A sensitivity interpretation of adjoint variables in optimal design

The adjoint method of computing derivatives of cost and constraint functions with respect to design variables requires the calculation of certain adjoint variables. Until now, the adjoint variables have been looked upon only as some intermediate vectors needed to calculate design derivatives. In this paper, they are shown to have an important significance. They represent the sensitivity of the cost and constraint functions with respect to the loading or forcing function in the design problem. A sensitivity theorem for the adjoint variables is presented for structural, mechanical dynamic, and distributed parameter systems. These results offer some immediate practical advantages, such as a method for computing influence coefficients for structural systems, and a method for verifying (debugging) the analytical calculation of adjoint variables in development of a computer code.