This book gives a concise and gentle introduction to finite element programming in Python based on the popular FEniCS software library. FEniCS can be programmed in both C++ and Python, but this tutorial focuses exclusively on Python programming, since this is the simplest and most effective approach for beginners. It will also deliver high performance since FEniCS automatically delegates compute-intensive tasks to C++ by help of code generation. After having digested the examples in this tutorial, the reader should be able to learn more from the FEniCS documentation, the numerous demo programs that come with the software, and the comprehensive FEniCS book Automated Solution of Differential Equations by the Finite element Method [1]. This tutorial is a further development of the opening chapter in [1].
We thank Johan Hake, Kent-Andre Mardal, and Kristian Valen-Sendstad for many helpful discussions during the preparation of the first version of this tutorial for the FEniCS book [1]. We are particularly thankful to Professor Douglas Arnold for very valuable feedback on early versions of the text. Øystein Sørensen pointed out a lot of typos and contributed with many helpful comments. Many errors and typos were also reported by Mauricio Angeles, Ida Drøsdal, Miroslav Kuchta, Hans Ekkehard Plesser, Marie Rognes, and Hans Joachim Scroll. Ekkehard Ellmann as well as two anonymous reviewers provided a series of suggestions and improvements.
Special thank goes to Benjamin Kehlet for all his work with the mshr
tool and for quickly implementing our requests in this tutorial.
The FEniCS Project is a research and software project aiming at
creating mathematical methods and software for automated computational
mathematical modeling. This means creating easy, intuitive, efficient,
and flexible software for solving partial differential equations
(PDEs) using finite element methods. FEniCS was initially created in
2003 and is developed in collaboration between researchers from a
number of universities and research institutes around the world. For
more information about FEniCS and the latest updates of the FEniCS
software and this tutorial, visit the FEniCS web page at
http://fenicsproject.org.
FEniCS consists of a number of building blocks (software components) that together form the FEniCS software: DOLFIN, FFC, FIAT, UFL, mshr, and a few others. FEniCS users rarely need to think about this internal organization of FEniCS, but since even casual users may sometimes encounter the names of various FEniCS components, we briefly list the components and their main roles in FEniCS. DOLFIN is the computational high-performance C++ backend of FEniCS. DOLFIN implements data structures such as meshes, function spaces and functions, compute-intensive algorithms such as finite element assembly and mesh refinement, and interfaces to linear algebra solvers and data structures such as PETSc. DOLFIN also implements the FEniCS problem-solving environment in both C++ and Python. FFC is the code generation engine of FEniCS (the form compiler), responsible for generating efficient C++ code from high-level mathematical abstractions. FIAT is the finite element backend of FEniCS, responsible for generating finite element basis functions, UFL implements the abstract mathematical language by which users may express variational problems, and mshr provides FEniCS with mesh generation capabilities.
The goal of this tutorial is to introduce the concept of programming finite element solvers for PDEs and get you started with FEniCS through a series of simple examples that demonstrate
The mathematics of the illustrations is kept simple to better focus on FEniCS functionality and syntax. This means that we mostly use the Poisson equation and the time-dependent diffusion equation as model problems, often with input data adjusted such that we get a very simple solution that can be exactly reproduced by any standard finite element method over a uniform, structured mesh. This latter property greatly simplifies the verification of the implementations. Occasionally we insert a physically more relevant example to remind the reader that the step from solving a simple model problem to a challenging real-world problem is often quite short and easy with FEniCS.
Using FEniCS to solve PDEs may seem to require a thorough understanding of the abstract mathematical framework of the finite element method as well as expertise in Python programming. Nevertheless, it turns out that many users are able to pick up the fundamentals of finite elements and Python programming as they go along with this tutorial. Simply keep on reading and try out the examples. You will be amazed at how easy it is to solve PDEs with FEniCS!
Reading this tutorial obviously requires access to the FEniCS software. FEniCS is a complex software library, both in itself and due to its many dependencies to state-of-the-art open-source scientific software libraries. Manually building FEniCS and all its dependencies from source can thus be a daunting task. Even for an expert who knows exactly how to configure and build each component, a full build can literally take hours! In addition to the complexity of the software itself, there is an additional layer of complexity in how many different kinds of operating systems (GNU/Linux, Mac OS X, Windows) that may be running on a user's laptop or compute server, with different requirements for how to configure and build software.
For this reason, the FEniCS Project provides prebuilt packages to make the installation easy, fast, and foolproof.
http://fenicsproject.org/download.
