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

• Main Authors: Potthoff, Jonas, Wirth, Benedikt
• Format: Journal Article
• Language: English
• Published: 12-11-2021
• Subjects: Mathematics - Optimization and Control
• DOI: 10.48550/arxiv.2111.06910

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