The distance to cubic symmetry class as a polynomial optimization problem

TL;DR

This paper formulates the problem of finding the closest cubic symmetry tensor to a given elasticity tensor as a polynomial optimization problem, providing a quasi-analytical solution using Gröbner bases, applicable to elasticity and elasto-plasticity.

Contribution

It introduces a novel polynomial optimization approach for computing the nearest cubic symmetry tensor, leveraging algebraic properties of the symmetry stratum and Gröbner bases for solutions.

Findings

The distance to cubic symmetry can be formulated as a quadratic polynomial optimization problem.

A quasi-analytical solution using Gröbner bases is derived.

Method applies to both elasticity tensors and cubic Hill elasto-plasticity models.

Abstract

Generically, a fully measured elasticity tensor has no material symmetry. For single crystals with a cubic lattice, or for the aeronautics turbine blades superalloys such as Nickelbased CMSX-4, cubic symmetry is nevertheless expected. It is in practice necessary to compute the nearest cubic elasticity tensor to a given raw one. Mathematically formulated, the problem consists in finding the distance between a given tensor and the cubic symmetry stratum. It is known that closed symmetry strata (for any tensorial representation of the rotation group) are semialgebraic sets, defined by polynomial equations and inequalities. It has been recently shown that the closed cubic elasticity stratum is moreover algebraic, which means that it can be defined by polynomial equations only (without requirement to polynomial inequalities). We propose to make use of this mathematical property to formulate the distance to cubic symmetry problem as a polynomial (in fact quadratic) optimization problem, and to derive its quasi-analytical solution using the technique of Gr{\"o}bner bases. The proposed methodology also applies to cubic Hill elasto-plasticity (where two fourth-order constitutive tensors are involved).