# A Numerical Study on Computational Time Reversal for Structural Health Monitoring
## 1 Introduction  
The detection and localization of defects, based on measurements at a limited number of spatial points, falls into the category of inverse problems which are usually ill-posed and difficult to solve. Dynamical signals have the ability to detect hidden defects, and provided that measurements can be postprocessed effectively, are very useful for structural health monitoring purposes. The availability of cheap and interconnected sensors and data processing power facilitate practical applications. Structural health monitoring (SHM) problems are based on the postprocessing of suitable measurements in order to monitor the structural health of a structure. Warning, in case of significant changes from the healthy nominal model, is the first stage of SHM. The qualification and quantification of damages and other deviation from the nominal structural model are subsequent, more delicate tasks. They belong to the class of inverse and parameter identification problems, which lead to ill-posed mathematical problems that require delicate numerical processing tools. A general solution of inverse and parameter identification problems in mechanics can be based on the comparison of predictions based on suitably parametrized models with measurements, using optimization or soft computing, as can be seen in [1, 2, 3, 4], among others. A computational tool for solving a class of inverse wave (and/or vibration) problems is the time reversal (TR) technique originally introduced as a physical process [5]. Time reversal concepts have been raised in several scientific communities and seems to be an object of high interest for mathematicians, physicists and engineers. Time reversal is a technique that focuses waves onto a source or scatter by emitting a time-reversed analogue of the received wave field measured by an array of transducers. This refocusing is due to the reciprocity in space and reversibility in time for the wave field [6, 7]. The main idea behind TR is to send the recorded signals back into the medium but reversed in time. Due to the time-reversibility of the wave equation,this process created a back propagating wave that will focus on the original source location. A defect or damaged area can be understood to act as a secondary source and therefore the principle of TR can be used to find its location. The TR method has been used for the impact location determination in [8, 9] and for both impact location and force identification in [10, 11]. Lamb waves in connection with TR have been used in several works for the detection of micro-cracks in plates and composite structures [12, 13, 14, 15, 16, 17, 18, 19, 20]. Good experimental results have been reported in several publications like [11, 21]. Further information can be found in the review publications [22, 23, 24]. The technique has possible applications in difficult nonlinear identification problems, like the breathing unilateral crack identification [25]. Furthermore, it can be used within active SHM systems, incorporating modern deep learning techniques in order to solve difficult identification problems [26]. Frame structures that are excited by some loading on respective source points called \(x_{s}\) are considered in this paper. The response is collected on a number of points called receivers \(x_{r}\). Each receiver is defined by a pair of its spatial location and the direction on which response is recorded and then we use time-reversal as an imaging technique in order to detect and localize the defect. Receivers collect data for selected degrees of freedom (DOFs) on some nodes, that for the used beam elements, are the axial, transverse and rotational motions. For the source localization procedure, it is necessary to reverse the responses of every receiver in time and from now on the receivers operate as actuators because they transmit the reversal responses for each receiver/stimulator and save the results on the selected point to use for imaging. These points will be referred to as the imagers \(x_{i}\).