An empirical interpolation method for two-dimensional vector fields and vector measurements

This article presents two extensions of the empirical interpolation method (EIM) designed to deal with vector interpolation problems. These reduced-order modeling techniques are aimed at exploiting pointwise (vector) measurements to obtain the unknown field reconstruction and they are preferred to other, more efficient, techniques as the proper orthogonal decomposition (POD) because of their intrinsic capability to identify measurement positions and to perform field reconstruction. The “EIM-roto” method implements rotation matrix coefficients and should be intended as a composition of rotations and dilatations of the vector basis functions. The “EIM-orto” implements diagonal matrices coefficients and can be intended as the interpolation, component by component, of the unknown vector field, projected on a fixed reference system. The two techniques are tested over the lid-driven cavity benchmark, in laminar conditions. The results obtained on this study case highlight how the “EIM-orto” interpolation does not allow a reliable reconstructions, while the “EIM-roto” interpolation allows reconstruction performances close to the POD ones (here used as reference method). In particular, the worst reconstruction error, that is, the maximum L² − norm of the residuals, decreases exponentially, reaching 5% with 25 basis functions. This result can be consider satisfactory, considering the nature of the problem.