A discretize-then-map approach for the treatment of parameterized geometries in model order reduction

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