MULTISCALE MODELING OF SOLIDIFICATION OF MULTICOMPONENT ALLOYS

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MULTISCALE MODELING OF SOLIDIFICATION OF
MULTICOMPONENT ALLOYS
LIJIAN TAN Presentation for Thesis Defense
(B-exam) Date 22 May 2007 Sibley School of
Mechanical and Aerospace Engineering Cornell
University
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ACKNOWLEDGEMENTS

• SPECIAL COMMITTEE
• Prof. Nicholas Zabaras, M A.E., Cornell
  University
• Prof. Subrata Mukherjee, T A.M., Cornell
  University
• Prof. Stephen Vavasis, C.S., Cornell University
• Prof. Doug James, C.S., Cornell University

• FUNDING SOURCES
• National Aeronautics and Space Administration
  (NASA), Department
• of Energy (DoE)
• Sibley School of Mechanical Aerospace
  Engineering
• Cornell Theory Center (CTC)

Materials Process Design and Control Laboratory
(MPDC)
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OUTLINE OF THE PRESENTATION

• Introduction alloy solidification processes.
• Micro-scale mathematical model
• Applications
• Interaction between multiple dendrites during
  solidification
• Multi-scale modeling of solidification
• Suggestions for future study

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Introduction and objectives of the current
research
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Introduction
Castings since 5500 BC
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Microstructure
Will it break?
Different microstructures
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Alloy solidification process
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Micro-scale mathematical model
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Mathematical model
Two main difficulties

• Applying boundary conditions on interface for
  heat transfer, fluid flow and solute transport.
• Multiple moving interfaces (multiple
  phases/crystals).

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Complexity of the moving interface

• Jump in temperature gradient governs interface
  motion

• Gibbs-Thomson relation

• No slip condition for flow

• Solute rejection flux

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Level Set Method
History Devised by Sethian and Osher (1988) as
a mathematical tool for computing interface
propagation.
Level set variable is simply distance to interface
We pay additional storage and extra computation
time to maintain the above signed distance by
solving
Advantage is that we get extra information
(distance to interface). This information helps
to compute interfacial geometric quantities,
define a novel model, doing adaptive meshing, and
etc.
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Present Model
Assumption 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.
This assumption allows us to use the volume
averaging technique.
(N. Zabaras and D. Samanta, 2004)
Dont need to worry about boundary conditions of
flow and solute any more!
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Extended Stefan Condition
Gibbs-Thomson condition has to be satisfied (one
of the major difficulties)
Do not want to apply this directly, because any
scheme with essential boundary condition is
numerically not energy conserving. Introduce
another assumption
Assumption 2 The solid-liquid interface
temperature is allowed to vary from the
equilibrium temperature in a way governed by
Temperature boundary condition is automatically
satisfied. Energy is numerically conserving!
Unknown parameter kN. How will selection of kN
affect the numerical solution?
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Numerical Solution For A Simple Problem
Steady state
Initial
If L1, C1
T0 Ice
kN0.001
kN1
kN1000
Conclusion Large kN converges to classical
Stefan problem.
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Stability Analysis
In the simple case of fixed heat fluxes,
interface temperature approaches equilibrium
temperature exponentially.
Stability requirement for this simple case is
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Convergence Behavior

• Benchmark problem

With a grid of 64by64, we get
Results using finer mesh are compared with
results from literature in the next slide.
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Convergence Behavior
Benchmark problem Crystal growth with
initial perturbation.
Triggavason (1996)
Our method
Osher (1997)
Different results obtained by researchers
suggest that this problem is nontrivial.
All the referred results are using sharp
interface model.
Energy conserving makes the difference!
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Computation Requirement
Tracking interface makes the difference!
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Mesh Anisotropy Study
Normal surface tension
Rotated surface tension
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Mesh Anisotropy Study
Crystal shape mainly determined by the
anisotropy in surface tension not the initial
perturbation.
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Applications

• Pure material
• Crystal growth with convection
• Binary alloy
• Multi-component alloy

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Effects of Undercooling
(1) A small change in under-cooling will
lead to a drastic change of tip velocity.
(consistent with the solvability theory) (2)
Increased undercooling leads to sharper dendrite
shape and more obvious secondary dendrites.
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Extension to three dimension crystal growth

• Applicable to low under-cooling (at previously
  unreachable range using phase field method, Ref.
  Karma 2000) with a moderate grid.

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Crystal Growth with Convection
Velocity of inlet flow at top 0.035
Pr23.1 Other Conditions are the same as the
previous 2d diffusion benchmark problem.
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Crystal Growth with Convection in 3D

• Similar to the 2D case, crystal tips will tilt
  in the upstream direction.
• Distribute work and storage. (12 processors are
  used in the below example)

Thermal boundary layer
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Alloy solidification
For alloys, uniform mesh doesnt work very well
due to the huge difference between thermal
boundary layer and solute boundary layer.

