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<span id="reissnermindlindg"></span><h1>Reissner-Mindlin plate with a Discontinuous-Galerkin approach<a class="headerlink" href="#reissner-mindlin-plate-with-a-discontinuous-galerkin-approach" title="Permalink to this headline"></a></h1>
<div class="section" id="introduction">
<h2>Introduction<a class="headerlink" href="#introduction" title="Permalink to this headline"></a></h2>
<p>This program solves the Reissner-Mindlin plate equations on the unit
square with uniform transverse loading and clamped boundary conditions.
The corresponding file can be obtained from <a class="reference download internal" href="../../_downloads/reissner_mindlin_dg.py" download=""><code class="xref download docutils literal"><span class="pre">reissner_mindlin_dg.py</span></code></a>.</p>
<p>It uses a Discontinuous Galerkin interpolation for the rotation field to
remove shear-locking issues in the thin plate limit. Details of the formulation
can be found in <a class="reference internal" href="#han2011" id="id1">[HAN2011]</a>.</p>
<p>The solution for <span class="math">\(\theta_x\)</span> on the middle line of equation <span class="math">\(y=0.5\)</span>
will look as follows for 10 elements and a stabilization parameter <span class="math">\(s=1\)</span>:</p>
<a class="reference internal image-reference" href="../../_images/dg_rotation_N10_s1.png"><img alt="../../_images/dg_rotation_N10_s1.png" src="../../_images/dg_rotation_N10_s1.png" style="width: 546.15px; height: 340.5px;" /></a>
</div>
<div class="section" id="implementation">
<h2>Implementation<a class="headerlink" href="#implementation" title="Permalink to this headline"></a></h2>
<p>Material properties and loading are the same as in <a class="reference internal" href="reissner_mindlin_quads.py.html#reissnermindlinquads"><span class="std std-ref">Reissner-Mindlin plate with Quadrilaterals</span></a>:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">__future__</span> <span class="k">import</span> <span class="n">print_function</span>
<span class="kn">from</span> <span class="nn">fenics</span> <span class="k">import</span> <span class="o">*</span>

