Commit 779039e4 authored by Jeremy BLEYER's avatar Jeremy BLEYER

Published buckling example

parent b7bef5d1
......@@ -156,7 +156,7 @@ for i in range(N_eig):
# Initialize function and assign eigenvector (renormalize by stiffness matrix)
eigenmode = Function(V,name="Eigenvector "+str(i))
eigenmode.vector()[:] = rx/omega
eigenmode.vector()[:] = rx
# The beam analytical solution is obtained using the eigenfrequencies of a clamped
# beam in bending given by :math:`\omega_n = \alpha_n^2\sqrt{\dfrac{EI}{\rho S L^4}}`
......
......@@ -156,7 +156,7 @@ real and complex part of the eigenvector)::
# Initialize function and assign eigenvector (renormalize by stiffness matrix)
eigenmode = Function(V,name="Eigenvector "+str(i))
eigenmode.vector()[:] = rx/omega
eigenmode.vector()[:] = rx
The beam analytical solution is obtained using the eigenfrequencies of a clamped
beam in bending given by :math:`\omega_n = \alpha_n^2\sqrt{\dfrac{EI}{\rho S L^4}}`
......
......@@ -15,7 +15,7 @@ Contents:
linear_problems
homogenization
nonlinear_problems
demo/reissner_mindlin/reissner_mindlin.rst
beams_and_plates
tips_and_tricks
......
......@@ -24,7 +24,7 @@
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......@@ -35,7 +35,7 @@
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......@@ -431,7 +431,8 @@ when considering a zero hardening modulus.</p>
<li class="toctree-l2 current"><a class="current reference internal" href="#">Elasto-plastic analysis of a 2D von Mises material</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="../reissner_mindlin/reissner_mindlin.html">Reissner-Mindlin plates</a><ul>
<li class="toctree-l1"><a class="reference internal" href="../../beams_and_plates.html">Beams and plates</a><ul>
<li class="toctree-l2"><a class="reference internal" href="../timoshenko/beam_buckling.html">Eulerian buckling of a beam</a></li>
<li class="toctree-l2"><a class="reference internal" href="../reissner_mindlin/reissner_mindlin_quads.py.html">Reissner-Mindlin plate with Quadrilaterals</a></li>
<li class="toctree-l2"><a class="reference internal" href="../reissner_mindlin/reissner_mindlin_dg.py.html">Reissner-Mindlin plate with a Discontinuous-Galerkin approach</a></li>
</ul>
......@@ -462,7 +463,7 @@ when considering a zero hardening modulus.</p>
<div role="navigation" aria-label="related navigaton">
<a href="../../nonlinear_problems.html" title="Nonlinear problems in solid mechanics"
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<a href="../../genindex.html" title="General Index"
>index</a>
......
......@@ -891,7 +891,8 @@ div.nboutput div.output_area.stderr {
<li class="toctree-l2"><a class="reference internal" href="../2D_plasticity/vonMises_plasticity.py.html">Elasto-plastic analysis of a 2D von Mises material</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="../reissner_mindlin/reissner_mindlin.html">Reissner-Mindlin plates</a><ul>
<li class="toctree-l1"><a class="reference internal" href="../../beams_and_plates.html">Beams and plates</a><ul>
<li class="toctree-l2"><a class="reference internal" href="../timoshenko/beam_buckling.html">Eulerian buckling of a beam</a></li>
<li class="toctree-l2"><a class="reference internal" href="../reissner_mindlin/reissner_mindlin_quads.py.html">Reissner-Mindlin plate with Quadrilaterals</a></li>
<li class="toctree-l2"><a class="reference internal" href="../reissner_mindlin/reissner_mindlin_dg.py.html">Reissner-Mindlin plate with a Discontinuous-Galerkin approach</a></li>
</ul>
......
