2

121

2)(

2

21 march 2012 33

Illustration for ADI

X-direction implicit Y-direction implicit

i = 1 2 hellip m

1

njiT

Step I Step II

j = 1

n

2

hellip

1 2 hellip m

11

njiT

11

njiT

j = 1

n

2hellip

21

njiT

21

1

njiT

21

1

njiT

21 march 2012 34

Analysis of ADI Method

mm

Tridiagonal Matrix

Time complexity O(N)

2xnxm = 2nm =2N

2 steps n matrices tridaigonal matrix

X-direction implicit

21

njiT

i = 1 2 hellip mj = 1

n

2

hellip

21

1

njiT

21

1

njiT

ADVANTAGE over conditionally stable

Reduce simulation time

Good accuracy

Size of geometrical feature may be typical of wavelengh

Good geometrical flexibility to allow with corner or high curvature than fdtd

Advantages of the FDTD method over other methodsAdvantages of the FDTD method over other methods

bull It is conceptually simpleIt is conceptually simple

bull The algorithm does not require the formulation of integral equation and relatively complex The algorithm does not require the formulation of integral equation and relatively complex

scatters can be treated without inversion of large matrices scatters can be treated without inversion of large matrices

bull It is simple to implement for complicated inhomogeneous conducting or dielectric structures It is simple to implement for complicated inhomogeneous conducting or dielectric structures

because constitutive parameters can be assigned to each lattice pointbecause constitutive parameters can be assigned to each lattice point

bull Its computer memory requirement is not prohibitive for many complex structures of interest Its computer memory requirement is not prohibitive for many complex structures of interest

bull The algorithm make use of the memory in a simple sequential orderThe algorithm make use of the memory in a simple sequential order

bull It is much easier to obtain frequency domain data from time domain results than the converse It is much easier to obtain frequency domain data from time domain results than the converse

Thus it is more convenient to obtain frequency domain results via time domain when many Thus it is more convenient to obtain frequency domain results via time domain when many

frequencies are involvedfrequencies are involved

Include examples of unconditionally stable fdtd and prove the advantages

Disadvantages Disadvantages

bull Its implementation necessitates modeling object as well as its surroundings Thus the Its implementation necessitates modeling object as well as its surroundings Thus the

required program execution time may be excessiverequired program execution time may be excessive

bull Its accuracy is at least one order of magnitude worse than that of the method of Its accuracy is at least one order of magnitude worse than that of the method of

moments for example moments for example

bull Since the computational meshes are rectangular in shape they do not conform the Since the computational meshes are rectangular in shape they do not conform the

scatterers with curved surfaces as is the case of the cylindrical or spherical boundary scatterers with curved surfaces as is the case of the cylindrical or spherical boundary

Its computer memory requirement is not prohibitive for many complex structures of Its computer memory requirement is not prohibitive for many complex structures of

interest interest

bull As in all finite difference algorithms the field quantities are only known at grid nodes As in all finite difference algorithms the field quantities are only known at grid nodes

Applications of FDTD method

1048708Electromagnetic scattering amp antenna design1048708EMCEMI design1048708Simulation of wave propagation problem1048708Solving partial defferential equation 1048708Waveguide analysis

Stripline amp microstrip line analysishellipetc hellip

Referenceshelliphelliphellip1-Electromagnetic simulation using FDTD method

ldquoDennis M Sillevanrdquo IEEE press series on rf amp mivcrowave technology ldquoRoger DPollard amp Richard

Bootanrdquoseries

2Understanding the FDTD method ldquoJohn BSchneiderrdquo may 8 2011

3 3 D -ADI -Mehod-unconditionaly stable time domain algorithom for solving maxwell equations

ldquoIEEE Transaction on microwave theory amp techniques vol-48No10 october 2010 ldquoTakefumi

IEEE

4High order split step unconditionally stable FDTD method ampnumerical analysis ldquoIEEE

Transactions on antinna amp propagationvol 59 no9sept2011rdquo Yong-Dan Kong amp Qing xin chu

senior memberIEEE

5Genaral Finite DifferenceSchemes for Heat equations ldquoIndian J Pure Applemath

10(2)209-222Febuary 1979rdquoby PS Jain ampDN Holla department of methemetics

IITB 400076

  • Unconditionally Stable FDTD Methods
  • Outlineshelliphelliphelliphellip
  • Outlineshellip
  • Slide 4
  • Computational Electromagnetics
  • Slide 6
  • FDTD Overview ndash Updating Equations
  • Slide 8
  • Slide 9
  • Finite Difference Time Domain Method
  • The Yee Discretization (1966)
  • Slide 12
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Slide 17
  • Slide 18
  • Slide 19
  • Space Domain Discretization
  • Finite-Difference Formulation of the Heat Conduction on a Chip
  • Time domain discretization
  • Slide 23
  • Slide 24
  • Simple Explicit Method
  • Simple Implicit Method
  • Crank-Nicolson Method
  • Analysis of Crank-Nicolson Method
  • Alternating Direction Implicit Method
  • Slide 30
  • Peaceman-Rachford Algorithm
  • Douglas-Gunn Algorithm
  • Illustration for ADI
  • Analysis of ADI Method
  • Slide 35
  • Slide 36
  • Slide 37
  • Slide 38
  • Referenceshelliphelliphellip
  • Slide 40
Page 32: Unconditionally stable fdtd methods

