L1 regularization approach to structural damage detection using frequency data

Structural damage often occurs only at several locations that exhibit stiffness reduction at sparse elements out of the large total number of elements in the entire structure. The conventional vibration-based damage detection methods employ a so-called l2 regularization approach in model updating. This generally leads to the damaged elements distributed to numerous elements, which does not represent the actual case. A new l1 regularization approach is developed to detect structural damage using the first few frequency data. The technique is based on the sparse recovery theory that a sparse vector can be successfully recovered using a small number of measurement data. One advantage of using frequency data is that the first few frequencies can be measured more accurately and conveniently than mode shapes and other modal properties. A cantilever beam is utilized to demonstrate the effectiveness of the proposed method. Only the first six modal frequencies are required to detect two damaged elements among 90 finite beam elements, which cannot be achieved using the conventional damage detection approach. The effects of measurement number, damage severity, number of damage, and noise level on damage detection results are also studied through a numerical example. The advantage of the new regularization approach over the conventional one is finally interpreted.