The challenge of this PhD thesis is to implement adaptive mesh refinement methods for non-linear 3D solids mechanics adapted to parallel computers.

This research topic is proposed as part of the NumPEx (Digital for Exascale) Priority Research Programs and Equipment (PEPR). It is part of the Exa-MA (Methods and Algorithms for Exascale) Targeted Project. The PhD will take place at CEA Cadarache, within the Institute for Research on Nuclear Energy Systems for Low-Carbon Energy Production (IRESNE), as part of the PLEIADES software platform development team, which specializes in fuel behavior simulation and multi-scale numerical methods.

In finite element simulation, adaptive mesh refinement (AMR) has become an essential tool for performing accurate calculations with a controlled number of unknowns. The phenomena to be taken into account, particularly in solids mechanics, are often complex and non-linear: contact between deformable solids, viscoplastic behaviour, cracking, etc. Furthermore, these phenomena require intrinsically 3D modelling. Thus, the number of unknowns to be taken into account requires the use of parallel solvers. One of the current computational challenges is therefore to combine adaptive mesh refinement methods and nonlinear solid mechanics for deployment on parallel computers.

The first research topic of this PhD thesis concerns the development of a local mesh refinement method (of block-structured type) for non-linear mechanics, with dynamic mesh adaptation. We will therefore focus on projection operators to obtain an accurate dynamic AMR solution during the evolution of refined areas.

The other area of research will focus on the effective treatment of contact between deformable solids in a parallel environment. This will involve extending previous work, which was limited to matching contact meshes, to the case of arbitrary contact geometries (node-to-surface algorithm).

The preferred development environment will be the MFEM tool. Finite element management and dynamic re-evaluation of adaptive meshes require assessing (and probably improving) the efficiency of the data structures involved. Large 3D calculations will be performed on national supercomputers using thousands of computing cores.
his will ensure that the solutions implemented can be scaled up to tens of thousands of cores.