Discussion of the Accuracy of the Multi-Phase-Field Approach to Simulate Grain Growth with Anisotropic Grain Boundary Properties

摘要

Well-defined benchmark problems based on simple geometries and idealized assumptions are extremely useful, because they offer a precise analytical solution as reference for quantitative validation of alternative numerical simulation approaches. In a recently published paper, different phase-field approaches for anisotropic grain growth were validated by application to a tri-crystal benchmark problem. It was concluded that the multi-phase-field (MPF) approach by Steinbach and Pezzola can only be applied to predict anisotropic grain growth, if an error of around 10% is accepted. An extended phase-field approach with adjusted higher-order energy terms was claimed to allow for prediction with significantly improved accuracy. However, a wide-spread approximate solution to the tri-crystal problem was used as reference to validate the phase-field results. As the mathematical inaccuracy of this approximation widely exceeds the evaluated inaccuracy of the phase-field results, the conclusions of this validation have to be questioned. The present paper provides the accurate analytical solution to the tri-crystal problem and discusses the implications of the approximation on the accuracy evaluation. For a reevaluation, a series of own MPF simulations were performed. Comparison with the derived analytical solution proves the high-accuracy of the MPF formulation and demonstrates that additional higher-order energy terms in the free energy functional are not required, but can result in considerable deviation from the targeted sharp interface solution.