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,¹⁾ 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 Pezzola²⁾ can only be applied to predict anisotropic grain growth, if an error of around 10% is accepted. An extended phase-field approach³⁾ 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 MPF²⁾ 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.

基于简单几何形状和理想假设的定义明确的基准测试问题非常有用，因为它们提供了精确的分析解决方案，可作为定量验证替代数值模拟方法的参考。在最近发表的论文中，^1）通过应用于三晶体基准问题验证了各向异性晶粒生长的不同相场方法。结论是，如果接受大约10％的误差，Steinbach和Pezzola ^2）的多相场（MPF）方法只能用于预测各向异性晶粒的生长。扩展的相场方法^3）声称具有调整后的高阶能量项可以进​​行准确度大大提高的预测。然而，广泛使用的三晶体问题近似解被用作验证相场结果的参考。由于这种近似的数学误差大大超过了相场结果的估计误差，因此必须对这种验证的结论提出质疑。本文为三晶体问题提供了精确的解析解，并讨论了近似对精度评估的意义。为了进行重新评估，我们进行了一系列自己的MPF模拟。与派生的分析解决方案进行比较证明了MPF ²的高精度 公式，并证明在自由能函数中不需要其他高阶能量项，但是会导致与目标尖锐的界面解决方案有相当大的偏差。