Commit e9ee2373 authored by Jeremy BLEYER's avatar Jeremy BLEYER

Added contact penalty example and licenses

parent 53f479f5
#
# .. # gedit: set fileencoding=utf8 :
# .. raw:: html
#
# <a rel="license" href="http://creativecommons.org/licenses/by-sa/4.0/"><p align="center"><img alt="Creative Commons License" style="border-width:0" src="https://i.creativecommons.org/l/by-sa/4.0/88x31.png"/></a><br />This work is licensed under a <a rel="license" href="http://creativecommons.org/licenses/by-sa/4.0/">Creative Commons Attribution-ShareAlike 4.0 International License</a></p>
#
# .. _LinearElasticity2D:
#
# .. _LinearElasticity2D:
#
# =========================
# 2D linear elasticity
......
#
raw:: html
<a rel="license" href="http://creativecommons.org/licenses/by-sa/4.0/"><p align="center"><img alt="Creative Commons License" style="border-width:0" src="https://i.creativecommons.org/l/by-sa/4.0/88x31.png"/></a><br />This work is licensed under a <a rel="license" href="http://creativecommons.org/licenses/by-sa/4.0/">Creative Commons Attribution-ShareAlike 4.0 International License</a></p>
# .. _ModalAnalysis:
#
# ==========================================
......
#
# .. # gedit: set fileencoding=utf8 :
# .. raw:: html
#
# <a rel="license" href="http://creativecommons.org/licenses/by-sa/4.0/"><p align="center"><img alt="Creative Commons License" style="border-width:0" src="https://i.creativecommons.org/l/by-sa/4.0/88x31.png"/></a><br />This work is licensed under a <a rel="license" href="http://creativecommons.org/licenses/by-sa/4.0/">Creative Commons Attribution-ShareAlike 4.0 International License</a></p>
#
# .. _OrthotropicElasticity:
#
......@@ -11,10 +14,10 @@
# Introduction
# ------------
#
# In this numerical tour, we will show how to tackle the case of orthotropic elasticity (in a 2D setting). The corresponding file can be obtained from
# In this numerical tour, we will show how to tackle the case of orthotropic elasticity (in a 2D setting). The corresponding file can be obtained from
# :download:`orthotropic_elasticity.py`.
#
# We consider here the case of a square plate perforated by a circular hole of
# We consider here the case of a square plate perforated by a circular hole of
# radius :math:`R`, the plate dimension is :math:`2L\times 2L` with :math:`L \gg R`
# Only the top-right quarter of the plate will be considered. Loading will consist
# of a uniform traction on the top/bottom boundaries, symmetry conditions will also
......@@ -27,7 +30,7 @@ from mshr import *
L, R = 1., 0.1
N = 50 # mesh density
domain = Rectangle(Point(0.,0.), Point(L, L)) - Circle(Point(0., 0.), R)
mesh = generate_mesh(domain, N)
......@@ -35,7 +38,7 @@ mesh = generate_mesh(domain, N)
