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CEM in action
Computed surface currents on prototype military aircraft at 100MHzThe plane wave is incident from left to right at nose on incidence.The currents re-radiate back to the source radar (and so can bedetected)
83 Camaro at 1 GHz
• Irradiation of a 83 Camaro at 1 GHz by a Hertzian dipole.
Inlet Scattering
Simulation Measurement
> 2,000,000 unknowns
Corrugated Horn Antenna
Microstrip Antenna Array
Current distribution
Radiation patterns
Time Varying Current Distribution
EMP
Microwave pulse penetrating a missile radome containing a hornantenna. Wave is from right to left at 15° from boresight.
Broadband Analysis of Wave Interactions with Nonlinear Electronic Circuitry
25 cm
25 cm
5 cm
17.5 cm
10 cm
1 cm
20 cm
4.5 cm6 cm
xy
z
k̂
excE
0.5 cm
15 cm
1 cm
y
500 500 500 500
Voltages on the varistors
0 2 4 6 8x 10
-3
-1.5
-1
-0.5
0
0.5
1
1.5line1line2line3
Vo l
t ag e
(kV
)
( )t s
EM solvers permit analysis of wave broadband EMC/EMI phenomena, and the assessment of electronic upset and terrorism scenarios
Scattering at 3 GHz from Full Fighter Plane (fast solvers)
Bistatic RCS of VFY218 at 3 GHz8 processors of SGI Origin 2000# of Unknowns N = 2 millions
FIESLUDCG
Memory Matrix-fill LUD One-RHS (GB) (days) (years) (hrs)
5 0.1 932,000 600.0 200 432,000 600.0 500
AZ
Computational Electromagnetics
computationalelectromagnetics
High frequencyrigorous methods
IE DE
MoMFDTDTLM
field basedcurrent based
GO/GTD PO/PTD
TD FD TD FD
VM
FEM
Computational Electromagnetics
Electromagnetic problems are mostly described by three methods:
Differential Equations (DE) Finite difference (FD, FDTD)Integral Equations (IE) Method of Moments (MoM)Minimization of a functional (VM) Finite Element (FEM)
Theoreticaleffort
less more
Computationaleffort
more less
Fields• Fields: A space (and time) varying
quantity– Static field: space varying only– Time varying field: space and time varying– Scalar field: Magnitude varies in space (and
time)– Vector field: Magnitude & direction varies in
space (and time)
Moving Fields…... Electromagnetic waves
Time Harmonic Fields
• Fields that vary periodically (sinusoidally) with time
Time Harmonic Scalar Fields
PhasorTransform
P
Real, time harmonic
scalar
ComplexNumber (Phasor)
Maxwell’s Equations in Differential Form
mB
D
Jt
DH
Mt
BE
Faraday’s Law
Ampere’s Law
Gauss’s Law
Gauss’s Magnetic Law
Faraday’s Law
sdBt
ldE
t
BE
c s
S
C
t
B
E
Ampere’s Law
sc ssdJsdD
tldH
t
DJH
t
D
J
J
H
H
Gauss’s Law
v totsQdvsdD
D
totQ
D
Gauss’s Magnetic Law
0
0
ssdB
B
B
“all the flow of B entering the volume V must leave the volume”
ms
m
QsdB
B
(no magnetic charges!)
CONSTITUTIVE RELATIONS
EJ
HB
ED
c
r o=permittivity (F/m)
o=8.854 x 10-12 (F/m)
r o=permeability (H/m)
o=4 x 10-7 (H/m)
=conductivity (S/m)
POWER and ENERGY
0,0][
]2
1[,]
2
1[
)(
2
22
vdv ii
vevm
ss
dvEPdvJEP
dvEWdvHW
dsHEP
diems PPWt
Wt
P
Stored magnetic power (W)
Stored electric power (W)
Supplied power (W)
Dissipated power (W)
What is this term?
POWER and ENERGY
0,0][
]2
1[,]
2
1[
)(
2
22
vdv ii
vevm
ss
dvEPdvJEP
dvEWdvHW
dsHEP
diems PPWt
Wt
P
Stored magnetic power (W)
Stored electric power (W)
Supplied power (W)
Dissipated power (W)
What is this term?
Ps = power exiting the volume through radiation
HES
W/m2 Poynting vector
TIME HARMONIC EM FIELDS
]),,(~
Re[),,,(
)),,(cos(),,(),,,(tj
o
ezyxEtzyxE
zyxtzyxEtzyxE
Assume all sources have a sinusoidal time dependence and all materialsproperties are linear. Since Maxwell’s equations are linear all electricand magnetic fields must also have the same sinusoidal time dependence.They can be written for the electric field as:
),,(~
zyxE is a complex function of space (phasor) called the time-harmonic electricfield. All field values and sources can be represented by their time-harmonic form.