<span class="n">E</span> <span class="o">=</span> <span class="n">Constant</span><span class="p">(</span><span class="mf">1e3</span><span class="p">)</span>
<span class="n">nu</span> <span class="o">=</span> <span class="n">Constant</span><span class="p">(</span><span class="mf">0.3</span><span class="p">)</span>
<span class="n">thick</span> <span class="o">=</span> <span class="n">Constant</span><span class="p">(</span><span class="mf">1e-2</span><span class="p">)</span>
<span class="n">D</span> <span class="o">=</span> <span class="n">E</span><span class="o">*</span><span class="n">thick</span><span class="o">**</span><span class="mi">3</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">nu</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="mf">12.</span>
<span class="n">F</span> <span class="o">=</span> <span class="n">E</span><span class="o">/</span><span class="mi">2</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">nu</span><span class="p">)</span><span class="o">*</span><span class="n">thick</span><span class="o">*</span><span class="mf">5.</span><span class="o">/</span><span class="mf">6.</span>
<span class="n">f</span> <span class="o">=</span> <span class="n">Constant</span><span class="p">(</span><span class="o">-</span><span class="n">thick</span><span class="o">**</span><span class="mi">3</span><span class="p">)</span>
</pre></div>
</div>
<p>The unit square mesh is here divided in triangles and we get the facet MeshFunction for the integration measure <span class="math">\(\text{d}s\)</span>:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="n">N</span> <span class="o">=</span> <span class="mi">40</span>
<span class="n">mesh</span> <span class="o">=</span> <span class="n">UnitSquareMesh</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">facets</span> <span class="o">=</span> <span class="n">MeshFunction</span><span class="p">(</span><span class="s2">&quot;size_t&quot;</span><span class="p">,</span> <span class="n">mesh</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">facets</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">ds</span> <span class="o">=</span> <span class="n">Measure</span><span class="p">(</span><span class="s2">&quot;ds&quot;</span><span class="p">,</span> <span class="n">subdomain_data</span><span class="o">=</span><span class="n">facets</span><span class="p">)</span>
</pre></div>
</div>
<p>Continuous interpolation using of degree 2 is chosen for the deflection <span class="math">\(w\)</span>
whereas the rotation field <span class="math">\(\underline{\theta}\)</span> is discretized using discontinuous linear polynomials:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="n">We</span> <span class="o">=</span> <span class="n">FiniteElement</span><span class="p">(</span><span class="s2">&quot;Lagrange&quot;</span><span class="p">,</span> <span class="n">mesh</span><span class="o">.</span><span class="n">ufl_cell</span><span class="p">(),</span> <span class="mi">2</span><span class="p">)</span>
<span class="n">Te</span> <span class="o">=</span> <span class="n">VectorElement</span><span class="p">(</span><span class="s2">&quot;DG&quot;</span><span class="p">,</span> <span class="n">mesh</span><span class="o">.</span><span class="n">ufl_cell</span><span class="p">(),</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">V</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="n">MixedElement</span><span class="p">([</span><span class="n">We</span><span class="p">,</span> <span class="n">Te</span><span class="p">]))</span>
</pre></div>
</div>
<p>Clamped boundary conditions on the lateral boundary are defined as:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">border</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="n">bc</span> <span class="o">=</span>  <span class="p">[</span><span class="n">DirichletBC</span><span class="p">(</span><span class="n">V</span><span class="o">.</span><span class="n">sub</span><span class="p">(</span><span class="mi">0</span><span class="p">),</span> <span class="n">Constant</span><span class="p">(</span><span class="mf">0.</span><span class="p">),</span> <span class="n">border</span><span class="p">)]</span>
</pre></div>
</div>
<p>Standard part of the variational form is the same (without full integration):</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">strain2voigt</span><span class="p">(</span><span class="n">eps</span><span class="p">):</span>
    <span class="k">return</span> <span class="n">as_vector</span><span class="p">([</span><span class="n">eps</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="n">eps</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="mi">2</span><span class="o">*</span><span class="n">eps</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="k">def</span> <span class="nf">voigt2stress</span><span class="p">(</span><span class="n">S</span><span class="p">):</span>
    <span class="k">return</span> <span class="n">as_tensor</span><span class="p">([[</span><span class="n">S</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">S</span><span class="p">[</span><span class="mi">2</span><span class="p">]],</span> <span class="p">[</span><span class="n">S</span><span class="p">[</span><span class="mi">2</span><span class="p">],</span> <span class="n">S</span><span class="p">[</span><span class="mi">1</span><span class="p">]]])</span>
<span class="k">def</span> <span class="nf">curv</span><span class="p">(</span><span class="n">u</span><span class="p">):</span>
    <span class="p">(</span><span class="n">w</span><span class="p">,</span> <span class="n">theta</span><span class="p">)</span> <span class="o">=</span> <span class="n">split</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>
    <span class="k">return</span> <span class="n">sym</span><span class="p">(</span><span class="n">grad</span><span class="p">(</span><span class="n">theta</span><span class="p">))</span>
<span class="k">def</span> <span class="nf">shear_strain</span><span class="p">(</span><span class="n">u</span><span class="p">):</span>
    <span class="p">(</span><span class="n">w</span><span class="p">,</span> <span class="n">theta</span><span class="p">)</span> <span class="o">=</span> <span class="n">split</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>
    <span class="k">return</span> <span class="n">theta</span><span class="o">-</span><span class="n">grad</span><span class="p">(</span><span class="n">w</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">bending_moment</span><span class="p">(</span><span class="n">u</span><span class="p">):</span>
    <span class="n">DD</span> <span class="o">=</span> <span class="n">as_tensor</span><span class="p">([[</span><span class="n">D</span><span class="p">,</span> <span class="n">nu</span><span class="o">*</span><span class="n">D</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="n">nu</span><span class="o">*</span><span class="n">D</span><span class="p">,</span> <span class="n">D</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">0</span><span class="p">,</span> <span class="n">D</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">nu</span><span class="p">)</span><span class="o">/</span><span class="mf">2.</span><span class="p">]])</span>
    <span class="k">return</span> <span class="n">voigt2stress</span><span class="p">(</span><span class="n">dot</span><span class="p">(</span><span class="n">DD</span><span class="p">,</span><span class="n">strain2voigt</span><span class="p">(</span><span class="n">curv</span><span class="p">(</span><span class="n">u</span><span class="p">))))</span>
<span class="k">def</span> <span class="nf">shear_force</span><span class="p">(</span><span class="n">u</span><span class="p">):</span>
    <span class="k">return</span> <span class="n">F</span><span class="o">*</span><span class="n">shear_strain</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>

<span class="n">u</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="n">V</span><span class="p">)</span>
<span class="n">u_</span> <span class="o">=</span> <span class="n">TestFunction</span><span class="p">(</span><span class="n">V</span><span class="p">)</span>
<span class="n">du</span> <span class="o">=</span> <span class="n">TrialFunction</span><span class="p">(</span><span class="n">V</span><span class="p">)</span>


