Reference solution
=====================


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


* In thermal: we solve the stationary thermal problem:


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**Note:**

*The boundary conditions chosen here are not those necessary for the homogenization method: we would in fact find*

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*everywhere.*

The solution is then (verifying the conditions defined in [:ref:`§1.3 <§1.3>`]):

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*The potential energy is then in equilibrium:*

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.. _RefImage_Object_7.svg:


* In mechanics: we solve the elastostatic problem:


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for cases:


+----------------------------------------------------------------------------------+----------------------------------------------------------------------------------+----------------------------------------------------------------------------------+
|3D loading                                                                        |3D loading                                                                        |2D loading                                                                        |
+                                                                                  +                                                                                  +                                                                                  +
| membrane                                                                         | of bending                                                                       | plane constraints                                                                |
+                                                                                  +                                                                                  +                                                                                  +
|                                                                                  |                                                                                  |                                                                                  |
+----------------------------------------------------------------------------------+----------------------------------------------------------------------------------+----------------------------------------------------------------------------------+
|                                                                                  |                                                                                  |                                                                                  |
+ .. image:: images/Object_9.svg                                                   + .. image:: images/Object_10.svg                                                  + .. image:: images/Object_11.svg                                                  +
|    :width: 398                                                                   |    :width: 38                                                                    |    :width: 97                                                                    |
+    :height: 74                                                                   +    :height: 15                                                                   +    :height: 26                                                                   +
|                                                                                  |                                                                                  |                                                                                  |
+                                                                                  +                                                                                  +                                                                                  +
|                                                                                  |                                                                                  |                                                                                  |
+----------------------------------------------------------------------------------+----------------------------------------------------------------------------------+----------------------------------------------------------------------------------+


The solutions are:


*in 3D,**membrane** loading and isotropic elasticity:

    .. image:: images/Object_12.svg
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    .. _RefImage_Object_12.svg:

the potential energy at equilibrium is:


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.. _RefImage_Object_13.svg:


*in 3D,**membrane** loading and orthotropic elasticity:

    .. image:: images/Object_14.svg
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        :height: 74


    .. _RefImage_Object_14.svg:

with

    .. image: images/1000028e000069d500005421a41e8a62 BA96A976 .svg
        :width: 398
        :height: 74


    .. _refImage_1000028e000069d500005421a41e8a62 BA96A976 .svg:

either

    .. image: images/1000050e000069d500003e1d3b7c319 DB7227D2C .svg
        :width: 398
        :height: 74


    .. _refImage_1000050e000069d500003e1d3b7c319 DB7227D2C .svg:

because the local coordinate system is not confused with the global coordinate system (the nautical angles are all equal to 90°).


.. image:: images/100016AA000069D500001389E75473AD8F381DD8.svg
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    :height: 74


.. _RefImage_100016AA000069D500001389E75473AD8F381DD8.svg:

where

.. image:: images/Object_15.svg
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.. _RefImage_Object_15.svg:

and

.. image:: images/10000476000069D500001020293ECDA7851C214D.svg
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    :height: 74


.. _RefImage_10000476000069D500001020293ECDA7851C214D.svg:


*in 3D, loading**flexion**:

    .. image:: images/Object_16.svg
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    .. _RefImage_Object_16.svg:

;


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.. _RefImage_Object_17.svg:


*in 3D,**flexure load** and orthotropic elasticity:

    .. image:: images/Object_18.svg
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    .. _RefImage_Object_18.svg:

with

    .. image:: images/Object_19.svg
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either

    .. image:: images/Object_20.svg
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    .. _RefImage_Object_20.svg:

because the local coordinate system is not confused with the global coordinate system (the nautical angles are all equal to 90°).


.. image:: images/Object_21.svg
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    :height: 74


.. _RefImage_Object_21.svg:

where

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and

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.. _RefImage_Object_23.svg:


* in 2D, plane loading:

    .. image:: images/Object_24.svg
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        :height: 74


    .. _RefImage_Object_24.svg:

;


.. image:: images/Object_25.svg
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.. _RefImage_Object_25.svg: