Multiphysics Problems via the Cell Method: The Role of Tonti Diagrams

A common structure of several physical laws emerges naturally from the Tonti diagrams of different physical theories so that topological operators can be built only once and used to assemble the stiffness matrices and the coupling terms of the various problems. This process is known in algebraic topology as coboundary process and is presented as the theoretical background for solving multiphysics problems. The main contribution of this paper is to show that the discrete setting provided by Tonti diagrams not only allows to define discrete counterparts of the differential operators and constituive matrices, but that the same matrices can be used to set up the coupling terms in multiphysics problem formulations. The proposed method is compared with a commercial code on an electro-thermo-mechanical benchmark.