Optimal control in fluid flow problems with POD applications to FEM solutions

Download
2018
Evcin, Cansu
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 elements for the velocity and the linear elements 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. 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. Besides, providing a user-friendly framework, computational costs are minimized.

Suggestions

FEM solutions of magnetohydrodynamic and biomagnetic fluid flows in channels
Türk, Önder; Tezer Sezgin, Münevver; Department of Scientific Computing (2014)
In this thesis, solutions to steady and unsteady flow problems of incompressible viscous fluids are obtained numerically. In computational aspects, the primary focus is on the finite element analysis, however, spectral collocation and boundary element methods are also employed. The two-dimensional Navier-Stokes (N-S) equations in stream function-vorticity form are solved by using both finite element method (FEM) and Chebyshev spectral collocation method (CSCM). The accuracy of the FEM and CSCM methodologies...
Local improvements to reduced-order approximations of PDE-constrained optimization problems
Akman, Tuğba; Karasözen, Bülent; Department of Scientific Computing (2015)
Optimal control problems (OCPs) governed partial differential equations (PDEs) arise in environmental control problems, optimal control of fluid flow, petroleum reservoir simulation, laser surface hardening of steel, parameter estimation and in many other applications. Although the OCPs governed by elliptic and parabolic problems are investigated theoretically and numerically in several papers, the studies concerning the optimal control of evolutionary diffusion-convection-reaction (DCR) equation and Burger...
Implicit lattice boltzmann method for laminar/turbulent flows
Çevik, Fatih; Albayrak, Kahraman; Department of Mechanical Engineering (2016)
Lattice Boltzmann Method is an alternative computational method for fluid physics problems. The development of the method started in the late 1980s and early 1990s. Various numerical schemes like stream and collide, finite difference, finite element and finite volume schemes are used to solve the discrete Lattice Boltzmann Equation. Almost all of the numerical schemes in the literature are explicit schemes to exploit the natural features of the discrete Lattice Boltzmann Equation like parallelism and easy c...
Direct Calculation of Entropy Generation by Solving Reynolds-Averaged Entropy Transport Equation in an Air-Cooled Turbine Cascade
Orhan, Omer Emre; Uzol, Oğuz (2012-06-15)
This paper presents an implementation of directly solving Reynolds-Averaged Entropy Transport equation as a part of the CFD solution to predict entropy generation rates in a two-dimensional turbine blade stator section. The Reynolds Averaged Entropy Transport and the necessary modeling. equations are implemented to a commercial CFD solver as a User Defined Scalar (UDS). The results are compared with those obtained by post-processing the temperature and velocity fields obtained by solving full Navier-Stokes ...
Incompressible flow simulations using least squares spectral element method on adaptively refined triangular grids
Akdağ, Osman; Sert, Cüneyt; Department of Mechanical Engineering (2012)
The main purpose of this study is to develop a flow solver that employs triangular grids to solve two-dimensional, viscous, laminar, steady, incompressible flows. The flow solver is based on Least Squares Spectral Element Method (LSSEM). It has p-type adaptive mesh refinement/coarsening capability and supports p-type nonconforming element interfaces. To validate the developed flow solver several benchmark problems are studied and successful results are obtained. The performances of two different triangular ...
Citation Formats
C. Evcin, “Optimal control in fluid flow problems with POD applications to FEM solutions,” Ph.D. - Doctoral Program, Middle East Technical University, 2018.