α-NEM and model reduction: Two new and powerful numerical techniques to describe flows involving short fibers suspensions - Chinesta, Francisco; Cueto, Elias; Ryckelynck, David; Ammar, Amine (European Journal of Computational Mechanics, 2005)
Numerical modeling of non-Newtonian flows typically involves the coupling between the equations of motion characterized by an elliptic character, and the fluid constitutive equation, which is an advection equation linked to the fluid history. In this work we propose a coupling between the natural element method which provides the capabilities of Lagrangian models to describe the flow front tracking as well as to treat the convection terms related to the fluid microstructure evolution – without the mesh quality requirements characteristics of the standard finite elements method – with a new approximation of the Fokker-Planck equation. This approximation is efficient and accurate, and is based on the use of an adaptive model reduction which couples the proper orthogonal decomposition (Karhunen-Loève) with an approximation basis enrichment based on the use of the Krylov subspaces, for describing the microstructure evolution.