Benchmark solutions
======================

Calculation method used for reference solutions
----------------------------------------------------------

Modeling A
~~~~~~~~~~~~~~~

The equations governing behavior are nonlinear differential equations. To validate the answer obtained with *Code_Aster* in non-linear statics, an integration by a Runge-Kutta method is carried out with a tool external to *Code_Aster*.

The comparison is carried out on the displacement and on the effort, for the 4 types of discrete elements.


.. image:: images/10000201000005FE000003D2E5308AB1F2B5F4F6.png
    :width: 5.9055in
    :height: 3.7445in


.. _RefImage_10000201000005FE000003D2E5308AB1F2B5F4F6.png:

**Figure** 2.1.1-a **: Force displacement curve, modeling A.**

The answer given by*Code_Aster* is tested for the following values:


.. csv-table::

    "**Moment**", "**Movement** :math:`{U}_{x}` ", "**Force** :math:`{F}_{x}`"
    "2.00E-02", "5.877852523E-02", "2.187710580E+00"
    "4.00E-02", "9.510565163E-02", "2.829192223E+00"
    "6.00E-02", "9.510565163E-02", "2.035749590E+00"
    "8.00E-02", "5.877852523E-02", "2.402408962E-01"
    "1.00E-01", "-1.653950414E-16", "-1.851221553E+00"
    "1.32E-01", "-8.443279255E-02", "-3.445042947E+00"
    "2.00E-01", "4.196133458E-16", "1.745702939E+00"
    "2.32E-01", "8.443279255E-02", "3.409095131E+00"
    "2.68E-01", "8.443279255E-02", "1.626471785E+00"
    "3.16E-01", "-4.817536741E-02", "-2.962435650E+00"
    "3.56E-01", "-9.8228725072507E-02", "-2.590008311E+00"
    "4.12E-01", "3.681245527E-02", "2.724835444E+00"
    "4.36E-01", "9.048270525E-02", "3.394150679E+00"
    "5.20E-01", "-5.877852523E-02", "-3.151025904E+00"
    "6.24E-01", "6.845471059E-02", "3.289283317E+00"
    "7.16E-01", "-4.817536741E-02", "-2.962278876E+00"
    "8.00E-01", "1.678385621E-15", "1.750844985E+00"
    "8.16E-01", "4.817536741E-02", "2.962278875E+00"
    "8.48E-01", "9.980267284E-02", "3.047135026E+00"
    "9.40E-01", "-9.510565163E-02", "-3.326860603E+00"
    "9.68E-01", "-8.443279255E-02", "-1.627037269E+00"
    "1.00E+00", "-1.224606354E-16", "1.750844985E+00"


**Table** 2.1.1-a **: Displacement and Forces, modeling A.**


B modeling
~~~~~~~~~~~~~~~

This non-linear static modeling makes it possible to test, in addition to the law of behavior, the dissipation during a stabilized cyclic loading. The dissipation is compared to a theoretical value obtained in the particular case :math:`{\alpha }_{3}=1.0`.

Note: For a cyclic loading with :math:`{\alpha }_{3}\ne 1` the theoretical calculation of the dissipation is not available, except for a stabilized cycle.

The answer given by*Code_Aster* is tested for the following values:

.. csv-table::

