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FEniCS101.html
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<!DOCTYPE html>
<html>
<head><meta charset="utf-8" />
<title>FEniCS101</title>
<script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.0.3/jquery.min.js"></script>
<style type="text/css">
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<h1 id="FEniCS101-Tutorial">FEniCS101 Tutorial<a class="anchor-link" href="#FEniCS101-Tutorial">¶</a></h1><p>In this tutorial we consider the boundary value problem (BVP)</p>
\begin{eqnarray*}
- \nabla \cdot (k \nabla u) = f & \text{ in } \Omega,\\
u = u_0 & \text{ on } \Gamma_D = \Gamma_{\rm left} \bigcup \Gamma_{\rm right},\\
k \frac{\partial u}{\partial {\bf{n}}} = \sigma & \text{ on } \Gamma_N = \Gamma_{\rm top} \bigcup \Gamma_{\rm bottom},
\end{eqnarray*}<p>where $\Omega = (0,1) \times (0,1)$, $\Gamma_D$ and and $\Gamma_N$ are the union of
the left and right, and top and bottom boundaries of $\Omega$,
respectively.</p>
<p>Here
\begin{eqnarray<em>}
k(x,y) = 1 & \text{ on } \Omega\
f(x,y) = \left(4\pi^2+\frac{\pi^2}{4}\right)\sin(2 \pi x) \sin\left(\frac{\pi}{2} y\right) & \text{ on } \Omega\
u<em>0(x,y) = 0 & \text{ on } \Gamma_D, \
\sigma(x) = \left{ \begin{array}{l} -\frac{\pi}{2}\sin(2 \pi x) \ 0 \end{array}\right.
& \begin{array}{l} \text{ on } \Gamma</em>{\rm bottom},\ \text{ on } \Gamma_{\rm top}.\end{array}
\end{eqnarray</em>}</p>
<p>The exact solution is
$$ u_e(x,y) = \sin(2\pi x)\sin\left(\frac{\pi}{2}y\right). $$</p>
<h2 id="Weak-formulation">Weak formulation<a class="anchor-link" href="#Weak-formulation">¶</a></h2><p>Let us define the Hilbert spaces $V_{u_0}, V_0 \in \Omega$ as
$$ V_{u_0} := \left\{ v \in H^1(\Omega) \text{ s. t. } v = u_0 \text{ on } \Gamma_D \right\},$$
$$ V_{0} := \left\{ v \in H^1(\Omega) \text{ s. t. } v = 0 \text{ on } \Gamma_D \right\}.$$</p>
<p>To obtain the weak formulation, we multiply the PDE by an arbitrary function $v \in V_0$ and integrate over the domain $\Omega$ leading to</p>
$$ -\int_{\Omega} \nabla \cdot (k \nabla u) v \, dx = \int_\Omega f v \, dx\quad \forall \; v \in V_0. $$<p>Then, integration by parts the non-conforming term gives</p>
$$ \int_{\Omega} k \nabla u \cdot \nabla v \, dx - \int_{\partial \Omega} k \frac{\partial u}{\partial {\bf n} } v \, ds = \int_\Omega f v \, dx \quad \forall \; v \in V_0. $$<p>Finally by recalling that $ v = 0 $ on $\Gamma_D$ and that $k \frac{\partial u}{\partial {\bf n} } = \sigma $ on $\Gamma_N$, we find the weak formulation:</p>
<p><em>Find </em> $u \in V_{u_0}$ <em>such that</em>
$$ \int_{\Omega} k \nabla u \cdot \nabla v \, dx = \int_\Omega f v \, dx + \int_{\Gamma_N} \sigma v \, ds \quad \forall \; v \in V_0. $$</p>
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<h2 id="1.-Load-modules">1. Load modules<a class="anchor-link" href="#1.-Load-modules">¶</a></h2><p>To start we load the following modules:</p>
<ul>
<li><p>dolfin: the python/C++ interface to FEniCS</p>
</li>
<li><p><a href="https://docs.python.org/2/library/math.html">math</a>: the python module for mathematical functions</p>
</li>
<li><p><a href="http://www.numpy.org/">numpy</a>: a python package for linear algebra</p>
</li>
<li><p><a href="http://matplotlib.org/">matplotlib</a>: a python package used for plotting the results</p>
</li>
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<div class=" highlight hl-ipython2"><pre><span class="kn">from</span> <span class="nn">dolfin</span> <span class="kn">import</span> <span class="o">*</span>
<span class="kn">import</span> <span class="nn">math</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="kn">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">logging</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="kn">as</span> <span class="nn">plt</span>
<span class="o">%</span><span class="k">matplotlib</span> inline
<span class="kn">import</span> <span class="nn">nb</span>
<span class="n">logging</span><span class="o">.</span><span class="n">getLogger</span><span class="p">(</span><span class="s">'FFC'</span><span class="p">)</span><span class="o">.</span><span class="n">setLevel</span><span class="p">(</span><span class="n">logging</span><span class="o">.</span><span class="n">WARNING</span><span class="p">)</span>
<span class="n">logging</span><span class="o">.</span><span class="n">getLogger</span><span class="p">(</span><span class="s">'UFL'</span><span class="p">)</span><span class="o">.</span><span class="n">setLevel</span><span class="p">(</span><span class="n">logging</span><span class="o">.</span><span class="n">WARNING</span><span class="p">)</span>
