Numerical treatment of reactive diffusion using the discontinuous Galerkin method

Abstract

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.