    "**Moment**", "**Movement** :math:`{U}_{x}` ", "**Force** :math:`{F}_{x}`"
    "2.00E-02", "5.877852523E-02", "2.160195640E+00"
    "4.00E-02", "9.510565163E-02", "2.849834733E+00"
    "6.00E-02", "9.510565163E-02", "2.052734480E+00"
    "8.00E-02", "5.877852523E-02", "2.258915314E-01"
    "1.00E-01", "-1.653950414E-16", "-1.838798378E+00"
    "1.32E-01", "-8.443279255E-02", "-3.611426479E+00"
    "2.00E-01", "4.195726882E-16", "1.674446965E+00"
    "2.32E-01", "8.443279255E-02", "3.535539017E+00"
    "2.68E-01", "8.443279255E-02", "1.730277335E+00"
    "3.16E-01", "-4.817536741E-02", "-2.984761046E+00"
    "3.56E-01", "-9.8228725072507E-02", "-2.752278435E+00"
    "4.12E-01", "3.681245527E-02", "2.719185079E+00"
    "4.36E-01", "9.048270525E-02", "3.544941424E+00"
    "5.20E-01", "-5.87785252523E-02", "-3.201565830E+00"
    "6.24E-01", "6.845471059E-02", "3.368686714E+00"
    "7.16E-01", "-4.817536741E-02", "-2.983942123E+00"
    "8.00E-01", "1.678385621E-15", "1.687931415E+00"
    "8.16E-01", "4.817536741E-02", "2.983942066E+00"
    "8.48E-01", "9.980267284E-02", "3.223403140E+00"
    "9.40E-01", "-9.510565163E-02", "-3.492301297E+00"
    "9.68E-01", "-8.443279255E-02", "-1.732887550E+00"
    "1.00E+00", "-1.224606354E-16", "1.687931421E+00"


**Table** 2.1.2-a: Displacement and Forces, modeling B.

The calculation of the dissipation over a stabilized cycle is obtained by integrating the equations of the system in the particular case where :math:`{\alpha }_{3}=1`.

On a stabilized cycle, for :math:`{\alpha }_{3}=1`, the dissipation value is:

 :math:`\Delta D=\frac{\pi \mathrm{.}{U}_{0}^{2}\mathrm{.}{E}_{1}^{2}\mathrm{.}{E}_{3}^{2}\mathrm{.}\omega \mathrm{.}{C}_{3}}{{\omega }^{2}\mathrm{.}{C}_{3}^{2}\mathrm{.}{({E}_{1}+{E}_{2}+{E}_{3})}^{2}+{({E}_{1}+{E}_{2})}^{2}\mathrm{.}{E}_{3}^{2}}` [:ref:`éq2.1.2-1 <éq2.1.2-1>`]


C modeling
~~~~~~~~~~~~~~~

This static nonlinear modeling makes it possible to test the law of behavior during a creep test. Displacement is imposed and remains constant: :math:`{U}_{0}=0.1`. The response of the law of behavior as well as the dissipation are compared to the theoretical values obtained in the particular case :math:`{\alpha }_{3}=0.5`.

The integrated differential equations in the particular case of constant :math:`U` and :math:`{\alpha }_{3}=0.5` give the equations of effort and dissipation as a function of time:

 :math:`F(t)=\frac{{U}_{0}\mathrm{.}{E}_{1}\mathrm{.}({\mathrm{AA}}_{s}+{\mathrm{BB}}_{s}\mathrm{.}{E}_{2}\mathrm{.}t)}{{({E}_{3}+{E}_{2}+{E}_{1})}^{2}\mathrm{.}{C}_{3}^{2}+{\mathrm{BB}}_{s}\mathrm{.}({E}_{2}+{E}_{1})\mathrm{.}t}` [:ref:`éq2.1.3-1 <éq2.1.3-1>`]

 :math:`D(t)=\frac{{U}_{0}^{3}.{E}_{1}^{3}.{E}_{3}^{3}}{2.({E}_{3}+{E}_{2}+{E}_{1})}.\mathit{t.}\frac{(2.{\mathit{AA}}_{e}+{\mathit{BB}}_{e}.t)}{{({\mathit{AA}}_{e}+{\mathit{BB}}_{e}.t)}^{2}}` [:ref:`éq2.1.3-2 <éq2.1.3-2>`]

with :math:`\{\begin{array}{}{\mathrm{AA}}_{s}=({E}_{3}+{E}_{2})\mathrm{.}({E}_{1}+{E}_{2}+{E}_{3})\mathrm{.}{C}_{3}^{2}\\ {\mathrm{BB}}_{s}={U}_{0}\mathrm{.}{E}_{1}\mathrm{.}{E}_{3}^{2}\end{array}\{\begin{array}{}{\mathrm{AA}}_{e}={({E}_{3}+{E}_{2}+{E}_{1})}^{2}\mathrm{.}{C}_{3}^{2}\\ {\mathrm{BB}}_{e}={U}_{0}\mathrm{.}{E}_{1}\mathrm{.}{E}_{3}^{2}\mathrm{.}({E}_{2}+{E}_{1})\end{array}`


D modeling
~~~~~~~~~~~~~~~

