Electromagnetic Course

1. Engineering Electromagnetics
Poisson’s & Laplace’s Equations
Nguyễn Công Phương

2. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn
Contents
I. Introduction
II. Vector Analysis
III. Coulomb’s Law & Electric Field Intensity
IV. Electric Flux Density, Gauss’ Law & Divergence
V. Energy & Potential
VI. Current & Conductors
VII. Dielectrics & Capacitance
VIII.Poisson’s & Laplace’s Equations
IX. The Steady Magnetic Field
X. Magnetic Forces & Inductance
XI. Time – Varying Fields & Maxwell’s Equations
XII. Transmission Lines
XIII. The Uniform Plane Wave
XIV. Plane Wave Reflection & Dispersion
XV. Guided Waves & Radiation
2

3. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn
Poisson’s & Laplace’s Equations
1. Poisson’s Equation
2. Laplace’s Equation
3. Uniqueness Theorem
4. Examples of the Solution of Laplace’s Equation
5. Examples of the Solution of Poisson’s Equation
6. Product Solution of Laplace’s Equation
7. Numerical Methods
3

4. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn
Poisson's Equation (1)
0
D E
ε
=
Gauss’s Law: v
ρ
∇ =
.D
( ) ( ) v
V
ε ε ρ
→ ∇ = ∇ = −∇ ∇ =
.D . E .
V
= −∇
E
Gradient: v
V
ρ
ε
→ ∇ ∇ = −
.
(Poisson's Equation)
x y z
V V V
V
x y z
∂ ∂ ∂
∇ = + +
∂ ∂ ∂
a a a
y
x z
A
A A
x y z
∂
∂ ∂
∇ = + +
∂ ∂ ∂
.A
2 2 2
2 2 2
.
y
x z
V
V V V V V
V
x x y y z z x y z
∂
 
∂ ∂
∂ ∂ ∂ ∂ ∂ ∂
   
→ ∇∇ = + + = + +
   
 
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
 
   
4

5. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn
Poisson's Equation (2)
v
V
ρ
ε
∇ ∇ = −
.
Define
2
∇ ∇ = ∇
.
2 2 2
2 2 2
.
V V V
V
x y z
∂ ∂ ∂
∇ ∇ = + +
∂ ∂ ∂
2 2 2
2
2 2 2
v
V V V
V
x y z
ρ
ε
∂ ∂ ∂
∇ = + + = −
∂ ∂ ∂
(rectangular)
2 2
2 2 2
1 1 v
V V V
z
ρ
ρ
ρ ρ ρ ρ ϕ ε
 
∂ ∂ ∂ ∂
+ + = −
 
∂ ∂ ∂ ∂
 
2
2
2 2 2 2 2
1 1 1
sin
sin sin
v
V V V
r
r r r r r
ρ
θ
θ θ θ θ ϕ ε
∂ ∂ ∂ ∂ ∂
   
+ + = −
   
∂ ∂ ∂ ∂ ∂
   
(cylindrical)
(spherical)
5

6. Poisson’s Equation (3)
Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn
Ex.
Find the Laplacian of the following scalar fields:
2 3
3
) 2
cos2
)
20sin
)
a A xy z
b B
c C
r
ϕ
ρ
θ
=
=
=
6

7. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn
Poisson’s & Laplace’s Equations
1. Poisson’s Equation
2. Laplace’s Equation
3. Uniqueness Theorem
4. Examples of the Solution of Laplace’s Equation
5. Examples of the Solution of Poisson’s Equation
6. Product Solution of Laplace’s Equation
7. Numerical Methods
7

8. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn
Laplace’s Equation
0
v
ρ =
2 2 2
2
2 2 2
v
V V V
V
x y z
ρ
ε
∂ ∂ ∂
∇ = + + = −
∂ ∂ ∂
(Laplace’s equation, rectangular)
2 2
2 2 2
1 1
0
V V V
z
ρ
ρ ρ ρ ρ ϕ
 
∂ ∂ ∂ ∂
+ + =
 
∂ ∂ ∂ ∂
 
2
2
2 2 2 2 2
1 1 1
sin 0
sin sin
V V V
r
r r r r r
θ
θ θ θ θ ϕ
∂ ∂ ∂ ∂ ∂
   
+ + =
   
∂ ∂ ∂ ∂ ∂
   
(cylindrical)
(spherical)
2 2 2
2
2 2 2
0
V V V
V
x y z
∂ ∂ ∂
∇ = + + =
∂ ∂ ∂
Poisson's Equation:
8

9. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn
Poisson’s & Laplace’s Equations
1. Poisson’s Equation
2. Laplace’s Equation
3. Uniqueness Theorem
4. Examples of the Solution of Laplace’s Equation
5. Examples of the Solution of Poisson’s Equation
6. Product Solution of Laplace’s Equation
7. Numerical Methods
9

10. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn
Uniqueness Theorem (1)
2 2 2
2
2 2 2
0
V V V
V
x y z
∂ ∂ ∂
∇ = + + =
∂ ∂ ∂
Assume two solutions V1 & V2, :
2
1 0
V
∇ =
2
2 0
V
∇ =
2
1 2
( ) 0
V V
→ ∇ − =
Assume the boundary condition Vb 1 2
b b b
V V V
→ = =
( ) ( ) ( )
. D .D D.
V V V
∇ = ∇ + ∇
1 2 1 2 1 2 1 2
1 2 1 2
[( ) ( )] ( )[ ( )]
( ) ( )
. .
.
V V V V V V V V
V V V V
→ ∇ − ∇ − = − ∇ ∇ − +
+∇ − ∇ −
1 2
V V V
= −
1 2
( )
D V V
= ∇ −
10

11. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn
Uniqueness Theorem (2)
1 2 1 2 1 2 1 2 1 2 1 2
[( ) ( )] ( )[ ( )] ( ) ( )
. . .
V V V V V V V V V V V V
∇ − ∇ − = − ∇ ∇ − +∇ − ∇ −
1 2 1 2 1 2 1 2
1 2 1 2
[( ) ( )] ( )[ ( )]
( ) ( )
V V
V
V V V V dv V V V V dv
V V V V dv
→ ∇ − ∇ − = − ∇ ∇ − +
+ ∇ − ∇ −
 

. .
.
. .
S V
d dv
= ∇
 
D S D
Divergence theorem:
1 2 1 2 1 2 1 2
[( ) ( )] [( ) ( )]
b b b b
V S
V V V V dv V V V V d
→ ∇ − ∇ − = − ∇ −
 
. . S
1 2
b b b
V V V
= =
1 2 1 2
[( ) ( )] 0
V
V V V V dv
→ ∇ − ∇ − =
 .
1 2 1 2 1 2 1 2
0 ( )[ ( )] ( ) ( )
V V
V V V V dv V V V V dv
→ = − ∇ ∇ − + ∇ − ∇ −
 
