### Abstract

We consider the optimization of the topology and geometry of an elastic structure O ⊂ℝ^{2}subjected to a fixed boundary load, i.e. we aim to minimize a weighted sum of material volume Vol(O), structure perimeter Per(O) and structure compliance Comp(O) (which is the work done by the load). As a first simple and instructive case, this paper treats the situation of an imposed uniform uniaxial tension load in two dimensions. If the weight ε of the perimeter is small, optimal geometries exhibit very fine-scale structure which cannot be resolved by numerical optimization. Instead, we prove how the minimum energy scales in ε, which involves the construction of a family of near-optimal geometries and thus provides qualitative insights. The construction is based on a classical branching procedure with some features unique to compliance minimization. The proof of the energy scaling also requires an ansatzindependent lower bound, which we derive once via a classical convex duality argument (which is restricted to two dimensions and the uniaxial load) and once via a Fourier-based refinement of the Hashin.Shtrikman bounds for the effective elastic moduli of composite materials. We also highlight the close relation to and the differences from shape optimization with a scalar PDE-constraint and a link to the pattern formation observed in intermediate states of type-I superconductors.

Original language | English (US) |
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Article number | 20140432 |

Journal | Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences |

Volume | 470 |

Issue number | 2170 |

DOIs | |

State | Published - Oct 8 2014 |

### Keywords

- Energy scaling law
- Pattern formation
- Shape optimization

### ASJC Scopus subject areas

- Mathematics(all)
- Engineering(all)
- Physics and Astronomy(all)