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.
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polynomial optimization, Lasserre's method, semidefinite
programming, distance to a symmetry class, cubic symmetry,
elasticity, piezoelectricity, semialgebraic and real
algebraic geometry