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Efficient Computation of Magnetic Polarizability Tensor Spectral Signatures for Object Characterisation in Metal Detection

James Elgy, Paul D. Ledger

TL;DR

The paper tackles the computational bottleneck in obtaining magnetic polarizability tensor (MPT) spectral signatures for object identification in metal detection by combining hp-FEM discretisation with prismatic boundary layers and a POD-based reduced-order model (PODP). It develops three computational pathways (Integral Method, Full Matrix Method, and Matrix Method) to accelerate online MPT evaluations while preserving accuracy, and it introduces an adaptive, error-driven strategy for selecting frequency snapshots. The authors demonstrate substantial speedups and accuracy gains across representative geometries (sphere, disks) and a realistic multi-material cleaver, enabling the construction of large, reliable dictionaries for object classification. The work provides practical guidelines for boundary-layer design, adaptive snapshot selection, and offers open-source software to facilitate deployment in real-world metal-detection applications, potentially reducing false positives and negatives in UXO clearance, recycling, and security contexts.

Abstract

Purpose: Magnetic polarizability tensors (MPTs) provide an economical characterisation of conducting magnetic metallic objects and their spectral signature can aid in the solution of metal detection inverse problems, such as scrap metal sorting, searching for unexploded ordnance in areas of former conflict, and security screening at event venues and transport hubs. In this work, the authors discuss methods for efficiently building large dictionaries for classification approaches. Design/methodology/approach: Previous work has established explicit formulae for MPT coefficients, underpinned by a rigorous mathematical theory. To assist with the efficient computation of MPTs at differing parameters and objects of interest this work applies new observations about the way the MPT coefficients can be computed. Furthermore, the authors discuss discretisation strategies for hp-finite elements on meshes of unstructured tetrahedra combined with prismatic boundary layer elements for resolving thin skin depths and using an adaptive proper orthogonal decomposition (POD) reduced order modelling methodology to accelerate computations for varying parameters. Findings: The success of the proposed methodologies is demonstrated using a series of examples. A significant reduction in computational effort is observed across all examples. The authors identify and recommend a simple discretisation strategy, and improved accuracy is obtained using adaptive POD. Originality: The authors present novel computations, timings, and error certificates of MPT characterisations of realistic objects made of magnetic materials. A novel postprocessing implementation is introduced, and an adaptive POD algorithm is demonstrated.

Efficient Computation of Magnetic Polarizability Tensor Spectral Signatures for Object Characterisation in Metal Detection

TL;DR

The paper tackles the computational bottleneck in obtaining magnetic polarizability tensor (MPT) spectral signatures for object identification in metal detection by combining hp-FEM discretisation with prismatic boundary layers and a POD-based reduced-order model (PODP). It develops three computational pathways (Integral Method, Full Matrix Method, and Matrix Method) to accelerate online MPT evaluations while preserving accuracy, and it introduces an adaptive, error-driven strategy for selecting frequency snapshots. The authors demonstrate substantial speedups and accuracy gains across representative geometries (sphere, disks) and a realistic multi-material cleaver, enabling the construction of large, reliable dictionaries for object classification. The work provides practical guidelines for boundary-layer design, adaptive snapshot selection, and offers open-source software to facilitate deployment in real-world metal-detection applications, potentially reducing false positives and negatives in UXO clearance, recycling, and security contexts.

Abstract

Purpose: Magnetic polarizability tensors (MPTs) provide an economical characterisation of conducting magnetic metallic objects and their spectral signature can aid in the solution of metal detection inverse problems, such as scrap metal sorting, searching for unexploded ordnance in areas of former conflict, and security screening at event venues and transport hubs. In this work, the authors discuss methods for efficiently building large dictionaries for classification approaches. Design/methodology/approach: Previous work has established explicit formulae for MPT coefficients, underpinned by a rigorous mathematical theory. To assist with the efficient computation of MPTs at differing parameters and objects of interest this work applies new observations about the way the MPT coefficients can be computed. Furthermore, the authors discuss discretisation strategies for hp-finite elements on meshes of unstructured tetrahedra combined with prismatic boundary layer elements for resolving thin skin depths and using an adaptive proper orthogonal decomposition (POD) reduced order modelling methodology to accelerate computations for varying parameters. Findings: The success of the proposed methodologies is demonstrated using a series of examples. A significant reduction in computational effort is observed across all examples. The authors identify and recommend a simple discretisation strategy, and improved accuracy is obtained using adaptive POD. Originality: The authors present novel computations, timings, and error certificates of MPT characterisations of realistic objects made of magnetic materials. A novel postprocessing implementation is introduced, and an adaptive POD algorithm is demonstrated.
Paper Structure (18 sections, 1 theorem, 30 equations, 18 figures, 1 algorithm)

This paper contains 18 sections, 1 theorem, 30 equations, 18 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation
  3. Magnetic Polarizability Tensor Object Characterisation
  4. Finite Element Discretisation
  5. Improved Efficiency for the Calculation of the MPT Spectral Signature
  6. PODP Approach
  7. Off-line Stage
  8. On-line Stage
  9. Improved Computational Efficiency for the PODP Prediction of the MPT Spectral Signature
  10. Adaptive Selection of Frequency Snapshots for the Off-line Stage
  11. Computational Resources and Software
  12. A Recipe for Choosing the Number and Thicknesses of Prismatic Boundary Layers
  13. Numerical Examples
  14. Conducting Sphere
  15. Conducting and Magnetic Disks
  16. ...and 3 more sections

Key Result

Lemma 4.2

An a-posteriori error estimate for the tensor coefficients computed using PODP is where $Y^{(hp)}: = Y^\varepsilon \cap W^{(hp)}$ and $\alpha_{LB}$ is a lower bound on a stability constant.

Figures (18)

  • Figure 1: Illustrative diagram of a ($a$) handheld and ($b$) walk through metal detector indicating the coil arrays for the transmitting (green) and receiving (red) coils. The object $B_\alpha$ and its material properties are also indicated.
  • Figure 2: Proposals for boundary layer thickness for the simple function $y=e^{-x}$ showing the thicknesses for $L=3$ layers of elements in terms of the non-dimensional skin depth $\tau$ being for $(a)$ "uniform" distribution, $(b)$ "geometric decreasing" distribution, and $(c)$ "geometric increasing" distribution.
  • Figure 3: Magnetic conducting sphere: Showing the effect of $p$--refinement on $E = \lVert \mathcal{M}^{hp} - \mathcal{M}\rVert_F / \lVert\mathcal{M}\rVert_F$ for the different prismatic layer strategies with respect to number of degrees of freedom (left column) and computational time (right column) for $\mu_r =1 \, (a, b), \, 16 \, (c,d),$ and $64\, (e,f)$.
  • Figure 4: Magnetic conducting sphere: Showing a comparison between the original IM and the new faster FMM and MM approaches for the calculation of the MPT spectral signature using PODP $(a)$$(\tilde{\mathcal{R}})_{ij}$ and $(b)$$({\mathcal{I}})_{ij}$.
  • Figure 5: Magnetic conducting sphere: A comparison between time and memory usage for ($a$) IM, ($b$) FMM, and ($c$) MM methods showing substantial speed up for the on--line stage of POD. Timings are further broken down into common parts of the problem to correlate memory usage with specific tasks.
  • ...and 13 more figures

Theorems & Definitions (5)

  • Remark 3.1
  • Remark 4.1
  • Lemma 4.2: Wilson, Ledger Wilson2021
  • Remark 6.1
  • Remark 6.2