materials process design and control laboratory multiscale modeling of alloy solidification...
TRANSCRIPT
![Page 1: Materials Process Design and Control Laboratory MULTISCALE MODELING OF ALLOY SOLIDIFICATION PROCESSES LIJIAN TAN Presentation for Admission to candidacy](https://reader036.vdocuments.us/reader036/viewer/2022081516/56649dea5503460f94ae51b5/html5/thumbnails/1.jpg)
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
MULTISCALE MODELING OF ALLOY SOLIDIFICATION PROCESSES
LIJIAN TANPresentation for Admission to candidacy
examinationDate: 11 May 2006
Sibley School of Mechanical and Aerospace Engineering
Cornell University
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Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
FUNDING SOURCES: National Aeronautics and Space Administration (NASA), Department
of Energy (DoE), Aluminum Corporation of America (ALCOA) Cornell Theory Center (CTC) Sibley School of Mechanical & Aerospace Engineering
Materials Process Design and Control Laboratory (MPDC)
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Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
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OUTLINE OF THE PRESENTATION
Brief review of alloy solidification process
Macro-scale Model
Meso-scale Model
Ongoing and future work
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Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Review of alloy solidification process
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Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Piping Micro shrinkage
Macroshrinkage
Shrinkage porosity (Ref. EPFL, Switzerland)
DEFECTS DURING ALLOY SOLIDIFICATIONSub-surface liquation and crack formation
(Ref. ALCOA corp.)
Non-uniform growth and microstructure(Ref. ALCOA corp.)
Freckle formation(Ref. Beckermann)
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Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
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MULTISCALE NATURE OF ALLOY SOLIDIFICATION
solid Mushy zone liquid ~10-1 - 100 m
(b) Meso/micro scale
~ 10-4 – 10-5m
solid
liquid(a) Macro scale (mushy zone is represented by volume fraction)
q g
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Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Macro-scale solidification modeling
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Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
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CCOORRNNEELLLL U N I V E R S I T Y
2
0
2
2
( ) 0,
( )( ) [ ( ( ) ( ))]
,
( ) ,
( ) (
(1 )
)
Tl l l
lgl
l s
l l l lii i i
ll l
ll
l
l
l
t
pp
t
gK
Tc c T k T L
tC
C D Ct
v
v vvv I v v
ve
v
v
VOLUME AVERAGING MODEL
Governing equations
Legacy code developed by Deep Samanta (former MPDC graduate student) to solve this system based on stabilized FEM method.
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Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
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PREVIOUS WORK
L. Tan and N. Zabaras, "A thermomechanical study of the effects of mold topography on the solidification of Aluminum alloys", Materials Science and Engineering: A, Vol. 404, pp. 197-207, 2005
D. Samanta and N. Zabaras, "Macrosegregation in the solidification of Aluminum Alloys on uneven surfaces", International Journal of Heat and Mass Transfer, Vol. 48, pp. 4541-4556, 2005
D. Samanta and N. Zabaras, "Modeling melt convection in solidification processes with stabilized finite element techniques", International Journal for Numerical Methods in Engineering, Vol. 64, pp. 1769-1799, 2005
B. Ganapathysubramanian and N. Zabaras, "On the control of solidification of conducting materials using magnetic fields and magnetic field gradients", International Journal of Heat and Mass Transfer, Vol. 48, pp. 4174-4189, 2005
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. 1103-1138, 2004
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Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
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Lever Rule : (Infinite back-diffusion)
Scheil Rule :(Zero back-diffusion)
• These relationships are based on phase diagrams and certain assumption of the diffusion in solid phase (infinite back-diffusion or zero back-diffusion).
• These relationships only give the upper bound and the lower bound.According to our experience, numerical results using Scheil rule is closer to experimental results. But neither of them is accurate.
• We can only turn to meso-scale modeling for better accuracy.
11
1liq
lp m
T Tf
k T T
æ ö- ÷ç ÷= - ç ÷ç ÷ç- -è ø
1
1pkm
lliq m
T Tf
T T
-æ ö- ÷ç ÷ç= ÷ç ÷÷-çè ø
COMPUTE VOLUME FRACTION
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Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
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CCOORRNNEELLLL U N I V E R S I T Y
Meso-scale solidification modeling
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Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
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CCOORRNNEELLLL U N I V E R S I T Y
GOVERNING EQUATIONS
2
2
2
( , ) 0, ,
( , )( , ) ( , ) ( , ) ( , ) , ,
( , )( , ), ,
( , )( , ) ( , ), ,
( , )( , )
l
l l
s s s s
l l l l
ll lii i
t x
tt t p t t b
t
T tc k T t
tT t
c T t k T tt
C tC t D C
t
v x
v xv x v x I v x v x x
xx x
xv x x x
xv x ( , ), , 2,3,... .l l
i t i n x x
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
( ) /( ), s l s slV q q L on
Jump in temperature gradient governs interface motion
No slip condition for flow
0, slon v
Gibbs-Thomson relation
( ) ( ) , l slI m c VT T mC V on n n
Solute rejection flux
(1 ) , l
l i l slii p i
CD k C V on
nn
Requires curvature computation at the moving interface!
