Non-linear static analysis with the Newton-Raphson method
=========================================================

The geometrically non-linear models, based on Donnell's equations, provide the
internal force vector `\{F_{int}\}`, calculated with :meth:`.Shell.calc_fint`,
and the tangent stiffness matrix

.. math::

    [K_T] = [K_C] + [K_G]

calculated with :meth:`.Shell.calc_kT`, or with :meth:`.Shell.calc_kC` and
:meth:`.Shell.calc_kG` using ``NLgeom=True``. The tangent stiffness matrix is
the exact derivative of the internal force vector, such that the
Newton-Raphson iterations converge quadratically.

The example below applies a compression of about 37 times the first linear
buckling load in a single step, deep in the postbuckling regime, starting from
the linear solution, and verifies the order of convergence of the iterations:

.. literalinclude:: ../../tests/tests_shell/test_nonlinear.py
    :pyobject: test_nonlinear

The consistency between the tangent stiffness matrix and the internal force
vector is verified by a directional Taylor test, where the error of the linear
approximation of `\{F_{int}\}` must decrease quadratically with the step size:

.. literalinclude:: ../../tests/tests_shell/test_tangent_consistency.py
    :pyobject: test_tangent_is_derivative_of_fint

The methods ``calc_fext``, ``calc_fint``, ``calc_kC`` and ``calc_kG`` of
:class:`.Shell` and :class:`.MultiDomain` have the signatures required by
:class:`structsolve.Analysis`, which implements the Newton-Raphson method with
load control and the arc-length methods, see :doc:`ex_arc_length`.