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\begin{titlepage}
\begin{center}

% Upper part of the page. The '~' is needed because \\
% only works if a paragraph has started.

\textsc{\LARGE Google Summer of Code 2013}\\[0.5cm]

\textsc{\Large GNU-Octave}\\[2.5cm]

%\includegraphics[width=1.\textwidth]{./octave-header.png}~\\[1.5cm]

{ \huge \bfseries Fem-fenics \\[0.5cm] }
{ \Large \bfseries Genaral Purpose Finite Element Library for GNU-Octave \\[2.4cm] }
% Author and supervisor
%\begin{minipage}{0.4\textwidth}
%\begin{flushleft} \large
%\emph{Author:}\\
Marco \textsc{Vassallo}\\[0.5cm]
%matr. 780787
%\end{flushleft}
%\end{minipage}
%\begin{minipage}{0.4\textwidth}
%\begin{flushright} \large
%\emph{Supervisor:} \\
%Dr.~Carlo \textsc{de Falco}
%\end{flushright}
%\end{minipage}

\vfill

% Bottom of the page
{\large Version 0.0 \\[0.5cm]}
{\large \today}

\end{center}
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\frontmatter

\tableofcontents

\mainmatter

\chapter{Introduction}
Fem-fenics is an open source package (pkg) for the resolution of partial differential equations with Octave.
The project has been developed during the Google Summer of Code 2013 with the help and the sustain of the GNU-Octave 
community under the supervision of prof. De Falco.

The report is structured as follows:
\begin{itemize}
  \item in chapter \ref{intr} we provide a simple reference guide for beginners
  \item in chapter \ref{impl} is presented a detailed explanation of the relevant parts of the program. In this way, the 
  interested reader can see what there is ``behind'' and expecially anyone interested in it can learn quickly how
  it is possible to extend the code and contribute to the project.
  \item in chapter \ref{exem} more examples are provided. For a lot of them, we present the octave script 
  alongside the code for Fenics (in C++ and/or Python) in order to provide the user with a quick reference
  guide.
\end{itemize}

If you think that going inside the report could be boring, it is available a wiki at
\begin{center}
\url{http://wiki.octave.org/Fem-fenics}
\end{center}
while if you want to see how the project has grown during the time you can give a look at
\begin{center}
\url{http://gedeone-gsoc.blogspot.com/}
\end{center}
The API is available as Appendix \ref{app} but also at the following address
\begin{center}
\url{http://octave.sourceforge.net/fem-fenics/overview.html}
\end{center}
and if you would like to contribute to the project or give a look to the source code
you can clone it from the following repository using mercurial
\begin{center}
\url{http://sourceforge.net/p/octave/fem-fenics/} .
\end{center}

\chapter{Introduction to Fem-fenics}\label{intr} 

\section{Installation}
Fem-fenics is an external package for Octave, which means that it can be installed only once that Octave has been
successfully installed  on the PC. Furthermore, as Fem-fenics is based on Fenics,
it is also needed a running version of the latter. They can be easily installed following the guidelines provided
on the official Octave \cite{instoctave} and Fenics \cite{instfenics} websites.
Once that Octave and Fenics are correctly installed, to install Fem-fenics open Octave (which now is provided with a new
amazing GUI) and type

\begin{verbatim}
 >> pkg install fem-fenics -forge
\end{verbatim}

That's all! For any problem during the installation don't hesitate to contact us.
To be sure that everything is working fine, load the fem-fenics pkg and run
one of the examples provided within the package:

\begin{verbatim}
 >> pkg load fem-fenics
 >> femfenics_examples()
\end{verbatim}

For a description of the examples, look at chapter \ref{exem}.


\paragraph*{NOTE} For completing the installation process successfully,
the form compiler FFC and the header file dolfin.h should also be available on the machine.
They are managed automatically by Fenics if it is installed as a binary package or with Dorsal.
If it has been done manually, please be sure that they are available before starting the
installation of Fem-fenics.

\section{General layout and first example}\label{genlayout}

A generic problem has to be solved in two steps:
\begin{enumerate}
\item a \textbf{.ufl file} where the abstract problem is described: this file has to be written in Unified Form Language (UFL),
       which is a domain specific language for defining discrete variational forms and functionals in a notation 
       close to pen-and-paper formulation. UFL is easy to learn, and the User manual provides explanations
       and examples \cite{ufl}.
\item a script file \textbf{.m} where the abstract problem is imported and a specific problem is implemented and solved:
      this is the script file where the fem-fenics functions described in the following chapters are used.
\end{enumerate}

We provide immediately a simple example in order to familiarize the user with the code.

\paragraph{The Poisson equation}
In this example, we show how it is possible to solve the Poisson equation with mixed Boundary Conditions.
If we indicate with $\Omega$ the domain and with $\Gamma = \Gamma_{N} \cup \Gamma_{D}$ the
boundaries, the problem can be expressed as
\begin{align*}
  \Delta u &= f \qquad \text{on } \Omega \\
  u &= 0 \qquad \text{on } \Gamma_{D} \\
  \nabla u \cdot n &= g \qquad \text{on } \Gamma_{N}
\end{align*}
where $f, \, g$ are data which represent the source and the flux
of the scalar variable $u$.
A possible variational formulation of the problem is: \\
find $u \in H_{0, \Gamma_{D}}^{1} :$
\begin{align*}
  a(u, v) &= L(v) \qquad \forall v \in H_{0, \Gamma_{D}}^{1} \\
  a(u, v) &= \int_{\Omega} \nabla u \cdot \nabla v \\
  L(v)    &= \int_{\Omega} f v + \int_{\Gamma_{N}} g v \\
\end{align*}

The abstract problem can thus be written in the \verb|Poisson.ufl| file immediately.
The only thing that has to be specified at this stage is the space of Finite Elements
used for the discretization of $H_{0, \Gamma_{D}}^{1}$. In this example,
we choose the space of continuous lagrangian polynomial of degree one
\begin{lstlisting}[numbers=none]
 FiniteElement("Lagrange", triangle, 1)
\end{lstlisting}
but many more possibilities are available.
\subparagraph{Poisson.ufl}

\begin{lstlisting}
element = FiniteElement("Lagrange", triangle, 1)

u = TrialFunction(element)
v = TestFunction(element)

f = Coefficient(element)
g = Coefficient(element)

a = inner(grad(u), grad(v))*dx
L = f*v*dx + g*v*ds
\end{lstlisting}

It is always a good idea to check if the ufl code is correctly written before importing it into Octave. Typing
\begin{lstlisting}[numbers=none, language = Octave]
 >> ffc -l dolfin Poisson.ufl
\end{lstlisting} 
in the shell shouldn't produce any error.

We can now implement and solve a specific instance of the Poisson problem with Octave.
The parameters are set as follow
\begin{itemize}
 \item $\Omega = [0, 1]\times[0, 1]$
 \item $\Gamma_{D} = {(0, y) \cup(1, y)} \ \subset \partial\Omega$
 \item $\Gamma_{N} = {(x, 0) \cup(x, 1)} \ \subset \partial\Omega$
 \item $f = 10 \exp \dfrac{(x-0.5)^{2} + (y-0.5)^{2}}{0.02}$
 \item $g = \sin(5x)$
\end{itemize}

As a first thing we need to load into Octave the pkgs previously installed
\begin{lstlisting}[numbers=none, language = Octave]
    pkg load fem-fenics msh
\end{lstlisting}
The ufl file can thus be imported inside Octave. For every specific element defined inside the ufl file
there is a specific function which stores it for later use
\begin{itemize}
    \item \verb$ufl_import_FunctionSpace ('Poisson')$ is a function which looks for the finite element 
    space defined inside the file called Poisson.ufl; if everything is ok, it generates a function
    which we will use later
    \item \verb$ufl_import_BilinearForm ('Poisson')$ is a function which looks for the rhs of the 
    equation, i.e. for the bilinear form defined inside Poisson.ufl
    \item \verb$ufl_import_LinearForm ('Poisson')$ is a function which looks for the linear 
    form.
\end{itemize}

In some cases one could be interested in using these functions separately but if, 
as in our example, all the three elements are defined in the same ufl file (and only in this case), 
the \verb$import_ufl_Problem ('Poisson')$ can be used, which generates at once all 
the three functions described above

\begin{lstlisting}[numbers=none, language = Octave]
    ufl_import_Problem ('Poisson');
\end{lstlisting}

To set the concrete elements which define the problem, 
the first things to do is to create a mesh. 
It can be managed easily using the msh pkg. For a structured squared mesh
\begin{lstlisting}[numbers=none, language = Octave]
    x = y = linspace (0, 1, 33);
    msho = msh2m_structured_mesh (x, y, 1, 1:4);
\end{lstlisting}
Once that the mesh is available, we can thus initialize the 
Fem-fenics mesh using the function \verb$Mesh ()$:
\begin{lstlisting}[numbers=none, language = Octave]
    mesh = Mesh (msho);
\end{lstlisting}

