Numerical modeling of highly nonlinear phenomena in heterogeneous materials and domains
Résumé
Recent research activities on numerical modeling of highly nonlinear phenomena in heterogeneous materials and domains are presented. The two studied phenomena are single-phase or two-phase flows where the heterogeneity is due to obstacles of a very small size compared to the simulation domain, and damage and fracture of structures where it is the mesostructure and/or the microstructure of the material that is heterogeneous.
In both cases, numerical modeling relies on a representation of the heterogeneity and an additional interface, which is the liquid/gas interface in two-phase flows, and the crack in fracture problems. These simulations are computationally demanding because these phenomena are nonlinear and also because there are interfaces at a very fine scale. This is the reason why it is suitable to accelerate simulations by model order reduction and multiscale modeling.
The originality of this work mainly consists in the development of advanced numerical methods. Heterogeneity discretization is achieved either by finite element mesh generation and adaption or voxel meshes and the fast Fourier transform based numerical method. Interfaces are modeled in flows using level-set functions and a quadratic finite element interpolation, while cracks are modeled in fracture problems using a phase-field approach. Model order reduction methods are developed by borrowing techniques from data science and deep learning, but some fundamental principles from mechanics are still enforced. Flow and fracture problems that are rarely dealt with in the literature are tackled thanks to multiscale modeling.