MULTISCALE COMPUTATIONAL MODELING OF ALLOY SOLIDIFICATION PROCESSES

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MULTISCALE COMPUTATIONAL MODELING OF ALLOY
SOLIDIFICATION PROCESSES
Lijian Tan and Prof. Nicholas Zabaras
Materials Process Design and Control
Laboratory Sibley School of Mechanical and
Aerospace Engineering169 Frank H. T. Rhodes
Hall Cornell University Ithaca, NY
14853-3801 Email zabaras_at_cornell.edu
lt69_at_cornell.edu URL http//mpdc.mae.cornell.edu
/
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OUTLINE OF THE PRESENTATION

• Brief review of alloy solidification process
• Brief review of macro-scale model
• Review of meso-scale model
• Multi-scale modeling
• Adaptive meshing
• Database approach

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MULTISCALE NATURE OF SOLIDIFICATION
liquid
10-1 - 100 m
Mushy zone
solid
q
g
(a) Macro scale (mushy zone is represented by
volume fraction)
liquid
10-4 10-5m
(b) Meso/micro scale
solid
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MACRO SCALE MODELING

• Governing equations

• Volume fraction is related with microstructure.
• Lever rule or Scheil rule is used to obtain
  volume fraction.
• Meso scale modeling is required to obtain volume
  fraction more accurately.

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MESO SCALE MODELING

• A moving solid-liquid interface
• Presence of fluid flow
• Heat transfer
• Solute transport

Although equations are even simpler than
macro-scale model, numerically it is very hard to
handle due to the moving solid-liquid interface.
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ISSUES RELATED WITH MOVING INTERFACE

• Jump in temperature gradient governs interface
  motion

• No slip condition for flow

Requires curvature computation at the moving
interface!

• Gibbs-Thomson relation

• Solute rejection flux

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MESO SCALE MODELS
Techniques for handling moving interface

• Cellular automata
• Phase field method
• Front tracking
• Level set method

Level set method
Devised by SethianOsher
Signed distance
Interface motion (level set equation)
Ref. S. Osher 1997,
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OUR WORK WITH LEVEL SET METHOD

• L. Tan and N. Zabaras, "A level set simulation of
  dendritic solidification of multi-component
  alloys", Journal of Computational Physics, in
  press
• N. Zabaras, B. Ganapathysubramanian and L. Tan,
  "Modeling dendritic solidification with melt
  convection using the extended finite element
  method (XFEM) and level set methods", Journal of
  Computational Physics, in press.
• L. 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, Vol. 211, pp.
  36-63, 2006

Comparing with the other method in literature,
our method has better convergence, less
computational requirement. Our method can also be
easily extended to 3D.
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COVERGENCE BEHAVIOR
Triggavason (1996)
Benchmark problem Crystal growth with
initial perturbation.
Osher (1997)
Our method
Different results obtained by researchers
suggest that this problem is nontrivial.
All the referred results are using sharp
interface model.
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COMPUTATIONAL REQUIREMENT
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EXTEND TO 3D
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MULTISCALE MODELING WITH ADAPTIVE MESH
Solution changes rapidly only near solid-liquid
interface. Still solve the meso-scale model, but
with refined element only near interface. When
combine adaptive meshing with parallel computing,
we may be able to solve some macro-scale problems
using the meso-scale model.
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SIMPLE BINARY ALLOY GROWTH EXAMPLE
Le10 (boundary layer differ by 10 times)
Micro-segregation can be observed in the crystal
maximum liquid concentration about 0.05.
(compares well with Ref Heinrich 2003)
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COMBINE WITH PARALLEL COMPUTING
Domain decomposition Colored by process id
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APPLY TO MACROSCALE PROBLEM
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APPLY TO MACROSCALE PROBLEM
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IDEA OF DATA BASE APPROACH
Even with adaptive meshing and parallel
computation, solving very large problem with a
meso scale model is not realistic.
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DETAILS OF THE DATA BASE APPROACH

• Macro scale Model the evolution of T
  (temperature), fl (liquid volume fraction), and
  two state variables s1 (grain size), s2 (grain
  type).

• Model the unknowns (fl, s1 and s2) with the
  following form.

R,G are the cooling rate and temperature
gradient, defined at each point with value equal
to the temperature rate and gradient when fl
reaches 0.
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WHY LINK THE TWO SCALES IN THIS WAY

• Volume fraction is mainly related with
  temperature and microstructure type.
• Studies (experimental and numerical) have shown
  that microstructure type is determined by cooling
  rate and temperature gradient for a given alloy.
• All variables (T, fl, R, G, s1, s2) have direct
  physical meaning, so data can be easily obtained
  from meso scale simulation results. No
  optimization is required to fit parameters.
• In our numerical example, the following
  simplification is used.

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LINK BETWEEN TWO SCALES
Macro scale
Linking hypothesis Relations between macroscopic
variables follow the same relations between the
averaged microscopic variables in meso scale e.g.
Meso scale

• Macro scale and meso scale are decoupled.
• Accuracy of this approach relies on the accuracy
  of data obtained from meso scale simulation and
  the way for modeling and analyzing the data.

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PROBLEM CONSIDERED
Domain 10cm by 10cm Alloy
Al - 0.3Cu Superheat 30K Nucli density
106/m3 Cooling rate 0.5K/s
Want to know the solidification time, and the
final microstructure.
Use symmetry, around 2 million elements are
required to solve microstructure evolution using
the meso scale, which takes 24 hours.
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DATA GENERATION
mean of interception length is a function of
angle s1 (grain size) mean of interception
length for all angles s2 (structure type) ratio
of smallest mean interception length to largest
mean interception length
Heyns interception measure
Adiabatic
For each point, extract information of (R, s1,
s2) For each point every 100 time step, extract
information of (T, fl). Remember from which run
case and which position the data is generated. It
will be useful to give an idea what (s1, s2)
means.
Cooling rate 0.4K/s, 0.6K/s
2 hours for each case
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OBTAINED MICROSTRUCTURE INFORMATION
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COMPARE WITH MESO SCALE SIMULATION
Volume fraction
Predicted structure at P Q
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COMPARE WITH STATISTICAL RESULT
Meso scale model
Data base approach
Grain size (s1)
Microstructure type (s2)
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COMPARE WITH MACRO SCALE MODEL

• Solidification time

Lever Rule
420s
420s
Scheil Rule
423s
Data base approach
Both macro modeling (with Lever rule or Scheil
rule) and the data base approach gives
approximate total solidification time. The true
solution (meso-scale simulation result) is about
428s.

• Liquid fraction

Lever Rule
Database
Volume fraction
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COMPUTATION REQUIREMENT
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THANK YOU FOR YOUR ATTENTION