Dealing with corners and changing geometry - The Boundary Element Method with Programming - page 328

d
d

[
[

Right sub region: [[ 1

The weakly singular part of the integral of Kernel

e
xx

' is for example

1

2

§·

1 d [

¦ ³

e
xx

'

UCC N [ ln PJ , [ [ d [ (12.5)

¨ ©¹

1 n i

r d [

r 1

1

For straight elements the radius r can be expressed in the following way

Ld

[

ra [ w ea (12.6)

2 d [

The factor a is a constant and depends on d [[ and the element length L . Because
of this, Equation (12.5) can be split into two parts. The first part can be integrated by the
Quadrature of Gauss-Laguerre and the second one can be evaluated regularly

§

1 1

·

§· [ [

2

1 d §· 1 d

¦ ³

e
xx

³

' UCC N [ ln J [ d N [ [ ln J [ d [ (12.7)

¨ ¸

¨¸

¨ ©¹

1 n n

[ d [ a d [

©¹

© ¹

r 1

0 0

The integration limits of the second integral in equation (12.7) are from 0 to 1 and
when using normal Gauss Quadrature, the integral has to be transformed to the limits
1 to

1 as follows

1 1

1 d [ 1 d [ [ d

§· §·

³ ³

N [ ln J [ d [ N [ ln P , [ J [ d [ (12.8)

¨¸ ¨¸

n n i

a d [ a d [ [ d

©¹ ©¹

0 1

The transformation from [ to [ is independent of the location of the collocation
point and the sub region. The transformation has the following form:

[ 1 d

d

[ 1

[ and

2 2

[

12.2.3 Numerical implementation

Two dimensional discontinuous elements (linear and quadratic) are implemented in the
single region program prog71_discont and in the multi region program prog111_discont.