linear_elasticity/linear_elastic.py

Description

Linear elasticity with given displacements.

Find \ul{u} such that:

\int_{\Omega} D_{ijkl}\ e_{ij}(\ul{v}) e_{kl}(\ul{u})
= 0
\;, \quad \forall \ul{v} \;,

where

D_{ijkl} = \mu (\delta_{ik} \delta_{jl}+\delta_{il} \delta_{jk}) +
\lambda \ \delta_{ij} \delta_{kl}
\;.

This example models a cylinder that is fixed at one end while the second end has a specified displacement of 0.01 in the x direction (this boundary condition is named 'Displaced'). There is also a specified displacement of 0.005 in the z direction for points in the region labeled 'SomewhereTop'. This boundary condition is named 'PerturbedSurface'. The region 'SomewhereTop' is specified as those vertices for which:

(z > 0.017) & (x > 0.03) & (x <  0.07)

The displacement field (three DOFs/node) in the 'Omega region' is approximated using P1 (four-node tetrahedral) finite elements. The material is linear elastic and its properties are specified as Lamé parameters \lambda and \mu (see http://en.wikipedia.org/wiki/Lam%C3%A9_parameters)

The output is the displacement for each vertex, saved by default to cylinder.vtk. View the results using:

$ ./postproc.py cylinder.vtk --wireframe -b --only-names=u -d'u,plot_displacements,rel_scaling=1'
../../_images/linear_elasticity-linear_elastic.png

source code

# -*- coding: utf-8 -*-
r"""
Linear elasticity with given displacements.

Find :math:`\ul{u}` such that:

.. math::
    \int_{\Omega} D_{ijkl}\ e_{ij}(\ul{v}) e_{kl}(\ul{u})
    = 0
    \;, \quad \forall \ul{v} \;,

where

.. math::
    D_{ijkl} = \mu (\delta_{ik} \delta_{jl}+\delta_{il} \delta_{jk}) +
    \lambda \ \delta_{ij} \delta_{kl}
    \;.

This example models a cylinder that is fixed at one end while the second end
has a specified displacement of 0.01 in the x direction (this boundary
condition is named ``'Displaced'``). There is also a specified displacement of
0.005 in the z direction for points in the region labeled
``'SomewhereTop'``. This boundary condition is named
``'PerturbedSurface'``. The region ``'SomewhereTop'`` is specified as those
vertices for which::

    (z > 0.017) & (x > 0.03) & (x <  0.07)

The displacement field (three DOFs/node) in the ``'Omega region'`` is
approximated using P1 (four-node tetrahedral) finite elements. The material is
linear elastic and its properties are specified as Lamé parameters
:math:`\lambda` and :math:`\mu` (see
http://en.wikipedia.org/wiki/Lam%C3%A9_parameters)

The output is the displacement for each vertex, saved by default to
cylinder.vtk. View the results using::

  $ ./postproc.py cylinder.vtk --wireframe -b --only-names=u -d'u,plot_displacements,rel_scaling=1'
"""
from __future__ import absolute_import
from sfepy import data_dir
from sfepy.mechanics.matcoefs import stiffness_from_lame

filename_mesh = data_dir + '/meshes/3d/cylinder.mesh'

regions = {
    'Omega' : 'all',
    'Left' : ('vertices in (x < 0.001)', 'facet'),
    'Right' : ('vertices in (x > 0.099)', 'facet'),
    'SomewhereTop' : ('vertices in (z > 0.017) & (x > 0.03) & (x < 0.07)',
                      'vertex'),
}

materials = {
    'solid' : ({'D': stiffness_from_lame(dim=3, lam=1e1, mu=1e0)},),
}

fields = {
    'displacement': ('real', 'vector', 'Omega', 1),
}

integrals = {
    'i' : 1,
}

variables = {
    'u' : ('unknown field', 'displacement', 0),
    'v' : ('test field', 'displacement', 'u'),
}

ebcs = {
    'Fixed' : ('Left', {'u.all' : 0.0}),
    'Displaced' : ('Right', {'u.0' : 0.01, 'u.[1,2]' : 0.0}),
    'PerturbedSurface' : ('SomewhereTop', {'u.2' : 0.005}),
}

equations = {
    'balance_of_forces' :
    """dw_lin_elastic.i.Omega(solid.D, v, u) = 0""",
}

solvers = {
    'ls' : ('ls.scipy_direct', {}),
    'newton' : ('nls.newton', {
        'i_max'      : 1,
        'eps_a'      : 1e-10,
    }),
}