The development of the TrefRzian methodology for engineering design analysis

This paper reviews the developments in the Trefftzian Methodology, which have been undertaken by the authors at the University of Sheffield during the past fifteen years and the application of these developments to engineering design analysis. Initially, in the late 197Os, this work concentrated on the Direct Boundary Element Method (DBEM) and the Indirect Boundary Element Method (IBEM). Unfortunately these methods, as they are normally formulated, give rise to singular integrals, which require special mathematical treatment, when the source and field points coincide on the boundary of the component being analysed. These singular integrals can however be eliminated by placing the source boundary outside the domain of the problem being analysed so that the field and source points never coincide. This technique is known as either the Regular Direct Boundary Element Method (RDBEM) or the Regular Indirect Boundary Element Method (RIBEM). In a further development of the RIBEM, based on the Trefftz Method, the continuous distribution of sources is replaced with sources distributed at discrete points on the source boundary. This modified Trefftz Method eliminates the integrations in the solution procedure, it provides a series solution in terms of the fundamental solution of the problem being analysed and is referred to as the Indirect Discrete Boundary Method (IDBM). The emergence of the IDBM provided the opportunity to develop a combined Boundary Element Finite Element technique which enables these methods to be used simultaneously in a single calculation, thereby exploiting their strengths and minimising their weaknesses. A number of case studies will be discussed in the paper to illustrate the developments in the Trefftzian Methodology and its application to engineering design analysis.