<span class="n">set_log_active</span><span class="p">(</span><span class="bp">False</span><span class="p">)</span>
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<h2 id="2.-Define-the-mesh-and-the-finite-element-space">2. Define the mesh and the finite element space<a class="anchor-link" href="#2.-Define-the-mesh-and-the-finite-element-space">¶</a></h2><p>We construct a triangulation (mesh) $\mathcal{T}_h$ of the computational domain $\Omega := [0, 1]^2$ with <code>n</code> elements in each direction.</p>
<p>On the mesh $\mathcal{T}_h$, we then define the finite element space $V_h \subset H^1(\Omega)$ consisting of globally continuous piecewise polinomials functions. The <code>degree</code> variable defines the polinomial degree.</p>
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<div class=" highlight hl-ipython2"><pre><span class="n">n</span> <span class="o">=</span> <span class="mi">16</span>
<span class="n">degree</span> <span class="o">=</span> <span class="mi">1</span>
<span class="n">mesh</span> <span class="o">=</span> <span class="n">RectangleMesh</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
<span class="n">nb</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">mesh</span><span class="p">)</span>
<span class="n">Vh</span> <span class="o">=</span> <span class="n">FunctionSpace</span><span class="p">(</span><span class="n">mesh</span><span class="p">,</span> <span class="s">'Lagrange'</span><span class="p">,</span> <span class="n">degree</span><span class="p">)</span>
<span class="k">print</span> <span class="s">"dim(Vh) = "</span><span class="p">,</span> <span class="n">Vh</span><span class="o">.</span><span class="n">dim</span><span class="p">()</span>
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<pre>dim(Vh) = 289
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<h2 id="3.-Define-boundary-labels">3. Define boundary labels<a class="anchor-link" href="#3.-Define-boundary-labels">¶</a></h2><p>To partition the boundary of $\Omega$ in the subdomains $\Gamma_{\rm top}$, $\Gamma_{\rm bottom}$, $\Gamma_{\rm left}$, $\Gamma_{\rm right}$ we assign a unique label <code>boundary_parts</code> to each of part of $\partial \Omega$.</p>
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<div class=" highlight hl-ipython2"><pre><span class="k">class</span> <span class="nc">TopBoundary</span><span class="p">(</span><span class="n">SubDomain</span><span class="p">):</span>
<span class="k">def</span> <span class="nf">inside</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">on_boundary</span><span class="p">):</span>
<span class="k">return</span> <span class="n">on_boundary</span> <span class="ow">and</span> <span class="nb">abs</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o"><</span> <span class="n">DOLFIN_EPS</span>
<span class="k">class</span> <span class="nc">BottomBoundary</span><span class="p">(</span><span class="n">SubDomain</span><span class="p">):</span>
<span class="k">def</span> <span class="nf">inside</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">on_boundary</span><span class="p">):</span>
<span class="k">return</span> <span class="n">on_boundary</span> <span class="ow">and</span> <span class="nb">abs</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span> <span class="o"><</span> <span class="n">DOLFIN_EPS</span>
<span class="k">class</span> <span class="nc">LeftBoundary</span><span class="p">(</span><span class="n">SubDomain</span><span class="p">):</span>
<span class="k">def</span> <span class="nf">inside</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">on_boundary</span><span class="p">):</span>
<span class="k">return</span> <span class="n">on_boundary</span> <span class="ow">and</span> <span class="nb">abs</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="o"><</span> <span class="n">DOLFIN_EPS</span>
<span class="k">class</span> <span class="nc">RightBoundary</span><span class="p">(</span><span class="n">SubDomain</span><span class="p">):</span>
<span class="k">def</span> <span class="nf">inside</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">on_boundary</span><span class="p">):</span>
<span class="k">return</span> <span class="n">on_boundary</span> <span class="ow">and</span> <span class="nb">abs</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o"><</span> <span class="n">DOLFIN_EPS</span>
<span class="n">boundary_parts</span> <span class="o">=</span> <span class="n">FacetFunction</span><span class="p">(</span><span class="s">"size_t"</span><span class="p">,</span> <span class="n">mesh</span><span class="p">)</span>
