Application of the level-set method to a mixed-mode driven Stefan problem in 2 $$D$$ and 3 $$D$$
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Application of the level-set method to a mixed-mode driven Stefan problem in 2D and 3D D. den Ouden · A. Segal · F. J. Vermolen · L. Zhao · C. Vuik · J. Sietsma
Received: 28 September 2012 / Accepted: 20 November 2012 / Published online: 2 December 2012 © Springer-Verlag Wien 2012
Abstract This study focusses on the growth of small precipitates within a matrix phase (see also den Ouden et al., Comput Mater Sci 50:2397–2410, 2011). The growth of a precipitate is assumed to be affected by the concentration gradients of a single chemical element within the matrix phase at the precipitate/matrix boundary and by an interface reaction, resulting into a mixed-mode formulation of the boundary condition on the precipitate/matrix interface. Within the matrix phase we assume that the standard diffusion equation applies to the concentration of the considered chemical element. The formulated Stefan problem is solved using a level-set method (J Comput Phys 79:12–49, 1988) by introducing a time-dependent signed-distance function for which the zero level-set describes the precipitate/matrix interface. All appearing hyperbolic partial differential equations are discretised by the use of Streamline-Upwind Petrov– Galerkin finite-element techniques (Comput Vis Sci 3:93–101, 2000). All level-set related equations are solved on a background mesh, which is enriched with interface nodes located on the zero-level of the signed-distance function. The diffusion equation is solved in the diffusive phase. Simulations with the implemented methods for the
In memory of our late colleague Jeroen Colijn. D. den Ouden (B) · L. Zhao Materials innovation institute, Mekelweg 2, 2628 CD Delft, The Netherlands e-mail: [email protected] A. Segal · F. J. Vermolen · C. Vuik Delft University of Technology, DIAM, Mekelweg 4, 2628 CD Delft, The Netherlands J. Sietsma Delft University of Technology, MS&E, Mekelweg 2, 2628 CD Delft, The Netherlands
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growth of various precipitate shapes show that the methods employed in this study correctly capture the evolution of the precipitate/matrix interface including topological changes. At the final stage of growth/dissolution physical equilibrium is attained. We also observe that our solutions show mass conservation as the time-step and elementsize tend to zero. Keywords Level-set · Finite-element · Stefan problem · Diffusion · Precipitate growth Mathematics Subject Classification (2000) 80A22 · 82C24
35K57 · 65M60 · 74A50 · 74N25 ·
1 Introduction Metalworking of alloys is a widely used and complex process that involves several physical phenomena, such as dislocation movement, grain recrystallisation and secondary phase precipitation [14], that influence the workability and applicability of the object. These phenomena and their influences have been studied and documented using mainly “trial and error”-based experiments and by experience. These experimentally obtained results could be verified by an analytical investigation of the studied aspects and hence they improve the understandin
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