An iterative approach to the solution of an inverse problem in linear elasticity

Abstract

This paper presents an iterative alternating algorithm for solving an inverse problem in linear elasticity. A relaxation procedure is developed in order to increase the rate of convergence of the algorithm and two selection criteria for the variable relaxation factors are provided. The boundary element method is used in order to implement numerically the constructing algorithm. We discuss this implementation, mention the use of Krylov methods to solve the obtained linear algebraic systems of equations and investigate the convergence and the stability when the data is perturbed by noise.

Quantitative elastography, solving the inverse elasticity problem using the Gauss-Newton method.

“…When the technique is applied to the surgical environment a different approach for the definition of the BC should be adopted. Recently a new work has been conducted to estimate the BC for an IEP Ellabib and Nachaoui [4] . In the future this algorithm will be used because is more suitable for medical application where the real boundary conditions are not easy to define.…”

A relaxation method of an alternating iterative MFS algorithm for the Cauchy problem associated with the two‐dimensional modified Helmholtz equation

“…Consequently, an iterative procedure, which provides the selection of the optimal regularization parameter, occurs within each step of the iterative algorithm of Kozlov et al [12] and hence the computational cost of the iterative MFS-based algorithm is increased. To overcome this inconvenience and encouraged by the recent findings of Johansson and Marin [32], as well as similar results obtained for the Cauchy problem associated with the Poisson equation [55], the Laplace equation [56,57], and the Cauchy-Navier system of elasticity [58,59], we decided to employ the two relaxation procedures, as proposed and investigated using the BEM in [32], for the iterative MFS-based algorithm implemented by Marin [52] and study the influence of the relaxation parameter upon the rate of convergence of the modified method. The efficiency of these relaxation procedures is tested for Cauchy problems associated with the two-dimensional modified Helmholtz operator in simply and doubly connected domains with smooth or piecewise smooth boundaries.…”

Boundary element methods for boundary condition inverse problems in elasticity using PCGM and CGM regularization

“…A numerical iterative BEM for solving Cauchy problem in elasticity was developed by Ellabib et al The accuracy was improved by the use of automatic selection of relaxation parameter. It was concluded that the approach produced accurate, convergent and stable solution with respect to increasing the number of elements and decreasing the amount of noise [15]. Huang and Shih applied the conjugate gradient and regularization methods together with the boundary element method to estimate the unknown boundary displacement and traction conditions.…”