Example 3


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

For a perturbation of 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 \frac{1}{4} \frac{1}{3} \frac{1}{2} 1-\frac{1}{2} 1-\frac{1}{3} 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 \frac{1}{3} \frac{1}{2} 1-\frac{1}{2} 1-\frac{1}{3}
volume: 1.2136e-02 1.5528e-02 1.5528e-02 2.4875e-02


Optimal solutions: (script)

../../_images/example03.png ../../_images/example03_interior.png

Robust solutions: (script)

../../_images/example03_robust.png ../../_images/example03_robust_interior.png

Redundant solutions: (script)

\left(\frac{1}{4}\right)-redundant \left(\frac{1}{3}\right)-redundant
../../_images/example03_redundant_1div4.png ../../_images/example03_redundant_1div3.png
\left(\frac{1}{2}\right)-redundant \left(1-\frac{1}{2}\right)-redundant
../../_images/example03_redundant_1div2.png ../../_images/example03_redundant_1minus1div2.png
\left(1-\frac{1}{3}\right)-redundant \left(1-\frac{1}{4}\right)-redundant
../../_images/example03_redundant_1div3.png ../../_images/example03_redundant_1minus1div4.png

Robust and redundant solutions:

robust and \left(\frac{1}{3}\right)-redundant: (script) robust and \left(\frac{1}{2}\right)-redundant: (script)
../../_images/example03_redundant_1div3_robust.png ../../_images/example03_redundant_1div2_robust.png
robust and \left(1-\frac{1}{2}\right)-redundant: (script) robust and \left(1-\frac{1}{3}\right)-redundant: (script)
../../_images/example03_redundant_1minus1div2_robust.png ../../_images/example03_redundant_1minus1div3_robust.png

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