Optimal Control of Temperature in Fluid Flow Using the Second Order Adjoint Equation

The purpose of this paper is an optimal control problem of temperature using the Newton based method and the finite element method. This method is based on the first and second order adjoint technique allowing to obtained a better approximation to the Newton line search direction. The formulation is based on the optimal control theory in which the performance function is expressed by the computed and target temperatures. The optimal value can be obtained by minimizing the performance function. The gradient of the performance function is obtained by the first order adjoint equation. The Hessian vector product is calculated by the second order adjoint equation. The thermal fluid flow is described by the Boussinesq approximated incompressible Navier-Stokes equation and the energy equation is analyzed. The Newton based method is employed as a minimization technique. The BFGS, the DFP and the Broyden family method are applied to calculate the Hessian matrix which is used in the Newton based method. The traction given by the perturbation for the Lagrange multiplier is used in the BFGS method, the DFP and the Broyden family Method. Keyword : Finite Element Method, Newton Based Method, First Order Adjoint, Second Order Adjoint, Hessian V ector Product, BFGS Method, DFP Method, Broyden Family Method.