Missions

The proposed project is part of a research programme in solid mechanics at the interface with other disciplines. It focuses mainly on multi-scale methods in mechanics, i.e. the theory of homogenization and its numerical implementation in scientific computing tools for engineering. Positioned at the crossroads of several disciplines, including applied mathematics and electromagnetism, our aim is to take an original and innovative look at a number of problems affecting multi-scale methods now widely used in solid mechanics. By building bridges with the fields of photonics, plasmonics and the mathematical study of integral operators, we aim at proposing a spectral theory that will make it possible to improve the modelling and performance of numerical homogenization methods in mechanics. This will enable us to make progress in understanding and predicting the behaviour of complex heterogeneous materials such as composites, which are high-performance materials now widely used in fields such as aeronautics and space, sustainable construction and energy production.

In solid mechanics, the numerical homogenization methods based on the Fast Fourier Transform (FFT) from microstructure images constitute competitive approaches for tackling small-scale material problems. These approaches rely on a volume integral formulation of the equations governing the local problem in a representative elementary volume V. This paved the way for numerous developments and numerical tools widely used today in computational homogenization. The formulation used is based on the so-called Lippmann-Schwinger integral equation, an equation found in numerous fields such as the study of diffraction in quantum mechanics, wave propagation in electromagnetism but also in classical mechanics (acoustic or elastic). This equation governs the deformation solution in V for a given macroscopic loading. In the linear case, it is expressed using the Green's operator associated with a homogeneous reference medium and the fluctuation of the properties of the material considered around this reference. The introduction of this comparison medium is a necessity for the implementation of FFT-based algorithms. However, such a rewriting of the local mechanics problem introduces a material fluctuation, a term that induces a fictitious change of sign.

Activités

In this context, this project focuses on the integral operator involved in the equation above and on its properties. The proposed research programme is divided into two main axes:
- Lippmann-Schwinger equation and boundary integral operators. The first axis of research aims to gain a better understanding of the exotic phenomena generated by the introduction of the reference medium, which leads to the emergence of surface resonances, called plasmons in the field of electromagnetism, at the interfaces between the various material constituents. Proving the existence of such resonances in relation to the properties of the integral operator considered, characterizing them theoretically and demonstrating them numerically will be an important first step in this project. This type of phenomenon is well known in the field of electromagnetism, albeit in a completely different context (at the interface of a metallic material and a dielectric material) and related to sign changing parameters. The link will be made on the basis of the study of the volume integral formulation used in mechanics and some known boundary integral operators, such as the so-called Neumann-Poincaré operator.
- Scientific computing: The second area of research will focus on scientific computing tools. The FFT numerical simulation code for homogenization is the CRAFT code https://lma-software-craft.cnrs.fr. In connection with the previous axis, it is expected that this type of tool, dedicated to solving the equation above from images, will be compared with the specific tool Inti.jl https://github.com/IntegralEquations/Inti.jl dedicated to the discretisation and high-order approximation of boundary integral operators in complex 2D and 3D geometries.

Compétences

The prospective student is expected to hold a Ph.D. at the time of appointment with a strong background in applied mathematics, scientific computing, theoretical or computational mechanics, spectral theory.

Contexte de travail

The team formed for this project is multidisciplinary and involves three laboratories jointly affiliated to CNRS and Aix-Marseille University. The members of this project are:
Cédric Bellis and Hervé Moulinec at the Laboratory of Mechanics and Acoustics (LMA), Florian Monteghetti at the Institute of Mathematics of Marseille, Maxence Cassier and Alice Vanel at Fresnel Institute. The successful candidate will join the LMA and interact will all members of the project.