Optimal Control for Burgers Flow Using the Discontinuous Galerkin Method

The coupling of accurate computational fluid dynamics analysis with optimal control theory has the potential to advance active flow-control for complex fluid systems. In this paper, an optimal control framework for the viscous Burgers equation is constructed based on the Discontinuous Galerkin Method (DGM). A DGM discretization has several potential advantages for optimization studies including high formal accuracy and convenient local refinement. The Burgers equation is solved numerically using a DGM spectral-element discretization for spatial terms and fourthorder Runge-Kutta time integration and the control is updated using a nonlinear Conjugate Gradient method. Initial results for both distributed and boundary control (Dirichlet and Neumann) are presented using a continuous-adjoint formulation. In the future, this DGM formulation will be applied to Euler and Navier–Stokes problems to develop active flow-control strategies for aeroacoustic applications. Introduction The numerical solution of optimal control problems governed by the unsteady compressible Navier-Stokes equations is a challenging problem that requires careful mathematical formulation, accurate state solution, efficient gradient computation, and convergent optimization algorithms. As a simplified model of the Navier–Stokes (NS) equation, the one-dimensional Burgers equation represents many of the properties of NS equations, such as nonlinear convection and viscous diffusion leading to shock waves and boundary layers. Given this, the viscous Burgers equation has received significant attention [1, 2] and recent research has focused on the control of Burgers flow as a model for control of Navier-Stokes flows [3, 4]. To meet the challenges associated with optimal control of unsteady flow, we have developed a new computational framework based on the discontinuous Galerkin method (DGM) that allows for spectral accuracy on unstructured grids with the ability to use local hp-refinement. These capabilities will be of particular importance for large-scale optimal control for ∗Ph.D. Candidate, Department of Mechanical Engineering and Materials Science, [email protected] †Assistant Professor, Mechanical Engineering and Materials Science, [email protected], member AIAA. Copyright c © 2003 by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. complex fluid flows. This paper presents our preliminary efforts in this direction by applying DGM to the solution of optimal control problems for flows governed by the viscous Burgers equation. Since the number of control variables is large in the problems that we target, an adjoint equation is utilized to efficiently evaluate the gradient of our objective functional with-respect-to the control. In general, there are two approaches to adjoint-based gradient evaluation: the optimize-then-discretize approach and discretize-thenoptimize approach. One of the goals of our research is to evaluate and compare these two approaches for formulating and solving optimal control problems using DGM. In this paper, we focus on the optimize-then-discretize approach by presenting a discussion of the problem formulation, implementation, and preliminary results. Problem Formulation Governing equations The Burgers equation is given by ∂u ∂t + 1 2 ∂u2 ∂x − ν ∂ 2u ∂x2 = f +Φ (1) with boundary conditions u(0, t) = φL u,x(L, t) = φR (2) and initial condition u(x, 0) = u0(x) (3) in which Φ is the distributed control, and φL and φR are the boundary controls, with the spatial domain Ω = [0, L]. Here, we set the source term, f = 0. Objective functional For the problems in this paper, the objective functional is defined as J = 2 ∫ t0+T