21 march 2012 32

Douglas-Gunn Algorithm

bull Step I

bull Step II

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

12

1

22

gc

tTrTT

rTT

p

nyy

nnxxnn

22

122

1

)(2

gc

tTT

rTT

rTT

p

nnyynnxxnn

12

2

121

2)(

2

21 march 2012 33

Illustration for ADI

X-direction implicit Y-direction implicit

i = 1 2 hellip m

1

njiT

Step I Step II

j = 1

n

2

hellip

1 2 hellip m

11

njiT

11

njiT

j = 1

n

2hellip

21

njiT

21

1

njiT

21

1

njiT

21 march 2012 34

Analysis of ADI Method

mm

Tridiagonal Matrix

Time complexity O(N)

2xnxm = 2nm =2N

2 steps n matrices tridaigonal matrix

X-direction implicit

21

njiT

i = 1 2 hellip mj = 1

n

2

hellip

21

1

njiT

21

1

njiT

ADVANTAGE over conditionally stable

Reduce simulation time

Good accuracy

Size of geometrical feature may be typical of wavelengh

Good geometrical flexibility to allow with corner or high curvature than fdtd

Advantages of the FDTD method over other methodsAdvantages of the FDTD method over other methods

bull It is conceptually simpleIt is conceptually simple

bull The algorithm does not require the formulation of integral equation and relatively complex The algorithm does not require the formulation of integral equation and relatively complex

scatters can be treated without inversion of large matrices scatters can be treated without inversion of large matrices

bull It is simple to implement for complicated inhomogeneous conducting or dielectric structures It is simple to implement for complicated inhomogeneous conducting or dielectric structures

because constitutive parameters can be assigned to each lattice pointbecause constitutive parameters can be assigned to each lattice point

bull Its computer memory requirement is not prohibitive for many complex structures of interest Its computer memory requirement is not prohibitive for many complex structures of interest

bull The algorithm make use of the memory in a simple sequential orderThe algorithm make use of the memory in a simple sequential order

bull It is much easier to obtain frequency domain data from time domain results than the converse It is much easier to obtain frequency domain data from time domain results than the converse

Thus it is more convenient to obtain frequency domain results via time domain when many Thus it is more convenient to obtain frequency domain results via time domain when many

frequencies are involvedfrequencies are involved

Include examples of unconditionally stable fdtd and prove the advantages

Disadvantages Disadvantages

bull Its implementation necessitates modeling object as well as its surroundings Thus the Its implementation necessitates modeling object as well as its surroundings Thus the

required program execution time may be excessiverequired program execution time may be excessive

bull Its accuracy is at least one order of magnitude worse than that of the method of Its accuracy is at least one order of magnitude worse than that of the method of

moments for example moments for example

bull Since the computational meshes are rectangular in shape they do not conform the Since the computational meshes are rectangular in shape they do not conform the

scatterers with curved surfaces as is the case of the cylindrical or spherical boundary scatterers with curved surfaces as is the case of the cylindrical or spherical boundary

Its computer memory requirement is not prohibitive for many complex structures of Its computer memory requirement is not prohibitive for many complex structures of

interest interest

bull As in all finite difference algorithms the field quantities are only known at grid nodes As in all finite difference algorithms the field quantities are only known at grid nodes

Applications of FDTD method

1048708Electromagnetic scattering amp antenna design1048708EMCEMI design1048708Simulation of wave propagation problem1048708Solving partial defferential equation 1048708Waveguide analysis

Stripline amp microstrip line analysishellipetc hellip

Referenceshelliphelliphellip1-Electromagnetic simulation using FDTD method

ldquoDennis M Sillevanrdquo IEEE press series on rf amp mivcrowave technology ldquoRoger DPollard amp Richard

Bootanrdquoseries

2Understanding the FDTD method ldquoJohn BSchneiderrdquo may 8 2011

3 3 D -ADI -Mehod-unconditionaly stable time domain algorithom for solving maxwell equations

ldquoIEEE Transaction on microwave theory amp techniques vol-48No10 october 2010 ldquoTakefumi

IEEE

4High order split step unconditionally stable FDTD method ampnumerical analysis ldquoIEEE

Transactions on antinna amp propagationvol 59 no9sept2011rdquo Yong-Dan Kong amp Qing xin chu

senior memberIEEE

5Genaral Finite DifferenceSchemes for Heat equations ldquoIndian J Pure Applemath

10(2)209-222Febuary 1979rdquoby PS Jain ampDN Holla department of methemetics

IITB 400076

  • Unconditionally Stable FDTD Methods
  • Outlineshelliphelliphelliphellip
  • Outlineshellip
  • Slide 4
  • Computational Electromagnetics
  • Slide 6
  • FDTD Overview ndash Updating Equations
  • Slide 8
  • Slide 9
  • Finite Difference Time Domain Method
  • The Yee Discretization (1966)
  • Slide 12
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Slide 17
  • Slide 18
  • Slide 19
  • Space Domain Discretization
  • Finite-Difference Formulation of the Heat Conduction on a Chip
  • Time domain discretization
  • Slide 23
  • Slide 24
  • Simple Explicit Method
  • Simple Implicit Method
  • Crank-Nicolson Method
  • Analysis of Crank-Nicolson Method
  • Alternating Direction Implicit Method
  • Slide 30
  • Peaceman-Rachford Algorithm
  • Douglas-Gunn Algorithm
  • Illustration for ADI
  • Analysis of ADI Method
  • Slide 35
  • Slide 36
  • Slide 37
  • Slide 38
  • Referenceshelliphelliphellip
  • Slide 40
Page 33: Unconditionally stable fdtd methods

