Properties of generalized magnetic polarizability tensors

Ledger, Paul

Abstract

We present new alternative complete asymptotic expansions for the time harmonic low‐frequency magnetic field perturbation caused by the presence of a conducting permeable object as its size tends to zero for the eddy current approximation of the Maxwell system. Our new alternative formulations enable a natural extension of the well‐known rank 2 magnetic polarizability tensor (MPT) object characterization to higher order tensor descriptions by introducing generalized MPTs (GMPTs) using multi‐indices. In particular, we identify the magnetostatic contribution, provide new results on the symmetries of GMPTs, derive explicit formulae for the real and imaginary parts of GMPT coefficients and also describe the spectral behavior of GMPT coefficients. We also introduce the concept of harmonic GMPTs (HGMPTs) that have fewer coefficients than other GMPT descriptions of the same order. We describe the scaling, translation and rotational properties of HGMPTs and describe an approach for obtaining those HGMPT coefficients that are invariant under the action of a symmetry group. Such an approach is one candidate for selecting features in object classification for hidden object identification using HGMPTs.

• Journal Article Type: Article
• Acceptance Date: Oct 19, 2022
• Online Publication Date: Nov 19, 2022
• Publication Date: Mar 30, 2023
• Publicly Available Date: May 30, 2023
• Journal: Mathematical Methods in the Applied Sciences
• Print ISSN: 0170-4214
• Electronic ISSN: 1099-1476
• Publisher: Wiley
• Peer Reviewed: Peer Reviewed
• Volume: 46
• Issue: 5
• Pages: 5604-5631
• DOI: https://doi.org/10.1002/mma.8856
• Keywords: asymptotic analysis; eddy current; inverse problems; magnetic polarizability tensor; metal detection; spectral problems
• Publisher URL: https://onlinelibrary.wiley.com/doi/10.1002/mma.8856