This paper deals with the use of the topological derivative as an analysis tool for structural health monitoring, to locate the presence of flaws in a homogeneous material plate that is subject to guided elastic waves excitation. Using a numerical solver to compute the response of the system and defining a scalar objective function that measures the least squares difference between the measured and calculated signals, the topological derivative somehow describes the sensitivity of the objective function to localized perturbations of the material properties due to the presence of defects. Thus, defects are guessed to be located near the topological derivative peaks. This is somehow related to the minimization of the objective function and uses the whole physics of the problem (to compute the objective function), instead of a smaller amount of physical information, as conventional methods do. Here, we reconstruct small defects via the topological derivative by using multi-frequency synthetic data, for several representative configurations of the actuators and sensors, and several defect locations. Among these, some fairly demanding configurations are considered that are not accessible to conventional methods, such as actuators and sensors located very close to the plate boundary, and defects located beyond both elongated through-slits and elongated inclusions of a different material.

The topological derivative tool is applied here in structural health monitoring (SHM) problems to locate small defects in a material plate with complex geometry that is subject to permanent multifrequency guided waves excitation. Compared to more standard SHM methods, based in measuring the time-lag between emitted and received propagative pulses plus some postprocessing, the topological derivative somehow compares the measured and computed (solving the full elasto-dynamic equations) response of the damaged plate, instead of relying on only the time of flight of the wave. Thus, the method profits the knowledge behind the physics of the problem and can cope with scenarios in which classical methods give poor results. The authors of this paper have already used the topological derivative in rectangular plates with constant thickness, but with defects consisting simply in both through slits and inclusions of a different material, and actuators/sensors located near the boundary, which makes very difficult to use standard SHM methods. This is an extension of the method, also considering the much more difficult to analyze case of plates with variable thickness and complex (non-rectangular) planform.