The Boundary Element Method - Mechanics of Structures: Variational and Computational Methods - page 558

and g is the Galerkin vector. It can be verified that a vector S has the property

2 S

∇× ( ∇× S ) =∇ ( ∇· S ) −∇

or

2 S

∇ =∇ ( ∇· ) −∇× ( ∇× )

S S

Substitute Eqs. (9.55) and (9.56) into this identity

4

div g

π
λ

4 π u =∇ −∇× [4 π( ∇× g ) ]

or

1
λ ∇

u = div g −∇× ( ∇× g ) (9.57)

2 g , Eq. (9.57) can be written as

Since ∇× ( ∇× g ) =∇ div g −∇

div g

1

2

u = ∇ −

− ν) ∇

(9.58a)

2 ( 1

or

1

u i = g i,kk − g k,ik (9.58b)

2 ( 1 − ν)

This is the desired property of the Galerkin vector.
Substitution of Eq. (9.58) into Eq. (9.53) results in a biharmonic equation

p V

4 g 4 g i +

∇ +

G =

0 or ∇ p Vi / G = 0 (9.59)

This is the equilibrium equation expressed in terms of the Galerkin vector. Once the expres-
sions for the components of the Galerkin vector are ob tained, the fundamental solutions
can be found. Equation (9.59) in component form with p V = a 3 ] T [ δ(ξ ,x ) a 1 δ(ξ ,x ) a 2 δ(ξ ,x )

becomes

1
G δ(ξ

4 g i +

∇ ,x ) a i = 0 (9.60)

Here g i has replaced g i of Eq. (9.59) because p Vi has been replaced by p ∗ Vi . The solution of
Eq. (9.60) leads to the desired fundamental solutions.
Write Eq. (9.60) as

1
G δ(ξ

2 F i +

∇ ,x ) a i = 0 (9.61)

with

2 g i

F i =∇

The solution to Eq. (9.61) is [Haberman, 1987]

1

2 g i =

F i =∇

rG a i

(9.62)

4 π