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

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.

Source: arXiv:2602.23180v1 - http://arxiv.org/abs/2602.23180v1 PDF: https://arxiv.org/pdf/2602.23180v1 Original Link: http://arxiv.org/abs/2602.23180v1