An Enthalpy Based Scheme for Simulating Dendritic Growth V.R. Voller 4d o (blue) 3.25d o (black) 2.5d o (red) Dendrite shape with 3 grid sizes shows reasonable independence  = 0.05, T 0 = Dimensionless time  = 6000 Solvabillity (kim et al) Long term tip dynamics approaches theory  = 0.25,  = 0.75 BUT results begin to deteriorate if grid is made smaller !! Grid Dependence-Tip velocity  = 0.05, T 0 = Dimensionless time  = 37,600 Black-- Phase field and Level set from Kim, Goldenfeld and Dantzig Red-- Current work Comparison with other Methods Tip velocity The Solid color is solved with a 45 deg twist on the anisotropy and then twisted back—the white line is with the normal anisotropy Note: Different “smear” parameters are used in 0 0 and 45 0 case Tip position with time Dimensionless time  = 6000  = 0.05, T 0 = Not perfect: In 45 0 case the tip velocity at time 6000 (slope of line) is below the theoretical limit. Low Grid Anisotropy Surface of seed is undercooled Problem Growing a solid seed in an undercooeld binary alloy melt T 0 < 0 Governing equations potential: continuous at interface crystal anisotropy solute undercooling curvature This work is based on an original idea of Tacke 1988— modified here to allow for anisotropy and solute resulting in crystal growth f = liqu. frac. Enthalpy Solute Conservation Problem and Solution Use FIXED finite difference grid Initial condition— rectangle of side 3  x Solve for H and C Explicitly BUT If 0 < f < 1 THEN Iterate until Calculate  and  from current f filed Calculate Undercooling T i Update If f = 0 or f = 1 set Solution Problem: range of cells with 0 < f < 1 restricted to width of one cell Accuracy in curvature calc? Remedial scheme: smear out f value, e.g., Remedial Scheme: Use nine volume stencil to calculate derivatives Tricks and Devices Numerical Considerations When cell first reaches f = 0 “infect” every fully liquid neighboring cell (f =1) with a small solid “seed” Like a CA RULE Testing Comparison with one-d Analytical Solution Constant T i, C i k = 0.1, Mc = 0.1, T 0 = -.5, Le = 1.0 Concentration and Temperature at dimensionless time t =100 Symbol-numeric sol. Front Movement Red-line Numeric sol. Covers analytical Results Effect of Lewis Number k = 0.15, Mc = 0.1, T 0 = -.55, Le = 20.0  = 0.02,  x = 2.5d 0 Concentration field at time  = 30,000 Profile along dashed line Concentration k = 0.15, Mc = 0.1, T 0 = -.65  = 0.05,  x =3.333d 0 All predictions at time  =6000  = 0.05, T 0 = time  = 6000  = 0.25,  = 0.75,  x =4d 0 FAST-CPU This On This In 60 seconds ! Comments The proposed enthalpy method is EASY to program and can RAPIDLY produce ACCURATE RESULTS consistent with known growth characteristics. The approach can account for growth in undercooled ALLOYS A DRAWBACK GRID dependence can be limited BUT choice of “smearing” parameters is ad-hoc. Talking Point: Method has features of both CA, Level Set and Phase Fields. How is the enthalpy method related to these alternative methods ?