A semidefinite programming approach for the projection onto the cone of negative semidefinite symmetric tensors with applications to solid mechanics

TL;DR

This paper introduces a semidefinite programming algorithm for projecting symmetric tensors onto the cone of negative semidefinite tensors, with applications in solid mechanics, particularly for modeling masonry-like materials.

Contribution

It develops a novel primal-dual interior point method for tensor projection problems, extending semidefinite programming techniques to tensor spaces in solid mechanics.

Findings

Algorithm effectively computes tensor projections in solid mechanics models.

Numerical experiments demonstrate the method's efficiency and robustness.

Applications to masonry-like materials show practical relevance.

Abstract

We propose an algorithm for computing the projection of a symmetric second-order tensor onto the cone of negative semidefinite symmetric tensors with respect to the inner product defined by an assigned positive definite symmetric fourth-order tensor C. The projection problem is written as a semidefinite programming problem and an algorithm based on a primal-dual path-following interior point method coupled with a Mehrotra's predictor-corrector approach is proposed. Implementations based on direct methods are theoretically and numerically investigated taking into account tensors C arising in the modelling of masonry-like materials.