21 march 2012 33

Illustration for ADI

X-direction implicit Y-direction implicit

i = 1 2 hellip m

1

njiT

Step I Step II

j = 1

n

2

hellip

1 2 hellip m

11

njiT

11

njiT

j = 1

n

2hellip

21

njiT

21

1

njiT

21

1

njiT

21 march 2012 34

Analysis of ADI Method

mm

Tridiagonal Matrix

Time complexity O(N)

2xnxm = 2nm =2N

2 steps n matrices tridaigonal matrix

X-direction implicit

21

njiT

i = 1 2 hellip mj = 1

n

2

hellip

21

1

njiT

21

1

njiT

ADVANTAGE over conditionally stable

Reduce simulation time

Good accuracy

Size of geometrical feature may be typical of wavelengh

Good geometrical flexibility to allow with corner or high curvature than fdtd

Advantages of the FDTD method over other methodsAdvantages of the FDTD method over other methods

bull It is conceptually simpleIt is conceptually simple

bull The algorithm does not require the formulation of integral equation and relatively complex The algorithm does not require the formulation of integral equation and relatively complex

scatters can be treated without inversion of large matrices scatters can be treated without inversion of large matrices

bull It is simple to implement for complicated inhomogeneous conducting or dielectric structures It is simple to implement for complicated inhomogeneous conducting or dielectric structures

because constitutive parameters can be assigned to each lattice pointbecause constitutive parameters can be assigned to each lattice point

bull Its computer memory requirement is not prohibitive for many complex structures of interest Its computer memory requirement is not prohibitive for many complex structures of interest

bull The algorithm make use of the memory in a simple sequential orderThe algorithm make use of the memory in a simple sequential order

bull It is much easier to obtain frequency domain data from time domain results than the converse It is much easier to obtain frequency domain data from time domain results than the converse

Thus it is more convenient to obtain frequency domain results via time domain when many Thus it is more convenient to obtain frequency domain results via time domain when many

frequencies are involvedfrequencies are involved

Include examples of unconditionally stable fdtd and prove the advantages

Disadvantages Disadvantages

bull Its implementation necessitates modeling object as well as its surroundings Thus the Its implementation necessitates modeling object as well as its surroundings Thus the

required program execution time may be excessiverequired program execution time may be excessive

bull Its accuracy is at least one order of magnitude worse than that of the method of Its accuracy is at least one order of magnitude worse than that of the method of

moments for example moments for example

bull Since the computational meshes are rectangular in shape they do not conform the Since the computational meshes are rectangular in shape they do not conform the

scatterers with curved surfaces as is the case of the cylindrical or spherical boundary scatterers with curved surfaces as is the case of the cylindrical or spherical boundary

Its computer memory requirement is not prohibitive for many complex structures of Its computer memory requirement is not prohibitive for many complex structures of

interest interest

bull As in all finite difference algorithms the field quantities are only known at grid nodes As in all finite difference algorithms the field quantities are only known at grid nodes

Applications of FDTD method

1048708Electromagnetic scattering amp antenna design1048708EMCEMI design1048708Simulation of wave propagation problem1048708Solving partial defferential equation 1048708Waveguide analysis

Stripline amp microstrip line analysishellipetc hellip

Referenceshelliphelliphellip1-Electromagnetic simulation using FDTD method

ldquoDennis M Sillevanrdquo IEEE press series on rf amp mivcrowave technology ldquoRoger DPollard amp Richard

Bootanrdquoseries

2Understanding the FDTD method ldquoJohn BSchneiderrdquo may 8 2011

3 3 D -ADI -Mehod-unconditionaly stable time domain algorithom for solving maxwell equations

ldquoIEEE Transaction on microwave theory amp techniques vol-48No10 october 2010 ldquoTakefumi

IEEE

4High order split step unconditionally stable FDTD method ampnumerical analysis ldquoIEEE

Transactions on antinna amp propagationvol 59 no9sept2011rdquo Yong-Dan Kong amp Qing xin chu

senior memberIEEE

5Genaral Finite DifferenceSchemes for Heat equations ldquoIndian J Pure Applemath

10(2)209-222Febuary 1979rdquoby PS Jain ampDN Holla department of methemetics

IITB 400076

  • Unconditionally Stable FDTD Methods
  • Outlineshelliphelliphelliphellip
  • Outlineshellip
  • Slide 4
  • Computational Electromagnetics
  • Slide 6
  • FDTD Overview ndash Updating Equations
  • Slide 8
  • Slide 9
  • Finite Difference Time Domain Method
  • The Yee Discretization (1966)
  • Slide 12
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Slide 17
  • Slide 18
  • Slide 19
  • Space Domain Discretization
  • Finite-Difference Formulation of the Heat Conduction on a Chip
  • Time domain discretization
  • Slide 23
  • Slide 24
  • Simple Explicit Method
  • Simple Implicit Method
  • Crank-Nicolson Method
  • Analysis of Crank-Nicolson Method
  • Alternating Direction Implicit Method
  • Slide 30
  • Peaceman-Rachford Algorithm
  • Douglas-Gunn Algorithm
  • Illustration for ADI
  • Analysis of ADI Method
  • Slide 35
  • Slide 36
  • Slide 37
  • Slide 38
  • Referenceshelliphelliphellip
  • Slide 40
Page 34: Unconditionally stable fdtd methods

21 march 2012 34

Analysis of ADI Method

mm

Tridiagonal Matrix

Time complexity O(N)

