Müller-Breslau's Principle in Adjoint Design Sensitivity Analysis

The adjoint variable method is now a well-known approach for calculating design sensitivity coefficients in structural systems. The adjoint variable method has its origins in optimal control theory. For structures, however, the adjoint variables have important significance. Past results have shown the adjoint displacements to be sensitivity coefficients with respect to the load vector. For linear functions, the adjoint displacements have been shown to be influence coefficients associated with the function, an important finding in locating worstcase positions of moving loads in bridge design. In this paper, further interpretations are presented. Specifically, the adjoint load is interpreted physically and is shown to be an initial strain and/or initial displacement given to the structure. This fact, together with previous findings, has been related to the classical Müller-Breslau principle in mechanics. Finally, a proof of equivalence is provided between the aforementioned adjoint variable method derived from optimal control theory and the adjoint-structure approach published recently. This equivalence is also evident from the adjoint load interpretations given in the paper.