# Constitutive relation
# ---------------------
#
# Constitutive relations will be defined using an engineering (or Voigt) notation (i.e.
# Constitutive relations will be defined using an engineering (or Voigt) notation (i.e.
# second order tensors will be written as a vector of their components) contrary
# to the :ref:`LinearElasticity2D` example which used an intrinsic notation. In
# the material frame, which is assumed to coincide here with the global :math:`(Oxy)`
......@@ -46,13 +49,13 @@ mesh = generate_mesh(domain, N)
# .. math::
# \begin{Bmatrix} \varepsilon_{xx} \\ \varepsilon_{yy} \\ 2\varepsilon_{xy}
# \end{Bmatrix} = \begin{bmatrix} 1/E_x & -\nu_{xy}/E_x & 0\\
# -\nu_{yx}/E_y & 1/E_y & 0 \\ 0 & 0 & 1/G_{xy} \end{bmatrix}\begin{Bmatrix}
# -\nu_{yx}/E_y & 1/E_y & 0 \\ 0 & 0 & 1/G_{xy} \end{bmatrix}\begin{Bmatrix}
# \sigma_{xx} \\ \sigma_{yy} \\ \sigma_{xy}
# \end{Bmatrix}
#
# with :math:`E_x, E_y` the two Young's moduli in the orthotropy directions, :math:`\nu_{xy}`
# the in-plane Poisson ration (with the following relation ensuring the constitutive
# relation symmetry :math:`\nu_{yx}=\nu_{xy}E_y/E_x`) and :math:`G_{xy}` being the
# relation symmetry :math:`\nu_{yx}=\nu_{xy}E_y/E_x`) and :math:`G_{xy}` being the
# shear modulus. This relation needs to be inverted to obtain the stress components as a function
# of the strain components :math:`\boldsymbol{\sigma}=\mathbf{C}\boldsymbol{\varepsilon}` with
# :math:`\mathbf{C}=\mathbf{S}^{-1}`::
......@@ -89,7 +92,7 @@ def sigma(v):
# Problem position and resolution
# --------------------------------
#
# Different parts of the quarter plate boundaries are now defined as well as the
# Different parts of the quarter plate boundaries are now defined as well as the
# exterior integration measure ``ds``::
class Top(SubDomain):
......@@ -127,7 +130,7 @@ a = inner(sigma(du), eps(u_))*dx
T = Constant((0, 1e-3))
l = dot(T, u_)*ds(1)
# Symmetric boundary conditions are applied on the ``Top`` and ``Left`` boundaries
# Symmetric boundary conditions are applied on the ``Top`` and ``Left`` boundaries
# and the problem is solved::
# symmetry boundary conditions
......
#
raw:: html
<a rel="license" href="http://creativecommons.org/licenses/by-sa/4.0/"><p align="center"><img alt="Creative Commons License" style="border-width:0" src="https://i.creativecommons.org/l/by-sa/4.0/88x31.png"/></a><br />This work is licensed under a <a rel="license" href="http://creativecommons.org/licenses/by-sa/4.0/">Creative Commons Attribution-ShareAlike 4.0 International License</a></p>
# .. _ReissnerMindlinDG:
#
# ==============================================================
......
#
raw:: html
<a rel="license" href="http://creativecommons.org/licenses/by-sa/4.0/"><p align="center"><img alt="Creative Commons License" style="border-width:0" src="https://i.creativecommons.org/l/by-sa/4.0/88x31.png"/></a><br />This work is licensed under a <a rel="license" href="http://creativecommons.org/licenses/by-sa/4.0/">Creative Commons Attribution-ShareAlike 4.0 International License</a></p>
# .. _ReissnerMindlinQuads:
#
# ==========================================
......
.. _vonMisesPlasticity:
.. raw:: html
<a rel="license" href="http://creativecommons.org/licenses/by-sa/4.0/"><p align="center"><img alt="Creative Commons License" style="border-width:0" src="https://i.creativecommons.org/l/by-sa/4.0/88x31.png"/></a><br />This work is licensed under a <a rel="license" href="http://creativecommons.org/licenses/by-sa/4.0/">Creative Commons Attribution-ShareAlike 4.0 International License</a></p>
==================================================
Elasto-plastic analysis of a 2D von Mises material
==================================================
......
.. # gedit: set fileencoding=utf8 :
.. raw:: html
<a rel="license" href="http://creativecommons.org/licenses/by-sa/4.0/"><p align="center"><img alt="Creative Commons License" style="border-width:0" src="https://i.creativecommons.org/l/by-sa/4.0/88x31.png"/></a><br />This work is licensed under a <a rel="license" href="http://creativecommons.org/licenses/by-sa/4.0/">Creative Commons Attribution-ShareAlike 4.0 International License</a></p>
.. _LinearElasticity2D:
.. _LinearElasticity2D:
=========================
2D linear elasticity
......
.. # gedit: set fileencoding=utf8 :
.. raw:: html
<a rel="license" href="http://creativecommons.org/licenses/by-sa/4.0/"><p align="center"><img alt="Creative Commons License" style="border-width:0" src="https://i.creativecommons.org/l/by-sa/4.0/88x31.png"/></a><br />This work is licensed under a <a rel="license" href="http://creativecommons.org/licenses/by-sa/4.0/">Creative Commons Attribution-ShareAlike 4.0 International License</a></p>
.. _OrthotropicElasticity:
......@@ -11,10 +14,10 @@
Introduction
------------
In this numerical tour, we will show how to tackle the case of orthotropic elasticity (in a 2D setting). The corresponding file can be obtained from
In this numerical tour, we will show how to tackle the case of orthotropic elasticity (in a 2D setting). The corresponding file can be obtained from
:download:`orthotropic_elasticity.py`.
