• Tonnoir, Antoine

Abstract

This thesis is motivated by the numerical simulation of Non Destructive Testing by ultrasonic waves. It aims at designing a method to compute by Finite Element (EF) the diffraction of elastic waves in time-harmonic regime by a bounded defect in an anisotropic plate. The goal is to take into account an infinite plate and to restrict the FE calculations to a bounded area. This point is difficult due to the anisotropy and, in particular, methods such as perfectly matched layers fail.In this thesis, we have mainly considered two-dimensional cases that enabled us to implement the main ingredients of a method designed for the three-dimensional case of the plate. The first part deals with the diffractionproblem in an infinite strip. The classical approach consists in writing transparent conditions by matching on a boundary the displacement and the axial stress using a modal expansion in the safe part of the plate, and the FE representation in the perturbed area. We have shown the interest of imposing these matching conditions on two separated boundaries, by introducing an overlap between the modal domain and the FE domain. Thus, we can take advantage of the bi-orthogonality relations valid for general anisotropy, and also improve the rate of convergence of iterative methods of resolution. In the second part, that represents the main part of the thesis, we discuss the diffraction problem in an anisotropic medium infinite in the two directions.The key idea is that we can express the solution (via the Fourier transform) in a half-plane given its trace on the boundary. Therefore, the approach consists in coupling several analytical representationsof the solution in half-planes surrounding the defect (at least 3) with the FE representation. The difficulty is to ensure that all these representations match, in particular in the infinite intersections of the half-planes. It leads to a formulation which couples, via integral operators, the solution in a bounded domain including the defect, and its traces on the edge of the half-planes. The approximation releases a truncation and a discretization both in space and Fourier variables.For each of these two parts, the methods have been implemented and validated with a C++ code developed during the thesis, first in the scalar acoustic case, and then in the elastic case. 

Topics

• impedance spectroscopy
• simulation
• anisotropic
• defect
• ultrasonic
• two-dimensional