Benchmark solution
=====================


Calculation method used for the reference solution
--------------------------------------------------------

The search for the transitory answer to this problem with non-proportional damping, and where rigid modes are not fixed, can be carried out by numerical integration in real space:


.. image:: images/Object_4.svg
    :width: 240
    :height: 25


.. _RefImage_Object_4.svg:

.

To do this, the response was calculated using two industrial codes:


.. csv-table::

    "* PERMAS:", "Newmark integration diagram (:math:`\alpha =\mathrm{0,25}`, :math:`\delta =\mathrm{0,5}`), :math:`\Delta t={10}^{-\mathrm{4s}}`,"
    "", "Integration diagram with Hermite cubic interpolation [:ref:`bib1 <bib1>`], :math:`\Delta t={10}^{-\mathrm{4s}}`,"
    "", ""
    "* ABAQUS:", "Hilber-Hughes-Taylor integration diagram [:ref:`bib2 <bib2>`] (:math:`\alpha =-\mathrm{0,05}`), :math:`\Delta t={10}^{-\mathrm{4s}}`,"


and the improved :math:`\beta` -Newmark integration method [:ref:`bib3 <bib3>`]:


.. image:: images/Object_5.svg
    :width: 240
    :height: 25


.. _RefImage_Object_5.svg:

where :math:`n`, :math:`n+1`, :math:`n+2` respectively designate the calculations performed at times :math:`{t}_{n}`, :math:`{t}_{n+1}={t}_{n}+\Delta t` and :math:`{t}_{n+2}={t}_{n}+2\Delta t` where :math:`\Delta t` is the time increment used.

To start, we take:


*

    .. image:: images/Object_6.svg
        :width: 240
        :height: 25


    .. _RefImage_Object_6.svg:


*

    .. image:: images/Object_7.svg
        :width: 240
        :height: 25


    .. _RefImage_Object_7.svg:

The adopted time step is :math:`\Delta t={10}^{-\mathrm{5s}}`.


Benchmark results
----------------------

Displacement, speed and acceleration of point :math:`{P}_{3}`.

Displacement differential between points :math:`{P}_{3}` and :math:`{P}_{1}`.


Uncertainty about the solution
---------------------------

Average of digital solutions.


Bibliographical references
---------------------------


1. J.H. ARGYRIS, P.C. DUNNE and T. ANGELOPOULOS "Non-linear oscillations using the finite technical element" Comp. Meth. Call. Mech. Engng., Vol.2, 1972, pp. 203-254


2. H.M. HILBER, T.J.R. HUGHES and R.L. TAYLOR "Improved numerical dissipation for time integration algorithms in structural dynamics" Earthquake Engineering and Structural Dynamics, Vol.5, 1977, 1977, pp. 283-292


3. N.M. NEWMARK "A method of computation for structural dynamics" Proceeding ASCE J.Eng.Mech. Div E-3, July 1959, pp. 67-94