2xnxm = 2nm =2N

2 steps n matrices tridaigonal matrix

X-direction implicit

21

njiT

i = 1 2 hellip mj = 1

n

2

hellip

21

1

njiT

21

1

njiT

ADVANTAGE over conditionally stable

Reduce simulation time

Good accuracy

Size of geometrical feature may be typical of wavelengh

Good geometrical flexibility to allow with corner or high curvature than fdtd

Advantages of the FDTD method over other methodsAdvantages of the FDTD method over other methods

bull It is conceptually simpleIt is conceptually simple

bull The algorithm does not require the formulation of integral equation and relatively complex The algorithm does not require the formulation of integral equation and relatively complex

scatters can be treated without inversion of large matrices scatters can be treated without inversion of large matrices

bull It is simple to implement for complicated inhomogeneous conducting or dielectric structures It is simple to implement for complicated inhomogeneous conducting or dielectric structures

because constitutive parameters can be assigned to each lattice pointbecause constitutive parameters can be assigned to each lattice point

bull Its computer memory requirement is not prohibitive for many complex structures of interest Its computer memory requirement is not prohibitive for many complex structures of interest

bull The algorithm make use of the memory in a simple sequential orderThe algorithm make use of the memory in a simple sequential order

bull It is much easier to obtain frequency domain data from time domain results than the converse It is much easier to obtain frequency domain data from time domain results than the converse

Thus it is more convenient to obtain frequency domain results via time domain when many Thus it is more convenient to obtain frequency domain results via time domain when many

frequencies are involvedfrequencies are involved

Include examples of unconditionally stable fdtd and prove the advantages

Disadvantages Disadvantages

bull Its implementation necessitates modeling object as well as its surroundings Thus the Its implementation necessitates modeling object as well as its surroundings Thus the

required program execution time may be excessiverequired program execution time may be excessive

bull Its accuracy is at least one order of magnitude worse than that of the method of Its accuracy is at least one order of magnitude worse than that of the method of

moments for example moments for example

bull Since the computational meshes are rectangular in shape they do not conform the Since the computational meshes are rectangular in shape they do not conform the

scatterers with curved surfaces as is the case of the cylindrical or spherical boundary scatterers with curved surfaces as is the case of the cylindrical or spherical boundary

Its computer memory requirement is not prohibitive for many complex structures of Its computer memory requirement is not prohibitive for many complex structures of

interest interest

bull As in all finite difference algorithms the field quantities are only known at grid nodes As in all finite difference algorithms the field quantities are only known at grid nodes

Applications of FDTD method

1048708Electromagnetic scattering amp antenna design1048708EMCEMI design1048708Simulation of wave propagation problem1048708Solving partial defferential equation 1048708Waveguide analysis

Stripline amp microstrip line analysishellipetc hellip

Referenceshelliphelliphellip1-Electromagnetic simulation using FDTD method

ldquoDennis M Sillevanrdquo IEEE press series on rf amp mivcrowave technology ldquoRoger DPollard amp Richard

Bootanrdquoseries

2Understanding the FDTD method ldquoJohn BSchneiderrdquo may 8 2011

3 3 D -ADI -Mehod-unconditionaly stable time domain algorithom for solving maxwell equations

ldquoIEEE Transaction on microwave theory amp techniques vol-48No10 october 2010 ldquoTakefumi

IEEE

4High order split step unconditionally stable FDTD method ampnumerical analysis ldquoIEEE

Transactions on antinna amp propagationvol 59 no9sept2011rdquo Yong-Dan Kong amp Qing xin chu

senior memberIEEE

5Genaral Finite DifferenceSchemes for Heat equations ldquoIndian J Pure Applemath

10(2)209-222Febuary 1979rdquoby PS Jain ampDN Holla department of methemetics

IITB 400076

  • Unconditionally Stable FDTD Methods
  • Outlineshelliphelliphelliphellip
  • Outlineshellip
  • Slide 4
  • Computational Electromagnetics
  • Slide 6
  • FDTD Overview ndash Updating Equations
  • Slide 8
  • Slide 9
  • Finite Difference Time Domain Method
  • The Yee Discretization (1966)
  • Slide 12
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Slide 17
  • Slide 18
  • Slide 19
  • Space Domain Discretization
  • Finite-Difference Formulation of the Heat Conduction on a Chip
  • Time domain discretization
  • Slide 23
  • Slide 24
  • Simple Explicit Method
  • Simple Implicit Method
  • Crank-Nicolson Method
  • Analysis of Crank-Nicolson Method
  • Alternating Direction Implicit Method
  • Slide 30
  • Peaceman-Rachford Algorithm
  • Douglas-Gunn Algorithm
  • Illustration for ADI
  • Analysis of ADI Method
  • Slide 35
  • Slide 36
  • Slide 37
  • Slide 38
  • Referenceshelliphelliphellip
  • Slide 40
Page 35: Unconditionally stable fdtd methods

ADVANTAGE over conditionally stable

Reduce simulation time

Good accuracy

Size of geometrical feature may be typical of wavelengh

Good geometrical flexibility to allow with corner or high curvature than fdtd

Advantages of the FDTD method over other methodsAdvantages of the FDTD method over other methods

bull It is conceptually simpleIt is conceptually simple

bull The algorithm does not require the formulation of integral equation and relatively complex The algorithm does not require the formulation of integral equation and relatively complex

scatters can be treated without inversion of large matrices scatters can be treated without inversion of large matrices

bull It is simple to implement for complicated inhomogeneous conducting or dielectric structures It is simple to implement for complicated inhomogeneous conducting or dielectric structures

because constitutive parameters can be assigned to each lattice pointbecause constitutive parameters can be assigned to each lattice point

bull Its computer memory requirement is not prohibitive for many complex structures of interest Its computer memory requirement is not prohibitive for many complex structures of interest

bull The algorithm make use of the memory in a simple sequential orderThe algorithm make use of the memory in a simple sequential order

bull It is much easier to obtain frequency domain data from time domain results than the converse It is much easier to obtain frequency domain data from time domain results than the converse

Thus it is more convenient to obtain frequency domain results via time domain when many Thus it is more convenient to obtain frequency domain results via time domain when many

frequencies are involvedfrequencies are involved

Include examples of unconditionally stable fdtd and prove the advantages

Disadvantages Disadvantages

bull Its implementation necessitates modeling object as well as its surroundings Thus the Its implementation necessitates modeling object as well as its surroundings Thus the

required program execution time may be excessiverequired program execution time may be excessive

bull Its accuracy is at least one order of magnitude worse than that of the method of Its accuracy is at least one order of magnitude worse than that of the method of

moments for example moments for example

bull Since the computational meshes are rectangular in shape they do not conform the Since the computational meshes are rectangular in shape they do not conform the

scatterers with curved surfaces as is the case of the cylindrical or spherical boundary scatterers with curved surfaces as is the case of the cylindrical or spherical boundary

Its computer memory requirement is not prohibitive for many complex structures of Its computer memory requirement is not prohibitive for many complex structures of

interest interest

bull As in all finite difference algorithms the field quantities are only known at grid nodes As in all finite difference algorithms the field quantities are only known at grid nodes

Applications of FDTD method

1048708Electromagnetic scattering amp antenna design1048708EMCEMI design1048708Simulation of wave propagation problem1048708Solving partial defferential equation 1048708Waveguide analysis

Stripline amp microstrip line analysishellipetc hellip

Referenceshelliphelliphellip1-Electromagnetic simulation using FDTD method

ldquoDennis M Sillevanrdquo IEEE press series on rf amp mivcrowave technology ldquoRoger DPollard amp Richard

Bootanrdquoseries

2Understanding the FDTD method ldquoJohn BSchneiderrdquo may 8 2011

3 3 D -ADI -Mehod-unconditionaly stable time domain algorithom for solving maxwell equations

ldquoIEEE Transaction on microwave theory amp techniques vol-48No10 october 2010 ldquoTakefumi

IEEE

4High order split step unconditionally stable FDTD method ampnumerical analysis ldquoIEEE

Transactions on antinna amp propagationvol 59 no9sept2011rdquo Yong-Dan Kong amp Qing xin chu

senior memberIEEE

5Genaral Finite DifferenceSchemes for Heat equations ldquoIndian J Pure Applemath

10(2)209-222Febuary 1979rdquoby PS Jain ampDN Holla department of methemetics

IITB 400076

  • Unconditionally Stable FDTD Methods
  • Outlineshelliphelliphelliphellip
  • Outlineshellip
  • Slide 4
  • Computational Electromagnetics
  • Slide 6
  • FDTD Overview ndash Updating Equations
  • Slide 8
  • Slide 9
  • Finite Difference Time Domain Method
  • The Yee Discretization (1966)
  • Slide 12
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Slide 17
  • Slide 18
  • Slide 19
  • Space Domain Discretization
  • Finite-Difference Formulation of the Heat Conduction on a Chip
  • Time domain discretization
  • Slide 23
  • Slide 24
  • Simple Explicit Method
  • Simple Implicit Method
  • Crank-Nicolson Method
  • Analysis of Crank-Nicolson Method
  • Alternating Direction Implicit Method
  • Slide 30
  • Peaceman-Rachford Algorithm
  • Douglas-Gunn Algorithm
  • Illustration for ADI
  • Analysis of ADI Method
  • Slide 35
  • Slide 36
  • Slide 37
  • Slide 38
  • Referenceshelliphelliphellip
  • Slide 40
Page 36: Unconditionally stable fdtd methods

Advantages of the FDTD method over other methodsAdvantages of the FDTD method over other methods

bull It is conceptually simpleIt is conceptually simple

bull The algorithm does not require the formulation of integral equation and relatively complex The algorithm does not require the formulation of integral equation and relatively complex

scatters can be treated without inversion of large matrices scatters can be treated without inversion of large matrices

bull It is simple to implement for complicated inhomogeneous conducting or dielectric structures It is simple to implement for complicated inhomogeneous conducting or dielectric structures

because constitutive parameters can be assigned to each lattice pointbecause constitutive parameters can be assigned to each lattice point

bull Its computer memory requirement is not prohibitive for many complex structures of interest Its computer memory requirement is not prohibitive for many complex structures of interest

bull The algorithm make use of the memory in a simple sequential orderThe algorithm make use of the memory in a simple sequential order

bull It is much easier to obtain frequency domain data from time domain results than the converse It is much easier to obtain frequency domain data from time domain results than the converse

Thus it is more convenient to obtain frequency domain results via time domain when many Thus it is more convenient to obtain frequency domain results via time domain when many

frequencies are involvedfrequencies are involved

Include examples of unconditionally stable fdtd and prove the advantages

Disadvantages Disadvantages

bull Its implementation necessitates modeling object as well as its surroundings Thus the Its implementation necessitates modeling object as well as its surroundings Thus the

required program execution time may be excessiverequired program execution time may be excessive

bull Its accuracy is at least one order of magnitude worse than that of the method of Its accuracy is at least one order of magnitude worse than that of the method of

moments for example moments for example

bull Since the computational meshes are rectangular in shape they do not conform the Since the computational meshes are rectangular in shape they do not conform the

scatterers with curved surfaces as is the case of the cylindrical or spherical boundary scatterers with curved surfaces as is the case of the cylindrical or spherical boundary

