anastruct.fem.system_components.solver
======================================

.. py:module:: anastruct.fem.system_components.solver


Functions
---------

.. autoapisummary::

   anastruct.fem.system_components.solver.stiffness_adaptation
   anastruct.fem.system_components.solver.det_linear_buckling
   anastruct.fem.system_components.solver.geometrically_non_linear


Module Contents
---------------

.. py:function:: stiffness_adaptation(system: anastruct.fem.system.SystemElements, verbosity: int, max_iter: int) -> numpy.ndarray

   Non linear solver for the nodes by adapting the stiffness of the elements (nodes).

   :return: Vector with displacements.


.. py:function:: det_linear_buckling(system: anastruct.fem.system.SystemElements) -> float

   Determine linear buckling by solving the generalized eigenvalue problem (k -λkg)x = 0.

   geometrical stiffness matrix at buckling point: Kg = f(N_max)
   1st order forces: N0
   Nmax = λN0
   Kg(Nmax) = λ(Kg(N0) = λKg0

   2nd order analysis is solved by:
   (K + λKg0)U = F

   We are interested in the point that there is nog additional load F and displacement U
   is possible.
   (K + λKg0)ΔU = ΔF = 0
   (K + λKg0) = 0

   Is the generalized eigenvalue problem:
   (A - λB)x = 0

   :return: The factor the loads can be increased until the structure fails due to buckling.


.. py:function:: geometrically_non_linear(system: anastruct.fem.system.SystemElements, verbosity: int = 0, return_buckling_factor: bool = True, discretize_kwargs: Optional[dict] = None) -> Optional[float]

   :param system: (SystemElements)
   :param verbosity: (int)
   :param return_buckling_factor: (bool)
   :param discretize_kwargs: (dict) Containing the kwargs passed to the discretize function
   :param discretize: (function) discretize function.
   :return: buckling_factor: (flt) The factor the loads can be increased until the structure
                             fails due to buckling.