Multiple regions - The Boundary Element Method with Programming - page 286

11
Multiple regions

Imagination is more important than knowledge..
A. Einstein

11.1 INTRODUCTION

The solution procedures described so far are only applicable to homogeneous domains,
as the fundamental solutions used assume that material properties do not change inside
the domain being analysed. There are many instances, however, where this assumption
does not hold. For example, in a soil or rock mass, the modulus of elasticity may change
with depth or there might be various layers/inclusions with different properties. For
some special types of heterogeneity it is possible to derive fundamental solutions, for
example, if the material properties change in a simplified way (linear increase with
depth). However, such fundamental solutions are often complicated and the
programming effort significant 1 .
In cases where we have layers or zones of different materials, however, we can
develop special solution methods based on the fundamental solutions for homogeneous
materials in Chapter 4. The basic idea is to consider a number of regions which are
connected to each other, much like pieces of a puzzle. Each region is treated in the same
way as discussed previously but can now be assigned different material properties. With
this method we are able to solve piecewise homogeneous material problems. As we will
see later, the method also allows simulating contact and cracking propagation problems.
Since at the interfaces between the regions both t and u are not known, the number of
unknowns is increased and additional equations are required to solve the problem. These
equations can be obtained from the conditions of equilibrium and compatibility at the