Its computer memory requirement is not prohibitive for many complex structures of Its computer memory requirement is not prohibitive for many complex structures of

interest interest

bull As in all finite difference algorithms the field quantities are only known at grid nodes As in all finite difference algorithms the field quantities are only known at grid nodes

Applications of FDTD method

1048708Electromagnetic scattering amp antenna design1048708EMCEMI design1048708Simulation of wave propagation problem1048708Solving partial defferential equation 1048708Waveguide analysis

Stripline amp microstrip line analysishellipetc hellip

Referenceshelliphelliphellip1-Electromagnetic simulation using FDTD method

ldquoDennis M Sillevanrdquo IEEE press series on rf amp mivcrowave technology ldquoRoger DPollard amp Richard

Bootanrdquoseries

2Understanding the FDTD method ldquoJohn BSchneiderrdquo may 8 2011

3 3 D -ADI -Mehod-unconditionaly stable time domain algorithom for solving maxwell equations

ldquoIEEE Transaction on microwave theory amp techniques vol-48No10 october 2010 ldquoTakefumi

IEEE

4High order split step unconditionally stable FDTD method ampnumerical analysis ldquoIEEE

Transactions on antinna amp propagationvol 59 no9sept2011rdquo Yong-Dan Kong amp Qing xin chu

senior memberIEEE

5Genaral Finite DifferenceSchemes for Heat equations ldquoIndian J Pure Applemath

10(2)209-222Febuary 1979rdquoby PS Jain ampDN Holla department of methemetics

IITB 400076

  • Unconditionally Stable FDTD Methods
  • Outlineshelliphelliphelliphellip
  • Outlineshellip
  • Slide 4
  • Computational Electromagnetics
  • Slide 6
  • FDTD Overview ndash Updating Equations
  • Slide 8
  • Slide 9
  • Finite Difference Time Domain Method
  • The Yee Discretization (1966)
  • Slide 12
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Slide 17
  • Slide 18
  • Slide 19
  • Space Domain Discretization
  • Finite-Difference Formulation of the Heat Conduction on a Chip
  • Time domain discretization
  • Slide 23
  • Slide 24
  • Simple Explicit Method
  • Simple Implicit Method
  • Crank-Nicolson Method
  • Analysis of Crank-Nicolson Method
  • Alternating Direction Implicit Method
  • Slide 30
  • Peaceman-Rachford Algorithm
  • Douglas-Gunn Algorithm
  • Illustration for ADI
  • Analysis of ADI Method
  • Slide 35
  • Slide 36
  • Slide 37
  • Slide 38
  • Referenceshelliphelliphellip
  • Slide 40
Page 37: Unconditionally stable fdtd methods

Disadvantages Disadvantages

bull Its implementation necessitates modeling object as well as its surroundings Thus the Its implementation necessitates modeling object as well as its surroundings Thus the

required program execution time may be excessiverequired program execution time may be excessive

bull Its accuracy is at least one order of magnitude worse than that of the method of Its accuracy is at least one order of magnitude worse than that of the method of

moments for example moments for example

bull Since the computational meshes are rectangular in shape they do not conform the Since the computational meshes are rectangular in shape they do not conform the

scatterers with curved surfaces as is the case of the cylindrical or spherical boundary scatterers with curved surfaces as is the case of the cylindrical or spherical boundary

Its computer memory requirement is not prohibitive for many complex structures of Its computer memory requirement is not prohibitive for many complex structures of

interest interest

bull As in all finite difference algorithms the field quantities are only known at grid nodes As in all finite difference algorithms the field quantities are only known at grid nodes

Applications of FDTD method

1048708Electromagnetic scattering amp antenna design1048708EMCEMI design1048708Simulation of wave propagation problem1048708Solving partial defferential equation 1048708Waveguide analysis

Stripline amp microstrip line analysishellipetc hellip

Referenceshelliphelliphellip1-Electromagnetic simulation using FDTD method

ldquoDennis M Sillevanrdquo IEEE press series on rf amp mivcrowave technology ldquoRoger DPollard amp Richard

Bootanrdquoseries

2Understanding the FDTD method ldquoJohn BSchneiderrdquo may 8 2011

3 3 D -ADI -Mehod-unconditionaly stable time domain algorithom for solving maxwell equations

ldquoIEEE Transaction on microwave theory amp techniques vol-48No10 october 2010 ldquoTakefumi

IEEE

4High order split step unconditionally stable FDTD method ampnumerical analysis ldquoIEEE

Transactions on antinna amp propagationvol 59 no9sept2011rdquo Yong-Dan Kong amp Qing xin chu

senior memberIEEE

5Genaral Finite DifferenceSchemes for Heat equations ldquoIndian J Pure Applemath

10(2)209-222Febuary 1979rdquoby PS Jain ampDN Holla department of methemetics

IITB 400076

  • Unconditionally Stable FDTD Methods
  • Outlineshelliphelliphelliphellip
  • Outlineshellip
  • Slide 4
  • Computational Electromagnetics
  • Slide 6
  • FDTD Overview ndash Updating Equations
  • Slide 8
  • Slide 9
  • Finite Difference Time Domain Method
  • The Yee Discretization (1966)
  • Slide 12
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Slide 17
  • Slide 18
  • Slide 19
  • Space Domain Discretization
  • Finite-Difference Formulation of the Heat Conduction on a Chip
  • Time domain discretization
  • Slide 23
  • Slide 24
  • Simple Explicit Method
  • Simple Implicit Method
  • Crank-Nicolson Method
  • Analysis of Crank-Nicolson Method
  • Alternating Direction Implicit Method
  • Slide 30
  • Peaceman-Rachford Algorithm
  • Douglas-Gunn Algorithm
  • Illustration for ADI
  • Analysis of ADI Method
  • Slide 35
  • Slide 36
  • Slide 37
  • Slide 38
  • Referenceshelliphelliphellip
  • Slide 40
Page 38: Unconditionally stable fdtd methods

