Design derivatives of eigenvalues and eigenfunctions for self-adjoint distributed parameter systems

Abstract

In this paper, analytic expressions are obtained for the design derivatives of eigenvalues and eigenfunctions of self-adjoint linear distributed parameter systems. Explicit treatment of boundary conditions is avoided by casting the eigenvalue equation into integral form. Results are expressed in terms of the linear operators defining the eigenvalue problem, and are therefore quite general. Sufficiency conditions appropriate to structural op- timization of eigenvalues are obtained. ERIVATIVES of eigenvalues and eigenvectors with respect to the design parameters are useful, if not essen- tial, for design sensitivity and structural optimization studies. In one of the earlier works to address this question, Fox and Kapoor1 restrict their discussion to self-adjoint discrete systems. Plaut and Huseyin2 extend this approach to non-self-adjoint discrete systems by treating the associated adjoint eigenvalue problem. For distributed parameter systems, design derivatives of eigenvalues were first encountered in optimization studies. Prager and Taylor3 generalized several earlier studies4'5 using Rayleigh's principle. This approach, however, is not suitable for determining the design derivative of eigenfunctions, or for calculating higher order derivatives of eigenvalues. Finally, Rayleigh's principle is not appropriate for the determina- tion of conditions, necessary or sufficient, to ensure the ex- istence of the design derivatives of eigenvalues and eigenfunctions. The foregoing considerations were recently addressed by Haug and Rousselet6 using a functional analysis approach. In this paper, a relatively simple method is used to determine explicit results for the design derivatives of eigenvalues and eigenfunctions. Self-adjoint operator equations are cast into their integral form using Green's function. Since the design derivative of Green's functional is explicitly known,7 it is a straightforward matter to differentiate the eigenvalue equa- tion with respect to the design parameters to obtain an equa- tion for the variations of the eigenvalues and eigenfunctions. The solution to this equation is explicitly obtained by recognizing that the eigenfunctions form a complete set.