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

Perla Azzi, Rodrigue Desmorat, Boris Kolev and Fabien Priziac

Vol. 11 (2023), No. 3, 393–428
Abstract

We give a detailed description of a polynomial optimization method allowing one 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.

Keywords
polynomial optimization, Lasserre's method, semidefinite programming, distance to a symmetry class, cubic symmetry, elasticity, piezoelectricity, semialgebraic and real algebraic geometry
Mathematical Subject Classification
Primary: 14P10, 74B05, 74E10, 90C22, 90C23
Milestones
Received: 8 February 2023
Revised: 16 July 2023
Accepted: 13 August 2023
Published: 15 November 2023

Communicated by Francesco dell'Isola
Authors
Perla Azzi
Sorbonne Université
Institut de Mathématiques de Jussieu-Paris Rive Gauche
4 place Jussieu, 75005
Paris
France
Rodrigue Desmorat
Université Paris-Saclay
CentraleSupélec, ENS Paris-Saclay, CNRS
LMPS - Laboratoire de Mécanique Paris-Saclay
91190, Gif-sur-Yvette
France
Boris Kolev
Université Paris-Saclay
CentraleSupélec, ENS Paris-Saclay, CNRS
LMPS - Laboratoire de Mécanique Paris-Saclay
91190, Gif-sur-Yvette
France
Fabien Priziac
Université Bretagne Sud
CNRS UMR 6205, LMBA
F-56000 Vannes
France