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Presentation

The direct and optimal control solution of the laminar, fully developed, steady MHD flow
of an incompressible, electrically conducting fluid in a duct is considered together with the
heat transfer. The flow is driven by a constant pressure gradient and an external uniform
magnetic field. The fluid viscosity is either temperature dependent, varying exponentially, or it depends on the flow in the case of power law fluid; and the viscous and Joule dissipations are taken into consideration. The coupled nonlinear set of momentum and energy equations are solved by using Finite Element Method with the implementation of the Newton’s method for nonlinearity. In this respect, direct FEM solutions are obtained for various values of the problem parameters to ensure the sound structure of the underlying scheme. The FEM results obtained in this study are not only in good agreement with, but also extends, the results in [1]. The aim of this study is to investigate the problem of controlling the steady flow by using the physically significant parameters of the problem as control variables: Hartmann number (Ha), Brinkmann number (Br), Hall parameter (m) and viscosity parameter (B) in the case of temperature dependent viscosity. For the case of power law fluid the control parameters are Ha, Br and the flow index (n). The control problem is solved by the discretize-then-optimize approach [2] with a gradient based algorithm. Starting with an initial estimate the optimization loop to calculate new estimates for optimal solution repeated until the norm of the gradient of the reduced cost function is less than a predefined tolerance. Control variables are considered as single and pairwise as well. It is observed that controls with multiple control variables require relatively more number of iterations than the one with a single control parameter. The most costly one is observed as the case with the pair (m, B) since they have contrary effects on the fluid for the temperature dependent viscosity case.
Numerical results ensure that the proposed control approach is effective at driving the flow to prescribed velocity profiles as well as isolines.
1. M. E. Sayed-Ahmed. Numerical solution of power law fluids flow and heat transfer with a magnetic field in a rectangular duct. International Communications in Heat and Mass Transfer, 33 (2006), 1165-1176.
2. M. Hinze and R. Pinnau and M. Ulbrich and S. Ulbrich. Optimization with PDE Constraints. Springer, 2009.

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The combined effect of viscous and Ohmic dissipations on unsteady, laminar magneto convection fully developed flow in a vertical rectangular duct considering the effects of heat source/sink is investigated. Finite element method based on Galerkin weighted residual approach is used to solve two dimensional governing momentum and energy equations for unsteady, magneto convection flow in a vertical rectangular duct. The investigations are conducted for the effects of various flow parameters such as buoyancy parameter N, Hartmann number M, aspect ratio A, circuit parameter E and heat source/sink parameter λ. The results indicate that the flow pattern and the temperature field are significantly dependent on the above mentioned parameters. It is shown that buoyancy parameter, Hartmann number and aspect ratio parameter increase both the velocity and temperature for open circuit (E ≠ 0) but decrease for short circuit (E = 0). © 2015, Journal of Applied Science and Engineering. All rights reserved.

The issue of minimizing turbulence in an evolutionary Navier-Stokes flow is addressed from the point of view of optimal control. We derive theoretical results for various physical situations: distributed control, Bénard-type problems with boundary control, and flow in a channel. For each case that we consider, our results include the formulation of the problem as an optimal control problem and proof of the existence of an optimal control (which is not expected to be unique). Finally, we describe a numerical algorithm based on the gradient method for the corresponding cost function. For readers who are not interested in the mathematical details and the mathematical justifications, a nontechnical description of our results is included in Section 5.

We examine the optimal control of stationary thermally convected uid ows from the the-oretical and numerical point of view. We use thermal convection as control mechanism, that is, control is eeected through the temperature on part of the boundary. Control problems are formulated as constrained minimization problem. Existence of optimal control is given and a rst order necessary conditions of opti-mality from which optimal solutions can be obtained is established. We develop numerical methods to solve the necessary conditions of optimality and present computational results for control of cavity and channel type ows showing the feasibility of the proposed approach.

An analysis is performed to study the MHD free convection flow in a vertical rectangular duct for laminar and fully developed
regime taking into consideration the effects of Ohmic heating and viscous dissipation. Numerical solutions are found using
finite difference method of second-order accuracy. The effects of various physical parameters such as Hartmann number, aspect
ratio, buoyancy parameter and circuit parameter are presented graphically. It is found that as Hartmann number, buoyancy parameter
and aspect ratio increase, the upward and downward flow rates are increased for open circuit but decrease for short circuit.

