Different approaches to the development of reduced-order models for NS equations

Michel Bergmann 1 Andrea Ferrero 1 Angelo Iollo 1
1 MEMPHIS - Modeling Enablers for Multi-PHysics and InteractionS
IMB - Institut de Mathématiques de Bordeaux, Inria Bordeaux - Sud-Ouest
Abstract : In this contribution we explore some numerical alternatives to derive efficient and robust low-order models of the Navier–Stokes equations. The first considered approach is based on an hybrid CFD-ROM approach in which the ROM is used to define the boundary conditions of a CFD simulation. This makes possible to reduce significantly the size of the domain studied by the CFD solver. An alternative approach, based on residual minimization, is presented in the following. We start from the fact that classical Galerkin or Petrov-Galerkin approaches for ROM can be derived in the context of a residual minimization method similar to variational multi scale modelling , VMS [1]. Based on this, we introduce a residual minimization scheme that directly includes VMS stabilizing terms in the low-order model as proposed in [2], [3]. Here, however, the unknowns of the minimization problem are the union of the coefficients of a modal representation of the solution and of the physical unknowns at certain collocation points [4]. The modal representation is typically based on empirical eigenfunctions obtained bye proper-orthogonal decomposition, whereas the residual at collocation points are obtained by an adaptive discretization. Examples relative to model problems and moderately complex flows will be presented. [1] Hugues TJR, Feijoo G, Mazzei L, Quincy JB. The variational multiscale method—a paradigm for computational mechanics. Computer Methods in Applied Mechanics and Engineering 1998; 166:3–24. [2] Bergmann M, Bruneau C, Iollo A. Enablers for robust pod models. Journal of Computational Physics 2009; 228(2):516–538. [3] Weller J., Lombardi E., Bergmann, Iollo A. Numerical methods for low-order modeling of fluid flows based on POD Int. J. Numer. Meth. Fluids 2009. [4] Buffoni, M., Telib, H., & Iollo, A. (2009). Iterative methods for model reduction by domain decomposition. Computers & Fluids, 38(6), 1160–1167. http://doi.org/10.1016/j.compfluid.2008.11.008
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Michel Bergmann, Andrea Ferrero, Angelo Iollo. Different approaches to the development of reduced-order models for NS equations. ALOP Workshop: Reduced Order Models in Optimization , Sep 2016, Trier, Germany. ⟨hal-01405487⟩



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