materials process design and control laboratory sibley school of mechanical and aerospace...

CCOORRNNEELLLL U N I V E R S I T Y


Materials Process Design and Control Laboratory

Materials Process Design and Control LaboratorySibley School of Mechanical and Aerospace Engineering

169 Frank H. T. Rhodes HallCornell University

Ithaca, NY 14853-3801

Email: [email protected] [email protected] URL: http://mpdc.mae.cornell.edu

Lijian Tan andLijian Tan and

Nicholas ZabarasNicholas Zabaras

Level set method for simulating Level set method for simulating multi-phase multi-component multi-phase multi-component

dendritic solidificationdendritic solidification



Outline


• Brief introduction

• Level set method & Mathematical model for multi-phase multi-

component solidification systems

• Numerical examples

• Conclusions and future work



Background


10 m



Phase field method


fM

t

22 2 2

surface energy bulk energy

| | (1 )2 8

Wf

2 2 21(1 )

2

W

M t

General kinematics equationApproximating the free energy using as

Major difficulty: Parameter identification

History: First developed by J. Langer (1978) as a computational technique to solve Stefan problems for pure materials

Ideas: (1) enthalpy method (2) Cahn-Allen equation

Phase field variable: (1) no direct physical meaning (2) can describe the real world when

Easy to implement (coding), major success in the last two decades



Front tracking method


Major difficulty: Difficult for 3D and multiphase

Ideas:(1) Uses markers to represent interface(2) Markers are moved using velocity computed from Stefan equation

Sharp interface model Uses directly thermodynamic data

Computationally difficult to implement



Level set method


| |,

( , )

( , ) 0

( , )

| | 0t

d x t x

x t x

d x t x

F

n n

History: Devised by Sethian and Osher (1988) as a simple and versatile method for computing and analyzing the motion of an interface in two or three dimensions.

Advantage: Interfacial geometric quantities can be easily calculated using signed distance.



Multiphase solidification system


We use a signed distance function for each phase.

( , )

( , ) 0

( , )

d x t x

x 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



Multiphase solidification system


We use a signed distance function for each phase.

( , )

( , ) 0

( , )

d x t x

x 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



Level set equation


Level set equation: | | 0t F

Stabilized Galerkin form:

1

| | 0| |

el

e

Ne

t ee

FF d

Semi-descretized form:

1 1

1 1

1

( ) 0

, | |

| | , | || |

el el

e

e

el el

e e

el

e

GLS GLS SC

N NT e T e e

GLSe e

N NT e T e e

GLSe e

NT e

SCe

M M f f f

M N Nd M N F Nd

f N F d f N F F d

f N d

43

6

10 for triple points10 ,

| | 10 otherwise

eee

e e

hh

F h



Reinitialize


0

2 20

(1 )t

Iterative method:

Fast matching method:

1

,1 ,1

,2 ,2

A A A A A A

B B B B B B

x y x y

x y x y

1 1 1

1 1 2

( )

1 ( ) 1

i ij j ij ij jj j j

ij ij ji j j

n M M M

M M



Reinitialize


l

P

( )l ( , )

( , ) 0

( , )

d x t x

x t x

d x t x

| | 0V

At interface V V

: 0 , l lAt P

1: Find , so that ,

2 : Compute , For all , 2

Step

Step err err



Computational techniques


Narrow band:Adaptive meshing:

1 The level set equation is solved on a narrow band. 2 Re-initialization, heat transfer and solute transport is performed in the whole domain using adaptive meshing based on the distance from the interface.

Dantzig et al. (1996)



Diffused interface


• For numerical convenience, we assume phase change occurs in a diffused zone of width 2w that is symmetric around phase boundary.

Diffused interface feature (Convenience of whole domain method)

1

( , ) 0

( , ) 0.5 [ , ]2

w

x t w

x t w ww

This diffused interface allows us to use whole domain method conveniently for heat transfer and fluid flow as shown in these two figures.

