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
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
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
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
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
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
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
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
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
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
(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
(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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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