......@@ -241,7 +241,8 @@ writing/reading. Prefered output format is now .xdmf:</p>
<li class="toctree-l2"><a class="reference internal" href="../2D_plasticity/vonMises_plasticity.py.html">Elasto-plastic analysis of a 2D von Mises material</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="../reissner_mindlin/reissner_mindlin.html">Reissner-Mindlin plates</a><ul>
<li class="toctree-l1"><a class="reference internal" href="../../beams_and_plates.html">Beams and plates</a><ul>
<li class="toctree-l2"><a class="reference internal" href="../timoshenko/beam_buckling.html">Eulerian buckling of a beam</a></li>
<li class="toctree-l2"><a class="reference internal" href="../reissner_mindlin/reissner_mindlin_quads.py.html">Reissner-Mindlin plate with Quadrilaterals</a></li>
<li class="toctree-l2"><a class="reference internal" href="../reissner_mindlin/reissner_mindlin_dg.py.html">Reissner-Mindlin plate with a Discontinuous-Galerkin approach</a></li>
</ul>
......
......@@ -2826,7 +2826,8 @@ if (IPython.notebook.kernel != null) {
<li class="toctree-l2"><a class="reference internal" href="../2D_plasticity/vonMises_plasticity.py.html">Elasto-plastic analysis of a 2D von Mises material</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="../reissner_mindlin/reissner_mindlin.html">Reissner-Mindlin plates</a><ul>
<li class="toctree-l1"><a class="reference internal" href="../../beams_and_plates.html">Beams and plates</a><ul>
<li class="toctree-l2"><a class="reference internal" href="../timoshenko/beam_buckling.html">Eulerian buckling of a beam</a></li>
<li class="toctree-l2"><a class="reference internal" href="../reissner_mindlin/reissner_mindlin_quads.py.html">Reissner-Mindlin plate with Quadrilaterals</a></li>
<li class="toctree-l2"><a class="reference internal" href="../reissner_mindlin/reissner_mindlin_dg.py.html">Reissner-Mindlin plate with a Discontinuous-Galerkin approach</a></li>
</ul>
......
......@@ -219,7 +219,8 @@ that:</p>
<li class="toctree-l2"><a class="reference internal" href="../2D_plasticity/vonMises_plasticity.py.html">Elasto-plastic analysis of a 2D von Mises material</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="../reissner_mindlin/reissner_mindlin.html">Reissner-Mindlin plates</a><ul>
<li class="toctree-l1"><a class="reference internal" href="../../beams_and_plates.html">Beams and plates</a><ul>
<li class="toctree-l2"><a class="reference internal" href="../timoshenko/beam_buckling.html">Eulerian buckling of a beam</a></li>
<li class="toctree-l2"><a class="reference internal" href="../reissner_mindlin/reissner_mindlin_quads.py.html">Reissner-Mindlin plate with Quadrilaterals</a></li>
<li class="toctree-l2"><a class="reference internal" href="../reissner_mindlin/reissner_mindlin_dg.py.html">Reissner-Mindlin plate with a Discontinuous-Galerkin approach</a></li>
</ul>
......
......@@ -188,7 +188,7 @@ real and complex part of the eigenvector):</p>
<span class="c1"># Initialize function and assign eigenvector (renormalize by stiffness matrix)</span>
<span class="n">eigenmode</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">name</span><span class="o">=</span><span class="s2">&quot;Eigenvector &quot;</span><span class="o">+</span><span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">))</span>
<span class="n">eigenmode</span><span class="o">.</span><span class="n">vector</span><span class="p">()[:]</span> <span class="o">=</span> <span class="n">rx</span><span class="o">/</span><span class="n">omega</span>
<span class="n">eigenmode</span><span class="o">.</span><span class="n">vector</span><span class="p">()[:]</span> <span class="o">=</span> <span class="n">rx</span>
</pre></div>
</div>
<p>The beam analytical solution is obtained using the eigenfrequencies of a clamped
......@@ -282,7 +282,8 @@ and the beam theory eigenfrequencies :</p>
<li class="toctree-l2"><a class="reference internal" href="../2D_plasticity/vonMises_plasticity.py.html">Elasto-plastic analysis of a 2D von Mises material</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="../reissner_mindlin/reissner_mindlin.html">Reissner-Mindlin plates</a><ul>
<li class="toctree-l1"><a class="reference internal" href="../../beams_and_plates.html">Beams and plates</a><ul>
<li class="toctree-l2"><a class="reference internal" href="../timoshenko/beam_buckling.html">Eulerian buckling of a beam</a></li>
<li class="toctree-l2"><a class="reference internal" href="../reissner_mindlin/reissner_mindlin_quads.py.html">Reissner-Mindlin plate with Quadrilaterals</a></li>
<li class="toctree-l2"><a class="reference internal" href="../reissner_mindlin/reissner_mindlin_dg.py.html">Reissner-Mindlin plate with a Discontinuous-Galerkin approach</a></li>
</ul>
......