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Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
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Cellular automata
Phase field method
Front tracking
Level set method
Techniques for handling moving interface
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Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
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CELLULAR AUTOMATA METHOD
Numerically, phase transformation is just change of 0 to 1. The interface phase growth is model as a probability of change 0 to 1.
Ref. Kremeyer (1998)
Use 0/1 to represent liquid/solid for each pixel.
Advantages
CA method might be the most widely used method till now, mainly because it is very easy to implement.
Disadvantage
However, the model is so different from the original governing equations. Even for some simple problems, it deviates from the analytical solution significantly.
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Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
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2 2 21(1 )
2
W
M t
Review paper by Botteringer, Warren, Beckermann, Karma (2002)
Ref Langer (1978)
Advantages
Still easy to implement, also widely used.
No essential boundary conditions (global energy conserving)
Disadvantages
A number of parameters in the phase field equation of little or no physical meaning. To determine their values is not a trivial task.
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. (Due to the large gradient of phase field variable within the diffused interface.)
PHASE FIELD METHOD
This method diffuses the sharp interface into one with a certain width and
uses a variable varying from 0 to 1 to approximately represent interface.
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FRONT TRACKING METHODRef. Tryggvason (1996), J. Heinrich (2001)
Advantages
Solving sharp interface model directly (physics clear in governing eqs.)
Disadvantages
BC scheme not strictly energy conserving
Curvature computation (from mark pts) complicated
Difficult to implement
Ideas:(1) Uses markers to represent interface(2) Markers are moved using velocity computed from Stefan equation
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LEVEL SET METHOD
Interface motion (level set equation)
Ref. S. Osher 1997,
| | 0t V
( , )
( , ) 0
( , )
d x t x
x t x
d x t x
Devised by Sethian&Osher
Advantages Interface geometries can be easily and accurately computed.
Level set equation well studied (FDM with higher order accuracy, FEM)
Disadvantage: Application of boundary conditions still not easy (most applications are restricted to pure materials without melt convection).
Signed distance
Introduced to this area by J. Dantzig 2000, R. Fedkiw 2003 etc.
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OUR WORK WITH LEVEL SET METHOD
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
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(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
PRESENT MODEL BASED ON LEVEL SET METHOD Assumptions
0, ( , )
( , ) 1, ( , )
0.5 (2 ), ( , ) [ , ]
x t w
x t x t w
w x t w w
*( )sl
sl slIN I
dTk T T
dt Instead of forcing
*sl sl
IT T
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CCOORRNNEELLLL U N I V E R S I T Y
EXTENDED STEFAN CONDITION
*( )sl
sl slIN I
dTk T T
dt
*( )s l
sl sss Is
lsN
wq qV
L
ck T T
L
Without applying essential boundary condition on the moving interface, the numerical scheme satisfies energy conservation. This leads to both accuracy and convenience.
Only one parameter kN needs to be specified. How will the selection of kN affect the numerical solution?
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CCOORRNNEELLLL U N I V E R S I T Y
NUMERICAL STUDY FOR A SIMPLE CASEPick up a very simple system
Initially with left half ice and right half water; whole domain is under-cooled at temperature -0.5.At steady state: Whole domain is ice with temperature 0.
Effect of kN
X
T
-100 -50 0 50 100-0.5
-0.45
-0.4
-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
t=1080
t=3.9
t=197
t=21547
t=4273
t=8122
t=12327
t=47
X
T
-100 -50 0 50 100-0.5
-0.45
-0.4
-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
t=1072
t=3.5
t=197
t=21250
t=4204
t=8018
t=12178
t=48
X
T
-100 -50 0 50 100-0.5
-0.45
-0.4
-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
t=1062
t=22981
t=4362
t=8320
t=12365
t=305
kN=0.001 kN=1 kN=1000Temperature distribution in the domain at various time.
Conclusion: Large kN converges to classical Stefan problem.
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STABILITY ANALYSIS
*( )sl
sl slIN I
dTk T T
dt
In the simple case of fixed heat fluxes, interface temperature approaches equilibrium temperature exponentially.
