Parametric vector flows for registration fields in bounded domains with applications to nonlinear interpolation of shock-dominated flows

We present a registration procedure for parametric model order reduction (MOR) in two- and three-dimensional bounded domains. In the MOR framework, registration methods exploit solution snapshots to identify a parametric coordinate transformation that improves the approximation of the solution set through linear subspaces. For each training parameter, optimization-based (or variational) registration methods minimize a target function that measures the alignment of the coherent structures of interest (e.g., shocks, shear layers, cracks) for different parameter values, over a family of bijections of the computational domain \Omega. We consider diffeomorphisms that are vector flows of given velocity fields with vanishing normal component on \partial \Omega; we rely on a sensor to extract appropriate point clouds from the solution snapshots and we develop an expectation–maximization procedure to simultaneously solve the point cloud matching problem and to determine the velocity (and thus the bijection); finally, we combine our registration method with the nonlinear interpolation technique of Iollo and Taddei (2022) to perform accurate interpolations of fluid dynamic fields in the presence of shocks. Numerical results for a two-dimensional inviscid transonic flow past symmetric and asymmetric NACA airfoils and a three-dimensional viscous transonic flow past an ONERA M6 wing illustrate the many elements of the methodology and demonstrate the effectiveness of nonlinear interpolation for shock-dominated fields.