Stabilized Reduced Basis Approximation of Incompressible Three-Dimensional Navier-Stokes Equations in Parametrized Deformed Domains

In this work we are interested in the numerical solution of the steady incompressible Navier-Stokes equations for fluid flow in pipes with varying curvatures and cross-sections. We intend to compute a reduced basis approximation of the solution, employing the geometry as a parameter in the reduced basis method. This has previously been done in a spectral element $P_{{ \mathcal{N}}} - P_{{ \mathcal{N}}-2}$ setting in two dimensions for the steady Stokes equations. To compute the necessary basis-functions in the reduced basis method, we propose to use a stabilized P 1−P 1 finite element method for solving the Navier-Stokes equations on different geometries. By employing the same stabilization in the reduced basis approximation, we avoid having to enrich the velocity basis in order to satisfy the inf-sup condition. This reduces the complexity of the reduced basis method for the Navier-Stokes problem, while keeping its good approximation properties. We prove the well posedness of the reduced problem and present numerical results for selected parameter dependent three dimensional pipe

Stabilized reduced basis approximation of incompressible three-dimensional Navier–Stokes equations in parametrized deformed domains

In this work we are interested in the numerical solution of the steady incompressible Navier-Stokes equations for fluid flow in pipes with varying curvatures and cross-sections. We intend to compute a reduced basis approximation of the solution, employing the geometry as a parameter in the reduced basis method. This has previously been done in a spectral element $P_{N} - P_{N-2}$ setting in two dimensions for the steady Stokes equations. To compute the necessary basis-functions in the reduced basis method, we propose to use a stabilized $P_1 - P_1$ finite element method for solving the Navier-Stokes equations on different geometries. By employing the same stabilization in the reduced basis approximation, we avoid having to enrich the velocity basis in order to satisfy the inf-sup condition. This reduces the complexity of the reduced basis method for the Navier-Stokes problem, while keeping its good approximation properties. We prove the well posedness of the reduced problem and present numerical results for selected parameter dependent three dimensional pipes

Model Order Reduction in Fluid Dynamics: Challenges and Perspectives

This chapter reviews techniques of model reduction of fluid dynamics systems. Fluid systems are known to be difficult to reduce efficiently due to several reasons. First of all, they exhibit strong nonlinearities — which are mainly related either to nonlinear convection terms and/or some geometric variability — that often cannot be treated by simple linearization. Additional difficulties arise when attempting model reduction of unsteady flows, especially when long-term transient behavior needs to be accurately predicted using reduced order models and more complex features, such as turbulence or multiphysics phenomena, have to be taken into consideration. We first discuss some general principles that apply to many parametric model order reduction problems, then we apply them on steady and unsteady viscous flows modelled by the incompressible Navier-Stokes equations. We address questions of inf-sup stability, certification through error estimation, computational issues and — in the unsteady case — long-time stability of the reduced model. Moreover, we provide an extensive list of literature references

Reduced Basis Methods for the Solution of Parametrized PDEs in Repetitive and Complex Networks with Application to CFD

The objective of this work is to develop a numerical framework to perform rapid and reliable simulations for solving parametric problems in domains represented by networks and to extend the classical reduced basis method. Aimed at this scope, we propose two original methodological approaches for the approximation of partial differential equations in domains made up by repetitive parametrized geometries where topological features are recurrent: the reduced basis hybrid method (RBHM) and the reduced basis-domain decomposition-finite element (RDF) method. The common paradigm of these methods is the consideration that the blocks composing the computational domain are topologically similar to a few reference shapes. On the latter, we compute representative solutions, corresponding to the same governing partial differential equations, but for different values of some parameters of interest and representing, for example, the deformation of the blocks. A new desired solution for a new deformed domain is recovered as projection on the reduced spaces built by the previously precomputed solutions and the continuity of the solution across subdomain interfaces is guaranteed by suitable coupling conditions. The different choices for the reduced spaces and coupling conditions adopted characterize one method with respect to the other one. The geometrical parametrization of the considered domains, by transfinite maps, induces non-affine parameter dependence: an empirical interpolation technique is used to recover an approximate affine parameter dependence and a sub–sequent offline/online decomposition of the reduced basis procedure. This computational decomposition yields a considerable reduction of the problem complexity. Results computed on some combinations of 2D and 3D geometries, representing cardiovascular networks, show the flexibility and the advantages of the proposed methods in terms of reduced computational costs and complexities. The computational time with these new approaches is, in general, much reduced with respect to a classical finite element method on the whole domain but also only marginally slower than a classical reduced basis approach on the whole domain. However, these approaches decrease drastically the offline time to pre-compute the reduced basis by splitting the total number of parameters characterizing the problem into smaller subsets for each reference block, moreover they allow to considerably increase the geometrical flexibility and versatility