Interior Control of the Heat Equation with the Steepest Descent Method and OpenFOAM

This is an OpenFOAM solver for the distributed control of the heat equation through the minimization problem

subject to the state equation

with and .

The cost functional gradient is

where solves the adjoint problem

In the steepest descent method the cost gradient is used to update the control as

with sufficiently small.

Getting Started

The solver must be compiled in the terminal. It is advisable to first clean previous compilations with

wclean

and then use

wmake

Prerequisites

OpenFOAM C++ library must be installed in order to compile the code.

The OpenFOAM distribution provided by the OpenFOAM Foundation was used.

Running a Case

In order to run the solver move to the case folder heatAdjoinFoamCase and type in the command line

./Allprepare

heatAdjointFoam

• The heatAdjointFoam solver has been tested in a square domain with zero Dirichlet boundary conditions, , and , , . The target function is .

• The heatAdjointFoam solver has been tested in a square domain with zero Dirichlet boundary conditions, , and , , . The target function is .

Warning

It might be needed to use

sed -i -e 's/\r$//' filename

and

chmod +x filename

in order to be able to execute

./filename