Mathematics

 Hierarchical Model (HiMod) reduction is a model reduction technique specifically developed for problems featuring a prevailing directionality, typical in fields such as hemodynamics, acoustic wave propagation, or the optimization of industrial pipelines and circuits. Although these problems exhibit a dominant flow or transport direction, they remain fully three-dimensional in nature, making full-order simulations often prohibitively expensive. The idea behind HiMod is to exploit the separation of variables principle: the solution is approximated by combining a standard Finite Element (FE) discretization along the main direction with a modal expansion in the transverse ones. This leads to a reduced system of coupled 1D problems that significantly lowers computational cost, while retaining a high degree of accuracy, as confirmed by a range of successful applications. In this presentation, the main principles behind the HiMod framework are introduced. Then, we focus on recent developments for HiMod application to the Stokes problem, which is crucial for fluid dynamics modeling. In particular, we present the well-posedness analysis for the HiMod formulation of the Stokes problem, along with a new family of modal basis functions derived from a dedicated Sturm–Liouville eigenvalue problem. Although the validation of these new bases is still ongoing, preliminary results are promising, highlighting the potential of HiMod as a versatile and efficient tool for future engineering applications.