In this paper we demonstrate a new technique for deriving discrete adjoint
and tangent linear models of finite element models. The technique is
significantly more efficient and automatic than standard algorithmic
differentiation techniques. The approach relies on a high-level symbolic
representation of the forward problem. In contrast to developing a model
directly in Fortran or C++, high-level systems allow the developer to express
the variational problems to be solved in near-mathematical notation. As such,
these systems have a key advantage: since the mathematical structure of the
problem is preserved, they are more amenable to automated analysis and
manipulation. The framework introduced here is implemented in a freely
available software package named dolfin-adjoint, based on the FEniCS Project.
Our approach to automated adjoint derivation relies on run-time annotation of
the temporal structure of the model, and employs the FEniCS finite element form
compiler to automatically generate the low-level code for the derived models.
The approach requires only trivial changes to a large class of forward models,
including complicated time-dependent nonlinear models. The adjoint model
automatically employs optimal checkpointing schemes to mitigate storage
requirements for nonlinear models, without any user management or intervention.
Furthermore, both the tangent linear and adjoint models naturally work in
parallel, without any need to differentiate through calls to MPI or to parse
OpenMP directives. The generality, applicability and efficiency of the approach
are demonstrated with examples from a wide range of scientific applications.

The paper presents a new affine invariant theory on asymptotic mesh independence of Newton's method for discretized nonlinear operator equations. Compared to earlier attempts, the new approach is both much simpler and more intuitive from the algorithmic point of view. The theory is exemplified at finite element methods for elliptic PDE problems.

This paper studies the effect of variable viscosity on the transient Couette flow of dusty fluid with heat transfer between parallel plates. The fluid is acted upon by a constant pressure gradient and an external uniform magnetic field is applied perpendicular to the plates. The parallel plates are assumed to be porous and subjected to a uniform suction from above and injection from below. The upper plate is moving with a uniform velocity while the lower is kept stationary. The governing nonlinear partial differential equations are solved numerically and some important effects for the variable viscosity and the uniform magnetic field on the transient flow and heat transfer of both the fluid and dust particles are indicated.

The design of efficient fins for reducing cost, space, materials and the energy consumption for heat removal and refrigeration applications is a great challenge. In this paper, consideration is given to the constructal (optimal) design of multi-scale annular fins that are attached to a pin fin. The geometrical scales of the assembly are relaxed to reach the maximal heat transfer removed by the assembly of the fins, subject to the space and materials constraints. Based on a one-dimensional model, analytical solution is performed to deliver the amount of heat transfer density. The direct search method and the Genetic Algorithm (GA) are used to optimize the geometric configuration of the assembly. The optimization results indicate that the increment in the number of geometrical variables of the assembly remarkably enhances the thermal performance of the assembly, however, it imposes excess complexity in the optimization process. The design guidelines for the design of multi-scale fins provided after optimizations, can be useful in the design of variant cooling devices in practical engineering applications.

Keeping the excess temperature of heat generating mediums and circuitry devices below an allowable level is a great challenge. To facilitate the heat transfer process, it is significant to improve and optimize the geometric structure of the cooling tools. The present paper deals with the shape optimization of a tree-shaped inverted fin ('cavity') that is penetrated into a heat generating body and extracts the generated heat from the hot piece to a cold environment. The objective is to obtain the optimal geometry of the tree-shaped cavity that minimizes the peak (maximum) temperature of the medium subject to the volume (space) constraint. A numerical solution is performed to deliver the temperature field within the medium and the direct search method is used to reach the optimal situation. The optimization results indicate that the final (optimized) tree-shaped configuration performs the best over the existing single-root cavities introduced in the open literature. For example, the maximum excess temperature calculated in the final tree-shaped cavity with four branches was found about 50% lower compared to the best configuration investigated in the open literature.

p>This thesis is concerned with the automation of solving optimisation problems constrained by partial differential equations (PDEs). Gradient-based optimisation algorithms are the key to solve optimisation problems of practical interest. The required derivatives can be efficiently computed with the adjoint approach. However, current methods for the development of adjoint models often require a significant amount of effort and expertise, in particular for non-linear time-dependent problems.
This work presents a new high-level reinterpretation of algorithmic differentiation to develop adjoint models. This reinterpretation considers the discrete system as a sequence of equation solves. Applying this approach to a general finite-element framework results in an automatic and robust way of deriving and solving adjoint models. This drastically reduces the development effort compared to traditional methods.
Based on this result, a new framework for rapidly defining and solving optimisation problems constrained by PDEs is developed. The user specifies the discrete optimisation problem in a compact high-level language that resembles the mathematical structure of the underlying system. All remaining steps, including parameter updates, PDE solves and derivative computations, are performed without user intervention. The framework can be applied to a wide range of governing PDEs, and interfaces to various gradient-free and gradient-based optimisation algorithms.
The capabilities of this framework are demonstrated through the application to two PDE-constrained optimisation problems. The first is concerned with the optimal layout of turbines in tidal stream farms; this optimisation problem is one of the main challenges facing the marine renewable energy industry. The second application applies data assimilation to reconstruct the profile of tsunami waves based on inundation observations. This provides the first step towards the general reconstruction of tsunami signals from satellite information.</p

