Chapter 3
The Discrete Helmholtz Operators

This Chapter considers the general problem of constructing methods for discretising the integral operators, as required in the application of collocation to integral equation formulations of the Helmholtz equation. The Chapter is important only if the precise algorithms used to compute the discrete operators need to be known or the reader wishes to develop programs for solving Helmholtz problems that are not covered in the text. Readers wishing to go directly to the solution of interior, exterior or modal analysis problems may omit this Chapter.

Linear Helmholtz or Helmholtz problems obviously give rise to a range of integral equation formulations, for example depending on whether the Helmholtz field lies in an interior or exterior domain. However, the integral equations that arise in all problems in the same spatial dimensions contain similar integral operators. For example a computational method for evaluating the discrete from of the integral operators in an three-dimensional exterior Helmholtz problem can also be used in any other three-dimensional Helmholtz problem.

For each dimensional space it is possible to develop a module for computing the discrete form of the integral operators that is common to the interior, exterior and modal analysis subroutines. The purpose of this Section is to show how the discrete forms of the integral operators in the three spatial dimensions are computed and to introduce the subroutines H2LC, H3LC and H3ALC that have been developed in order to carry this out [33]. The naming of the subroutines is such that the H represents Helmholtz, the 2, 3, 3A identifies the dimensionality, the LC represents the linear boundary approximation with a constant function approximation on each panel.