Transient analysis
==================

Time-marching response :math:`\mathbf{M}\,
\ddot{\mathbf{U}} + \mathbf{C}\, \dot{\mathbf{U}} +
\mathbf{K}\, \mathbf{U} = \mathbf{F}(t)`, driven by the
upcoming :meth:`Model.transient_solve` API.

.. note::

    The transient solver path is partial as of the current
    release.  Modal-superposition transient on a
    pre-computed eigen basis is shipped via the harmonic /
    transient roadmap (TA-14 / TA-16).  Direct time-stepping
    on the full DOF system (Newmark-β / generalised-α) is
    on the roadmap.

Approach: Newmark-β family
--------------------------

For time step :math:`\Delta t` and Newmark parameters
:math:`(\beta, \gamma)`, the implicit one-step update solves

.. math::

    \tilde{\mathbf{K}}\, \mathbf{U}_{n + 1} =
    \tilde{\mathbf{F}}_{n + 1},

with

.. math::

    \tilde{\mathbf{K}} =
    \mathbf{K}
    + \frac{\gamma}{\beta\, \Delta t}\, \mathbf{C}
    + \frac{1}{\beta\, \Delta t^{2}}\, \mathbf{M}.

Standard parameter choices:

* :math:`(\beta, \gamma) = (1/4, 1/2)` — the classical
  trapezoidal / **average-acceleration** rule.
  Unconditionally stable, second-order accurate, no numerical
  damping.  Default.
* :math:`(\beta, \gamma) = (1/6, 1/2)` — **linear-
  acceleration** rule.  Conditionally stable
  (:math:`\Delta t \le 0.55\, T_n`); often preferred for short
  simulations where the highest-frequency content is benign.

The factor of :math:`\tilde{\mathbf{K}}` is **shared across
time steps** for a constant-:math:`\Delta t` solve — the
linear backend (see :doc:`linear_backends`) factors it once
and re-uses the factor for every step.

Modal superposition (TA-14 path)
--------------------------------

For damped systems where a modal basis already exists
(:meth:`Model.modal_solve`), the **modal-superposition**
path projects the full system onto the first :math:`r` modes:

.. math::

    \mathbf{U}(t) =
    \boldsymbol{\Phi}_{r}\, \boldsymbol{\eta}(t),
    \qquad
    \ddot{\eta}_{i}
    + 2 \zeta_{i}\, \omega_{i}\, \dot{\eta}_{i}
    + \omega_{i}^{2}\, \eta_{i}
    = \boldsymbol{\phi}_{i}^{\!\top}\, \mathbf{F}(t).

The :math:`r` decoupled modal SDOFs are integrated cheaply
(closed-form for harmonic, RK / Newmark-β per mode for
arbitrary forcing).  Reconstruct the full DOF response via
:math:`\mathbf{U}(t) = \boldsymbol{\Phi}_{r}\,
\boldsymbol{\eta}(t)` for plotting / stress recovery.

Damping
-------

Rayleigh damping
:math:`\mathbf{C} = \alpha\, \mathbf{M} + \beta\, \mathbf{K}`
is the practitioner default; the
:math:`(\alpha, \beta)` constants are chosen to hit a target
damping ratio at two reference frequencies.  Modal-
superposition supports per-mode damping ratios :math:`\zeta_i`
directly (no Rayleigh fit needed).

Public API (forthcoming)
------------------------

* :meth:`Model.transient_solve` — direct time-stepping (TA-16
  roadmap).
* :class:`HarmonicResult` /
  :class:`TransientResult` — modal-superposition response
  containers (TA-14 shipping).

References
----------

* Newmark, N. M. (1959) "A method of computation for
  structural dynamics," *J. Eng. Mech.* 85 (3), 67–94.
* Hilber, H. M., Hughes, T. J. R. and Taylor, R. L. (1977)
  "Improved numerical dissipation for time integration
  algorithms in structural dynamics," *EESD* 5 (3), 283–292
  (HHT-α / generalised-α family).
* Chung, J. and Hulbert, G. M. (1993) "A time integration
  algorithm for structural dynamics with improved numerical
  dissipation: the generalized-α method," *J. Appl. Mech.* 60
  (2), 371–375.
* Bathe, K.-J. (2014) *Finite Element Procedures*, 2nd ed.,
  §9 (direct time integration), §12 (mode superposition).
* Cook, R. D., Malkus, D. S., Plesha, M. E., Witt, R. J.
  (2002) *Concepts and Applications of Finite Element
  Analysis*, 4th ed., Wiley, §11 (modal analysis), §13
  (transient response).
* Hughes, T. J. R. (2000) *The Finite Element Method —
  Linear Static and Dynamic Finite Element Analysis*, Dover,
  §9 (algorithms for hyperbolic and parabolic problems).