Design Sensitivity Analysis in Structural Mechanics.II. Eigenvalue Variations

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Abstract
The dependence of eigenvalues of boundary-value operators of structural mechanics on design variables that specify material properties and distribution is characterized. Prototype problems considered include vibration of strings, membranes, beams, plates, and plane elastic slabs and buckling of beams. Symmetry and positive definiteness properties of the elliptic differential operators that govern system response are used to show that the eigenvalues depend continuously on design. Further, it is shown that simple eigenvalues are Frechet differentiable with respect to design, but that repeated eigenvalues can only be expected to be Gateaux (directionally) differentiable with respect to design. The latter fact is shown to have substantial consequences in classes of optimal design problems in which the fundamental eigenvalue is known to be repeated at an optimum design. Explicit and computable formulas for derivatives (first variation) of both simple and repeated eigenvalues of each of the prototype problems are presented.

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