V12/13 - Institut für Allgemeine Elektrotechnik, Uni Rostock

iae.uni.rostock.de

V12/13 - Institut für Allgemeine Elektrotechnik, Uni Rostock

Boundary Element Method

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1


Free Space Green‘s Functions

Dirac‘s function:

⎧δi




⎩δ

i,

, j

j

=

0

= ∞

for

for

i

i


=

j

j


2 G = −δ i,j

source point

r i,j

G

1

=

4⋅

π⋅ r

i,j

1 ⎛

G = ln⎜

2⋅

π


1

r

i,j

in




3D

in

2D

x

z j r i r field point

y

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2


Electrostatics

Gauss‘ law:

Scalar potential:

r r

D = ε ⋅ E

divD = ρ

E r

= gradφ

φ

i

=

N


j=

1

q 1

q

j

4⋅

π⋅ r

r i,j

i, j

⋅ε

Identity:

divgradφ = ∇

2

φ

z

i r

φ i

ρ

∇ 2 φ = −

ε

φ(i)

r

=


V

ρ


4⋅

π⋅ r ⋅ε

dV

x

y

r i,j

q2

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3


Ampre‘s law:

Magnetostatics (1)

curl H

r

= r

J

Magnetic vector potential:

r r

B = µ ⋅ H

Identity:

r

curlcurlA =

r r

B = curlA

graddiv A

2

− ∇

r

Coulomb‘s gauge condition: div A = 0

A


2

r

A

r

= −µ ⋅ J

r r

A(i)

=

µ

4⋅

V

r r

J( j)

r

π∫


i,j

dV

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4


Magnetostatics (2)

r r

A(i)

=

µ

4⋅

V

r r

J( j)

r

π∫


i,j

dV

J ≠

0

A z

field point

J z

A x

A y


⎪A



⎨A



⎪A

⎪⎩

x

y

z

r

(i)

r

(i)

r

(i)

=

=

=

µ

4⋅

π

µ

4⋅

π

µ

4⋅

π


V


V


V

r

Jx

( j)

⋅ dV

ri,j

r

J y(

j)

⋅ dV

ri,j

r

Jz(

j)

⋅ dV

r

i,j

x

z

r

j

J x

y

J y

r

i

r i,j

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5


Calculating of volume integrals


⎪A



⎨A



⎪A

⎪⎩

x

y

z

r

(i)

r

(i)

r

(i)

=

=

=

µ

4⋅

π

µ

4⋅

π

µ

4⋅

π


V


V


V

r

Jx

( j)

⋅ dV

ri,j

r

J y(

j)

⋅ dV

ri,j

r

Jz(

j)

⋅ dV

r

i,j

J ≠ 0

J x

J z

r i,j

J y

A x

A z

A y

I

=

x

+ ∆ y + ∆z

+ ∆

0

∫ ∫ ∫

0

0

f (x, y,z)dxdydz


V ⋅f (x0 + ∆,

y0

+ ∆,z0

+ ∆)

x

0

y

0

z

0

Gaussian

Quadrature:

I

=

1

1

1

∫∫∫

f (x, y,z)dxdydz ≅

∑∑∑

−1

−1

− 1

k= 1 j= 1 i=

1

n

n

n

f (x

i

, y

i

,z

i

) ⋅ w

i

⋅ w

j

⋅ w

k

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6


Biot-Savart‘s law

Magnetic vector potential:

r r

B = ∇× A

r

B(i)

=

µ 0

4⋅

π


V

r r

J( j) × 1r

2

r

i,j


dV

r

J ⋅dV

=

r

J ⋅dS⋅dl

=

I⋅dl

r

B(i)

=

µ 0

4⋅

π


L

I⋅

( dl(j) × 1r

)

⋅ dL

r

2

i,j

r

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7


Laplace‘s equation

2

∇ φ =

0

in

Ω

n r

Γ 2

2

∇ φ =

0

Γ 1

⎧ φ = φ



∂φ

⎪ = q

⎩∂n

on

on

Γ

Γ

1

2

x

z

y

Ω

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8


Differential methods versus integral ones

Differential model

Integral model

(homogeneous domain)

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9


Direct Boundary Element Method (1)