We consider here the case of a square plate perforated by a circular hole of
We consider here the case of a square plate perforated by a circular hole of
radius :math:`R`, the plate dimension is :math:`2L\times 2L` with :math:`L \gg R`
Only the top-right quarter of the plate will be considered. Loading will consist
of a uniform traction on the top/bottom boundaries, symmetry conditions will also
......@@ -27,7 +30,7 @@ between a rectangle and a circle::
L, R = 1., 0.1
N = 50 # mesh density
domain = Rectangle(Point(0.,0.), Point(L, L)) - Circle(Point(0., 0.), R)
mesh = generate_mesh(domain, N)
......@@ -35,7 +38,7 @@ between a rectangle and a circle::
Constitutive relation
---------------------
Constitutive relations will be defined using an engineering (or Voigt) notation (i.e.
Constitutive relations will be defined using an engineering (or Voigt) notation (i.e.
second order tensors will be written as a vector of their components) contrary
to the :ref:`LinearElasticity2D` example which used an intrinsic notation. In
the material frame, which is assumed to coincide here with the global :math:`(Oxy)`
......@@ -46,13 +49,13 @@ frame, the orthotropic constitutive law writes :math:`\boldsymbol{\varepsilon}=\
.. math::
\begin{Bmatrix} \varepsilon_{xx} \\ \varepsilon_{yy} \\ 2\varepsilon_{xy}
\end{Bmatrix} = \begin{bmatrix} 1/E_x & -\nu_{xy}/E_x & 0\\
-\nu_{yx}/E_y & 1/E_y & 0 \\ 0 & 0 & 1/G_{xy} \end{bmatrix}\begin{Bmatrix}
-\nu_{yx}/E_y & 1/E_y & 0 \\ 0 & 0 & 1/G_{xy} \end{bmatrix}\begin{Bmatrix}
\sigma_{xx} \\ \sigma_{yy} \\ \sigma_{xy}
\end{Bmatrix}
with :math:`E_x, E_y` the two Young's moduli in the orthotropy directions, :math:`\nu_{xy}`
the in-plane Poisson ration (with the following relation ensuring the constitutive
relation symmetry :math:`\nu_{yx}=\nu_{xy}E_y/E_x`) and :math:`G_{xy}` being the
relation symmetry :math:`\nu_{yx}=\nu_{xy}E_y/E_x`) and :math:`G_{xy}` being the
shear modulus. This relation needs to be inverted to obtain the stress components as a function
of the strain components :math:`\boldsymbol{\sigma}=\mathbf{C}\boldsymbol{\varepsilon}` with
:math:`\mathbf{C}=\mathbf{S}^{-1}`::
......@@ -89,7 +92,7 @@ second-order tensor or using the Voigt engineering notation::
Problem position and resolution
--------------------------------
Different parts of the quarter plate boundaries are now defined as well as the
Different parts of the quarter plate boundaries are now defined as well as the
exterior integration measure ``ds``::
class Top(SubDomain):
......@@ -101,7 +104,7 @@ exterior integration measure ``ds``::
class Bottom(SubDomain):
def inside(self, x, on_boundary):
return near(x[1],0) and on_boundary
# exterior facets MeshFunction
facets = MeshFunction("size_t", mesh, 1)
facets.set_all(0)
......@@ -122,12 +125,12 @@ on the top boundary::
u_ = TestFunction(V)
u = Function(V, name='Displacement')
a = inner(sigma(du), eps(u_))*dx
# uniform traction on top boundary
T = Constant((0, 1e-3))
l = dot(T, u_)*ds(1)
Symmetric boundary conditions are applied on the ``Top`` and ``Left`` boundaries
Symmetric boundary conditions are applied on the ``Top`` and ``Left`` boundaries
and the problem is solved::
# symmetry boundary conditions
......@@ -135,12 +138,12 @@ and the problem is solved::
DirichletBC(V.sub(1), Constant(0.), facets, 3)]
solve(a == l, u, bc)
import matplotlib.pyplot as plt
p = plot(sigma(u)[1,1]/T[1], mode='color')
plt.colorbar(p)
plt.title(r"$\sigma_{yy}$",fontsize=26)
The :math:`\sigma_{xx}` and :math:`\sigma_{yy}` components should look like
that:
......
