Comput Mech (2014) 54:443–459 DOI 10.1007/s00466-014-0996-2 ORIGINAL PAPER Time reversal for crack identification Eyal Amitt · Dan Givoli · Eli Turkel Received: 5 August 2013 / Accepted: 30 January 2014 / Published online: 20 February 2014 © Springer-Verlag Berlin Heidelberg 2014 Abstract A general computational methodology is proposed for identifying cracks in structures. It is based on a time reversal (TR) technique and on the notion of refocus- ing. In the proposed procedure, a known source generates waves in the structure, and the time-varying response of the structure is measured only at certain points and times. In an industrial application this step is performed experimentally, but in the present study it is emulated numerically. Relying on a computational model of the structure and on the mea- sured signals, a TR solution is obtained for each assumed set of crack parameters. This amounts to evolving the solu- tion backward in time, till the initiation time of the orig- inal source. The crack identification is based on seeking, among all crack candidates, the crack which yields the best wave refocusing at the true source location. To test the pro- posed methodology, a simple rectangular membrane model governed by the 2D time-dependent scalar wave equation is employed. Finite element discretization of the structure and an explicit time-stepping scheme are used. The perfor- mance of the method is tested under various conditions and with various amounts of partial information. Its sensitivity to noise and to perturbations in the material properties is also investigated. E. Amitt · D. Givoli (B ) Department of Aerospace Engineering, Technion — Israel Institute of Technology, Haifa 32000, Israel e-mail: givolid@aerodyne.technion.ac.il E. Amitt e-mail: amitt.eyal@gmail.com E. Turkel Department of Applied Mathematics, School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel e-mail: turkel@post.tau.ac.il Keywords Time reversal · Crack identification · Non-destructive resting · Refocusing · Waves · Inverse problems 1 Introduction Non-destructive testing (NDT) (also known as Non- destructive evaluation (NDE)) is a collection of tools for controlling the structural integrity of structures, and is done routinely in the aircraft industry. Ultrasonic wave techniques are dominant in this area. See, e.g., the book [1] and the more recent review paper [2]. Standard NDT has the capability to detect damage but usually not to provide detailed information on the damage parameters. Hence, in recent years computa- tional methods have been combined with physical NDT to yield better identification. The diagrams in Fig. 1 illustrate the differences between a standard NDT system and a model- based one. The mathematical problem of crack identification, and of identification of other types of damage, falls into the category of inverse problems. See, e.g., the work surveyed in [3, 4]. One technique for the computational solution of inverse problems is that of time reversal (TR). TR was invented origi- nally as a physical (experimental) technique by Fink et al. [5]. It has been employed in various application fields involving wave propagation, as biomedical imaging [6], solid-earth geophysics [7, 8] and oceanography [9]. This procedure is based on the reversibility property of wave propagation phe- nomena in non-dissipative media. A consequence of this property is that one can “time-reverse” developed signals, by letting them propagate back in time to the location of the source that emitted them originally. If the original source is very local this process is called refocusing. TR can be used as a computational tool to obtain refocusing, e.g., for 123