A volume of fluid approach for crystal growth simulation

A new approach to simulating the dendritic growth of pure metals, based on a recent volume of fluid (VOF) method with PLIC (piecewise linear interface calculation) reconstruction of the interface, is presented. The energy equation is solved using a diffuse-interface method, which avoids the need to apply the thermal boundary conditions directly at the solid front. The thermal gradients at both sides of the interface, which are needed to obtain the front velocity, are calculated with the aid of a distance function to the reconstructed interface. The advection equation of a discretized solid fraction function is solved using the unsplit VOF advection method proposed by Lopez et al. [J. Comput. Phys. 195 (2004) 718-742] (extended to three dimensions by Hernandez et al. [Int. J. Numer. Methods Fluids 58 (2008) 897-921]), and the interface curvature is computed using an improved height function technique, which provides second-order accuracy. The proposed methodology is assessed by comparing the numerical results with analytical solutions and with results obtained by different authors for the formation of complex dendritic structures in two and three dimensions.

[1]  Gretar Tryggvason,et al.  Numerical simulation of dendritic solidification with convection: three-dimensional flow , 2004 .

[2]  Li-Tien Cheng,et al.  A second-order-accurate symmetric discretization of the Poisson equation on irregular domains , 2002 .

[3]  A. Karma,et al.  Phase-Field Simulation of Solidification , 2002 .

[4]  S. Zaleski,et al.  DIRECT NUMERICAL SIMULATION OF FREE-SURFACE AND INTERFACIAL FLOW , 1999 .

[5]  S. Cummins,et al.  Estimating curvature from volume fractions , 2005 .

[6]  Juan C. Heinrich,et al.  Numerical approximation of a thermally driven interface using finite elements , 2003 .

[7]  M. Rappaz,et al.  A Pseudo-Front Tracking Technique for the Modelling of Solidification Microstructures in Multi-Component Alloys , 2002 .

[8]  D. Juric,et al.  A Front-Tracking Method for Dendritic Solidification , 1996 .

[9]  W. Shyy,et al.  Computation of Solid-Liquid Phase Fronts in the Sharp Interface Limit on Fixed Grids , 1999 .

[10]  Nicholas Zabaras,et al.  A stabilized volume‐averaging finite element method for flow in porous media and binary alloy solidification processes , 2004 .

[11]  A. Karma,et al.  Regular Article: Modeling Melt Convection in Phase-Field Simulations of Solidification , 1999 .

[12]  Vaughan R Voller,et al.  An enthalpy method for modeling dendritic growth in a binary alloy , 2008 .

[13]  Dantzig,et al.  Computation of dendritic microstructures using a level set method , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[14]  Markus Bussmann,et al.  Adaptive VOF with curvature‐based refinement , 2007 .

[15]  A. Karma,et al.  Quantitative phase-field modeling of dendritic growth in two and three dimensions , 1996 .

[16]  F. Faura,et al.  A new volume of fluid method in three dimensions—Part II: Piecewise‐planar interface reconstruction with cubic‐Bézier fit , 2008 .

[17]  F. Faura,et al.  An improved height function technique for computing interface curvature from volume fractions , 2009 .

[18]  Juan C. Heinrich,et al.  Front-tracking finite element method for dendritic solidification: 765 , 2001 .

[19]  J. Warren,et al.  Prediction of dendritic growth and microsegregation patterns in a binary alloy using the phase-field method , 1995 .

[20]  M. Sussman,et al.  A Coupled Level Set and Volume-of-Fluid Method for Computing 3D and Axisymmetric Incompressible Two-Phase Flows , 2000 .

[21]  S. Osher,et al.  A Simple Level Set Method for Solving Stefan Problems , 1997, Journal of Computational Physics.

[22]  Department of Physics,et al.  EFFICIENT COMPUTATION OF DENDRITIC MICROSTRUCTURES USING ADAPTIVE MESH REFINEMENT , 1998 .

[23]  A. D. Solomon,et al.  Mathematical Modeling Of Melting And Freezing Processes , 1992 .

[24]  F. Faura,et al.  A new volume of fluid method in three dimensions—Part I: Multidimensional advection method with face‐matched flux polyhedra , 2008 .

[25]  J. Sethian,et al.  Crystal growth and dendritic solidification , 1992 .

[26]  Xiaofeng Yang,et al.  An adaptive coupled level-set/volume-of-fluid interface capturing method for unstructured triangular grids , 2006, J. Comput. Phys..

[27]  Carlos Alberto Brebbia,et al.  Finite Difference Enthalpy Methods for Dendritic Growth , 1991 .

[28]  T. Ménard,et al.  Coupling level set/VOF/ghost fluid methods: Validation and application to 3D simulation of the primary break-up of a liquid jet , 2007 .

[29]  J. López,et al.  A volume of fluid method based on multidimensional advection and spline interface reconstruction , 2004 .

[30]  Julio Hernández,et al.  Analytical and geometrical tools for 3D volume of fluid methods in general grids , 2008, J. Comput. Phys..

[31]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[32]  Gretar Tryggvason,et al.  Numerical simulation of dendritic solidification with convection: two-dimensional geometry , 2002 .

[33]  Pradip Dutta,et al.  An Enthalpy Model for Simulation of Dendritic Growth , 2006 .

[34]  R. Almgren Variational algorithms and pattern formation in dendritic solidification , 1993 .

[35]  Juan C. Heinrich,et al.  Numerical simulation of crystal growth in three dimensions using a sharp‐interface finite element method , 2007 .

[36]  N. Zabaras,et al.  A level set simulation of dendritic solidification with combined features of front-tracking and fixed-domain methods , 2006 .