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

Communications on Analysis and Computation · 2026-01-01

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.

Well-posedness for the biharmonic scattering problem for a penetrable obstacle

ArXiv.org · 2025-06-11

We address the direct scattering problem for a penetrable obstacle in an infinite elastic two--dimensional Kirchhoff--Love plate. Under the assumption that the plate's thickness is small relative to the wavelength of the incident wave, the propagation of perturbations on the plate is governed by the two-dimensional biharmonic wave equation, which we study in the frequency domain. With the help of an operator factorization, the scattering problem is analyzed from the perspective of a coupled boundary value problem involving the Helmholtz and modified Helmholtz equations. Well-posedness and reciprocity relations for the problem are established. Numerical examples for some special cases are provided to validate the theoretical findings.

The anisotropic interior transmission eigenvalue problem with a conductive boundary

Communications on Analysis and Computation · 2025-01-01 · 1 citations

In this paper, we study the transmission eigenvalue problem for an anisotropic material with a conductive boundary. We prove that the transmission eigenvalues for this problem exist and are, at most, a discrete set. We also study the dependence of the transmission eigenvalues on the physical parameters and prove that the first transmission eigenvalue is monotonic. We then consider the limiting behavior of the transmission eigenvalues as the conductive boundary parameter $ \eta $ vanishes or goes to infinity in magnitude. Finally, we provide numerical examples on three domains to demonstrate our theoretical results.

Direct sampling for recovering a clamped cavity from the biharmonic far-field data

Inverse Problems · 2025-02-21 · 6 citations

Abstract This paper concerns the inverse shape problem of recovering an unknown clamped cavity embedded in a thin infinite plate. The model problem is assumed to be governed by the two-dimensional biharmonic wave equation in the frequency domain. Based on the far-field data, a resolution analysis is conducted for cavity recovery via the direct sampling method. The Funk–Hecke integral identity is employed to analyze the performance of two imaging functions. Our analysis demonstrates that the same imaging functions commonly used for acoustic inverse shape problems are applicable to the biharmonic wave context. This work presents the first extension of direct sampling methods to biharmonic waves using far-field data. Numerical examples are provided to illustrate the effectiveness of these imaging functions in recovering a clamped cavity.

Existence of transmission eigenvalues for biharmonic scattering by a clamped planar region

ArXiv.org · 2025-08-28

In this paper, we study the so-called clamped transmission eigenvalue problem. This is a new transmission eigenvalue problem that is derived from the scattering of an impenetrable clamped obstacle in a thin elastic plate. The scattering problem is modeled by a biharmonic wave operator given by the Kirchhoff--Love infinite plate problem in the frequency domain. These scattering problems have not been studied to the extent of other models. Unlike other transmission eigenvalue problems, the problem studied here is a system of homogeneous PDEs defined in all of $\mathbb{R}^2$. This provides unique analytical and computational difficulties when studying the clamped transmission eigenvalue problem. We are able to prove that there exist infinitely many real clamped transmission eigenvalues. This is done by studying the equivalent variational formulation. We also investigate the relationship of the clamped transmission eigenvalues to the Dirichlet and Neumann eigenvalues of the negative Laplacian for the bounded scattering obstacle.

Sampling methods for the inverse cavity scattering problem of biharmonic waves

Inverse Problems · 2025-12-18 · 1 citations

Abstract This paper addresses the inverse problem of qualitatively recovering a clamped cavity in a thin elastic plate using far-field measurements. We present a strengthened analysis of the linear sampling method (LSM) by carefully examining the range of the far-field operator and employing the reciprocity relation of the biharmonic far-field pattern. In addition, we implement both the LSM for reconstructing the cavity and the extended sampling method for localizing the cavity under limited-aperture data. Numerical experiments demonstrate the effectiveness and robustness of both methods.

Sampling Methods for Recovering Buried Corroded Boundaries from Partial Electrostatic Cauchy Data

SIAM Journal on Applied Mathematics · 2025-09-26

Analysis of the transmission eigenvalue problem for biharmonic scattering considering penetrable scatterers

ArXiv.org · 2025-10-09

In this paper, we provide an analytical study of the transmission eigenvalue problem in the context of biharmonic scattering with a penetrable obstacle. We will assume that the underlying physical model is given by an infinite elastic two--dimensional Kirchhoff--Love plate in $\mathbb{R}^2$, where the plate's thickness is small relative to the wavelength of the incident wave. In previous studies, transmission eigenvalues have been studied for acoustic scattering, whereas in this case, we consider biharmonic scattering. We prove the existence and discreteness of the transmission eigenvalues as well as study the dependence on the refractive index. We are able to prove the monotonicity of the first transmission eigenvalue with respect to the refractive index. Lastly, we provide numerical experiments to validate the theoretical work.

Existence of transmission eigenvalues for biharmonic scattering by a clamped planar region

Inverse Problems · 2025-11-28 · 2 citations

Abstract In this paper, we study the so-called clamped transmission eigenvalue problem. This is a new transmission eigenvalue problem that is derived from the scattering of an impenetrable clamped obstacle in a thin elastic plate. The scattering problem is modeled by a biharmonic wave operator given by the Kirchhoff–Love infinite plate problem in the frequency domain. These scattering problems have not been studied to the extent of other models. Unlike other transmission eigenvalue problems, the problem studied here is a system of homogeneous PDEs defined in all of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> . This provides unique analytical and computational difficulties when studying the clamped transmission eigenvalue problem. We are able to prove that there exist infinitely many real clamped transmission eigenvalues. This is done by studying the equivalent variational formulation. We also investigate the relationship of the clamped transmission eigenvalues to the Dirichlet and Neumann eigenvalues of the negative Laplacian for the bounded scattering obstacle.

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

ArXiv.org · 2025-11-07

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.