]),,(~Re[),,,(
]),,(~
Re[),,,(
]),,(~
Re[),,,(
]),,(~
Re[),,,(
]),,(~
Re[),,,(
]),,(~
Re[),,,(
tj
tj
tj
tj
tj
tj
ezyxtzyx
ezyxJtzyxJ
ezyxBtzyxB
ezyxHtzyxH
ezyxDtzyxD
ezyxEtzyxE
)sin()cos( tjte tj Euler’s Formula
PROPERTIES OF TIME HARMONIC FIELDS
]),,(~
[Re[]]),,(~
[Re[ tjtj ezyxEjezyxEt
]),,(~
[Re[1
]),,(~
[Re[ tjtj ezyxEj
dtezyxE
Time derivative:
Time integration:
TIME HARMONIC MAXWELL’S EQUATIONS
tj
mtj
tjtj
tjtjtj
tjtjtj
eeB
eeD
eJeDt
eH
eMeBt
eE
~Re
~Re
~Re~
Re
~Re
~Re
~Re
~Re
~Re
~Re
mB
D
Jt
DH
Mt
BE
mB
D
JDjH
MBjE
~~
~~
~~~
~~~
Employing the derivative property results in the following set of equations:
TIME HARMONIC EM FIELDSBOUNDARY CONDITIONS AND CONSTITUTIVE PROPERTIES
The constitutive properties and boundary conditions are very similarfor the time harmonic form:
0)~~
(ˆ
~)~~
(ˆ
~)
~~(ˆ
0)~~
(ˆ
12
12
12
12
BBn
DDn
JHHn
EEn
s
s
EJ
HB
ED
c~~
~~
~~
Constitutive Properties
General Boundary Conditions
0~
ˆ
~~ˆ
~~ˆ
0~
ˆ
2
2
2
2
Bn
Dn
JHn
En
s
s
PEC Boundary Conditions
TIME HARMONIC EM FIELDSIMPEDANCE BOUNDARY CONDITIONS
If one of the material at an interface is a good conductor but of finiteconductivity it is useful to define an impedance boundary condition:
HnjHnZJZE
jjXRZ
ssst
sss
~ˆ
2)1(
~ˆ
~~
2)1(
1,
2,
1>> 2
POWER and ENERGY: TIME HARMONIC
0~
2
1,0]
~~2
1[
]~
4
1[,]
~4
1[
)~~
(
2*
22
*
vdv ii
vevm
ss
dvEPdvJEP
dvEWdvHW
dsHEP
diems PPWWjP )(2
Time average magneticenergy (J)
Time average electric energy (J)
Supplied complex power (W)
Dissipated real power (W)Time average exiting power
CONTINUITY OF CURRENT LAW
JDt
Jt
DH
][][)(
0
B
D
Jt
DH
t
BE
0)( A
vector identity
JDt
][0
Jt
][0
tJ
jJ
time harmonic
SUMMARY
mBD
Jt
DHM
t
BE
mBD
JDjHMBjE
~~~~
~~~~~~
0)~~
(ˆ~)~~
(ˆ
~)
~~(ˆ0)
~~(ˆ
1212
1212
BBnDDn
JHHnEEn
s
s
0)(ˆ)(ˆ
)(ˆ0)(ˆ
1212
1212
BBnDDn
JHHnEEn
s
s
EJ
HB
ED
c~~
~~
~~
EJ
HB
ED
c
2
)1( jjXRZ sss
0,0][
]2
1[,]
2
1[
)(
2
22
vdv ii
vevm
ss
dvEPdvJEP
dvEWdvHW
dsHEP
0~
2
1,0]
~~2
1[
]~
4
1[,]
~4
1[
)~~
(
2*
22
*
vdv ii
vevm
ss
dvEPdvJEP
dvEWdvHW
dsHEP
Frequency DomainTime Domain
Wave Equation
0
B
E
JEt
EH
t
HE
Ht
E
Htt
HE
)(][
t
J
t
E
t
EE
JEt
E
tE
2
2
AAA
2)( Vector Identity
t
J
t
E
t
EEE
2
22)(
t
J
t
E
t
EE
2
221
Time Dependent Homogenous Wave Equation (E-Field)
1
2
22
t
J
t
E
t
EE
Wave EquationSource-Free Time Dependent Homogenous Wave Equation (E-Field)
1
2
22
t
J
t
E
t
EE
0,0 J
Source Free
02
22
t
E
t
EE
Source-Free Lossless Time Dependent Homogenous Wave Equation (E-Field)
0Lossless
02
22
t
EE
Wave EquationSource-Free Time Dependent Homogenous Wave Equation (H-Field)
1
2
22 J
t
H
t
HH
0,0 J
Source Free 02
22
t
H
t
HH
0,0,0 J
Source Free and Lossless 02
22
t
HH
Wave Equation: Time Harmonic
1
2
22
t
J
t
E
t
EE
0,0 J
Source Free
02
22
t
E
t
EE
0Lossless
02
22
t
EE
Time Domain Frequency Domain
~1~~~~ 22 JEjEE
0,0 J
Source Free
0~~~ 22 EjEE