To initialize the functional space, we have to specify as argument only the fem-fenics mesh,
because the finite element type and the polynomial degree have already been specified in the ufl file:
\begin{lstlisting}[numbers=none, language = Octave]
    V = FunctionSpace('Poisson', mesh);
\end{lstlisting}
Essential BC can now be applied using \verb$DirichletBC ()$; this function receives as argument the functional space,
a function handle which specifies the value to set, and the label of the sides where the BC applies.
In this case, homogenous boundary conditions hold on the left and right side of the square
\begin{lstlisting}[numbers=none, language = Octave]
    bc = DirichletBC(V, @(x, y) 0.0, [2; 4]);
\end{lstlisting}
The last thing to do before solving the problem, is to set the coefficients specified
in the ufl file. 
To set them, the function \verb$Expression ()$ can be used passing as argument a string 
which specifies the name of the coefficient
(it is important that they are called in the same way as in the ufl file: 
the source term 'f' and the normal flux 'g'),
and a function handle with the value prescribed:
\begin{lstlisting}[numbers=none, language = Octave]
    ff = Expression ('f', 
          @(x,y) 10*exp(-((x - 0.5)^2 + (y - 0.5)^2) / 0.02));
    gg = Expression ('g', @(x,y) sin (5.0 * x));
\end{lstlisting}
Another possibility for dealing with the coefficients defined in the ufl file would be to use 
the function \verb$Constant ()$ or \verb$Function ()$.
The coefficients can thus be used together with the FunctionSpace to set 
the Bilinear and the Linear form
\begin{lstlisting}[numbers=none, language = Octave]
    a = BilinearForm ('Poisson', V, V);
    L = LinearForm ('Poisson', V, ff, gg);
\end{lstlisting}
The discretized representation of our operator is obtained using the 
functions \verb$assemble ()$ or \verb$assemble_system ()$, which also allow
to specify the BC(s) to apply
\begin{lstlisting}[numbers=none, language = Octave]
    [A, b] = assemble_system (a, L, bc);
\end{lstlisting}
Here A is a sparse matrix and b is a column vector. All the 
functionalities available within Octave can now be exploited to solve the linear system. 
The easisest possibility is the backslash command:
\begin{lstlisting}[numbers=none, language = Octave]
    u = A \ b;
\end{lstlisting}
Once that the solution has been obtained, the \verb$u$ vector is converted into a 
Fem-fenics function and plotted \verb$plot ()$ or saved \verb$save ()$ in the vtu
format
\begin{lstlisting}[numbers=none, language = Octave]
    u = Function ('u', V, sol);
    save (u, 'poisson')
    plot (u);
\end{lstlisting}

The complete code for the Poisson problem is reported below, while
in figure \ref{Poissonfig} is presented the output.
\begin{figure}
 \begin{center}
  \includegraphics[height=7 cm,keepaspectratio=true]{./Fem-fenics_poisson.png}
   \caption{The result for the Poisson equation}
   \label{Poissonfig}
  \end{center}
\end{figure}

\subparagraph{Poisson.m}
\begin{lstlisting}[language=Octave]
#load the pkg and import the ufl problem
pkg load fem-fenics msh
import_ufl_Problem ('Poisson')
 
# Create the mesh and define function space
x = y = linspace (0, 1, 33);
mesh = Mesh(msh2m_structured_mesh (x, y, 1, 1:4));
V = FunctionSpace('Poisson', mesh);

# Define boundary condition and source term
bc = DirichletBC(V, @(x, y) 0.0, [2;4]);
ff = Expression ('f', @(x,y) 10*exp(-((x - 0.5)^2 + (y - 0.5)^2) / 0.02));
gg = Expression ('g', @(x,y) sin (5.0 * x));

#Create the Bilinear and the Linear form
a = BilinearForm ('Poisson', V, V);
L = LinearForm ('Poisson', V, ff, gg);
 
#Extract the matrix and compute the solution
[A, b] = assemble_system (a, L, bc);
sol = A \ b;
u = Function ('u', V, sol);
 
# Save solution in VTK format and plot it
save (u, 'poisson')
plot (u);

\end{lstlisting}



\chapter{Implementation}\label{impl} 

Fem-fenics aims to fill a gap in Octave: even if there are packages for the creation of mesh \cite{msh},
for the postprocessing of data \cite{fpl} and for the resolution of some specific pde \cite{secs1d} \cite{bim},
no general purpose finite element library is available.

The goal of the project is thus to provide a package which can be used to solve user defined problems 
and which is able to exploit the functionality provided with Octave.

\begin{figure}
 \begin{center}
  \includegraphics[height=10 cm,keepaspectratio=true]{./code_layout.png}
   \caption{General layout of the package}
   \label{Codelayout}
  \end{center}
\end{figure}

Instead of writing a library starting from scratch, an interface to one of the finite element library
which are already available has been created.
Among the many libraries taken into account, the one which was best suited for our
purposes seemed to be the FEniCS project. It ``is a collection of free, open source, software
components with the common goal to enable automated solution of pde.''
In particular, Dolfin is the C++/Python interface of FEniCS, providing a consistent Problem
Solving Environment for ODE and PDE. The idea has been to create wrappers in Octave for C++ Dolfin,
in a similar way to what it has been done for Python.
This is a very natural choice, because Octave is mainly written in script language 
and in C++. It is in fact possible to implement an Octave interpreter function in C++ through the
native oct-file interface or, conversely, to use Octave's Matrix/Array Classes in a C++ application
\cite{whatoctave}. 

The works can be summarized as follows (fig. \ref{Codelayout}):

the elements already available in Octave for the resolution of PDE (Mesh and Linear
Algebra) have been exploited, and wrappers to the other FEniCS functions added.
To allow exchanges between these programs, the necessary functions
for converting an Octave mesh/matrix into a FEniCS one and viceversa have been written.

Two main ideas have guided us throughout the realization of the pkg:
\begin{itemize}
  \item keep the syntax as close as possible to the original one in Fenics (Python)
  \item make the interface as simple as possible.
\end{itemize}

\section{General layout of a class}\label{class}
Seven new classes are implemented for dealing with FEniCS objects and for using them inside Octave:
\begin{itemize}
  \item \textbf{boundarycondition} stores and builds a dolfin::DirichletBC
  \item \textbf{coefficient} stores an expression object which is used for the
  evaluation of user defined values
  \item \textbf{expression} is needed for internal use only as explained below
  \item \textbf{form} stores a general dolfin::Form and can be used both for
  a dolfn::BilinearForm and for a dolfin::LinearForm
  \item \textbf{function} for the dolfin::Function objects
  \item \textbf{functionspace} stores the user defined FunctionSpace
  \item \textbf{mesh} converts a PDE-tool like mesh structure in a dolfin::Mesh
\end{itemize}

The classes are written with the ``usual'' C++ style, but they need to be derived publicly
from octave\_base\_value and to be added to the Octave interpreter \cite{whatoctave}.
When a type is used for the first time during a session, it is also temporarily
registered in the interpreter after all the other basic types (int, double, ...).

The general layout of a class can thus be kept simple and with the main purpose of storing
the associated FEniCS objects, which is done throughout 
boost::shared\_ptr< > to the corresponding FEniCS type. 
All the classes also implement at least two constructors: 
a default one which is necessary to register a type in the Octave interpreter,
and a constructor which takes as argument the corresponding dolfin type.

As an example, the form class implementation follows, while classes which differ from the general
layout are presented below in more details.

\begin{lstlisting}
#ifndef _FORM_OCTAVE_
#define _FORM_OCTAVE_

#include <memory>
#include <vector>
#include <dolfin.h>
#include <octave/oct.h>

class form : public octave_base_value
{

 public:

  form () : octave_base_value () {}

  form (const dolfin::Form _frm)
    : octave_base_value (), frm (new dolfin::Form (_frm)) {}

  form (boost::shared_ptr <const dolfin::Form> _frm)
    : octave_base_value (), frm (_frm) {}

  void
  print (std::ostream& os, bool pr_as_read_syntax = false) const
    {  
       os << "Form " << ": is a form of rank " << frm->rank ()
       << " with " << frm->num_coefficients () 
       << " coefficients" << std::endl; 
    }

  ~form(void) {}

  bool is_defined (void) const { return true; }

  const dolfin::Form & get_form (void) const { return (*frm); }

  const boost::shared_ptr <const dolfin::Form> & 
  get_pform (void) const { return frm; }

 private:

  boost::shared_ptr <const dolfin::Form> frm;

  DECLARE_OCTAVE_ALLOCATOR;
  DECLARE_OV_TYPEID_FUNCTIONS_AND_DATA;

};

static bool form_type_loaded = false;

DEFINE_OCTAVE_ALLOCATOR (form);
DEFINE_OV_TYPEID_FUNCTIONS_AND_DATA (form, "form", "form");
#endif

\end{lstlisting}

\subsection{Shared pointer}
In all the classes presented above, the private members are stored using 
a boost::shared\_ptr< >  to the corresponding FEniCS type.
This is done because we have to refer in several places to resources which are built dynamically 
 and we want that they are destroyed only
when the last reference is destroyed \cite{Formaggia_smart}.
For example, if we have two different functional spaces in the same problem, 
like with Navier-Stokes for the velocity and the pressure, the mesh is shared between 
them and no one has its own copy.
Furthermore, they are widely supported inside DOLFIN, and it can thus be avoided to have a copy of the
same object for FEniCS and another one for DOLFIN: there is just one copy which is shared between DOLFIN
and FEniCS.


