  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.