0Lossless
0~~ 22 EE
“Helmholtz Equation”
MOST POPULAR COMPUTATIONALELECTROMAGNETICS ALGORITHMS
• FINITE DIFFERENCE (FD) METHODSExample: Finite difference time domain (FDTD)
• INTEGRAL EQUATION METHODS (IE)Example: Method of Moments (MoM)
• VARIATIONAL METHODSExample: Finite element method (FEM)
Numerical Differentiation“FINITE DIFFERENCES”
Introduction to differentiation
• Conventional Calculus
– The operation of diff. of a function is a well-defined procedure
– The operations highly depend on the form of the function involved
– Many different types of rules are needed for different functions
– For some complex function it can be very difficult to find closed form solutions
• Numerical differentiation
– Is a technique for approximating the derivative of functions by employing only arithmetic operations (e.g., addition, subtraction, multiplication, and division)
– Commonly known as “finite differences”
Taylor SeriesProblem: For a smooth function f(x),
Given: Values of f(xi) and its derivatives at xi
Find out: Value of f(x) in terms of f(xi), f(xi), f(xi), ….
x
yf(x)
f(xi)
xi
Taylor’s TheoremIf the function f and its n+1 derivatives are continuous on an interval containing xi and x, then the value of the function f at x is given by
nn
ii
n
ii
ii
iii
Rxxn
xf
xxxf
xxxf
xxxfxfxf
)(!
)(...
)(!3
)()(
!2
)(''))((')()(
)(
3)3(
2
Finite Difference Approximationsof the First Derivative using the Taylor Series
(forward difference)
x
yf(x)
f(xi)
xi xi+1
f(xi+1)
h
Assume we can expand a function f(x) into a Taylor Series about the point xi+1
nn
iii
n
iii
iii
iiiii
Rxxn
xf
xxxf
xxxf
xxxfxfxf
)(!
)(...
)(!3
)()(
!2
)(''))((')()(
1
)(
31
)3(2
111
h
Finite Difference Approximationsof the First Derivative using the Taylor Series (forward
difference)Assume we can expand a function f(x) into a Taylor Series about the point xi+1
ni
nii
iii hn
xfh
xfh
xfhxfxfxf
!
)(
!3
)(
!2
)(")(')()(
)(3
)3(2
1
h
xfxfxf iii
)()()(' 1
Ignore all of these terms
1)(
2)3(
1
!
)(
!3
)(
!2
)(")()()(' ni
niiii
i hn
xfh
xfh
xf
h
xfxfxf
Finite Difference Approximationsof the First Derivative using the Taylor
Series (forward difference)
h
xfxfxf iii
)()()(' 1
x
yf(x)
f(xi)
xi xi+1
f(xi+1)
h
Finite Difference Approximationsof the First Derivative using the
forward difference: What is the error?
)()()(
)(' 1 hOh
xfxfxf iii
The first term we ignored is of power h1. This is defined as first order accurate.
1)(
2)3(
1
!
)(
!3
)(
!2
)(")()()(' ni
niiii
i hn
xfh
xfh
xf
h
xfxfxf
)()('
)()( 1
hOh
fxf
xfxff
ii
iii
First forwarddifference
Finite Difference Approximationsof the First Derivative using the Taylor
Series (backward difference)
x
yf(x)
f(xi-1)
xi-1 xi
f(xi)
h
Assume we can expand a function f(x) into a Taylor Series about the point xi-1
nn
iii
n
iii
iii
iiiii
Rxxn
xf
xxxf
xxxf
xxxfxfxf
)(!
)(...