......@@ -2162,7 +2162,8 @@ if (IPython.notebook.kernel != null) {
<li class="toctree-l2"><a class="reference internal" href="../2D_plasticity/vonMises_plasticity.py.html">Elasto-plastic analysis of a 2D von Mises material</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="../reissner_mindlin/reissner_mindlin.html">Reissner-Mindlin plates</a><ul>
<li class="toctree-l1"><a class="reference internal" href="../../beams_and_plates.html">Beams and plates</a><ul>
<li class="toctree-l2"><a class="reference internal" href="../timoshenko/beam_buckling.html">Eulerian buckling of a beam</a></li>
<li class="toctree-l2"><a class="reference internal" href="../reissner_mindlin/reissner_mindlin_quads.py.html">Reissner-Mindlin plate with Quadrilaterals</a></li>
<li class="toctree-l2"><a class="reference internal" href="../reissner_mindlin/reissner_mindlin_dg.py.html">Reissner-Mindlin plate with a Discontinuous-Galerkin approach</a></li>
</ul>
......
......@@ -24,8 +24,6 @@
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......@@ -33,10 +31,6 @@
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......@@ -69,7 +63,7 @@
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<h3>Table Of Contents</h3>
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......@@ -93,7 +87,8 @@
<li class="toctree-l2"><a class="reference internal" href="../2D_plasticity/vonMises_plasticity.py.html">Elasto-plastic analysis of a 2D von Mises material</a></li>
</ul>
</li>
<li class="toctree-l1 current"><a class="current reference internal" href="#">Reissner-Mindlin plates</a><ul>
<li class="toctree-l1"><a class="reference internal" href="../../beams_and_plates.html">Beams and plates</a><ul>
<li class="toctree-l2"><a class="reference internal" href="../timoshenko/beam_buckling.html">Eulerian buckling of a beam</a></li>
<li class="toctree-l2"><a class="reference internal" href="reissner_mindlin_quads.py.html">Reissner-Mindlin plate with Quadrilaterals</a></li>
<li class="toctree-l2"><a class="reference internal" href="reissner_mindlin_dg.py.html">Reissner-Mindlin plate with a Discontinuous-Galerkin approach</a></li>
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......@@ -122,10 +117,6 @@
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......
......@@ -216,7 +216,8 @@ so that no term arise in the linear functional):</p>
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<li class="toctree-l2"><a class="reference internal" href="reissner_mindlin_quads.py.html">Reissner-Mindlin plate with Quadrilaterals</a></li>
<li class="toctree-l2 current"><a class="current reference internal" href="#">Reissner-Mindlin plate with a Discontinuous-Galerkin approach</a></li>
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......
......@@ -25,7 +25,7 @@
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......@@ -209,7 +209,8 @@ the thin plate solution.</p>
<li class="toctree-l2"><a class="reference internal" href="../2D_plasticity/vonMises_plasticity.py.html">Elasto-plastic analysis of a 2D von Mises material</a></li>
</ul>
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<li class="toctree-l2"><a class="reference internal" href="../timoshenko/beam_buckling.html">Eulerian buckling of a beam</a></li>
<li class="toctree-l2 current"><a class="current reference internal" href="#">Reissner-Mindlin plate with Quadrilaterals</a></li>
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......@@ -238,7 +239,7 @@ the thin plate solution.</p>
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......
