Hierarchy of bounds in free orthotropic material optimization: From convex relaxations to Hashin-Shtrikman via sequential global programming

TL;DR

This paper develops a hierarchy of bounds for free orthotropic material optimization, moving from convex relaxations to Hashin-Shtrikman bounds, and demonstrates their effectiveness through theoretical analysis and numerical experiments.

Contribution

It introduces a realizability-aware hierarchy of bounds for orthotropic material optimization, including new formulations and solution methods, bridging the gap between convex relaxations and Hashin-Shtrikman bounds.

Findings

Voigt set is tighter than zeroth-order for intermediate volume fractions.

Hashin-Shtrikman bounds are tight in single-load cases and provide lower bounds in multi-load scenarios.

Numerical results align with theoretical compliance hierarchy and match laminate references.

Abstract

We study free orthotropic material optimization for two-dimensional plane-stress compliance minimization with two well-ordered isotropic phases, motivated by the gap between tensors admissible in classical free-material optimization and tensors realizable by composites. To reduce this gap, we construct a hierarchy of realizability-aware admissible sets induced by zeroth-order, Voigt, and Hashin--Shtrikman (HS) energy bounds, moving from convex relaxations to a tighter nonconvex model. In the convex zeroth-order and Voigt settings, the Voigt set is strictly tighter for intermediate volume fractions and coincides with the zeroth-order set at pure-phase endpoints, and the Voigt model reduces to an isotropic variable-thickness-sheet formulation. In the single-loadcase continuum zeroth-order problem, at least one optimal solution can be chosen orthotropic. For HS constraints, we rewrite the bound as a Voigt term minus a nonnegative correction, clarifying strict tightening for interior volume fractions and local nonconvexity. We further prove that the convex hull of the HS feasible set equals the Voigt set and derive reduced formulations via active-constraint analysis and explicit elementwise volume characterization, including reductions specialized to orthotropic effective tensors. In the single-loadcase continuum setting, the HS relaxation is tight with the Allaire--Kohn relaxed problem, attained in the relaxation sense by sequential laminates, whereas in generic multi-loadcase settings it provides a lower bound on optimal compliance over general microstructures. The resulting nonconvex orthotropic HS problem is solved by sequential global programming, and numerical results confirm the predicted compliance hierarchy and show close agreement with finite-rank laminate references.