Identification of Localized Damage in Structures Using Highly Incomplete Modal Information

The objective of this paper is to provide a new theoretical basis to localize and quantify sparse damage in a structure based on highly incomplete modal information. Although a large number of papers have been written on the subject, this paper offers a new perspective on the problem by proposing the L-1 norm minimization criteria, in contrast to the more traditional L-2 (Euclidean) norm minimization criterion. The proposed L-1 norm approach enables accurate and robust examination of a number of potential damage locations much larger than the number of frequencies used in the formulation of the modal sensitivity matrix. In addition, it is shown that L-1 minimization leads to sparse solutions, this is in contrast with the L-2 criteria, which leads to disperse solutions. The computational effort necessary to solve the L-1 optimization is significantly larger than in the traditional Euclidean norm and requires the use of convex optimization algorithms. However, given the results that can be obtained, the computational effort is justified. The efficacy of the proposed framework is demonstrated in detecting sparse damage in a simulated 21 degree of freedom shear building structure.