Solution of the inverse elastography problem for parametric classes of inclusions with a posteriori error estimate
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.
scientific article; zbMATH DE number 6914525