\subsection{The mesh class}
In addition to usual methods, the mesh class implemens functionalities which allow to deal with meshes 
as they are currently available with the msh pkg, i.e. in the (p, e, t) format, and in Fenics, i.e. 
in the xml Dolfin format.
It is therefore necessary to have two different constructors
\begin{lstlisting}[numbers=none]
 mesh (Array<double>& p, Array<octave_idx_type>& e, 
       Array<octave_idx_type>& t);
       
 mesh (std::string _filename)
    : octave_base_value (), pmsh (new dolfin::Mesh(_filename)) {}
\end{lstlisting}

where the first one accepts as input a mesh in (p, e, t) format and converts it into a xml one, while
the latter loads the mesh stored in the \_filename.xml file.

The constructors are used within the Mesh () function, which therefore accepts as argument 
either a mesh generated within the msh pkg or a string with the name of the file
where the dolfin mesh is stored.

Furthermore, if a mesh is stored in another different format,
the program dolfin-convert can try to convert it to the dolfin xml format.
For example, for a mesh generated with Metis:
\begin{lstlisting}[numbers=none, language=bash]
    Shell:
      >> dolfin-convert msh.gra msh.xml
\end{lstlisting}
and then inside the Octave script:
\begin{lstlisting}[numbers=none, language=Octave]
    mesh = Mesh ('msh.xml');
\end{lstlisting}
Before exploring the code in more details, the main differences between the two storing formats are presented using 
the very simple, but rather instructive, example of a unit square mesh with just two elements, fig. \ref{mesh}.

\paragraph{pet}

\begin{figure}
 \begin{center}
  \includegraphics[height=5 cm,keepaspectratio=true]{./mesh_1.png}
   \caption{The (very) simple mesh for our example}
   \label{mesh}
  \end{center}
\end{figure}

A mesh is represented using the three matrices $p$, $e$, $t$, and, using msh, 
we can easily obtain the mesh for our example typing
\begin{lstlisting}[numbers=none, language=octave]
 mesh = msh2m_structured_mesh ([0 1], [0 1], 1, [11 12 12 13])
\end{lstlisting}

The matrix $p$ stores information about the coordinates of the vertices
\begin{mdframed}[backgroundcolor=LightBlue, outerlinewidth=0.25pt,linecolor=LightBlue]
\begin{lstlisting}[numbers=none, language=octave]
 >> mesh.p 
\end{lstlisting}

$
\begin{array}{rrrrl}
           \; 0  & 0 &  1 &  1 & \quad \text{x-coordinates} \\
           \; 0  & 1 &  0 &  1 & \quad \text{y-coordinates} \\
\end{array}
$
\end{mdframed}
Thus the vertex in the $n^{th}$ column is labelled as the vertex number $n$, and so on.
     
The matrix $t$ stores information about the connectivity
\begin{mdframed}[backgroundcolor=LightBlue, outerlinewidth=0.25pt,linecolor=LightBlue]
\begin{lstlisting}[numbers=none, language=octave]
 >> mesh.t
\end{lstlisting}

$
\begin{array}{rrl}
          \; 1  & 1 &  \quad \text{number of the first vertex of the element}  \\
          \; 3  & 4 &  \quad \text{number of the second vertex of the element}  \\
          \; 4  & 2 &  \quad \text{number of the third vertex of the element}  \\
          \; 0  & 0 &   \\           
\end{array}
$
\end{mdframed}                       
The first element is thus the one obtained connecting vertices 1-3-4 and so on.

The matrix $e$ stores information related to every side edge, 
like the number of the vertices of the boundary elements,  
and the number of the geometrical border containing the edge,
which is a convenient way to deal with boundary conditions in a problem.
\begin{mdframed}[backgroundcolor=LightBlue, outerlinewidth=0.25pt,linecolor=LightBlue]
\begin{lstlisting}[numbers=none, language=octave]
 >> mesh.e
\end{lstlisting} 
$
\begin{array}{rrrrl}
           \; 1  & 3 & 2 & 1 & \quad \text{first vertex of the side edge}   \\
           \; 3  & 4 & 4 & 2 & \quad \text{second vertex of the side edge} \\
           \; 0  & 0 & 0 & 0 &   \text{} \\
           \; 0  & 0 & 0 & 0 &    \text{} \\
           \; 11  & 12 & 12 & 13 & \quad \text{label of the geometrical border containing the edge}  \\
           \; 0  & 0 & 0 & 0 &  \text{}\\
           \; 1  & 1 & 1 & 1 &  \text{} \\
\end{array}
$
\end{mdframed}
The side edge between vertex 1-3 is labelled 11, between 3-4 is 12...

\paragraph{dolfin xml} A mesh is an object of the dolfin::Mesh class which stores information only about
the coordinates of the vertices (like $p$) and the information about the connectivity (like $t$). 
A mesh can thus be manipulated using the functions and the methods of the class, which are presented below.
Instead, the information about boundaries is not directly stored in the mesh.
The mesh used in the example is stored as

\begin{lstlisting}[numbers=none, language=xml]
<?xml version="1.0"?>
<dolfin xmlns:dolfin="http://fenicsproject.org">
  <mesh celltype="triangle" dim="2">
    <vertices size="4">
      <vertex index="0" x="0.000e+00" y="0.000e+00" />
      <vertex index="1" x="0.000e+00" y="1.000e+00" />
      <vertex index="2" x="1.000e+00" y="0.000e+00" />
      <vertex index="3" x="1.000e+00" y="1.000e+00" />
    </vertices>
    <cells size="2">
      <triangle index="0" v0="0" v1="2" v2="3" />
      <triangle index="1" v0="0" v1="1" v2="3" />
    </cells>
  </mesh>
</dolfin>
\end{lstlisting}

\paragraph{Conversion between the formats}
The first necessary step in our way to a package which links Octave and FEniCS
is to convert a mesh from the $(p, e, t)$ format
into the dolfin xml one. 
Furthermore, as dolfin provides methods and functions which 
allow to manipulate a mesh and which don't have a conterpart in the msh pkg,
we have also created wrappers for them (specifically for mesh::refine).

As it has been shown above, the main difference between $(p, e, t)$ and DOLFIN $xml$ 
is the way in which the boundaries are distinguished.
The former stores all the information in the $e$ matrix, while the latter uses
the functions and the methods of the dolfin::mesh class to set/get information about a mesh.
The most useful classes available in dolfin are recalled
\begin{itemize}
\item \textbf{MeshIterator}
To know whether an edge belongs or not to the boundary, we can iterate over all the
edges of our mesh using the classes provided by DOLFIN:
\begin{lstlisting}[numbers=none]
  for (dolfin::FacetIterator f (mesh); ! f.end (); ++f)
    {
      if ((*f).exterior () == true)
        {
          //do something with the boundary cells
        }  
    }
\end{lstlisting}

\item \textbf{MeshFunction}
To store data related to a mesh, dolfin provides the template class MeshFunctions.
"A MeshFunction is a function that can be evaluated at a set of mesh entities. 
A MeshFunction is discrete and is only defined at the set of mesh entities of a fixed topological dimension.
A MeshFunction may for example be used to store a global numbering scheme for the entities of a (parallel) mesh,
marking sub domains or boolean markers for mesh refinement." \cite{meshfunction}
For example, in the function \verb$mshm_refine$ of the msh package, the list of cells to be refined 
is stored as a MeshFunction, which for every cell says whether or not it has to be refined:
\begin{lstlisting}[numbers=none]
  dolfin::CellFunction<bool> cell_markers (mesh);
  cell_markers.set_all (false);

  for (octave_idx_type i = 0;
       i < cells_to_refine.length (); ++i)
    cell_markers.set_value (cells_to_refine (i) , true);
\end{lstlisting}


\item \textbf{MeshValueCollection} "It differs from the MeshFunction class in two ways. 
First, data do not need to be associated with all entities (only a subset). 
Second, data are associated with entities through the corresponding cell index and local entity number
(relative to the cell), not by global entity index, which means that data may be stored robustly to file."\cite{meshvalue}
It is thus obvious that it is better to use the MeshValueCollection whenever saving or writing a mesh.
\end{itemize}

The container classes presented above can be used by their own, 
but to set/get data from a mesh it is better to use the methods provided by the classes:
\begin{itemize}
\item \textbf{MeshDomains} "The class MeshDomains stores the division of a Mesh into subdomains.
For each topological dimension 0 <= d <= D, where D is the topological dimension of the Mesh, 
a set of integer markers are stored for a subset of the entities of dimension d, 
indicating for each entity in the subset the number of the subdomain. 
It should be noted that the subset does not need to contain all entities of any given dimension;
entities not contained in the subset are “unmarked”." \cite{meshdomain}
\item \textbf{MeshData} "The class MeshData is a container for auxiliary mesh data,
represented either as MeshFunction over topological mesh entities, arrays or maps.
Each dataset is identified by a unique user-specified string." \cite{meshdata}
\end{itemize}

\subparagraph{Geometry from (p, e, t) to dolfin xml}
Converting the vertices and cells from (p, e, t) in the xml format can be done using the
dolfin editor, while caution has to be taken for storing information associated with boundaries
and subdomains, as presented in the next paragraph.
\begin{lstlisting}[numbers=none]
      dolfin::MeshEditor editor;
      boost::shared_ptr<dolfin::Mesh> msh (new dolfin::Mesh ());
      editor.open (*msh, D, D);
      editor.init_vertices (p.cols ());
      editor.init_cells (t.cols ());

      if (D == 2)
        {
          for (uint i = 0; i < p.cols (); ++i)
            editor.add_vertex (i,
                               p.xelem (0, i),
                               p.xelem (1, i));

          for (uint i = 0; i < t.cols (); ++i)
            editor.add_cell (i,
                             t.xelem (0, i) - 1,
                             t.xelem (1, i) - 1,
                             t.xelem (2, i) - 1);
        }

      if (D == 3)
        {
          ...
        }

      editor.close ();

