Development Of Dual BEM In TAIWAN

Abstract

In this lecture, the development of dual BEM is reviewed since its appearance in 1986 by Hong and Chen. Roles of hypersigularity are also examined in the computational mechanics. A novel method using SVD for problems with degenerate boundaries was proposed without employing hypersingular formulation and subdomain approach in 2003. Some traps of BEM in engineering applications, degenerate scale, spurious eigenvalues and fictitious frequencies, are investigated. We provide a perspective on the nonuniqueness and its treatment, including degenerate boundary, degenerate scale, spurious eigensolution, fictitious frequency and corner problems as shown in Table 1, in the boundary integral formulation and boundary element method (BEM). All the nonuniqueness problems originate from the rank deficiency in the influence matrix. Both the Fredholm alternative theorem and singular value decomposition (SVD) technique are employed to study the nonuniqueness problems. Based on the circulant properties and degenerate kernels as analytical tools for circular and annular cases, mathematical analysis can be done. Updating terms and updating documents of the SVD technique are utilized. The roles of right and left unitary vectors of SVD in BEM and their relations to true, spurious and fictitious modes are examined by using the Fredholm alternative theorem. Two methods, CHIEF and CHEEF concept, for dealing with the nonuniqueness problems in BEM are proposed. Null fields of nonuniqueness problems are shown in Table 2. The related works since 1984 are shown in Table 3. More than one hundred references of papers by NTOU/MSV group are contained.