.. raw:: html
<a rel="license" href="http://creativecommons.org/licenses/by-sa/4.0/"><p align="center"><img alt="Creative Commons License" style="border-width:0" src="https://i.creativecommons.org/l/by-sa/4.0/88x31.png"/></a><br />This work is licensed under a <a rel="license" href="http://creativecommons.org/licenses/by-sa/4.0/">Creative Commons Attribution-ShareAlike 4.0 International License</a></p>
.. _ModalAnalysis:
......
.. raw:: html
<a rel="license" href="http://creativecommons.org/licenses/by-sa/4.0/"><p align="center"><img alt="Creative Commons License" style="border-width:0" src="https://i.creativecommons.org/l/by-sa/4.0/88x31.png"/></a><br />This work is licensed under a <a rel="license" href="http://creativecommons.org/licenses/by-sa/4.0/">Creative Commons Attribution-ShareAlike 4.0 International License</a></p>
.. _ReissnerMindlinDG:
......
.. raw:: html
<a rel="license" href="http://creativecommons.org/licenses/by-sa/4.0/"><p align="center"><img alt="Creative Commons License" style="border-width:0" src="https://i.creativecommons.org/l/by-sa/4.0/88x31.png"/></a><br />This work is licensed under a <a rel="license" href="http://creativecommons.org/licenses/by-sa/4.0/">Creative Commons Attribution-ShareAlike 4.0 International License</a></p>
.. _ReissnerMindlinQuads:
......
......@@ -11,10 +11,10 @@ What is it about ?
------------------------
These numerical tours will introduce you to a wide variety of topics in
computational continuum and structural mechanics using the finite element software FEniCS.
These numerical tours will introduce you to a wide variety of topics in
computational continuum and structural mechanics using the finite element software FEniCS.
Many covered topics can be considered as standard and will help the reader in
getting started with FEniCS using solid mechanics examples.
getting started with FEniCS using solid mechanics examples.
Other topics will also be more exploratory and will reflect currently investigated research topics,
illustrating the versatility of FEniCS.
......@@ -22,33 +22,40 @@ illustrating the versatility of FEniCS.
The full set of demos can be obtained from the *COmputational MEchanics Toolbox* (COMET) available at
https://gitlab.enpc.fr/jeremy.bleyer/comet-fenics.
--------------------
Citing and license
--------------------
If you find these demos useful for your research work, please consider citing them using the following
Zenodo DOI https://doi.org/10.5281/zenodo.1287832
.. code-block:: none
@article{bleyer2018numericaltours,
title={Numerical Tours of Computational Mechanics with FEniCS},
DOI={10.5281/zenodo.1287832},
publisher={Zenodo},
author={Jeremy Bleyer},
@article{bleyer2018numericaltours,
title={Numerical Tours of Computational Mechanics with FEniCS},
DOI={10.5281/zenodo.1287832},
publisher={Zenodo},
author={Jeremy Bleyer},
year={2018}}
All this work is licensed under the `Creative Commons Attribution-ShareAlike 4.0 International License <http://creativecommons.org/licenses/by-sa/4.0/>`_ |license|.
.. |license| image:: https://i.creativecommons.org/l/by-sa/4.0/88x31.png
-----------------------
How do I get started ?
How do I get started ?
-----------------------
You can find instructions on how to install FEniCS on the FEniCS project website
http://fenicsproject.org. In the following numerical tours, we will use the
You can find instructions on how to install FEniCS on the FEniCS project website
http://fenicsproject.org. In the following numerical tours, we will use the
Python interface for the different FEniCS scripts. These demos have been written
using FEniCS 2017.2.0 but many should work with older versions.
FEniCS is also distributed along with an important number of documented or
undocumented examples, some of them will be revisited in these tours but do not
FEniCS is also distributed along with an important number of documented or
undocumented examples, some of them will be revisited in these tours but do not
hesitate over looking at other interesting examples.
In the following, we will assume that readers possess basic knowledge of FEniCS commands.
In the following, we will assume that readers possess basic knowledge of FEniCS commands.
In particular, we advise you to go first through the documentation and tutorials https://fenicsproject.org/tutorial/
if this is not the case.
......@@ -56,9 +63,9 @@ if this is not the case.
