Design Sensitivity Analysis in Structural Mechanics. I. Static Response Variations

The dependence of the solution of boundary-value problems of structural mechanics on design variables that specify material properties and distribution is characterized. Prototype problems treated include beams, plates, and plane elastic solids. Symmetry and positive definiteness properties of the elliptic differential operators that govern system response are used to show that their inverses, hence the displacement fields, are Frechet differentiable with respect to design variables. Formulas for the derivatives are given and used to obtain computable formulas for design sensitivity coefficients (first variation) of integrals that arise in optimal design formulations. The results establish an extension of the concept of “well-posed” problems of structural mechanics to include continuity (in fact, differentiability” of static structural response with respect to distributed design variables and design parameters