Example of a mast


In this example of a mast are 72 nodes \left(a,b,c\right) \in \R^3, a,b \in \{0,\frac{1}{7},\frac{2}{7}\}, c \in \{\frac{x}{7}:x=0,1,\ldots,7\} measured in meter and 503 potential bars between every two not fixed nodes with a maximal distance of \frac{\sqrt{3}}{7} — long bars which are located alongside several shorter bars are ignored. The supports are in the nodes \left(a,b,0\right), a,b \in \{0,\frac{2}{7}\} which are fixed in every direction. In the node \left(\frac{1}{7},\frac{1}{7},1\right) acts a force of 10^5 \, \mbox{N} in the negative z-direction.

The main problem is implemented as a script. The other scripts are linked by the pictures.

For a perturbation of 50 \, \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 solution.

Comparison of the volumes

  optimal robust
volume: 1.4000e-03 3.4975e-03

Comparison of the volumes for different redundancies:

redundancy \frac{1}{3} \frac{1}{2} 1-\frac{1}{2} 1-\frac{1}{3}
volume: 2.3000e-03 3.0000e-03 3.0000e-03 4.6000e-03

Comparison of the volumes for different redundancies and robustness:

redundancy \frac{1}{2} 1-\frac{1}{2}
volume: 8.0805e-03 8.0805e-03


Optimal solution (script) Robust solution (script)
../../_images/example05.png ../../_images/example05_robust.png

Redundant solutions: (script)

\left(\frac{1}{3}\right)-redundant \left(\frac{1}{2}\right)-redundant
../../_images/example05_redundant_1div3.png ../../_images/example05_redundant_1div2.png
\left(1-\frac{1}{2}\right)-redundant \left(1-\frac{1}{3}\right)-redundant
../../_images/example05_redundant_1minus1div2.png ../../_images/example05_redundant_1minus1div3.png

Robust and redundant solutions:

robust and \left(\frac{1}{2}\right)-redundant: (script) robust and \left(1-\frac{1}{2}\right)-redundant: (script)
../../_images/example05_redundant_1div2_robust.png ../../_images/example05_redundant_1minus1div2_robust.png

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