The Boundary Element Method - Mechanics of Structures: Variational and Computational Methods - page 540

FIGURE 9.2
Hemispherical surface with radius .

Thus moving to the boundary leads to no contribution from the second integral on the

right-hand side of Eq. (9.23).
In the case of the third integral of Eq. (9.23),

ξ

uq ∗ dS uq ∗ dS uq ∗ dS = lim

→

+ lim

→

(9.26)

0 0

S S − S S

and

uq ∗ dS uq ∗ dS

lim
→

=

0

S − S S

r r . Then, with r

i ,q ∗ can be written

For a hemispherical surface of Fig. 9.2, a i = = ,r i =

as

1

u ∗
∂

1 2 3

q ∗ = ∂

n = ∂

=−

i a i

3 =− 3

=−

+ +

i i

∂ n

4

From the expressions for

1 , 2 , and 3 of Fig. 9.2, it can be seen that

2
1

2
2

2
3

2

+ + =

Thus, for a hemispherical surface,

u ∗
∂

q ∗ = ∂

1

n =−

2

Then the second integral on the right-hand side of Eq. (9.26) becomes

dS dS u

1 1

uq ∗ dS

= − = −

lim
→

lim
→

u lim

→

2 2

0 0 0

S S S