Sensitivity-free damage identification based on incomplete modal data, sparse regularization and alternating minimization approach

Abstract The incomplete modal data is mostly measured for experimental and engineering structures, however, its application into structural damage identification suffers from the drawback that the amount of the incomplete modal data is often insufficient, rendering the identification very sensitive to the measurement noise. Aiming to overcome this drawback, this paper proposes a new damage identification approach that combines the incomplete modal data with the sparse regularization. The realization of the proposed damage identification approach is mainly threefold: (a) The first is the establishment of a new goal function which is decoupled with respect to the damage parameters. To this end, the decomposition of the stiffness matrix so that the damage parameters are contained in a diagonal matrix must be introduced. (b) The second is the application of the alternating minimization approach to get the solution of the new goal function. (c) The third is the development of a novel and simple threshold setting method to properly determine the sparse regularization parameter. The feature of the proposed damage identification approach lies in that the sensitivity analysis is not involved and the exact orders of the modal data are not demanded. Numerical and experimental examples are conducted to verify the proposed damage identification approach. As a result, the proposed new goal function along with the sparse regularization indeed improves the accuracy and robustness for damage identification, in comparison to the cases when no regularization is used with the new goal function and when the Tikhonov regularization is incorporated into the conventional nonlinear least-squares goal function. Moreover, the proposed damage identification approach can work for various types of structures and even for only frequency measurements.