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

Published buckling example

parent b7bef5d1
......@@ -10,8 +10,8 @@
# -------------
#
# This program performs a dynamic modal analysis of an elastic cantilever beam
# represented by a 3D solid continuum. The eigenmodes are computed using the
# **SLEPcEigensolver** and compared against an analytical solution of beam theory.
# represented by a 3D solid continuum. The eigenmodes are computed using the
# **SLEPcEigensolver** and compared against an analytical solution of beam theory.
# The corresponding file can be obtained from :download:`cantilever_modal.py`.
#
#
......@@ -20,14 +20,14 @@
# .. image:: vibration_modes.gif
# :scale: 80 %
#
# The first two fundamental modes are on top with bending along the weak axis (left) and along
# the strong axis (right), the next two modes are at the bottom.
# The first two fundamental modes are on top with bending along the weak axis (left) and along
# the strong axis (right), the next two modes are at the bottom.
#
# ---------------
# Implementation
# ---------------
#
# After importing the relevant modules, the geometry of a beam of length :math:`L=20`
# After importing the relevant modules, the geometry of a beam of length :math:`L=20`
# and rectangular section of size :math:`B\times H` with :math:`B=0.5, H=1` is first defined::
from fenics import *
......@@ -43,7 +43,7 @@ mesh = BoxMesh(Point(0.,0.,0.),Point(L,B,H), Nx, Ny, Nz)
# Material parameters and elastic constitutive relations are classical (here we
# take :math:`\nu=0`) and we also introduce the material density :math:`\rho` for
# take :math:`\nu=0`) and we also introduce the material density :math:`\rho` for
# later definition of the mass matrix::
E, nu = 1e5, 0.
......@@ -59,7 +59,7 @@ def sigma(v):
dim = v.geometric_dimension()
return 2.0*mu*eps(v) + lmbda*tr(eps(v))*Identity(dim)
# Standard FunctionSpace is defined and boundary conditions correspond to a
# Standard FunctionSpace is defined and boundary conditions correspond to a
# fully clamped support at :math:`x=0`::
V = VectorFunctionSpace(mesh, 'Lagrange', degree=1)
......@@ -73,7 +73,7 @@ def left(x, on_boundary):
bc = DirichletBC(V, Constant((0.,0.,0.)), left)
# The system stiffness matrix :math:`[K]` and mass matrix :math:`[M]` are
# The system stiffness matrix :math:`[K]` and mass matrix :math:`[M]` are
# respectively obtained from assembling the corresponding variational forms::
k_form = inner(sigma(du),eps(u_))*dx
......@@ -86,14 +86,14 @@ m_form = rho*dot(du,u_)*dx
M = PETScMatrix()
assemble(m_form, tensor=M)
# Matrices :math:`[K]` and :math:`[M]` are first defined as PETSc Matrix and
# Matrices :math:`[K]` and :math:`[M]` are first defined as PETSc Matrix and
# forms are assembled into it to ensure that they have the right type.
# Note that boundary conditions have been applied to the stiffness matrix using
# ``assemble_system`` so as to preserve symmetry (a dummy ``l_form`` and right-hand side
# vector have been introduced to call this function).
# vector have been introduced to call this function).
#
#
# Modal dynamic analysis consists in solving the following generalized
# Modal dynamic analysis consists in solving the following generalized
# eigenvalue problem :math:`[K]\{U\}=\lambda[M]\{U\}` where the eigenvalue
# is related to the eigenfrequency :math:`\lambda=\omega^2`. This problem
# can be solved using the ``SLEPcEigenSolver``. ::
......@@ -121,7 +121,7 @@ eigensolver.parameters['spectral_shift'] = 0.
# and extract the corresponding eigenpair (first two arguments of ``get_eigenpair``
# correspond to the real and complex part of the eigenvalue, the last two to the
# real and complex part of the eigenvector)::
N_eig = 6 # number of eigenvalues
print "Computing %i first eigenvalues..." % N_eig
eigensolver.solve(N_eig)
......@@ -141,26 +141,26 @@ file_results.parameters["functions_share_mesh"] = True
for i in range(N_eig):
# Extract eigenpair
r, c, rx, cx = eigensolver.get_eigenpair(i)
# 3D eigenfrequency
freq_3D = sqrt(r)/2/pi
# Beam eigenfrequency
if i % 2 == 0: # exact solution should correspond to weak axis bending
I_bend = H*B**3/12.
else: #exact solution should correspond to strong axis bending
I_bend = B*H**3/12.