. .
11

12. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn
Uniqueness Theorem (3)
1 2 1 2 1 2 1 2
( )[ ( )] ( ) ( ) 0
V V
V V V V dv V V V V dv
− ∇ ∇ − + ∇ − ∇ − =
 
. .
2
1 2 1 2
( ) ( ) 0
. V V V V
∇ ∇ − = ∇ − =
1 2 1 2
( ) ( ) 0
V
V V V V dv
→ ∇ − ∇ − =
 . [ ]
2
1 2
( )
V
V V dv
= ∇ −

[ ]
2
1 2
( ) 0
V V
→ ∇ − =
[ ]
2
1 2
( ) 0
V V
∇ − ≥
1 2 const
V V
→ − =
1 2
( ) 0
V V
→ ∇ − =
x y z
V V V
V
x y z
∂ ∂ ∂
∇ = + +
∂ ∂ ∂
a a a
At boundary V1 = Vb1, V2 = Vb2
→ const = Vb1 – Vb2 = 0
1 2
b b b
V V V
= =
V1 = V2
12

13. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn
Poisson’s & Laplace’s Equations
1. Poisson’s Equation
2. Laplace’s Equation
3. Uniqueness Theorem
4. Examples of the Solution of Laplace’s Equation
5. Examples of the Solution of Poisson’s Equation
6. Product Solution of Laplace’s Equation
7. Numerical Methods
13

14. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn
Examples of the Solution of Laplace’s Equation (1)
Assume V = V(x)
2 2 2
2
2 2 2
0
V V V
V
x y z
∂ ∂ ∂
∇ = + + =
∂ ∂ ∂
2
2
0
d V
dx
→ = V Ax B
→ = +
1
1
x x
V V
=
=
2
2
x x
V V
=
=
1 2
1 2
2 1 1 2
1 2
V V
A
x x
V x V x
B
x x
−

=
 −

→ 
−
 =
 −

1 2 2 1
1 2
( ) ( )
V x x V x x
V
x x
− − −
→ =
−
0
0
x
V =
=
0
x d
V V
=
=
0
V x
V
d
→ =
Ex. 1
14

15. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn
Examples of the Solution of Laplace’s Equation (2)
Conductor surface
Conductor surface
x = d
x = 0
x
V = V(x)
0
0
x
V =
=
0
x d
V V
=
=
0
V x
V
d
→ =
E V
= −∇
0
E ax
V
d
→ = −
D E
ε
=
0
D ax
V
d
ε
→ = −
0
0
D D a
S x
x
V
d
ε
=
→ = = − 0
N
V
D
d
ε
→ = − 0
S N
V
D
d
ρ ε
→ = = −
0
V S
d
ε
= −
0
S
S S
V
Q dS dS
d
ε
ρ
−
→ = =
  0
Q
C
V
→ =
S
d
ε
=
Ex. 1
15

16. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn
Examples of the Solution of Laplace’s Equation (3)
Assume V = V(ρ) (cylindrical)
2 2
2
2 2 2
1 1
0
V V V
V
z
ρ
ρ ρ ρ ρ ϕ
 
∂ ∂ ∂ ∂
∇ = + + =
 
∂ ∂ ∂ ∂
 
1
0
V
ρ
ρ ρ ρ
 
∂ ∂
→ =
 
∂ ∂
 
1
0
d dV
d d
ρ
ρ ρ ρ
 
→ =
 
 
0
d dV
d d
ρ
ρ ρ
 
→ =
 
 
dV
A
d
ρ
ρ
→ =
ln
V A B
ρ
→ = +
0
ln
a
V A a B V
ρ =
= + =
ln 0 ( )
b
V A b B b a
ρ =
= + = >
0
0
ln ln
ln
ln ln
V
A
a b
V b
B
a b

=

 −
→ 
 = −
 −

0
ln( / )
ln( / )
b
V V
b a
ρ
→ =
Ex. 2
16

17. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn
Examples of the Solution of Laplace’s Equation (4)
Assume V = V(ρ) (cylindrical)
2 2
2
2 2 2
1 1
0
V V V
V
z
ρ
ρ ρ ρ ρ ϕ
 
∂ ∂ ∂ ∂
∇ = + + =
 
∂ ∂ ∂ ∂
 
0
ln( / )
ln( / )
b
V V
b a
ρ
→ =
0
ln( / )
E a
V
V
b a
ρ
ρ
→ = −∇ =
0
( )
ln( / )
N a S
V
D
a b a
ρ
ε
ρ
=
→ = =
0 2
ln( / )
S
S
V aL
Q dS
a b a
ε π
ρ
→ = =
 0
2
ln( / )
Q L
C
V b a
ε π
→ = =
Ex. 2
17

18. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn
Examples of the Solution of Laplace’s Equation (5)
Assume V = V(φ) (cylindrical)
2 2
2
2 2 2
1 1
0
V V V
V
z
ρ
ρ ρ ρ ρ ϕ
 
∂ ∂ ∂ ∂
∇ = + + =
 
∂ ∂ ∂ ∂
 
2
2 2
1
0
V
ρ ϕ
∂
→ =
∂
2
2
0
V
ϕ
∂
→ =
∂
V A B
ϕ
→ = +
0
0
V B
ϕ=
= =
0
V A B V
ϕ α
α
=
= + =
0
0
B
V
A
α
=


→ 
=


0
V V
ϕ
α
→ =
0
E a
V
V ϕ
αρ
→ = −∇ = −
Ex. 3
z
α
Air gap
18

19. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn
Examples of the Solution of Laplace’s Equation (6)
Ex. 4
Assume V = V(θ) (spherical)
2
2 2
2 2 2 2 2
1 1 1
sin 0
sin sin
V V V
V r
r r r r r
θ
θ θ θ θ ϕ
∂ ∂ ∂ ∂ ∂
   
∇ = + + =
   
∂ ∂ ∂ ∂ ∂
   
2
1
sin 0
sin
V
r
θ
θ θ θ
∂ ∂
 
→ =
 
∂ ∂
  sin 0
V
θ
θ θ
∂ ∂
 
→ =
 
∂ ∂
 
sin
dV
A
d
θ
θ
→ =
sin
d
dV A
θ
θ
→ = ln tan
2
A B
θ
 
= +
 
 
sin
d
V A B
θ
θ
→ = +

Assume r ≠ 0; θ ≠ 0; θ ≠ π
19

20. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn
Examples of the Solution of Laplace’s Equation (7)
Ex. 4
Assume V = V(θ) ln tan
2
V A B
θ
 
→ = +
 
 
0
ln tan
2
ln tan
2
V V
θ
α
 
 
 
→ =
 
 
 
