The aim of this paper is to show that, for the eddy current model, the leading order term for the perturbation in the magnetic field, due to the presence of a small conducting magnetic inclusion, can be expressed in terms of a symmetric rank 2 polarization tensor. This tensor contains information about the shape and material properties of the object and is independent of position. We apply a recently derived asymptotic formula for the perturbed magnetic field, due to the presence of a conducting inclusion, which is expressed in terms of a new class of rank 4 polarization tensors (

SUMMARYWe propose a new computational framework for the treatment of acousto-magneto-mechanical coupling that arises in low-frequency electro-magneto-mechanical systems such as magnetic resonance imaging scanners. Our transient Newton-Raphson strategy involves the solution of a monolithic system obtained from the linearisation of the coupled system of equations. Moreover, this framework, in the case of excitation from static and harmonic current sources, allows us to propose a simple linearised system and rigorously motivate a single-step strategy for understanding the response of systems under different frequencies of excitation. Motivated by the need to solve industrial problems rapidly, we restrict ourselves to solving problems consisting of axisymmetric geometries and current sources. Our treatment also discusses in detail the computational requirements for the solution of these coupled problems on unbounded domains and the accurate discretisation of the fields using hp -finite elements. We include a set of academic and industrially relevant examples to benchmark and illustrate our approach.

The measurement of time-harmonic perturbed field data, at a range of frequencies, is beneficial for practical metal detection, where the goal is to locate and identify hidden targets. In particular, these benefits are realised when frequency-dependent magnetic polarizability tensors (MPTs) are used to provide an economical characterisation of conducting permeable objects, and a dictionary-based classifier is employed. However, despite the advantages shown in dictionary-based classifiers, the behaviour of the MPT coefficients with frequency is not properly understood. In this paper, we rigorously analyse, for the first time, the spectral properties of the coefficients of the MPT. This analysis has the potential to improve existing algorithms and design new approaches for object location and identification in metal detection. Our analysis also enables the response transient response from a conducting permeable object to be predicted for more general forms of excitation. KEYWORDSasymptotic analysis, eddy current, inverse problems, magnetic polarizability tensor, metal detection, spectral problems MSC CLASSIFICATION 35R30; 35B30 INTRODUCTIONIn metal detection, there is considerable interest in being able to locate and identify conducting permeable objects from the measurements of mutual inductance between a transmitting and a measurement coil. Applications include security screening, archaeology excavations, ensuring food safety as well as the search for land mines, and unexploded ordnance. There are also closely related topics such as magnetic induction tomography for medical imaging and eddy current testing for monitoring the corrosion of steel reinforcement in concrete structures.Within the metal detection community, magnetic polarizability tensors (MPTs) have attracted considerable interest to assist with the identification of objects when the transmitting coil is excited by a sinusoidal signal, eg, 1-7 Engineers believe that a rank 2 MPT provides an economical characterisation of a conducting permeable object that is invariant of position. An asymptotic formula providing the leading order term for the perturbed magnetic field due to the presence of a small conducting permeable object has been obtained by Ammari, Chen, Chen, Garnier, and Volkov, 8 which characterises the object in terms of a rank 4 tensor. We have shown that this simplifies for orthonormal coordinates and allows an object to be characterised by a complex symmetric rank 2 MPT, with an explicit formula for its coefficients, thus, justifying the This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

The aim of this paper is provide new insights into the properties of the rank 2 polarizability tensor } } M proposed in (P.D. Ledger and W.R.B. Lionheart Characterising the shape and material properties of hidden targets from magnetic induction data, IMA Journal of Applied Mathematics, doi: 10.1093/imamat/hxv015) for describing the perturbation in the magnetic field caused by the presence of a conducting object in the eddy current regime. In particular, we explore its connection with the magnetic polarizability tensor and the Pólya-Szegö tensor and how, by introducing new splittings of } } M, they form a family of rank 2 tensors for describing the response from different categories of conducting (permeable) objects. We include new bounds on the invariants of the Pólya-Szegö tensor and expressions for the low frequency and high conductivity limiting coefficients of } } M. We show, for the high conductivity case (and for frequencies at the limit of the quasi-static approximation), that it is important to consider whether the object is simply or multiply connected but, for the low frequency case, the coefficients are independent of the connectedness of the object. Furthermore, we explore the frequency response of the coefficients of } } M for a range of simply and multiply connected objects.

This work considers the accurate and efficient finite element simulation of threedimensional eddy current problems. We review the application of H and A based formulations for multiply connected domains for the cases where the conductor has a handle and/or a hole. We focus on an hierarchical hp-finite element discretization of the A based formulation that is gauged by regularization. Based on an explicit kernel splitting of the underlying hp-finite element basis, we present a novel preconditioning technique for eddy current problems. We demonstrate its validity on multiply connected domains and include a series of numerical examples to show the effectiveness of the proposed approach.