Materials Process Design and Control Laboratory Sibley School of Mechanical and Aerospace Engineering 169 Frank H. T. Rhodes Hall Cornell University Ithaca, NY URL: Nicholas Zabaras and Lijian Tan An energy conserving level set simulation of dendritic solidification including the effects of melt convection An energy conserving level set simulation of dendritic solidification including the effects of melt convection

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Materials Process Design and Control Laboratory  Background  Issues with the moving boundary in problem definition  Some existing techniques to handle the moving boundary  Present model based on level set method  Necessity to track the interface  Numerical scheme for fluid flow  Adaptive meshing and its effect  Convergence study and mesh anisotropy study  Pure material crystal growth without fluid flow  Pure material crystal growth with fluid flow  Binary alloy crystal growth  Extensions to multi-component systems–a brief discussion Outline

Background Materials Process Design and Control Laboratory  Microstructure determines mechanical properties  Available thermo-dynamic data to simulate microstructure

Problem definition Materials Process Design and Control Laboratory  Presence of fluid flow  Heat transfer  Solute transport  A moving solid-liquid interface

Materials Process Design and Control Laboratory Issues with the moving boundary  Jump in temperature gradient governs interface motion  No slip condition for equal densities or shrinkage driven flow for variable densities  Gibbs-Thomson relation  Solute rejection flux Requires curvature computation at the moving interface!

Existing techniques to handle the moving boundary Materials Process Design and Control Laboratory Review paper by Botteringer, Warren, Beckermann, Karma (2002)  Phase field method Ref Langer (1978)  Advantages  No essential boundary conditions  (global energy conserving)  Whole domain method, one phase field equation takes all of the physics on the moving interface  (Easy to implement)  Disadvantages  Difficult for parameter identification  Require huge grid to be consistent with the sharp interface model Typical grid sizes: 400 × 400 Karma (1998), 800×800 Goldenfeld (2001), 3000 × 3000 Beckermann (2005), etc.

Materials Process Design and Control Laboratory  Front tracking method Ref. Tryggvason (1996), J. Heinrich (2001)  Advantages  Solving sharp interface model directly  (physics clear in governing eqs.)  Separate mesh for solid and liquid  (easy application of various conditions at moving boundary)  Disadvantages  Essbc  scheme not strictly energy conserving  Need remeshing at every step.  Curvature computation (from mark pts) complicated  Difficult to implement (6 conditions in 2D+ splitting/merging) Ideas: (1) Uses markers to represent interface (2) Markers are moved using velocity computed from Stefan equation Existing techniques to handle the moving boundary

Interface motion (level set equation) Materials Process Design and Control Laboratory Ref. S. Osher 1997, J. Dantzig 2000, R. Fedkiw 2003 etc.  Level set method Devised by Sethian&Osher  Advantages  Interface geometries can be easily and accurately computed with  Level set equation well studied (FDM with high order accuracy, FEM)  Disadvantages  No separate mesh for solid and liquid making the application of boundary conditions still not easy (current applications are restricted to pure materials without melt convection) Existing techniques to handle the moving boundary Signed distance

Present model based on level set method Materials Process Design and Control Laboratory (1) Solidification occurs in a diffused zone of width 2w that is symmetric around the zero level set. A phase volume fraction can be defined accordingly. (2) The mean solid-liquid interface temperature in the freezing zone of width 2w is allowed to vary from the equilibrium temperature in a way governed by (3) The diffused interface is treated as a porous medium with Kozeny-Carman approximation for the permeability.  Assumptions

Materials Process Design and Control Laboratory Present model based on level set method  Governing equations  Volume averaging techniques can be utilized based on the previous diffused interface assumptions.

Stabilized finite element method for flow Materials Process Design and Control Laboratory Stabilized equal-order velocity-pressure formulation for fluid flow Derived from SUPG/PSPG formulation Additional stabilizing term for Darcy drag force incorporated  The diffused interface is treated as a porous medium with Kozeny- Carman approximation for the permeability.  Our previous work (N. Zabaras and D. Samanta, 2004) is used to solve the volume averaged momentum equations.  Galerkin formulation for the fluid flow problem.

