Advanced topics - The Boundary Element Method with Programming - page 465

18.2 HETEROGENEOUS DOMAINS

18.2.1 Theory

The approach is first explained on a problem with 2 different properties but it will
become obvious that the method will also work for a general heterogeneous domain. The
example in Figure 18.1 shows a problem of a tunnel being excavated in a domain with 2
different materials (this is a pure Neumann problem). We could solve this with a multi-
region approach but here we choose a different method.

Figure 18.1 Example with heterogeneous domain

The idea is to start with an analysis that assumes that the whole domain has the same
properties (

E Q ) which are represented by the constitutive matrix

D . Hence we solve 11

the following system of equations (see also 7.2)

> @ ^` ^`

Tu F

(18.1)

Next we compute displacements at the boundary nodes and via post-processing the
strains, İ , at the cell nodes

P

³ ³

İ P S P , Q t Q dS R P , Q u Q dS (18.2)

a a a

S S

We find that when we compute stresses these should be computed according to: