Mathematical analysis and simulation of fixed point formulation of Cauchy problem in linear elasticity

Abdellatif Ellabib , Abdeljalil Nachaoui (1) , Abdessamad Ousaadane

1 LMJL - Laboratoire de Mathématiques Jean Leray

In this work, an inverse problem in linear elasticity is considered, it is about reconstructing the unknown boundary conditions on a part of the boundary based on the other boundaries. A methodology based on the domain decomposition operating mode is opted by constructing a Steklov–Poincaré kind’s operator. This allows us to reformulate our inverse problem into a fixed point one involving a Steklov kind’s operator, the existence of the fixed point problem is shown using the topological degree of Leray–Schauder. The proposed approach offers the opportunity to exploit domain decomposition methods for solving this inverse problem. Finally, a numerical study of this problem using the boundary element method is presented. The obtained numerical results show the efficiency of the proposed approach.

Mots clés: Cauchy inverse problem Elasticity equation Steklov–Poincaré operator Fixed point theory Degree of Leray–Schauder Boundary elements method

Domaines: Analyse numérique [math.NA]; Optimisation et contrôle [math.OC]

Abdellatif Ellabib, Abdeljalil Nachaoui, Abdessamad Ousaadane. Mathematical analysis and simulation of fixed point formulation of Cauchy problem in linear elasticity. Mathematics and Computers in Simulation, 2021, 187, pp.231-247. ⟨10.1016/j.matcom.2021.02.020⟩. ⟨hal-03174129⟩