Stability requirement for this simple case is
2tkN
Although this is only for a very simple case, we find that selection of
is stable for most problems of interest.
tkN 1
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NUMERICAL CONVERGENCE STUDY
Infinite corner problem (2D with analytical solution)
0
0.25, 0.3
1 inL T
T applied at two boundary sides
After the mesh is refined to 20by20, the error reduces almost quadratically.
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CCOORRNNEELLLL U N I V E R S I T Y
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Initial crystal shape (0.1 0.02cos 4 )cos
(0.1 0.02cos 4 )sin
x
y
Domain size [ 2, 2] [ 2,2]
Initial temperature ( ,0) 0
( ,0) 0.5 s
T x x
T x x
Boundary conditions adiabatic
With a grid of 64by64, we get
: 0.002
: 0.002
Surface tension
Kinetic undercooling coeff
Results using finer mesh are compared with results from literature in the next slide.
Benchmark problem
COVERGENCE BEHAVIOR OF VARIOUS METHODS
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CCOORRNNEELLLL U N I V E R S I T Y
COVERGENCE BEHAVIOR OF VARIOUS METHODS
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Our method Osher (1997)
Top 400 400
Middle 200 200
Bottom 100 100
Different results obtained by researchers suggest that this problem is nontrivial.
All the referred results are using sharp interface model.
Triggavason (1996)
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Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
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CCOORRNNEELLLL U N I V E R S I T Y
APPLICATIONS
Application: Modeling dendritic solidification (pure materials & binary alloy)
2D crystal growth benchmark problem (pure material) 3D crystal growth benchmark problem (pure material) Effects of fluid flow on crystal growth (pure material)
Adaptive mesh technique (required for alloys) 2D crystal growth benchmark problem (alloy) 3D crystal growth benchmark problem (alloy) Effects of fluid flow on crystal growth (alloy)
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2D CRYSTAL GROWTH BENCHMARK PROBLEM
(1) A small change in under-cooling will lead to a drastic change of tip velocity. (consistent with the solvability theory)
(2) An increase of diffusion coefficient in the liquid region tends to make the tip sharper. Effects of solid diffusion coefficient are not obvious.
* 0 0
30 0.55 / 0.65
: 400 400, 1 15 cos(4 ) , 0.5, 0.05, 1
Crystal with initial radius growing in domain with initial undercooling
domain T d d other parameters normalized to
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. . ,
. (2002)
Y T Kim
N Goldenfield
& (1998)Karma Rapel
0 100 200 300 4000
50
100
150
200
250
300
350
400
~ 3000hours on DEC Alpha
:~ 20Mesh element size
CPUT 1 2~ minute on a GHz PC
Our diffused interface model with tracking of interface
Phase field model without tracking of interface
:1Mesh element size
:~ 270node no :~ 160000node no
COMPARISON WITH PHASE FIELD METHOD
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
MESH ANISOTROPY STUDY
* 0 1 15 cos(4 )T d * 0 1 15 cos(4( ))4
T d
Rotated surface tensionNormal surface tension
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Top results from Heinrich (2003)
Bottom results from our method
483 2
:
(0.1 0.02cos 4 )cos
(0.1 0.02cos 4 )sin
:
0.001{1 0.4[ sin 3( ) 1]}
: 0.8
Initial shape
x
y
Surface tension
Undercooling
4 6 fold initial crystal grow with fold Surface tension
400 400Mesh 800 800Mesh
Crystal growth mainly determined by surface tension not initial perturbation.
MESH ANISOTROPY STUDY
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Using a larger domain, perturbations/ second arm dendrites will be developed.
0.55T 0.80T
FORMATION OF SECONDARY DENDRITE ARMS
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EXTENSION TO THREE DIMENSION CRYSTAL GROWTH
Applicable to 3d with high under-cooling using a coarse mesh. 4 4 4
* 1 2 3
3
0.5(1 0.05(4( ) 3)) , 0.55
[ 400,400] , 60 60 60
T n n n T
Domain size Mesh size
Temperature and crystal shape at time t=105
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Applicable to low under-cooling (at previously unreachable range using phase field method, Ref. Karma 2000) with a moderate grid.
4 4 4* 1 2 3
3
(1 3 )(1 4 ( ) /(1 3 )) , 0.025, 0.05
[ 20000,20000] , 120 120 120
T n n n T
Domain size Mesh size
CRYSTAL GROWTH AT LOW UNDERCOOLING
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CRYSTAL GROWTH WITH CONVECTION
Velocity of inlet flow at top: 0.035 Pr=23.1
Other Conditions are the same as the previous 2d diffusion benchmark problem.
