On the Rank-deficiency Problems in Boundary Integral Formulation Using the Fredholm Alternative Theorem and Singular Value Decomposition Technique

Abstract

We provide a perspective on the degenerate problems, including degenerate boundary, degenerate scale, spurious eigensolution and fictitious frequency, in the boundary integral formulation. All the degenerate 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 degenerate problems. 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. A unified method for dealing with the degenerate problem in BEM is proposed. Several examples are demonstrated to check the validity of the unified formulation.