The fitness function determines the "goodness" of a
given design based on a weighted combination of full weight of the structure, the amount
of hinges and some geometrical restrictions:
fitness = f (weight) * f (geometry) ,
where
f (weight) - dependence upon weight, it may be linear;
f (geometry) - necessary geometrical and
physical restrictions.
Usual methods are applied to determine geometrical stability,
stresses and cross section areas for compressed, tensile and zero steel bars
with flexibility limits (compressed bars = 120, tensile bars = 200 and zero
bars = 150).
Usual - it means standard methods that engineers use to calculate
the stresses and necessary cross-section areas of bars of steel trusses.
This solution is rather general - hinges are scattered at
random with random ties between them. Structure may have any number of
support ties at any position. Mutations are applied to the position of
hinges and ties. Finite element method is used to calculate displacements
and stresses for any negative or zero degree of freedom.
Loads are shown on the trusses and there may be any number
of load combinations - like it's in real construction design.
Flexure limit is taken 1/1000 of span.
In all cases fitness function k and weight of the best truss
is compared to the weight of base steel standard model (the first picture in
some examples). The best structures among population of 5 islands each with
the subpopulation size of 100 members with 1000 generations are shown.
Optimal trusses are shown (blue - tensile bars, red - compressed bars, black
– zero bars). Optimal – not the best, it’s always possible to get better solution.
There are possible two ways: to choose a shape with given
ties and optimize it by evolutionary algorithm (as it’s shown on Figure
2) or to use random position of
hinges with random ties between them for every member of first generation.
It’s evolution and every new start of the computer – new
results, with some differences bigger or smaller from time to time.
Sure, not all best structures may be used - there is a possibility
to choose among the best in every island.
Though initial structures have different negative or zero
degree of freedom (SS on the pictures) the best results of evolution
in most cases are statically determinate structures.
