hermit.fenics.kinematics
========================

.. py:module:: hermit.fenics.kinematics

.. autoapi-nested-parse::

   Basis transformations and Voigt notation for the R-M shell.

   Ported from ``femo_alpha/rm_shell/linear_shell_fenicsx/kinematics.py`` (dev_coupling).
   Unchanged across DOLFINx 0.9 / 0.11 -- these are pure UFL. Note there is *no* ``gradx`` / ``J`` /
   ``F``: geometry sensitivity comes from differentiating w.r.t. ``SpatialCoordinate``
   directly, so all gradients here are plain ``ufl.grad`` on the (possibly moved) mesh.

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Functions
---------

.. autoapisummary::

   hermit.fenics.kinematics.congruent_transform
   hermit.fenics.kinematics.constant_normal_bending_curvature
   hermit.fenics.kinematics.global_to_local_inplane
   hermit.fenics.kinematics.gradv_local
   hermit.fenics.kinematics.local_basis_inplane
   hermit.fenics.kinematics.membrane_strain
   hermit.fenics.kinematics.orientation_cos_sin
   hermit.fenics.kinematics.shear_rotation
   hermit.fenics.kinematics.strain2D_local_to_global
   hermit.fenics.kinematics.strain_rotation
   hermit.fenics.kinematics.unit
   hermit.fenics.kinematics.vec2D_local_to_global
   hermit.fenics.kinematics.voigt2D
   hermit.fenics.kinematics.voigt3D


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

.. py:function:: congruent_transform(T, X)

   
   ``T^T X T`` for square UFL matrices (``T`` orthogonal, ``X`` symmetric) --
   rotates a laminate-axes constitutive matrix into the element frame,
   ``A_element = Teps^T A Teps`` (and ``As_element = R^T As R``). Index notation
   (not ``ufl.transpose(T) * X * T``) so it is unambiguous for any square size.
















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.. py:function:: constant_normal_bending_curvature(theta, E01)

   
   Shallow-shell curvature with the cell normal held constant.

   Only the rotation gradient enters. In particular, a rigid rotation is a
   constant ``theta`` and therefore has zero curvature structurally, including on
   a warped quadrilateral cell.















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.. py:function:: global_to_local_inplane(E0, E1)

   
   2x3 change-of-basis matrix T; T[i, j] is the j-th component of the i-th basis vector.
















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.. py:function:: gradv_local(gradv_global, T)

   
   In-plane components of a rank-2 tensor in the local orthonormal frame.
















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.. py:function:: local_basis_inplane(mesh)

   
   Local orthonormal shell basis (E0, E1 in-plane; E2 = element normal).

   E0 is aligned with the 0-th parametric coordinate direction.















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.. py:function:: membrane_strain(u_mid, E01)

   
   Mid-surface membrane strain in the local in-plane frame.
















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.. py:function:: orientation_cos_sin(fiber_angle, fiber_direction, E0, E1, E2)

   
   ``(cos theta, sin theta)`` of the in-plane angle from the element ``E0`` to the
   fibre direction -- from ``fiber_angle`` directly, or projected from a global
   ``fiber_direction`` vector (no ``atan2``: it is more expensive and puts a branch
   cut at +/-pi right in the differentiated path). Exactly one of ``fiber_angle`` /
   ``fiber_direction`` must be given.

   Degenerate only when ``fiber_direction`` is parallel to the shell normal ``E2``
   (the tangent-plane projection vanishes) -- a UFL expression cannot raise, so
   callers must guard that numerically, with concrete values, before building the
   form.















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.. py:function:: shear_rotation(c, s)

   
   R(theta) -- 2x2 transverse-shear rotation, element -> laminate axes, standard
   FSDT Voigt (xz, yz) ordering: ``gam_L = R @ gamma``.
















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.. py:function:: strain2D_local_to_global(strain_local, T)

   
   2x2 local in-plane strain -> 3x3 global Cartesian tensor.
















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.. py:function:: strain_rotation(c, s)

   
   Teps(theta) -- 3x3 engineering-Voigt in-plane strain/curvature rotation,
   element -> laminate axes: ``eps_L = Teps @ voigt2D(eps)``. ``Teps^T == Tsig^-1``
   for this convention (the identity ``hermit.csdl_helpers.rotate_abd`` also uses).
















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.. py:function:: unit(v)

   
   Normalize the vector ``v``.
















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.. py:function:: vec2D_local_to_global(v_local, T)

   
   length-2 local in-plane vector -> length-3 global Cartesian vector.
















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.. py:function:: voigt2D(T, strain=True)

   
   2x2 symmetric tensor -> Voigt vector. ``strain`` doubles the shear component.
















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.. py:function:: voigt3D(T, strain=True)

   
   3x3 symmetric tensor -> length-6 Voigt vector ``[xx, yy, zz, yz, xz, xy]``.
   ``strain`` doubles the three shear components (engineering convention, matching
   :func:`voigt2D`).
















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