cem in action computed surface currents on prototype military aircraft at 100mhz the plane wave is...
TRANSCRIPT
![Page 1: CEM in action Computed surface currents on prototype military aircraft at 100MHz The plane wave is incident from left to right at nose on incidence. The](https://reader036.vdocuments.us/reader036/viewer/2022062323/5697bfc71a28abf838ca8096/html5/thumbnails/1.jpg)
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)
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83 Camaro at 1 GHz
• Irradiation of a 83 Camaro at 1 GHz by a Hertzian dipole.
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Inlet Scattering
Simulation Measurement
> 2,000,000 unknowns
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Corrugated Horn Antenna
![Page 5: CEM in action Computed surface currents on prototype military aircraft at 100MHz The plane wave is incident from left to right at nose on incidence. The](https://reader036.vdocuments.us/reader036/viewer/2022062323/5697bfc71a28abf838ca8096/html5/thumbnails/5.jpg)
Microstrip Antenna Array
Current distribution
Radiation patterns
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Time Varying Current Distribution
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EMP
Microwave pulse penetrating a missile radome containing a hornantenna. Wave is from right to left at 15° from boresight.
![Page 8: CEM in action Computed surface currents on prototype military aircraft at 100MHz The plane wave is incident from left to right at nose on incidence. The](https://reader036.vdocuments.us/reader036/viewer/2022062323/5697bfc71a28abf838ca8096/html5/thumbnails/8.jpg)
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
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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
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Computational Electromagnetics
computationalelectromagnetics
High frequencyrigorous methods
IE DE
MoMFDTDTLM
field basedcurrent based
GO/GTD PO/PTD
TD FD TD FD
VM
FEM
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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
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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
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Time Harmonic Fields
• Fields that vary periodically (sinusoidally) with time
Time Harmonic Scalar Fields
PhasorTransform
P
Real, time harmonic
scalar
ComplexNumber (Phasor)
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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
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Faraday’s Law
sdBt
ldE
t
BE
c s
S
C
t
B
E
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Ampere’s Law
sc ssdJsdD
tldH
t
DJH
t
D
J
J
H
H
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Gauss’s Law
v totsQdvsdD
D
totQ
D
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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!)
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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)
![Page 25: CEM in action Computed surface currents on prototype military aircraft at 100MHz The plane wave is incident from left to right at nose on incidence. The](https://reader036.vdocuments.us/reader036/viewer/2022062323/5697bfc71a28abf838ca8096/html5/thumbnails/25.jpg)
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?
![Page 26: CEM in action Computed surface currents on prototype military aircraft at 100MHz The plane wave is incident from left to right at nose on incidence. The](https://reader036.vdocuments.us/reader036/viewer/2022062323/5697bfc71a28abf838ca8096/html5/thumbnails/26.jpg)
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
![Page 27: CEM in action Computed surface currents on prototype military aircraft at 100MHz The plane wave is incident from left to right at nose on incidence. The](https://reader036.vdocuments.us/reader036/viewer/2022062323/5697bfc71a28abf838ca8096/html5/thumbnails/27.jpg)
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
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PROPERTIES OF TIME HARMONIC FIELDS
]),,(~
[Re[]]),,(~
[Re[ tjtj ezyxEjezyxEt
]),,(~
[Re[1
]),,(~
[Re[ tjtj ezyxEj
dtezyxE
Time derivative:
Time integration:
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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:
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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
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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
![Page 32: CEM in action Computed surface currents on prototype military aircraft at 100MHz The plane wave is incident from left to right at nose on incidence. The](https://reader036.vdocuments.us/reader036/viewer/2022062323/5697bfc71a28abf838ca8096/html5/thumbnails/32.jpg)
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
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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
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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
![Page 35: CEM in action Computed surface currents on prototype military aircraft at 100MHz The plane wave is incident from left to right at nose on incidence. The](https://reader036.vdocuments.us/reader036/viewer/2022062323/5697bfc71a28abf838ca8096/html5/thumbnails/35.jpg)
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
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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
![Page 37: CEM in action Computed surface currents on prototype military aircraft at 100MHz The plane wave is incident from left to right at nose on incidence. The](https://reader036.vdocuments.us/reader036/viewer/2022062323/5697bfc71a28abf838ca8096/html5/thumbnails/37.jpg)
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
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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”
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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)
![Page 40: CEM in action Computed surface currents on prototype military aircraft at 100MHz The plane wave is incident from left to right at nose on incidence. The](https://reader036.vdocuments.us/reader036/viewer/2022062323/5697bfc71a28abf838ca8096/html5/thumbnails/40.jpg)
Numerical Differentiation“FINITE DIFFERENCES”
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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”
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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
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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
![Page 44: CEM in action Computed surface currents on prototype military aircraft at 100MHz The plane wave is incident from left to right at nose on incidence. The](https://reader036.vdocuments.us/reader036/viewer/2022062323/5697bfc71a28abf838ca8096/html5/thumbnails/44.jpg)
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
![Page 45: CEM in action Computed surface currents on prototype military aircraft at 100MHz The plane wave is incident from left to right at nose on incidence. The](https://reader036.vdocuments.us/reader036/viewer/2022062323/5697bfc71a28abf838ca8096/html5/thumbnails/45.jpg)
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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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Forward finite-difference formulas
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Centered finite difference formulas
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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
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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
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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
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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
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Partial First Derivatives
Problem: FIND ?),(
?),(
y
yxuor
x
yxu
recall:
h
xfxfxf iii 2
)()()(' 11
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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?
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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,,
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Partial Second DerivativesProblem: FIND ?
),(?
),(2
2
2
2
y
yxuor
x
yxu
recall:
211 )()(2)(
)(h
xfxfxfxf iiii
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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
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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
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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
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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
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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
![Page 73: CEM in action Computed surface currents on prototype military aircraft at 100MHz The plane wave is incident from left to right at nose on incidence. The](https://reader036.vdocuments.us/reader036/viewer/2022062323/5697bfc71a28abf838ca8096/html5/thumbnails/73.jpg)
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
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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
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Electrostatic Example using FD
MATLAB CODE EXAMPLE