Solução de um problema de autovalor especial gerado pela formulação autorregularizada do método dos elementos de contorno com interpolação direta

Abstract

The present work aims to solve an eigenvalue problem obtained after the development of the Helmholtz Equation in its integral form and its later discretization following the methodology associated with the Direct Interpolation Boundary Elements Method. In order to eliminate the singularities eliminating mathematical procedure based on the Hadammard’s Regularization, a new integral equation was generated based on the use of a more elaborate fundamental solution. However, the discrete form of this formulation, named self-regulated form, generated matrices in addition to those usually obtained by the most common numerical methods. Unlike the response problems that are solved by a scanning, in which the mentioned formulation presented good results and easy operation, the calculation of natural frequencies now becomes quite complex and unorthodox, as the associated eigenvalue problem becomes a fourth order problem. Thus, solving this type of problem requires a different approach, where laborious mathematical manipulation and some approximations will be necessary. In this context, the generalization of the Przemieniecki Proposition stands out, because it is well known in the treatment of damped vibration problems, aiming to write the matrix equation in an accessible form for its computational resolution.