2

∇ φ =

0

in

Ω

⎧ φ = φ



∂φ

⎪ = q

⎩∂n

on

on

Γ

Γ

1

2

weighted residual

equation:


Ω

w

⋅∇

2

φ

dΩ =


Γ

2




∂φ

∂n


q


⎟⋅


w dΓ −


Γ

1

∂w

∂n

( φ − φ) ⋅ dΓ

Green‘s second

theorem:


Ω

w

⋅∇

2

φ dΩ =


Ω

φ ⋅∇

2

w dΩ +


Γ




∂φ

∂w⎞

w⋅

−φ

⋅ ⎟

∂n

∂n



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10


Direct Boundary Element Method (2)


Ω

φ⋅∇

2

w dΩ +


Γ




∂φ

w ⋅

∂n

∂w

− φ⋅

∂n




=


Γ

2




∂φ

∂n

− q


⎟⋅


w dΓ −


Γ

1

∂w

∂n

( φ − φ) ⋅ dΓ


Ω

φ⋅∇

2

w dΩ = −


Γ

2

w ⋅ q

dΓ +


Γ

2

∂w

φ⋅

∂n

dΓ −


Γ

1

∂φ

w ⋅

∂n

dΓ +


Γ

1

∂w

φ ⋅

∂n



Ω

φ⋅∇

2

w dΩ =


Γ

∂w

φ⋅

∂n

dΓ −


Γ

∂φ

w ⋅

∂n


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11


Direct Boundary Element Method (3)


Ω

φ⋅∇

2

w dΩ =


Γ

∂w

φ⋅

∂n

dΓ −


Γ

∂φ

w ⋅

∂n



2

w = −δ i,

j

φ*

=

φ*

=

1

4⋅

π⋅ r

i,j

1 ⎛

ln⎜

2⋅

π


1

r

i,j

in




3D

in

2D

q* =

∂φ*

δn

φ =

∫φ

j

Ω

⋅δ

i i,

j


φ

i

=


Γ

∂φ j

φ* ⋅ Γ −∫

i,j d q * i,j ⋅φ j

∂n

Γ


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12


Direct Boundary Element Method (4)

φ

i

=


Γ

∂φ j

φ* ⋅ Γ −∫

i,j d q * i,j ⋅φ j

∂n

Γ


φ*

=

1

4⋅

π⋅ r

i,j

in

3D

φ

j1

∂φ

,

∂n

φ

j1

j2

∂φ

,

∂n

j2

φ

φ i

j3

∂φ

,

∂n

j3

φ*

=

1 ⎛

ln⎜

2⋅

π


∂φ*

q* =

δn

1

r

i,j




in

2D

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13


Base equation of Boundary Element Method (5)

c

i

⋅φ

i

=


Γ

φ*

i,j

∂φ j


∂n

dΓ −


Γ

q *

i,j

⋅φ

j


c i

=

⎧1


⎪1


⎪2


⎩0

for i inside Ω

for i on Γ

for i outside Ω

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14


Discretisation of boundary (1)

2D BEM models:

constat (zero order)

elements:

linear (first order)

elements:

quadratic (second order)

elements:

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15


Discretisation of boundary (2)

1

⋅φi

2

=


Γ

φ*

i,j

∂φ j


∂n

dΓ −


Γ

q *

i,j

⋅φ

j


r

i

1

⋅φi

2

1

2

G

⋅φ

i

=

=

n


j=

1

n


j=

1

=

∫φ

∂φ

j

∂n

∂φ

j

∂n

i , j * i,

j

Γ

j



Γ

j

⋅G


φ*

i,j

i,j



dΓ −

n


j=

1

φ

j

n


j=

1

⋅Ĥ

=


q

Γ

j

φ

i,j

i , j * i,

j

j



Γ

j

q *


i,j


r

j

φ*

=

r i,j

1 ⎛

ln⎜

2⋅

π


∂φ*

q* =

δn

1

r

i,j




in

2D

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16


17

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Matrix equation






































∂ ∂ ∂

=

n

n

n

n

n

2

2

1

1

2

1

.

.

