A unified approach for high order sensitivity analysis

摘要

Many engineering problems require solving PDEs by means of numerical methods (type FEM/BEM) which sensitivity analysis entails taking derivatives of functions denned through integration. In sizing optimization problems, the integration domains are fixed, what enables the regular use of analytical sensitivity techniques. In shape optimization problems, the integration domains are nevertheless variable. This fact causes some cumbersome difficulties, that have traditionally been overcome by means of finite difference approximations. Three kinds of analytical approaches have been proposed for computing sensitivity derivatives in shape optimization problems. The first is based on differentiation of the final discretized equations. The second is based on variation of the continuum equations and on the concept of material derivative. The third is based upon the existence of a mapping that links the material space with a fixed space of reference coordinates. 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 unified procedure for integration in manifolds. Our aim is to obtain a single, unified, compact procedure to compute arbitrarily high order directional derivatives of the objective function and the constraints in FEM/BEM shape optimization problems. Special care has been taken on heading for easy-to-compute recurrent expressions. The proposed scheme is basically independent from the specific form of the state equations, and can be applied to both, direct and adjoint state formulations. Thus, its numerical implementation in current engineering codes is straightforward. An application example is finally presented.