S. S. Ravindran

A reduced-order approach for optimal control of fluids using proper orthogonal decomposition

In this article, a reduced-order modeling approach, suitable for active control of fluid dynamical systems, based on proper orthogonal decomposition (POD) is presented. The rationale behind the reduced-order modeling is that numerical simulation of Navier–Stokes equations is still too costly for the purpose of optimization and control of unsteady flows. The possibility of obtaining reduced-order models that reduce the computational complexity associated with the Navier–Stokes equations is examined while capturing the essential dynamics by using the POD. The POD allows the extraction of a reduced set of basis functions, perhaps just a few, from a computational or experimental database through an eigenvalue analysis. The solution is then obtained as a linear combination of this reduced set of basis functions by means of Galerkin projection. This makes it attractive for optimal control and estimation of systems governed by partial differential equations (PDEs). It is used here in active control of fluid flows governed by the Navier–Stokes equations. In particular, flow over a backward-facing step is considered. Reduced-order models/low-dimensional dynamical models for this system are obtained using POD basis functions (global) from the finite element discretizations of the Navier–Stokes equations. Their effectiveness in flow control applications is shown on a recirculation control problem using blowing on the channel boundary. Implementational issues are discussed and numerical experiments are presented. Copyright

Reduced-Order Adaptive Controllers for Fluid Flows Using POD

This article presents a reduced-order adaptive controller design for fluid flows. Frequently, reduced-order models are derived from low-order bases computed by applying proper orthogonal decomposition (POD) on an a priori ensemble of data of the Navier–Stokes model. This reduced-order model is then used to derive a reduced-order controller. The approach discussed here differ from these approaches. It uses an adaptive procedure that improves the reduced-order model by successively updating the ensemble of data. The idea is to begin with an ensemble to form a reduced-order control problem. The resulting control is then applied back to the Navier–Stokes model to generate a new ensemble. This new ensemble then replaces the previous ensemble to derive a new reduced-order model. This iteration is repeated until convergence is achieved. The adaptive reduced-order controllers effectiveness in flow control applications is shown on a recirculation control problem in channel flow using blowing (actuation) on the boundary. Optimal placement for actuators is explored. Numerical implementations and results are provided illustrating the various issues discussed.

Optimal Control of Thermally Convected Fluid Flows

We examine the optimal control of stationary thermally convected fluid flows from the theoretical and numerical point of view. We use thermal convection as control mechanism; that is, control is effected through the temperature on part of the boundary. Control problems are formulated as constrained minimization problems. Existence of optimal control is given and a first-order necessary condition of optimality from which optimal solutions can be obtained is established. We develop numerical methods to solve the necessary condition of optimality and present computational results for control of cavity- and channel-type flows showing the feasibility of the proposed approach.

Adaptive Reduced-Order Controllers for a Thermal Flow System Using Proper Orthogonal Decomposition

An adaptive reduced-order controller design is presented for flow control using proper orthogonal decomposition (POD). In reduced-order controller design, the idea is to start with an ensemble of data obtained from numerical simulation of the underlying partial differential equations (PDEs). POD is then used to obtain a reduced set of basis functions which is then used to derive a reduced-order model of the PDEs via Galerkin projection. This reduced-order model allows us to derive a reduced-order controller. However, it is not clear, a priori, what is the best way to obtain an ensemble of data that would give basis functions that represent the influence of the control action on the system. In this paper we explore an adaptive procedure for reduced-order controller design that improves the reduced-order model by successively updating the ensemble of data during the optimization iterations. We illustrate this method on a control problem in thermal flow system modeled by a thermally coupled Navier--Stokes equations. Numerical results are presented for a vorticity regulation problem in fluid flows using boundary temperature as control mechanism. Through our numerical experiments we demonstrate the feasibility and applicability of the adaptive reduced-order controllers.

A Reduced Basis Method for Control Problems Governed by PDEs

This article presents a reduced basis method for constructing a reduced order system for control problems governed by nonlinear partial differential equations. The major advantage of the reduced basis method over others based on finite element, finite difference or spectral method is that it may capture the essential property of solutions with very few basis elements. The feasibility of this method is demonstrated for boundary control problems modeled by the incompressible Navier-Stokes and related equations with the boundary temperature control and boundary electromagnetic control in channel flows.

