Optimal fine-scale structures in compliance minimization for a uniaxial load.

We consider the optimization of the topology and geometry of an elastic structure [Formula: see text] subjected to a fixed boundary load, i.e. we aim to minimize a weighted sum of material volume [Formula: see text], structure perimeter [Formula: see text] and structure compliance [Formula: see text] (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 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 , 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 ansatz-independent 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.