Dealing with corners and changing geometry - The Boundary Element Method with Programming - page 325

We now investigate the other possibilities for the location of

P .

P is located at one of the discontinuous nodes inside the element and the shape
function value at this node is zero

Since

i NOr

() the singularity of the kernel shape function product cancels out if

Q [ approaches

P . The integral of the kernel shape function product remains regular,

but the fact, that

P is located inside the element and the integrand is discontinuous

at

P , requires a splitting of the integration region into two sub regions.

U

N

2

1

1 3 2

P i

d d

1 2

[

1 1

[

1 1

[

1 1

Figure 12.2

P is located at local node 1 and the shape function value is zero

Integration when

The integrals of equation (12.1) have to be evaluated separately and added for the 2 sub
regions

1 1

2

d [

2

d [

¦ ¦

e

³

e

³

' U N [ U P , [[ [ J d , ' T N [ [[ [ T P , J d (12.3)

ni n i ni n i

d [ d [

r 1 r 1

1 1

The local coordinate system is changed from [ to [ for the left and the right sub
region in the following way: