1. GROWING NUMERICAL CRYSTALS Vaughan Voller, and Man Liang, University of Minnesota We revisit an analytical solution in Carslaw and Jaeger for the solidification of an an under-cooled melt in a cylindrical geometry. We show that when the one-d axi-symmetry is exploited a fixed grid enthalpy model Produces excellent results. BUT—when a 2-D Cartesian solution is sort—”exotic” numerical crystals grow. This IS NOT A numerical crystal

2. The Carslaw and Jaeger Solution for a cylindrical sold seed in an under-cooled melt Consider a LIQUID melt infinite in extent At temperature T< 0 BELOW Freezing Temp At time t = 0 a solid seed at temperature T = 0 is placed in the center This sets up a temp gradient that favors the growth of the solid

3. Similarity Solution Exponential integral Also develop similarity solutions for planar and spherical case Assume radius grows as Then With Found from

4. Enthalpy Solution in Cylindrical Cordiantes Assume an arbitrary thin diffuse interface where liquid fraction Define Throughout Domain a single governing Eq Numerical Solution Very Straight-forward

5. If Initially seed Set Transition: When

6. R(t) Excellent agreement with analytical when predicting growth R(t)

7. Similarity and enthalpy solutions can be extended to account for a binary alloy and a spherical seed Concentration and Temperature Profiles for spherical seed at time Time 20, Le= 50Time 250, Le= 2

8. If A 2-D Cartesian application of enthalpy model Start with a single solid cell When cell finishes freezing “infect” -- seed liquid cells in mane compass directions

9. Initial-Seed Infection This choice will grow a fairly nice four-fold symmetry dendritic crystal is a stable configuration (1)where is the anisotropy (2)Why is growth stable (no surface tension of kinetic surface under-cooling) Pleasing at first!!! But not physically reasonable 1.The initial seed, grid geometry and infection routine introduce artificial anisotropy 2.The grid size enforces a stable configuration—largest microstructure has to be at grid size

10. Initial-Seed Infection Demonstration of artificial anisotropy induced by seed and infection routine Similarity Solution

11. Similarity Solution Serious codes impose anisotropy-and include surface tension and kinetic effects But the choice of seed shape and grid can (will) cause artificial anisotropic effects With the Cartesian grid Hard to avoid non-cylindrical perturbations Which will always locate in a region favorable for growth If imposed anisotropy is weak this feature will swamp Physical effect and lead to a Numerical Crystal Numerical Crystal Can The similarity solution be used to test the intrinsic grid anisotropy in numerical crystal growth simulators ?

12. Conclusions: 1. Growth of a cylindrical solid seed in an undercooled binary alloy melt can (in the absence of imposed anisotropy, surface tension and kinetic effects) be resolved with a similarity solution and axisymmetric enthalpy code 2. On Cartesian structured grid however the enthalpy method breaksdown and—due to artificial grid anisotropy grows NUMERICAL crystals stabilized by grid cell size. 3. Similarity solution stringent test of ability Of a given method to suppress grid anisotropy

13. What about unstructured meshes ? We get a sea-weed pattern Can we use this as a CA solver for channels in a delta