Applications of FDTD method

1048708Electromagnetic scattering amp antenna design1048708EMCEMI design1048708Simulation of wave propagation problem1048708Solving partial defferential equation 1048708Waveguide analysis

Stripline amp microstrip line analysishellipetc hellip

Referenceshelliphelliphellip1-Electromagnetic simulation using FDTD method

ldquoDennis M Sillevanrdquo IEEE press series on rf amp mivcrowave technology ldquoRoger DPollard amp Richard

Bootanrdquoseries

2Understanding the FDTD method ldquoJohn BSchneiderrdquo may 8 2011

3 3 D -ADI -Mehod-unconditionaly stable time domain algorithom for solving maxwell equations

ldquoIEEE Transaction on microwave theory amp techniques vol-48No10 october 2010 ldquoTakefumi

IEEE

4High order split step unconditionally stable FDTD method ampnumerical analysis ldquoIEEE

Transactions on antinna amp propagationvol 59 no9sept2011rdquo Yong-Dan Kong amp Qing xin chu

senior memberIEEE

5Genaral Finite DifferenceSchemes for Heat equations ldquoIndian J Pure Applemath

10(2)209-222Febuary 1979rdquoby PS Jain ampDN Holla department of methemetics

IITB 400076

  • Unconditionally Stable FDTD Methods
  • Outlineshelliphelliphelliphellip
  • Outlineshellip
  • Slide 4
  • Computational Electromagnetics
  • Slide 6
  • FDTD Overview ndash Updating Equations
  • Slide 8
  • Slide 9
  • Finite Difference Time Domain Method
  • The Yee Discretization (1966)
  • Slide 12
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Slide 17
  • Slide 18
  • Slide 19
  • Space Domain Discretization
  • Finite-Difference Formulation of the Heat Conduction on a Chip
  • Time domain discretization
  • Slide 23
  • Slide 24
  • Simple Explicit Method
  • Simple Implicit Method
  • Crank-Nicolson Method
  • Analysis of Crank-Nicolson Method
  • Alternating Direction Implicit Method
  • Slide 30
  • Peaceman-Rachford Algorithm
  • Douglas-Gunn Algorithm
  • Illustration for ADI
  • Analysis of ADI Method
  • Slide 35
  • Slide 36
  • Slide 37
  • Slide 38
  • Referenceshelliphelliphellip
  • Slide 40
Page 39: Unconditionally stable fdtd methods

Referenceshelliphelliphellip1-Electromagnetic simulation using FDTD method

ldquoDennis M Sillevanrdquo IEEE press series on rf amp mivcrowave technology ldquoRoger DPollard amp Richard

Bootanrdquoseries

2Understanding the FDTD method ldquoJohn BSchneiderrdquo may 8 2011

3 3 D -ADI -Mehod-unconditionaly stable time domain algorithom for solving maxwell equations

ldquoIEEE Transaction on microwave theory amp techniques vol-48No10 october 2010 ldquoTakefumi

IEEE

4High order split step unconditionally stable FDTD method ampnumerical analysis ldquoIEEE

Transactions on antinna amp propagationvol 59 no9sept2011rdquo Yong-Dan Kong amp Qing xin chu

senior memberIEEE

5Genaral Finite DifferenceSchemes for Heat equations ldquoIndian J Pure Applemath

10(2)209-222Febuary 1979rdquoby PS Jain ampDN Holla department of methemetics

IITB 400076

  • Unconditionally Stable FDTD Methods
  • Outlineshelliphelliphelliphellip
  • Outlineshellip
  • Slide 4
  • Computational Electromagnetics
  • Slide 6
  • FDTD Overview ndash Updating Equations
  • Slide 8
  • Slide 9
  • Finite Difference Time Domain Method
  • The Yee Discretization (1966)
  • Slide 12
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Slide 17
  • Slide 18
  • Slide 19
  • Space Domain Discretization
  • Finite-Difference Formulation of the Heat Conduction on a Chip
  • Time domain discretization
  • Slide 23
  • Slide 24
  • Simple Explicit Method
  • Simple Implicit Method
  • Crank-Nicolson Method
  • Analysis of Crank-Nicolson Method
  • Alternating Direction Implicit Method
  • Slide 30
  • Peaceman-Rachford Algorithm
  • Douglas-Gunn Algorithm
  • Illustration for ADI
  • Analysis of ADI Method
  • Slide 35
  • Slide 36
  • Slide 37
  • Slide 38
  • Referenceshelliphelliphellip
  • Slide 40
Page 40: Unconditionally stable fdtd methods

4High order split step unconditionally stable FDTD method ampnumerical analysis ldquoIEEE

Transactions on antinna amp propagationvol 59 no9sept2011rdquo Yong-Dan Kong amp Qing xin chu

senior memberIEEE

5Genaral Finite DifferenceSchemes for Heat equations ldquoIndian J Pure Applemath

