Semidefinite relaxations for nonlinear elasticity with energies convex in the Cauchy-Green strain tensor

TL;DR

This paper develops a semidefinite relaxation approach for nonlinear elasticity problems with energies convex in the Cauchy-Green strain tensor, ensuring convergence and exactness of the relaxation under certain conditions.

Contribution

It proves the absence of relaxation gaps and the convergence of the Lasserre hierarchy for a class of nonlinear elasticity energies, enabling efficient mesh-free computations.

Findings

No relaxation gap between non-convex and convex formulations.

Lasserre hierarchy converges and is exact under specific conditions.

Provides a mesh-free numerical method avoiding mesh artifacts.

Abstract

In nonlinear elasticity, finding the deformation of a material which minimizes a given stored energy density is a challenging calculus of variations problem which may fail to have minimizers: the energy optimal material forms infinitely fine microstructures (wrinkles) rather than deforming smoothly. In the case where the energy function is non-convex but frame indifferent and convex with respect to the Cauchy-Green strain tensor, we use the standard Le Dret-Raoult semidefinite projection formula for the quasiconvex envelope of the energy function to prove that there is no relaxation gap between the original non-convex calculus of variations problem and its linear moment formulation based on occupation measures. This implies convergence of the Lasserre moment-sum-of-squares (SOS) hierarchy and provides a computationally efficient, mesh-free numerical method that, unlike the finite element method, avoids undesirable mesh-dependent artifacts. Under the additional condition that the boundary condition is linear and the function is SOS convex in the strain tensor, we show that the first relaxation of the Lasserre hierarchy is exact. In other words, computing the quasiconvex envelope at a point boils down to solving a small convex semidefinite optimization problem.