Polyconvexity with Moments and Sums of Squares
Giovanni Fantuzzi1, Didier Henrion2, Martin Kružík3, Ajay Murali1, Stephan Weis4*
1 University of Erlangen–Nuremberg, Department Mathematik, Erlangen, Germany; 2 University of Toulouse, LAAS-CNRS, Toulouse, France; 3 Czech Academy of Sciences, UTIA, Prague, Czechia; 4 Czech Technical University in Prague, Faculty of Electrical Engineering, Prague, Czechia
Keywords: elasticity, polyconvexity, sum of squares, polynomial optimization
Energy densities that are polyconvex are popular in material modeling and nonlinear elasticity. They can describe physically meaningful properties while guaranteeing mathematical well-posedness. High compression can be prevented by penalizing small determinants of the deformation gradient. Energy-minimizing deformations exists under growth conditions at infinity, given suitable boundary conditions. If the energy density is not polyconvex, then its polyconvex envelope is a relaxation that still respects physics. Yet, computing the polyconvex envelope pointwise is generally difficult. Existing numerical approaches employ an infinite-dimensional moment problem and face significant challenges. We propose a numerical method to compute the polyconvex envelope of polynomial energy densities, based on the moment sum-of-squares hierarchy from polynomial optimization. Furthermore, we present a sufficient conditions to certify polyconvexity via sum-of-squares technology. A preprint [1] is available at https://arxiv.org/abs/2604.11124.
References
- G. Fantuzzi, D. Henrion, M. Kružík, A. Murali, and S. Weis, Polyconvexity with moments and sums of squares, 2026, https://arxiv.org/abs/2604.11124.