High Order Shape Design Sensitivity: A Unified Approach

Abstract

[Abstract] Three basic analytical approaches have been proposed for the calculation of sen- sitivity derivatives in shape optimization problems. The first approach is based on differentiation of the discretized equations [1-3]. The second approach is based on variation of the continuum equations [1,4,5] and on the concept of material deriva- tive. The third approach [6] is based upon the existence of a transformation that links the material coordinate system with a fixed reference coordinate system. This is not restrictive, since such a transformation is inherent to FEM and BEM imple- mentations. In this paper we present a generalization of the latter approach on the basis of a generic unified procedure for integration in manifolds. Our aim is to obtain a single, unified, compact expression to compute arbitrarily high order directional deriva- tives, independently of the dimension of the material coordinates system and of the dimension of the elements. Special care has been taken on giving the final 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.