The Boundary Element Method - Mechanics of Structures: Variational and Computational Methods - page 562

9.3.3 Integral Equation Formulation

Substitute Eqs. (9.52), (9.68), and (9.77) into Eq. (9.50) to obtain

p Vi u ji p i u ji p i u ji − u i a i + a j dV =− a j dS − a j dS

V S p S u

u i p ji a j dS u i p ji a j dS + +

S p S u

Exchange the positions of i, j in the above integrals to find

p Vj u ij a i dV p j u ij a i dS p j u ij a i dS

− u i a i + =− −

V S p S u

u j p ij a i dS u j p ij a i dS + + (9.79)

S p S u

The a i 's can be factored out of Eq. (9.79) with the result

p Vj u ij dV u ij p j dS u ij p j dS − u i (ξ ) + =− −

V S p S u

p ij u j dS p ij u j dS

+ + (9.80a)

S p S u

or

p ij u j dS u ij p j dS p Vj u ij dV

u i (ξ ) + = + (9.80b)

S S V

where S S p ,u j and p j are the displacements and the tractions on the whole boundary.

Equation ( 9.8 0) is known as Somigliana's 3 identity . It should be observed that the integral
containing p Vj does not involve any unknowns.
Since the quantities in the integrals all refer to a point x inside the body or on the boundary,
i.e.,

= S u +

p Vj u ij dV u ij (ξ = p Vj ( x ) ,x ) dV ( x )

u ij p j dS u ij (ξ = ,x ) p j ( x ) dS ( x )

p ij u j dS p ij (ξ

= ) ( ) ( )

,x u j x dS x

Eq. (9.80) is a relationship between the displacement at a point ξ inside the boundary and

the displacements u j ( on the boundary. Equation (9.80) is the theory

of elasticity equivalence of Eq. (9.9) for beams. These are the integral equations on which
the boundary element method is based. In Eq. (9.80) the unknowns appear both inside and
outside the boundary integrals, a property that characterizes a Fredholm integral equa-
tion of the second kind. The aim of the boundary element method is first to calculate the
unknown displacements and tractions on the boundary, then compute the displacements
inside the boundary, and, finally, calculate the stresses.

x ) and tractions p j ( x )

3 Carlos Somigliana (1860-1955) was an Italian physicist who specialized in geophysics. In 1892, he began a career
as a professor of mathematical physics in Pavia and Turin.