)(!3
)()(
!2
)(''))((')()(
1
)(
31
)3(2
111
-h
Finite Difference Approximationsof the First Derivative using the Taylor Series
(backward difference)
ni
nii
iii hn
xfh
xfh
xfhxfxfxf
!
)(
!3
)(
!2
)(")(')()(
)(3
)3(2
1
Ignore all of these terms
1)(
2)3(
1
!
)(
!3
)(
!2
)(")()()(' ni
niiii
i hn
xfh
xfh
xf
h
xfxfxf
)()()(
)(' 1 hOh
xfxfxf iii
)()('
)()( 1
hOh
fxf
xfxff
ii
iii
First backwarddifference
Finite Difference Approximationsof the First Derivative using the Taylor
Series (backward difference)
x
yf(x)
f(xi-1)
xi-1 xi
f(xi)
h
)()()(
)(' 1 hOh
xfxfxf iii
Finite Difference Approximationsof the Second Derivative using the Taylor Series
(forward difference)
y
x
f(x)
f(xi)
xi xi+1
f(xi+1)
h
xi+2
f(xi+2)
ni
nii
iii hn
xfh
xfh
xfhxfxfxf
!
)(
!3
)(
!2
)(")(')()(
)(3
)3(2
1
nni
nii
iii hn
xfh
xfh
xfhxfxfxf 2
!
)(8
!3
)(4
!2
)("2)(')()(
)(3
)3(2
2
(1)
(2)
(2)-2* (1)
)()()(2)(
)(" )3(2
112i
iiii xhf
h
xfxfxfxf
Finite Difference Approximationsof the Second Derivative using the Taylor Series
(forward difference)
y
x
f(x)
f(xi)
xi xi+1
f(xi+1)
h
xi+2
f(xi+2)
)()()(2)(
)(" )3(2
112i
iiii xhf
h
xfxfxfxf
)()(
)()("22
2
hOh
fhO
h
fxf iii
)(2
2
hOh
f
dx
fd in
xx
n
i
Recursive formula forany order derivative
Higher Order Finite Difference Approximations
)()()(2)(
)(" )3(2
112i
iiii xhf
h
xfxfxfxf
1)(
2)3(
1
!
)(
!3
)(
!2
)(")()()(' ni
niiii
i hn
xfh
xfh
xf
h
xfxfxf
1)(
2)3(
)3(12
1
!
)(
!3
)(
!2
...)()()(2)(
)()()('
nin
i
iiii
iii
hn
xfh
xf
hxhf
hxfxfxf
h
xfxfxf
...)('''32
)(3)(4)()('
212
xfh
h
xfxfxfxf iiii
)(2
)(3)(4)()(' 212 hO
h
xfxfxfxf iiii
Centered Difference Approximation
)(2
)()()(' 211 hO
h
xfxfxf iii
3
)3(2
1 !3
)(
!2
)(")(')()( h
xfh
xfhxfxfxf ii
iii
3
)3(2
1 !3
)(
!2
)(")(')()( h
xfh
xfhxfxfxf ii
iii
(1)
(2)
(1)-(2) 3
)3(
11 !3
)(2)('2)()( h
xfhxfxfxf i
iii
2)3(
11
!3
)(2
2
)()()(' h
xf
h
xfxfxf iiii
Finite Difference Approximationsof the First Derivative using the Taylor
Series (central difference)
x
yf(x)
f(xi-1)
xi-1 xi
f(xi)
h
xi+1
f(xi+1)
)(2
)()()(' 211 hO
h
xfxfxf iii
Second Derivative Centered Difference Approximation (central
difference)
)()()(2)(
)( 22
11 hOh
xfxfxfxf iiii
3
)3(2
1 !3
)(
!2
)(")(')()( h
xfh
xfhxfxfxf ii
iii
3
)3(2
1 !3
)(
!2
)(")(')()( h
xfh
xfhxfxfxf ii
iii
(1)
(2)
(1)+(2) 4
)4(2
11 !4
)(2)()(2)()( h
xfhxfxfxfxf i
iiii
2)4(
211
!4
)(2
)()(2)()( h
xf
h
xfxfxfxf iiiii
Using Taylor Series Expansions we found the following finite-differences
equations
)()()(
)(' 1 hOh
xfxfxf iii
FORWARD DIFFERENCE
)()()(
)(' 1 hOh
xfxfxf iii
BACKWARD DIFFERENCE
)(2
)()()(' 211 hO
h
xfxfxf iii
CENTRAL DIFFERENCE
)()()(2)(
)( 22
11 hOh
xfxfxfxf iiii
CENTRAL DIFFERENCE
Forward finite-difference formulas
Centered finite difference formulas
Finite Difference Approx. Partial DerivativesProblem: Given a function u(x,y) of two independent
