#### Vol. 5, No. 4, 2010

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Energy-minimizing openings around a fixed hole in an elastic plate

### Shmuel Vigdergauz

Vol. 5 (2010), No. 4, 661–677
##### Abstract

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 $15%$$19%$ less energy, depending on the hole spacing and the central hole shape.

##### Keywords
plane elasticity problem, Kolosov–Muskhelishvili potentials, shape optimization, effective energy, extremal elastic structures, genetic algorithm