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CCOORRNNEELLLL U N I V E R S I T Y
Materials Process Design and Control Laboratory
CRYSTAL GROWTH WITH CONVECTION
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)
For alloys, uniform mesh doesn’t work very well due to the huge difference between thermal boundary layer and solute boundary layer.
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Difference between thermal boundary layer and solute boundary layer
~ 100Lewis number LeD
ADAPTIVE MESHING
Tree type data structure for mesh refinement
Coarsen
Refine
Implemented for both 2D and 3D.
Coupled with domain decomposition.
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Initial crystal shape (0.1 0.02cos 4 )cos
(0.1 0.02cos 4 )sin
x
y
Domain size [ 10,10] [ 10,10]
Initial temperature ( ,0) 0
( ,0) 0.5 s
T x x
T x x
Boundary conditions no heat/solute flux
Initial concentration ( ,0) 2.2
( ,0) 2.2 sp
C x x
C x k x
: 0.035
: 0.312
: 0.1
: 0
Liquidus slop
Partition coefficient
Liquid mass diffusivity
Solid mass diffusivity
Surface ten
: 0.002
: 0.002
: 1
: 0.002
sion
Kinematic undercooling coeff
Thermal conductivity
Latent heat
-10 -5 0 5 10-10
-5
0
5
10
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
SIMPLE ADAPTIVE MESH TEST PROBLEM
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Le=10 (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)
( )
Adaptive mesh for solute concentration
Color of mesh represents concentration
RESULTS USING ADAPTIVE MESHING
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EFFECTS OF REFINEMENT CRITERION
( )
( )
(| |)
e
e
T
C
e
Error e T d tol
Error e C d tol
h element size upper bound
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.
: 256 256Full mesh
Element size invisible
no variations seen in most of the elements here
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3D CRYSTAL GROWH (Ni-Cu Alloy)
Ni-Cu alloy Copper concentration 0.40831 at.frac.Domain: a cube with side length 35m
Difficulties in this problemHigh under-cooling: 226 KHigh solidification speedHigh Lewis number: 14,860
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3D CRYSTAL GROWH (Ni-Cu Alloy)
3 million elements (without adaptive meshing 200 million elements)
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UnfinishedColored by process id
DOMAIN DECOMPOSTION
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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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CONCLUSION
Successfully applied level set method to single crystal growth (pure material and alloys).
This new method is accurate and efficient.
Our method is a very promising method for this purpose. But can we do multi-scale modeling with this method? Multi-scale modeling is my ongoing and planned research work.
However, we need to compute thousands of dendrites in practice!
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Adaptive meshing is one option for multi-scale modeling of solidification.
Estimation of computational effort:
: 0.1
: 10
~ 100 (2 ),
1000 (3 )
Length Scale m
x m
DOF Million d
Billion d
4
:
~ 2 10
CFLCFL V t x
steps
4 million elements if a full mesh is used
2D promising3D impossible
Necessary for validation purpose!
ADAPTIVE MESHING
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DATA BASE APPROACHA number of runs in the meso-scale
volume fraction and macroscopic variables
Data base
For the interested problem, use the same macro scale model, but with volume fraction from interpolation (not Scheil/Lever rule).
Been used in the solid group, Terada and Kikuchi (1995), Lee and Ghosh (1999).
Advantages:(1) Very fast.(2) No need to do new developments in coding (non-intrusive).
Disadvantage:May only be applicable to certain problems.
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SUBGRID MODEL
Meso-scale grid for each element
Macro-scale grid for casting object
Simple average based on volume fraction
Homogenization(shape, orientation)
Boundary condition for temperature, solute, and fluid flow
Time step in meso-scale
CA Level set
Nucleation
Binary alloy Multi-component
Widely used, but CA method is the underlying micro-scale model due to the huge computational requirement for other methods.
Moving of nuclei under convection
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COMPARISONS
Strategy Storage Computational work
Accuracy
Adaptive meshing
One huge matrix Very huge Very good
Sub grid model
A number of small matrices
Moderate Good
Data base approach
Just one small matrix
Very little (?)
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COMPUTATIONAL ISSUES
• Most alloy used in industry is multi-component and usually with multiple solid phases. Limited multi-component capability has been developed using multiphase level set method. But triple points and computational efficiency are still unsolved issues.
• Nucleation needs to implemented in the meso scale model.
• Mechanical properties prediction is the final goal.
X xx = FXx = FX
y = FY + w
N
n
0
10
20
30
40
50
60
0.000 0.010 0.020 0.030 0.040 0.050 0.060
Equivalent plastic strain
Equi
vale
nt s
tress
(MPa
)
Simple shear
Plane strain compression
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CCOORRNNEELLLL U N I V E R S I T Y
THANK YOU FOR YOURATTENTION