Consequently, a phase fraction can be defined as



Heat transfer & fluid flow


Heat transfer and fluid flow can be modeled using volume averaging

2

2

20

( )

( )( ) [ ( ( ) ( ))]

(1 )

, ,

l l

Tl

l l l l l

ll l g

l l

Tc c v T k T h

t

pp

t

gK

c c k k

v vvv v

ve

• For heat transfer: Temperature on the interface is not applied as an essential boundary condition to guarantee energy conservation of the numerical scheme (The Gibbs-Thomson relation is weakly forced by adjusting the growth velocity of phases)

• For fluid flow: The diffused interface is treated as a porous medium using a Kozeny-Carman approximation (This is only to avoid applying the no slip condition.)



Numerical scheme for fluid flow


•Stabilized equal-order velocity-pressure formulation for fluid flow

•Derived from SUPG/PSPG formulation

•Additional stabilizing term for Darcy drag force incorporated

Galerkin formulation for the fluid flow problem



Numerical scheme for fluid flow


Stabilized formulation for the fluid flow problem

Advection stabilizing term

Darcy dragstabilizing term

Pressurestabilizing term Diffusion

stabilizing term



Stabilizing parameters for fluid flow


advective

viscous

Darcy

Stabilizing terms Stabilizing parameters

continuity

•Convective and pressure stabilizing terms modified form of SUPG/PSPG terms

•Darcy stabilizing term obtained by least squares, necessary for convergence

•Viscous term with second derivatives neglected

•A fifth continuity stabilizing term added to the stabilized formulation

pressure



Solute redistribution


Solute is diffused from places with high chemical potential to places with low chemical potential. Particularly, solute rejection is because chemical potential is higher in solid phase than in liquid phase.

For a multi-phase multi-component system, it is only necessary to determine

( , ,..., , , )i i il C T

( ) for 1,2,...i

i i iCv C D i n

t

Solute transport in a system with n component can then be modeled as:

Solute transport should also be compatible with 1i

i

C

( , ,..., , , )i i ilD D C T



Solute redistribution


A Bl lD D D

(1 )( , , , )

( )

A BA

l A B Al

CC

(1 )( , , , )

( )

B AB

l B B Al

CC

(1) Define the chemical potential equal to the concentration in the coexistence liquid phase. (2) Within each phase, chemical potential is only related with concentration

Assumptions:



Interface kinematics


Equilibrium temperature: Given chemical potential (or concentration) of all components , we can get the equilibrium temperature from phase diagram.

( 50%, 50%) 1326Al Sil l lT C C K

* 1( ,... ) ( ) ( )nVT T V n n

Gibbs-Thomson relation (Incorporate surface tension and kinetic effects)



Interface kinetics


Interface velocity can then be derived from energy conservation at diffused interface assuming the interface temperature approaches equilibrium temperature exponentially with a form similar to Newton’s Cooling law.

*( )( )

( ) ( ) N I

q q c c wV k T T

h h h h

Loops (augmentations) may be necessary to make interface temperature equal to equilibrium temperature.

Given V

| | 0t F

Interface position

1

( , ) 0

( , ) 0.5 [ , ]2

w

x t w

x t w ww

Phase fractions

2

...

( )( ) ...

...

l l

l l

ii

Tc c v T

t

t

Cv C

t

v vv



Nucleation


The nucleation rate is proportional to the number of critical clusters with an adsorption rate and has the form of

exp( )nn l

B

GN N

k T

The number of clusters with n atoms in equilibrium is

A new phase is generated through nucleation process.

0

0 exp( )n d

B

G GI I

k T

A schematic of nucleation in eutectic growth is shown in the bottom left figure.

As the interface of alpha phase becomes unstable, solute B becomes richer and richer in these valleys. When the solute concentration of component B beyond a certain point, a beta phase will be nucleated in the valley and keep growing. For simplicity, we currently only considered this type of nucleation in our numerical examples.



Numerical examples


• Pure material

• Binary alloy

• Eutectic growth

• Ternary alloy (Single phase multi-component alloy)

• 3D examples with fluid flow



Numerical examples (pure material case 1)


-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

: 1

: 1

Surface tension

Kinematic undercooling coeff

Thermal conductivity

Latent heat

Results using finer mesh are compared with other researcher’s results in the next slide.





-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

400 400

200 200

100 100

Our methodOsher et. (1997)

Triggavason et. (1996)

Heinrich et. (2003)





483 26 : 0.001{1 0.4[ sin 3( ) 1]}

: 0.8

fold Surface tension

Undercooling

Heinrich et al.

(2003)

Our method





0 100 200 300 4000

50

100

150

200

250

300

350

400

:1Mesh size

. . , . ,

2002

Y T Kim N Goldenfield

Physical Review E

: 1

: 1

: 1

: 1

Density

Specific heat


Latent heat

: 400 400

: 0.55

: 30

domain

initial undercooling

initial interface position R

& 1998, . Re . , 3000Karma Rapel Phy v E performed on DEC Alpha

CPU : 20, T 1 2Mesh size ~ minute on a GHz PC



Numerical examples (binary alloy case)


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 not 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


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



Numerical examples (binary alloy case)


Solute concentration Adaptive Mesh

Results are presented only within domain [0,2]by[0,2]. Micro-segregation can be observed in crystal.



Numerical examples (Pb-Sb binary alloy)


Pb-Sb alloy dendritic growth

0

0

: 0.24mm 0.80mm

: 10K/mm

:

: 2.2

:

m l

domain

initial temperature gradient

initial temperature at left side T m C

initial concentration C

initial interface position

l

0

0.1K/s

k 10K/mm

x

cooling rate at left side :

heat flux at right side :

-5 3: 1.04 10 kg/mm

: 1.51J/kg K

: 0.030J/s mm K

: 0.016J/s mm K

Density

Specific heat

Solid thermal conductivity

Liquid thermal conductivity

Latent

3 2

2

: 29775J/kg

: 1.13 10 mm /s

: 0mm /s

: 6.829K/wt%

:

heat

Liquid mass diffusivity

Solid mass diffusivity

Liquidus slop

Partition coefficient

0.312

C0.02801970.02671890.02541810.02411730.02281650.02151570.02021490.01891410.01761330.01631250.01501170.01371090.01241010.01110930.00980847



Numerical examples (Eutectic growth)


C0.8185390.7556580.6927770.6298960.5670150.5041340.501720.5006210.5001980.5001070.4998940.4991880.4989260.4978660.4412540.3783730.3154920.252611



Numerical examples (Ternary alloy Ni-5.8%Al-15.2%Ta)


Important parameters

Insulated boundaries on the rest of faces

ux = uz = 0

ux =

uz =

0ux

= u

z =

0

ux = uz = 0

T/t = r

T/

x =

0 T/x =

0

T/z = G

C/x =

0C/

x =

0

T(x,z,0) = T0 + Gz

C(x,z,0) = C0



Numerical examples (Ternary alloy Ni-5.8%Al-15.2%Ta)


0 0.5 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

C0

0.0800.0780.0750.0730.0700.0680.0650.0630.0600.0580.0550.0530.0500.0480.045

0 0.5 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

C1

0.2230.2150.2070.1990.1910.1840.1760.1680.1600.1520.1440.1360.1280.1210.113

(a) Interface position(b) Al concentration(c) Ta concentration

Pattern of concentration for Al and Ta are the similar due to the assumption of equal diffusion coefficient in liquid for both component and similar partition coefficient.

( a ) ( b ) ( c )



Two dimension crystal growth with convection


With fluid flow, the crystal tips will tilt in the upstream direction.

• Whole domain method with convective and pressure stabilizing terms (SUPG/PSPG)

• The diffused interface is treated as a porous medium using a Kozeny-Carman approximation. (Using stabilize term DSPG )



Three dimension crystal growth with convection


As in the 2D case, the crystal tips will tilt in the upstream direction.

Low undercooling High undercooling



Conclusions and future work


• A level set method is implemented using the finite element technique for multi-phase evolution.

• Fast marching is implemented for re-initialization.

• Techniques such as narrow band computation and adaptive meshing is implemented with the aid of signed distance.

• A volume averaging model with diffused interface is used for heat transfer, fluid flow and solute transport.

• Because of the energy-conserving features of the diffused interface model, our model compares well with sharp interface models, and shows improvement over the phase field method with a much coarser mesh.

• The numerical algorithm is tested for 2d and 3d solidification of pure materials with convection, binary alloy solidification, and eutectic growth.

• We are currently extending this framework to various practical alloy systems and plan for developing multiscale solidification design algorithms for explicit control of the microstructure and mechanical properties.

http://mpdc.mae.cornell.edu/

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