POINT_MASS — single-node lumped translational mass
==================================================

A diagonal 3 × 3 contribution to the global mass matrix at a
single grid node.  Translational-only (3 DOFs per node);
no stiffness contribution.

* **Spec:** ``ELEMENTS.POINT_MASS``

.. raw:: html

    <span class="accuracy-badge accuracy-pass">
    Verified vs single-DOF spring-mass oscillator natural frequency
    </span>

Cross-vendor mapping
--------------------

==========================  ==============================  ============================================
Solver                       Element name                    Notes
==========================  ==============================  ============================================
femorph-solver               ``ELEMENTS.POINT_MASS``         translational only; rotational inertia not modelled
ANSYS Mechanical APDL        ``MASS21``                      KEYOPT(3)=2 → translational only (matches our model)
NX / MSC Nastran             ``CONM1`` / ``CONM2``           CONM2 = standard concentrated mass with inertia tensor
Abaqus                       ``MASS``                        scalar mass element
LS-DYNA                      ``ELEMENT_MASS``                lumped translational mass on a node
==========================  ==============================  ============================================

Restrictions
------------

Use a different approach when:

* **Rotational inertia** is needed (e.g., flywheels, eccentric
  masses with offset CG) — currently un-modelled.  Workaround:
  attach the mass via a stiff BEAM2 stub from the rotation axis.
* **Anisotropic mass** (different in each direction) — the
  scalar :math:`m\, \mathbf{I}` form is isotropic by construction;
  use multiple coupled springs + masses if directional inertia
  matters.

Mass
----

The full element contribution to the global mass matrix is

.. math::

    \mathbf{M}^{\mathrm{loc}} =
    m\, \mathbf{I}_{3 \times 3},

acting only on the three translational DOFs of the host node;
zero on all rotational DOFs (this kernel does not model
rotational inertia).  The diagonal structure makes assembly
trivial — no Gauss quadrature, no shape-function evaluation.

Stiffness
---------

None.  POINT_MASS is a mass-only kernel.

Real constants
--------------

* ``REAL[0]`` — :math:`m`, the lumped mass in kg.

Verification cross-references
-----------------------------

* :ref:`sphx_glr_gallery_elements_mass21_example_mass21.py` — spring + lumped mass single-DOF oscillator,
  :math:`\omega = \sqrt{k / m}`.

Implementation: :mod:`femorph_solver.elements.point_mass`.

References
----------

* Cook, R. D., Malkus, D. S., Plesha, M. E., Witt, R. J.
  (2002) *Concepts and Applications of Finite Element
  Analysis*, 4th ed., Wiley, §11.3 (lumped mass).
* Bathe, K.-J. (2014) *Finite Element Procedures*, 2nd ed.,
  §11.4.1 (concentrated-mass elements).