Example 3
=========

  .. image:: apng/groundstructure.png
     :width: 7.0cm

This is a :math:`3`-dimensional example with the ground structure of :math:`27` nodes :math:`\left(a,b,c\right) \in \R^3$, $a,b,c \in \{1,2,3\}` measured in meter and :math:`274` potential bars between every two not fixed nodes -- long bars which are located alongside several shorter bars are ignored. The supports are in the nodes :math:`\left(1,a,b\right)$, $a,b \in \{1,2,3\}` which are fixed in every direction. In the node :math:`\left(3,2,1\right)` acts a force of :math:`4\cdot10^5 \, \mbox{N}` in the negative :math:`z`-direction.

For a perturbation of :math:`10 \, \mbox{\%}` in every direction of the force, and scaling to obtain the same maximal quantities as in the unperturbed case, our robust optimization method produces the robust optimal solutions. In contrast to the optimal solution the robust solution use the third dimension of the space.


Comparison of the volumes
-------------------------

+---------+----------------+----------------+
|         | optimal        | robust         |
+---------+----------------+----------------+
| volume: | 6.0000e-03     | 6.5822e-03     |
+---------+----------------+----------------+


Comparison of the volumes for different redundancies:

+------------+---------------------+---------------------+---------------------+-----------------------+-----------------------+-----------------------+
| redundancy | :math:`\frac{1}{4}` | :math:`\frac{1}{3}` | :math:`\frac{1}{2}` | :math:`1-\frac{1}{2}` | :math:`1-\frac{1}{3}` | :math:`1-\frac{1}{4}` |
+------------+---------------------+---------------------+---------------------+-----------------------+-----------------------+-----------------------+
| volume:    | 8.6667e-03          | 9.5000e-03          | 1.2000e-02          | 1.2000e-02            | 1.9000e-02            | 2.6000e-02            |
+------------+---------------------+---------------------+---------------------+-----------------------+-----------------------+-----------------------+

Comparison of the volumes for different redundancies and robustness:

+------------+---------------------+---------------------+-----------------------+-----------------------+
| redundancy | :math:`\frac{1}{3}` | :math:`\frac{1}{2}` | :math:`1-\frac{1}{2}` | :math:`1-\frac{1}{3}` |
+------------+---------------------+---------------------+-----------------------+-----------------------+
| volume:    | 1.2136e-02          | 1.5528e-02          | 1.5528e-02            | 2.4875e-02            |
+------------+---------------------+---------------------+-----------------------+-----------------------+

Visualizations
--------------

Optimal solutions: (:download:`script <batch_example03.py>`)

+-------------------------------+----------------------------------------+
| .. image:: apng/example03.png | .. image:: apng/example03_interior.png |
|    :width: 7.0cm              |    :width: 7.0cm                       |
+-------------------------------+----------------------------------------+

Robust solutions: (:download:`script <batch_example03_robust.py>`)

+--------------------------------------+-----------------------------------------------+
| .. image:: apng/example03_robust.png | .. image:: apng/example03_robust_interior.png |
|    :width: 7.0cm                     |    :width: 7.0cm                              |
+--------------------------------------+-----------------------------------------------+

Redundant solutions: (:download:`script <batch_example03_redundant.py>`)

+-----------------------------------------------+-----------------------------------------------------+
| :math:`\left(\frac{1}{4}\right)`-redundant    | :math:`\left(\frac{1}{3}\right)`-redundant          |
+-----------------------------------------------+-----------------------------------------------------+
| .. image:: apng/example03_redundant_1div4.png | .. image:: apng/example03_redundant_1div3.png       |
|    :width: 7.0cm                              |    :width: 7.0cm                                    |
+-----------------------------------------------+-----------------------------------------------------+

+-----------------------------------------------+-----------------------------------------------------+
| :math:`\left(\frac{1}{2}\right)`-redundant    | :math:`\left(1-\frac{1}{2}\right)`-redundant        |
+-----------------------------------------------+-----------------------------------------------------+
| .. image:: apng/example03_redundant_1div2.png | .. image:: apng/example03_redundant_1minus1div2.png |
|    :width: 7.0cm                              |    :width: 7.0cm                                    |
+-----------------------------------------------+-----------------------------------------------------+

+-----------------------------------------------+-----------------------------------------------------+
| :math:`\left(1-\frac{1}{3}\right)`-redundant  | :math:`\left(1-\frac{1}{4}\right)`-redundant        |
+-----------------------------------------------+-----------------------------------------------------+
| .. image:: apng/example03_redundant_1div3.png | .. image:: apng/example03_redundant_1minus1div4.png |
|    :width: 7.0cm                              |    :width: 7.0cm                                    |
+-----------------------------------------------+-----------------------------------------------------+

Robust and redundant solutions:

+-------------------------------------------------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------------------------------+
| robust and :math:`\left(\frac{1}{3}\right)`-redundant: (:download:`script <batch_example03_redundant_1div3_robust.py>`) | robust and :math:`\left(\frac{1}{2}\right)`-redundant: (:download:`script <batch_example03_redundant_1div2_robust.py>`) | 
+-------------------------------------------------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------------------------------+
| .. image:: apng/example03_redundant_1div3_robust.png                                                                    |  .. image:: apng/example03_redundant_1div2_robust.png                                                                   |
|    :width: 7.0cm                                                                                                        |     :width: 7.0cm                                                                                                       |
+-------------------------------------------------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------------------------------+

+---------------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------------+
| robust and :math:`\left(1-\frac{1}{2}\right)`-redundant: (:download:`script <batch_example03_redundant_1minus1div2_robust.py>`) | robust and :math:`\left(1-\frac{1}{3}\right)`-redundant: (:download:`script <batch_example03_redundant_1minus1div3_robust.py>`) |
+---------------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------------+
|  .. image:: apng/example03_redundant_1minus1div2_robust.png                                                                     |   .. image:: apng/example03_redundant_1minus1div3_robust.png                                                                    |
|     :width: 7.0cm                                                                                                               |      :width: 7.0cm                                                                                                              |
+---------------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------------+

Python scripts
--------------
* :download:`batch_example03.py`
* :download:`batch_example03_robust.py`
* :download:`batch_example03_redundant.py`
* :download:`batch_example03_redundant_1div3_robust.py`
* :download:`batch_example03_redundant_1div2_robust.py`
* :download:`batch_example03_redundant_1minus1div2_robust.py`
* :download:`batch_example03_redundant_1minus1div3_robust.py`