VDM-based damage identification in structures and systems
Unlike the majority of the LF SHM methods, the SHM procedure based on the Virtual Distortion Method [18] does not require modal analysis. It can be classified as a model updating method however using a strain-based objective function in optimization.The Virtual Distortion Method emerged as a tool of fast reanalysis [19] in structural mechanics. It was effectively used for problems where local modifications needed to be analyzed quickly. Good examples can be the progressive collapse analysis or optimal remodeling of structures [20].
The core of VDM-based numerical algorithms is an influence matrix. It contains a series of responses of structural members to prescribed loads e.g. self-equilibrated pair of forces (equivalent to initial strain) or unequilibrated external force. An influence vector defines how local modification introduced to one member impacts the behavior of the whole structure. Full influence matrix is an assembly of all influence vectors. The influence matrix may be considered both in a static or dynamic versions.
With the advent of SHM ideas, the VDM has been extended to a dynamic version [21] using an analogy to linear time-invariant systems, in which the output is a convolution of input and the transfer function. Consequently the modification-modeling response in VDM is a convolution of virtual distortions (design variables) and the influence matrix.
As VDM is analytically formulated, an accurate sensitivity analysis is feasible. The objective function is a squared difference between the measured and simulated strain responses. The inverse analysis is performed by employing sensitivity-based optimization routines. The analysis can be run both in the time and frequency domains [22]. In the time domain a few evenly distributed sensors are enough to obtain a satisfactory result. The number of sensors in the frequency domain should practically correspond to the number of structural elements in order to compensate for the fact that the analysis is quasi-static. The measurement noise obviously has an evil influence on results but does not disturb the overall interpretation of damage identification.
Thanks to analogies between engineering systems, modeled by graphs, adaptations of VDM to identification of leakages in water distribution networks [23] and losses of conductance in electrical networks [24] are possible.
References
- [18] Holnicki-Szulc J. (Ed.) (2008): Smart Technologies for Safety Engineering. Wiley, ISBN 978-0-470-05846-6
- [19] Akgun MA, Garcelon JH, Haftka RT (2001) Fast exact linear and non-linear structural reanalysis and the Sherman-Morrison-Woodbury formulas. Int J Numer Methods Eng 50:1587-1606
- [20] Kolakowski P., Holnicki-Szulc J. (1997) Optimal Remodelling of Truss Structures - Simulation by Virtual Distortions, Computer Assisted Mechanics and Engineering Sciences, 4(97), pp. 257-281
- [21] Kolakowski P., Zielinski T. G., Holnicki-Szulc J. (2004): Damage Identification by the Dynamic Virtual Distortion Method, Journal of Intelligent Material Systems and Structures, vol. 15, issue 6, pp. 479-493
- [22] Swiercz A., Kolakowski P., Holnicki-Szulc J. (2008): Damage Identification in Skeletal Structures Using the Virtual Distortion Method in Frequency Domain, Mechanical Systems and Signal Processing, vol. 22, issue 8, pp. 1826-1839
- [23] Holnicki-Szulc J., Kolakowski P., Nasher N. (2005) Leakage Detection in Water Networks, Journal of Intelligent Material Systems and Structures, vol. 16, issue 3, pp. 207-219
- [24] Kokot M., Holnicki-Szulc J. (2009) Defect Identification in Electrical Circuits via the Virtual Distortion Method. Part 1: Steady-state Case, Journal of Intelligent Material Systems and Structures, 20 (12): 1465-1473