# An epicycle method for elasticity limit calculations

## TL;DR

This paper introduces a novel computational method inspired by the dimer technique to efficiently determine the elasticity limit of solids by locating the onset of mechanical instability, even in complex systems.

## Contribution

A new nested optimization approach that combines a dimer-like search with energy minimization to find the mechanical instability point in high-dimensional systems.

## Key findings

- Successfully applied to model systems and electronic structure calculations.
- Efficiently locates the elasticity limit without exhaustive Hessian computations.
- Applicable to low-symmetry and complex materials.

## Abstract

The task of finding the smallest energy needed to bring a solid to its onset of mechanical instability arises in many problems in materials science, from the determination of the elasticity limit to the consistent assignment of free energies to mechanically unstable phases. However, unless the space of possible deformations is low-dimensional and a priori known, this problem is numerically difficult, as it involves minimizing a function under a constraint on its Hessian, which is computionally prohibitive to obtain in low symmetry systems, especially if electronic structure calculations are used. We propose a method that is inspired by the well-known dimer method for saddle point searches but that adds the necessary ingredients to solve for the lowest onset of mechanical instability. The method consists of two nested optimization problems. The inner one involves a dimer-like construction to find the direction of smallest curvature as well as the gradient of this curvature function. The outer optimization then minimizes energy using the result of the inner optimization problem to constrain the search to the hypersurface enclosing all points of zero minimum curvature. Example applications to both model systems and electronic structure calculations are given.