About the author
----------------------
`Jeremy Bleyer <https://sites.google.com/site/bleyerjeremy/>`_ is a researcher
in Solid and Structural Mechanics at `Laboratoire Navier <http://navier.enpc.fr>`_,
a joint research unit of `Ecole Nationale des Ponts et Chaussées <http://www.enpc.fr>`_,
`Jeremy Bleyer <https://sites.google.com/site/bleyerjeremy/>`_ is a researcher
in Solid and Structural Mechanics at `Laboratoire Navier <http://navier.enpc.fr>`_,
a joint research unit of `Ecole Nationale des Ponts et Chaussées <http://www.enpc.fr>`_,
`IFSTTAR <http://www.ifsttar.fr>`_ and `CNRS <http://www.cnrs.fr>`_ (UMR 8205).
email: jeremy.bleyer@enpc.fr
......
......@@ -51,8 +51,8 @@
<div class="bodywrapper">
<div class="body" role="main">
<div class="section" id="elasto-plastic-analysis-of-a-2d-von-mises-material">
<span id="vonmisesplasticity"></span><h1>Elasto-plastic analysis of a 2D von Mises material<a class="headerlink" href="#elasto-plastic-analysis-of-a-2d-von-mises-material" title="Permalink to this headline"></a></h1>
<a rel="license" href="http://creativecommons.org/licenses/by-sa/4.0/"><p align="center"><img alt="Creative Commons License" style="border-width:0" src="https://i.creativecommons.org/l/by-sa/4.0/88x31.png"/></a><br />This work is licensed under a <a rel="license" href="http://creativecommons.org/licenses/by-sa/4.0/">Creative Commons Attribution-ShareAlike 4.0 International License</a></p><div class="section" id="elasto-plastic-analysis-of-a-2d-von-mises-material">
<h1>Elasto-plastic analysis of a 2D von Mises material<a class="headerlink" href="#elasto-plastic-analysis-of-a-2d-von-mises-material" 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 example is concerned with the incremental analysis of an elasto-plastic
......@@ -410,6 +410,7 @@ when considering a zero hardening modulus.</p>
<ul class="current">
<li class="toctree-l1"><a class="reference internal" href="../../intro.html">Introduction</a><ul>
<li class="toctree-l2"><a class="reference internal" href="../../intro.html#what-is-it-about">What is it about ?</a></li>
<li class="toctree-l2"><a class="reference internal" href="../../intro.html#citing-and-license">Citing and license</a></li>
<li class="toctree-l2"><a class="reference internal" href="../../intro.html#how-do-i-get-started">How do I get started ?</a></li>
<li class="toctree-l2"><a class="reference internal" href="../../intro.html#about-the-author">About the author</a></li>
</ul>
......
......@@ -51,7 +51,7 @@
<div class="bodywrapper">
<div class="body" role="main">
<div class="section" id="d-linear-elasticity">
<a rel="license" href="http://creativecommons.org/licenses/by-sa/4.0/"><p align="center"><img alt="Creative Commons License" style="border-width:0" src="https://i.creativecommons.org/l/by-sa/4.0/88x31.png"/></a><br />This work is licensed under a <a rel="license" href="http://creativecommons.org/licenses/by-sa/4.0/">Creative Commons Attribution-ShareAlike 4.0 International License</a></p><div class="section" id="d-linear-elasticity">
<span id="linearelasticity2d"></span><h1>2D linear elasticity<a class="headerlink" href="#d-linear-elasticity" 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>
......@@ -220,6 +220,7 @@ writing/reading. Prefered output format is now .xdmf:</p>
<ul class="current">
<li class="toctree-l1"><a class="reference internal" href="../../intro.html">Introduction</a><ul>
<li class="toctree-l2"><a class="reference internal" href="../../intro.html#what-is-it-about">What is it about ?</a></li>
<li class="toctree-l2"><a class="reference internal" href="../../intro.html#citing-and-license">Citing and license</a></li>
<li class="toctree-l2"><a class="reference internal" href="../../intro.html#how-do-i-get-started">How do I get started ?</a></li>
<li class="toctree-l2"><a class="reference internal" href="../../intro.html#about-the-author">About the author</a></li>
</ul>
......