\end{lstlisting}

\subparagraph{Subdomain markers: from (p, e, t) to dolfin xml}
There are no fundamental differences between the 2D and 3D case,
and they are thus treated together referring to the 
general dimension D.
The subdomain information is contained in the t matrix,
and it is temporarily copied to a MeshValueCollection.
For every column of the $t$ matrix, i.e. for every element of the mesh, 
we have to look for the corresponding element in the DOLFIN mesh. 
We use the class MeshIterator for moving around on the DOLFIN mesh:

\begin{lstlisting}[numbers=none]
  dolfin::MeshValueCollection<uint> my_cell_marker (D);
  
  for (uint i = 0; i < num_cells; ++i)
    dolfin::Vertex v (mesh, t(0, i));
      for (dolfin::CellIterator f (v); ! f.end (); ++f)
        {
          if ((*f) == all_vertices_in_the_ith_column)
            {
              my_cell_marker.set_value
                ((*f).index (), t(last_row, i), mesh);
              break;
            }
         }
\end{lstlisting}

The \verb$all_vertices_in_the_ith_column$ is just like a pseudo code: 
we have to be sure that the Cell pointed by f is the one corresponding 
to the $i^{th}$ column of the matrix, checking the vertices one-by-one:

in $2D$ the cell is a triangle, and we thus have to check $3$ vertices. 
As we don't know the order in which vertices are visited, we have to check all the $3! = 6$ 
different combinations:
\begin{lstlisting}[numbers=none]
          ...
          if ((*f).entities(0)[0] == t(0, i)
              && (*f).entities(0)[1] == t(1, i) 
              && (*f).entities(0)[2] == t(2, i)
              || ...  check the other 5 possibilities... )
         ....
\end{lstlisting}

where the \verb$entities(std::size_t dim)$ method returns an array with the indexes of the elements
of dimension dim. Thus we use $dim = 0$ as we are looking for vertices.

In the $3D$ case, our cell is a tetrahedron, and we have to check all the $4! = 24$ possibilities,
each of which is composed by $4$ assertions; in total we have almost one hundred conditions!

Now that the information is stored in our function, it can be associated to the mesh

\begin{lstlisting}[numbers=none]
    *(mesh.domains ().markers (D)) = my_marked_cell;
\end{lstlisting}

\subparagraph{Subdomain markers: from dolfin xml to (p, e, t)}

In the DOLFIN .xml file, the information is stored like:
\begin{lstlisting}[numbers=none]
 ...
<mesh_value_collection name="m" type="uint" dim="2" size="2">
     <value cell_index="0" local_entity="0" value="1"/>
     <value cell_index="1" local_entity="0" value="2"/>
 ...
\end{lstlisting}

When the file is read using DOLFIN, the information is automatically associated with the mesh 
as a MeshValueCollection named \verb$cell_domains$, which can be accessed to extract the information
using the MeshDomains class.
Obviously we have to be sure that the information is available within 
the file that we are reading, and that it is related to Cell, i.e. to elements of dimension D,
before it is associated to the last row of the $t$ matrix:

\begin{lstlisting}[numbers=none]
  dolfin::MeshFunction<uint> my_cell_marker;
  if (! mesh.domains ().is_empty ())
    if (mesh.domains ().num_marked (D) != 0)
      my_cell_marker = *(mesh.domains ().cell_domains ());
      
    for (j = 0; j < t.cols (); ++j)
      t(D + 1, j) = my_cell_marker[j];
\end{lstlisting}

\subparagraph{Boundary Markers}
For boundary markers, things work in a similar way, as long as we remember that we are working with objects of 
dimension D - 1.
In this case, the main difference is in the .xml file: it is no longer enough to say 
to what cell element the label is referred to, but we have to specify to which $D - 1$ 
entity (a side or a face) the label is referred.
For example:

\begin{lstlisting}[numbers=none]
    ....
    mesh_value_collection name="m" type="uint" dim="1" size="4">
         <value cell_index="0" local_entity="0" value="12"/>
         <value cell_index="0" local_entity="2" value="11"/>
         <value cell_index="1" local_entity="0" value="12"/>
         <value cell_index="1" local_entity="2" value="13"/>  
    ...
\end{lstlisting}

The cell number $"0"$ is a triangle, 
and to the \verb$local_entity$ number $"0"$, i.e. to the side number $"0"$, 
is associated the label $"12"$, while to the side number $"2"$ is associated the label $"11"$.
To the side number $"1"$, there are no labels associated.
The number of the \verb$local_entity$ refers to the enumeration of the reference element.
In any case, it is DOLFIN which takes care of the conversion of indeces from this format to the usual one,
and we can thus use methods and functions as explained for the subdomain markers.

\subparagraph{Mesh refine}
Now that it is possible to convert meshes between Octave and DOLFIN,
the functions available in the dolfin::mesh class can be used to improve the 
functionality of the msh package.
For the moment, it has been added the possibility of refining a mesh,
either uniformly or specifying the list of the vertices we want to be refined.
The function is now part of the msh pkg\cite{msh}, and a more detailed desciption has been 
provided previously \cite{refine}.



\subsection{The functionspace class}
A dolfin::FunctionSpace is defined by specifying a mesh and the type of the finite element which we want to use. 
The mesh is handled as presented above, while the FE are specified inside the .ufl file. Possible choices are
\cite{logg2012automated}:

\begin{center}
\begin{tabular}{  l | l }
  \hline
    \textbf{Finite Element Space} & \textbf{Symbol} \\ \hline \hline
   

    Argyris                  &    ARG * \\ \hline
    Arnold–Winther           &    AW * \\ \hline
    Brezzi–Douglas–Marini    & BDM\\ \hline
    Crouzeix–Raviart         &      CR\\ \hline
    Discontinuous Lagrange   & DG\\ \hline
    Hermite                  &        HER*\\ \hline
    Lagrange                 &        CG\\ \hline
    Mardal–Tai–Winther       &  MTW *\\ \hline
    Morley                   &            MOR*\\ \hline
    Nédélec 1st kind H (curl)   &  N1curl\\ \hline
    Nédélec 2nd kind H (curl)   & N2curl\\ \hline
    Raviart–Thomas              &    RT\\
\end{tabular} 

\end{center}

where the Finite Elements denoted with * are not yet fully supported inside FEniCS.

\section{General layout of a function}
There are two general kinds of functions in the code: functions which create an abstract problem
(wrappers to UFL) and
functions which create the specific instance of a problem and discretize it (wrapper to DOLFIN).
\section{Wrappers to UFL}
As stated in section \ref{genlayout}, a problem is divided in two files: a \texttt{.ufl} file 
where the abstract problem is described in Unified Form Language (UFL),
and a script file \texttt{.m} where a specific problem is implemented and solved.
We suppose that they are called \texttt{Poisson.ufl} and \texttt{Poisson.m} .
In order to use the information stored in the UFL file, i.e. the bilinear and the linear form, 
they have to be ``imported'' inside Octave. This is done using the 
functions \texttt{import\_ufl\_BilinearForm, import\_ufl\_LinearForm, ...} .
\subsection{Generation of code on the fly}
When a UFL file is compiled using the ffc compiler, a header file \texttt{Poisson.h} is generated.
In this header file, it is defined the Poisson class, which derives from dolfin::Form,
and the constructor for the bilinear and linear form are set.
This file is thus available only at compilation time, but it has to be included somehow
in the wrapper function for the Bilinear and the Linear form.
An easy solution would have been to write a set of pre established problems where the user could only
change the values of the coefficient for a specific problem; 
but, as we want to let the user free to write his own 
variational problem, a different approach has been adopted.
The \texttt{ufl} file is compiled at run time and generates its header file.
Then, a Poisson.cc file is written from a template which takes as input the name 
of the header file and is compiled including the Poisson.h file;
now the corresponding Octave functions for the specific problem are available and
will be later used from
\texttt{BilinearForm, LinearForm, FunctionSpace, ...} .
As an example it is presented the import\_ufl\_BilinearForm function.

\begin{lstlisting}[language=Octave]
 function import_ufl_BilinearForm (var_prob)

 ...
 
  %the function which writes the var_prob.cc file (see below)
  generate_rhs (var_prob);
  
  %the function which writes the makefile
  generate_makefile (var_prob, private);

  % the makefile is executed in a terminal:
  % 1) generate the header file from ufl
  %  ffc -l dolfin var_prob.ufl
  % 2) compile the var_prob.cc
  %  mkoctfile var_prob.cc -I.
  system (sprintf ("make -f Makefile_%s rhs", var_prob));
  
  ...

endfunction
\end{lstlisting}

\begin{lstlisting}[language=Octave]
function output = generate_rhs (ufl_name)

  STRING ="
  #include "@@UFL_NAME@@.h"

  ...

