Theory of Boundary Eigensolutions in Engineering Mechanics

Abstract

A theory of boundary eigensolutions is presented for boundary value problems in engineering mechanics. While the theory is quite general, the presentation here is restricted to potential problems. Contrary to the traditional approach, the eigenproblem is formed by inserting the eigenparameter, along with a positive weight function, into the boundary condition. The resulting spectra are real and the eigenfunctions are mutually orthogonal on the boundary, thus providing a basis for solutions. The weight function permits effective treatment of nonsmooth problems associated with cracks, notches and mixed boundary conditions. Several ideas related to the convergence characteristics are also introduced. Furthermore, the connection is made to integral equation methods and variational methods. This paves the way toward the development of new computational formulations for finite element and boundary element methods. Two numerical examples are included to illustrate the applicability.