An iterative method for the Cauchy problem in linear elasticity with fading regularization effect

Abstract

In this paper, an iterative method for solving the Cauchy problem in linear elasticity is introduced. This problem consists in recovering missing data (displacements and forces) on some parts of a domain boundary from the knowledge of overspecified data (displacements and forces) on the remaining parts. The algorithm reads as a least square fitting of the given data, with a regularization term whose effect fades as the iterations go on. So the algorithm converges to the solution of the Cauchy problem. Numerical simulations using the finite element method highlight the algorithm's efficiency, accuracy, robustness to noisy data as well as its ability to deblur noisy data.

Inverse Cauchy method with conformal mapping: Application to latent flatness defect detection during rolling process

“…For example, Delvare et al (2002) proposed an iterative boundary-element method for an inverse Cauchy problem related to the Laplace's equation. Marin and Lesnic (2004) proposed an inverse method based on fundamental solutions for the Cauchy problem in two dimensions related to isotropic elasticity, Andrieux and Baranger (2008) proposed an energy error-based method for 3D inverse Cauchy problems (both displacement and surface tractions are known for a part of the boundary and the remaining part should be determined), and Delvare et al (2010) also proposed an iterative 3D inverse method for Cauchy problems. Liu (2008) developed an inverse method for the Laplace equation using overdetermined data on a part of a circle and recovering data on the remaining part.…”

An Invariant Method of Fundamental Solutions for the Cauchy Problem in Two-Dimensional Isotropic Linear Elasticity

“…Comino et al [6] proposed the alternating iterative method [13] to solve the Cauchy problem in two dimensional anisotropic elasticity, and the boundary element method (BEM) has been used for the numerical implementation. Delvare et al [7] gave a least square fitting method to solve the Cauchy problem by solving a sequence of optimization problems under equality constraints. Durand et al [8] gave an iterative method for solving axisymmetric Cauchy problems in linear elasticity based the finite element method.…”

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

“…An iterative method for solving the Cauchy problem in linear elasticity was introduced. The algorithm read as a least square fitting of the given data, with a regularization term whose effect faded as the iterations went on [26]. An iterative method for solving the axisymmetric Cauchy problem in linear elasticity was presented.…”