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Characterising small objects in the regime between the eddy current model and wave propagation

Ledger, Paul David; Lionheart, William R B

Authors

William R B Lionheart



Abstract

Being able to characterise objects at low frequencies, but in situations where the modelling error in the eddy current approximation of the Maxwell system becomes large, is important for improving current metal detection technologies. Importantly, the modelling error becomes large as the frequency increases, but the accuracy of the eddy current model also depends on the object topology and on its materials, with the error being much larger for certain geome-tries compared to others of the same size and materials. Additionally, the eddy current model breaks down at much smaller frequencies for highly magnetic conducting materials compared to non-permeable objects (with similar conductivities, sizes and shapes) and, hence, characterising small magnetic objects made of permeable materials using the eddy current at typical frequencies of operation for a metal detector is not always possible. To address this, we derive a new asymptotic expansion for permeable highly conducting objects that is valid for small objects and holds not only for frequencies where the eddy current model is valid but also for situations where the eddy current modelling error becomes large and applying the eddy approximation would be invalid. The leading-order term we derive leads to new forms of object characterisations in terms of polarizability tensor object descriptions where the coefficients can be obtained from solving vectorial transmission problems. We expect these new characterisations to be important when considering objects at greater stand-off distance from the coils, which is important for safety critical applications, such as the identification of landmines, unexploded ordnance and concealed weapons. We also expect our results to be important when characterising artefacts of archaeological and forensic significance at greater depths than the eddy current model allows and to have further applications parking sensors and improving the detection of hidden, out-of-sight, metallic objects.

Journal Article Type Article
Acceptance Date Jul 9, 2023
Online Publication Date Aug 8, 2023
Publication Date Aug 8, 2023
Deposit Date Aug 8, 2023
Publicly Available Date Aug 9, 2023
Journal European Journal of Applied Mathematics
Print ISSN 0956-7925
Publisher Cambridge University Press
Peer Reviewed Peer Reviewed
Article Number PII S0956792523000207
Pages 1-24
DOI https://doi.org/10.1017/S0956792523000207
Keywords Asymptotic analysis; time harmonic Maxwell; eddy current; inverse problems; magnetic polarizability tensor; metal detection 2020 Mathematics Subject Classification: 35R30; 35Q61; 78A46 (Primary); 35B30; 78A25 (Secondary)

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