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Summary Conventional vibration‐based damage detection methods employ the Tikhonov regularization in model updating to deal with the problems of underdeterminacy and measurement noise. However, the Tikhonov regularization technique tends to provide over smooth solutions that the identified damage is distributed to many structural elements. This result does not match the sparsity property of the actual damage scenario, in which structural damage typically occurs at a small number of locations only in comparison with the total elements of the entire structure. In this study, an l1 regularization‐based model updating technique is developed by utilizing the sparsity of the structural damage. Both natural frequencies and mode shapes are employed during the model updating. A strategy of selecting the regularization parameter for the l1 regularization problem is also developed. A numerical and an experimental examples are utilized to demonstrate the effectiveness of the proposed damage detection method. The results showed that the proposed l1 regularization‐based method is able to locate and quantify the sparse damage correctly over a large number of elements. The effects of the mode number on the damage detection results are also investigated. The advantage of the present l1 regularization over the traditional l2 regularization method in damage detection is also demonstrated.

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Sparse recovery theory has been applied to damage detection by utilizing the sparsity feature of structural damage. The theory requires that the columns of the sensing matrix suffice certain independence criteria. In l 1 -regularized damage detection, the sensitivity matrix serves as the sensing matrix and is directly related to sensor locations. An optimal sensor placement technique is proposed such that the resulting sensitivity matrix is of the maximum independence in the columns or is of the least mutual coherence. Given a total number of sensors, the selection of sensor locations is a combinatorial problem. A genetic algorithm is thus used to solve this optimization problem, in which the mutual coherence of the sensitivity matrix is minimized. The obtained optimal sensor locations and associated sensitivity matrix are used in l 1 -regularized damage detection. An experimental cantilever beam and a three-storey frame are utilized to verify the effectiveness and reliability of the proposed sensor placement technique. Results show that using the modal data based on the optimal sensor placement can identify damage location and severity more accurately than using the ones based on uniformly selected sensor locations.

Summary Sparse Bayesian learning (SBL) methods have been developed and applied in the context of regression and classification, in which latent variables and hyperparameters are iteratively obtained using type‐II maximization likelihood. However, this method is ineffective in structural damage detection using modal parameters, which have a nonlinear relation with structural damage. Consequently, the analytical solution of the type‐II maximization likelihood is unavailable. In this study, an iterative expectation–maximization (EM) technique is employed to tackle the difficulty. During the iteration, structural damage and hyperparameters are updated through an expectation and maximization processes alternatively. Two sampling methods are utilized during the expectation procedure. Upon convergence, some hyperparameters approach infinity and the associated damage variables become zero, resulting in the sparsity of damage. Numerical and experimental examples demonstrate that the proposed SBL method can accurately locate and quantify the sparse damage. The proposed EM technique is easy to implement while also containing clear physical meaning.