<span class="n">boundary_parts</span><span class="o">.</span><span class="n">set_all</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="n">Gamma_top</span> <span class="o">=</span> <span class="n">TopBoundary</span><span class="p">()</span>
<span class="n">Gamma_top</span><span class="o">.</span><span class="n">mark</span><span class="p">(</span><span class="n">boundary_parts</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">Gamma_bottom</span> <span class="o">=</span> <span class="n">BottomBoundary</span><span class="p">()</span>
<span class="n">Gamma_bottom</span><span class="o">.</span><span class="n">mark</span><span class="p">(</span><span class="n">boundary_parts</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
<span class="n">Gamma_left</span> <span class="o">=</span> <span class="n">LeftBoundary</span><span class="p">()</span>
<span class="n">Gamma_left</span><span class="o">.</span><span class="n">mark</span><span class="p">(</span><span class="n">boundary_parts</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
<span class="n">Gamma_right</span> <span class="o">=</span> <span class="n">RightBoundary</span><span class="p">()</span>
<span class="n">Gamma_right</span><span class="o">.</span><span class="n">mark</span><span class="p">(</span><span class="n">boundary_parts</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
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<h2 id="4.-Define-the-coefficients-of-the-PDE-and-the-boundary-conditions">4. Define the coefficients of the PDE and the boundary conditions<a class="anchor-link" href="#4.-Define-the-coefficients-of-the-PDE-and-the-boundary-conditions">¶</a></h2><p>We first define the coefficients of the PDE using the <code>Constant</code> and <code>Expression</code> classes. <code>Constant</code> is used to define coefficients that do not depend on the space coordinates, <code>Expression</code> is used to define coefficients that are a known function of the space coordinates <code>x[0]</code> (x-axis direction) and <code>x[1]</code> (y-axis direction).</p>
<p>In the finite element method community, Dirichlet boundary conditions are also known as <em>essential</em> boundary conditions since they are imposed directly in the definition of the finite element space. In FEniCS, we use the class <code>DirichletBC</code> to indicate this type of condition.</p>
<p>On the other hand, Newman boundary conditions are also known as <em>natural</em> boundary conditions since they are weakly imposed as boundary integrals in the variational formulation (weak form). In FEniCS, we create a new boundary measure <code>ds[i]</code> to integrate over the portion of the boundary marked with label <code>i</code>.</p>
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<div class=" highlight hl-ipython2"><pre><span class="n">u_L</span> <span class="o">=</span> <span class="n">Constant</span><span class="p">(</span><span class="mf">0.</span><span class="p">)</span>
<span class="n">u_R</span> <span class="o">=</span> <span class="n">Constant</span><span class="p">(</span><span class="mf">0.</span><span class="p">)</span>
<span class="n">sigma_bottom</span> <span class="o">=</span> <span class="n">Expression</span><span class="p">(</span><span class="s">'-(pi/2.0)*sin(2*pi*x[0])'</span><span class="p">)</span>
<span class="n">sigma_top</span> <span class="o">=</span> <span class="n">Expression</span><span class="p">(</span><span class="s">'0'</span><span class="p">)</span>
<span class="n">f</span> <span class="o">=</span> <span class="n">Expression</span><span class="p">(</span><span class="s">'(4.0*pi*pi+pi*pi/4.0)*(sin(2*pi*x[0])*sin((pi/2.0)*x[1]))'</span><span class="p">)</span>
<span class="n">bcs</span> <span class="o">=</span> <span class="p">[</span><span class="n">DirichletBC</span><span class="p">(</span><span class="n">Vh</span><span class="p">,</span> <span class="n">u_L</span><span class="p">,</span> <span class="n">boundary_parts</span><span class="p">,</span> <span class="mi">3</span><span class="p">),</span>
<span class="n">DirichletBC</span><span class="p">(</span><span class="n">Vh</span><span class="p">,</span> <span class="n">u_R</span><span class="p">,</span> <span class="n">boundary_parts</span><span class="p">,</span> <span class="mi">4</span><span class="p">)]</span>