<span class="n">L</span> <span class="o">=</span> <span class="n">f</span><span class="o">*</span><span class="n">u_</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">dx</span>
<span class="n">a</span> <span class="o">=</span> <span class="n">inner</span><span class="p">(</span><span class="n">bending_moment</span><span class="p">(</span><span class="n">u_</span><span class="p">),</span> <span class="n">curv</span><span class="p">(</span><span class="n">du</span><span class="p">))</span><span class="o">*</span><span class="n">dx</span> <span class="o">+</span> <span class="n">dot</span><span class="p">(</span><span class="n">shear_force</span><span class="p">(</span><span class="n">u_</span><span class="p">),</span> <span class="n">shear_strain</span><span class="p">(</span><span class="n">du</span><span class="p">))</span><span class="o">*</span><span class="n">dx</span>
</pre></div>
</div>
<p>We then add the contribution of jumps in rotation across all internal facets plus
a stabilization term involing a user-defined parameter <span class="math">\(s\)</span>:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="n">n</span> <span class="o">=</span> <span class="n">FacetNormal</span><span class="p">(</span><span class="n">mesh</span><span class="p">)</span>
<span class="n">h</span> <span class="o">=</span> <span class="n">CellVolume</span><span class="p">(</span><span class="n">mesh</span><span class="p">)</span>
<span class="n">h_avg</span> <span class="o">=</span> <span class="p">(</span><span class="n">h</span><span class="p">(</span><span class="s1">&#39;+&#39;</span><span class="p">)</span><span class="o">+</span><span class="n">h</span><span class="p">(</span><span class="s1">&#39;-&#39;</span><span class="p">))</span><span class="o">/</span><span class="mi">2</span>
<span class="n">stabilization</span> <span class="o">=</span> <span class="n">Constant</span><span class="p">(</span><span class="mf">10.</span><span class="p">)</span>

<span class="p">(</span><span class="n">dw</span><span class="p">,</span> <span class="n">dtheta</span><span class="p">)</span> <span class="o">=</span> <span class="n">split</span><span class="p">(</span><span class="n">du</span><span class="p">)</span>
<span class="p">(</span><span class="n">w_</span><span class="p">,</span> <span class="n">theta_</span><span class="p">)</span> <span class="o">=</span> <span class="n">split</span><span class="p">(</span><span class="n">u_</span><span class="p">)</span>

<span class="n">a</span> <span class="o">-=</span> <span class="n">dot</span><span class="p">(</span><span class="n">avg</span><span class="p">(</span><span class="n">dot</span><span class="p">(</span><span class="n">bending_moment</span><span class="p">(</span><span class="n">u_</span><span class="p">),</span> <span class="n">n</span><span class="p">)),</span> <span class="n">jump</span><span class="p">(</span><span class="n">dtheta</span><span class="p">))</span><span class="o">*</span><span class="n">dS</span> <span class="o">+</span> <span class="n">dot</span><span class="p">(</span><span class="n">avg</span><span class="p">(</span><span class="n">dot</span><span class="p">(</span><span class="n">bending_moment</span><span class="p">(</span><span class="n">du</span><span class="p">),</span> <span class="n">n</span><span class="p">)),</span> <span class="n">jump</span><span class="p">(</span><span class="n">theta_</span><span class="p">))</span><span class="o">*</span><span class="n">dS</span> \
   <span class="o">-</span> <span class="n">stabilization</span><span class="o">*</span><span class="n">D</span><span class="o">/</span><span class="n">h_avg</span><span class="o">*</span><span class="n">dot</span><span class="p">(</span><span class="n">jump</span><span class="p">(</span><span class="n">theta_</span><span class="p">),</span> <span class="n">jump</span><span class="p">(</span><span class="n">dtheta</span><span class="p">))</span><span class="o">*</span><span class="n">dS</span>
</pre></div>
</div>
<p>Because of the clamped boundary conditions, we also need to add the corresponding
contributions of the external facets (the imposed rotation is zero on the boundary
so that no term arise in the linear functional):</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="n">a</span> <span class="o">-=</span> <span class="n">dot</span><span class="p">(</span><span class="n">dot</span><span class="p">(</span><span class="n">bending_moment</span><span class="p">(</span><span class="n">u_</span><span class="p">),</span> <span class="n">n</span><span class="p">),</span> <span class="n">dtheta</span><span class="p">)</span><span class="o">*</span><span class="n">ds</span> <span class="o">+</span> <span class="n">dot</span><span class="p">(</span><span class="n">dot</span><span class="p">(</span><span class="n">bending_moment</span><span class="p">(</span><span class="n">du</span><span class="p">),</span> <span class="n">n</span><span class="p">),</span> <span class="n">theta_</span><span class="p">)</span><span class="o">*</span><span class="n">ds</span> \
   <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">stabilization</span><span class="o">*</span><span class="n">D</span><span class="o">/</span><span class="n">h</span><span class="o">*</span><span class="n">dot</span><span class="p">(</span><span class="n">theta_</span><span class="p">,</span> <span class="n">dtheta</span><span class="p">)</span><span class="o">*</span><span class="n">ds</span>
</pre></div>
</div>
<p>We then solve for the solution and export the relevant fields to XDMF files</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="n">solve</span><span class="p">(</span><span class="n">a</span> <span class="o">==</span> <span class="n">L</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">bc</span><span class="p">)</span>

