Multiresolution Dynamic Mode Decomposition (mrDMD) of Elastic Waves for Damage Localisation in Piezoelectric Ceramic

The performance of piezoelectric sensors deteriorated due to the presence of defect, delamination, and corrosion that needed to be diagnosed for the effective implementation of the structural health monitoring (SHM) framework. A novel experimental approach based on Coulomb coupling is devised to visualise the interaction of ultrasonic waves with microscale defects in the Lead Zirconate Titanate (PZT). Multiresolution dynamic mode decomposition (mrDMD) technique in conjunction with image registration, and Kullback Leibler (KL) divergence is utilised to diagnose and localise the surface defect in the PZT. The mrDMD technique extracts the spatiotemporal coherent mode and provides an equation-free architecture to reconstruct underlying system dynamics. Additionally, due to the strong connection between mrDMD and Koopman operator theory, the proposed technique is well suited to resolve the nonlinear and dispersive interaction of elastic waves with boundaries and defects. The mrDMD sequentially decomposes the three-dimensional spatiotemporal data into low and high frequency modes. The spectral modes are sensitive to defects based on the scaling of wavelength with the size of the defect. The error due to offset and distortion was minimised with ad hoc image registration technique. Further, localisation and quantification of defect are performed by evaluating the distance metric of the probability distribution of coherent data of mrDMD acquired from healthy and defected samples. In the arena of big-data that is ubiquitous in SHM, the paper demonstrates an efficient damage localisation algorithm that explores the nonlinear system dynamics using spectral multi-mode resolution techniques by sensitising the damage features

Lie group valued Koopman eigenfunctions

Every flow on a topological space / manifold has associated to it a Koopman operator, which operates by time-shifts on various spaces of functions, such as $C^r$, $L^2$, or functions of bounded variation. An eigenfunction of the vector field (and thus for the Koopman operator) can be viewed as an $S^1$-valued function, which also plays the role of a semiconjugacy to a rigid rotation on $S^1$. This notion of Koopman eigenfunctions will be generalized to Lie-group valued eigenfunctions, and we will discuss the dynamical aspects of these functions. One of the tools that will be developed to aid the discussion, is a concept of exterior derivative for Lie group valued functions, which generalizes the notion of the differential $df$ of a real valued function $f$