Evaluation of free terms in hypersingular boundary integral equations
The accurate numerical solution of hypersingular boundary integral equations necessitates the precise evaluation of free terms, which are required to counter discontinuous and often unbounded behaviour of hypersingular integrals at a boundary. The common approach for the evaluation of free terms involves integration over a portion of a spherical shaped surface centred at a singularity and allowing the radius of the sphere to tend to zero. In this paper two alternative methods, which are shape invariant, are proposed and investigated for the determination of free terms. One approach, the point-limiting method, involves moving a singularity towards a shrinking integration domain at a faster rate than the domain shrinks. Issues surrounding the choice of approach and shrinkage rates, and path dependency are examined. A related approach, the boundary-limiting method, involves moving an invariant but shrinking boundary toward the singularity again at a faster rate than the shrinkage of the domain. The latter method can be viewed as a vanishing exclusion zone approach but the actual boundary shape is used for the boundary of the exclusion zone. Both these methods are shown to provide consistent answers and can be shown to be directly related to the result obtained by moving a singularity towards a boundary, i.e. by comparison with the direct method. Unlike the spherical approach the two methods involve integration over the actual boundary shape and consequently shape dependency is not a concern. A particular highlight of the point limiting approach, as a result of field approximations being restricted to the boundary, is the ability to obtain free terms in a mixed formulation without reference to the underpinning constitutive equations, which is not available to the spherical method. Focus in the paper is on the 2-D potential equation as this is shown to be sufficient to demonstrate the concepts involved.
free terms hypersingular integrals limit methods