Hyper-reduced order models for parametrized unsteady Navier-Stokes equations on domains with variable shape
Monday 4th February 2019
Dal Santo, N.; Manzoni, A.
In this work we set up a new, general and computationally efficient way to tackle parametrized fluid flows modeled through unsteady Navier-Stokes equations defined on domains with variable shape, when relying on the reduced basis method. We easily describe a domain by flexible boundary parametrizations, and generate domain (and mesh) deformations by means of a solid extension, obtained by solving an harmonic extension or a linear elasticity problem. The proposed procedure is built over a two-stages reduction: (i) first, we construct a reduced basis approximation for the mesh motion problem, irrespectively of the fluid flow problem we focus on; (ii) then, we generate a reduced basis approximation of the unsteady Navier-Stokes problem, relying on finite element snapshots evaluated over a set of reduced deformed configuration, and approximating both velocity and pressure fields simultaneously. To deal with unavoidable nonaffine parametric dependencies arising in both the mesh motion and the state problem, we apply a matrix version of the discrete empirical interpolation method, allowing to treat geometrical deformations in a non-intrusive, efficient and purely algebraic way. The same strategy is used to per- form hyper-reduction of nonlinear terms. To assess the numerical performances of the proposed technique, we address the solution of parametrized fluid flows where the parameters describe both the shape of the domain, and relevant physical features. Complex flow patterns such as the ones appearing in a patient specific carotid bifurcation are accurately approximated, as well as derived quantities of potential clinical interest.