A systematic approach to shape sensitivity analysis

Abstract

A fully detailed development of the domain parameterization method is presented for shape sensitivity analysis. It is shown that equivalent results are obtained from the material derivative method. In fact, the material derivative method may be viewed as a special case of the domain parameterization method which occurs when the reference configuration coincides with the body configuration. The method is illustrated for the Laplace problem in which explicit shape sensitivities are derived by the adjoint and direct differentiation methods. Both finite element and boundary element applications are discussed. The similarities between this approach and the isoparametric finite/boundary element method are transparent. In the finite element approach, it is shown that the sensitivity integrals may be transformed to the boundary (as is commonly done in the material derivative method) for the adjoint method, however, this does not seem possible for the direct differentiation method. Finally, in the boundary element approach, the sensitivities do not require the differentiation of the fundamental solutions.