freq_beam = alpha(i/2)**2*sqrt(E*I_bend/(rho*B*H*L**4))/2/pi
print("Solid FE: {0:8.5f} [Hz] Beam theory: {1:8.5f} [Hz]".format(freq_3D, freq_beam))
# 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}}`
# where :math:`S=BH` is the beam section, :math:`I` the bending inertia and
# where :math:`S=BH` is the beam section, :math:`I` the bending inertia and
# :math:`\alpha_n` is the solution of the following nonlinear equation:
#
# .. math::
......@@ -173,9 +173,9 @@ for i in range(N_eig):
# for the considered numerical values, the strong axis bending frequency will be twice that corresponsing
# to bending along the weak axis. The solution :math:`\alpha_n` are computed using the
# ``scipy.optimize.root`` function with initial guess given by :math:`(2n+1)\pi/2`.
#
#
# With ``Nx=400``, we obtain the following comparison between the FE eigenfrequencies
# and the beam theory eigenfrequencies :
# and the beam theory eigenfrequencies :
#
#
# ===== ============= =================
......@@ -184,11 +184,11 @@ for i in range(N_eig):
# # Solid FE [Hz] Beam theory [Hz]
# ===== ============= =================
# 1 2.04991 2.01925
# 2 4.04854 4.03850
# 2 4.04854 4.03850
# 3 12.81504 12.65443
# 4 25.12717 25.30886
# 4 25.12717 25.30886
# 5 35.74168 35.43277
# 6 66.94816 70.86554
# 6 66.94816 70.86554
# ===== ============= =================
#
#
......@@ -10,8 +10,8 @@ Introduction
-------------
This program performs a dynamic modal analysis of an elastic cantilever beam
represented by a 3D solid continuum. The eigenmodes are computed using the
**SLEPcEigensolver** and compared against an analytical solution of beam theory.
represented by a 3D solid continuum. The eigenmodes are computed using the
**SLEPcEigensolver** and compared against an analytical solution of beam theory.
The corresponding file can be obtained from :download:`cantilever_modal.py`.
......@@ -20,14 +20,14 @@ The first four eigenmodes of this demo will look as follows:
.. image:: vibration_modes.gif
:scale: 80 %
The first two fundamental modes are on top with bending along the weak axis (left) and along
the strong axis (right), the next two modes are at the bottom.
The first two fundamental modes are on top with bending along the weak axis (left) and along
the strong axis (right), the next two modes are at the bottom.
---------------
Implementation
---------------
After importing the relevant modules, the geometry of a beam of length :math:`L=20`
After importing the relevant modules, the geometry of a beam of length :math:`L=20`
and rectangular section of size :math:`B\times H` with :math:`B=0.5, H=1` is first defined::
from fenics import *
......@@ -43,7 +43,7 @@ and rectangular section of size :math:`B\times H` with :math:`B=0.5, H=1` is fir
Material parameters and elastic constitutive relations are classical (here we
take :math:`\nu=0`) and we also introduce the material density :math:`\rho` for
take :math:`\nu=0`) and we also introduce the material density :math:`\rho` for
later definition of the mass matrix::
E, nu = 1e5, 0.
......@@ -59,7 +59,7 @@ later definition of the mass matrix::
dim = v.geometric_dimension()
return 2.0*mu*eps(v) + lmbda*tr(eps(v))*Identity(dim)
Standard FunctionSpace is defined and boundary conditions correspond to a
Standard FunctionSpace is defined and boundary conditions correspond to a
fully clamped support at :math:`x=0`::
V = VectorFunctionSpace(mesh, 'Lagrange', degree=1)
......@@ -73,7 +73,7 @@ fully clamped support at :math:`x=0`::
bc = DirichletBC(V, Constant((0.,0.,0.)), left)
The system stiffness matrix :math:`[K]` and mass matrix :math:`[M]` are
The system stiffness matrix :math:`[K]` and mass matrix :math:`[M]` are
respectively obtained from assembling the corresponding variational forms::
k_form = inner(sigma(du),eps(u_))*dx
......@@ -86,14 +86,14 @@ respectively obtained from assembling the corresponding variational forms::
M = PETScMatrix()
assemble(m_form, tensor=M)
Matrices :math:`[K]` and :math:`[M]` are first defined as PETSc Matrix and
Matrices :math:`[K]` and :math:`[M]` are first defined as PETSc Matrix and
forms are assembled into it to ensure that they have the right type.