A modern solution to the challenge of software installation on diverse software platforms is to use so-called containers. The FEniCS Project provides custom-made containers that are controlled, consistent, and high-performance software environments for FEniCS programming. FEniCS containers work equally well on all operating systems, including Linux, Mac and Windows.
1: Running Docker containers on Mac and Windows involves a small performance overhead compared to running Docker containers on Linux. However, this performance penalty is typically small and is often compensated for by using the highly tuned and optimized version of FEniCS that comes with the official FEniCS containers, compared to building FEniCS and its dependencies from source on Mac or Windows.
To use FEniCS containers, you must first install the Docker platform. Docker installation is simple, just follow the instructions from the Docker web page. Once you have installed Docker, just copy the following line into a terminal window:
Terminal> curl -s https://get.fenicsproject.org | bash
The command above will install the program fenicsproject on your system. This
command lets you easily create FEniCS sessions (containers) on your
system:
Terminal> fenicsproject run
This command has several useful options, such as easily switching
between the latest release of FEniCS, the latest development version
and many more. To learn more, type fenicsproject help.
fenicsproject run, it will
automatically share your current working directory (the directory
from which you run the fenicsproject command) with the FEniCS
session. When the FEniCS session starts, it will automatically
enter into a directory named shared which will be identical with
your current working directory on your host system. This means that
you can easily edit files and write data inside the FEniCS session, and
the files will be directly accessible on your host system. It is
recommended that you edit your programs using your favorite editor
(such as Emacs or Vim) on your host system and use the FEniCS session
only to run your program(s).
For users of Ubuntu GNU/Linux, FEniCS can also be installed easily via
the standard Ubuntu package manager apt-get. Just copy the following
lines into a terminal window:
Terminal> sudo add-apt-repository ppa:fenics-packages/fenics
Terminal> sudo apt-get update
Terminal> sudo apt-get install fenics
Terminal> sudo apt-get dist-upgrade
This will add the FEniCS package archive (PPA) to your Ubuntu computer's list of software sources and then install FEniCS. This step will also automatically install packages for dependencies of FEniCS.
Once you have installed FEniCS, you should make a quick test to see that your installation works properly. To do this, type the following command in a FEniCS-enabled terminal:
2: For users of FEniCS containers, this means first
running the command fenicsproject run.
Terminal> python -c 'import fenics'
If all goes well, you should be able to run this command without any error message (or any other output).
In this tutorial, you will learn finite element and FEniCS programming
through a number of example programs that demonstrate both how to
solve particular PDEs using the finite element method, how to program
solvers in FEniCS, and how to create well-designed Python codes that
can later be extended to solve more complex problems. All
example programs are available from the web page of this book at
http://fenicsproject.org/tutorial. The programs as well as the
source code for this text can also be accessed directly from the Git
repository for this book.
While you can likely pick up basic Python programming by working through the examples in this tutorial, you may want to have some additional material on the Python language. A natural starting point for beginners is the classical Python Tutorial [2], or a tutorial geared towards scientific computing [3]. In the latter, you will also find pointers to other tutorials for scientific computing in Python. Among ordinary books we recommend the general introduction Dive into Python [4] as well as texts that focus on scientific computing with Python [5] [6] [7] [8] [9].
from __future__ import print_function
to enable the print function from Python 3 in Python 2. All
use of print in the programs in this tutorial consists of function
calls, like print('a:', a). Almost all other constructions are of
a form that looks the same in Python 2 and 3.
There exist many good books on the finite element method. The books typically fall in either of two categories: the abstract mathematical version of the method or the engineering "structural analysis" formulation. FEniCS builds heavily on concepts from the abstract mathematical exposition. The first author has in development a book [10] that explains all details of the finite element method in an intuitive way, though with the abstract mathematical formulations that FEniCS employ.
The finite element text by Larson and Bengzon [11] is our recommended introduction to the finite element method, with a mathematical notation that goes well with FEniCS. An easy-to-read book, which also provides a good general background for using FEniCS, is Gockenbach [12]. The book by Donea and Huerta [13] has a similar style, but aims at readers with interest in fluid flow problems. Hughes [14] is also recommended, especially for those interested in solid mechanics and heat transfer applications.
Readers with a background in the engineering "structural analysis" version of the finite element method may find Bickford [15] as an attractive bridge over to the abstract mathematical formulation that FEniCS builds upon. Those who have a weak background in differential equations in general should consult a more fundamental book, and Eriksson et al [16] is a very good choice. On the other hand, FEniCS users with a strong background in mathematics and interest in the mathematical properties of the finite element method, will appreciate the texts by Brenner and Scott [17], Braess [18], Ern and Guermond [19], Quarteroni and Valli [20], or Ciarlet [21].