Calibration and fast evaluation algorithms for homogeneous orthotropic polynomial yield functions

Homogeneous polynomial functions have the potential to provide a general modeling framework for yield surfaces in metal plasticity. They incorporate as particular cases many of the previously proposed yield functions and their fitting capabilities allow for capturing a wide range of yield surface shapes. And yet, there are still two unsolved problems which turn into major obstacles when it comes to actual implementations in both academic and industrial environments: The lack of a general optimization algorithm for the calibration of their parameters and the lack of an efficient computational scheme for their value, gradient and hessian. The difficulty of the first problem is two-fold, necessitating an adequate specification of the experimental input data set and satisfaction of the convexity constraint. The second problem is specific to all high degree polynomials and is comprised of issues such as numerical stability, precision and implementation efficiency. We present practical solutions to both problems: An optimization algorithm that reduces to solving a sequence of quadratic problems and a double Horner evaluation scheme that is optimal (featuring the least number of multiplications). The resulting modeling framework can account for arbitrary input data, experimental or from crystal plasticity predictions. As illustration we show new results regarding the relationship between generalized r-values and the earing profile of deep-drawn cylindrical cups. Practicality is demonstrated by the high level of automation of the entire workflow, from material parameters calibration to finite element simulations, and supporting code (Python scripts and constitutive subroutine) made available at https://github.com/stefanSCS/PolyN.