COMPUTATIONAL TECHNIQUES FOR THE

1
COMPUTATIONAL TECHNIQUES FOR THE ANALYSIS AND
CONTROL OF ALLOY SOLIDIFICATION PROCESSES
DEEP SAMANTA Presentation for Thesis Defense
(B-exam) Date 21 December 2005 Sibley School
of Mechanical and Aerospace Engineering Cornell
University
2
ACKNOWLEDGEMENTS

• SPECIAL COMMITTEE
• Prof. Nicholas Zabaras, M A.E., Cornell
  University
• Prof. Ruediger Dieckmann, M.S E., Cornell
  University
• Prof. Lance Collins, M A.E., 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)
3
OUTLINE OF THE PRESENTATION

• Introduction alloy solidification processes.
• Objectives of the current research.
• Numerical model of alloy solidification with
  external magnetic fields.
• FEM based computational techniques employed.
• Numerical Examples.
• Optimization problem for alloy solidification
  using magnetic fields.
• Surface defect formation in aluminum alloys.
• Exploring the role of mold surface topography.
• Numerical Examples (parametric analysis).
• Important observations and conclusions.
• Suggestions for future study.

4
Introduction and objectives of the current
research
5
INTRODUCTION
Phase change process involving more than one
chemical species
Alloy solidification
Appearance of solid and or crystalline phases

• Alloy Solidification Processes
• Used in industry to obtain near net shaped
  objects casting, welding,
• directional and rapid solidification etc.
• Highly coupled process that involves several
  underlying phenomena
• fluid flow, heat transfer, solute transfer,
  latent heat release and
• microstructure formation.
• Influences the underlying microstructure and
  properties of cast products.
• Most cast or solidified alloys characterized by
  defects.

6
INTRODUCTION
COMPLEXITIES IN ALLOY SOLIDIFICATION PROCESSES
Mushy zone
10-1 - 100 m
liquid
solid
qos
g
(a) Macroscopic scale
10-4 10-5m
liquid
(b) Microscopic scale
solid
7
INTRODUCTION
Non-equilibrium effects
Phase Change
Mass Transfer
Alloy solidification process
Fluid flow
Shrinkage
Heat Transfer
Deformation
Microstructure evolution
8
DEFECTS DURING ALLOY SOLIDIFICATION
Non uniform solute concentration in bulk
Macrosegregation

• Oriented parallel to the direction of gravity in
  directionally
• (vertically) solidified cast alloys.
• Concentration of solute element inside freckles
  varies a
• lot from the bulk.
• Serve as sites for fatigue cracks and other
  types of failure

Freckles defects
Close view of a freckle in a Nickel based
super-alloy blade
Freckles in a single crystal Nickel based
superalloy blade
Freckles in a cast ingot (Ref. Beckermann C.)
(Ref Beckermann C., 2000)
9
DEFECTS DURING ALLOY SOLIDIFICATION
(a) Macro-segregation patterns in a steel
ingots (b) Centerline segregation in
continuously cast steel (Ref Beckermann C.,
2000) (c) Freckle defects in directionally
solidified blades (Ref Tin and Pollock,
2004) (d) Freckle chain on the surface of a
single crystal superalloy casting (Ref. Spowart
and Mullens, 2003)
(b)
(a)
(c)
(d)
10
DEFECTS DURING ALLOY SOLIDIFICATION
Different types of shrinkage porosity (ref.
Calcom, EPFL, Switzerland)
Surface defects in casting (Ref. ALCOA corp.)
Macro shrinkage
Micro shrinkage
Piping
(a) Sub-surface liquation and crack formation on
top surface of a cast
Piping occurs during early stages
of solidification. Macro shrinkage - leads to
internal defects. Micro shrinkage occurs late
during solidification and between
solidifying dendrites.
(b) Non-uniform front and undesirable growth with
non-uniform thickness (left) and
non-uniform microstructure (right)
11
OBJECTIVES OF THIS RESEARCH
Large scale distribution of solute
Non uniform properties on macro scale
Macrosegregation
Thermosolutal convection
Development of freckles channels and other defects
Control of macro- segregation
Thermal and solutal buoyancy in the liquid and
mushy zones
Control or suppression of convection
Macrosegregation
Density variations in terrestrial gravity
conditions
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OBJECTIVES OF THIS RESEARCH

