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 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 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.