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