scholarly journals An asymptotic approach to inverse scattering problems on weakly nonlinear elastic rods

2002 ◽  
Vol 2 (8) ◽  
pp. 407-435 ◽  
Author(s):  
Shinuk Kim ◽  
Kevin L. Kreider
Keyword(s):  
Inverse Problems ◽  
Scattering Problem ◽  
Elastic Rods ◽  
Elastic Rod ◽  
Scattering Problems ◽  
Mathematical Basis ◽  
Cross Sectional ◽  
Weakly Nonlinear ◽  
The Time Domain ◽  
Nonlinear Elastic

Elastic wave propagation in weakly nonlinear elastic rods is considered in the time domain. The method of wave splitting is employed to formulate a standard scattering problem, forming the mathematical basis for both direct and inverse problems. A quasi-linear version of the Wendroff scheme (FDTD) is used to solve the direct problem. To solve the inverse problem, an asymptotic expansion is used for the wave field; this linearizes the order equations, allowing the use of standard numerical techniques. Analysis and numerical results are presented for three model inverse problems: (i) recovery of the nonlinear parameter in the stress-strain relation for a homogeneous elastic rod, (ii) recovery of the cross-sectional area for a homogeneous elastic rod, (iii) recovery of the elastic modulus for an inhomogeneous elastic rod.

scholarly journals Convergence of the uniaxial PML method for time-domain electromagnetic scattering problems

2021 ◽  
Author(s):  
Changkun Wei ◽  
Jiaqing Yang ◽  
Bo Zhang
Keyword(s):  
Time Domain ◽  
Electromagnetic Scattering ◽  
Scattering Problem ◽  
Three Dimensional ◽  
Scattering Problems ◽  
Laplace Transform Technique ◽  
The Laplace Transform ◽  
Time Domain Electromagnetic ◽  
Numer Anal ◽  
The Time Domain

In this paper, we propose and study the uniaxial perfectly matched layer (PML) method for three-dimensional time-domain electromagnetic scattering problems, which has a great advantage over the spherical one in dealing with problems involving anisotropic scatterers. The truncated uniaxial PML problem is proved to be well-posed and stable, based on the Laplace transform technique and the energy method. Moreover, the $L^2$-norm and $L^{\infty}$-norm error estimates in time are given between the solutions of the original scattering problem and the truncated PML problem, leading to the exponential convergence of the time-domain uniaxial PML method in terms of the thickness and absorbing parameters of the PML layer. The proof depends on the error analysis between the EtM operators for the original scattering problem and the truncated PML problem, which is different from our previous work (SIAM J. Numer. Anal. 58(3) (2020), 1918-1940).

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.

An optimum PML for scattering problems in the time domain

2013 ◽  
Vol 64 (2) ◽  
pp. 24502 ◽  
Author(s):  
Axel Modave ◽  
Abelin Kameni ◽  
Jonathan Lambrechts ◽  
Eric Delhez ◽  
Lionel Pichon ◽  
...  
Keyword(s):  
Time Domain ◽  
Scattering Problems ◽  
The Time Domain

scholarly journals Solitary waves on nonlinear elastic rods. II.

1987 ◽  
Vol 81 (6) ◽  
pp. 1718-1722 ◽  
Author(s):  
M. P. Soerensen ◽  
P. L. Christiansen ◽  
P. S. Lomdahl ◽  
O. Skovgaard
Keyword(s):  
Solitary Waves ◽  
Elastic Rods ◽  
Nonlinear Elastic

The shape of a Möbius band

1993 ◽  
Vol 440 (1908) ◽  
pp. 149-162 ◽  
Keyword(s):  
Aspect Ratio ◽  
Asymptotic Solution ◽  
Cross Section ◽  
Rectangular Cross Section ◽  
Elastic Rods ◽  
Large Deflections ◽  
Mechanical Equilibrium ◽  
Elastic Rod ◽  
Flexible Material ◽  
Möbius Band

