Numerical methods for low‐order modeling of fluid flows based on POD

Abstract

This report explores some numerical alternatives that can be exploited to derive efficient low-order models of the Navier-Stokes equations. It is shown that an optimal solution sampling can be derived using appropriate norms of the Navier-Stokes residuals. Then the classical Galerkin approach is derived in the context of a residual minimization method that is similar to variational multiscale modeling. Finally, calibration techniques are reviewed and applied to the computation of unsteady aerodynamic forces. Examples pertaining to both non-actuated and actuated flows are shown.

Model Order Reduction in Fluid Dynamics: Challenges and Perspectives

“…In this case, snapshots are typically chosen iteratively by measuring the error of the current ROM at different trial points of the parameter space, then computing snapshots at the parameter point where the maximum error (estimate) is obtained and adding them to the ROM, like in the greedy-RB algorithm, first proposed in [52], and now standard in the parametric model reduction community. For sampling in time POD-greedy strategies have been proposed for linear evolution equations in [56], the viscous nonlinear Burgers' equation in [93], and the Navier-Stokes equations in [127]. In [73] the authors derived sensitivity equations to measure the effect of adding new snapshots in the POD basis and use them to find optimal locations for new snapshots that minimize the error between the POD-solution and the trajectory of the full-order system.…”

POD‐identification reduced order model of linear transport equations for control purposes

“…In industry, however, the FV method is often used, by commercial software and open‐source codes, for the spatial discretization of the governing equations that describe the physical fluid model, as the method is robust and preserves locally the conservation laws . A set of reduced basis functions, containing the essential dynamics of the full order system, is often created with the proper orthogonal decomposition (POD) method . In addition, POD is commonly applied in fluid dynamics literature for this purpose, as it can also be applied to nonlinear models .…”

“…These POD basis functions are obtained through solving an eigenvalue problem on snapshots which are generated by sampling the full order model (FOM) at several moments in time. For unsteady problems, POD is typically combined with the Galerkin projection where the full order system is projected onto the low‐dimensional subspace of POD modes and the difference with the snapshots is minimized to obtain a system of time‐dependent coefficients, the reduced order model (ROM). However, the main issue of this reduced basis method is that knowledge of the solver's discretization and solution algorithm is required in order to perform the Galerkin projection and could therefore not be used for most (commercial) software.…”

Evaluation of Aerodynamic Loads via Reduced-Order Methodology

“…Intrusive methods elaborate on the mathematics of the projection step to reformulate the original system of partial differential equations (PDEs) as a system of ordinary differential equations (ODEs), with the modal coefficients as unknowns [3][4][5][6]. Nonintrusive approaches instead adopt interpolation methods or similar variants to build a response surface for each of the coefficients, avoiding the solution of the reduced system of ODEs [7][8][9].…”