Solution of the inverse elastography problem for parametric classes of inclusions with a posteriori error estimate
type: Item
instance of: scholarly article
title: Solution of the inverse elastography problem for parametric classes of inclusions with a posteriori error estimate
zbMATH Open document ID: 1397.65237
author: A. S. Leonov
author: Alexander N. Sharov
author: A. G. Yagola
publication date: 2018-08-07T00:00:00Z
review text: In the framework of the plain strain theory of linear elasticity, the authors consider the system of equations for vertical and horizontal displacements \(u(x,y), w(x,y)\), \((x,y)\in \Omega=[-a,a] \times [-b,b]\): \((1-\nu) (E u_x)_x+(0.5-\nu)(E u_y)_y+\nu (E w_y)_x+(0.5-\nu)(E w_x)_y=0\), \((0.5-\nu) (E u_y)_x+\nu(E u_x)_y+(0.5-\nu) (E w_x)_x+(1-\nu)(E w_y)_x=0\). The displacements are caused by surface forces \((f(x,y),g(x,y))\) applied to the boundary part \(\Gamma_1=[-a,a]\times \{ b \}\), therefore \((0.5-\nu) u_y+(0.5-\nu) w_x=E^{-1} (1+\nu) (1-2\nu) f\), \(\nu u_x+(1-\nu) w_y=E^{-1} (1+\nu) (1-2\nu) g\). Other boundary conditions are \(u_x=u_y=w_x=w_y=0\) for \((x,y)\in \Gamma_{2,4}\) and \(u=w=0\) for \((x,y) \in \Gamma_3\), where \(\Gamma_2=\{ a \} \times [-b,b]\), \(\Gamma_3=[-a,a] \times \{ -b \}\), \(\Gamma_4=\{ -a\} \times [-b,b]\). Here, the Poisson ratio \(\nu\) is constant and Young's modulus \(E=E(x,y)\) a function on \(\Omega\). The authors consider the inverse problem of recovering \(E(x,y)\) by the given vertical displacement \(w(x,y)\), \((x,y)\in \Omega\). To solve the problem, the authors apply a modification of the extending compacts method by V. K. Ivanov and I. N. Dombrovskaya.
reviewed by: Mikhail Yu. Kokurin
Mathematics Subject Classification ID: 65N20
Mathematics Subject Classification ID: 65N12
Mathematics Subject Classification ID: 65N21
Mathematics Subject Classification ID: 65N30
Mathematics Subject Classification ID: 74B10
Mathematics Subject Classification ID: 65N15
zbMATH DE Number: 6914525
zbMATH Keywords: elastography
zbMATH Keywords: Young's modulus
zbMATH Keywords: inverse problem
zbMATH Keywords: quasisolution
zbMATH Keywords: a posteriori error estimate
MaRDI profile type: Publication
full work available at URL: jiip-2017-0043
OpenAlex ID: W2757019705
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DOI: 10.1515/JIIP-2017-0043
published in: Journal of Inverse and Ill-posed Problems
description: scientific article; zbMATH DE number 6914525