variables how do we determine the derivative numerically (or more precisely PARTIAL DERIVATIVES) of u(x,y)
?),(
?),(
?),(
?),(
?),( 2
2
2
2
2
yx
yxUor
y
yxUor
x
yxUor
y
yxUor
x
yxU
Pretty much the same way
STEP #1: Discretize (or sample) U(x,y) on a 2D grid of evenly spaced points in the x-y plane
x axis
y axis
xi xi+1xi-1 xi+2
yj
yj+1
yj-1
yj-2
u(xi,yj) u(xi+1,yj)
u(xi,yj-1)
u(xi,yj+1)
u(xi-1,yj)
u(xi-1,yj+1)
u(xi-1,yj-1)
u(xi-1,yj-2) u(xi,yj-2)
u(xi+1,yj-1)
u(xi+1,yj-2)
u(xi+1,yj+1)
u(xi+2,yj)
u(xi+2,yj-1)
u(xi+2,yj-2)
u(xi+2,yj+1)
2D GRID
x axis
y axis
i i+1i-1 i+2
j
j+1
j-1
j-2
ui,j ui+1,jui-1,j
ui,j-1
ui,j+1
SHORT HAND NOTATION
Partial First Derivatives
Problem: FIND ?),(
?),(
y
yxuor
x
yxu
recall:
h
xfxfxf iii 2
)()()(' 11
Partial First Derivatives
Problem: FIND ?),(
?),(
y
yxuor
x
yxu
x
yxuyxu
x
yxu jijiji
2
),(),(),( 11
x
y
y
yxuyxu
y
yxu jijiji
2
),(),(),( 11
These are central difference formulas
Are these the only formulaswe could use?
Could we use forward or backwarddifference formulas?
Partial First Derivatives: short hand notation
Problem: FIND ?),(
?),(
y
yxuor
x
yxu
x
uu
x
u jijiji
2,1,1,
x
y y
uu
y
u jijiji
21,1,,
Partial Second DerivativesProblem: FIND ?
),(?
),(2
2
2
2
y
yxuor
x
yxu
recall:
211 )()(2)(
)(h
xfxfxfxf iiii
Partial Second Derivatives
Problem: FIND
2
11
2
2 ),(),(2),(),(
x
yxuyxuyxu
x
yxu jijijiji
x
y
?),(
?),(
2
2
2
2
y
yxuor
x
yxu
2
11
2
2 ),(),(2),(),(
y
yxuyxuyxu
y
yxu jijijiji
Partial Second Derivatives: short hand notation
Problem: FIND
2
,1,1,1
2
,2 2
x
uuu
x
u jijijiji
x
y
?),(
?),(
2
2
2
2
y
yxuor
x
yxu
2
1,,1,
2
,2 2
y
uuu
y
u jijijiji
FINITE DIFFERENCE ELECTROSTATICS
Electrostatics deals with voltages and charges that do no vary as a functionof time.
/),,(),,(2 zyxzyx Poisson’s equation
0),,(2 zyx Laplace’s equation
Where, is the electrical potential (voltage), is the charge density and is the permittivity.
E
o
1
2
3
FINITE DIFFERENCE ELECTROSTATICS: Example
0),(2 yx
Find(x,y) inside the box due to the voltages applied to its boundary. Thenfind the electric field strength in the box.
E
Electrostatic Example using FD
Problem: FIND
2
,1,,1
2
,2 2
xxjijijiji
x
y
0),(),(
2
2
2
2
y
yx
x
yx
2
1,,1,
2
,2 2
yyjijijiji
Electrostatic Example using FD
Problem: FIND
022
2
,1,,1
2
1,,1,
xyjijijijijiji
0),(),(
2
2
2
2
y
yx
x
yx
If x = y
jijijijiji
jijijijiji
jijijijijiji
,1,11,1,,
,,1,11,1,
,1,,11,,1,
4
1
04
022
Electrostatic Example using FD
Problem: FIND 0),(),(
2
2
2
2
y
yx
x
yx
jijijijiji ,1,11,1,, 4
1
Iterative solution technique:(1) Discretize domain into a grid of points(2) Set boundary values to the fixed boundary values(3) Set all interior nodes to some initial value (guess at it!)(4) Solve the FD equation at all interior nodes(5) Go back to step #4 until the solution stops changing(6) DONE
Electrostatic Example using FD
MATLAB CODE EXAMPLE