scholarly journals Interior Elastic Scattering by a Non-Penetrable Partially Coated Obstacle and Its Shape Recovering

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2485
Author(s):  
Angeliki Kaiafa ◽  
Vassilios Sevroglou
Keyword(s):  
Mixed Boundary ◽  
Variational Equation ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Point Of View ◽  
Inversion Algorithm ◽  
Direct Scattering ◽  
Time Harmonic ◽  
Navier Equation ◽  
Robin Boundary

In this paper, the interior elastic direct and inverse scattering problem of time-harmonic waves for a non-penetrable partially coated obstacle placed in a homogeneous and isotropic medium is studied. The scattering problem is formulated via the Navier equation, considering incident circular waves due to point-source fields, where the corresponding scattered data are measured on a closed curve inside the obstacle. Our model, from the mathematical point of view, is described by a mixed boundary value problem in which the scattered field satisfies mixed Dirichlet-Robin boundary conditions on the Lipschitz boundary of the obstacle. Using a variational equation method in an appropriate Sobolev space setting, uniqueness and existence results as well as stability ones are established. The corresponding inverse problem is also studied, and using some specific auxiliary integral operators an appropriate modified factorisation method is given. In addition, an inversion algorithm for shape recovering of the partially coated obstacle is presented and proved. Last but not least, useful remarks and conclusions concerning the direct scattering problem and its linchpin with the corresponding inverse one are given.

The inverse elastic scattering by an impenetrable obstacle and a crack

2020 ◽  
Author(s):  
Jianli Xiang ◽  
Guozheng Yan
Keyword(s):  
Mixed Type ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Field Operator ◽  
Integral Equation Method ◽  
Factorization Method ◽  
Boundary Integral ◽  
Far Field ◽  
Direct Scattering ◽  
Time Harmonic

Abstract This paper is concerned with the inverse scattering problem of time-harmonic elastic waves by a mixed-type scatterer, which is given as the union of an impenetrable obstacle and a crack. We develop the modified factorization method to determine the shape of the mixed-type scatterer from the far field data. However, the factorization of the far field operator $F$ is related to the boundary integral matrix operator $A$, which is obtained in the study of the direct scattering problem. So, in the first part, we show the well posedness of the direct scattering problem by the boundary integral equation method. Some numerical examples are presented at the end of the paper to demonstrate the feasibility and effectiveness of the inverse algorithm.

Time harmonic electromagnetic waves in an inhomogeneous medium

1990 ◽  
Vol 116 (3-4) ◽  
pp. 279-293 ◽  
Author(s):  
David Colton ◽  
Rainer Kress
Keyword(s):  
Inhomogeneous Medium ◽  
Electromagnetic Waves ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Impedance Boundary ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Direct Scattering ◽  
Time Harmonic ◽  
Regularity Results

SynopsisWe consider the scattering of time harmonic electromagnetic waves by an inhomogeneous medium of compact support, i.e. the permittivity ε = ε(x) and the conductivity σ = σ(x) are functions of x ∊ ℝ3. Existence, uniqueness and regularity results are established for the direct scattering problem. Then, based on existence and uniqueness results for the exterior and interior impedance boundary value problem, a method is presented for solving the inverse scattering problem.

Algorithms for solving scattering problems for the Manakov model of nonlinear Schrödinger equations

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Leonid L. Frumin
Keyword(s):  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Numerical Algorithms ◽  
Scattering Problems ◽  
Block Matrices ◽  
Nonlinear Schrödinger ◽  
Direct Scattering ◽  
Vector Case ◽  
Manakov Model ◽  
Vector Variant

AbstractWe introduce numerical algorithms for solving the inverse and direct scattering problems for the Manakov model of vector nonlinear Schrödinger equation. We have found an algebraic group of 4-block matrices with off-diagonal blocks consisting of special vector-like matrices for generalizing the scalar problem’s efficient numerical algorithms to the vector case. The inversion of block matrices of the discretized system of Gelfand–Levitan–Marchenko integral equations solves the inverse scattering problem using the vector variant the Toeplitz Inner Bordering algorithm of Levinson’s type. The reversal of steps of the inverse problem algorithm gives the solution of the direct scattering problem. Numerical tests confirm the proposed vector algorithms’ efficiency and stability. We also present an example of the algorithms’ application to simulate the Manakov vector solitons’ collision.

scholarly journals Microwave Imaging of Three-Dimensional Targets by Means of an Inexact-Newton-Based Inversion Algorithm

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Claudio Estatico ◽  
Matteo Pastorino ◽  
Andrea Randazzo
Keyword(s):  
Inverse Scattering ◽  
Scattering Problem ◽  
Three Dimensional ◽  
Inverse Scattering Problem ◽  
Microwave Imaging ◽  
Inexact Newton Method ◽  
Imaging Method ◽  
Inversion Algorithm ◽  
Three Dimensional Objects ◽  
Full Vector

A microwave imaging method previously developed for tomographic inspection of dielectric targets is extended to three-dimensional objects. The approach is based on the full vector equations of the electromagnetic inverse scattering problem. The ill-posedness of the problem is faced by the application of an inexact-Newton method. Preliminary reconstruction results are reported.

scholarly journals The determination of the surface impedance of an obstacle

1993 ◽  
Vol 36 (1) ◽  
pp. 1-15 ◽  
Author(s):  
Andrzej W. Kȩdzierawski
Keyword(s):  
Acoustic Waves ◽  
Surface Impedance ◽  
Scattering Problem ◽  
Three Dimensional ◽  
Inverse Scattering Problem ◽  
Far Field ◽  
Field Patterns ◽  
Time Harmonic ◽  
Scattered Fields

The inverse scattering problem we consider is to determine the surface impedance of a three-dimensional obstacle of known shape from a knowledge of the far-field patterns of the scattered fields corresponding to many incident time-harmonic plane acoustic waves. We solve this problem by using both the methods of Kirsch-Kress and Colton-Monk.

The electromagnetic inverse-scattering problem for partly coated Lipschitz domains

2004 ◽  
Vol 134 (4) ◽  
pp. 661-682 ◽  
Author(s):  
Fioralba Cakoni ◽  
David Colton ◽  
Peter Monk
Keyword(s):  
Inverse Scattering ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Scattered Wave ◽  
Field Pattern ◽  
Electromagnetic Plane Wave ◽  
Linear Sampling Method ◽  
Time Harmonic ◽  
Far Field Pattern ◽  
Electromagnetic Plane

We consider the inverse-scattering problem of determining the shape of a partly coated obstacle in R3 from a knowledge of the incident time-harmonic electromagnetic plane wave and the electric far-field pattern of the scattered wave. A justification is given of the linear sampling method in this case and numerical examples are provided showing the practicality of our method.

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