φ

φ

φ

φ

φ

φ

X






=

+


=

j

i

H

j

i

H

H

j

i

j

i

j

i

for

ˆ

2

1

for

ˆ

,

,

,

i,j

n

1

j

j

i,j

n

1

j

j

G

n

H



=

=



∂φ

=


φ

Q

G

Φ

H


=


F

X

A =


Calculation of integrals (i,i)

G

G

i,i

i,i

=

∫ φ *

Γ

i

i,i


=

∫φ*

dΓ = 2⋅∫

i,i

Γ

because:

i


⎛ 1

ln⎜

⎝ a ⋅ r

φ*

=

1

0




1 ⎛

ln⎜

2 ⋅π


1 ⎛ 1

⋅ln⎜


⎝ l⋅ξ

1

r i , j


⎟⋅





l dξ

in

⎡ ⎛ 1 ⎞⎤

dr = r ⋅


1+

ln⎜

⎟ ⎣ ⎝ a ⋅ r ⎠



2D

ξ = −1

local coordinate

system:

r

ξ = 0

= l⋅ξ

r

l

ξ =1

1

l ⎡ ⎛ ⎛ 1 ⎞⎞⎤

1

G i , i = ⋅ ⎢ξ

⋅⎜1+

ln⎜

⎟⎟

π


⎣ ⎝ ⎝ l ⋅ξ

⎠⎠⎦

0

l ⎡ ⎛ ⎞⎤

= ⋅


1+

ln⎜


π ⎣ ⎝ l ⎠



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18


Calculation of integrals (i,i)

Ĥ

i,i

q *

i,i

=

∫q *

Γ

i

i,i

∂φ*

=

∂n

i,i


ξ = −1

n r

ξ = 0

ξ =1

Ĥ i , i =

0

H

i,

j

=

⎧Hˆ

i,

j


⎨1

⎪ + Hˆ

⎩2

i,

j

for

for

i

i


=

j

j

H i , i =

1

2

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19


Calculation of integrals (i,j)



i,

j

=


q

i , j * i,

j

Γ

=


q * dΓ =


i,

j

Γ

j

j


Γ

j

∂φ

*

∂n

i,

j


i

d

r i,j

ξ = −1

ξ = 0

j

ξ =1


i,

j

=

l



Γ

j


∂r

⎡ ⎛

⎢ ⎜

1

ln

⎢⎣

⎝ ri

,

j

⎞⎤

∂ri

,

⎟⎥


⎠⎥⎦

∂n

j

l


= −


1


−1

d

r

i,

j

2

i , j



i,

j

l

= −


n


k=

1

w

k


r

d

i,

j

2

( ξk

)

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20


Calculation of integrals (i,j)

G

=

∫φ

i , j * i,

j

Γ

j


d

ξ = −1

ξ = 0

G

i,

j

=

∫φ

* dΓ =


i,

j

Γ

j

Γ

j

1 ⎛


1

⋅ ln


⎝ ri

,

j





i

r i,j

j

ξ = 1

G

i,

j

=

1

n

⎛ ⎞



1


l

ln dξ

= ⋅


⎝ ri

, j

1

⎠ 2π

k=

1

l


⎛ 1

wk

⋅ ln⎜


⎝ξk




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21


First order boundary elements (1)

c

i

⋅φ

i

=


Γ

φ*

i,j

∂φ j


∂n

dΓ −


Γ

q *

i,j

⋅φ

j


shape functions of element:



N


⎪N


1

2

1

( ξ ) = ⋅

2

1

( ξ ) = ⋅

2

⎪⎧

x(

ξ ) = N


⎪⎩ y(

ξ ) = N

1

1

( 1−ξ

)

( 1+

ξ )

( ξ ) ⋅ x

( ξ ) ⋅ y

1

1

+

+

N

N

2

2

( ξ ) ⋅ x

( ξ ) ⋅ y

2

2

ξ = −1

( x1,

y1)

ξ = 0

( x2,

y2

ξ =1

)

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22


First order boundary elements (2)

⎧φ

( ξ ) = N1(

ξ ) ⋅φ1

+ N2(

ξ ) ⋅φ2


⎨∂φ

∂φ1

∂φ2

( ξ ) = N1(

ξ ) ⋅ + N2(

ξ ) ⋅

⎪⎩ ∂n

∂n

∂n



φ(

ξ)

=




⎪∂φ

⎪ ( ξ)

=

⎪∂n

⎪⎩

[ N N ]

1

[ N N ]

