.. _linear_elasticity-rigid_lagrange:

linear_elasticity/rigid_lagrange.py
===================================

**Description**


Linear elasticity example with rigid-body constraints enforced via Lagrange
multipliers.

Find :math:`\ul{u}`, :math:`\ul{\bar{u}}`, :math:`\ul{\bar{\omega}}`,
:math:`\ul{\lambda}` such that:

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

    - \int_{\Gamma_R} \ul{\bar{v}} \ul{\lambda}
    = 0
    \;, \quad \forall \ul{\bar{v}} \;,

    - \int_{\Gamma_R} (\ul{\bar{q}} \times (\ul{x} - \ul{x_c}) \ul{\lambda}
    = 0
    \;, \quad \forall \ul{\bar{q}} \;,

    + \int_{\Gamma_R} \ul{\nu} \ul{u}
    - \int_{\Gamma_R} \ul{\nu} \ul{\bar{u}}
    - \int_{\Gamma_R} \ul{\nu} (\ul{\bar{\omega}} \times (\ul{x} - \ul{x_c})
    = 0
    \;, \quad \forall \ul{\eta} \;.

The displacements in the ``'Rigid'`` region are constrained to rigid body
motions. The rigid volume, centered at :math:`\ul{x_c}`, is not present in
the domain's mesh, the constraint is applied to its surface :math:`\Gamma_R`.

Usage Examples
--------------

- Run with the default parameters, visualize deformation with 10 times
  magnified displacements::

    sfepy-run sfepy/examples/linear_elasticity/rigid_lagrange.py
    sfepy-view output/rigid_lagrange/cut-cylinder.*.vtk -f u:wu:f10:p0 1:vw:wu:f10:p0 1:vw:p0

- View the Lagrange multipliers :math:`\ul{\lambda}` and the rigid body motion
  displacements :math:`\ul{\bar{u}}` and rotations :math:`\ul{\bar{\omega}}`::

    sfepy-view output/rigid_lagrange/cut-cylinder.*.vtk -f lam:wu:f10:p0 1:vw:wu:f10:p0 1:vw:p0
    sfepy-view output/rigid_lagrange/cut-cylinder.*.vtk -f ub:wu:f10:p0 1:vw:wu:f10:p0 1:vw:p0
    sfepy-view output/rigid_lagrange/cut-cylinder.*.vtk -f ob:wu:f10:p0 1:vw:wu:f10:p0 1:vw:p0

- Change elasticity parameters and field approximation orders::

    sfepy-run sfepy/examples/linear_elasticity/rigid_lagrange.py -d "E=1e9,nu=0.3,order_u=2,order_lam=0"


.. image:: /../doc/images/gallery/linear_elasticity-rigid_lagrange.png


:download:`source code </../sfepy/examples/linear_elasticity/rigid_lagrange.py>`

.. literalinclude:: /../sfepy/examples/linear_elasticity/rigid_lagrange.py