Follower pressure
=================

.. currentmodule:: bfsccylinder.bfsccylinder

Residual and tangent stiffness matrix
-------------------------------------

The residual `\boldsymbol{F}_{int}(\boldsymbol{u}) - \boldsymbol{F}_p(\boldsymbol{u})` and its Jacobian
`\boldsymbol{K}_{C0} + \boldsymbol{K}_{CNL} + \boldsymbol{K}_{GNL} + \boldsymbol{K}_G + \boldsymbol{K}_F` of one element, for both
kinematics, the modules of :doc:`kinematics` being selected by the
dictionary ``KINEMATICS``, with a pressure that varies over the element:

.. literalinclude:: ../../tests/test_follower_pressure.py
    :pyobject: make_element

.. literalinclude:: ../../tests/test_follower_pressure.py
    :pyobject: residual_and_tangent

The tangent stiffness matrix against a Taylor test of the residual:

.. literalinclude:: ../../tests/test_follower_pressure.py
    :pyobject: test_tangent_is_jacobian_of_residual

Linear buckling of a ring
-------------------------

A long cylinder under a uniform external pressure buckles as a ring. The
ring is modelled with one element along `x`, the mesh closed around the
circumference, and symmetry planes that keep the modes with an even number
of circumferential waves; see the docstring of
``tests/test_follower_pressure_validation.py``. The matrices of the model,
with `\boldsymbol{K}_G` of the linear static solution of the dead pressure load and
the load stiffness matrix `\boldsymbol{K}_{F_{ref}}` of the same pressure at
`\boldsymbol{u} = \boldsymbol{0}`:

.. literalinclude:: ../../tests/test_follower_pressure_validation.py
    :pyobject: cylinder_model

.. literalinclude:: ../../tests/test_follower_pressure_validation.py
    :pyobject: assemble

.. literalinclude:: ../../tests/test_follower_pressure_validation.py
    :pyobject: matrices

.. literalinclude:: ../../tests/test_follower_pressure_validation.py
    :pyobject: ring

The linear buckling problem `(\boldsymbol{K}_{C0} + \lambda (\boldsymbol{K}_G +
\boldsymbol{K}_{F_{ref}}))\boldsymbol{\phi} = \boldsymbol{0}` of the follower pressure, or without
`\boldsymbol{K}_{F_{ref}}` for the dead pressure, solved with ``structsolve.lb``,
with the unsymmetric solvers for the follower pressure:

.. literalinclude:: ../../tests/test_follower_pressure_validation.py
    :pyobject: buckling

.. literalinclude:: ../../tests/test_follower_pressure_validation.py
    :pyobject: ring_buckling_pressure

The buckling pressure against the closed-form solution of the ring with the
same strains and load terms, and against its inextensional limit, for both
kinematics and both pressures:

.. literalinclude:: ../../tests/test_follower_pressure_validation.py
    :pyobject: test_isotropic_ring