• Difference between thermal boundary layer and
  solute boundary layer

• Tree type data structure for mesh refinement

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Simple Adaptive Mesh Test Problem
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Results Using Adaptive Meshing
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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Effects Of Refinement Criterion
Interface position (curved interface) is the
solved variable in this problem. Carefully
choosing the refinement criterion leads to the
same solution using a full grid.
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Crystal Growth of Alloy in 3D
Ni-Cu alloy Copper concentration 0.40831
at.frac. Domain a cube with side length
35mm Difficulties in this problem High
under-cooling 226 K High solidification
speed High Lewis number 14,860
Simulation of crystal growth of alloy in 3D is
computationally very intensive. Our solution is
to use both techniques of domain decomposition
and adaptive meshing!
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Adaptive Domain Decomposition (Mesh Partition)
Mesh
Dual graph
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Technique Issues about Mesh Partition

• Efficient
• Require mesh partition very frequently
  (adaptive). Slow is unacceptable.
• Maintain neighboring information using link list,
  e.g. for a node, there is a link list for its
  neighboring elements, and a link list for its
  neighboring edges.
• Still linear in storage greatly speed up the
  mesh partition procedure.
• Parallel
• Keep data distributed, work distributed. (Need to
  handle huge data)
• Defined a global address (process id pointer)
• Batch way
• (From To) Message Type Message Length
  Message content
• Put all messages in a link list, and send them
  out together

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Demonstration of adaptive domain decomposition
Colored by process id
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3D CRYSTAL GROWH (Ni-Cu Alloy)
3 million elements (without adaptive meshing 200
million elements)
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3D CRYSTAL GROWTH WITH CONVECTION
Comparing with the pure material case, the growth
for alloy is much more unstable due to the
rejection of solution.
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Multi-component alloy system
Multi-phase system one liquid phase one or
more than one solid phases.
We use a signed distance function for each phase.

• Relation between the signed distances
• Exactly one signed distance would be negative
• The smallest positive signed distance has same
  absolute value of the negative signed distance

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Compute Eutectic Growth with Multiple Level Sets

• Stable growth with 4 seeds
• Unstable growth with 2 seeds
• Unstable to stable growth with 10 seeds

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Compute Peritectic Growth with Multiple Level Sets
Solute concentration for peritectic growth of Fe
0.3wt C alloy at time 0.6s, 1.5s, 1.8s, and
2.4s.
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Interaction between multiple crystals
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Handle Multiple Interfaces
Method 1 A signed distance function for each
phase.
Method 2 Markers to identify different region
Each color (orientation of the crystal) is used
as a marker.
Efficient, appropriate for hundreds of crystals.
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Crystal orientation
Different crystal orientation leads to different
growth velocity.
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Extension of crystal orientation
As a feature of level set method, interface
velocity must be evaluated at nodes near
interface on both sides. Crystal orientation
needs to be extended a certain distance away from
the crystal to the liquid region.
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Simple numerical study
The purpose of this study is to verify the
accuracy of using markers.
Growth of 9 initial seeds (circular shape) with
different orientation.
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Comparison with method using multiple level sets
Dashed line method with multiple level set
functions. Solid line method with a single level
set function (using markers).
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Nucleation model
Crystals are not nucleated simultaneously. To
simulate nucleation, we use the following model

• Nucleation sites density ?, location of each
  nucleation site totally random (uniformly
  distributed in the domain).
• Orientation angle orientation angle of each
  nucleation site totally random (uniformly
  distributed between 0 and 2p).
• Each nucleation site becomes an actual seed iff
  the required undercooling is satisfied. The
  required undercooling is modeled to be a fixed
  value or as a random variable.
• We assume the nucleation sites fixed (do sampling
  first and then run the micro-scale model
  deterministically).

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Signed distance change due to nucleation
We update the signed distance function at each
node y, after a circular seed with radius R0 is
generated at location xi.
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CET transition study of Al-3Cu alloy
Follow conditions in Beckermann (2006). Relation
between microstructure and processing parameters
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Randomness effects
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Interaction between a large number of crystals
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Multi-scale modeling
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An example which requires multi-scale modeling
Material properties
4
0
3
0
2
0
Boundary conditions
1
0
Initial condition
0
0
1
0
2
0
3
0
4
0
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Computational results using adaptive domain
decomposition
Computation time 2 days with 8 nodes (16
CPUs). Cannot wait so long! Can we obtain
results in a faster way (multi-scale modeling)?
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What we can expect from multi-scale modeling

• Of course, we cannot expect microscopic details.
  But
• We want to know macroscopic temperature,
  macroscopic concentration, liquid volume fraction.