<span class="p">(</span><span class="n">w</span><span class="p">,</span> <span class="n">theta</span><span class="p">)</span> <span class="o">=</span> <span class="n">split</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>

<span class="n">Vw</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="n">We</span><span class="p">)</span>
<span class="n">Vt</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="n">Te</span><span class="p">)</span>
<span class="n">ww</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="n">Vw</span><span class="p">,</span> <span class="n">name</span><span class="o">=</span><span class="s2">&quot;Deflection&quot;</span><span class="p">)</span>
<span class="n">tt</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="n">Vt</span><span class="p">,</span> <span class="n">name</span><span class="o">=</span><span class="s2">&quot;Rotation&quot;</span><span class="p">)</span>
<span class="n">ww</span><span class="o">.</span><span class="n">assign</span><span class="p">(</span><span class="n">project</span><span class="p">(</span><span class="n">w</span><span class="p">,</span> <span class="n">Vw</span><span class="p">))</span>
<span class="n">tt</span><span class="o">.</span><span class="n">assign</span><span class="p">(</span><span class="n">project</span><span class="p">(</span><span class="n">theta</span><span class="p">,</span> <span class="n">Vt</span><span class="p">))</span>

<span class="n">file_results</span> <span class="o">=</span> <span class="n">XDMFFile</span><span class="p">(</span><span class="s2">&quot;RM_DG_results.xdmf&quot;</span><span class="p">)</span>
<span class="n">file_results</span><span class="o">.</span><span class="n">parameters</span><span class="p">[</span><span class="s2">&quot;flush_output&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="kc">True</span>
<span class="n">file_results</span><span class="o">.</span><span class="n">parameters</span><span class="p">[</span><span class="s2">&quot;functions_share_mesh&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="kc">True</span>
<span class="n">file_results</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="n">ww</span><span class="p">,</span> <span class="mf">0.</span><span class="p">)</span>
<span class="n">file_results</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="n">tt</span><span class="p">,</span> <span class="mf">0.</span><span class="p">)</span>
</pre></div>
</div>
<p>The solution is compared to the Kirchhoff analytical solution:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Kirchhoff deflection:&quot;</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.265319087e-3</span><span class="o">*</span><span class="nb">float</span><span class="p">(</span><span class="n">f</span><span class="o">/</span><span class="n">D</span><span class="p">))</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Reissner-Mindlin FE deflection:&quot;</span><span class="p">,</span> <span class="o">-</span><span class="n">ww</span><span class="p">(</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">))</span>
</pre></div>
</div>
<p>For <span class="math">\(h=0.001\)</span> and 50 elements per side, one finds <span class="math">\(w_{FE} = 1.38322\text{e-5}\)</span>  against <span class="math">\(w_{\text{Kirchhoff}} = 1.38173\text{e-5}\)</span> for the thin plate solution.</p>
</div>
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<h2>References<a class="headerlink" href="#references" title="Permalink to this headline"></a></h2>
<table class="docutils citation" frame="void" id="han2011" rules="none">
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<tr><td class="label"><a class="fn-backref" href="#id1">[HAN2011]</a></td><td>Peter Hansbo, David Heintz, Mats G. Larson, A finite element method with discontinuous rotations for the Mindlin-Reissner plate model, <em>Computer Methods in Applied Mechanics and Engineering</em>, 200, 5-8, 2011, pp. 638-648, <a class="reference external" href="https://doi.org/10.1016/j.cma.2010.09.009">https://doi.org/10.1016/j.cma.2010.09.009</a>.</td></tr>
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