This work presents a new finite element variational formulation for the numerical treatment of diffusional phase transformations using the discontinuous Galerkin method (DGM). Steep concentration and property gradients near phase boundaries require particular focus on a sound numerical treatment. There are different ways to tackle this problem ranging from (i) the well-known phase field method (PFM) (Biner et al. in Programming phase-field modeling, Springer, Berlin, 2017, Emmerich in The diffuse interface approach in materials science: thermodynamic concepts and applications of phase-field models, Springer, Berlin, 2003), where the interface is described continuously to (ii) methods that allow sharp transitions at phase boundaries, such as reactive diffusion models (Svoboda and Fischer in Comput Mater Sci 127:136–140, 2017, 78:39–46, 2013, Svoboda et al. in Comput Mater Sci 95:309–315, 2014). Phase transformation problems with continuous property changes can be implemented using the continuous Galerkin method (GM). Sharp interface models, however, lead to stability problems with the GM. A method that is able to treat the features of sharp interface models is the discontinuous Galerkin method. This method is well understood for regular diffusion problems (Cockburn in ZAMM J Appl Math Mech 83(11):731–754, 2003). As will be shown, it is also particularly well suited to model phase transformations. We discuss the thermodynamic background by review of a multi-phase, binary system. A new DGM formulation for the phase transformation problem with sharp interfaces is then introduced. Finally, the derived method is used in a 2D microstructural evolution simulation that features a binary, three-phase system that also takes the vacancy mechanism of solid body diffusion into account.

这项工作提出了一种新的有限元变分公式，用于使用不连续伽辽金方法（DGM）对扩散相变进行数值处理。相边界附近的陡峭浓度和性​​质梯度需要特别关注合理的数值处理。有不同的方法可以解决这个问题，包括 (i) 众所周知的相场方法 (PFM)（Biner 等人，《编程相场建模》，Springer，柏林，2017 年，Emmerich，《材料科学中的扩散界面方法》） ：相场模型的热力学概念和应用，Springer，柏林，2003 年），其中接口连续描述为 (ii) 允许在相边界处急剧转变的方法，例如反应扩散模型（Comput Mater Sci 中的 Svoboda 和 Fischer） 127:136–140, 2017, 78:39–46, 2013, Svoboda 等人，Comput Mater Sci 95:309–315, 2014）。具有连续性质变化的相变问题可以使用连续伽辽金方法（GM）来实现。然而，尖锐的接口模型会导致 GM 的稳定性问题。一种能够处理尖锐界面模型特征的方法是间断伽辽金方法。对于常规扩散问题，这种方法很好理解（Cockburn in ZAMM J Appl Math Mech 83(11):731–754, 2003）。正如将要展示的，它也特别适合模型相变。我们通过回顾多相二元系统来讨论热力学背景。然后介绍了一种新的 DGM 公式，用于解决具有尖锐界面的相变问题。最后，导出的方法用于二维微观结构演化模拟，该模拟具有二元、三相系统，还考虑了固体扩散的空位机制。