10(2)209-222Febuary 1979rdquoby PS Jain ampDN Holla department of methemetics

IITB 400076

  • Unconditionally Stable FDTD Methods
  • Outlineshelliphelliphelliphellip
  • Outlineshellip
  • Slide 4
  • Computational Electromagnetics
  • Slide 6
  • FDTD Overview ndash Updating Equations
  • Slide 8
  • Slide 9
  • Finite Difference Time Domain Method
  • The Yee Discretization (1966)
  • Slide 12
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Slide 17
  • Slide 18
  • Slide 19
  • Space Domain Discretization
  • Finite-Difference Formulation of the Heat Conduction on a Chip
  • Time domain discretization
  • Slide 23
  • Slide 24
  • Simple Explicit Method
  • Simple Implicit Method
  • Crank-Nicolson Method
  • Analysis of Crank-Nicolson Method
  • Alternating Direction Implicit Method
  • Slide 30
  • Peaceman-Rachford Algorithm
  • Douglas-Gunn Algorithm
  • Illustration for ADI
  • Analysis of ADI Method
  • Slide 35
  • Slide 36
  • Slide 37
  • Slide 38
  • Referenceshelliphelliphellip
  • Slide 40

UNCONDITIONALLY STABLE TIME SPLITTING METHODS FOR THE ELECTROSTATIC …acumen.lib.ua.edu/content/u0015/0000001/0001874/u0015... · 2015-07-28 · UNCONDITIONALLY STABLE TIME SPLITTING

FDTD(Intro) (3)

An Explicit and Unconditionally Stable FDTD Method for the ...djiao/publications/DanJiao_Lossy_F… · Digital Object Identifier 10.1109/TAP.2015.2448751 an infinitely large time

Fdtd Stuff

Unconditionally Stable Laguerre Polynomial-Based ... · PDF fileThe weighted Laguerre polynomials ... Electromagnetic-thermal-mechanical problem ... FDTD method for RCS calculations,

ppt on fdtd

AN UNCONDITIONALLY STABLE NONSTANDARD …matwbn.icm.edu.pl/ksiazki/amc/amc23/amc2329.pdfAN UNCONDITIONALLY STABLE NONSTANDARD FINITE DIFFERENCE METHOD APPLIED TO A MATHEMATICAL MODEL

Sullivan FDTD

View PDF - Purdue Universityshen7/pub/JCP_dong.pdfSpectral element method Navier-Stokes equation 1. Introduction ABSTRACT We present an unconditionally stable splitting scheme for

Radiofrequency Electronic Systems (2018-2019)mwl.diet.uniroma1.it/people/pisa/RFELSYS/L00_INTRODUCTION_2019.pdf · on unconditionally stable transistors, lumped-distributed matching

An unconditionally stable semi-implicit FSI finite element ......AN UNCONDITIONALLY STABLE SEMI-IMPLICIT FSI FINITE ELEMENT METHOD ALEXANDER LOZOVSKIY y, MAXIM A. OLSHANSKII z, VICTORIA

A novel unconditionally stable explicit ... - Mianlun Zheng

Fast Design of Unconditionally Stable Power Amplifier ...robertmarks.org/REPRINTS/2017 Fast design of unconditionally stable... · based on large signal stability analysis including

fdtd modelling

Comput. Methods Appl. Mech. Engrg. - College of Engineering · 2019. 1. 1. · Time-integration Unconditionally stable Phase-field crystal abstract The phase field crystal equation

EMPro FDTD Simulation

Understanding FDTD Method

194988734 Katy Perry Unconditionally

Fdtd Method

An Alternative Explicit and Unconditionally Stable Time

A novel unconditionally stable explicit integration method ...scbms.whu.edu.cn/pdf/A Novel Unconditionally Stable... · explicit integration methods based on the discrete control

fdtd simulation

Decoupled, unconditionally stable, higher order ...heister/preprints... · We now prove unconditional stability and well-posedness for the Algorithm3.1. To simplify no-tation, denote

HIGH SPATIAL ORDER ENERGY STABLE FDTD METHODS FOR …lif/papers/paperOP_Yee.pdf · the finite difference time domain (FDTD) method developed by Kane Yee in 1969, called the Yee scheme

UNCONDITIONALLY STABLE SCHEMES FOR HIGHER ORDER …bertozzi/papers/CMS11.pdf · COMMUN. MATH. SCI. c 2011 International Press Vol. 9, No. 2, pp. 413–457 UNCONDITIONALLY STABLE SCHEMES

Provably Unconditionally Stable, Second-order Time ... · Provably Unconditionally Stable, Second-order Time-accurate, Mixed Variational Methods for Phase-field Models Hector Gomez

FDTD Paper 2008

PIERS 2010 FDTD

Unconditionally stable method and numerical …Unconditionally stable method and numerical solution of the hyperbolic phase- eld crystal equation P.K.Galenko∗ Friedrich-Schiller-Universit

Make him love unconditionally

Simple, Fast, and Unconditionally Stable Direct Nonlinear ...esrcen.com/Paper-HASTAG-2015-DNA.pdf · Simple, Fast, and Unconditionally Stable Direct Nonlinear Analysis using nathan-a

· PDF fileContents Preface xv ... 2 Unconditionally Stable Laguerre Polynomial-Based FDTD Method Bin Chen, ... 4.8 Thermal Sources 92 4.9 Summary 95

Love unconditionally

FDTD Project report

HIGH-ORDER UNCONDITIONALLY-STABLE FOUR-STEP ADI … · 2018. 1. 9. · High-order four-step ADI-FDTD methods in 3-D domains are presented in this paper. Firstly, based on the exponential