The Boundary Element Method - Mechanics of Structures: Variational and Computational Methods - page 557

of the sifting property of the Dirac delta function of Eq. (9.6) leads to the isolation of u i ,
and Eq. (9.50) will be an expression of the displacement u i in the domain in terms of the
boundary unknowns p i and u i .
It can be seen that σ ij in Eq. (9.52) are the stresses at point x due to a unit point load

in the direction of i . Hence, the displacements and tractions with superscript ∗ ,
which are called the fundamental solutions of the problem, in Eq. (9.50), are the quantities
at point x due to the same force. The fundamental solution u i can be obtained from the
displac e ment for m of the equilibrium equations, which for a solid subjected to a body force
p V

at point ξ

= [ p V 1 p V 2 p V 3 ] T = [ δ(ξ ,x ) a 1 δ(ξ ,x ) a 2 δ(ξ ,x ) a 3 ] T can be expressed as [Chapter 1,

Eq. (1.85)]

1 p Vi

G

2 u i +

∇ u k,ki + = 0

1 − 2 ν

or in vector form,

div u

1 p V

2

∇ +

ν ∇

+

G =

0 (9.53)

1 − 2

in which ∇ and div are the operators

∂
∂

∂ ∂ ∂

∂

∂
∂

∂
∂

∇= div =

x 1 + x 2 + x 1 ∂ x 2 ∂ x 3 x 3

and

[ u 1 u 2 u 3 ] T
A simple way to find the fundamental solution is to express Eq. (9.53) in terms of a
vector g , called the Galerkin vector . This procedure is treated in theory of elasticity texts
such as Boresi and Chong (1987). In order to find an expression for g ,define another vector
S

u =

S 3 ] T

= [ S 1 S 2 at each point P in a volume V with piecewise smooth surfaces such that

u
r

S = dV (9.54a)

V

where r is the distance from point P to the differential volume dV . In component form,

u i
r

S i = dV (9.54b)

V

It can be verified by differentiating Eq. (9.54) that [Brebbia, et al., 1984]
∇

2 S

=− π

4 u (9.55)

Since div S S (a vector) are independent quantities, i.e., a scalar

and a vector are not related, they can be assigned to any scalar and vector quantities. Let

( =∇ · S , a scalar) and ∇×

4 π

λ

∇· S =− div g (9.56a)

∇× S =− 4 π( ∇× g ) (9.56b)

where λ = 2 ( 1 − ν)/( 1 − 2 ν) , · and × are the dot and cross-product operators of vectors with

∂
∂

∂
∂

∂
∂

S 3 ] T

∇· S = · [ S 1 S 2

x 1 x 2 x 3

= ∂

S 1

x 1 + ∂

S 2

x 2 + ∂

S 3

∂ ∂ ∂ x 3

∂

S 3

x 2 − ∂

S 2 ∂ S 1

x 3 − ∂

S 3 ∂ S 2

x 1 − ∂

S 1

∇× S =

∂ ∂ ∂ ∂ ∂ ∂

x 3 x 1 x 2