materials process design and control laboratory multiscale modeling of solidification of...

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 SOLIDIFICATION OF MULTICOMPONENT ALLOYS

LIJIAN TANPresentation for Thesis Defense (B-exam)

Date: 22 May 2007

Sibley School of Mechanical and Aerospace EngineeringCornell 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 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)



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



Introduction and objectives of the current research



Introduction

Castings since 5500 BC…



Will it break?

Different microstructures

Microstructure



Alloy solidification process

solidMushy zone liquid ~10-1 - 100 m

(b) Microscopic scale

~ 10-4 – 10-5m

solid

liquid(a) Macroscopic scale

qos g



Micro-scale mathematical model



2

2

( , ) 0, ,( , ) ( , ) ( , )

( , ) ( , ) , ,

( , ) ( , ), ,

( , ) ( , ) ( , ), ,

( , )

l

l

l

s s s s

l l l l

li

t xt t t p

t

t t b

T tc k T tt

T tc T t k T tt

C tt

v xv x v x v x I

v x v x x

x x x

x v x x x

x v 2( , ) ( , ), , 2,3,... .l l l li i iC t D C t i n x x x

Two main difficulties

Mathematical model

Applying boundary conditions on interface for heat transfer, fluid flow and solute transport.

Multiple moving interfaces (multiple phases/crystals).



( ) /( ), s l s slV q q L on

Jump in temperature gradient governs interface motion

Gibbs-Thomson relation

0, slon v No slip condition for flow

* ( ) ( ) , l slI m c VT T T mC V on n n

Solute rejection flux

(1 ) , l

l i l slii p i

CD k C V on

n

n

Complexity of the moving interface



| |, n n

History: Devised by Sethian and Osher (1988) as a mathematical tool for computing interface propagation.

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.

( , ) ( , ) 0

( , )

d x t xx t x

d x t x

Level Set Method

| | 0t V

We pay additional storage and extra computation time to maintain the above signed distance by solving

Level set variable is simply distance to interface



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.

0, ( , )

( , ) 1, ( , )

0.5 (2 ), ( , ) [ , ]

x t w

x t x t w

w x t w w

2 2

20

( ) 0,

( ) (1 )( ) [ ( ( ) ( ))] ,

( ) ,

( ) ( )

ll T l l

gl l l l l l l

l s l

l l l l lii i i

tpp g

t KTc c T k T Lt

CC D C

t

v

v vv vv I v v e

v

v

This assumption allows us to use the volume averaging technique.

(N. Zabaras and D. Samanta, 2004)

Present Model

Don’t need to worry about boundary conditions of flow and solute any more!



*( )IN I

dT k T Tdt

*( )s l

sN Iss s

wckq q TL

TL

V

%

Unknown parameter kN. How will selection of kN affect the numerical solution?

Assumption 2: The solid-liquid interface temperature is allowed to vary from the equilibrium temperature in a way governed by

Gibbs-Thomson condition has to be satisfied (one of the major difficulties)

*IT T

Extended Stefan Condition

Do not want to apply this directly, because any scheme with essential boundary condition is numerically not energy conserving. Introduce another assumption:

Temperature boundary condition is automatically satisfied. Energy is numerically conserving!



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=1000

Conclusion: Large kN converges to classical Stefan problem.

T=-0.5 Ice T=-0.5 Water T=0 Ice

Initial Steady stateNumerical Solution For A Simple Problem

If L=1, C=1



*( )IN I

dT k T Tdt

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 all problems we have considered.

tkN 1

Stability Analysis



-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

xy

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 tensionKinetic undercooling coeff

Results using finer mesh are compared with results from literature in the next slide.

Benchmark problemConvergence Behavior



-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 400Middle 200 200Bottom 100 100

Different results obtained by researchers suggest that this problem is nontrivial.

All the referred results are using sharp interface model.

Triggavason (1996)

Benchmark problem: Crystal growth with initial perturbation.

Convergence Behavior

Energy conserving makes the difference!



* 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 undercoolingdomain T d d other parameters normalized to

. . , . (2002)

Y T KimN 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

Computation Requirement

Tracking interface makes the difference!



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

T d

Rotated surface tensionNormal surface tension

Mesh Anisotropy Study



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 shapexySurface tension

Undercooling

4 6 fold initial crystal grow with fold Surface tension

Crystal shape mainly determined by the anisotropy in surface tension not the initial perturbation.

Mesh Anisotropy Study



Applications

1. Pure material 2. Crystal growth with convection3. Binary alloy4. Multi-component alloy



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.

* 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 undercoolingdomain T d d other parameters normalized to



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

Extension to three dimension crystal growth



Velocity of inlet flow at top: 0.035 Pr=23.1

Other Conditions are the same as the previous 2d diffusion benchmark problem.

