Algorithms for bifurcation detection in solid deformation using eigenvalues

Bifurcation theory has long been used to signal the onset of localization. While analytical solutions exist for many special cases, in general numerical solutions are still used. Most such solutions search for a zero or negative determinant of the acoustic tensor. Here, we search for a minimum eigenvalue, simultaneously optimizing the normals and slip directions. This approach leads to a simple formulation amenable to solution by a Newton-Raphson scheme. We test various parameterizations of the normal and slip directions, with the Lagrange multiplier scheme being optimal. We also examine the distribution of the starting points to find the minimum number of points to robustly and efficiently solve the problem, optimizing with Platonic solids. Tests are carried out in a variety of tangent moduli with one, two, or multiple solutions. The algorithm is also implemented in a finite element code to detect localization for a hardening/softening material model.