The design of elastic structures to optimize the stress state of flat plates
with appropriately shaped construction holes is a problem of considerable
mathematical and industrial significance. This paper continues the shape
optimization study previously reported in this journal,
1:2 (2006), 307–406, for the
energy-minimizing single hole under remote shear, and in
3:7 (2008), 1341–1363
for two identical holes. Here, a challenging and more practical three-hole
arrangement is considered, where the central hole is fixed, while the two
identical side holes are varied not only in their shapes, but also in their
areas.
This twofold novelty is resolved by enhancing a standard genetic algorithm
combined with a general method of shape parametrization for multiconnected
regions. The method employs conformal mapping of the outside of each optimized
contour
separately onto the outside of a unit circle, as was first proposed in the 2008
paper. We show here that this approach has a significant computational
advantage over the common practice of mapping the entire domain under
consideration. The numerical simulations present in detail the influence of
sizes, shapes, and relative positions of the openings on the induced energy
increment and, to a much smaller extent, on the local stresses. The main result is
that, compared to a single hole, interacting optimal openings induce up to
–
less energy, depending on the hole spacing and the central hole shape.
