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


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

The search for the transitory answer to this problem with non-proportional damping can be carried out by numerical integration in real space:

:math:`[M]\{\ddot{{u}_{n}}\}+[C]\{\dot{{u}_{n}}\}+[K]\{{u}_{n}\}=\{F\}`

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


* PERMAS: Newmark integration diagram (:math:`\alpha =\mathrm{0,25}` and :math:`\delta =\mathrm{0,5}`) :math:`\Delta t={10}^{-\mathrm{4s}}`;


* ABAQUS: Hilbert-Hugues-Taylor integration diagram [:ref:`bib1 <bib1>`] (:math:`\alpha =–\mathrm{0,05}`) :math:`\Delta t={10}^{-\mathrm{4s}}`;

and the improved :math:`\beta` -Newmark integration method [:ref:`bib2 <bib2>`]:

:math:`\left[\frac{\mathrm{[}M\mathrm{]}}{\Delta {t}^{2}}+\frac{\mathrm{[}C\mathrm{]}}{2\Delta t}+\frac{\mathrm{[}K\mathrm{]}}{3}\right]\left\{{u}_{n+2}\right\}\mathrm{=}\left[\frac{\mathrm{\{}{F}_{n+2}\mathrm{\}}+\mathrm{\{}{F}_{n+1}\mathrm{\}}+\mathrm{\{}{F}_{n}\mathrm{\}}}{3}\right]+\left[\frac{2\mathrm{[}M\mathrm{]}}{\Delta {t}^{2}}\mathrm{-}\frac{\mathrm{[}K\mathrm{]}}{3}\right]\left\{{u}_{n+1}\right\}+\left[\frac{\mathrm{[}M\mathrm{]}}{\Delta {t}^{2}}+\frac{\mathrm{[}C\mathrm{]}}{2\Delta t}\mathrm{-}\frac{\mathrm{[}K\mathrm{]}}{3}\right]\left\{{u}_{n}\right\}`

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}\mathrm{=}{t}_{n}+2\Delta t` where :math:`\Delta t` is the time increment used.

To start, we take:


* :math:`{u}_{0}` and :math:`{u}_{-1}={u}_{0}-\Delta t\dot{{u}_{0}}`


* :math:`{F}_{-1}=2{F}_{0}-{F}_{1}`

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


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

Displacement and speed of the endpoint :math:`B`.


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

Average of digital solutions.


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


1. H.M. HILBERT, T.J.R HUGUES 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


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