Dual BEM since 1986

Abstract

In this talk, 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 the SVD (singular value decomposition) 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 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 techniques, 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. The three-years project on this topic supported by NSC Wu Ta-You Award is also shown in Table 4. Only key references of NTOU/MSV group are contained here.