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Hyper-reduced order models for parametrized unsteady Navier-Stokes equations on domains with variable shape

  • Niccolò Dal Santo
  • Andrea ManzoniEmail author
Article
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Part of the following topical collections:
  1. Model reduction of parametrized Systems

Abstract

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 a harmonic extension or a linear elasticity problem. The proposed procedure is built over a two-stage reduction: (i) first, we construct a reduced basis approximation for the mesh motion problem, irrespective 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 treating a wide range of geometrical deformations in an efficient and purely algebraic way. The same strategy is used to perform 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.

Keywords

Reduced basis method Parametrized PDEs Hyper-reduction techniques Navier-Stokes equations Computational fluid dynamics Varying domains 

Mathematics Subject Classification (2010)

65M60 65N12 76M10 

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Notes

Acknowledgments

We acknowledge the Swiss National Supercomputing Centre (CSCS) for providing us the CPU resources under project ID s796.

Funding information

The research of N. Dal Santo has been supported by the Swiss State Secretariat for Education, Research and Innovation (SERI), project No. C14.0068, in the framework of the COST action number TD1307.

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Authors and Affiliations

  1. 1.École Polytechnique Fédérale de Lausanne (EPFL)School of MathematicsLausanneSwitzerland
  2. 2.Department of Mathematics, MOX-Laboratory for Modeling and Scientific ComputingPolitecnico di MilanoMilanoItaly

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