Abstract. In this paper, we are interested in applying the reduced basis methodology (RBM) to steady-state natural convection problems. The latter has applications in many engineering domains and being able to apply the RBM would allow to gain huge computation savings when querying the model for many parameter evaluations. In this work, we focus on the order reduction of the modelin particular the handling of the non-linear terms, -as well as the design of the RBM computational framework and the requirements on high performance computing to treat 3D models using Feel++, a C++ open source library to solve partial differential equations. Numerical experiments are presented on 2D and 3D models.
IntroductionNowadays, in many application fields, engineering problems require accurate, reliable, and efficient evaluation of quantities of interest. Often, these quantities of interest depend on the solution of a parametrized partial differential equation where the -e.g. physical or geometrical -parameters are inputs of the model and the evaluation of quantities of interest are outputs -e.g. average values. In a real-time or many-query context, the reduced basis method (RBM) offers a rapid and reliable evaluation of the input-output relationship (see [PRV + 02, VPP03, VPRP03, PP04, QRM11, RHP07] for the methodology) for a large class of problems. In this paper, we are interested in studying the RBM applied to steady-state natural convection problems parametrized by the Grashof and Prandtl numbers, see also [VP05,Yan12]. Natural convection has applications in many engineering domains and being able to apply the RBM would allow to gain huge computation savings when querying the model the reduced model for many parameter evaluations. In this work, we focus on the order reduction of the model -in particular the handling of the non-linear terms, -as well as the design of the RBM computational framework and the requirements on high performance computing (HPC) to treat 3D models. Even though the model considered remains simple with respect to industrial applications, we tackle some of the main difficulties namely order reduction and computational costs. In this work, we underly the difficulties linked to the resolution of non-linear problems in a reduced space. We detail a projection technique that can be applied to any second order problem and that allows to solve it in the reduced space, without any projection onto the FE space. To the authors knowledge, previous works on the RB applied to non-linear steady Navier-Stokes equations concentrate on the choice of the basis and on error estimators, see [VP05,Yan12]. We propose here a reduced order method that can be applied to any non-linear problem and that does not depend on the choice of the basis. . Feel++ is a library to solve problems arising from partial differential equations (PDEs) with Galerkin methods, standard or generalized, continuous or discontinuous, from 1D to 3D, for low to high order approximations (including geometry). Among the many other Feel++ features, it pro...