1

2

2

⎡φ1


⎢ ⎥

⎢ ⎥

⎢⎣

φ2⎥⎦

⎡∂φ

⎢ ∂n


⎢∂φ

⎢⎣

∂n

1

2





⎥⎦

φ

∂φ

n

1 1

, ∂

ξ = −1

ξ = 0

ξ = 1

∂φ2

φ2, ∂n

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23


First order boundary elements (3)

c

i

⋅φ

i

=


Γ

φ*

i,j

∂φ j


∂n

dΓ −


Γ

q *

i,j

⋅φ

j


∫ φ ⋅ Γ =


j q * i,j d

Γ

j

Γ

j

⎡φ


[ ]

⎢ ⎥

⋅ Γ = [

(1) (2)

N N q * d h h ]

1

2


⎣⎢

φ

1

2


⎥⎦

i,j

i,j

i,j

⎡φ



⎢⎣

φ

1

2




⎥⎦

( 1) =


( 2)

hi , j N1

⋅ q * i,

j dΓ

h =


i , j N2

⋅ q * i,

j dΓ

Γ

j

Γ

j

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24


First order boundary elements (4)

c

i

⋅φ

i

=


Γ

φ*

i,j

∂φ j


∂n

dΓ −


Γ

q *

i,j

⋅φ

j



Γ

j

∂φ

j

∂n

⋅φ*

i,j

dΓ =


Γ

j

⎡∂φ

⎢ ∂n

1



[ ] [

(1) (2)

N N ⎢ ⎥ ⋅φ*

dΓ = g g ]

1

2

⎢∂φ

⎢⎣

∂n

2


⎥⎦

i,j

i,j

i,j

⎡∂φ

⎢ ∂n


⎢∂φ

⎢⎣

∂n

1

2





⎥⎦

( 1)

g , =


j N1

⋅φ

*

i i,

j


( 2)

g , =


j N2

⋅φ

*

i i,

j


Γ

j

Γ

j

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25


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First order boundary elements (5)

[ ] [ ]























φ

φ

φ



























∂φ


∂φ


∂φ

=

⋅φ

n

2

1

i,n

i,2

i,1

n

2

1

i,n

i,2

i,1

i

i

..

Ĥ

..

Ĥ

Ĥ

n

..

n

n

G

..

G

G

c




=

+


=

j

i

for

Ĥ

c

j

i

for

Ĥ

H

i,j

i

i,j

i,j



=

=


∂φ


=

⋅φ

n

1

j

j

i,j

n

1

j

j

i,j

n

G

H


27

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Postprocessing

Γ

⋅φ

Γ −


∂φ


φ

=

φ




Γ

Γ

d

q *

d

n

*

c

j

i,j

j

i,j

i

i








=

Ω

i

Γ

i

Ω

i

c i

outside

for

0

on

for

2

1

inside

for

1

i,j

n

1

j

j

i,j

n

1

j

j

i

Ĥ

G

n



=

=


φ




∂φ

=

φ

n

,

j1

j1


∂φ

φ

n

,

j3

j3


∂φ

φ

i

φ

n

,

j2

j2


∂φ

φ


3D boundary elements

boundary triangle elemens:

constat (zero order)

elements:

linear (first order)

elements:

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28


First order triagle element (1)

r r r r r r r r r

= ⋅η⋅1

r x ⋅1x

+ y ⋅1y

+ z ⋅1z

= x3

⋅1x

+ y3

⋅1y

+ z3

⋅1z

+ l1

⋅ξ⋅1ξ

+ l2

η

r

1

r

1

ξ

η

=

=

x

x

− x

r

y

− y

r

1

− z

1 3 1 3 1 3

⋅1x

+ ⋅ y + ⋅1z

l1

l1

l1

− x

r

y

− y

2 3 2 3 2 3

⋅1x

+ ⋅1y

+ ⋅1z

l1

l1

l1

r

z

z

− z

z

r

r

(0,0)

r

3

l 2

l 1

2

1

(0,1)

(1,0)

ξ

η

x

y

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29


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First order triagle element (2)

( ) [ ]

[ ]

[ ] z

y

x

z

z

z

z

z

y

y

y

y

y

x

x

x

x

x

r

1

)

(

)

(

1

)

(

)

(

1

)

(

)

