📝 SummarXiv

Abstract

A critical issue that affects engineers trying to assess the structural integrity of various infrastructures, such as metal rods or acoustic ducts, is the challenge of detecting internal fractures (defects). Traditionally, engineers depend on audible and visual aids to identify these fractures, as they do not physically dissect the object in question into multiple pieces to check for inconsistencies. This research introduces ideas towards the development of a robust strategy to image such defects using only a small set of minimal, non-invasive measurements. Assuming a one dimensional model (e.g. longitudinal waves in long and thin rods/acoustic ducts or transverse vibrations of strings), we make use of the continuous one-dimensional wave equation to model these physical phenomena and then employ specialized mathematical analysis tools (the Laplace transform and optimization) to introduce our defect imaging ideas. In particular, we will focus on the case of a long bar which is homogeneous throughout except in a small area where a defect in its Young's modulus is present. We will first demonstrate how the problem is equivalent to a spring-mass vibrational system, and then show how our imaging strategy makes use of the Laplace domain analytic map between the characteristics of the respective defect and the measurement data. More explicitly, we will utilize MATLAB (a platform for numerical computations) to collect synthetic data (computational alternative to real world measurements) for several scenarios with one defect of arbitrary location and stiffness. Subsequently, we will use this data along with our analytically developed map (between defect characteristics and measurements) to construct a residual function which, once optimized, will reveal the location and magnitude of the stiffness defect.