/ 2
0
V θ π
=
=
0 ( / 2)
V V
θ α
α π
=
= <
V = 0
V = V0
α
Air gap
0
1
sin ln tan
2
V V
V
r
r
θ θ
α
θ
θ
∂
→ = −∇ = − = −
∂  
 
 
E a a
0
sin ln tan
2
S N
V
D E
r
ε
ρ ε
α
α
→ = = = −
 
 
 
20

21. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn
Examples of the Solution of Laplace’s Equation (8)
Ex. 4
Assume V = V(θ)
0
0
2
ln tan
2
V
dr
πε
α
∞
−
=
 
 
 

0
sin ln tan
2
S
S S
V
Q dS dS
r
ε
ρ
α
α
→ = = −
 
 
 
 
2
0
0 0
sin
sin ln tan
2
V r d dr
Q
r
π
ε α ϕ
α
α
∞
−
→ =
 
 
 
 
sin
dS r d dr
α ϕ
=
0
sin ln tan
2
S
V
r
ε
ρ
α
α
→ = −
 
 
 
1
0
2
ln cot
2
Q r
C
V
πε
α
→ =
 
 
 
≐
V = 0
V = V0
α
Air gap
21

22. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn
Poisson’s & Laplace’s Equations
1. Poisson’s Equation
2. Laplace’s Equation
3. Uniqueness Theorem
4. Examples of the Solution of Laplace’s Equation
5. Examples of the Solution of Poisson’s Equation
6. Product Solution of Laplace’s Equation
7. Numerical Methods
22

23. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn
Examples of the Solution of
Poisson’s Equation (1)
2
0
2
2
sech th
v
d V x x
a a
dx
ρ
ε
→ = −
2
( sech ; th )
x x
x x x x
e e
x x
e e e e
−
− −
−
= =
+ +
0
2 sech th
v v
x x
a a
ρ ρ
=
x
dV
E
dx
= −
2 v
V
ρ
ε
∇ = −
Poisson’s Equation :
0
1
2
sech
v a
dV x
C
dx a
ρ
ε
→ = +
0
1
2
sech
v
x
a x
E C
a
ρ
ε
→ = − −
–5 –4 –3 –2 –1
1 2 3 4 5
0,5
1
–0,5
–1
/
x a
0
v
v
ρ
ρ
p-type n-type
v
ρ
–5 –4 –3 –2 –1
1 2 3 4 5
–0,5
–1
/
x a
0
2
x
v
E
a
ε
ρ
x
E
1 0
C
→ =
0
2
sech
v
x
a x
E
a
ρ
ε
→ = −
If x → ± ∞ then Ex → 0
23

24. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn
2
/
0
4
arctg
4
x a
v a
V e
ρ π
ε
 
→ = −
 
 
0
2 sech th
v v
x x
a a
ρ ρ
=
2 v
V
ρ
ε
∇ = −
2
/
0
2
4
arctg x a
v a
V e C
ρ
ε
→ = +
0
2
sech
v
x
a x
E
a
ρ
ε
→ = −
–5 –4 –3 –2 –1
1 2 3 4 5
–0,5
–1
/
x a
0
2
x
v
E
a
ε
ρ
V
–5 –4 –3 –2 –1
1 2 3 4 5
0,25
0,5
–0,25
–0,5
/
x a
2
0
2 v
V
a
πε
ρ
x
E
Poisson’s equation :
0
0
x
V =
=
Supp.
2
0
2
4
0
4
v a
C
ρ π
ε
→ = +
Examples of the Solution of
Poisson’s Equation (2)
–5 –4 –3 –2 –1
1 2 3 4 5
0,5
1
–0,5
–1
/
x a
0
v
v
ρ
ρ
p-type n-type
v
ρ
24

25. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn
2
/
0
4
arctg
4
x a
v a
V e
ρ π
ε
 
= −
 
 
2
0
0
2 v
x x
a
V V V
πρ
ε
→ ∞ → −∞
= − =
0 0 0
0
2 sech th 2 sech th 2
v v v v
V V
x x x x
Q dv dv S dx aS
a a a a
ρ ρ ρ ρ
∞
= = = =
  
0 0
2 v V
Q S
ρ ε
π
→ =
0
2 sech th
v v
x x
a a
ρ ρ
=
0
0
dV
dQ dQ
I C C
dt dt dV
= = → =
0
0
2 2
v S
C S
V a
ρ ε ε
π π
→ = =
Examples of the Solution of
Poisson’s Equation (3)
–5 –4 –3 –2 –1
1 2 3 4 5
0,5
1
–0,5
–1
/
x a
0
v
v
ρ
ρ
p-type n-type
v
ρ
25

26. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn
Poisson’s & Laplace’s Equations
1. Poisson’s Equation
2. Laplace’s Equation
3. Uniqueness Theorem
4. Examples of the Solution of Laplace’s Equation
5. Examples of the Solution of Poisson’s Equation
6. Product Solution of Laplace’s Equation
7. Numerical Methods
26

27. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn
Product Solution of Laplace’s Equation (1)
• Previous examples assumed that V varies with one of the
three coordinates
• The product solution can be used to solve for V(x, y)
• Assume V = XY, X = X(x), Y = Y(y)
2 2 2
2
2 2 2
0
V V V
V
x y z
∂ ∂ ∂
∇ = + + =
∂ ∂ ∂
2 2
2 2
0
V V
x y
∂ ∂
→ + =
∂ ∂
( , )
V V x y
=
2 2
2 2
0
X Y
Y X
x y
∂ ∂
→ + =
∂ ∂
2 2
2 2
0
d X d Y
Y X
dx dy
→ + =
27

28. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn
Product Solution of Laplace’s Equation (2)
2 2
2 2
0
d X d Y
Y X
dx dy
+ =
2 2
2 2
1 1
0
d X d Y
X dx Y dy
→ + =
2 2
2 2
1 1
d X d Y
X dx Y dy
→ = −
2
2
1 d X
X dx
involves no y
2
2
1 d Y
Y dy
− involves no x
2
2
2
2
2
2
1
1
d X
X dx
d Y
Y dy
α
α

=


→ 
− =


28

29. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn
Product Solution of Laplace’s Equation (3)
2
2
2
2
1
d dR
R d d
d
d
ρ
ρ α
ρ ρ
α
ϕ
  
=
  
  
→ 
Φ
− =
 Φ

( , ) ( ) ( )
V V R
ρ ϕ ρ ϕ
= = Φ
2 2
2 2 2
1 1
0
V V V
z
ρ
ρ ρ ρ ρ ϕ
 
∂ ∂ ∂ ∂
+ + =
 
∂ ∂ ∂ ∂
 
2
2
1
0
R
R
ρ
ρ
ρ ρ ϕ
 
∂ ∂ ∂ Φ
→ + =
 
∂ ∂ Φ ∂
 
29

30. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn
Product Solution of Laplace’s Equation (4)
2
2 2
2
2
2
( 1)
1 1
( 1)
tg
R R
n n
R R
n n
ρ ρ
ρ ρ
θ θ θ
 ∂ ∂
+ = +