(

,

3

2

3

1

3

3

2

3

1

3

3

2

3

1

3

r

r

r




+



+

+




+



+

+




+



+

=

η

ξ

η

ξ

η

ξ

η

ξ

r










+


+


=




+


+


=




+


+


=

3

2

1

3

2

1

3

2

1

)

(1

)

,

(

)

(1

)

,

(

)

(1

)

,

(

z

z

z

z

y

y

y

y

x

x

x

x

η

ξ

η

ξ

η

ξ

η

ξ

η

ξ

η

ξ

η

ξ

η

ξ

η

ξ


First order triagle element (3)

⎧ x(

ξ,

η)

= N


⎨y(

ξ,

η)

= N


⎩ z(

ξ,

η)

= N

⎧N


⎨N


⎩N

1

2

3

( ξ,

η)

( ξ,

η)

( ξ,

η)

1

1

1

= ξ

= η

( ξ,

η) ⋅ x + N ( ξ,

η) ⋅ x + N ( ξ,

η)

1

( ξ,

η) ⋅ y + N ( ξ,

η) ⋅ y + N ( ξ,

η)

1

( ξ,

η) ⋅ z + N ( ξ,

η) ⋅ z + N ( ξ,

η)

= 1−ξ

−η

1

2

2

2

2

2

2

3

3

3

⋅ x


y

⋅ z

3

3

3

FB Elektrotechnik und Informationstechnik, AG Computational Electrodynamics

31


First order triagle element (4)

φ(

ξ,

η)

=

[ N ( ξ,

η)

N ( ξ,

η)

N ( ξ,

η ]

1 2

3 )

⎡φ1


⎢ ⎥

⎢φ2


⎢ ⎥

⎣⎢

φ3

⎥⎦

∂φ3

φ3,

∂n

2

φ

∂φ

∂n

2 , 2

∂φ

( ξ,

η)

∂n

=

[ N ( ξ,

η)

N ( ξ,

η)

N ( ξ,

η ]

1 2

3 )

⎡∂φ1


⎢ ⎥

⎢∂ ∂ n

φ ⎥

2

⎢ ⎥

⎢ ⎥

⎢∂ ∂ n

φ3


⎢⎣

∂n

⎥⎦

3

1

φ

∂φ

n

1 1

, ∂

FB Elektrotechnik und Informationstechnik, AG Computational Electrodynamics

32


3D Model with first order triangle elements (1)

c

i

⋅φ

i

=

n

n

∂φ

∑∫

j

⋅φ*

∑∫

i,j dΓ − φ j ⋅q * i,j dΓ

∂n

j= 1 Γ j= 1 Γ

j

j

∫ φ ⋅ Γ =


j q * i,j d

Γ

j

Γ

j

⎡φ






[ ] [

(1) (2) (3)

N N N ⎢φ

⎥ ⋅q * dΓ = h h h ]

1

2

3



⎣⎢

φ

1

2

3

( 1) =


( 2)

hi , j N1

⋅ q * i,

j dΓ

=


( 3)

hi , j N2

⋅ q * i,

j dΓ

h =


i , j N3

⋅ q * i,

j dΓ

Γ

j

Γ

j



⎥⎦

i,j

i,j

i,j

Γ

j

i,j

⎡φ



⎢φ



⎣⎢

φ

1

2

3







⎥⎦

FB Elektrotechnik und Informationstechnik, AG Computational Electrodynamics

33


3D Model with first order triangle elements (2)

c

i

⋅φ

i

=

n

n

∂φ

∑∫

j

⋅φ*

∑∫

i,j dΓ − φ j ⋅q * i,j dΓ

∂n

j= 1 Γ j= 1 Γ

j

j


Γ

j

∂φ

j

∂n

⎡∂φ1


⎢ ⎥

⎢∂ ∂ n

φ ⎥

⋅φ

* Γ =


i , j d 1 2 3 ⎢ ⎥

⎢ ⎥

⎢∂ ∂ *

n

Γj

φ3


⎢⎣

∂n

⎥⎦

[ ] [

( 3)

N N N ⋅ dΓ = g g g ]

2

1) 2)

φ

,

i , j

(

i , j

i

j

(

i , j

⎡∂φ1


⎢ ⎥

⎢∂ ∂ n

φ ⎥

2

⎢ ⎥

⎢ ⎥

⎢∂ ∂ n

φ3


⎢⎣

∂n

⎥⎦

( 1)

g , =


j N1

⋅φ

*

i i,

j


( 2)

g , =


j N2

⋅φ

*

i i,

j


( 3)

g , =


j N3

⋅φ

*

i i,

j


Γ

j

Γ

j

Γ

j

FB Elektrotechnik und Informationstechnik, AG Computational Electrodynamics

34


Calculation of integrals in triangle element

Zienkiewicz formula:

∫∫

i j k i!

j!

k!