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)

Crystal Growth with Convection in 3D

Thermal boundary layer



Difference between thermal boundary layer and solute boundary layer ~ 100Lewis number Le

D

Alloy solidification

Tree type data structure for mesh refinement

Coarsen

Refine

For alloys, uniform mesh doesn’t work very well due to the huge difference between thermal boundary layer and solute boundary layer.



Initial crystal shape (0.1 0.02cos 4 )cos(0.1 0.02cos 4 )sin

xy

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 slopPartition coefficientLiquid mass diffusivitySolid mass diffusivitySurface ten

: 0.002 : 0.002

: 1 : 0.002

sionKinematic undercooling coeffThermal conductivityLatent 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 concentrationColor 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



Crystal Growth of Alloy in 3D

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

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!



Adaptive Domain Decomposition (Mesh Partition)

12

345

12

345

1

2

3

45

1

2

3

45

Mesh

Dual graph



Technique Issues about Mesh PartitionEfficient: 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



Colored by process id

Demonstration of adaptive domain decomposition



3D CRYSTAL GROWH (Ni-Cu Alloy)

3 million elements (without adaptive meshing 200 million elements)



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.



Multi-component alloy system

We use a signed distance function for each phase.

( , ) ( , ) 0

( , )

d x t xx t x

d x t x

: 0 , l lAt P

Multi-phase system: one liquid phase + one or more than one solid phases.

Relation between the signed distances:

(1) Exactly one signed distance would be negative

(2) The smallest positive signed distance has same absolute value of the negative signed distance

l

P

( )l



Stable growth with 4 seeds

Unstable growth with 2 seeds

Unstable to stable growth with 10 seeds

Compute Eutectic 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.

Compute Peritectic Growth with Multiple Level Sets



Interaction between multiple crystals



( , )

( , ) 0

( , )

d x t x

x t x

d x t x

l

P

( )l

Method 2: Markers to identify different region

Method 1: A signed distance function for each phase.

Each color (orientation of the crystal) is used as a marker.

Efficient, appropriate for hundreds of crystals.

Handle Multiple Interfaces



Different crystal orientation leads to different growth velocity.

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.

Extension of crystal orientation



The purpose of this study is to verify the accuracy of using markers.Simple numerical study

Growth of 9 initial seeds (circular shape) with different orientation.



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).



Nucleation modelCrystals are not nucleated simultaneously.

To simulate nucleation, we use the following model:

Nucleation sites: density ρ, 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 2π).

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).



Signed distance change due to nucleationWe update the signed distance function at each node y, after a circular seed with radius R0 is generated at location xi.



CET transition study of Al-3%Cu alloyFollow conditions in Beckermann (2006).

Relation between microstructure and processing parameters:



Randomness effects



Interaction between a large number of crystals



Multi-scale modeling



An example which requires multi-scale modeling

2

0

2* 3

: 100

( ) ~ (1.5,0.2 )1, 100

0, 0.1, 10, 0.1

1, 100{1 cos[4( )]}

n

n

m l p

m l l

Potential nucleation sites density

p Nc L

C m k

k Lem C I V

Material properties:

Boundary conditions:

Initial condition:

: : 50exp( /10) 40b

Right side AdiabaticOther sides T t

10T 0 10 20 30 400

10

20

30

40

CFL

Adaptive meshing with smallest 0.0098Adaptive time stepping with 0.125

x

~ 25 Millon elements~ 33,000 time steps



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)?



What we can expect from multi-scale modeling

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

( , ) ( , )

( , ) ( , )( , ) ( 0 | , )

T x t x t

C x t C x tf x t p x t

( ) ( , , , )x x C

Of course, we cannot expect microscopic details. But

We want to know macroscopic temperature, macroscopic concentration, liquid volume fraction.



Widely accepted assumptions

2Tc k T Lft

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.

0C C

Assumption 4: Volume fraction only depends on microstructure, and temperature.

,f f T

Assumption 3: Microstructure depends on macroscopic cooling history and thermal gradient history.

( , ) R G ,T Tt

R G



Macro-scale model

Temperature

Liquid volume fraction

Microstructure features

2Tc k T Lft

,f f T

( , ) R G

Unknown functions:

( , ) R G

First two equations coupled.

Microstructure features determined as a post-processing process.

,f f T

,T Tt

R G

Solve sample problems using the fully-resolved model (micro-scale model) to evaluate them!



Relevant sample problemsInfinite 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



Comparison of three involved models



Solution features of model MDefine solute features of model M to be the interface velocity and thermal gradient in the liquid at the time the interface passes through.

, lM M MF V G



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.