  DEFUN_DLD (@@UFL_NAME@@_BilinearForm, args, , ""A = fem_rhs_@@UFL_NAME@@ (FUNCTIONAL SPACE, COEFF)"")
  {
    ...
    
    const functionspace & fspo1
      = static_cast<const functionspace&> (args(0).get_rep ());
    const functionspace & fspo2
      = static_cast<const functionspace&> (args(1).get_rep ());

    const dolfin::FunctionSpace & U = fspo1.get_fsp ();
    const dolfin::FunctionSpace & V = fspo2.get_fsp ();
    @@UFL_NAME@@::BilinearForm a (U, V);

    ...
    

  }";

  STRING =  strrep (STRING, "@@UFL_NAME@@", ufl_name);

  fid = fopen (sprintf ("%s_BilinearForm.cc", ufl_name), 'w');
  fputs (fid, STRING);
  output = fclose (fid);

endfunction
\end{lstlisting}


\section{Wrappers to DOLFIN}
The general layout of a function is very simple and it is composed of 4 steps which we describe using an example:
\begin{lstlisting}
DEFUN_DLD (FunctionSpace, args, , "initialize a FunctionSpace from a mesh")
{
          // 1 read data
          const mesh & msho = static_cast<const mesh&> (args(0).get_rep ());
          
          // 2 extract the data stored in the Octave class as a DOLFIN object
          const dolfin::Mesh & mshd = msho.get_msh ();
          
          // 3 build a new object or extract the information needed using DOLFIN
          boost::shared_ptr <const dolfin::FunctionSpace> g (new Laplace::FunctionSpace (mshd));
          
          // 4 convert the new object from DOLFIN to Octave and return it
          octave_value retval = new functionspace(g);
          return retval;
}
\end{lstlisting}
All the functions presented above follow this general structure, and thus here we present 
in detail only functions which present some differences.

\subsection{DirichletBC and Coefficient}
 These two functions take as input a function handle which cannot be directly evaluated by
 a dolfin function to set, respectively, the value on the boundary or the value of the coefficient. 
 It has thus been derived from dolfin::Expression a class "expression" which has as private member 
 an octave function handle and which  overloads the function eval(). In this way, an object of 
 the class expression can be initialized throughout a function handle and can be used inside dolfin because 
 "it is" a dolfin::Expression
\begin{lstlisting}
class expression : public dolfin::Expression
{
  ...
  
  void 
  eval (dolfin::Array<double>& values,
        const dolfin::Array<double>& x) const
    {
      octave_value_list b;
      b.resize (x.size ());
      for (std::size_t i = 0; i < x.size (); ++i)
        b(i) = x[i];
      octave_value_list tmp = feval (f->function_value (), b);
      Array<double> res = tmp(0).array_value ();

      for (std::size_t i = 0; i < values.size (); ++i)
        values[i] = res(i);
    }

 private:
  octave_fcn_handle * f;
};


\end{lstlisting}

\paragraph{DirichletBC}
The BC are imposed directly to the mesh setting to zero all the off diagonal elements 
in the corresponding line. This means that we could loose the symmetry of the matrix, if any. 
To avoid this problem, instead of the method \verb$apply()$ it is possible to use the
function \verb$assemble_system()$ , which preserves the symmetry of the system but which needs to build 
together the lhs and the rhs.

\paragraph{Coefficient}
The coefficient of the variational problem can be specified using either a Coefficient 
or a Function. They are different objects which behave in different ways: a Coefficient, as exlained above, 
overloads the \verb$eval()$ method of the  dolfin::Expression class and it is evaluated at 
run time using the octave function \verb$feval()$. A Function instead doesn't need to be evaluated 
because it is assembled copying element-by-element the values contained in the input vector.

\subsection{Sparse Matrices}
The \texttt{assemble} function discretize the continuos problem and
return a sparse matrix. To deal with problems of big size, they are stored
using a compressed technique \cite{Formaggia_matr} both in DOLFIN and in Octave.
Unfortunately, DOLFIN uses row major orientation while Octave uses 
column major orientation. They have thus to be converted efficiently from
one type to the other.
\begin{lstlisting}[language=C++]
 
#include "form.h"
#include "boundarycondition.h"

DEFUN_DLD (assemble, args, nargout, " ")
{
  int nargin = args.length ();
  octave_value_list retval;

  if (! boundarycondition_type_loaded)
    {
      boundarycondition::register_type ();
      boundarycondition_type_loaded = true;
      mlock ();
    }

  if (! form_type_loaded)
    {
      form::register_type ();
      form_type_loaded = true;
      mlock ();
    }

  ...
  
  // Extract form object from the input
  const form & frm = static_cast<const form&> (args(0).get_rep ());

  const dolfin::Form & a = frm.get_form ();
  a.check ();

  ...
  
  // Assemble the Matrix in DOLFIN
  dolfin::parameters["linear_algebra_backend"] = "uBLAS";
  dolfin::Matrix A;
  dolfin::assemble (A, a);

  // Extract BC from input and apply BC
  ...
  const boundarycondition & bc
    = static_cast<const boundarycondition&> (args(i).get_rep ());

  const std::vector<boost::shared_ptr <const dolfin::DirichletBC> > 
    & pbc = bc.get_bc ();

  for (std::size_t j = 0; j < pbc.size (); ++j)
    pbc[j]->apply(A);


  // Get capacity of the dolfin sparse matrix
  boost::tuples::tuple<const std::size_t*, 
		       const std::size_t*, 
	               const double*, int> 
    aa = A.data ();

  // Create the Matrix for Octave

  ...
  
  for (std::size_t i = 0; i < nr; ++i)
    {
      A.getrow (i, cidx_tmp, data_tmp);
      nz += cidx_tmp.size ();

      for (octave_idx_type j = 0; 
	  j < cidx_tmp.size (); ++j)
	{
	  orow [ii + j] = i;
	  oc [ii + j] = cidx_tmp [j];
	  ov [ii + j] = data_tmp [j];
	}

      ii = nz;
    }

  dims(0) = ii;
  ridx.resize (dims);
  cidx.resize (dims);
  data.resize (dims);

  SparseMatrix sm (data, ridx, cidx, nr, nc);
  retval(0) = sm;

  ...
  
  return retval;
}
\end{lstlisting}


\subsection{Polymorphism}
The objects which belong to the new classes presented in section \ref{class}
have to overload some of the methods already available in Octave.
For example, we want to be able to \texttt{plot} a \texttt{Mesh} or a \texttt{function}, to \texttt{save} it
and to evaluate it at a specific point in the space (\texttt{feval}).
As Octave is a dynamically typed language it could be a difficult task to achieve, but hopefully
the Octave interpreter takes care of it and it is enough to put the polymorphic function
in a folder named as the type. For example, in the \texttt{@function} folder
inside the Fem-fenics directory we can find the \texttt{plot, save, feval} functions.

\iffalse

\paragraph{other function}


    SubSpace allows to extract a subspace from a vectorial one. 
    For example, if our space is P2 x P0 we can extract the one or 
    the other and then apply BC only where it is necessary.
    \verb$fem_eval$ takes as input a Function and a coordinate and returns a 
    vector representing the value of the function at this point.
    for dealing with form of rank 0, i.e. with functional, we have now 
    added the functions \verb$fem_create_functional$ to create it from a .ufl file. 
    We have thus extended the function assemble which returns the corresponding double value.
    \verb$plot_2d$ and \verb$plot_3d$: these functions allow us to plot a function specifying 
    a mesh and the value of the function at every node of the mesh. 
    This is something which could be useful also outside of fem-fenics.

\section{Implementation Details}
The relevant implementation details which the user should know are:

     We have split the construction of the form into two steps:

        We set all the coefficients of the form using the function which we create on the fly. 
        They will be named \verb$ProblemName_BilinearForm$ or \verb$ProblemName_LinearForm$.
        Then we apply specific BC to the form using the assemble() function and we get back the matrix. 
        If we are assembling the whole system and we want to keep the symmetry of the matrix (if any), 
        we can instead use the command \verb$assemble_system$ (). Finally, if we are solving a non-linear problem 
        and we need to apply essential BC, we should provide to the function also the vector with the 
        tentative solution in order to modify the entries corresponding to the boundary values. 
        This will be illustrated below in the HyperElasticity example.
\fi


\chapter{More Advanced Examples}\label{exem}
In this chapter more examples are provided.
At the beginning of each section, the problem is briefly presented and then
the Octave script for the resolution of the problem using Fem-fenics is presented alongside the code 
written in C++ and/or the Python.
For each problem, we refer the reader to the complete desciption on the FEniCS website.
\iffalse
In the following examples we can see directly in action the classes and the functions presented in the 
chapters before. A comparison with DOLFIN is given only for the first example, while more extensive case can
be found online. We do not report the code for all the examples but only the relevant parts.

 With the following examples, we can see directly in action the new features and understand how they work.