Numerical scheme for fluid flow Materials Process Design and Control Laboratory Advection stabilizing term Darcy drag stabilizing term Pressure stabilizing term Diffusion stabilizing term  Stabilized formulation for the fluid flow problem.

Stabilizing parameters for fluid flow Materials Process Design and Control Laboratory advective viscous Darcy Stabilizing terms Stabilizing parameters continuity Convective and pressure stabilizing terms modified form of SUPG/PSPG terms Darcy stabilizing term obtained by least squares, necessary for convergence Viscous term with second derivatives neglected A fifth continuity stabilizing term added to the stabilized formulation pressure

Compare with another method Materials Process Design and Control Laboratory  Another method using high viscosity in diffused interface region is introduced by Beckermann (1999).  Both method get almost zero velocity within the diffused interface (no slip condition).

Numerical techniques for level set computation Materials Process Design and Control Laboratory Level set equation  Stabilized Galerkin form  Semi-descretized form

Materials Process Design and Control Laboratory Present model based on level set method SOLUTION ALGORITHM AT EACH TIME STEP Construct a diffused interface All fields known at time t n Solve for the temperature field (energy equation) n = n +1 Adaptive meshing Solve for the concentration field (solute equation) Solve for the flow & pressure (momentum eq) Inner loop Adaptive time step (Determine the next time step size) Advance the time to t n+1 SALIENT FEATURES Interface geometries (normal, curvature) computed from signed distance. A diffused interface is constructed based on signed distance. Single set of transport equations for mass, momentum, energy and species transport based on volume averaging. Interface position tracked using the level set method (differs from other diffused interface models) Solve for the signed distance (level set equation)

Necessity to track the interface Materials Process Design and Control Laboratory Compare results of the benchmark problem in crystal growth : Our diffused-interface model with tracking of the interface Phase field model without tracking of the interface

Materials Process Design and Control Laboratory Numerical techniques for level set computation  Fast marching method  Iterative method Reinitialization

Materials Process Design and Control Laboratory Narrow band: Adaptive meshing: 1 Re-initialization, fluid flow, heat transfer and solute transport is performed in the whole domain using adaptive meshing based on the distance from the interface. 2 Level set equation is solved on a narrow band. Numerical techniques (adaptive & narrowband) Two reasons for adaptive meshing  Moving boundary  Difference between thermal boundary layer and solute boundary layer

Materials Process Design and Control Laboratory Numerical techniques (adaptive & narrowband)  Example  Data structure Coarsen Refine

Materials Process Design and Control Laboratory Numerical techniques (adaptive & narrowband) (1) Refine neighboring elements so that the rank difference less than 1 (2) Connecting node points  Generate conforming grid for FEM computation (1) (2)  Salient feature Much less computational task (adding or deleting elements, interpolation) than re-meshing. Suitable for transient problems

Materials Process Design and Control Laboratory Numerical techniques (adaptive & narrowband) (1) Use error estimator based on gradient of temperature (2) Use signed distance function (distance to interface)  Refine criterion (whether an element should be refined or coarsened)

Materials Process Design and Control Laboratory Effects of adaptive mesh Problem definition will be shown in later slides

Mesh anisotropy study Materials Process Design and Control Laboratory  Benchmark problem Rotated surface tension Normal surface tension

Materials Process Design and Control Laboratory Mesh anisotropy study Crystal growth mainly determined by surface tension not initial perturbation.

Materials Process Design and Control Laboratory Effects of mesh refinement Initial crystal shape Domain size Initial temperature Boundary conditions adiabatic With a grid of 64by64, we get Results using finer mesh are compared with other researcher’s -- results in the next slide.  Benchmark problem

Our method Osher (1997) Heinrich (2003) Materials Process Design and Control Laboratory Triggavason (1996) Different results obtained by researchers suggest that this problem is nontrivial. All the referred results are using sharp interface model. Effects of mesh refinement

Materials Process Design and Control Laboratory Convergence study  Infinite corner problem (2D with analytical solution) After the mesh is refined to 20by20, the error reduces almost quadratically.

