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We consider the shape and topology optimization problem to design a structurethat minimizes a weighted sum of material consumption and (linearly) elasticcompliance under a fixed given boundary load. As is well-known, this problem isin general not well-posed since its solution typically requires the use ofinfinitesimally fine microstructure. Therefore we examine the effect ofsingularly perturbing the problem by adding the structure perimeter to thecost. For a uniaxial and a shear load in two space dimensions, correspondingenergy scaling laws were already derived in the literature. This work nowderives the scaling law for the case of a uniaxial load in three spacedimensions, which can be considered the simplest three-dimensional setting. Inessence, it is expected (and confirmed in this article) that for a uniaxialload the compliance behaves almost like the dissipation in a scalar fluxproblem so that lower bounds from pattern analysis in superconductors candirectly be applied. The upper bounds though require nontrivial modificationsof the constructions known from superconductors. Those become necessary sincein elasticity one has the additional constraint of torque balance.

Optimal fine-scale structures in compliance minimization for a uniaxial load in three space dimensions