......@@ -217,7 +217,7 @@ div.nboutput div.output_area.stderr {
.ansi-bold { font-weight: bold; }
.ansi-underline { text-decoration: underline; }
</style>
<div class="admonition note">
<a rel="license" href="http://creativecommons.org/licenses/by-sa/4.0/"><p align="center"><img alt="Creative Commons License" style="border-width:0" src="https://i.creativecommons.org/l/by-sa/4.0/88x31.png"/></a><br />This work is licensed under a <a rel="license" href="http://creativecommons.org/licenses/by-sa/4.0/">Creative Commons Attribution-ShareAlike 4.0 International License</a></p><div class="admonition note">
<p class="admonition-title">The corresponding files can be obtained from:</p>
<ul class="simple">
<li>Jupyter Notebook: <a class="reference download internal" href="../../_downloads/axisymmetric_elasticity.ipynb" download=""><code class="xref download docutils literal"><span class="pre">axisymmetric_elasticity.ipynb</span></code></a></li>
......@@ -2805,6 +2805,7 @@ if (IPython.notebook.kernel != null) {
<ul class="current">
<li class="toctree-l1"><a class="reference internal" href="../../intro.html">Introduction</a><ul>
<li class="toctree-l2"><a class="reference internal" href="../../intro.html#what-is-it-about">What is it about ?</a></li>
<li class="toctree-l2"><a class="reference internal" href="../../intro.html#citing-and-license">Citing and license</a></li>
<li class="toctree-l2"><a class="reference internal" href="../../intro.html#how-do-i-get-started">How do I get started ?</a></li>
<li class="toctree-l2"><a class="reference internal" href="../../intro.html#about-the-author">About the author</a></li>
</ul>
......
......@@ -51,7 +51,7 @@
<div class="bodywrapper">
<div class="body" role="main">
<div class="section" id="orthotropic-linear-elasticity">
<a rel="license" href="http://creativecommons.org/licenses/by-sa/4.0/"><p align="center"><img alt="Creative Commons License" style="border-width:0" src="https://i.creativecommons.org/l/by-sa/4.0/88x31.png"/></a><br />This work is licensed under a <a rel="license" href="http://creativecommons.org/licenses/by-sa/4.0/">Creative Commons Attribution-ShareAlike 4.0 International License</a></p><div class="section" id="orthotropic-linear-elasticity">
<span id="orthotropicelasticity"></span><h1>Orthotropic linear elasticity<a class="headerlink" href="#orthotropic-linear-elasticity" 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>
......@@ -198,6 +198,7 @@ that:</p>
<ul class="current">
<li class="toctree-l1"><a class="reference internal" href="../../intro.html">Introduction</a><ul>
<li class="toctree-l2"><a class="reference internal" href="../../intro.html#what-is-it-about">What is it about ?</a></li>
<li class="toctree-l2"><a class="reference internal" href="../../intro.html#citing-and-license">Citing and license</a></li>
<li class="toctree-l2"><a class="reference internal" href="../../intro.html#how-do-i-get-started">How do I get started ?</a></li>
<li class="toctree-l2"><a class="reference internal" href="../../intro.html#about-the-author">About the author</a></li>
</ul>
......
......@@ -51,7 +51,7 @@
<div class="bodywrapper">
<div class="body" role="main">
<div class="section" id="modal-analysis-of-an-elastic-structure">
<a rel="license" href="http://creativecommons.org/licenses/by-sa/4.0/"><p align="center"><img alt="Creative Commons License" style="border-width:0" src="https://i.creativecommons.org/l/by-sa/4.0/88x31.png"/></a><br />This work is licensed under a <a rel="license" href="http://creativecommons.org/licenses/by-sa/4.0/">Creative Commons Attribution-ShareAlike 4.0 International License</a></p><div class="section" id="modal-analysis-of-an-elastic-structure">
<span id="modalanalysis"></span><h1>Modal analysis of an elastic structure<a class="headerlink" href="#modal-analysis-of-an-elastic-structure" 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>
......@@ -261,6 +261,7 @@ and the beam theory eigenfrequencies :</p>
<ul class="current">
<li class="toctree-l1"><a class="reference internal" href="../../intro.html">Introduction</a><ul>
<li class="toctree-l2"><a class="reference internal" href="../../intro.html#what-is-it-about">What is it about ?</a></li>
<li class="toctree-l2"><a class="reference internal" href="../../intro.html#citing-and-license">Citing and license</a></li>
<li class="toctree-l2"><a class="reference internal" href="../../intro.html#how-do-i-get-started">How do I get started ?</a></li>
<li class="toctree-l2"><a class="reference internal" href="../../intro.html#about-the-author">About the author</a></li>
</ul>
......