Note that boundary conditions have been applied to the stiffness matrix using
``assemble_system`` so as to preserve symmetry (a dummy ``l_form`` and right-hand side
vector have been introduced to call this function).
vector have been introduced to call this function).
Modal dynamic analysis consists in solving the following generalized
Modal dynamic analysis consists in solving the following generalized
eigenvalue problem :math:`[K]\{U\}=\lambda[M]\{U\}` where the eigenvalue
is related to the eigenfrequency :math:`\lambda=\omega^2`. This problem
can be solved using the ``SLEPcEigenSolver``. ::
......@@ -121,7 +121,7 @@ We now ask SLEPc to extract the first 6 eigenvalues by calling its solve functio
and extract the corresponding eigenpair (first two arguments of ``get_eigenpair``
correspond to the real and complex part of the eigenvalue, the last two to the
real and complex part of the eigenvector)::
N_eig = 6 # number of eigenvalues
print "Computing %i first eigenvalues..." % N_eig
eigensolver.solve(N_eig)
......@@ -136,31 +136,31 @@ real and complex part of the eigenvector)::
file_results = XDMFFile("modal_analysis.xdmf")
file_results.parameters["flush_output"] = True
file_results.parameters["functions_share_mesh"] = True
# Extraction
for i in range(N_eig):
# Extract eigenpair
r, c, rx, cx = eigensolver.get_eigenpair(i)
# 3D eigenfrequency
freq_3D = sqrt(r)/2/pi
# Beam eigenfrequency
if i % 2 == 0: # exact solution should correspond to weak axis bending
I_bend = H*B**3/12.
else: #exact solution should correspond to strong axis bending
I_bend = B*H**3/12.
freq_beam = alpha(i/2)**2*sqrt(E*I_bend/(rho*B*H*L**4))/2/pi
print("Solid FE: {0:8.5f} [Hz] Beam theory: {1:8.5f} [Hz]".format(freq_3D, freq_beam))
# 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}}`
where :math:`S=BH` is the beam section, :math:`I` the bending inertia and
where :math:`S=BH` is the beam section, :math:`I` the bending inertia and
:math:`\alpha_n` is the solution of the following nonlinear equation:
.. math::
......@@ -173,9 +173,9 @@ and the other along the strong axis (:math:`I=I_{\text{strong}} = BH^3/12`). Sin
for the considered numerical values, the strong axis bending frequency will be twice that corresponsing
to bending along the weak axis. The solution :math:`\alpha_n` are computed using the
``scipy.optimize.root`` function with initial guess given by :math:`(2n+1)\pi/2`.
With ``Nx=400``, we obtain the following comparison between the FE eigenfrequencies
and the beam theory eigenfrequencies :
and the beam theory eigenfrequencies :
===== ============= =================
......@@ -184,11 +184,11 @@ Mode Eigenfrequencies
# Solid FE [Hz] Beam theory [Hz]
===== ============= =================
1 2.04991 2.01925
2 4.04854 4.03850
2 4.04854 4.03850
3 12.81504 12.65443
4 25.12717 25.30886
4 25.12717 25.30886
5 35.74168 35.43277
6 66.94816 70.86554
6 66.94816 70.86554
===== ============= =================
......@@ -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 @@
<script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
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......@@ -35,7 +35,7 @@
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<a href="../../nonlinear_problems.html" title="Nonlinear problems in solid mechanics"
accesskey="P">previous</a> |
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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"
>previous</a> |
<a href="../reissner_mindlin/reissner_mindlin.html" title="Reissner-Mindlin plates"
<a href="../../beams_and_plates.html" title="Beams and plates"
>next</a> |
<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>
......
......@@ -23,9 +23,7 @@
<script type="text/javascript" src="../../_static/doctools.js"></script>
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......@@ -33,10 +31,6 @@
<div class="headertitle"><a
href="../../index.html">Numerical tours of continuum mechanics using FEniCS</a></div>
<div class="rel" role="navigation" aria-label="related navigation">
<a href="../2D_plasticity/vonMises_plasticity.py.html" title="Elasto-plastic analysis of a 2D von Mises material"
accesskey="P">previous</a> |
<a href="reissner_mindlin_quads.py.html" title="Reissner-Mindlin plate with Quadrilaterals"
accesskey="N">next</a> |
<a href="../../genindex.html" title="General Index"
accesskey="I">index</a>
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......@@ -69,7 +63,7 @@
</div>
<div class="sidebar">
<h3>Table Of Contents</h3>
<ul class="current">
<ul>
<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#how-do-i-get-started">How do I get started ?</a></li>
......@@ -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>
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