......@@ -4348,7 +4348,8 @@ if (IPython.notebook.kernel != null) {
<li class="toctree-l2"><a class="reference internal" href="../2D_plasticity/vonMises_plasticity.py.html">Elasto-plastic analysis of a 2D von Mises material</a></li>
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<li class="toctree-l1"><a class="reference internal" href="../reissner_mindlin/reissner_mindlin.html">Reissner-Mindlin plates</a><ul>
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<li class="toctree-l2"><a class="reference internal" href="../timoshenko/beam_buckling.html">Eulerian buckling of a beam</a></li>
<li class="toctree-l2"><a class="reference internal" href="../reissner_mindlin/reissner_mindlin_quads.py.html">Reissner-Mindlin plate with Quadrilaterals</a></li>
<li class="toctree-l2"><a class="reference internal" href="../reissner_mindlin/reissner_mindlin_dg.py.html">Reissner-Mindlin plate with a Discontinuous-Galerkin approach</a></li>
</ul>
......
......@@ -2912,7 +2912,8 @@ Mechanics and Engineering, 85(3), 349-365.
<li class="toctree-l2"><a class="reference internal" href="../2D_plasticity/vonMises_plasticity.py.html">Elasto-plastic analysis of a 2D von Mises material</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="../reissner_mindlin/reissner_mindlin.html">Reissner-Mindlin plates</a><ul>
<li class="toctree-l1"><a class="reference internal" href="../../beams_and_plates.html">Beams and plates</a><ul>
<li class="toctree-l2"><a class="reference internal" href="../timoshenko/beam_buckling.html">Eulerian buckling of a beam</a></li>
<li class="toctree-l2"><a class="reference internal" href="../reissner_mindlin/reissner_mindlin_quads.py.html">Reissner-Mindlin plate with Quadrilaterals</a></li>
<li class="toctree-l2"><a class="reference internal" href="../reissner_mindlin/reissner_mindlin_dg.py.html">Reissner-Mindlin plate with a Discontinuous-Galerkin approach</a></li>
</ul>
......
......@@ -84,7 +84,8 @@
<li class="toctree-l2"><a class="reference internal" href="demo/2D_plasticity/vonMises_plasticity.py.html">Elasto-plastic analysis of a 2D von Mises material</a></li>
</ul>
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<li class="toctree-l1"><a class="reference internal" href="demo/reissner_mindlin/reissner_mindlin.html">Reissner-Mindlin plates</a><ul>
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<li class="toctree-l2"><a class="reference internal" href="demo/timoshenko/beam_buckling.html">Eulerian buckling of a beam</a></li>
<li class="toctree-l2"><a class="reference internal" href="demo/reissner_mindlin/reissner_mindlin_quads.py.html">Reissner-Mindlin plate with Quadrilaterals</a></li>
<li class="toctree-l2"><a class="reference internal" href="demo/reissner_mindlin/reissner_mindlin_dg.py.html">Reissner-Mindlin plate with a Discontinuous-Galerkin approach</a></li>
</ul>
......
......@@ -92,7 +92,8 @@
<li class="toctree-l2"><a class="reference internal" href="demo/2D_plasticity/vonMises_plasticity.py.html">Elasto-plastic analysis of a 2D von Mises material</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="demo/reissner_mindlin/reissner_mindlin.html">Reissner-Mindlin plates</a><ul>
<li class="toctree-l1"><a class="reference internal" href="beams_and_plates.html">Beams and plates</a><ul>
<li class="toctree-l2"><a class="reference internal" href="demo/timoshenko/beam_buckling.html">Eulerian buckling of a beam</a></li>
<li class="toctree-l2"><a class="reference internal" href="demo/reissner_mindlin/reissner_mindlin_quads.py.html">Reissner-Mindlin plate with Quadrilaterals</a></li>
<li class="toctree-l2"><a class="reference internal" href="demo/reissner_mindlin/reissner_mindlin_dg.py.html">Reissner-Mindlin plate with a Discontinuous-Galerkin approach</a></li>
</ul>
......
