Topological derivative for a fast identification of short, linear perfectly conducting cracks with inaccurate background information

Won-Kwang Park; Won-Kwang Park

doi:10.3934/cam.2026009

Communications in Analysis and Mechanics

2026, Volume 18, Issue 1: 228-244. doi: 10.3934/cam.2026009

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Research article

Topological derivative for a fast identification of short, linear perfectly conducting cracks with inaccurate background information

Won-Kwang Park ^,

Department of Information Security, Cryptography, and Mathematics, Kookmin University, Seoul, 02707, Korea

Received: 04 August 2025 Revised: 12 October 2025 Accepted: 04 January 2026 Published: 16 March 2026
78A46

In this study, we consider a topological derivative-based imaging technique for the fast identification of short, linear perfectly conducting cracks completely embedded in a two-dimensional homogeneous domain with smooth boundary. Unlike conventional approaches, we assume that the background permittivity and permeability are unknown due to their dependence on frequency and temperature, and we propose a normalized imaging function to localize cracks. Despite inaccuracies in background parameters, application of the proposed imaging function enables recognition of the existence of cracks, but it is still impossible to identify accurate crack locations. Furthermore, the shift in crack localization of imaging results is significantly influenced by the applied background parameters. In order to theoretically explain this phenomenon, we show that the imaging function can be expressed in terms of the zero-order Bessel function of the first kind, the crack lengths, and the applied inaccurate background wavenumber corresponding to the applied inaccurate background permittivity and permeability. Various numerical simulation results with synthetic data polluted by random noise validate the theoretical results.
- topological derivative,
- inaccurate background information,
- Bessel function,
- numerical simulation results,
- perfectly conducting cracks
Citation: Won-Kwang Park. Topological derivative for a fast identification of short, linear perfectly conducting cracks with inaccurate background information[J]. Communications in Analysis and Mechanics, 2026, 18(1): 228-244. doi: 10.3934/cam.2026009

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Abstract

In this study, we consider a topological derivative-based imaging technique for the fast identification of short, linear perfectly conducting cracks completely embedded in a two-dimensional homogeneous domain with smooth boundary. Unlike conventional approaches, we assume that the background permittivity and permeability are unknown due to their dependence on frequency and temperature, and we propose a normalized imaging function to localize cracks. Despite inaccuracies in background parameters, application of the proposed imaging function enables recognition of the existence of cracks, but it is still impossible to identify accurate crack locations. Furthermore, the shift in crack localization of imaging results is significantly influenced by the applied background parameters. In order to theoretically explain this phenomenon, we show that the imaging function can be expressed in terms of the zero-order Bessel function of the first kind, the crack lengths, and the applied inaccurate background wavenumber corresponding to the applied inaccurate background permittivity and permeability. Various numerical simulation results with synthetic data polluted by random noise validate the theoretical results.

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