Discretisation and Interpolation - The Boundary Element Method with Programming - page 63

free-field, that is, numbers can be separated by blanks. After reading the general
information the incidences (connectivity) are read for all elements and stored in array
InciG. While reading this information, we find the largest node number, information
which we need for allocating the dimension for the array containing node coordinates
and which we do next before reading the node coordinates. We make use of the new
feature in F90, that allows the subscripts of an array to start with zero, because a
transition element that has the midside node missing will have a node number of 0 in
the incidences. We assign zero coordinates to node number 0.
We loop over all elements describing the boundary. For each element we get the
Gauss point coordinates and weightings, by a call to Gaus_coor, which correspond to the
integration order Specified by Intord. We then add all the Gauss point contributions, i.e.
the Jacobians computed (by a call to Normal_jac ) for each Gauss point multiplied by
the weighting. Note that there are two cases to be considered: for a one-dimensional
case, that is, if we work out a length of a curve, only one DO LOOP is required. For
two-dimensional cases, that is, when we work out surface areas, two nested DO LOOPS
are required (see equation 3.66).
3.12 CONCLUDING REMARKS
In this chapter we have dealt with methods for describing the geometry of a problem and
have concentrated on describing problem boundaries. The method consists of
subdividing the boundary into small elements and is commonly known as discretisation .
The concept of isoparametric elements was introduced, where we use interpolation
functions to describe the boundary surface in terms of nodal values and the variation of
known or unknown values. We have laid here the foundation for Chapter 6 (Boundary
Element Methods), where we will use the concepts described. We find that, once we use
this advanced discretisation method in the BEM, the analytical integration is no longer
possible. Therefore, we have also introduced the Gauss Quadrature method of numerical
integration, most commonly used in numerical work. For general purpose programs
using the isoparametric concept, the accuracy of the numerical integration will be
crucial. We have started here our process of building a Subroutine library which will be
needed later. A small program has been written which we can use to test the subroutines
and to do numerical experiments.
3.13 EXERCISES
Exercise 3.1
Using program Compute_area calculate the length of a quarter circle using:
(a) one linear element
(b) two linear elements
(c) one quadratic element