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


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

The following system is digitally integrated between :math:`t=0` and :math:`t=1`.

(Ordinary nonlinear differential system of 6 equations with 6 unknowns solved using library NAG by a 'Backward difference method')


.. image:: images/Object_2.svg
    :width: 584
    :height: 445


.. _RefImage_Object_2.svg:


.. image:: images/Object_3.svg
    :width: 584
    :height: 445


.. _RefImage_Object_3.svg:

with the initial conditions:


.. image:: images/Object_4.svg
    :width: 584
    :height: 445


.. _RefImage_Object_4.svg:


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

Values of

.. image:: images/Object_5.svg
    :width: 584
    :height: 445


.. _RefImage_Object_5.svg:

and

.. image:: images/Object_6.svg
    :width: 584
    :height: 445


.. _RefImage_Object_6.svg:

at the knots at :math:`t=\mathrm{1 }s`.


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

Library uncertainty NAG.


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


1. Library user manual NAG out of CRAY.


2. S. ANDRIEUX - P. SCHOENBERGER - S. TAHERI: A three dimensional cyclic constitutive law for metals with a semi-discrete memory variable - HI‑71/8147 (1992)


3. P. GEYER - J.M. PROIX - P. SCHOENBERGER - S. TAHERI: Modeling progressive deformation phenomena - Collection of internal notes from the DER 93 NB00153