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
. 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
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 potential flow, or any other problem for which we can supply a fundamental solution (see Chapter 18).