    Navier-Stokes: we learn how to deal with a vector-field problem and how we can save the solution using the 
    \verb$fem_save$ () function. We also use the fem pkg to generate a mesh using gmesh.
    Mixed-Poisson: we solve the Poisson problem presented in the previous posts using a mixed formulation, 
    and we see how we can extract a scalar field from a vector one.
    HyperElasticity: we exploit the fsolve () command to solve a non-linear problem. In particular, 
    we see how to use the assemble() function to apply BC also in this situation.
    Advection-Diffusion: we solve a time dependent problem using the lsode () command and save 
    the solution using the pkg flp.

or bim web-page, 
while here we highlight only the  implementation detail relevant for our pkg.
\fi
\section{Mixed Formulation for the Poisson Equation}
In this example the Poisson equation is solved with a 
''mixed approach'': it is usedthe stable FE space obtained using Brezzi-Douglas-Marini 
polynomial of order 1 and Dicontinuos element of order 0.
\begin{align*}
-\mathrm{div}\ ( \mathbf{\sigma} (x, y) ) ) &= f (x, y) & \quad \mbox{ in } \Omega \\
\sigma (x, y) &= \nabla u (x, y) & \quad \mbox{ in } \Omega \\
u(x, y) &= 0 & \quad \mbox{ on } \Gamma_D \\
(\sigma (x, y) )  \cdot \mathbf{n} &= \sin (5x) & \quad \mbox{ on } \Gamma_N
\end{align*}

A complete description of the problem is avilable on the Fenics website \cite{mixedpois}.
\begin{changemargin}{-1.5cm}{-1.5cm}
$\phantom {u}$
\begin{parcolumns}[colwidths={1=0.65\textwidth,2=0.65\textwidth}]{2}
\colchunk{\begin{lstlisting}[caption=Fem-fenics, language=Octave, numbers=none]{Name}
pkg load fem-fenics msh
import_ufl_Problem ('MixedPoisson')

# Create mesh
x = y = linspace (0, 1, 33);
mesh = Mesh(msh2m_structured_mesh (x, y, 1, 1:4));

# File MixedPoisson.ufl
#  BDM = FiniteElement("BDM", triangle, 1)
#  DG  = FiniteElement("DG", triangle, 0)
#  W = BDM * DG
V = FunctionSpace('MixedPoisson', mesh);

# Define trial and test function
# File MixedPoisson.ufl
#  (sigma, u) = TrialFunctions(W)
#  (tau, v)   = TestFunctions(W)
#  CG = FiniteElement("CG", triangle, 1)
#  f = Coefficient(CG)
f = Expression ('f', 
      @(x,y) 10*exp(-((x - 0.5)^2 + (y - 0.5)^2) / 0.02));

# Define variational form
# File MixedPoisson.ufl
#  a = (dot(sigma, tau) + div(tau)*u + div(sigma)*v)*dx
#  L = - f*v*dx
a = BilinearForm ('MixedPoisson', V, V);
L = LinearForm ('MixedPoisson', V, f);


















# Define essential boundary

bc1 = DirichletBC (SubSpace (V, 1), @(x,y) [0; -sin(5.0*x)], 1);
bc2 = DirichletBC (SubSpace (V, 1), @(x,y) [0;  sin(5.0*x)], 3);

# Compute solution
[A, b] = assemble_system (a, L, bc1, bc2);
sol = A \ b;
func = Function ('func', V, sol);

sigma = Function ('sigma', func, 1);
u = Function ('u', func, 2);

# Plot solution
plot (sigma);
plot (u);


#
\end{lstlisting}}

\colchunk{\begin{lstlisting}[caption=Python, language=Python, numbers=none]{Name}
from dolfin import *


# Create mesh
mesh = UnitSquareMesh(32, 32)


# Define function spaces and mixed (product) space
BDM = FunctionSpace(mesh, "BDM", 1)
DG = FunctionSpace(mesh, "DG", 0)
W = BDM * DG



# Define trial and test functions
(sigma, u) = TrialFunctions(W)
(tau, v) = TestFunctions(W)


f = Expression
    ("10*exp(-(pow(x[0] - 0.5, 2) + pow(x[1] - 0.5, 2)) / 0.02)")

# Define variational form

a = (dot(sigma, tau) + div(tau)*u + div(sigma)*v)*dx
L = - f*v*dx



# Define function G such that G \cdot n = g
class BoundarySource(Expression):
    def __init__(self, mesh):
        self.mesh = mesh
    def eval_cell(self, values, x, ufc_cell):
        cell = Cell(self.mesh, ufc_cell.index)
        n = cell.normal(ufc_cell.local_facet)
        g = sin(5*x[0])
        values[0] = g*n[0]
        values[1] = g*n[1]
    def value_shape(self):
        return (2,)

G = BoundarySource(mesh)

# Define essential boundary
def boundary(x):
    return x[1] < DOLFIN_EPS or x[1] > 1.0 - DOLFIN_EPS

bc = DirichletBC(W.sub(0), G, boundary)

# Compute solution
w = Function(W)
solve(a == L, w, bc)


(sigma, u) = w.split()


# Plot sigma and u
plot(sigma)
plot(u)
interactive()

# Copyright 2011, The FEniCS Project
\end{lstlisting}}

\colplacechunks
\end{parcolumns}
\end{changemargin} 

\section{Incompressible Navier-Stokes equation}
In this example the incompressible Navier-Stokes equation
\begin{align*}
\dfrac{\partial u}{\partial t} + (\mathbf u \cdot \mathrm{\nabla})  \mathbf u - \nu \Delta \mathbf u 
  + \nabla p &= f & \quad \mbox{ in } \Omega \\
\mathrm{\nabla} \cdot \mathbf u &= 0 & \quad \mbox{ in } \Omega \\
\end{align*}
are solved using the Chorin-Temam algorithm. The L-shaped domain $\Omega$ can be obtained using
the msh pkg.
\begin{lstlisting}[language=Octave]
name = [tmpnam ".geo"];
fid = fopen (name, "w");
fputs (fid,"Point (1)  = {0, 0, 0, 0.1};\n");
fputs (fid,"Point (2)  = {1, 0, 0, 0.1};\n");
fputs (fid,"Point (3)  = {1, 0.5, 0, 0.1};\n");
fputs (fid,"Point (4)  = {0.5, 0.5, 0, 0.1};\n");
fputs (fid,"Point (5) = {0.5, 1, 0, 0.1};\n");
fputs (fid,"Point (6) = {0, 1, 0,0.1};\n");
 
fputs (fid,"Line (1)  = {5, 6};\n");
fputs (fid,"Line (2) = {2, 3};\n");
 
fputs (fid,"Line(3) = {6,1,2};\n");
fputs (fid,"Line(4) = {5,4,3};\n");
fputs (fid,"Line Loop(7) = {3,2,-4,1};\n");
fputs (fid,"Plane Surface(8) = {7};\n");
fclose (fid);
msho = msh2m_gmsh (canonicalize_file_name (name)(1:end-4),...
                   "scale", 1,"clscale", .2);
unlink (canonicalize_file_name (name));
\end{lstlisting}

The flow is driven by an oscillating pressure $p_{in}(t) = \sin 3t$ at the inflow 
while the pressure is kept constant $p_{out} = 0$ at the outflow.
A complete description of the problem is avilable on the Fenics website \cite{navierstokes}.

\begin{changemargin}{-1.5cm}{-1.5cm}
$\phantom {u}$
\begin{parcolumns}[colwidths={1=0.65\textwidth,2=0.65\textwidth}]{2}
\colchunk{\begin{lstlisting}[caption=Fem-fenics, language=Octave, numbers=none]{Name}
pkg load fem-fenics msh
import_ufl_Problem ("TentativeVelocity");
import_ufl_Problem ("VelocityUpdate");
import_ufl_Problem ("PressureUpdate");

# We can either load the mesh from the file as in Dolfin but 
# we can also use the msh pkg to generate the L-shape domain
# as showed above

mesh = Mesh ('lshape.xml');

# Define function spaces (P2-P1). UFL file
#  V = VectorElement("CG", triangle, 2)
#  Q = FiniteElement("CG", triangle, 1)
V = FunctionSpace ('VelocityUpdate', mesh);
Q = FunctionSpace ('PressureUpdate', mesh);

# Define trial and test functions. From ufl file
#  u = TrialFunction(V)
#  p = TrialFunction(Q)
#  v = TestFunction(V)
#  q = TestFunction(Q)

# Set parameter values. From ufl file
#  nu = 0.01
dt = 0.01;
T = 3.;




# Define boundary conditions
noslip = DirichletBC (V, @(x,y) [0; 0], [3, 4]);






outflow = DirichletBC (Q, @(x,y) 0, 2);




# Create functions
u0 = Expression ('u0', @(x,y) [0; 0]);



# Define coefficients
k = Constant ('k', dt);
f = Constant ('f', [0; 0]);

# Tentative velocity step. From ufl file
#  eq = (1/k)*inner(u - u0, v)*dx + inner(grad(u0)*u0, v)*dx \
#       + nu*inner(grad(u), grad(v))*dx - inner(f, v)*dx
a1 = BilinearForm ('TentativeVelocity', V, V, k);

# Pressure update. From ufl file
#  a = inner(grad(p), grad(q))*dx
#  L = -(1/k)*div(u1)*q*dx
a2 = BilinearForm ('PressureUpdate', Q, Q);

# Velocity update
#  a = inner(u, v)*dx
#  L = inner(u1, v)*dx - k*inner(grad(p1), v)*dx
a3 = BilinearForm ('VelocityUpdate', V, V);

# Assemble matrices
A1 = assemble (a1, noslip);

A3 = assemble (a3, noslip);










# Time-stepping
t = dt; i = 0;
while t < T

  # Update pressure boundary condition
  inflow = DirichletBC (Q, @(x,y) sin(3.0*t), 1);

  # Compute tentative velocity step
  "Computing tentative velocity"
  L1 = LinearForm ('TentativeVelocity', V, k, u0, f);
  b1 = assemble (L1, noslip);
  utmp = A1 \ b1;
  u1 = Function ('u1', V, utmp);

  # Pressure correction
  "Computing pressure correction"
  L2 = LinearForm ('PressureUpdate', Q, u1, k);
  [A2, b2] = assemble_system (a2, L2, inflow, outflow);
  ptmp = A2 \ b2;
  p1 = Function ('p1', Q, ptmp);

  # Velocity correction
  "Computing velocity correction"
  L3 = LinearForm ('VelocityUpdate', V, k, u1, p1);
  b3 = assemble (L3, noslip);
  ut = A3 \ b3;
  u1 = Function ('u0', V, ut);

  # Plot solution
  plot (p1);
  plot (u1);

  # Save to file
  save (p1, sprintf ("p_%3.3d", ++i));
  save (u1, sprintf ("u_%3.3d", i));

  # Move to next time step
  u0 = u1;
  t += dt

end



#
\end{lstlisting}}

\colchunk{\begin{lstlisting}[caption=Python, language=Python, numbers=none]{Name}
from dolfin import *




# Load mesh from file





mesh = Mesh("lshape.xml")

# Define function spaces (P2-P1)


V = VectorFunctionSpace(mesh, "CG", 2)
Q = FunctionSpace(mesh, "CG", 1)

# Define trial and test functions

u = TrialFunction(V)
p = TrialFunction(Q)
v = TestFunction(V)
q = TestFunction(Q)

# Set parameter values
nu = 0.01
dt = 0.01
T = 3

# Define time-dependent pressure BC
p_in = Expression("sin(3.0*t)", t=0.0)

# Define boundary conditions
noslip  = DirichletBC(V, (0, 0),
           "on_boundary && \
           (x[0] < DOLFIN_EPS | x[1] < DOLFIN_EPS | \
           (x[0] > 0.5 - DOLFIN_EPS && x[1] > 0.5 - DOLFIN_EPS))")
inflow  = DirichletBC(Q, p_in, "x[1] > 1.0 - DOLFIN_EPS")
outflow = DirichletBC(Q, 0, "x[0] > 1.0 - DOLFIN_EPS")
bcu = [noslip]
bcp = [inflow, outflow]

# Create functions
u0 = Function(V)
u1 = Function(V)
p1 = Function(Q)

# Define coefficients
k = Constant(dt)
f = Constant((0, 0))

# Tentative velocity step
F1 = (1/k)*inner(u - u0, v)*dx + inner(grad(u0)*u0, v)*dx \
     + nu*inner(grad(u), grad(v))*dx - inner(f, v)*dx
a1 = lhs(F1)
L1 = rhs(F1)

# Pressure update
a2 = inner(grad(p), grad(q))*dx
L2 = -(1/k)*div(u1)*q*dx


# Velocity update
a3 = inner(u, v)*dx
L3 = inner(u1, v)*dx - k*inner(grad(p1), v)*dx


# Assemble matrices
A1 = assemble(a1)
A2 = assemble(a2)
A3 = assemble(a3)

# Use amg preconditioner if available
prec = "amg" if has_krylov_solver_preconditioner("amg") 
             else "default"

# Create files for storing solution
ufile = File("results/velocity.pvd")
pfile = File("results/pressure.pvd")

# Time-stepping
t = dt
while t < T + DOLFIN_EPS:

    # Update pressure boundary condition
    p_in.t = t

    
    # Compute tentative velocity step
    begin("Computing tentative velocity")
    b1 = assemble(L1)
    [bc.apply(A1, b1) for bc in bcu]
    solve(A1, u1.vector(), b1, "gmres", "default")
    end()

    # Pressure correction
    begin("Computing pressure correction")
    b2 = assemble(L2)
    [bc.apply(A2, b2) for bc in bcp]
    solve(A2, p1.vector(), b2, "gmres", prec)
    end()

    
    # Velocity correction
    begin("Computing velocity correction")
    b3 = assemble(L3)
    [bc.apply(A3, b3) for bc in bcu]
    solve(A3, u1.vector(), b3, "gmres", "default")
    end()

    # Plot solution
    plot(p1, title="Pressure", rescale=True)
    plot(u1, title="Velocity", rescale=True)

    # Save to file
    ufile << u1
    pfile << p1

    # Move to next time step
    u0.assign(u1)
    t += dt
    print "t =", t

# Hold plot
interactive()

# Copyright 2011, The FEniCS Project
\end{lstlisting}}

\colplacechunks
\end{parcolumns}
\end{changemargin}

\section{HyperElasticity}
This time we compare the code with the c++ version of DOLFIN. 
The problem for an elastic material can be expressed as a minimization problem
\begin{align*}
\min_{u \in V} \Pi\\
\Pi &= \int_{\Omega} \psi(u) \, {\rm d} x - \int_{\Omega} B \cdot u \, {\rm d} x - \int_{\partial\Omega} T \cdot u \, 
{\rm d} s\\
\end{align*}
where $\Pi$  is the total potential energy, $\psi$ is the elastic stored energy, $B$  is a body force and $T$
is a traction force.

A complete description of the problem is avilable on the Fenics website \cite{hyperelasticity}.
The final solution will look like in figure \ref{Hyp}.
\begin{figure}
 \begin{center}
  \includegraphics[height=6 cm,keepaspectratio=true]{./HyperElasticity.png}
   \caption{Solution of the HyperElasticity problem}
   \label{Hyp}
  \end{center}
\end{figure}

\begin{lstlisting}[caption=UFL code, language=Python, numbers=none]
# Function spaces
element = VectorElement("Lagrange", tetrahedron, 1)

# Trial and test functions
du = TrialFunction(element)     # Incremental displacement
v  = TestFunction(element)      # Test function

# Functions
u = Coefficient(element)        # Displacement from previous iteration
B = Coefficient(element)        # Body force per unit volume
T = Coefficient(element)        # Traction force on the boundary

# Kinematics
I = Identity(element.cell().d)  # Identity tensor
F = I + grad(u)                 # Deformation gradient
C = F.T*F                       # Right Cauchy-Green tensor

# Invariants of deformation tensors
Ic = tr(C)
J  = det(F)

# Elasticity parameters
mu    = Constant(tetrahedron)
lmbda = Constant(tetrahedron)

# Stored strain energy density (compressible neo-Hookean model)
psi = (mu/2)*(Ic - 3) - mu*ln(J) + (lmbda/2)*(ln(J))**2

# Total potential energy
Pi = psi*dx - inner(B, u)*dx - inner(T, u)*ds

# First variation of Pi (directional derivative about u in the direction of v)
F = derivative(Pi, u, v)

# Compute Jacobian of F
J = derivative(F, u, du)

# Copyright 2011, The FEniCS Project
\end{lstlisting}

\begin{changemargin}{-1.5cm}{-1.5cm}
$\phantom {u}$
\begin{parcolumns}[colwidths={1=0.65\textwidth,2=0.65\textwidth}]{2}
\colchunk{\begin{lstlisting}[caption=Fem-fenics, language=Octave, numbers=none]{Name}
pkg load fem-fenics msh
problem = 'HyperElasticity';
import_ufl_Problem (problem);


































































Rotation = @(x,y,z) ...
 [0; ...
 0.5*(0.5 + (y - 0.5)*cos(pi/3) - (z-0.5)*sin(pi/3) - y);...
 0.5*(0.5 + (y - 0.5)*sin(pi/3) + (z-0.5)*cos(pi/3) - z)];






# Create mesh and define function space
x = y = z = linspace (0, 1, 17);
mshd = Mesh (msh3m_structured_mesh (x, y, z, 1, 1:6));
V  = FunctionSpace (problem, mshd);







# Create Dirichlet boundary conditions
bcl = DirichletBC (V, @(x,y,z) [0; 0; 0], 1);
bcr = DirichletBC (V, Rotation, 2);
bcs = {bcl, bcr};

# Define source and boundary traction functions
B = Constant ('B', [0.0; -0.5; 0.0]);
T = Constant ('T', [0.1; 0.0; 0.0]);




# Set material parameters
E = 10.0;
nu = 0.3;
mu = Constant ('mu', E./(2*(1+nu)));
lmbda = Constant ('lmbda', E*nu./((1+nu)*(1-2*nu)));
u = Expression ('u', @(x,y,z) [0; 0; 0]);

# Create (linear) form defining (nonlinear) variational problem
L = ResidualForm (problem, V, mu, lmbda, B, T, u);







# Solve nonlinear variational problem F(u; v) = 0
u0 = assemble (L, bcs{:});
# Create function for the resolution of the NL problem
function [y, jac] = f (problem, xx, V, bc1, bc2, B, T, mu, lmbda)
  u = Function ('u', V, xx);
  a = JacobianForm (problem, V, mu, lmbda, u);
  L = ResidualForm (problem, V, mu, lmbda, B, T, u);
  if (nargout == 1)
    [y, xx] = assemble (L, xx, bc1, bc2);
  elseif (nargout == 2)
    [jac, y, xx] = assemble_system (a, L, xx, bc1, bc2);
  endif
endfunction

fs = @(xx) f (problem, xx, V, bcl, bcr, B, T, mu, lmbda);
[x, fval, info] = fsolve (fs, u0, optimset ("jacobian", "on"));
func = Function ('u', V, x);