Selection of sample problems , l

M M MF V G

/( ), /s lM M Mk c G G LV k

Chose a domain (rectangle is used) with initial and boundary condition form the following analytical solution:

( )1 exp , <

( )1 exp ,

sM M M

MM

lM M M

MM

G V x V t when x V tV

TG V x V t when x V t

V

Sample problem:



Multi-scale framework



Solve the previous problem

2

0

2* 3

: 100

( ) ~ (1.5,0.2 )1, 100

0, 0.1, 10, 0.1

1, 100{1 cos[4( )]}

n

n

m l p

m l l

Potential nucleation sites density

p Nc L

C m k

k Lem C I V

Material properties:

Boundary conditions:

Initial condition:

: : 50exp( /10) 40b

Right side AdiabaticOther sides T t

10T 0 10 20 30 400

10

20

30

40



Step 1: Get solution features of model M

MV

lMG

Plot solution features of model M for all nodes in the feature spaces

lMG

MV

10-2 10-1 100

0.02

0.04

0.06

0.08

0.10.12

lMG

MV

10-2 10-1 100

0.02

0.04

0.06

0.08

0.10.12



Step 2: Fully-resolved solutions of sample problems



Obtained liquid volume fraction



Use iterations to obtain temperature, volume fraction, microstructure features



Temperature at time 130

Macro-scale model result with Lever rule

Fully-resolved model results with different

sampling of nucleation sites.

Average

Data-base approach result



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



Predicted microstructure features

Results in rectangle: predicted microstructure Results in the middle: fully-resolved model resultsBlack solid line: predicted CET transition location



Solidification of Al-Cu alloy0q

650exp(- ) 320bT t

100mm100mm

( ,0) 970T x K

1%Al Cu

bT

bT

bT

Domain for sample problem (solve with fully-resolved model)

50 36 2 sd

120 2 sd

9.7sd m

2

3

2

0.14, 2.6 / .%, 3000 , 933.47 , 0.24 , 0.01,

2400 / , 1.06 /( ), 82.61 /( ), 397.5 / ,

7.5 , 1.25

p l l m c

n

k m K wt D m s T K K m

kg m c KJ kg K k W m K L KJ kg

T N K K



0 20 40 60 80 1000

20

40

60

80

100

0 20 40 60 80 1000

20

40

60

80

100 angle

4000380036003400320030002800260024002200200018001600140012001000

solidtime

0.0050.00450.0040.00350.0030.00250.0020.00150.0010.0005

MG MV

Step 1: Solution features of model M

0 0.002 0.004

1000

2000

3000

4000

0 0.002 0.004

1000

2000

3000

4000

1 2

3

45 6

7 8

910

11

MG

MV

MG

MV



0 0.002 0.004

1000

2000

3000

4000

Step 2: Fully-resolved solution of sample problems



Periodic boundary condition for the sample problem

Top half: results copied from belowBottom half: Computational domain

Periodic boundary condition to minimize effects of boundary on directional solidification solution



Temperature (K)

Liqu

idvo

lum

efra

ctio

n

915 920 925 9300

0.2

0.4

0.6

0.8

1 Sample 1Sample 2Sample 3Sample 4Sample 5Sample 6Sample 7Sample 8Lever rule

Lquid volume fraction for different microstructure features



Iterative process for convergence

Left half (black points):

results after iter 0.

Right half (green points):

results after iter 3.

angle

4000380036003400320030002800260024002200200018001600140012001000

solidtime

0.0050.00450.0040.00350.0030.00250.0020.00150.0010.0005

0 20 40 60 80 1000

20

40

60

80

100

0 20 40 60 80 1000

20

40

60

80

100

MG MV 0 0.002 0.004

1000

2000

3000

4000MV

MG



0 20 40 60 80 1000

20

40

60

80

100

945940935930925900850800750700650600550500450400350

Comparison with Lever rule (temperature at t=12.7s)

Left: Lever rule Right: Database approach



ABCD

A (95mm,75mm)B (90mm,75mm)C (75mm,75mm)D (60mm,80mm)

Microstructure in the domain

0 0.002 0.004

1000

2000

3000

4000

1 2

3

45 6

7 8

910

11

MG

MV

EF

G

H

E (90mm,10mm)F (80mm,20mm)G (65mm,35mm)H (50mm,50mm)

A

B

CD

E

F

G

H



ABCD

Fine columnar coarse columnar Equiaxed

Microstructure from side to center

A

B

C

D



Microstructure from corner to center

EF

G

H

Fine equiaxed Coarse equiaxed

E

F

G

H



Suggestions for future research



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.



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.



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.



Solid-Solid phase transformationIn 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.

Modeling solid-solid phase transformation after solidification and study of the properties of the final microstructure is an open area.



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