Optimal Boundary Control of the Incompressible Fluid Problems in Rotation form

Liu Liu

doi:10.3844/jmssp.2010.174.182

Abstract

Problem Statement: We consider the optimal boundary control of the linearized Navier-Stokes problem. Both the Stokes problem and Oseen problem in rotation form are considered. Approach: We use the Mark and Cell (MAC) discretization method to discretize the optimization problem with linear constraints including the Stokes problem and the Oseen problem in rotation from. Then Reduced Hessian methods are to solve the problem. Results: Numerical experimental results are performed for the boundary optimization problem with the Stokes constraints and Oseen constraints. All the computed solutions and the desired solutions are compared. Conclusions: The proposed reduced Hessian methods have a high accuracy obtaining the optimal boundary conditioning for the Stokes problem and the Oseen problem in rotation form.

Highlights

We study the incompressible viscous fluid problems with the following form:∂u − υ∆u + (u.∇)u + ∇p = f in Ω× (0,Γ] (1) ∂ t∇.u = 0 in Ω× (0,Γ] (2)Bu = g in ∂Ω× (0,Γ] (3)u(x,0) = u0 in Ω (4)Equation 1-4 is known as the Navier-Stokes equations
The resulting optimization problem written in constrained form is the following: Which leads to the equations: min 1 || Qx − d ||2 + 1 β || m ||2m = −PTA−1QTd
We have developed an algorithm to find out the optimal boundary conditions of linearized rotation form of the Navier-Stokes equations reduced

Summary

INTRODUCTION

We study the incompressible viscous fluid problems with the following form:. Equation 1-4 is known as the Navier-Stokes equations. A mathematical description of the problem is as follows: The air flow in the room can be governed by the Navier-Stokes equations subject to certain boundary conditions. & Stat., 6 (2): 174-182, 2010 behavior of the wind flow in the room which correspond the velocity u in the Navier-Stokes equation, which can be controlled by the condition imposed on the boundary. In this project, we are concerned with the boundary control of a flow process governed by the linear zed steady-state Navier-Stokes equations in rotation form. The optimal boundary problem governing the linearized rotation form of the Navier-Stokes equation is the following system: min 1 || Qx − d ||2 + 1 β || m ||2. Based on 17-19, we proceed to eliminate x, λ and solve for m:

MATERIALS AND METHODS

RESULTS

DISCUSSION

CONCLUSIONS