∂ ∂

→ 
∂ Θ ∂Θ
 + = − +
Θ ∂ Θ ∂

( , ) ( ) ( )
V V R
ρ θ ρ θ
= = Θ
2 2
2 2 2
2 1 1
0
tg
R R
R R
ρ ρ
ρ ρ θ θ θ
   
∂ ∂ ∂ Θ ∂Θ
→ + + + =
   
∂ ∂ Θ ∂ Θ ∂
   
2
2
2 2 2 2 2
1 1 1
sin 0
sin sin
V V V
r
r r r r r
θ
θ θ θ θ ϕ
∂ ∂ ∂ ∂ ∂
   
+ + =
   
∂ ∂ ∂ ∂ ∂
   
30

31. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn
Product Solution of Laplace’s Equation (5)
Ex.
2
2
2
2
1
d dR
R d d
d
d
ρ
ρ α
ρ ρ
α
ϕ
  
=
  
  
→ 
Φ
− =
 Φ

( , ) ( ) ( )
V V R
ρ ϕ ρ ϕ
= = Φ
( ) ( ); ( ) ( )
V V V V
ϕ ϕ ϕ π ϕ
= − = − −
( ) cos sin ,
Assume p p
A p B p p
ϕ ϕ ϕ α
Φ = + = ±
 
1
( ) cos , 1
A
ϕ ϕ α
→ Φ = =
Solve for the potential & EFI in the vicinity of a conducting cylinder (posses an
infinite length) in a uniform EFI E0. The permittivities of the environment & the
cylinder are ε1 & ε2 respectively. EFI is perpendicular to the cylinder’s axis.
31

32. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn
Product Solution of Laplace’s Equation (6)
2
2
2
2
1 d
d
d dR
R d d
α
ϕ
ρ
ρ α
ρ ρ
 Φ
− =

Φ


 
 =
 
  

2
2
k
k
k
k
k B
d dR
R d d B
ρ
ρ
ρ α
ρ ρ ρ
 
→ = =
 
 
1
α =
( )
Assume k
k
R B
ρ ρ
=
1
( ) cos , 1
A
ϕ ϕ α
→ Φ = =
1
k
→ = ±
1
1 1
( )
R B B
ρ ρ ρ
+ − −
→ = +
Ex.
32
Solve for the potential & EFI in the vicinity of a conducting cylinder (posses an
infinite length) in a uniform EFI E0. The permittivities of the environment & the
cylinder are ε1 & ε2 respectively. EFI is perpendicular to the cylinder’s axis.

33. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn
Product Solution of Laplace’s Equation (7)
2
2
1
2
2 1
1 1
1
( ) cos
( )
d
A
d
d dR
R B B
R d d
α ϕ ϕ
ϕ
ρ
ρ α ρ ρ ρ
ρ ρ
+ − −
 Φ
− = → Φ =

Φ


 
 = → = +
 
  

1
1 1 1 1
cos cos
V A B A B
ρ ϕ ρ ϕ
+ − −
→ = +
( , ) ( ) ( )
V V R
ρ ϕ ρ ϕ
= = Φ
1
cos cos
C C
ρ ϕ ρ ϕ
+ − −
= +
Ex.
33
Solve for the potential & EFI in the vicinity of a conducting cylinder (posses an
infinite length) in a uniform EFI E0. The permittivities of the environment & the
cylinder are ε1 & ε2 respectively. EFI is perpendicular to the cylinder’s axis.

34. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn
Product Solution of Laplace’s Equation (8)
1
1 1 1
cos cos
Exterior: V C C
ρ ϕ ρ ϕ
+ − −
= +
0 x
V E x
−∞ →−∞
=
1
2 2 2
cos cos
Interior: V C C
ρ ϕ ρ ϕ
+ − −
= +
1 1
, x
V V C x
θ π ρ
+
−∞ = →−∞ →−∞
= = −
x
y
E0
ρ
θ
ε1
ε2
1 0
C E
+
→ = −
2
2 0
0
gèc täa ®é
C
V V ρ
ρ
ρ
−
→
→
= = → ∞
2 0
C−
→ =
EFI is finite at the origin
Ex.
34
Solve for the potential & EFI in the vicinity of a conducting cylinder (posses an
infinite length) in a uniform EFI E0. The permittivities of the environment & the
cylinder are ε1 & ε2 respectively. EFI is perpendicular to the cylinder’s axis.

35. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn
Product Solution of Laplace’s Equation (9)
1
1 1 1
cos cos
Exterior: V C C
ρ ϕ ρ ϕ
+ − −
= +
1
2 2 2
cos cos
Interior: V C C
ρ ϕ ρ ϕ
+ − −
= +
x
y
E0
ρ
θ
ε1
ε2
1 0 2
, 0
C E C
+ −
= − =
1
1 0 1
2 2
cos cos
cos
V E C
V C
ρ ϕ ρ ϕ
ρ ϕ
− −
+
 = − +

→ 
=


Ex.
35
Solve for the potential & EFI in the vicinity of a conducting cylinder (posses an
infinite length) in a uniform EFI E0. The permittivities of the environment & the
cylinder are ε1 & ε2 respectively. EFI is perpendicular to the cylinder’s axis.

36. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn
Product Solution of Laplace’s Equation (10)
1
1 0 1
cos cos
V E C
ρ ϕ ρ ϕ
− −
= − +
2 2 cos
V C ρ ϕ
+
=
x
y
E0
ρ
θ
ε1
ε2
1 2
a a
V V
ρ ρ
= =
=
1
0 1 2
( )cos cos
E a C a C a
ϕ ϕ
− − +
→ − + =
a
1 2
1 2
a a
V V
ρ ρ
ε ε
ρ ρ
= =
∂ ∂
=
∂ ∂
2
1 0 1 2 2
( )cos cos
E C a C a
ε ϕ ε ϕ
− − +
→ − − =
Ex.
36
Solve for the potential & EFI in the vicinity of a conducting cylinder (posses an
infinite length) in a uniform EFI E0. The permittivities of the environment & the
cylinder are ε1 & ε2 respectively. EFI is perpendicular to the cylinder’s axis.

37. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn
Product Solution of Laplace’s Equation (11)
1
0 1 2
( )cos cos
E a C a C a
ϕ ϕ
− − +
− + =
2
1 0 1 2 2
( )cos cos
E C a C a
ε ϕ ε ϕ
− − +
− − =
2
1 2 1
1 0 2 0
1 2 1 2
2
,
C E a C E
ε ε ε
ε ε ε ε
− +
−
→ = − = −
+ +
2
1 2
1 0 2
1 2
1
2 0
1 2
1 cos ,
2
cos
as
as
a
V E a
V E a
ε ε
ρ ϕ ρ
ε ε ρ
ε
ρ ϕ ρ
ε ε
  
−
= − − ≥
  
+
  
→ 
 = − ≤
 +

1
1 0 1
cos cos
V E C
ρ ϕ ρ ϕ
− −
= − +
2 2 cos
V C ρ ϕ
+
=
Ex.
37
Solve for the potential & EFI in the vicinity of a conducting cylinder (posses an
infinite length) in a uniform EFI E0. The permittivities of the environment & the
cylinder are ε1 & ε2 respectively. EFI is perpendicular to the cylinder’s axis.

38. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn
Product Solution of Laplace’s Equation (12)
2
1 2
1 0 2
1 2
1
2 0
1 2
1 cos ,
2
cos
as
as
a
V E a
V E a
ε ε
ρ ϕ ρ
ε ε ρ
ε
ρ ϕ ρ
ε ε
  
−
= − − ≥
  
+
  

 = − ≤
 +

2 1 2 1
2 0 2 0
1 2 1 2
2 2
cos , sin
V V
E E E E
ρ ϕ
ε ε
ϕ ϕ
ρ ε ε ϕ ε ε
∂ ∂
= − = = − = −
∂ + ∂ +
2 2
1 1 2 1 1 2
1 0 1 0
2 2
1 2 1 2
1 cos , 1 sin
V V
a a
E E E E
ρ ϕ
ε ε ε ε
ϕ ϕ
ρ ε ε ρ ϕ ε ε ρ
   
∂ − ∂ −
= − = − = − = − −
   
∂ + ∂ +
   
1
2 2 0
1 2
2
z
E E E
ε
ε ε
→ = =
+
Ex.
38
Solve for the potential & EFI in the vicinity of a conducting cylinder (posses an
infinite length) in a uniform EFI E0. The permittivities of the environment & the
cylinder are ε1 & ε2 respectively. EFI is perpendicular to the cylinder’s axis.

39. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn
Poisson’s & Laplace’s Equations
1. Poisson’s Equation
2. Laplace’s Equation
3. Uniqueness Theorem
4. Examples of the Solution of Laplace’s Equation
5. Examples of the Solution of Poisson’s Equation
6. Product Solution of Laplace’s Equation
7. Numerical Methods
a. Finite Difference Method
b. Finite Element Method
39

40. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn
Finite Difference Method (1) x
y
h
h
V0 V1
V2
V3
V4
a
b
c
d
2 2 2
2
2 2 2
0
V V V
V
x y z
∂ ∂ ∂
∇ = + + =
∂ ∂ ∂
2 2
2 2
0
V V
x y
∂ ∂
→ + =
∂ ∂
( , )
V V x y
=
1 0
a
V V
V
x h
−
∂
≈
∂
2
1 0 0 3
2 2
V V V V
V
x h
− − +
∂
→ ≈
∂
0 3
c
V V
V
x h
−
∂
≈
∂
2
2
a c
V V
V x x
x h
∂ ∂
−
∂ ∂ ∂
≈
∂ 40

41. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn
Finite Difference Method (2) x
y
h
h
V0 V1
V2
V3
V4
2 2
2 2
0
V V
x y
∂ ∂
+ =
∂ ∂
2
2 0 0 4
2 2
V V V V
V
y h
− − +
∂
≈
∂
2
1 0 0 3
2 2
V V V V
V
x h
− − +
∂
≈
∂
2 2
1 2 3 4 0
2 2
4
0
V V V V V
V V
x y h
+ + + −
∂ ∂
→ + ≈ =
∂ ∂
( )
0 1 2 3 4
1
4
V V V V V
≈ + + +
→
41

42. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn
Finite Difference Method (3)
Ex. 1 V = 100
V = 0
V = 0
V = 0
Air gap Air gap
( )
0 1 2 3 4
1
4
V V V V V
= + + +
42
Ve
Vc
Va Vb
Vd Vf
Vg Vi
Vh
4 100 0
a d b
V V V
= + + +
4 100
b a e c
V V V V
= + + +
4 0 100
c b f
V V V
= + + +
4 0
d g e a
V V V V
= + + +
4 e b d h f
V V V V V
= + + +
4 0
f c e i
V V V V
= + + +
4 0 0
g d h
V V V
= + + +
4 0
h e g i
V V V V
= + + +
4 0 0
i f h
V V V
= + + +
42.86V
52.68V
42.86V
18.75V
25.00V
18,75V
7.14V
9.82 V
7.14V
a
b
c
d
e
f
g
h
i
V
V
V
V
V
V
V
V
V
 =

=

 =

=


→ =

 =

 =

=

 =

Method 1

43. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn
Finite Difference Method (4)
Ex. 1 V = 100
V = 0
V = 0
V = 0
Air gap Air gap
( )
0 1 2 3 4
1
4
V V V V V
= + + +
( )
1
0 100 0 0 25
4
+ + + =
( )
1
100 50 0 25 43.8
4
+ + + =
( )
1
0 25 0 0 6.2
4
+ + + =
( )
1
43.8 100 43.8 25 53.2
4
+ + + =
( )
1
25 43.8 0 6.2 18.8
4
+ + + = ( )
1
6.2 25 6.2 0 9.4
4
+ + + =
25
43.8
43.8 53.2
18.8 18.8
6.2 6.2
9.4
Step 1
43
Method 2

44. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn
Finite Difference Method (5)
V = 100
V = 0
V = 0
V = 0
Air gap Air gap
( )
0 1 2 3 4
1
4
V V V V V
= + + +
( )
1
53.2 100 0 18.8 43
4
+ + + =
( )
1
100 50 0 25 43.8
4
+ + + =
( )
1
43 100 43 25 52.8
4
+ + + =
( )
1
43.8 100 43.8 25 53.2
4
+ + + =
( )
1
25 43.8 0 6.2 18.8
4
+ + + =
25
43.8
43.8 53.2
18.8 18.8
6.2 6.2
9.4
( )
1
25 43 0 6.2 18.6
4
+ + + =
43 43
52.8
18.6 18.6
Step 2
Ex. 1
44
Method 2

45. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn
Finite Difference Method (6)
V = 100
V = 0
V = 0
V = 0
Air gap Air gap
( )
0 1 2 3 4
1
4
V V V V V
= + + +
25
43.8
43.8 53.2
18.8 18.8
6.2 6.2
9.4
43 43
52.8
18.6 18.6
( )
1
0 100 0 0 25
4
+ + + =
( )
1
0 25 0 0 6.2
4
+ + + =
( )
1
6.2 25 6.2 0 9.4
4
+ + + =
( )
1
18.6 52.8 18.6 9.4 24.9
4
+ + + =
( )
1
9.4 18.6 0 0 7.0
4
+ + + =
24.9
7.0 7.0
( )
1
7.0 25 7.0 0 9.8
4
+ + + =
9.8
Step 2
Ex. 1
45
Method 2

46. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn
Finite Difference Method (7)
V = 100
V = 0
V = 0
V = 0
Air gap Air gap
( )
0 1 2 3 4
1
4
V V V V V
= + + +
25
43.8
43.8 53.2
18.8 18.8
6.2 6.2
9.4
43 43
52.8
18.6 18.6
24.9
7.0 7.0
9.8
( )
1
52.8 100 0 18.6 42.9
4
+ + + =
( )
1
42.9 100 42.9 24.9 52.7
4
+ + + =
( )
1
24.9 42.9 0 7.0 18.7
4
+ + + =
( )
1
18.7 52.7 18.7 9.8 25.0
4
+ + + =
( )
1
9.8 18.7 0 0 7.1
4
+ + + = ( )
1
7.1 25 7.1 0 9.8
4
+ + + =
42.9 42.9
52.7
18.7 18.7
25.0
7.1 7.1
9.8
Step 3
Ex. 1
46
Method 2

47. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn
Finite Difference Method (8)
Ex. 2
0 V
0 V
0 V
0 V
10 V
20 V
4
1 2 3
5 6
7 8
9 10
47
1 2 4
4 10 0
V V V
= + + +
2 1 5 3
4 10
V V V V
= + + +
3 2 6
4 20 10
V V V
= + + +
4 1 5
4 0 0
V V V
= + + +
5 2 4 7 6
4V V V V V
= + + +
6 3 5 8
4 20
V V V V
= + + +
7 5 9 8
4 0
V V V V
= + + +
8 6 7 10
4 20
V V V V
= + + +
9 7 10
4 0 0
V V V
= + + +
10 8 9
4 0 20
V V V
= + + +
1
2
3
4
5
6
7
8
9
10
5.6423V
9.1735V
13.1111V
3.3957 V
7.9405V
13.2710 V
5.9219 V
12.0324V
3.7147 V
8.9368V
V
V
V
V
V
V
V
V
V
V
=


=

 =

=

 =

→ 
=

 =

=

 =

 =

Method 1

48. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn
Finite Difference Method (9)
Ex. 2
0 V
0 V
0 V
0 V
10 V
20 V
4
1 2 3
5 6
7 8
9 10
(0) (0) (0)
1 2 10
... 0
V V V
= = = =
( )
(1) (0) (0)
1 2 4
1
10 0 2.5000V
4
V V V
= + + + =
( )
(1) (0) (1) (0)
2 3 1 5
1
10 3.1250V
4
V V V V
= + + + =
( )
(1) (0) (1) (0)
7 8 5 9
...
1
0 0.2344V
4
...
V V V V
= + + + =
( )
(1) (1) (1)
10 8 9
1
20 0 6.7358V
4
V V V
= + + + =
48
Method 2

49. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn
Finite Difference Method (10)
Ex. 2
0 V
0 V
0 V
0 V
10 V
20 V
4
1 2 3
5 6
7 8
9 10
k
( )
1 (V)
k
V
( )
2 (V)
k
V
( )
3 (V)
k
V
( )
4 (V)
k
V
( )
5 (V)
k
V
( )
6 (V)
k
V
( )
7 (V)
k
V
( )
8 (V)
k
V
( )
9 (V)
k
V
( )
10 (V)
k
V
0 1
0
0
0
0
0
0
0
0
0
0
2.5000
3.1250
8.2813
0.6250
0.9375
7.3047
0.2344
6.8848
0.0586
6.7358
23
5.6429
9.1735
13.1111
3.3957
7.9405
13.2710
5.9219
13.0324
3.7147
8.9368
24
5.6429
9.1735
13.1111
3.3957
7.9405
13.2710
5.9219
13.0324
3.7147
8.9368
49
Method 2

50. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn
Poisson’s & Laplace’s Equations
1. Poisson’s Equation
2. Laplace’s Equation
3. Uniqueness Theorem
4. Examples of the Solution of Laplace’s Equation
5. Examples of the Solution of Poisson’s Equation
6. Product Solution of Laplace’s Equation
7. Numerical Methods
a. Finite Difference Method
b. Finite Element Method
50

51. Finite Element Method (1)
Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn
http://w ww.cosy.sbg.ac.at/~held/projects/mesh/mesh.html
51
http://imagine.inrialpes.fr/people/Francois.Faure/htmlCourses/FiniteElements.html
2
0
V
∇ =
( , )
V x y
1( , )
V x y
2
V
3
V
4
V

52. Finite Element Method (2)
Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn
http://illustrations.marin.ntnu.no/structures/analysis/FEM/theory/index.html
52
http://imagine.inrialpes.fr/people/Francois.Faure/htmlCourses/FiniteElements.html
2
0
V
∇ =
( , )
V x y
1( , )
V x y
2
V
3
V
4
V

53. Finite Element Method (3)
• Discretizing the solution region into a finite number of
elements,
• Obtaining governing equations for a typical elements,
• Combining all elements in the solution, &
• Solving the system of equations obtained.
Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 53
http://imagine.inrialpes.fr/people/Francois.Faure/htmlCourses/FiniteElements.html
2
0
V
∇ =
( , )
V x y
1( , )
V x y
2
V
3
V
4
V

54. Finite Element Method (4)
Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 54
1
e
V
2
e
V
3
e
V
1 1
( , )
x y
2 2
( , )
x y
3 3
( , )
x y
1
2
3
x
y
1
( , ) ( , )
N
e
e
V x y V x y
=
= 
( , )
e
V x y a bx cy
= + +
1 1 1
2 2 2
3 3 3
1
1
1
e
e
e
V x y a
V x y b
V x y c
     
     
=
     
     
     
1
1 1 1
2 2 2
3 3 3
1
1
1
e
e
e
a x y V
b x y V
c x y V
−
     
     
→ =
     
     
     
[ ]
1
2 3 3 2 3 1 1 3 1 2 2 1 1
2 3 3 1 1 2 2
1 1
3 2 1 3 2 1 3
2 2
3 3
( ) ( ) ( )
1
1 ( ) ( ) ( )
1
( ) ( ) ( )
1
1
e
e e
e
x y x y x y x y x y x y V
V x y y y y y y y V
x y
x x x x x x V
x y
x y
−
− − −
   
   
→ = − − −
   
   
− − −
   

55. Finite Element Method (5)
Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 55
[ ]
1
2 3 3 2 3 1 1 3 1 2 2 1 1
2 3 3 1 1 2 2
1 1
3 2 1 3 2 1 3
2 2
3 3
( ) ( ) ( )
1
1 ( ) ( ) ( )
1
( ) ( ) ( )
1
1
e
e e
e
x y x y x y x y x y x y V
V x y y y y y y y V
x y
x x x x x x V
x y
x y
−
− − −
   
