Variational principle for shape design sensitivity analysis

In this paper, the adjoint method of design sensitivity analysis is stated as a general variational principle that is applicable to a wide range of problems. The principle gives a very simple and straightforward method of obtaining the design sensitivity expression for a functional dependent on the state fields. The sensitivity expression involves certain adjoint fields and explicit design variations of the functional and the governing equations for the primary state fields. The principle is stated in terms of an augmented functional that is defined by adding to the response functional whose sensitivity is desired, the equilibrium equation of the primary problem. Then explicit design variation of the augmented functional gives the total design variation of the response functional. Stationarity of the augmented functional with respect to the state fields defines the adjoint problem for the response functional. The principle is proved and applied to several classes of linear and nonlinear problems, such as field problems, structural problems, and dynamic response problems. It is shown that the design sensitivity expressions derived in the literature using long procedures are obtained quite routinely by use of the principle.