Identification of Energies from Observations of Evolutions
Final Report Abstract
The second objective of the Project P7 of the first phase of SPP 1962 was related to the learning of energies in the context of quasi-static evolutions, focusing in particular on applications to phase-field models of fracture. The reliable numerical realization of the direct problem of simulating a (fracture) evolution is indispensable for solving the inverse problem of learning the energy. In the last decades the use of phase-field models in computational fracture mechanics has been constantly increasing and has found many interesting applications, such as hydraulic fracturing or fracture growth on thin structures. The advantage in using such models lies in their ability of handling the complexity of evolving cracks even in rather involved geometrical settings. Although existing numerical schemes provide qualitatively reasonable predictions of crack evolutions, they may still lack provable crucial physical features, such as damage irreversibility and energy conservation, and raise interesting questions regarding their relationship with the established theory of rate-independent processes. In this perspective, our project aims at more deeply investigating the convergence in space and time of the aforementioned numerical methods and providing a complete energetic description of the behavior of all the possible time-continuous limit evolutions. Our analysis will face three main issues: (i) the involved energies are not convex, (ii) the solutions of the related evolution problems exhibit time discontinuities, which correspond to catastrophic fracture growth, and (iii) the evolutions fulfill inequality constraints promoting irreversibility (a form of entropic solution). Therefore, it will be crucial to detect the behavior of solutions at discontinuity points, as well as to determine specific criteria according to which the observed evolutions experience such singularities.