......@@ -76,7 +76,8 @@
<li class="toctree-l2"><a class="reference internal" href="demo/2D_plasticity/vonMises_plasticity.py.html">Elasto-plastic analysis of a 2D von Mises material</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="demo/reissner_mindlin/reissner_mindlin.html">Reissner-Mindlin plates</a><ul>
<li class="toctree-l1"><a class="reference internal" href="beams_and_plates.html">Beams and plates</a><ul>
<li class="toctree-l2"><a class="reference internal" href="demo/timoshenko/beam_buckling.html">Eulerian buckling of a beam</a></li>
<li class="toctree-l2"><a class="reference internal" href="demo/reissner_mindlin/reissner_mindlin_quads.py.html">Reissner-Mindlin plate with Quadrilaterals</a></li>
<li class="toctree-l2"><a class="reference internal" href="demo/reissner_mindlin/reissner_mindlin_dg.py.html">Reissner-Mindlin plate with a Discontinuous-Galerkin approach</a></li>
</ul>
......@@ -120,7 +121,8 @@
<li class="toctree-l2"><a class="reference internal" href="demo/2D_plasticity/vonMises_plasticity.py.html">Elasto-plastic analysis of a 2D von Mises material</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="demo/reissner_mindlin/reissner_mindlin.html">Reissner-Mindlin plates</a><ul>
<li class="toctree-l1"><a class="reference internal" href="beams_and_plates.html">Beams and plates</a><ul>
<li class="toctree-l2"><a class="reference internal" href="demo/timoshenko/beam_buckling.html">Eulerian buckling of a beam</a></li>
<li class="toctree-l2"><a class="reference internal" href="demo/reissner_mindlin/reissner_mindlin_quads.py.html">Reissner-Mindlin plate with Quadrilaterals</a></li>
<li class="toctree-l2"><a class="reference internal" href="demo/reissner_mindlin/reissner_mindlin_dg.py.html">Reissner-Mindlin plate with a Discontinuous-Galerkin approach</a></li>
</ul>
......
......@@ -97,7 +97,8 @@
<li class="toctree-l2"><a class="reference internal" href="demo/2D_plasticity/vonMises_plasticity.py.html">Elasto-plastic analysis of a 2D von Mises material</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="demo/reissner_mindlin/reissner_mindlin.html">Reissner-Mindlin plates</a><ul>
<li class="toctree-l1"><a class="reference internal" href="beams_and_plates.html">Beams and plates</a><ul>
<li class="toctree-l2"><a class="reference internal" href="demo/timoshenko/beam_buckling.html">Eulerian buckling of a beam</a></li>
<li class="toctree-l2"><a class="reference internal" href="demo/reissner_mindlin/reissner_mindlin_quads.py.html">Reissner-Mindlin plate with Quadrilaterals</a></li>
<li class="toctree-l2"><a class="reference internal" href="demo/reissner_mindlin/reissner_mindlin_dg.py.html">Reissner-Mindlin plate with a Discontinuous-Galerkin approach</a></li>
</ul>
......
......@@ -92,7 +92,8 @@
<li class="toctree-l2"><a class="reference internal" href="demo/2D_plasticity/vonMises_plasticity.py.html">Elasto-plastic analysis of a 2D von Mises material</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="demo/reissner_mindlin/reissner_mindlin.html">Reissner-Mindlin plates</a><ul>
<li class="toctree-l1"><a class="reference internal" href="beams_and_plates.html">Beams and plates</a><ul>
<li class="toctree-l2"><a class="reference internal" href="demo/timoshenko/beam_buckling.html">Eulerian buckling of a beam</a></li>
<li class="toctree-l2"><a class="reference internal" href="demo/reissner_mindlin/reissner_mindlin_quads.py.html">Reissner-Mindlin plate with Quadrilaterals</a></li>
<li class="toctree-l2"><a class="reference internal" href="demo/reissner_mindlin/reissner_mindlin_dg.py.html">Reissner-Mindlin plate with a Discontinuous-Galerkin approach</a></li>
</ul>
......