......@@ -217,7 +217,7 @@ div.nboutput div.output_area.stderr {
.ansi-bold { font-weight: bold; }
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<p class="admonition-title">The corresponding files can be obtained from:</p>
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<li>Jupyter Notebook: <a class="reference download internal" href="../../_downloads/periodic_homog_elas.ipynb" download=""><code class="xref download docutils literal"><span class="pre">periodic_homog_elas.ipynb</span></code></a></li>
......@@ -2141,6 +2141,7 @@ if (IPython.notebook.kernel != null) {
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<li class="toctree-l1"><a class="reference internal" href="../../intro.html">Introduction</a><ul>
<li class="toctree-l2"><a class="reference internal" href="../../intro.html#what-is-it-about">What is it about ?</a></li>
<li class="toctree-l2"><a class="reference internal" href="../../intro.html#citing-and-license">Citing and license</a></li>
<li class="toctree-l2"><a class="reference internal" href="../../intro.html#how-do-i-get-started">How do I get started ?</a></li>
<li class="toctree-l2"><a class="reference internal" href="../../intro.html#about-the-author">About the author</a></li>
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......
......@@ -51,7 +51,7 @@
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<div class="body" role="main">
<div class="section" id="reissner-mindlin-plate-with-a-discontinuous-galerkin-approach">
<a rel="license" href="http://creativecommons.org/licenses/by-sa/4.0/"><p align="center"><img alt="Creative Commons License" style="border-width:0" src="https://i.creativecommons.org/l/by-sa/4.0/88x31.png"/></a><br />This work is licensed under a <a rel="license" href="http://creativecommons.org/licenses/by-sa/4.0/">Creative Commons Attribution-ShareAlike 4.0 International License</a></p><div class="section" id="reissner-mindlin-plate-with-a-discontinuous-galerkin-approach">
<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>
......@@ -195,6 +195,7 @@ so that no term arise in the linear functional):</p>
<ul class="current">
<li class="toctree-l1"><a class="reference internal" href="../../intro.html">Introduction</a><ul>
<li class="toctree-l2"><a class="reference internal" href="../../intro.html#what-is-it-about">What is it about ?</a></li>
<li class="toctree-l2"><a class="reference internal" href="../../intro.html#citing-and-license">Citing and license</a></li>
<li class="toctree-l2"><a class="reference internal" href="../../intro.html#how-do-i-get-started">How do I get started ?</a></li>
<li class="toctree-l2"><a class="reference internal" href="../../intro.html#about-the-author">About the author</a></li>
</ul>
......
......@@ -51,7 +51,7 @@
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<div class="body" role="main">
<div class="section" id="reissner-mindlin-plate-with-quadrilaterals">
<a rel="license" href="http://creativecommons.org/licenses/by-sa/4.0/"><p align="center"><img alt="Creative Commons License" style="border-width:0" src="https://i.creativecommons.org/l/by-sa/4.0/88x31.png"/></a><br />This work is licensed under a <a rel="license" href="http://creativecommons.org/licenses/by-sa/4.0/">Creative Commons Attribution-ShareAlike 4.0 International License</a></p><div class="section" id="reissner-mindlin-plate-with-quadrilaterals">
<span id="reissnermindlinquads"></span><h1>Reissner-Mindlin plate with Quadrilaterals<a class="headerlink" href="#reissner-mindlin-plate-with-quadrilaterals" 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>
......@@ -188,6 +188,7 @@ the thin plate solution.</p>
<ul class="current">
<li class="toctree-l1"><a class="reference internal" href="../../intro.html">Introduction</a><ul>
<li class="toctree-l2"><a class="reference internal" href="../../intro.html#what-is-it-about">What is it about ?</a></li>
<li class="toctree-l2"><a class="reference internal" href="../../intro.html#citing-and-license">Citing and license</a></li>
<li class="toctree-l2"><a class="reference internal" href="../../intro.html#how-do-i-get-started">How do I get started ?</a></li>
<li class="toctree-l2"><a class="reference internal" href="../../intro.html#about-the-author">About the author</a></li>
</ul>
......
