Assembly and Solution - The Boundary Element Method with Programming - page 166

the solution are usually much lower than the FEM. We will also see in chapter 11, that if
we introduce the concept of multiple boundary element regions sparsity is introduced to
the system of equations.
At the end of this chapter we will have all the procedures necessary for a general
purpose program, which can solve steady state problems in potential flow and elasticity.
The program, however, will only give us values of the unknown at the boundary. As
already pointed out, a special feature of the BEM is that results at any point inside the
domain can be computed with greater accuracy as a postprocessing exercise. This topic
will be dealt with in Chapter 9.

7.2 ASSEMBLY OF SYSTEM OF EQUATIONS

We start with potential problems. In the previous section we discussed the computation
of element contributions to equation (6.7), that is

E N E N

¦¦ ¦¦

e
ni

e e

ni

e

(7.1)

cu P ' T u ' U t

i

n n

e 1 n 1 e 1 n 1

We recall the notation used

Element number

p

(7.2)

e

' T Collocatio n po int

n i m

n

Node Number

For the solution of the system of equations it is convenient to replace the double sums
by a matrix multiplication of the type

> ^` > @ ^ `

(7.3)

' T u ' U t

where vectors { u }, { t } contain potential/temperature and fluxes respectively for all
nodes in global numbering system

` T

^` ^

uuu , , "

(7.4)

12

and [' T ], [' U ] are global coefficient matrices assembled by gathering element
contributions. In the global coefficient arrays, rows correspond to collocation points P i
and columns to the global node number. The gathering process is very similar to the
assembly process in the FEM, except that whole columns are added. For the gathering
process we need the Connectivity or Incidences of element e , which refer to the global
node numbers of the element.
Referring to the simple 2-D mesh with linear elements in Figure 7.1 the incidences of
are given in Table 7.1.