An inverse problem for Voronoi diagrams : a simplified model of non-destructive testing with ultrasonic array

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Bourne, David P. and Mulholland, Anthony J. and Sahu, Smita and Tant, Katherine M. M. (2020) An inverse problem for Voronoi diagrams : a simplified model of non-destructive testing with ultrasonic array. Mathematical Methods in the Applied Sciences, n/a. n/a. ISSN 0170-4214 (https://doi.org/10.1002/mma.6977)

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Abstract

In this paper, we study the inverse problem of recovering the spatially varying material properties of a solid polycrystalline object from ultrasonic travel time measurements taken between pairs of points lying on the domain boundary. We consider a medium of constant density in which the orientation of the material's lattice structure varies in a piecewise constant manner, generating locally anisotropic regions in which the wave speed varies according to the incident wave direction and the material's known slowness curve. This particular problem is inspired by current challenges faced by the ultrasonic non-destructive testing of polycrystalline solids. We model the geometry of the material using Voronoi tessellations and study two simplified inverse problems where we ignore wave refraction. In the first problem, the Voronoi geometry itself and the orientations associated to each region are unknowns. We solve this nonsmooth, nonconvex optimisation problem using a multistart non-linear least squares method. Good reconstructions are achieved, but the method is shown to be sensitive to the addition of noise. The second problem considers the reconstruction of the orientations on a fixed square mesh. This is a smooth optimisation problem but with a much larger number of degrees of freedom. We prove that the orientations can be determined uniquely given enough boundary measurements and provide a numerical method that is more stable with respect to the addition of noise.

ORCID iDs

Bourne, David P., Mulholland, Anthony J., Sahu, Smita and Tant, Katherine M. M. ORCID logoORCID: https://orcid.org/0000-0003-4345-7054;
  • Item type:Article
    ID code:74206
    Dates:
    Date
    Event
    11 November 2020
    Published
    11 November 2020
    Published Online
    1 October 2020
    Accepted
    31 January 2020
    Submitted
    Subjects:Science > Mathematics
    Department:Faculty of Science > Mathematics and Statistics
    Depositing user: Pure Administrator
    Date deposited:12 Oct 2020 10:02
    Last modified:12 Dec 2024 09:20
    URI:https://strathprints.strath.ac.uk/id/eprint/74206
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