Grant Details

In this project we will work on the modeling and the simulation of large, complex systems described in terms of eigenvalue problems associated with partial differential equations. We aim at reducing the complexity of the problem in the spirit of Reduced Order Methods (ROMs). ROMs rely on projecting the Full Order Models (FOMs) onto smaller subspaces. An expensive offline preprocessing phase is followed by a fast online phase capable of delivering a solution in real-time. The analysis of ROMs for eigenvalue problems is still at a preliminary stage and the available results show the potential and the difficulties of different approaches. Following our exploratory GRG2020 grant, we will develop new techniques and apply them to solid mechanics. We start from linear elasticity and the challenges associated with problems in mixed form and with the imposition of the symmetry of the stress at discrete level. We then move to nonlinear solid mechanics where we consider the vibration of hyperelastic structures. The challenges related to this problem start from the FOM and from the deep understanding of the nonlinear stress-strain relationship. We plan to investigate these aspects in detail and to explore the use of ROM for reducing the costs of the numerical simulations. Further critical challenges come from the study of heterogeneous materials. The heterogeneities can be modeled either as a parametric eigenvalue problem with deterministic parameters of with stochastic parameters. It is crucial in this case to be able to model correctly multiscale data beyond periodicity and scale separation. Another possible approach for modeling the heterogeneities consists in immersed methods which can be used in presence of biphasic systems with embedded interfaces such as vascularized or biological tissues or fiber reinforced materials. These problems are characterized by some of the intrinsic dimensions of the immersed phase that is orders of magnitude smaller than the system scale, making the resolution of the full geometrical complexity unfeasible. The final phase of the project will be devoted to the study of metameterials. Our investigations will deal with challenges related to the well-posedness of the problem, depending on the parameters describing the materials.