Materials Process Design and Control Laboratory MULTISCALE MODELING OF ALLOY SOLIDIFICATION...

Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory

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




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)




OUTLINE OF THE PRESENTATION

Brief review of alloy solidification process

Macro-scale Model

Meso-scale Model

Ongoing and future work




Review of alloy solidification process




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)




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




Macro-scale solidification modeling




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.




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




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




Meso-scale solidification modeling




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.




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!




Cellular automata

Phase field method

Front tracking

Level set method

Techniques for handling moving interface




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.




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.




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




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.




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




(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




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?




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.




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




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.




-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




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)




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)




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




. . ,

. (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




MESH ANISOTROPY STUDY

* 0 1 15 cos(4 )T d * 0 1 15 cos(4( ))4

T d

Rotated surface tensionNormal surface tension




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




Using a larger domain, perturbations/ second arm dendrites will be developed.

0.55T 0.80T

FORMATION OF SECONDARY DENDRITE ARMS




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




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




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.




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.




Materials Process Design and Control Laboratory

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.




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




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




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




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




3D CRYSTAL GROWH (Ni-Cu Alloy)

3 million elements (without adaptive meshing 200 million elements)




UnfinishedColored by process id

DOMAIN DECOMPOSTION




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.




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!




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




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.




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




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 (?)




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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