# Save solution in VTK format
save (func, 'displacement');


# Plot solution
plot (func);




#
\end{lstlisting}}

\colchunk{\begin{lstlisting}[caption=C++, language=C++, numbers=none]{Name}
#include <dolfin.h>
#include "HyperElasticity.h"

using namespace dolfin;

// Sub domain for clamp at left end
class Left : public SubDomain
{
  bool inside(const Array<double>& x, bool on_boundary) const
  {
    return (std::abs(x[0]) < DOLFIN_EPS) && on_boundary;
  }
};

// Sub domain for rotation at right end
class Right : public SubDomain
{
  bool inside(const Array<double>& x, bool on_boundary) const
  {
    return (std::abs(x[0] - 1.0) < DOLFIN_EPS) && on_boundary;
  }
};

// Dirichlet boundary condition for clamp at left end
class Clamp : public Expression
{
public:

  Clamp() : Expression(3) {}

  void eval(Array<double>& values, const Array<double>& x) const
  {
    values[0] = 0.0;
    values[1] = 0.0;
    values[2] = 0.0;
  }

};

// Dirichlet boundary condition for rotation at right end
class Rotation : public Expression
{
public:

  Rotation() : Expression(3) {}

  void eval(Array<double>& values, const Array<double>& x) const
  {
    const double scale = 0.5;

    // Center of rotation
    const double y0 = 0.5;
    const double z0 = 0.5;

    // Large angle of rotation (60 degrees)
    double theta = 1.04719755;

    // New coordinates
    double y = y0 + (x[1]-y0)*cos(theta) - (x[2]-z0)*sin(theta);
    double z = z0 + (x[1]-y0)*sin(theta) + (x[2]-z0)*cos(theta);

    // Rotate at right end
    values[0] = 0.0;
    values[1] = scale*(y - x[1]);
    values[2] = scale*(z - x[2]);
  }

};

int main()
{
  // Create mesh and define function space
  UnitCubeMesh mesh (16, 16, 16);
  HyperElasticity::FunctionSpace V(mesh);

  // Define Dirichlet boundaries
  Left left;
  Right right;

  // Define Dirichlet boundary functions
  Clamp c;
  Rotation r;

  // Create Dirichlet boundary conditions
  DirichletBC bcl(V, c, left);
  DirichletBC bcr(V, r, right);
  std::vector<const BoundaryCondition*> bcs;
  bcs.push_back(&bcl); bcs.push_back(&bcr);

  // Define source and boundary traction functions
  Constant B(0.0, -0.5, 0.0);
  Constant T(0.1,  0.0, 0.0);

  // Define solution function
  Function u(V);

  // Set material parameters
  const double E  = 10.0;
  const double nu = 0.3;
  Constant mu(E/(2*(1 + nu)));
  Constant lambda(E*nu/((1 + nu)*(1 - 2*nu)));


  
  // Create (linear) form defining (nonlinear) variational problem
  HyperElasticity::ResidualForm F(V);
  F.mu = mu; F.lmbda = lambda; F.B = B; F.T = T; F.u = u;

  // Create jacobian dF = F' (for use in nonlinear solver).
  HyperElasticity::JacobianForm J(V, V);
  J.mu = mu; J.lmbda = lambda; J.u = u;

  // Solve nonlinear variational problem F(u; v) = 0
  solve(F == 0, u, bcs, J);












  
  
  
  
  
  
  



  // Save solution in VTK format
  File file("displacement.pvd");
  file << u;

  // Plot solution
  plot(u);
  interactive();

  return 0;
}
# Copyright 2011, The FEniCS Project
\end{lstlisting}}

\colplacechunks
\end{parcolumns}
\end{changemargin}

\iffalse
\section{Fictitious Domain}
A penalization method to take into account obstacles in incompressible viscous flows
\fi
\newpage 

\backmatter

\appendix
\chapter{API reference}\label{app}

\section{Import problem defined with ufl}
\subsection*{import\_ufl\_BilinearForm}
\subimport{latex/}{API/import_ufl_BilinearForm.tex}
\subsection*{import\_ufl\_LinearForm}
\subimport{latex/}{API/import_ufl_LinearForm.tex}
 \subsection*{ import\_ufl\_Functional}
 \subimport{latex/}{API/import_ufl_Functional.tex}
 \subsection*{ import\_ufl\_FunctionSpace}
  \subimport{latex/}{API/import_ufl_FunctionSpace.tex}
 \subsection*{ import\_ufl\_Problem}
   \subimport{latex/}{API/import_ufl_Problem.tex}
\section{Problem geometry and FE space}
 \subsection*{ Mesh}
 \subimport{latex/}{API/Mesh.tex}
 \subsection*{ FunctionSpace}
  \subimport{latex/}{API/FunctionSpace.tex}
 \subsection*{ SubSpace}
   \subimport{latex/}{API/SubSpace.tex}
\section{Problem variables}
 \subsection*{ Constant}
   \subimport{latex/}{API/Constant.tex}
 \subsection*{ Expression}
   \subimport{latex/}{API/Expression.tex}
 \subsection*{ Function}
  \subimport{latex/}{API/Function.tex}
 \subsection*{ DirichletBC}
  \subimport{latex/}{API/DirichletBC.tex}
\section{Definition of the abstract Variational problem}
 \subsection*{ BilinearForm}
  \subimport{latex/}{API/BilinearForm.tex}
\subsection*{  LinearForm}
  \subimport{latex/}{API/LinearForm.tex}
\subsection*{  ResidualForm}
  \subimport{latex/}{API/ResidualForm.tex}
 \subsection*{ JacobianForm}
   \subimport{latex/}{API/JacobianForm.tex}
 \subsection*{ Functional}
      \subimport{latex/}{API/Functional.tex}
\section{Creation of the discretized problem}
\subsection*{  assemble}
\subimport{latex/}{API/assemble.tex}
\subsection*{  assemble\_system}
\subimport{latex/}{API/assemble_system.tex}
\section{Post processing}
\subsection*{  @function/save}
\subimport{latex/}{API/save.tex}
\subsection*{  @function/plot}
\subimport{latex/}{API/plot.tex}
\subsection*{  @mesh/plot}
\subimport{latex/}{API/plot_m.tex}
\subsection*{  @function/feval}
\subimport{latex/}{API/feval.tex}




\chapter{Autoconf and Automake}
 In this section we want to discuss how we can write a config.ac and a Makefile.in files which:
\begin{itemize}
    \item check if a program is available and stop if it is not
    \item check if a header file is available and issue a warning if not, but go ahead with the compilation
\end{itemize}

To reach this goal, we need two components:

\paragraph{configure.ac} Is a file which checks whether the program/header is available or not 
and sets consequently the values of some variables.
\begin{lstlisting}[language=make]
    # Checks if the program mkoctfile is available and sets the variable HAVE_MKOCTFILE consequently
    AC_CHECK_PROG([HAVE_MKOCTFILE], [mkoctfile], [yes], [no])
    # if mkoctfile is not available, it issues an error and stops the compilation
    if [test $HAVE_MKOCTFILE = "no"]; then 
      AC_MSG_ERROR([mkoctfile required to install $PACKAGE_NAME])
    fi

    #Checks if the header dolfin.h is available; if it is available, the value of the ac_dolfin_cpp_flags is substituted with -DHAVE_DOLFIN_H, otherwise it is left empty and a warning message is printed
    AC_CHECK_HEADER([dolfin.h],
      [AC_SUBST(ac_dolfin_cpp_flags,-DHAVE_DOLFIN_H)  AC_SUBST(ac_dolfin_ld_flags,-ldolfin)],
      [AC_MSG_WARN([dolfin headers could not be found, some functionalities will be disabled, don't worry your package will still be working, though.])] ).

    # It generates the Makefile, using the template described below
    AC_CONFIG_FILES([Makefile])
\end{lstlisting} 
\paragraph{Makefile.ac} This file is a template for the Makefile, which will be automatically generated when the configure.ac 
file is executed. The values of the variable \verb$ac_dolfin_cpp_flags$ and \verb$ac_dolfin_ld_flags$ are substituted with the 
results obtained above:
\begin{lstlisting}[language=make]
    CPPFLAGS += @ac_dolfin_cpp_flags@
    LDFLAGS += @ac_dolfin_ld_flags@
\end{lstlisting}

In this way, if dolfin.h is available, CPPFLAGS contains also the flag -DHAVE\_DOLFIN\_H.

\paragraph {program.cc}  Our .cc program, should thus include the header dolfin.h only if 
\verb$-DHAVE_DOLFIN_H$ is defined at compilation time.
For example

\begin{lstlisting}
    #ifdef HAVE_DOLFIN_H
    #include <dolfin.h> 
    #endif
    int main ()
    {  

    #ifndef HAVE_DOLFIN_H
        error("program: the program was built without support for dolfin");
    #else 
      /* Body of your function */
    #endif
     return 0;
    }

\end{lstlisting} 
\iffalse
\paragraph {Warning} If in the Makefile.in you write something like
\begin{lstlisting}[language=make]
    HAVE_DOLFIN_H = @HAVE_DOLFIN_H@  
    ifdef HAVE_DOLFIN_H   
      CPPFLAGS += -DHAVE_DOLFIN_H  
      LIBS += -ldolfin
    endif
 \end{lstlisting} 
 it doesn't work because the variable \verb$HAVE_DOLFIN_H$ seems to be always defined, even if the header is not available.
\fi

\bibliographystyle{unsrt} 
\bibliography{doc}
\end{document}