N1

⋅N

2

⋅ N3

dS =

⋅ 2∆

2 !

( i + j + k + )

Gaussian quadrature for triangles:

1 1−

N1

n

I = f 1 2 3 1 2 i 1i

2i,

3i

∫∫ ∑

=

0

0

( N , N , N ) dN dN = w ⋅f

( N , N N )

i

1

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35


Poisson‘s equation

2

∇ φ = − f

in

Ω

n r

f ≠ 0

Γ 1

⎧ φ = φ


⎨∂φ

⎪ = q

⎩∂n

on

on

Γ

Γ

1

2

z

Γ 2

Ω

x

y


Ω

2

w⋅∇

φ dΩ +


Ω

w⋅

f

dΩ =


Γ

2




∂φ


∂n

q


⎟⋅


w dΓ −


Γ

1

∂w

∂n

( φ −φ

) ⋅ dΓ

FB Elektrotechnik und Informationstechnik, AG Computational Electrodynamics

36


Direct BEM for Poisson‘s equation (1)


Ω

φ ⋅∇

2

w dΩ +


Γ




∂φ

∂w⎞

w⋅

−φ

⋅ ⎟ +

∂n

∂n



Ω

w⋅

f

dΩ =

=


Γ

2




∂φ


∂n

q


⎟⋅


w dΓ −


Γ

1

∂w

∂n

( φ −φ

) ⋅ dΓ


φ ⋅∇

2

w dΩ +


w⋅

f

dΩ =

Ω

Ω

= −


Γ

2

w⋅

q

dΓ +


Γ

2

∂w

φ ⋅

∂n

dΓ −


Γ

1

∂φ

w⋅

dΓ +

∂n


Γ

1

∂w

φ ⋅

∂n


FB Elektrotechnik und Informationstechnik, AG Computational Electrodynamics

37


Direct BEM for Poisson‘s equation (2)


Ω

φ ⋅∇

2

w dΩ +


Ω

w⋅

f

dΩ =


Γ

∂w

φ ⋅

∂n

dΓ −


Γ

∂φ

w⋅


∂n

⎧δi




⎩δ

i,

, j

j

=

0

= ∞

φ =

∫φ

j

Ω

⋅δ

i i,

j

for

for

i

i



=

j

j

1

φ*

=

4 ⋅π


r i, j

1 ⎛

φ*

= ln⎜

2 ⋅π


1

r i , j

in




3D

in

2D


2

w = −δ i,

j

FB Elektrotechnik und Informationstechnik, AG Computational Electrodynamics

38


Direct BEM for Poisson‘s equation (3)

∂φ

−∫

⋅ Ω =


j

φ f d ⋅ dΓ −∫

i φ * i, j φ * i,

j

q * i,

∂n

Ω

Γ

Γ

j

⋅φ

j


c

i

∂φ

⋅ −∫

⋅ Ω =


j

φ f d ⋅ dΓ −∫

i φ * i, j φ * i,

j

q * i,

∂n

Ω

Γ

Γ

j

⋅φ

j


c i

=

⎧1


⎪1


⎪2


⎩0

for i inside Ω

for i on Γ

for i outside Ω

FB Elektrotechnik und Informationstechnik, AG Computational Electrodynamics

39


Direct BEM for Poisson‘s equation (4)

