Three basic analytical approaches have been proposed for the calculation of sensitivity derivatives in shape optimization problems. The ®rst approach is based on dierentiation of the discretised equations. The second approach is based on variation of the continuum equations and on the concept of material derivative. The third approach is based upon the existence of a transformation that links the material coordinate system with a ®xed reference coordinate system. This is not restrictive, since such a transformation is inherent to FEM and BEM implementations.
In this paper, we present a generalization of the latter approach on the basis of a generic uni®ed procedure for integration in manifolds. Our aim is to obtain a single, uni®ed, compact expression to compute arbitrarily high order directional derivatives, independent of the dimension of the material coordinates system and of the dimension of the elements. Special care has been taken on giving the ®nal results in terms of easy-to-compute expressions, and special emphasis has been made in holding recurrence and simplicity of intermediate operations. The proposed scheme does not depend on any particular form of the state equations, and can be applied to both, direct and adjoint state formulations. Thus, its numerical implementation in standard engineering codes should be considered as a straightforward process. As an example, a second order sensitivity analysis is applied to the solution of a 3D shape design optimization problem.