Cansu Evcin, Ph.D.

Department of Scientific Computing

September 2018

*Supervisor*: Ömür Uğur (Institute of Applied Mathematics, Middle East Technical University, Ankara)

### Abstract

This study investigates the numerical solutions of optimal control problems constrained by the partial differential equations (PDEs) of laminar fluid flows and heat transfer with the model order reduction (MOR). This is achieved by the three objectives of the thesis: obtaining accurate solutions, controlling the dynamics of the fluid and reducing the computational cost.

Fluids exposed to an external magnetic field and the heat transfer are governed by the Magnetohydrodynamics (MHD) and energy equations. Considering an advanced physical systems with a temperature dependent viscosity such as chemical reactors, their control has significant importance and becomes one of the major subject of this thesis. Furthermore, power-law fluid flow, which describes the dynamics for non-Newtonian fluids such as polymer solutions, is considered as an optimal control problem for the characterization of these fluids as shear-thinning or shear-thickening.

Simulations of solutions of the fluid flows and heat transfer equations are carried out by the finite element method (FEM). First of all, FEM solution of the Navier- Stokes (N-S) equations with an exact solution is obtained for the validation of the method using quadratic-linear elements for the velocity-pressure formulation. On the other hand, considering the coupled non-linearity of the MHD flow and heat transfer equations with temperature dependent viscosity, quadratic elements are used for both velocity and temperature. Moreover, for the power-law fluid flows, due to the fact that equations are decoupled and the temperature equation is linear, quadratic element for the velocity and the linear element for the temperature are considered.

Solutions of the optimal control problems are attained by employing the adjoint method within the discretize-then-optimize approach. While the control of N-S equations are studied with a distributed force function, control of the MHD flow and power-law fluid flow is attained by using the problem parameters as control variables.

Computational cost and data storage problems arise with implementation of the optimal control strategies. Thus, computing resources are optimized by performing MOR using the Proper Orthogonal Decomposition (POD) method to obtain a reduced order model (ROM). The system dynamics is transferred by POD bases using the sample solutions (snapshots) for various values of the parameters. Setting up a user-friendly framework for the development of the ROM is also provided to help reduce the discretization procedure of the system of equations in ROM.

Consequently, the dynamics of the fluid flows and heat transfer are well identified by applying FEM and their control are successfully achieved by the optimal control using the parameters of the problems as control variables. As providing a user-friendly framework, computational costs are minimized.

*Keywords*: optimal control, finite element method, proper orthogonal decomposition, magnetohydrodynamics