Why use optimization?
As stated, the structures being considered in this paper are formed using bending elements, whose stiffness is a function of the moment of inertia of each structural member. Resulting is an engineering design problem with two main variables; a set of design variables (moment of inertia values) and a set of analysis values (measured displacements). Unfortunately, without an entire set of measured displacements at all DOF, there is not enough information to determine the design variables directly from the analysis values. As such, an optimization algorithm must be utilized to determine all unknown design variables. This can be accomplished by using constrained nonlinear optimization.
Fundamental to the optimization process is the force displacement relationship,
P = Ku ,
where P is the static load vector applied to the structure, K is the global stiffness matrix, and u is the displacement vector of the structure. With knowledge of the healthy stiffness matrix of a structure, a set of displacements for all DOF can be calculated for any set of applied static loads. Next, by applying the same static loads to the present, damaged structure and measuring strategic displacements, insight can be gained as to which structural elements have reduced in stiffness. A general explanation of the optimization routine to gain this insight will be explained next.