Analysis of Robust Asymptotic Methods In numerical Simulation in mechanics – ARAMIS

Numerical simulation of multiscale phenomena is an important challenge in industry. Performing such a simulation at a low computational cost, typically on a laptop and without spending too much time on mesh generation, is therefore a challenge for scientists. Our group aims at developing methods and academic codes to that end. The presence of defects modifies the physical behavior of the material. For instance, a concrete beam will be weakened by impacts and the maximum load it can bear is decreased. Usually, the modeling of such situations leads to a system of partial differential equations set on the real geometry of the object. It is necessary to refine the computational mesh refinement down to the defect length scale to obtain a good accuracy of the simulation in the situations involving different scales. This requirement can greatly increase the computational cost and complicate the meshing procedure especially in three-dimensional analysis. The aim of this project is to propose an alternative approach close to methods of extended finite elements: we take the defect into account through its impact on the solution by a localized enrichment by the space of functions used for the discretization of the boundary value problem. This adapted enrichment is obtained by an asymptotic analysis of the solution of the problem posed in the continuous frame. The computational strategy we want to develop is the following: numerical computation of the first corrector and enrichment of the finite element space by this corrector.