Lacking Data Recovery via Partially Overdetermined Boundary Conditions in Linear Elasticity

Abstract

This work focuses on the sub-Cauchy problem for linear elasticity in two dimentional case. Solving such a problem may be formulated as follows: given the displacement and one component of the traction in a given part of the boundary of the elastic body, reconstruct the displacement field in all the domain. Author propose herein, an iterative method borrowed from the domain decomposition communauty to solve the sub-Cauchy problem. Numerical results highlight the efficiency of the proposed method.

Voids identification from partially overspecified boundary data

“…In order to numerically recover the shear stress, we resort to the same approach proposed in [8] that we briefly present herein. This part is concerned with the Steklov-Poincaré operator carried out to solve the sub-Cauchy problem (11) which is nothing more than a data completion problem.…”

“…Motivated by the recent results of [8] obtained to recover lacking boundary data via partially overdetermined boundary data and results of [5] obtained to identify cavities from overdetermined boundary data; both results obtained in linear elasticity, we are concerned in this work with a geometrical inverse problem related to the identification of cavities in mechanical structures from partially overdetermined boundary data.…”

“…The present research aims to develop an iterative method based on coupling the Steklov Poincaré operator and the shape gradient approach combined with the level set method through the minimization of an energy-like functional [4,5,6,8,23,22] to numerically solve the inverse problem. However, it should be noted that some results related to the detection of cracks [7,15] and obstacles [13] for the Laplace problem are achieved through iterative methods in the case of a Cauchy problem.…”

Convergence study and regularizing property of a modified Robin–Robin method for the Cauchy problem in linear elasticity

“…The quasi-reversibility method was also investigated [10,26]. We can refer also to the method of fundamental solution [11,42], variational Steklov-Poincaré theory [7,8], alternating iterative methods [13,37,38,46], optimal control formulation [12], regularization methods [6,45] and boundary element method [31,32].…”