Dual reciprocity boundary element formulation for potential problems with arbitrarily distributed sources

Abstract

Rapid advances have been made on the application of Boundary Elements since the first book on this topic appeared in 19781 . The method is now used not only for linear problems but also for time dependent and non-linear applications 2. One of the main drawbacks of the method however is that in many cases these solutions require discretization of the internal domain into cells followed by integration over the volume. Because of this, early applications of the technique concentrated in developing alternative approaches, based on the use of time and space dependent fundamental solutions and referring the time integration always to the original time, to follow the evolutionary process using boundary only integrations. This approach was applied by Wrobel and Brebbia for parabolic problems 3 and by Mansur and Brebbia 4 for hyperbolic cases. The main difficulty in this case is that every new time step represents a complete new integration for the initial time making the approach very expensive for all but some cases where the domain tends to infinity. In addition it is difficult to extend the approach to deal with cases such as non-linearities and to modify codes written in this way without introduction of internal cells. In 1982 Brebbia and Nardini proposed an alternative approach which was presented at the International Conference on Soil Dynamics and Earthquake Engineering held at Southampton University and published in the journal of the same name in 19835 . The originality of the new method consists of using approximations to the time dependent functions which enable the corresponding volume integrals to be transferred into boundary integrals only. Nardini and Brebbia explained the new procedure in detail in a chapter written in 19856 for a State of the Art series and further applications were presented, also for elastodynamics during the Variational Methods Conference held at Southampton in the same year 7. The hyperbolic elastodynamics case was obviously the most difficult application and once this had been mastered, the solution of parabolic cases was comparatively easy 8. Transient thermal analysis has now been implemented into an operational code 9 which is used by industry and the results tend to validate the accuracy of the technique which is now called the Dual Reciprocity method. The present paper discusses a further extension of the approach to cases for which domain sources are considered to be arbitrarily distributed within the domain. Numerical results are now being obtained and will be the subject of a subsequent publication.