First- and second-order shape sensitivity analysis of structures

Abstract

The first- and second-order sensitivities with respect to varying structural shape are discussed for an arbitrary stress, strain and displacement functional. It is assumed that only the traction-free boundary of a structure can undergo the shape modification described by a set of shape design parameters. The first derivatives of a functional with respect to these parameters are derived using both the direct and adjoint approaches. Next the second derivatives are obtained using the mixed approach in which both the direct and adjoint first-order solutions are used. The general results are particularized for the case of complementary and potential energy of a structure. Some simple examples illustrate the theory presented.