A Reduced-Order Method for Simulation and Control of Fluid Flows

Abstract

This article presents a reduced-order modeling approach for simulation and control of viscous incompressible flows. The reduced-order models suitable for control and which capture the essential physics are developed using the reduced-basis method. The so-called Lagrange approach is used to define reduced bases and the basis functions in this approach are obtained from the numerical solutions. The feasibility of this method for flow control is demonstrated on boundary control problems in closed cavity and in wall-bounded channel flows. Control action is effected through boundary surface movement on a part of the solid wall. Our formulation of the reduced-order method applied to flow control problems leads to a constrained minimization problem and is solved by applying Newton-like methods to the necessary conditions of optimality. Through our computational experiments we demonstrate the feasibility and applicability of the reduced-order method for simulation and control of fluid flows. c 1998 Academic PressKey Words: reduced-basis method; reduced-order modeling; Navier-Stokes equations; finite element; optimal control.

Certified Reduced Basis Methods for Parametrized Partial Differential Equations

“…Furthermore, a number of different techniques for the identification of different reduced basis approximation spaces, exemplified by Taylor and Lagrange expansions [125] and more recently also Hermite [70] expansions, have emerged [79]. Further early work focused on different applications and specific classes of equations, e.g., fluid dynamics and the incompressible Navier-Stokes equations [54, 69,70,71,123].…”

Computational Reduction for Parametrized PDEs: Strategies and Applications

“…Early applications to problems arising in computational fluid dynamics are described in [24,43], while recent applications dealing with Stokes and Navier-Stokes equations are presented e.g. in [14,34,44,50,54].…”

“…In particular, we consider the flow around a symmetric airfoil profile parametrized w.r.t. thickness µ 1 ∈ [4,24] and the angle of attack µ 2 ∈ [0.01, π/4]; the profile is rotated according to µ 2 , while the external boundaries remain fixed and the inflow velocity is parallel to the x axis (see Fig. 1).…”

Optimal snapshot location for computing POD basis functions

“…For the case of linear dynamics, POD-model reduction is closely related to the well-known balanced truncation method [14,19]. An important alternative to POD model reduction is given by reduced basis approximations, we refer to [7,10,18,20] and the references given there. In [7,20] a reduced model is constructed for a parameter dependent family of large scale problems by an iterative procedure that adds new basis variables on the basis of a greedy algorithm.…”