1

2

1

2

∂φ

⋅ −∫

⋅ Ω =


j

φ f d ⋅ dΓ −∫

i φ * i, j φ * i,

j

q * i,

∂n

Ω

⋅φ

− B


j=

1

∂n


Γ

j

Γ

n

n

∂φ

j

i i = ⋅ φ * i,

j dΓ − φ j ⋅ q * i,

j


j=

1

Γ


Γ

j

j

⋅φ

j



1

2

⋅φ


i

n

n

∂φ

j

B i = ⋅Gi,

j − φ j ⋅Hˆ

i,

j


j=

1

∂n


j=

1

G

=

∫φ

i , j * i,

j



=


q

i , j * i,

j


Γ

j

Γ

j

FB Elektrotechnik und Informationstechnik, AG Computational Electrodynamics

40


Direct BEM for Poisson‘s equation (5)

B

i

=

∫ φ *

Ω

i, j

1

φ*

=

4 ⋅π


⋅f

r i, j


in

3D

φ

j1

∂φ

,

∂n

j1

2

∇ φ = − f

f ≠ 0

in

Ω

φ i

φ*

=

1 ⎛

ln⎜

2 ⋅π


1

r i , j

Electrostatics:




in

2D

φ

j2

∂φ

,

∂n

j2

φ

j3

∂φ

,

∂n

j3

ρ

∇ 2 φ = −

ε

φ(i)

r

=


V

ρ


4⋅

π⋅ r ⋅ε

dV

FB Elektrotechnik und Informationstechnik, AG Computational Electrodynamics

41


Direct BEM for Poisson‘s equation (6)

H

i,

j


⎧Hˆ

i,

j


= ⎨1

⎪ + Hˆ

⎩2

i,

j

for

for

n

n

∂φ

j

B i = ⋅Gi,

j − φ j ⋅Hi,

j


j=

1

∂n

− B = G ⋅Q

− H ⋅Φ

A ⋅ X =

F

i

i


=

j

j


j=

1

X

⎡ φ1


⎢ φ2


⎢ .


⎢ φn1

=


∂φ


1


⎢ ∂ ∂ n

φ2

⎢ ∂n



.

⎢∂φn

⎢⎣

∂n

2
















⎥⎦

FB Elektrotechnik und Informationstechnik, AG Computational Electrodynamics

42


Calculating integrals of internal elements

B

i

=

∫ φ *

i, j

⋅f


φ*

=

1

4 ⋅π


r i, j

in

3D

Ω

φ*

=

1 ⎛

ln⎜

2 ⋅π


1

r i , j




in

2D

Magnetostatics:


2

r

A

r

= −µ ⋅ J

r r

A(i)

=

µ

4⋅

V

r r

J( j)

r

π∫


i,j

dV

Gaussian

Quadrature:

I

=

1

1

1

∫∫∫

f (x, y,z)dxdydz ≅

∑∑∑

−1

−1

− 1

k= 1 j= 1 i=

1

n

n

n

f (x

i

, y

i

,z

i

) ⋅ w

i

⋅ w

j

⋅ w

k

FB Elektrotechnik und Informationstechnik, AG Computational Electrodynamics

43


Postprocessing

φ

i

=

B

i

+

n


j=

1

∂φ

j

∂n

⋅G

i,j


n


j=

1

φ

j

⋅Ĥ

i,j

φ

j1

∂φ

,

∂n

j1

B

i

=

∫ φ *

i, j

⋅f


f ≠ 0

φ i

Ω

φ

j2

∂φ

,

∂n

j2

φ

j3

∂φ

,

∂n

j3

FB Elektrotechnik und Informationstechnik, AG Computational Electrodynamics

44


First order elements (Poisson‘s equation)

c

i

∂φ

⋅ −∫

⋅ Ω =


j

φ f d ⋅ dΓ −∫

i φ * i, j φ * i,

j

q * i,

∂n

Ω

Γ

Γ

j

⋅φ

j




N


⎪N


1

2

1

( ξ ) = ⋅

2

1

( ξ ) = ⋅

2

( 1−ξ

)

( 1+

ξ )

⎪⎧

x(

ξ ) =


⎪⎩ y(

ξ ) =

N

N

1

1

( ξ ) ⋅

( ξ ) ⋅

x

y

1

1

+

+

N

N

2

2

( ξ ) ⋅ x

( ξ ) ⋅ y

2

2

FB Elektrotechnik und Informationstechnik, AG Computational Electrodynamics

45


Subregions (1)