......@@ -217,7 +217,7 @@ div.nboutput div.output_area.stderr {
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<a rel="license" href="http://creativecommons.org/licenses/by-sa/4.0/"><p align="center"><img alt="Creative Commons License" style="border-width:0" src="https://i.creativecommons.org/l/by-sa/4.0/88x31.png"/></a><br />This work is licensed under a <a rel="license" href="http://creativecommons.org/licenses/by-sa/4.0/">Creative Commons Attribution-ShareAlike 4.0 International License</a></p><div class="admonition note">
<p class="admonition-title">The corresponding files can be obtained from:</p>
<ul class="simple">
<li>Jupyter Notebook: <a class="reference download internal" href="../../_downloads/thermoelasticity.ipynb" download=""><code class="xref download docutils literal"><span class="pre">thermoelasticity.ipynb</span></code></a></li>
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<li class="toctree-l2"><a class="reference internal" href="../../intro.html#what-is-it-about">What is it about ?</a></li>
<li class="toctree-l2"><a class="reference internal" href="../../intro.html#citing-and-license">Citing and license</a></li>
<li class="toctree-l2"><a class="reference internal" href="../../intro.html#how-do-i-get-started">How do I get started ?</a></li>
<li class="toctree-l2"><a class="reference internal" href="../../intro.html#about-the-author">About the author</a></li>
</ul>
......
......@@ -217,7 +217,7 @@ div.nboutput div.output_area.stderr {
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<div class="admonition note">
<a rel="license" href="http://creativecommons.org/licenses/by-sa/4.0/"><p align="center"><img alt="Creative Commons License" style="border-width:0" src="https://i.creativecommons.org/l/by-sa/4.0/88x31.png"/></a><br />This work is licensed under a <a rel="license" href="http://creativecommons.org/licenses/by-sa/4.0/">Creative Commons Attribution-ShareAlike 4.0 International License</a></p><div class="admonition note">
<p class="admonition-title">The corresponding files can be obtained from:</p>
<ul class="simple">
<li>Jupyter Notebook: <a class="reference download internal" href="../../_downloads/thermoelasticity_transient.ipynb" download=""><code class="xref download docutils literal"><span class="pre">thermoelasticity_transient.ipynb</span></code></a></li>
......@@ -2891,6 +2891,7 @@ Mechanics and Engineering, 85(3), 349-365.
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<li class="toctree-l1"><a class="reference internal" href="../../intro.html">Introduction</a><ul>
<li class="toctree-l2"><a class="reference internal" href="../../intro.html#what-is-it-about">What is it about ?</a></li>
<li class="toctree-l2"><a class="reference internal" href="../../intro.html#citing-and-license">Citing and license</a></li>
<li class="toctree-l2"><a class="reference internal" href="../../intro.html#how-do-i-get-started">How do I get started ?</a></li>
<li class="toctree-l2"><a class="reference internal" href="../../intro.html#about-the-author">About the author</a></li>
</ul>
......
......@@ -63,6 +63,7 @@
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<li class="toctree-l1"><a class="reference internal" href="intro.html">Introduction</a><ul>
<li class="toctree-l2"><a class="reference internal" href="intro.html#what-is-it-about">What is it about ?</a></li>
<li class="toctree-l2"><a class="reference internal" href="intro.html#citing-and-license">Citing and license</a></li>
<li class="toctree-l2"><a class="reference internal" href="intro.html#how-do-i-get-started">How do I get started ?</a></li>
<li class="toctree-l2"><a class="reference internal" href="intro.html#about-the-author">About the author</a></li>
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......
......@@ -71,6 +71,7 @@
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<li class="toctree-l1"><a class="reference internal" href="intro.html">Introduction</a><ul>
<li class="toctree-l2"><a class="reference internal" href="intro.html#what-is-it-about">What is it about ?</a></li>
<li class="toctree-l2"><a class="reference internal" href="intro.html#citing-and-license">Citing and license</a></li>
<li class="toctree-l2"><a class="reference internal" href="intro.html#how-do-i-get-started">How do I get started ?</a></li>
<li class="toctree-l2"><a class="reference internal" href="intro.html#about-the-author">About the author</a></li>
</ul>
......
......@@ -55,6 +55,7 @@
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<li class="toctree-l1"><a class="reference internal" href="intro.html">Introduction</a><ul>
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<li class="toctree-l2"><a class="reference internal" href="intro.html#citing-and-license">Citing and license</a></li>
<li class="toctree-l2"><a class="reference internal" href="intro.html#how-do-i-get-started">How do I get started ?</a></li>
<li class="toctree-l2"><a class="reference internal" href="intro.html#about-the-author">About the author</a></li>
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......@@ -103,6 +104,7 @@
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<li class="toctree-l1"><a class="reference internal" href="intro.html">Introduction</a><ul>