<span class="n">ds</span> <span class="o">=</span> <span class="n">Measure</span><span class="p">(</span><span class="s">"ds"</span><span class="p">,</span> <span class="n">subdomain_data</span><span class="o">=</span><span class="n">boundary_parts</span><span class="p">)</span>
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<h2 id="5.-Define-and-solve-the-variational-problem">5. Define and solve the variational problem<a class="anchor-link" href="#5.-Define-and-solve-the-variational-problem">¶</a></h2><p>We also define two special types of functions: the <code>TrialFunction</code> <code>u</code> and the <code>TestFunction</code> <code>v</code>. These special types of function are used by <code>FEniCS</code> to generate the finite element vectors and matrices which stem from the weak formulation of the PDE.</p>
<p>More specifically, by denoting by $\left[{\phi_i(x)}\right]_{i=1}^{{\rm dim}(V_h)}$ the finite element basis for the space $V_h$, a function $u_h \in V_h$ can be written as
$$ u_h = \sum_{i=1}^{{\rm dim}(V_h)} {\rm u}_i \phi_i(x), $$
where ${\rm u}_i$ represents the coefficients in the finite element expansion of $u_h$.</p>
<p>We then define</p>
<ul>
<li><p>the bilinear form $a(u_h, v_h) = \int_\Omega \nabla u_h \cdot \nabla v_h dx $;</p>
</li>
<li><p>the linear form $L(v_h) = \int_\Omega f v_h dx + + \int_{\Gamma_{\rm top}} \sigma_{\rm top} v_h ds \int_{\Gamma_{\rm bottom}} \sigma_{\rm bottom} v_h ds $.</p>
</li>
</ul>
<p>We can then solve the variational problem</p>
<p><em>Find </em>$u_h \in V_h$<em> such that</em>
$$ a(u_h, v_h) = L(v_h) \quad \forall\, v_h \in V_h $$</p>
<p>using directly the built-in <code>solve</code> method in FEniCS.</p>
<p><strong>NOTE:</strong> As an alternative one can also assemble the finite element matrix $A$ and the right hand side $b$ that stems from the discretization of $a$ and $L$, and then solve the linear system
$$ A {\rm u} = {\rm b}, $$
where</p>
<ul>
<li><p>${\rm u}$ is the vector collecting the coefficient of the finite element expasion of $u_h$,</p>
</li>
<li><p>the entries of the matrix A are such that $A_{ij} = a(\phi_j, \phi_i)$,</p>
</li>
<li><p>the entries of the right hand side b are such that $b_i = L(\phi_i)$.</p>
</li>
</ul>
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<div class=" highlight hl-ipython2"><pre><span class="n">u</span> <span class="o">=</span> <span class="n">TrialFunction</span><span class="p">(</span><span class="n">Vh</span><span class="p">)</span>
<span class="n">v</span> <span class="o">=</span> <span class="n">TestFunction</span><span class="p">(</span><span class="n">Vh</span><span class="p">)</span>
<span class="n">a</span> <span class="o">=</span> <span class="n">inner</span><span class="p">(</span><span class="n">nabla_grad</span><span class="p">(</span><span class="n">u</span><span class="p">),</span> <span class="n">nabla_grad</span><span class="p">(</span><span class="n">v</span><span class="p">))</span><span class="o">*</span><span class="n">dx</span>
<span class="n">L</span> <span class="o">=</span> <span class="n">f</span><span class="o">*</span><span class="n">v</span><span class="o">*</span><span class="n">dx</span> <span class="o">+</span> <span class="n">sigma_top</span><span class="o">*</span><span class="n">v</span><span class="o">*</span><span class="n">ds</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="n">sigma_bottom</span><span class="o">*</span><span class="n">v</span><span class="o">*</span><span class="n">ds</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
<span class="n">uh</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="n">Vh</span><span class="p">)</span>
<span class="c">#solve(a == L, uh, bcs=bcs)</span>
<span class="n">A</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">assemble_system</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">L</span><span class="p">,</span> <span class="n">bcs</span><span class="o">=</span><span class="n">bcs</span><span class="p">)</span>
<span class="n">solve</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">uh</span><span class="o">.</span><span class="n">vector</span><span class="p">(),</span> <span class="n">b</span><span class="p">,</span> <span class="s">"cg"</span><span class="p">)</span>
<span class="n">nb</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">uh</span><span class="p">)</span>
</pre></div>
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<div class="output_area"><div class="prompt output_prompt">Out[5]:</div>
<div class="output_text output_subarea output_execute_result">
<pre><matplotlib.collections.TriMesh at 0x11664fd10></pre>
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