@incollection{LuRuToVa17,
author = {L{\"u}then, Nora and Rumpf, Martin and T{\"o}lkes, Sascha and Vantzos, Orestis},
title = {Branching Structures in Elastic Shape Optimization},
year = {2018},
editor = {Schulz, Volker and Seck, Diaraf},
bookTitle = {Shape Optimization, Homogenization and Optimal Control : DFG-AIMS workshop held at the AIMS Center Senegal, March 13-16, 2017},
publisher = {Birkh\"auser, Cham},
pages = {213--225},
abstract = {Fine scale elastic structures are widespread in nature, for instances
in plants or bones, whenever stiffness and low weight are required.
These patterns frequently refine towards a Dirichlet boundary to
ensure an effective load transfer. The paper discusses the optimization
of such supporting structures in a specific class of domain patterns
in 2D, which composes of periodic and branching period transitions
on subdomain facets. These investigations can be considered as a
case study to display examples of optimal branching domain patterns.
In explicit, a rectangular domain is decomposed into rectangular
subdomains, which share facets with neighbouring subdomains or with
facets which split on one side into equally sized facets of two different
subdomains. On each subdomain one considers an elastic material phase
with stiff elasticity coefficients and an approximate void phase
with orders of magnitude softer material. For given load on the outer
domain boundary, which is distributed on a prescribed fine scale
pattern representing the contact area of the shape, the interior
elastic phase is optimized with respect to the compliance cost. The
elastic stress is supposed to be continuous on the domain and a stress
based finite volume discretization is used for the optimization.
If in one direction equally sized subdomains with equal adjacent
subdomain topology line up, these subdomains are consider as equal
copies including the enforced boundary conditions for the stress
and form a locally periodic substructure. An alternating descent
algorithm is employed for a discrete characteristic function describing
the stiff elastic subset on the subdomains and the solution of the
elastic state equation. Numerical experiments are shown for compression
and shear load on the boundary of a quadratic domain.},
arxiv = {https://arxiv.org/abs/1711.03850},
eprint = {1711.03850},
pdf = {http://numod.ins.uni-bonn.de/research/papers/public/LuRuToVa17.pdf},
doi = {10.1007/978-3-319-90469-6_11},
volume = {169},
series = {International Series of Numerical Mathematics},
}