Lagrangian approach to design sensitivity analysis

A Lagrangian approach to design sensitivity analysis is presented. The final equations obtained by the Lagrangian approach are identical to those obtained by the adjoint method reviewed in the literature. The difference lies in the approach taken to derive these equations. The Lagrangian approach is identical for different categories of design problems, as is demonstrated by considering structural, dynamic, distributed-parameter, and shape optimal design problems. The Lagrangian approach exposes the fact that the "adjoint variables" referred to in the literature are, in fact, the Lagrange multipliers associated with the state equations, and the "adjoint equations" are the classical Euler-Lagrange equations. The clearer understanding that is obtained by this approach leads to some immediate practical advantages, and opens up some new areas for research.