Factorization method for the biharmonic scattering problem for an absorbing penetrable scatterer

TL;DR

The work develops a rigorous factorization method for inverse biharmonic scattering by an absorbing penetrable scatterer in a thin Kirchhoff–Love plate, modeled by $(\Delta^2-\kappa^4 n(x))u=0$ in $\mathbb{R}^2$. It proves a factorization $F=H^{*}TH$ of the far-field operator and a range-based criterion: $z\in D$ iff $\phi_z$ lies in the range of $H^{*}$, leveraging the coercivity of $\Im(T)$ under the absorbing condition. The paper also numerically assesses the Born approximation for weak scatterers and demonstrates accurate reconstructions for star-, kite-shaped cavities and two disks, using a regularized factorization approach on the far-field data. This non-iterative method provides a spectrally grounded tool for identifying transmission-type inclusions in Kirchhoff–Love plates, with potential applications in vibration control and structural health monitoring.

Abstract

This work extends the factorization method to the inverse scattering problem of reconstructing the shape and location of an absorbing penetrable scatterer embedded in a thin infinite elastic (Kirchhoff--Love) plate. With the assumption that the plate thickness is small compared to the wavelength of the incident wave, the propagation of flexural perturbations is modeled by the two--dimensional biharmonic wave equation in the frequency domain. Within this setting, we provide a rigorous justification of the factorization method and demonstrate that it yields a binary criterion for distinguishing whether a sampling point lies inside or outside the scatterer, using only the spectral data of the far--field operator. In addition, we numerically analyze the Born approximation for weak scatterers in this biharmonic scattering context and compute the relative error against exact far--field data for sample weak scatterers, thereby quantifying its validity as a limited but useful approximation.

Figures (9)

• Figure 1: Exact and Born-approximated far-field matrices for scattering by a disk of radius $\epsilon=0.5$ with complex refractive index $n=1.0+0.5\text{i}$ at wavenumber $\kappa=2$. The comparison illustrates the close approximation with a slight deviation of the Born approximation from the exact solution in the presence of moderate absorption.
• Figure 2: Comparison of the reconstructed penetrable disk $D=B_{\epsilon}$ using far-field data for the disk scatterer with radius $\epsilon = 0.5$, wavenumber $\kappa = 2$, and $n=1.0+0.5\text{i}$.
• Figure 3: Reconstruction of the star--shaped penetrable cavity using the factorization method for two different wavenumbers: $\kappa = 2\pi$ (left) and $\kappa = 3\pi$ (right) with a constant contrast $n=2.5+0.5\text{i}$.
• Figure 4: Reconstruction of the star--shaped penetrable cavity using the factorization method for $\kappa = 2\pi$ with noise added. Left: 5% noise ($\delta = 0.05$). Right: 10% noise ($\delta = 0.10$).
• Figure 5: Reconstruction of the shifted star--shaped penetrable cavity using the factorization method for $\kappa = 2\pi$ with $5\%$ noise and index of refraction $n=2.5+0.5\text{i}$. Left: Scatterer shifted to $(-0.5, 0.5)$. Right: Scatterer shifted to $(1,1.5)$.

Theorems & Definitions (8)

• Lemma 3.1
• Theorem 3.2
• Theorem 3.3
• proof
• Theorem 3.4
• proof
• Corollary 3.5
• Corollary 3.6