......@@ -107,7 +107,8 @@
<li class="toctree-l2"><a class="reference internal" href="demo/2D_plasticity/vonMises_plasticity.py.html">Elasto-plastic analysis of a 2D von Mises material</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="demo/reissner_mindlin/reissner_mindlin.html">Reissner-Mindlin plates</a><ul>
<li class="toctree-l1"><a class="reference internal" href="beams_and_plates.html">Beams and plates</a><ul>
<li class="toctree-l2"><a class="reference internal" href="demo/timoshenko/beam_buckling.html">Eulerian buckling of a beam</a></li>
<li class="toctree-l2"><a class="reference internal" href="demo/reissner_mindlin/reissner_mindlin_quads.py.html">Reissner-Mindlin plate with Quadrilaterals</a></li>
<li class="toctree-l2"><a class="reference internal" href="demo/reissner_mindlin/reissner_mindlin_dg.py.html">Reissner-Mindlin plate with a Discontinuous-Galerkin approach</a></li>
</ul>
......
This source diff could not be displayed because it is too large. You can view the blob instead.
.. Numerical tours of continuum mechanics using FEniCS documentation master file, created by
sphinx-quickstart on Wed Jun 8 21:25:10 2016.
You can adapt this file completely to your liking, but it should at least
contain the root `toctree` directive.
Beams and plates
=================
Contents:
.. toctree::
:maxdepth: 1
demo/timoshenko/beam_buckling.ipynb
demo/reissner_mindlin/reissner_mindlin_quads.py.rst
demo/reissner_mindlin/reissner_mindlin_dg.py.rst
......@@ -15,7 +15,7 @@ Contents:
linear_problems
homogenization
nonlinear_problems
demo/reissner_mindlin/reissner_mindlin.rst
beams_and_plates
tips_and_tricks
......
......@@ -21,7 +21,9 @@
},
{
"cell_type": "raw",
"metadata": {},
"metadata": {
"raw_mimetype": "text/restructuredtext"
},
"source": [
"For issues related to shear-locking and reduced integration formulation, we refer to the :ref:`ReissnerMindlinQuads` tour."
]
......@@ -67,28 +69,18 @@
},
{
"cell_type": "raw",
"metadata": {},
"metadata": {
"raw_mimetype": "text/restructuredtext"
},
"source": [
"As in the :ref:`ModalAnalysis` tour, a dummy linear form `l_form` is used to call the `assemble_system` function which retains the symmetric structure of the associated matrix when imposing boundary conditions. Here, we will consider clamped conditions on the left side :math:`x=0` and simple supports on the right side :math:`x=L`."
]
},
{
"cell_type": "code",
"execution_count": 73,
"execution_count": 78,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(<dolfin.cpp.la.PETScMatrix; proxy of <Swig Object of type 'std::shared_ptr< dolfin::PETScMatrix > *' at 0x7f9df3cf8780> >,\n",
" <dolfin.cpp.la.Vector; proxy of <Swig Object of type 'std::shared_ptr< dolfin::Vector > *' at 0x7f9df3cf8ae0> >)"
]
},
"execution_count": 73,
"metadata": {},
"output_type": "execute_result"
}
],
"outputs": [],
"source": [
"def both_ends(x, on_boundary):\n",
" return on_boundary\n",
......@@ -986,7 +978,8 @@
" r, c, rx, cx = eigensolver.get_eigenpair(i)\n",
" \n",
" critical_load_an = alpha(i+1)**2*float(EI/N0)/L**2\n",
" print(\"Exact: {0:>10.5f} FE: {1:>10.5f} Rel. gap {2:1.2f}%%\".format(critical_load_an, r, 100*(r/critical_load_an-1)))\n",
" print(\"Exact: {0:>10.5f} FE: {1:>10.5f} Rel. gap {2:1.2f}%%\".format(\n",
" critical_load_an, r, 100*(r/critical_load_an-1)))\n",
" \n",
" # Initialize function and assign eigenvector (renormalize by stiffness matrix)\n",
" eigenmode = Function(V,name=\"Eigenvector \"+str(i))\n",
......@@ -1009,11 +1002,19 @@
"F_n = (\\alpha_n)^2 \\dfrac{EI}{L^2} \\quad \\text{with }\\alpha_n \\text{ solutions to } \\tan(\\alpha) = \\alpha\n",
"\\end{equation}\n",
"\n",
"It can, in particular, be observed that the displacement-based FE solution overestimates the exact buckling load and that the error increases with the order of the buckling load."
