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.

Carslaw, H. S., 1950, An Introduction to the Theory of Fourier’s Series and Integrals, Dover, New York.
Courant, R., and Hilbert, D., 1953, Methods of Mathematical Physics, John Wiley and Sons, New York.
Morse, P. M., and Feshbach, H., 1953, Methods of Theoretical Physics, McGraw-Hill, New York.
Tolstov, G. P., 1962, Fourier Series, Dover, New York.
Lanczos, C., 1966, Discourse on Fourier Series, Oliver & Boyd, Edinburgh.
Hadjesfandiari, A. R., 1998, “Theoretical and Computational Concepts in Engineering Mechanics,” Ph.D. dissertation, University at Buffalo, State University of New York, Buffalo, NY.
Hilbert, D., 1912, Grundzu¨ge einer allgemeinen Theorie der linearen Integralgleichungen, B. G. Teubner, Leipzig.
Shubin, M. A., 1987, Pseudodifferential Operators and Spectral Theory, Springer-Verlag, Berlin.
Grubb, G., 1996, Functional Calculus of Pseudodifferential Boundary Problems, Birkhauser, Boston.
Banerjee, P. K., 1994, The Boundary Element Methods in Engineering, McGraw-Hill, London.
Hadjesfandiari, A. R., and Dargush, G. F., 2000, “Computational Mechanics Based on the Theory of Boundary Eigensolutions,” Int. J. Numer. Methods Eng., in press.
Bathe, K. J., 1996, Finite Element Procedures, Prentice-Hall, Englewood Cliffs, NJ.
Sneddon, I. N., 1966, Mixed Boundary Value Problems in Potential Theory, North Holland, Amsterdam.
Muskhelishvili, N. I., 1953, Some Basic Problems of the Mathematical Theory of Elasticity, P. Noordhoff, Groningen, Holland.
Noble, B., 1958, Methods Based on the Wiener-Hopf Technique for the Solution of Partial Differential Equations, Pergamon, New York.
You do not currently have access to this content.