Distance to a constitutive tensor isotropy stratum by Lasserre polynomial optimization method

TL;DR

This paper presents a polynomial optimization method based on Lasserre's approach to determine the closest isotropy stratum of a given tensor in continuum mechanics, useful for material characterization.

Contribution

It applies Lasserre's polynomial optimization technique to compute the distance from an experimental tensor to a specific isotropy stratum, leveraging the semialgebraic nature of these sets.

Findings

Successfully applies polynomial optimization to continuum mechanics tensors.

Provides a method to compute tensor distances to isotropy strata.

Enhances material analysis through advanced optimization techniques.

Abstract

We give a detailed description of a polynomial optimization method allowing to solve a problem in continuum mechanics: the determination of the elasticity or the piezoelectricity tensor of a specific isotropy stratum the closest to a given experimental tensor, and the calculation of the distance to the given tensor from the considered isotropy stratum. We take advantage of the fact that the isotropy strata are semialgebraic sets to show that the method, developed by Lasserre and coworkers which consists in solving polynomial optimization problems with semialgebraic constraints, successfully applies.