Boundary Element Methods–Numerical Implementation - The Boundary Element Method with Programming - page 125

6
Boundary Element Methods -
Numerical Implementation

There is nothing more powerful
than an idea whose time has come

V. Hugo

6.1 INTRODUCTION

In the previous chapter we derived boundary integral equations relating the known
boundary conditions to the unknowns. For practical problems, these integral equations
can only be solved numerically. The simplest numerical implementation is using line
elements, where the knowns and unknowns are assumed to be constant inside the
element. In this case, the integral equation can be written as the sum of integrals over
elements. The integrals over the elements can then be evaluated analytically. In the
previous chapter we have presented constant elements for the solution of two-
dimensional potential problems only. The analytical evaluation over elements would
become quite cumbersome for two- and three-dimensional elasticity problems. Constant
elements were used in the early days of the development, where the method was known
under the name Boundary Integral Equation (BIE) Method 1 . This is similar to the
development of the FEM, where triangular and tetrahedral elements, with exact
integration, were used in the early days. In 1968, Ergatoudis and Irons 2 suggested that
isoparametric finite elements and numerical integration could be used to obtain better
results, with fewer elements. The concept of higher order elements and numerical
integration is very appealing to engineers because it alleviates the need for tedious
analytical integration and, more importantly, it allows the writing of general purpose
software with a choice of element types. Indeed, this concept will allow us to develop
one single program to solve two- and three-dimensional problems in elasticity and