Reduced order model approaches for predicting the magnetic polarizability tensor for multiple parameters of interest

Abstract

The magnetic polarizability tensor (MPT) is an economical characterisation of a conducting magnetic object, which can assist with identifying hidden targets in metal detection. The MPT’s coefficients depend on multiple parameters of interest including the object shape, size, electrical conductivity, magnetic permeability, and the frequency of excitation. The computation of the coefficients follow from post-processing an eddy current transmission problem solved numerically using high-order finite elements. To reduce the computational cost of constructing these characterisations for multiple different parameters, we compare three methods by which the MPT can be efficiently calculated for two-dimensional parameter sets, with different levels of code invasiveness. We compare, with numerical examples, a neural network regression of MPT eigenvalues with a projection-based reduced order model (ROM) and a neural network enhanced ROM (POD–NN) for predicting MPT coefficients.

Computations and measurements of the magnetic polarizability tensor characterisation of highly conducting and magnetic objects

“…For steel, the conductivity is well defined as 1 × 10 6 ≤ σ * ≤ 1 × 10 7 S/m, however the relative permeability can vary substantially by several orders of magnitude. For this reason, we consider a range of μ r drawn from a normal distribution μr∼N((μr)m,(μr)s) with (μr)m=200 and (μr)s=50 and note that high values of μ r are not expected to result in significant change in the MPT spectral signature (Elgy and Ledger, 2023b). To ensure that accurate results are obtained, we introduce two thin layers of prismatic elements of thickness δ / α and 2 δ / α units resulting in a mesh with 12 479 prisms and 27 478 unstructured tetrahedra.…”

“…and ðμ r Þ s ¼ 50 and note that high values of μ r are not expected to result in significant change in the MPT spectral signature (Elgy and Ledger, 2023b). To ensure that accurate results are obtained, we introduce two thin layers of prismatic elements of thickness δ/α and 2δ/α units resulting in a mesh with 12 479 prisms and 27 478 unstructured tetrahedra.…”

“…An efficient procedure for computing the MPT spectral signature of a conducting object has been proposed by Wilson and Ledger (2021) and a revised and improved implementation by Elgy and Ledger (2023b) is in the updated [1] (InitialRelease branch), using version 6.2.2204 of the higher order finite library (NGSolve, n.d; Schöberl, 2014; Zaglmayr, 2006), and will be used throughout the work to generate the numerical results. The approach computes discrete approximations to bold-italicθi(1) and bold-italicθi(0) for a small number of sets of problem parameters using with a H (curl) conforming discretisation on unstructured (tetrahedral) grids (Schöberl and Zaglmayr, 2005; NGSolve, n.d; Schöberl, 2014; Zaglmayr, 2006) and the mesher.…”

“…Computational procedures based on the finite element method with a high order edge element (H(curl) conforming) discretisation have been described for calculating the MPT characterisation of conducting objects (Ledger and Lionheart, 2015). In addition, an efficient procedure for computing the MPT spectral signature object characterisation based on a reduced order model approach, using proper orthogonal decomposition (POD), has been developed (Wilson and Ledger, 2021) and extended to higher dimensional parameter spaces (Elgy and Ledger, 2023b). Dictionaries of MPT spectral signature characterisations of threat and non-threat objects have been developed (Ledger et al, 2021) and used as a basis for machine learning classifiers (Ledger et al, 2022) to distinguish between threat and non-threat objects using simulated data.…”

Resolving full-wave through-wall transmission effects in multi-static synthetic aperture radar

“…One answer to the computational cost has been the use of reduced order models (ROMs), in which a low-order surrogate model is developed which still encapsulates the degrees of freedom of the full physics (wave) model [20,35,41]. ROMs have been applied extensively to inverse scattering problems as a means to deal with the nonlinearity due to multiple scattering in various settings by Borcea et al [8][9][10][11], as well as more recently to the case of monostatic (i.e.…”