This study proposes the dual reciprocity boundary element (DRBEM) solution for full magnetohydrodynamics (MHD) equations in a lid-driven square cavity. MHD equations are coupled with the heat transfer equation by means of the Boussinesq approximation. Induced magnetic field is also taken into consideration. The governing equations in terms of stream function, temperature, induced magnetic field components, and vorticity are solved employing DRBEM in space together with the implicit backward Euler formula for the time derivatives. The use of DRBEM with linear boundary elements which is a boundary discretization method enables one to obtain small sized linear systems. This makes the whole procedure computationally efficient and cheap. The results are depicted with respect to varying physical parameters such as Prandtl (0.005 ≤ Pr ≤ 1), Reynolds (100 ≤ Re ≤ 2500), magnetic Reynolds (1 ≤ Rem ≤ 100), Hartmann (10 ≤ Ha ≤ 100) and Rayleigh (10 ≤ Ra ≤ 106) numbers for discussing the effect of each parameter on the flow and temperature behaviors of the fluid. It is found that an increase in Ha slows down the fluid motion and heat transfer becomes conductive. Centered square blockage causes secondary flows on its left and right even for small Re. Strong temperature gradients occur around the blockage and near the moving lid for increasing values of Ra.

The dual reciprocity boundary element method (DRBEM) and the differential quadrature method (DQM) are applied to solve the 2D, unsteady natural convection flow in enclosures under an externally applied magnetic field. Vorticity transport and energy equations are transformed to modified Helmholtz equations by utilising forward difference with relaxation parameters for the time derivatives, and approximating also Laplacian terms at two consecutive time levels. Thus, the need of another time integration scheme and very small time increment is eliminated. Inhomogeneities in modified Helmholtz equations are approximated with two types of radial basis functions. Solutions are obtained with both DRBEM and DQM for Ra and Ha values up to 10<sup align="right"> 6 </sup> and 300, respectively, and compared. DRBEM and DQM give almost the same accuracy, but DQM uses considerably small number of grid points resulting with less computational work.

Purpose
– The purpose of the paper is to obtain finite element method (FEM) solution of steady, laminar, natural convection flow in inclined enclosures in the presence of an oblique magnetic field. The momentum equations include the magnetic effect, and the induced magnetic field due to the motion of the electrically conducting fluid is neglected. Quadratic triangular elements are used to ensure accurate approximation for second order derivatives of stream function appearing in the vorticity equation.
Design/methodology/approach
– Governing equations in terms of stream function and vorticity are solved by FEM using quadratic triangular elements. Vorticity boundary conditions are obtained through Taylor series expansion of stream function equation by using more interior stream function values to improve the accuracy. Isothermally heated or cooled and/or adiabatic conditions for the temperature are imposed. Results are obtained for Rayleigh number values and Hartmann number values up to 1000000 and 100, respectively.
Findings
– It is observed that streamlines form a thin boundary layer close to the heated walls as Ha increases. The same effect is seen in the vorticity contours, and isotherms are not affected much. As Ra increases streamlines are deformed moving from the heated walls through cooled walls. Vorticity starts to develop boundary layers close to heated and adjacent walls. Isotherms are pushed towards the sinusoidally heated wall whereas in the case of linearly heated left and bottom walls they expand towards cooled part of the cavity as Ra increases.
Originality/value
– The application of FEM with quadratic elements for solving natural convection flow problem under the effect of a magnetic field is new in the sense that the results are obtained for large values of Rayleigh and Hartmann numbers.

An optimal control problem for the equations governing the stationary problem of magnetohydrodynamics (MHD) is considered. Control mechanisms by external and injected currents and magnetic fields are treated. An optimal control problem is formulated. First order necessary and second order sufficient conditions are developed. An operator splitting scheme for the numerical solution of the MHD state equations is analyzed.

The unsteady magnetohydrodynamic flow of a dusty fluid and heat transfer between parallel plates in which the electrically conducting fluid has temperature-dependent viscosity is studied. Both the fluid and the dust particles are governed by the coupled set of momentum and energy equations. The Chebyshev spectral method in space and implicit backward difference in time procedure is presented, introducing physically Navier-slip conditions for both the fluid and dust particle velocities. The Hartmann number, viscosity parameter, and Navier-slip parameter influences on the flow and temperature are simulated.

In the present paper the laminar fully developed MHD flow and heat transfer through a rectangular duct of a viscous incompressible electrically conducting fluid is studied. A constant pressure gradient and an external uniform magnetic field are applied and the Hall effect is taken into consideration. The fluid viscosity is assumed to be temperature dependent with the assumption of constant wall heat flux axially and constant wall temperature peripherally. A numerical solution for the governing non-linear partial differential equations is obtained. The effect of the Hall term and the variable viscosity on the velocity and temperature fields is examined.

