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

TL;DR

The paper addresses fast identification of short, linear perfectly conducting cracks embedded in a 2D domain when background parameters $\varepsilon_{\mathrm{b}}$ and $\mu_{\mathrm{b}}$ are uncertain. It introduces a topological-derivative-based imaging function $\mathfrak{F}(\mathbf{z};k_{\mathrm{a}})$ built from a misfit with an inaccurate background wavenumber $k_{\mathrm{a}}$, and shows that the imaging functional can be represented in terms of the zero-order Bessel function $J_0$ and crack lengths, enabling crack existence detection even under parametric uncertainty. A key insight is that maxima of $\mathfrak{F}$ occur at shifted locations $\mathbf{z}$ satisfying $|k\mathbf{x}_m - k_{\mathrm{a}}\mathbf{z}|=0$, i.e., $\mathbf{z} \approx (k/k_{\mathrm{a}})\mathbf{x}_m$, which explains observed mislocalizations when background data are inaccurate. Numerical simulations with synthetic, noisy boundary data validate the theory and illustrate how the shift depends on the ratio $k/k_{\mathrm{a}}$ and on the relative background parameter values, highlighting both the method’s robustness for crack existence and its sensitivity for precise localization. Future work includes extending to three dimensions and developing strategies to recover background parameters from data.

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 to recognize the existence of crack 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 simulations results with synthetic data polluted by random noise validate the theoretical results.

Figures (8)

• Figure 1: Permittivity values of blood (left), fat (center), and water (right) between $f=0.5GHz$ and $f=1.5GHz$.
• Figure 2: Illustration of $\partial\Omega_{\mathop{\mathrm{sing}}\limits}$ and $\partial\Omega_{\mathop{\mathrm{reg}}\limits}$.
• Figure 3: (Property \ref{['property2']}) Illustration of simulation results. Black-colored straight lines are true cracks, blue- and red-colored straight lines are retrieved cracks when $\varepsilon_{\mathrm{a}}<\varepsilon_{\mathrm{b}}$ and $\varepsilon_{\mathrm{a}}>\varepsilon_{\mathrm{b}}$, respectively.
• Figure 4: Illustration of simulation results. Black-colored straight lines are true cracks, blue- and red-colored straight lines are retrieved cracks when $\varepsilon_{\mathrm{a}}>\varepsilon_{\mathrm{b}}$ and $\varepsilon_{\mathrm{a}}<\varepsilon_{\mathrm{b}}$, respectively.
• Figure 5: (Example \ref{['Ex1']}) Maps of $\mathfrak{F}(\mathbf{z};k_{\mathrm{a}})$ with various $\varepsilon_{\mathrm{a}}$ at $f=875MHz$.

Theorems & Definitions (8)

• Lemma 2.1
• Lemma 3.1: Asymptotic expansion formula
• Theorem 3.1
• proof
• Example 4.1: Single crack centered at the origin
• Example 4.2: Two cracks whose centers are located on the axis
• Example 4.3: Three cracks with same length
• Example 4.4: Three cracks with difference lengths