We present a general approach for the treatment of parameterized geometries in projection-based model order reduction. During the offline stage, given (i) a family of parameterized domains ${ Omega_{mu}: mu in mathcal{P} } subset mathbb{R}^D$ where $mu in mathcal{P} subset mathbb{R}^P$ denotes a vector of parameters, (ii) a parameterized mapping ${Phi}_{mu}$ between a reference domain $Omega$ and the parameter-dependent domain $Omega_{mu}$, and (iii) a finite element triangulation of $Omega$, we resort to an empirical quadrature procedure to select a subset of the elements of the grid. During the online stage, we first use the mapping to move the nodes of the selected elements and then we use standard element-wise residual evaluation routines to evaluate the residual and possibly its Jacobian. We discuss how to devise an online-efficient reduced-order model and we discuss the differences with the more standard map-then-discretize approach (e.g., Rozza, Huynh, Patera, ACME, 2007); in particular, we show how the discretize-then-map framework greatly simplifies the implementation of the reduced-order model. We apply our approach to a two-dimensional potential flow problem past a parameterized airfoil, and to the two-dimensional RANS simulations of the flow past the Ahmed body.
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