"In particular, it can be observed that the displacement-based FE solution overestimates the exact buckling load and that the error increases with the order of the buckling load."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
}
],
"metadata": {
"celltoolbar": "Raw Cell Format",
"kernelspec": {
"display_name": "Python 2",
"language": "python",
......
......@@ -21,7 +21,9 @@
},
{
"cell_type": "raw",
"metadata": {},
"metadata": {
"raw_mimetype": "text/restructuredtext"
},
"source": [
"For issues related to shear-locking and reduced integration formulation, we refer to the :ref:`ReissnerMindlinQuads` tour."
]
......@@ -67,28 +69,18 @@
},
{
"cell_type": "raw",
"metadata": {},
"metadata": {
"raw_mimetype": "text/restructuredtext"
},
"source": [
"As in the :ref:`ModalAnalysis` tour, a dummy linear form `l_form` is used to call the `assemble_system` function which retains the symmetric structure of the associated matrix when imposing boundary conditions. Here, we will consider clamped conditions on the left side :math:`x=0` and simple supports on the right side :math:`x=L`."
]
},
{
"cell_type": "code",
"execution_count": 73,
"execution_count": 78,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(<dolfin.cpp.la.PETScMatrix; proxy of <Swig Object of type 'std::shared_ptr< dolfin::PETScMatrix > *' at 0x7f9df3cf8780> >,\n",
" <dolfin.cpp.la.Vector; proxy of <Swig Object of type 'std::shared_ptr< dolfin::Vector > *' at 0x7f9df3cf8ae0> >)"
]
},
"execution_count": 73,
"metadata": {},
"output_type": "execute_result"
}
],
"outputs": [],
"source": [
"def both_ends(x, on_boundary):\n",
" return on_boundary\n",
......@@ -150,7 +142,7 @@
},
{
"cell_type": "code",
"execution_count": 71,
"execution_count": 79,
"metadata": {},
"outputs": [
{
......@@ -954,9 +946,9 @@
"output_type": "stream",
"text": [
"Critical buckling loads:\n",
"Exact: 0.31800 FE: 0.31805 Rel. gap 0.01%%\n",
"Exact: 0.93995 FE: 0.94033 Rel. gap 0.04%%\n",
"Exact: 1.87267 FE: 1.87415 Rel. gap 0.08%%\n"
"Exact: 0.00032 FE: 0.31805 Rel. gap 99914.49%%\n",
"Exact: 0.00094 FE: 0.94033 Rel. gap 99940.30%%\n",
"Exact: 0.00187 FE: 1.87415 Rel. gap 99978.99%%\n"
]
}
],
......@@ -986,7 +978,8 @@
" r, c, rx, cx = eigensolver.get_eigenpair(i)\n",
" \n",
" critical_load_an = alpha(i+1)**2*float(EI/N0)/L**2\n",
" print(\"Exact: {0:>10.5f} FE: {1:>10.5f} Rel. gap {2:1.2f}%%\".format(critical_load_an, r, 100*(r/critical_load_an-1)))\n",
" print(\"Exact: {0:>10.5f} FE: {1:>10.5f} Rel. gap {2:1.2f}%%\".format(\n",
" critical_load_an, r, 100*(r/critical_load_an-1)))\n",
" \n",
" # Initialize function and assign eigenvector (renormalize by stiffness matrix)\n",
" eigenmode = Function(V,name=\"Eigenvector \"+str(i))\n",
......@@ -1009,11 +1002,19 @@
"F_n = (\\alpha_n)^2 \\dfrac{EI}{L^2} \\quad \\text{with }\\alpha_n \\text{ solutions to } \\tan(\\alpha) = \\alpha\n",
"\\end{equation}\n",
"\n",
"It can, in particular, be observed that the displacement-based FE solution overestimates the exact buckling load and that the error increases with the order of the buckling load."
"In particular, it can be observed that the displacement-based FE solution overestimates the exact buckling load and that the error increases with the order of the buckling load."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
}
],
"metadata": {
"celltoolbar": "Raw Cell Format",
"kernelspec": {
"display_name": "Python 2",
"language": "python",
......
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