   
= − − −
   
   
− − −
   
3
1
( , )
e i ei
i
V x y V
α
=
= 
1 2 3 3 2 2 3 3 2
2 3 1 1 3 3 1 1 3
3 1 2 2 1 1 2 2 1
2 1 3 1 3 1 2 1
1
[( ) ( ) ( ) ],
2
1
[( ) ( ) ( ) ],
2
1
[( ) ( ) ( ) ],
2
1
[( )( ) ( )( )]
2
x y x y y y x x x y
A
x y x y y y x x x y
A
x y x y y y x x x y
A
A x x y y x x y y
α
α
α
= − + − + −
= − + − + −
= − + − + −
= − − − − −

56. Finite Element Method (6)
Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 56
2
0
V
∇ = 0
ax b
+ =
2
0
2
d ax
bx c
dx
 
+ + =
 
 
2
1
2
e e
S
W E dS
ε
= 
V
= −∇
E
2
1
2
e e
S
W V dS
ε
→ = ∇

3
1
( , )
e i ei
i
V x y V
α
=
= 
3
1
e ei i
i
V V α
=
→ ∇ = ∇

3 3
1 1
1
( )( )
2
e ei i j ej
S
i j
W V dS V
ε α α
= =
 
→ = ∇ ∇
 
 

57. Finite Element Method (7)
Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 57
3 3
1 1
1
( )( )
2
e ei i j ej
S
i j
W V dS V
ε α α
= =
 
= ∇ ∇
 
 
( )
( )( )
e
ij i j
S
C dS
α α
= ∇ ∇

[ ]
1
2
3
e
e e
e
V
V V
V
 
 
=  
 
 
( ) ( ) ( )
11 12 13
( ) ( ) ( ) ( )
21 22 23
( ) ( ) ( )
31 32 33
e e e
e e e e
e e e
C C C
C C C C
C C C
 
 
  =  
 
 
 
[ ] [ ]
( )
1
2
T e
e e e
W V C V
ε  
→ =  

58. Finite Element Method (8)
Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 58
( )
( )( )
e
ij i j
S
C dS
α α
= ∇ ∇

( )
12 1 2
( )( )
e
S
C dS
α α
= ∇ ∇

1 2 3 3 2 2 3 3 2
2 3 1 1 3 3 1 1 3
2 1 3 1 3 1 2 1
1
[( ) ( ) ( ) ],
2
1
[( ) ( ) ( ) ],
2
1
[( )( ) ( )( )]
2
x y x y y y x x x y
A
x y x y y y x x x y
A
A x x y y x x y y
α
α
= − + − + −
= − + − + −
= − − − − −
( )
12 2 3 3 1 3 2 1 3
1
[( )( ) ( )( )]
4
e
C y y y y x x x x
A
→ = − − + − −

59. Finite Element Method (9)
Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 59
( )
12 2 3 3 1 3 2 1 3
1
[( )( ) ( )( )]
4
e
C y y y y x x x x
A
= − − + − −
( )
13 2 3 1 2 3 2 2 1
( )
23 3 1 1 2 1 3 2 1
( ) 2 2
11 2 3 3 2
( ) 2 2
22 3 1 1 3
( ) 2 2
33 1 2 2 1
( ) ( ) ( ) ( )
21 12 31 13 32
1
[( )( ) ( )( )]
4
1
[( )( ) ( )( )]
4
1
[( ) ( ) ]
4
1
[( ) ( ) ]
4
1
[( ) ( ) ]
4
, ,
e
e
e
e
e
e e e e
C y y y y x x x x
A
C y y y y x x x x
A
C y y x x
A
C y y x x
A
C y y x x
A
C C C C C
= − − + − −
= − − + − −
= − + −
= − + −
= − + −
= = ( ) ( )
23
e e
C
=
1 2 3 2 3 1 3 1 2
1 3 2 2 1 3 3 2 1
, ,
, ,
P y y P y y P y y
Q x x Q x x Q x x
= − = − = −
= − = − = −
( )
2 3 3 2
1
( )
4
2
e
ij i j i j
C PP QQ
A
PQ PQ
A
→ = +
−
=

60. Finite Element Method (3)
• Discretizing the solution region into a finite number of
elements,
• Obtaining governing equations for a typical elements,
• Combining all elements in the solution, &
• Solving the system of equations obtained.
Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 60
http://imagine.inrialpes.fr/people/Francois.Faure/htmlCourses/FiniteElements.html
2
0
V
∇ =
( , )
V x y
1( , )
V x y
2
V
3
V
4
V

61. Finite Element Method (10)
Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 61
[ ] [ ]
( )
1
2
T e
e e e
W V C V
ε  
=  
[ ] [ ][ ]
1
1
2
N
T
e
e
W W V C V
ε
=
= =

[ ]
1
2
3
n
V
V
V V
V
 
 
 
 
=
 
 
 
 
⋮

62. Finite Element Method (11)
Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 62
3
2
1
1
1
1
2
2
2
3
3 3
1
2 4
3
5
[ ] [ ][ ]
1
2
T
W V C V
ε
=
[ ]
11 12 13 14 15
21 22 23 24 25
31 32 33 34 35
41 42 43 44 45
51 52 53 54 55
C C C C C
C C C C C
C C C C C C
C C C C C
C C C C C
 
 
 
 
=
 
 
 
  (1) (2)
11 11 11
C C C
= +
(1)
22 33
C C
=
(1) (2) (3)
44 22 33 33
C C C C
= + +
(1) (2)
14 41 12 13
C C C C
= = +
23 32 0
C C
= =
[ ]
(1) (2) (1) (2) (1) (2)
11 11 13 12 12 13
(1) (1) (1)
31 33 32
(2) (2) (3) (2) (3) (3)
21 22 11 23 13 12
(1) (2) (1) (2) (3) (1) (2) (3) (3)
21 31 23 32 31 22 33 33 32
(3) (3) (3)
21 23 22
0
0 0
0
0 0
C C C C C C
C C C
C C C C C C C
C C C C C C C C C
C C C
 
+ +
 
 
 
= + +
 
+ + + +
 
 
 

63. Finite Element Method (3)
• Discretizing the solution region into a finite number of
elements,
• Obtaining governing equations for a typical elements,
• Combining all elements in the solution, &
• Solving the system of equations obtained.
Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 63
http://imagine.inrialpes.fr/people/Francois.Faure/htmlCourses/FiniteElements.html
2
0
V
∇ =
( , )
V x y
1( , )
V x y
2
V
3
V
4
V

