# Branching Structures in Elastic Shape Optimization

• Nora Lüthen
• Martin Rumpf
• Sascha Tölkes
• Orestis Vantzos
Chapter
Part of the International Series of Numerical Mathematics book series (ISNM, volume 169)

## 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.

## Keywords

Shape optimization Elasticity Branching patterns

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© Springer International Publishing AG, part of Springer Nature 2018

## Authors and Affiliations

• Nora Lüthen
• 1
• Martin Rumpf
• 2
Email author
• Sascha Tölkes
• 2
• Orestis Vantzos
• 3
1. 1.Departement Maschinenbau und VerfahrenstechnikETH ZürichZürichSwitzerland
2. 2.Institut für Numerische SimulationUniversität BonnBonnGermany
3. 3.Department of MathematicsIsrael Institute of TechnologyHaifaIsrael