Parameterized reduced-order-modeling based on matrix manifolds

Applying dimension reduction techniques to experimental and numerical simulation results is a well-known approach for extracting the characteristic features of flow fields and developing reduced-order models (ROMs). Especially, developing a ROM based on the Galerkin projection-based method using proper orthogonal decomposition (POD) has been intensively studied. However, the classical ROM often fails to reconstruct the flow field for parameters (e.g., Reynolds number, Mach number, and airfoil shape) different from the training point. As a result, developing a robust parametrized ROM is one of the major challenges. In this study, an interpolation method on the Grassmann manifold was developed to estimate the subspace spanned by appropriate POD modes for a given parameter. The flow fields around a rotating cylinder were reconstructed using the developed ROM framework for a wide range of parameters, including Reynolds number and rotational speed. While the classical ROM is unable to reconstruct the flow field for parameters not included in the training dataset, the developed ROM was able to reproduce the flow field while maintaining a low subspace dimension. Furthermore, we analyze the dependency of the flow field with respect to the parameter from the perspective of the geometrical characteristics of the Grassmann manifold.