Materials Process Design and Control Laboratory Numerical examples

Materials Process Design and Control Laboratory Pure material crystal growth without convection (1) A small change of under-cooling will lead to a drastic change of tip velocity (consistent with the solvability theory) (2) An increase of diffusion coefficient in liquid region tends to make the tip sharper. Effects of solid diffusion coefficient not obvious.  benchmark problem in crystal growth

Materials Process Design and Control Laboratory Pure material crystal growth without convection Using a larger domain, perturbations/ second arm dendrites will be developed.

Materials Process Design and Control Laboratory Pure material crystal growth without convection  Applicable to 3D with high under-cooling using a coarse mesh. Temperature and crystal shape at time t=105

Materials Process Design and Control Laboratory Pure material crystal growth without convection  Applicable to low under-cooling (at previously unreachable range using phase field method, Ref. Karma 2000) with a moderate grid.

Materials Process Design and Control Laboratory Crystal growth example with fluid flow  2D Benchmark problem Velocity of i nlet flow at top: Pr=23.1 Other conditions the same as the previous 2d diffusion benchmark problem. Temperature Stream function

Materials Process Design and Control Laboratory Crystal growth example with fluid flow  With fluid flow, the crystal tips will tilt in the upstream direction.  Steady growth velocity of tips normal to the flow close to the velocity in the diffusion case.  Velocity approximately zero in the diffused interface by treating it as a porous medium.

Materials Process Design and Control Laboratory 3D Crystal growth example with fluid flow Low undercooling High undercooling  Similar to the 2D case, the crystal tips will tilt in the upstream direction.

Binary alloy crystal growth Materials Process Design and Control Laboratory Initial crystal shape Domain size Initial temperature Boundary conditions no heat/solute flux Initial concentration

Materials Process Design and Control Laboratory Le=10 (boundary layers differ by 10 times)  require adaptive mesh. Mesh coarsening also important to keep dof as small as possible. Micro-segregation can be observed in crystal; maximum liquid concentration about 0.05 (compares well with Heinrich et al. 2003) Binary alloy crystal growth

Pb-Sb binary alloy Materials Process Design and Control Laboratory Pb-Sb alloy dendritic growth

Extensions to multi-component systems Materials Process Design and Control Laboratory  Most alloy in practice are multi-component with multi-phases.  Individual signed distance functions for different phases. (A set of level set equations.)  Re-Initialize scheme for multi-phase variables. Demo of the re- initialize scheme to handle gap or overlap.

Extensions to multi-component systems Materials Process Design and Control Laboratory  Example of eutectic growth

(Tertiary alloy Ni-5.8%Al- 15.2%Ta) Materials Process Design and Control Laboratory Important parameters Insulated boundaries on the rest of faces u x = u z = 0  T/  t = r  T/  x = 0  T/  z = G  C/  x = 0 T(x,z,0) = T 0 + Gz C(x,z,0) = C 0 Extensions to multi-component systems

Numerical examples (Tertiary alloy Ni-5.8%Al-15.2%Ta) Materials Process Design and Control Laboratory (a)Interface position (b) Al concentration (c) Ta concentration Patterns of concentration for Al and Ta are similar due to the assumpti- on of equal diffusion coefficients in liquid for both components and similar partition coefficients. ( a ) ( b ) ( c )

Future work & Related Publications Materials Process Design and Control Laboratory  Extending this framework to various practical alloy systems.  Coupling the meso-scale with macro-scale.  Developing multi-scale solidification design algorithms for explicit control of the microstructure and mechanical properties N. Zabaras and D. Samanta, "A stabilized volume-averaging finite element method for flow in porous media and binary alloy solidification processes", International Journal for Numerical Methods in Engineering, Vol. 60/6, pp , 2004 Lijian Tan and N. Zabaras, "A level set simulation of dendritic solidification with combined features of front tracking and fixed domain methods", Journal of Computational Physics, in press N. Zabaras and B. Ganapathysubramanian, "Modeling dendritic solidification with melt convection using the extended finite element method (XFEM) and level set methods", Journal of Computational Physics, submitted for publication.

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