Non-cooperative finite element games

This work proposes an approach to using the non-cooperative game theory for solving finite element problems. For this, the concept of generalized Nash equilibrium is applied to finite elements implying that each element is treated as a non-cooperative "player" in a larger finite element "game". The aim of the approach is for all players to reach a Nash equilibrium ensuring that the entire discretization is at a minimum with respect to the decision variables considered. The approach is numerically demonstrated by investigating a nonlinear elasticity problem formulated as a finite element game. It is shown that the approach matches analytical solutions in linear elasticity and is convergent to a prescribed precision for two-player nonlinear problems. (C) 2021 IMACS. Published by Elsevier B.V. All rights reserved.