The steady laminar flow and heat transfer of an incompressible, electrically conducting, power law non-Newtonian fluids in a rectangular duct are studied in the presence of an external uniform magnetic field. The momentum and energy equations are solved iteratively using a finite difference method. Two cases of the thermal boundary conditions are considered; (1) T thermal boundary condition “constant temperature at the wall” and (2) H2 thermal boundary condition “constant heat flux at the wall”. The viscous and Joule dissipations are taken into consideration in the energy equation. A numerical solution for the governing partial differential equations is developed and the influence of the magnetic field on the velocity distribution, the friction factor and the average Nusselt number are discussed.

In the present paper, a plate and fin heat exchanger is considered and air, as an ideal gas, is defined in both sides of the heat exchanger as the working fluid. Several geometric variables within the logical constraints are considered as optimization parameters. Two different objective functions including the total rate of heat transfer and the total annual cost of the system are defined. Since mentioned objectives are conflicting, no single solution can well-satisfy both objective functions simultaneously. In other words, any attempt to increase the value of the total rate of heat transfer leads to the higher total cost of the system which is certainly undesirable. Therefore, multi-objective optimization using genetic algorithm is utilized in order to achieve a set of optimal solutions, each of which is a trade-off between objectives and can satisfy both objective functions in an appropriate level. The main advantage of this work is providing a set of optimal solutions each of which can be selected by the designer based on the project’s limits and the available investment. A sensitivity analysis is also presented in order to investigate the effect of some geometric parameters on each objective functions.Highlights► We have optimized geometric parameters of a plate and frame heat exchanger. ► Total rate of heat transfer and the total annual cost are considered as objectives. ► Mentioned objectives are conflicting. ► Multi-objective genetic algorithm is utilized to optimize the system. ► Set of optimal solutions is generated each of which is a trade-off between objectives.

This paper deals with some optimal control problems associated with the equations of steady-state, incompressible magnetohydrodynamics. These problems have direct applications to nuclear reactor technology, magnetic propulsion devices, and design of electromagnetic pumps. These problems are first put into an appropriate mathematical formulation. Then the existence of optimal solutions is proved. The use of Lagrange multiplier techniques is justified and an optimality system of equations is derived. The theory is applied to an example.

This paper studies the effect of variable viscosity on the transient flow of dusty fluid with heat transfer. The fluid is acted upon by a constant pressure gradient, and an external uniform magnetic field is applied perpendieular to the plates. The governing nonlinear partial differential equations are solved numerically, and some important effects for the variable viscosity and the uniform magnetic field on the transient flow and heat transfer of both the fluid and dust particles are indicated.

For flow inside a four-to-one contraction domain, we minimize the vortex that occurs in the corner region by controlling the heat flux along the corner boundary. The problem of matching a desired temperature along the outflow boundary is also considered. The energy equation is coupled with the mass, momentum, and constitutive equations through the assumption that viscosity depends on temperature. The latter three equations are a non-isothermal version of the three-field Stokes–Oldroyd model, formulated to have the same dependent variable set as the equations governing viscoelastic flow. The state and adjoint equations are solved using the finite element method. Previous efforts in optimal control of fluid flows assume a temperature-dependent Newtonian viscosity when describing the model equations, but make the simplifying assumption of a constant Newtonian viscosity when carrying out computations. This assumption is not made in the current work.

Analysis and discretization of an optimal control problem for the time-periodic MHD equations

- M D Gunzburger
- C Trencha

M.D. Gunzburger, C. Trencha, Analysis and discretization of an optimal control problem for the time-periodic MHD equations, Math. Anal. Appl. 308
(2005) 440-446.

Analysis of Optimal Control Problems for the Incompressible MHD Equations and Implementation in a Finite Element Multiphysics Code

- G Bornia

G. Bornia, Analysis of Optimal Control Problems for the Incompressible MHD Equations and Implementation in a Finite Element Multiphysics
Code (Ph.D. thesis), Alma Mater Studiorum -University of Bologna, 2012.

Introducing a ψ-shaped cavity for cooling a heat generating medium

- M R Hajmohammadi

M.R. Hajmohammadi, Introducing a ψ-shaped cavity for cooling a heat generating medium, Int. J. Therm. Sci. 121 (2017) 204-212.

- L Dragos

L. Dragos, Magnetofluid Dynamics, Abacus Press, 1975.

Automated Solution of Differential Equations by the Finite Element Method

- A Logg
- K A Mardal
- G Wells

A. Logg, K.A. Mardal, G. Wells, Automated Solution of Differential Equations by the Finite Element Method, Lecture Notes in Computational Science
and Engineering, Springer, 2012.