Boundary Element Methods–Numerical Implementation - The Boundary Element Method with Programming - page 141

To evaluate the first integral in equation (6.8) we must substitute for r as a function of [.
For a linear element we may simply write r= J [ and obtain

1

§ ·

1 1 d [

e
ni

³

' UN [ ln J d [

¨ ¸ n

2 S k J [ d [

© ¹

0

(6.38)

1 1

1 §· 1 d [ 1 §· 1 d [

³ ³

N [ ln J d [ N [ ln J d [

¨¸ ¨ ©¹ n n

2 S k [ d [ 2 S k J d [

©¹

0 0

N 2

U

U

N 1

[

P i

P i

[

1

2

2

1

3 [

[

3

[ 1

[ 1

[ 1 [ 0

[ 1 [ 0

[ 0

[ 1

1

[ 1 [

1

[ 0 [

2

2

U

N 3

[ [

P i 3

1 [ 2

[ 0

[ 1 [ 1

[ 0

[ 1

[ 1

Figure 6.10 Integration when P i and n coincide

The first integral may be evaluated with Gauss-Laguerre:

1

1 § 1 · d [ e

ni

³

' U N [ ln J [ d [

© ¹

n

2 S k [ d [

(6.39)

0

M

d [

1

¦

| N [ J [ W

n m m m

2 S k d [

m 1