Boundary Element Methods–Numerical Implementation - The Boundary Element Method with Programming - page 129

where, for two-dimensional problems

³ ³

(6.11)

e
ni

e
ni

' U N [ U P , [ dS [ , ' T N [ T P , [ dS [

n i n i

S S

e e

For three-dimensional problems

e
ni

³

' U N [ , U P , [ , dS [ ,

n i

(6.12)

S

e

e
ni

³

' T N [ , T P , [ , dS [ ,

n i

S

e

where S e is the element area and [K are the intrinsic coordinates.
Since there are two or three integral equations per location P i , we now get 2 I or 3 I
equations depending on the Cartesian dimension. As we will see later in the section on
assembly, Equation (6.10) can be written in matrix form, where coefficients are
assembled in a similar way as in the FEM. For this it is convenient to store the
coefficients for element into arrays ['U] e and ['T] e . For potential problems we have for
example

o
'

elem nodes
UU

' !

!

ª º

11 21

« »

> @

e

' ' ' U U U p coll pnts .

(6.13)

« »

12 22

« »

# # !

¬ ¼

The arrays are of size N x I , where N is the number of element nodes and I is the
number of collocation points. For elasticity problems, the arrays are of size 2N x 2I , for
two-dimensional problems and 3N x 3I , for three-dimensional problems. In the following
section we will deal with the numerical integration of Kernel shape function products
over elements.

6.3

INTEGRATION OF KERNEL SHAPE
FUNCTION PRODUCTS

The evaluation of integrals (6.8) or (6.12) over isoparametric elements is probably the
most crucial aspect of the numerical implementation of BEM and this is much more
involved than in the FEM. The problem lies in the fact that the functions which have to
be integrated exhibit singularities at certain points in the elements. Here we first discuss
the treatment of “improper” integrals that exist as Cauchy principal values and then
discuss the numerical treatment of the other integrals.