Reduced numerical modeling of flows involving liquid-crystalline polymers

Kinetic theory models involving the Fokker–Planck equation can be accurately discretized using a mesh support (Finite Elements, Finite Differences, Finite Volumes, Spectral Techniques, …). However, these techniques involve a high number of approximation functions. In the finite element framework, widely used in complex flow simulations, each approximation function has only local support and is related to a node that defines the associated degree of freedom. In the technique proposed here, a reduced approximation basis is constructed. The new shape functions have extended support and are defined in the whole domain in an appropriate manner (the most characteristic functions related to the model solution). Thus, the number of degrees of freedom involved in the solution of the Fokker–Planck equation is very significantly reduced. The construction of those new approximation functions is done with an ‘a priori’ approach, which combines a basis reduction (using the Karhunen–Loève decomposition) with a basis enrichment based on the use of some Krylov subspaces. This paper analyzes the application of model reduction to the simulation of non-linear kinetic theory models involving complex behaviors, such as those coming from stability analysis, complex geometries and coupled models. We apply our model reduction approach to the Doi's classical constitutive equation for viscoelasticity of liquid-crystalline polymer.