64. Finite Element Method (12)
Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 64
0
ax b
+ =
2
0
2
d ax
bx c
dx
 
+ + =
 
 
2
0
V
∇ =
[ ] [ ][ ]
1
2
T
W V C V
ε
=
1 2
0 0, 1, 2,...,
n k
W W W W
k n
V V V V
∂ ∂ ∂ ∂
= = = = ↔ = =
∂ ∂ ∂ ∂
⋯
1,
1 n
k i ki
i i k
kk
V VC
C = ≠
→ = − 

65. Finite Element Method (13)
Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 65
Ex.
0
V =
1
2
3
x
y 4
100
V =
2
1
1
2
3
1
2
4
3
2
4
1
2
3
Node 1 2 3 4
x 0.5 3.1 5.0 2.8
y 1.0 0.4 1.7 2.0
( ) 2 3 3 2
1 2 3 2 3 1 3 1 2
1 3 2 2 1 3 3 2 1
1
( ),
4 2
, ,
, ,
e
ij i j i j
P Q PQ
C PP QQ A
A
P y y P y y P y y
Q x x Q x x Q x x
−
= + =
= − = − = −
= − = − = −
Element 1:
1 2 3
0.4 2.0 1.6; 2.0 1.0 1.0; 1.0 0.4 0.6
P P P
= − = − = − = = − =
1 2 3
2.8 3.1 0.3; 0.5 2.8 2.3; 3.1 0.5 2.6
Q Q Q
= − = − = − = − = − =
1.0 2.6 0.6( 2.3)
1.99
2
A
× − −
= =
(1) 1 2 1 2
12
( 1.6)1.0 ( 0.3)( 2.3)
0.1143
4 1.99 4 1.99
PP Q Q
C
+ − + − −
= = = −
× ×
(1)
0.3329 0.1143 0.2186
0.1143 0.7902 0.6759
0.2186 0.6759 0.8945
C
− −
 
 
  = − −
   
 
− −
 

66. Finite Element Method (14)
Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 66
Ex.
0
V =
1
2
3
x
y 4
100
V =
Node 1 2 3 4
x 0.5 3.1 5.0 2.8
y 1.0 0.4 1.7 2.0
2
1
1
2
3
1
2
4
3
2
4
1
2
3
( ) 2 3 3 2
1 2 3 2 3 1 3 1 2
1 3 2 2 1 3 3 2 1
1
( ),
4 2
, ,
, ,
e
ij i j i j
P Q PQ
C PP QQ A
A
P y y P y y P y y
Q x x Q x x Q x x
−
= + =
= − = − = −
= − = − = −
Element 2:
1 2 3
1.7 2.0 0.3; 2.0 0.4 1.6; 0.4 1.7 1.3
P P P
= − = − = − = = − = −
1 2 3
2.8 5 2.2; 3.1 2.8 0.3; 5.0 3.1 1.9
Q Q Q
= − = − = − = = − =
1.6 1.9 ( 1.3)0.3
1.715
2
A
× − −
= =
(2) 2 2 2 2
22
1.6 1.6 0.3 0.3
0.3863
4 1.715 4 1.715
P P Q Q
C
+ × + ×
= = =
× ×
(2)
0.7187 0.1662 0.5525
0.1662 0.3863 0.2201
0.5525 0.2201 0.7726
C
− −
 
 
  = − −
   
 
− −
 

67. Finite Element Method (15)
Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 67
Ex.
0
V =
1
2
3
x
y 4
100
V =
(1)
(2)
0.3329 0.1143 0.2186
0.1143 0.7902 0.6759
0.2186 0.6759 0.8945
0.7187 0.1662 0.5525
0.1662 0.3863 0.2201
0.5525 0.2201 0.7726
C
C
− −
 
 
  = − −
   
 
− −
 
− −
 
 
  = − −
   
 
− −
 
[ ]
(1) (1) (1)
11 12 13
(1) (1) (2) (2) (1) (2)
21 22 11 12 23 13
(2) (2) (2)
21 22 23
(1) (1) (2) (2) (1) (2)
31 32 31 32 33 33
0
0
C C C
C C C C C C
C
C C C
C C C C C C
 
 
+ +
 
=
 
 
+ +
 
 
1
2
3
1
2
3
0.3329 0.1143 0 0.2186
0.1143 1.5089 0.1662 1.2284
0 0.1662 0.3863 0.2201
0.2186 1.2284 0.2201 1.6671
− −
 
 
− − −
 
=
 
− −
 
− − −
 

68. Finite Element Method (16)
Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 68
Ex.
0
V =
1
2
3
x
y 4
100
V =
[ ]
0.3329 0.1143 0 0.2186
0.1143 1.5089 0.1662 1.2284
0 0.1662 0.3863 0.2201
0.2186 1.2284 0.2201 1.6671
C
− −
 
 
− − −
 
=
 
− −
 
− − −
 
1,
1 n
k i ki
i i k
kk
V VC
C = ≠
= − 
1
2
3
1
2
3
2 1 12 3 32 4 42
22
4 1 14 2 24 3 34
44
1
( )
1
( )
V VC V C V C
C
V VC V C V C
C

= − + +


→ 
 = − + +


( 1) ( ) ( )
2 4 4
( 1) ( ) ( )
4 2 2
1
[(0( 0.1143) 100( 0.1662) ( 1.2284)] 11.0146 0.8141
1.5089
1
[0( 0.2186) ( 1.2284) 100( 0.2201)] 13.2026 0.7368
1.6671
k k k
k k k
V V V
V V V
+
+

= − − + − + − = +


→ 
 = − − + − + − = +



69. Finite Element Method (17)
Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 69
Ex.
0
V =
1
2
3
x
y 4
100
V =
( 1) ( )
2 4
( 1) ( )
4 2
11.0146 0.8141
13.2026 0.7368
k k
k k
V V
V V
+
+
 = +


= +


(0) (0)
2 4
0 100
50
2
V V
+
= = =
1
2
3
1
2
3
(1) (0)
2 4
(1) (0)
4 2
11.0146 0.8141 11.0146 0.8141 50 51.7196
13.2026 0.7368 13.2026 0.7368 50 50.0426
V V
V V
 = + = + × =


= + = + × =


(2) (1)
2 4
(2) (1)
4 2
11.0146 0.8141 11.0146 0.8141 50.0426 51.7543
13.2026 0.7368 13.2026 0.7368 51.7196 51.3096
V V
V V
 = + = + × =


= + = + × =


(3) (2)
2 4
(3) (2)
4 2
11.0146 0.8141 11.0146 0.8141 51.3096 52.7857
13.2026 0.7368 13.2026 0.7368 51.7543 51.3352
V V
V V
 = + = + × =


= + = + × =



70. Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 70
Q 1 2
2
4
R
Q Q
R
πε
=
F a 2
4
R
Q
R
πε
=
E a ε
=
D E
.
W Q d
= − E L .
V d
= −E L
Q
C
V
=
dQ
I
dt
=
V
R
I
= 2 v
V
ρ
ε
∇ = −