The equations governing behavior are nonlinear differential equations. To validate the answer obtained with *Code_Aster* in non-linear statics, an integration by a Runge-Kutta method is carried out with a tool external to *Code_Aster*.

The comparison is carried out on the displacement and on the effort.


.. image:: images/100002010000052D0000042C70180983A2C042CE.png
    :width: 5.9055in
    :height: 4.7598in


.. _RefImage_100002010000052D0000042C70180983A2C042CE.png:

**Figure** 2.1.4-a **: Force displacement curve, modeling D.**

The answer given by*Code_Aster* is tested for the following values:

.. csv-table::

    "**Moment**", "**Movement** :math:`{U}_{x}` ", "**Force** :math:`{F}_{x}`"
    "4.000E-03", "1.2533323356430E-02", "1.3901305564654E+00"
    "4.800E-02", "9.9802672872842827E-02", "1.5399690347096E+00"
    "1.000E-01", "-1.6539504141266E-16", "-2.9840799981192E+00"
    "1.360E-01", "-9.048270505246602E-02", "-2.2555706075403E+00"
    "2.040E-01", "1.2533323356431E-02", "2.9999350282465E+00"
    "2.480E-01", "9.9802672872842827E-02", "1.5401915597398E+00"
    "3.040E-01", "-1.25333233532356431E-02", "-2.9999350282852E+00"
    "3.480E-01", "-9.9802672872842827E-02", "-1.5401915597074E+00"
    "4.040E-01", "1.2533323356431E-02", "2.9999350282970E+00"
    "5.000E-01", "-1.0045133128078E-15", "-2.9840798812719E+00"
    "5.600E-01", "-9.5105651629515E-02", "-4.1551773591104E-01"
    "6.000E-01", "1.3475548801822E-15", "2.9840798812750E+00"
    "6.400E-01", "9.510565162951629516E-02", "2.0490126532863E+00"
    "7.040E-01", "-1.25333233523356432E-02", "-2.9999350283063E+00"
    "7.480E-01", "-9.9802672872842827E-02", "-1.5401915596821E+00"
    "8.040E-01", "1.2533323356432E-02", "2.9999350283073E+00"
    "8.480E-01", "9.9802672872842827E-02", "1.5401915596806E+00"
    "9.040E-01", "-1.253332335323356432E-02", "-2.9999350283079E+00"
    "9.480E-01", "-9.9802672872842827E-02", "-1.5401915596795E+00"
    "1.000E+00", "-1.2240642527361E-16", "2.9840798812793E+00"


**Table** 2.1.4-a: Displacement and Forces, modeling D.


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

Modeling A
~~~~~~~~~~~~~~~

*For the effort response, displacement:*

The reference solution is obtained by numerical integration of a nonlinear differential system.


B modeling
~~~~~~~~~~~~~~~

*For the effort response, displacement:*

The reference solution is obtained by numerical integration of a differential system, with a 5th order Runge-Kutta method.

*For dissipation:*

No uncertainty, the solution is analytical.


C modeling
~~~~~~~~~~~~~~~

*For the effort response, displacement:*

No uncertainty, the solution is analytical.

*For dissipation:*

No uncertainty, the solution is analytical.


D modeling
~~~~~~~~~~~~~~~

*For the effort response, displacement:*

The reference solution is obtained by numerical integration of a differential system.