A Penalized Neumann Control Approach for Solving an Optimal Dirichlet Control Problem for the Navier--Stokes Equations

We introduce a penalized Neumann boundary control approach for solving an optimal Dirichlet boundary control problem associated with the two- or three-dimensional steady-state Navier--Stokes equations. We prove the convergence of the solutions of the penalized Neumann control problem, the suboptimality of the limit, and the optimality of the limit under further restrictions on the data. We describe the numerical algorithm for solving the penalized Neumann control problem and report some numerical results.

Reduced Basis Method for Optimal Control of Unsteady Viscous Flows

Abstract In this article we discuss the reduced basis method (RBM) for optimal control of unsteady viscous flows. RBM is a reduction method in which one can achieve the versatility of the finite element method or another for that matter and gain significant reduction in the number of degrees of freedom. The essential idea in this method is to define a reduced order subspace spanned by few basis elements and then obtain the solution via a Galerkin projection. We present several ways to define this subspace. Feasibility of the approach is demonstrated on two boundary control problems in cavity and wall bounded channel flows. Control action is effected through boundary surface movement on part of the solid wall. Application of RBM to the control problems leads to finite dimensional optimal control problems which are solved using Newtons method. Through computational experiments we demonstrate the feasibility and applicability of the reduced basis method for control of unsteady viscous flows.

Numerical Approximation of Optimal Flow Control Problems by a Penalty Method: Error Estimates and Numerical Results

The purpose of this paper is to present numerically convenient approaches to solve optimal Dirichlet control problems governed by the steady Navier--Stokes equations. We will examine a penalized Neumann control approach for solving Dirichlet control problems from numerical and computational points of view. The control is affected by the suction or injection of fluid through the boundary or by boundary surface movements in the tangential direction. The control objective is to minimize the vorticity in the flow or to drive the velocity field to a desired one. We develop sequential quadratic programming methods to solve these optimal control problems. The effectiveness of the optimal control techniques in flow controls and the feasibility of the proposed penalized Neumann control approaches for flow control problems are demonstrated by numerical experiments for a viscous, incompressible fluid flow in a two-dimensional channel and in a cavity geometry.

Control of flow separation over a forward-facing step by model reduction

Abstract Design and implementation of reduced-order optimal controller for flow separation are investigated. The reduced-order controller design is based on proper orthogonal decomposition (POD) and Galerkin projection. Detailed finite element simulations are performed, and POD is applied to the resulting data to extract the most energetic eigenmodes. These global eigenmodes are then used in conjunction with the Galerkin projection to obtain a reduced-order (low-dimensional) model. Reduced-order models are not only attractive for real-time control computation but also crucial for detailed stability and bifurcation analysis. We investigate the design of optimal controller for flow separation in a channel with this model where actuation is performed on a small part of the boundary. Two different types of surface actuation are considered––tangential blowing and suction through a single slot. Best locations for actuation and adaptive procedure for reduced-order model are explored. The proposed approaches are evaluated by investigating the control of flow separation over a forward-facing step channel. Our methods are found to be efficient and fast, and our methods demonstrate a significant reduction in computational time and feasibility. Dramatic separation delays are observed on all cases. It is found that the tangential blowing is more efficient in mitigating flow separation and reducing wake spread.

Numerical Solution of Optimal Distributed Control Problems for Incompressible Flows

Abstract We study the numerical solution of optimal control problems associated with the two-dimensional viscous incompressible flows which are governed by the Navier-Stokes equations. Although the techniques apply to more general settings, the presentation is confined to the objectives of minimizing the vorticity in the steady-state case with distributed controls and tracking the velocity field in the nonstationary case with piecewise distributed controls. In the steady-state case, we develop a systematic way to use the Lagrange multiplier rules to derive an optimality system of equations from which an optimal solution can be computed; finite element methods are used to find approximate solutions for the optimality system of equations. In the time-dependent case, a piecewise-in-time optimal control approach is proposed and the fully discrete approximation algorithm for solving the piecewise optimal control problem is defined. Numrical results are presented for both the steady-state and time-dependent opt...