1. Numerical Simulation of Dendritic Solidification

2. Outline
Problem statement
Solution strategies
Governing equations
FEM formulation
Numerical techniques
Results
Conclusion

3. Problem Statement T < TM TW < TM TW < TM T > TM Solid
Liquid
T < TM
TW < TM
TW < TM S
Liquid
Solid
T > TM S
When the solidification front grows into a superheated liquid, the interface is stable, heat is conducted through the solid
When the solidification front grows into an undercooled melt, the interface is unstable, heat is conducted through the liquid

4. Solution Strategies
Most analytical solutions are limited to parabolic
cylinder or a paraboloid of revolution
Experimental studies are usually limited to transparent
materials with ideal thermal properties such as succinonitrile
Numerical solution can change the various material
characteristics and modify the governing equations to gain
more insight to the actual underlying physics

5. Governing Equations
Heat conduction is the dominant mode of thermal transport:
The interface temperature is modified by local capillary effects:
The second boundary conditions on the interface preserves the
sensible heat transported away from the interface and the latent
heat of fusion released at the interface:

6. Finite Element Formulation
For the energy equation in each domain,
the Galerkin Weighted Residual finite element method formulation can be expresses as:

7. Finite Element Formulation - Continued
Since a moving mesh is used to track the interface, the motion of the coordinate system must be incorporated:
The FEM formulation of the heat equation can be rewritten as:

8. The time derivative is treated in standard finite difference form,
and the heat equation is evaluated at time:
with:

9. Finite Element Formulation - Continued
The interface motion balances the heat diffused from the front and the heat liberated at the front:
Numerically:

10. Numerical Simulation Domain
As the solidification
front advances into the
undercooled melt,
unstable perturbations
emerge and develop into
dendrite
Infinitesimal perturbation
is seeded in the middle of
the interface
Boundaries are extended far
enough to simulate a relatively
unconfined environment y
Free surface x
T = 0
T = DT
Frozen
region
Unfrozen
region

11. Simulation Results
Sequences of Dendrite Growth from a Crystallization Seed

12. Finite Element Mesh
Liquid
Solid

13. Numerical Technique -- Interface node adjustment
Coarse Interface Resolution
Refined Interface Resolution

14. Numerical Technique --Remeshing
Distorted Mesh
Remeshed Mesh

15. Numerical Technique -- Solution Mappingk P i j

16. Simulation Results
--Thermal Field in the Vicinity of the Dendrite Tip

17. Simulation Results -- Isotherm Coarsening o C .01 -.02 -.06 -.09 -.12
-.16
-.19
-.23
-.26
-.29
-.33
-.36
-.40
-.43
-.47
-.50 I J K
0.1 mm

18. Side Branch Steering
Steering direction
0.02mm

19. Simulation Results Scaling Relationship of Dendrite Size with the
Critical Instability
Wavelength
--- Dimensional
0.1 mm
DT: 5.92
DT: 10

20. Simulation Results
Scaling Relationship of Dendrite Size with the Critical Instability Wavelength
--- Non-dimensional 10
DT: 5.92
(DU: 0.25)
DT: 10
(DU: 0.42)

21. Simulation Results
Remelting
Remelted Branch Tip

22. Simulation Results
-- Dendrite Tip Velocity and Tip Radius Relationship 1
Experimental Data 2
Simulated Data 1 - 1 2 1 - 2
Dendrite Tip Velocity (cm/s) 2 1 - 3 2 1 - 4
Simulated undercooling range: oC
Experimental undercooling range: oC 2 1 - 5 1 - 4 2 3 4 5 1 - 3 2 3 4 5 1 - 2 2 3 4 5 1 - 1
Dendrite Tip Radius (cm)

23. Conclusions
A numerical model was developed for dendritic growth simulation
during solidification based on the physics governing the conduction
of heat.
The model accounts for interfacial curvature, surface energy,
and the latent heat of fusion released during solidification.
The moving interface was tracked with moving mesh, automatic
remeshing and refinement techniques.
The simulated dendrite growth characteristics agree with experimental
observation:
dendrite growth tip velocity and radius relationship
dendrite size scaling relationship
side branch competition and remelting, isothermal coarsening.