Variational formulation for Granular Contact Dynamics simulation via the Alternating Direction Method of Multipliers

The subject of this thesis is the development of a calculation software for the numerical analysis and the dynamic simulation of a granular flow. One of the major problems encountered in the dynamic analysis of the mechanical behaviour of a granular medium is the enormous computational time taken to reach the solution of the problem. Following studies that have verified the effectiveness of the implicit formulation proposed by the Granular Contact Dynamics approach, the idea of this thesis arises from the desire to apply the Alternating Direction Method of Multipliers for the optimization of the solution, a parallelizable algorithm already validated in similar contexts. The main part of the work consisted in the realization of the program using the Python programming language. During the process particular importance was given to computational optimization, and each part of the program has been designed to handle large scale problems. To generate the starting conditions we have implemented algorithms both for importing and for generating a model, and we have implemented methods for the introduction and management of the static and kinematic boundaries. As for the solution algorithm we reviewed the mathematical model of the GCD: the solution of the problem leads to the application of the principle of minimum of the total potential energy associated with the system. We introduced the augmented Lagrangian: its minimization, with respect to one of the primary variables and assuming the other unknowns as constants, constitutes the core of the ADMM. The software has been included within mechpy, an open source platform for the development of unconventional finite element formulations, and is able to manage both two-dimensional and three-dimensional models. The results are very promising: the output of the simulations has been compared with experimental results, and the noticeable correspondence validates the software functionality.