The shape of a Möbius band made of a flexible material, such as paper, is determined. The band is represented as a bent, twisted elastic rod with a rectangular cross-section. Its mechanical equilibrium is governed by the Kirchhoff–Love equations for the large deflections of elastic rods. These are solved numerically for various values of the aspect ratio of the cross-section, and an asymptotic solution is found for large values of this ratio. The resulting shape is shown to agree well with that of a band made from a strip of plastic.

scholarly journals GLOBAL SOLUTIONS OF THE EQUATION OF THE KIRCHHOFF ELASTIC ROD IN SPACE FORMS

2012 ◽  
Vol 88 (1) ◽  
pp. 70-80 ◽  
Author(s):  
SATOSHI KAWAKUBO
Keyword(s):  
Space Form ◽  
Global Solutions ◽  
Infinite Length ◽  
Elastic Rods ◽  
Elastic Rod ◽  
Lagrange Equations ◽  
Initial Value ◽  
Space Forms ◽  
Equilibrium Configurations ◽  
Rod Of Finite Length

AbstractThe Kirchhoff elastic rod is one of the mathematical models of equilibrium configurations of thin elastic rods, and is defined to be a solution of the Euler–Lagrange equations associated to the energy with the effect of bending and twisting. In this paper, we consider Kirchhoff elastic rods in a space form. In particular, we give the existence and uniqueness of global solutions of the initial-value problem for the Euler–Lagrange equations. This implies that an arbitrary Kirchhoff elastic rod of finite length extends to that of infinite length.

scholarly journals Role of Optical Coherence Tomography on Corneal Surface Laser Ablation

2012 ◽  
Vol 2012 ◽  
pp. 1-6 ◽  
Author(s):  
Bruna V. Ventura ◽  
Haroldo V. Moraes ◽  
Newton Kara-Junior ◽  
Marcony R. Santhiago
Keyword(s):  
Optical Coherence Tomography ◽  
Laser Ablation ◽  
Imaging Modality ◽  
Corneal Thickness ◽  
Optical Coherence ◽  
Corneal Surface ◽  
Cross Sectional ◽  
Surface Laser ◽  
The Time Domain

This paper focuses on reviewing the roles of optical coherence tomography (OCT) on corneal surface laser ablation procedures. OCT is an optical imaging modality that uses low-coherence interferometry to provide noninvasive cross-sectional imaging of tissue microstructurein vivo.There are two types of OCTs, each with transverse and axial spatial resolutions of a few micrometers: the time-domain and the fourier-domain OCTs. Both have been increasingly used by refractive surgeons and have specific advantages. Which of the current imaging instruments is a better choice depends on the specific application. In laserin situkeratomileusis (LASIK) and in excimer laser phototherapeutic keratectomy (PTK), OCT can be used to assess corneal characteristics and guide treatment decisions. OCT accurately measures central corneal thickness, evaluates the regularity of LASIK flaps, and quantifies flap and residual stromal bed thickness. When evaluating the ablation depth accuracy by subtracting preoperative from postoperative measurements, OCT pachymetry correlates well with laser ablation settings. In addition, OCT can be used to provide precise information on the morphology and depth of corneal pathologic abnormalities, such as corneal degenerations, dystrophies, and opacities, correlating with histopathologic findings.

Uniqueness and stability of the recovery of an absorbing obstacle from a knowledge of its scattering resonances

2000 ◽  
Vol 130 (1) ◽  
pp. 63-80
Author(s):  
C. Labreuche
Keyword(s):  
A Priori ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Large Error ◽  
Small Error ◽  
Compact Set ◽  
Far Field ◽  
Scattering Problems ◽  
Resonant Frequencies ◽  
Ill Posed

In a previous paper, I investigated the use (for the inverse scattering problem) of the resonant frequencies and the associated eigen far-fields. I showed that the shape of a sound soft obstacle is uniquely determined by a knowledge of one resonant frequency and one associated eigen far-field. Inverse obstacle scattering problems are ill-posed in the sense that a small error in the measurement may imply a large error in the reconstruction. This is contrary to the idea of continuity. I proved that, by adding some a priori information, the reconstruction becomes continuous. More precisely, continuity holds if we assume that the obstacle lies a fixed and known compact set.The goal of this paper is to extend these results to the case of absorbing obstacles.

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