RheoSuNN project brings together researchers in the domains of mathematics, numerical analysis, HPC and mechanics, interested in numerical simulations of dense suspensions of rigid particles embedded in a Stokes fluid. The scientific program of this project is based on the following objective: achieve a numerical rheological study in order to understand the interphase stress contribution to the total fluid stress. The last year of the project is dedicated to this task. It consists in a challenging study, for which no result is available (neither experimental, nor theoretical, nor numerical). The key point to make a breakthrough in the understanding of the interphase stress is to compute the velocity and pressure fields in the whole fluid domain for a dense suspension, to model carefully the multi-body lubrication and contact interactions and their feedback on the flow. From a computational point of view, it leads to suspension simulations with density around 50%, containing up to about one million of suspended particles. The corresponding mesh size for the direct fluid solver is around 500 x 2 000 x 10 000. Despite the wide range of available techniques, there is a great need of designing new mathematical models and numerical methods to achieve this numerical study. The project is based on two existing codes developed by members of the project: a fluid/particle solver (CAFES) and a contact solver (SCoPI). To reach the target, the project first two years are dedicated to the two codes coupling, together with their optimisation and the design of new numerical methods to be able to take into account stiff close interactions and their effects on the whole flow, for any shape of particles. This leads to highly coupled fluid/particle problems to be solved in an implicit way, in presence of singularities. This problem need fine numerical expertise to be treated in an efficient way. We plan to deliver a code including the implicit methods developed in the project. In order to run simulations for real physical configurations, it is crucial to develop a numerical code which is optimised and able to run with good scalability on the national and European HPC infrastructures proposed by GENCI and PRACE. This code will allow to reach more precise and dense simulations than the existing ones and will enable the study of open questions about the rheology of suspension. The code we develop allows to deal with non-spherical particles (more precisely with superellipsoids) while taking lubrication into account with retroaction on the flow, which is not possible with the other existing codes. As a consequence, even though we focus in this project on the interphase stress, it opens the opportunity for numerous other interesting and original numerical rheological studies of non-spherical particles suspensions (like rods, fibres or micro-swimmers). Moreover, taking in a very fine way lubrication phenomenon into account, it can be a reference code, providing precise results to study new rheological phenomenon or to compute reference solutions to be compared to other methods. It can also be a first step towards direct numerical simulation of suspensions of particles embedded in non-Newtonian fluids or of active suspensions which are two very active domains of research.