• Microstructure features are often of interest,
  e.g. 1st/2nd arm spacing, Heyns interception
  measure, etc. Let us denote these features as

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Widely accepted assumptions
Assumption 1 Without convection, macroscopic
temperature can be modeled as
Assumption 2 At a reasonably high solidification
speed and without fluid flow, macroscopic
concentration constant.
Assumption 3 Microstructure depends on
macroscopic cooling history and thermal
gradient history.
Assumption 4 Volume fraction only depends on
microstructure, and temperature.
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Macro-scale model
Temperature
Liquid volume fraction
Microstructure features
First two equations coupled. Microstructure
features determined as a post-processing process.
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Relevant sample problems
Infinite number of sample problems can be
selected. How to select the ones related to our
problem of interest is the key! Use a very
simple model to find relevant sample
problems. Model M (1)
treat material as pure material (sharp and stable
interface) (2) do not model
nucleation
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Comparison of three involved models
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Solution features of model M
Define solute features of model M to be the
interface velocity and thermal gradient in the
liquid at the time the interface passes through.
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Selection of sample problems
Given any solution feature of model M, we can
find a problem, such that features of model M
for this problem equals to the given solution
feature.
Sample problem
Chose a domain (rectangle is used) with initial
and boundary condition form the following
analytical solution
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Multi-scale framework
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Solve the previous problem
Material properties
4
0
3
0
2
0
Boundary conditions
1
0
Initial condition
0
0
1
0
2
0
3
0
4
0
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Step 1 Get solution features of model M
Plot solution features of model M for all nodes
in the feature spaces
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Step 2 Fully-resolved solutions of sample
problems
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Obtained liquid volume fraction
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Use iterations to obtain temperature, volume
fraction, microstructure features
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Temperature at time 130
Data-base approach result
Macro-scale model result with Lever rule
Fully-resolved model results with different
sampling of nucleation sites.
Average
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Liquid volume fraction at time 130
Left temperature field and volume fraction
contours (0.95 and 0.05) Right volume fraction
contour on top of fully-resolved model interface
position
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Predicted microstructure features
Results in rectangle predicted microstructure
Results in the middle fully-resolved model
results Black solid line predicted CET
transition location
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Solidification of Al-Cu alloy
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Step 1 Solution features of model M
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Step 2 Fully-resolved solution of sample problems
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Periodic boundary condition for the sample problem
Top half results copied from below Bottom half
Computational domain Periodic boundary condition
to minimize effects of boundary on directional
solidification solution
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Lquid volume fraction for different
microstructure features
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Iterative process for convergence
Left half (black points) results after iter 0.
Right half (green points) results after iter
3.
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Comparison with Lever rule (temperature at
t12.7s)
Left Lever rule Right Database approach
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Microstructure in the domain
E
D
A
C
A
B
F
B
H
H
G
F
G
C
E
D
A (95mm,75mm) B (90mm,75mm) C (75mm,75mm) D
(60mm,80mm)
E (90mm,10mm) F (80mm,20mm) G (65mm,35mm) H
(50mm,50mm)
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Microstructure from side to center
A
D
B
C
A
B
C
D
Fine columnar ? coarse columnar ? Equiaxed
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Microstructure from corner to center
E
F
H
G
G
F
E
H
Fine equiaxed ? Coarse equiaxed
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Suggestions for future research
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Consider flow effects in the multi-scale model
The computationally efficient model we used to
identify relevant sample problems (with its
analytical solution) is not applicable for
problems with convection effects. Extension of
the current technique or other techniques would
be necessary to efficiently consider convection
effects in a multi-scale framework.
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Modeling fluid structure interaction in
micro-scale
In our current micro-scale model, the crystal is
assumed static after nucleation. In reality, the
crystals would move, rotate, compact and break
into fragments. Recently, there are lots of
advances in fluid-structure interaction. These
advances can be used to improve the micro-scale
model.
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Atomic scale computation
Our current micro-scale model relies on input
from phase-diagram and a few parameters to mimic
the crystal orientation anisotropy, surface
tension effects, kinetic under-cooling effects
and nucleation. Computation in the atomic scale
(not continuum any more) and related multi-scale
techniques to use atomic scale computation
results are of great significance.
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Solid-Solid phase transformation
In our current model, only liquid to solid phase
transformation is considered. After this phase
transformation, solid-solid transformation is
also very crucial to the final microstructure. M
odeling solid-solid phase transformation after
solidification and study of the properties of the
final microstructure is an open area.
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THANK YOU FOR YOUR ATTENTION