H ⋅Φ

=

G ⋅Q

⎡Φ

[

1 1

] ⋅ ⎢ ⎥ = [

1 1

H H G G ]

a


⎣Φ

⎡Φ

1

1

a

[

2 2

] ⋅ ⎢ ⎥ = [

2 2

H H G G ]

a


⎣Φ

2

2

a







a

a

⎡Q

⋅ ⎢


⎢Q

1

1

a

⎡Q

⋅ ⎢


⎢Q

2

2

a









Ω

Γ

Γ

2

1 Ω 2

1

Ω

Γ

Γ

2

1 Ω 2

1 Γa

Γa

FB Elektrotechnik und Informationstechnik, AG Computational Electrodynamics

46


Subregions (2)

Φ =

a

= Φa

2

Φ

a

1 a

1

a

2

a

Q = −Q

=

Q

⎡H



⎣ 0

1

H

H

1

a

2

a

0

H

2

⎡Φ

⎤ ⎢

⎥ ⋅ ⎢Φ

⎥ ⎢




Φ

1

a

2







=

⎡G



⎣ 0

1

G

1

a

− G

2

a

0

G

2

⎡G

⎤ ⎢

⎥ ⋅ ⎢G

⎥ ⎢




G

1

a

2







H ⋅Φ

=

G ⋅Q

FB Elektrotechnik und Informationstechnik, AG Computational Electrodynamics

47


Indirect Boundary Element Method (1)

Fredholm equation:

φ*

=

1

4 ⋅π


r i, j

in

3D

φ

i

=


Γ

σ φ

j

* i, j


φ*

=

1 ⎛

ln⎜

2 ⋅π


1

r i , j




in

2D

where σ is density of single-layer potential

σ j2

φ i

σ j1

σ j3

FB Elektrotechnik und Informationstechnik, AG Computational Electrodynamics

48


Indirect Boundary Element Method (2)

φ

i

=


Γ

σ φ

j

* i, j


φ*

=

1

4 ⋅π


r i, j

in

3D

q

i

∂φi

1

= = − ⋅σ +


i σ jq *

∂n

2

Γ

i, j


φ*

=

1 ⎛

ln⎜

2 ⋅π


1

r i , j




in

2D

φ

q

i

i

=

= −

N


j=

1

1

2

σ

j

⋅σ

⋅G

i

+

i,j

N


j=

1

σ

j

⋅Ĥ

i,j

φ

q

i

i

=

=

N


j=

1

N


j=

1

σ

σ

j

j

⋅G

⋅H

i,j

i,j

FB Elektrotechnik und Informationstechnik, AG Computational Electrodynamics

49


Indirect Boundary Element Method (3)

φ

i

=

N


j=

1

σ

j

⋅G

i,j

φ i

σ j2

q

i

=

N


j=

1

σ

j

⋅H

i,j

σ j1

f ≠ 0

σ j3

FB Elektrotechnik und Informationstechnik, AG Computational Electrodynamics

50


Coupling FEM with BEM

r =1

Γ

µ Γ

µ r ≠ 1

r ≠ 1

µ r =1

Γ

Γ

µ FEM model

BEM model

⎧ φ = φ



∂φ

⎪ = q

⎩∂n

on

on

Γ

Γ

1

2

FB Elektrotechnik und Informationstechnik, AG Computational Electrodynamics

51


Comparing BEM to FEM

Finite Element Method

widely applicable

discretisation of whole domain

solution in whole domain

sparse matrix

not easy to solve open

boundary problems

Boundary Element Method

not all problems can be solved

discretisation of boundary

of homogeneous domain

soution at boundary and next

inside domain

full matrix

open boundary problems can be

solved easly

FB Elektrotechnik und Informationstechnik, AG Computational Electrodynamics

52


References

• C.A. Brebbia, J.C.F. Telles L.C. Wrobel „Boundary Element

Techniques – Theory and Applications in Engineering“, Springer-

Verlang

• M. Ameen „Boundary Element Analysis – Theory and Programming“,

Alpha Science International

• G. Beer „Programming the Boundary Element Method – An

Itroduction for Engineers“, John Wiley & Sons

• www.integratesoft.com

• http://www.boundary-element-method.com

• http://tabula.rutgers.edu/EJBE/

• http://www.olemiss.edu/sciencenet/benet/

FB Elektrotechnik und Informationstechnik, AG Computational Electrodynamics

53

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