• MEANS OF SUPPRESSING CONVECTION
• Control the boundary heat flux
• Multiple-zone controllable furnace design
• Rotation of the furnace
• Micro-gravity growth
• Electromagnetic fields

• Constant magnetic fields
• Time varying fields
• Rotating magnetic fields
• Combination of magnetic field and field
  gradients

13
OBJECTIVES OF THIS RESEARCH
Engineered mold surface (Ref. ALCOA Corp.)
Uniform front growth (left) and uniform
microstructure (right) obtained using grooved
molds

• In industry, the mold surface is pre-machined to
  control heat extraction in directional
• solidification
• This periodic groove surface topography allows
  multi-directional heat flow on the
• metal-mold interface
• However, the wavelengths should be with the
  appropriate value to obtain anticipated
• benefits.

14
OBJECTIVES OF THIS RESEARCH
Alloy solidification process
Formation of various defects
Material, monetary and energy losses

• Exploring methods to reduce defects during alloy
  solidification processes.
• Developing a computational framework for modeling
  alloy solidification processes.
• Studying the role of convection on
  macrosegregation.
• Employing constant or time varying magnetic
  fields to reduce macrosegregation based defects.
• Designing appropriate mold surface topographies
  to reduce surface defects in alloys.

15
Numerical model of alloy solidification underthe
influence of magnetic fields
16
PREVIOUS WORK

• Effect of magnetic field on transport phenomena
  in Bridgman crystal growth Oreper et al.
• (1984) and Motakef (1990).
• Numerical study of convection in the horizontal
  Bridgman configuration under the influence
• of constant magnetic fields Ben Hadid et al.
  (1997).
• Simulation of freckles during directional
  solidification of binary and multicomponent
  alloys
• Poirier, Felliceli and Heinrich (1997-04).
• Effects of low magnetic fields on the
  solidification of a Pb-Sn alloy in terrestrial
  gravity
• conditions Prescott and Incropera (1993).
• Effect of magnetic gradient fields on Rayleigh
  Benard convection in water and oxygen
• Tagawa et al.(2002-04). Suppression of
  thermosolutal convection by exploiting the
• temperature/composition dependence of magnetic
  susceptibility Evans (2000).
• Solidification of metals and alloys with
  negligible mushy zone under the influence of
  magnetic
• fields and gradients Control of solidification
  of conducting and non conducting materials
• using tailored magnetic fields
  B.Ganapathysubramanian and Zabaras (2004-05)

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PROBLEM DEFINITION
B
g
SOLID
qs
MELT
ql
Mushy zone

• Solidification of a metallic alloy inside a
  cavity in terrestrial gravity conditions
• heat removal from left
• Strong thermosolutal convection present drives
  convection during solidification
• Application of magnetic field on an electrically
  conducting moving fluid produces
• additional body force Lorentz force.
• This force is used for damping flow during
  solidification of electrically
• conducting metals and alloys.
• The main aim of the current study is to
  investigate its effect on macro-
• - segregation during alloy solidification.

18
NUMERICAL MODEL
SALIENT FEATURES
Microscopic transport equations

• Single domain model based on volume averaging is
  used.
• Single set of transport equations for mass,
  momentum, energy
• and species transport.
• Individual phase boundaries are not explicitly
  tracked.
• Complex geometrical modeling of interfaces
  avoided.
• Single grid used with a single set of boundary
  conditions.
• Solidification microstructures are not modeled
  here and
• empirical relationships used for drag force due
  to permeability.

wk
Volume- averaging process
dAk
(Ref Gray et al., 1977)
Macroscopic governing equations
19
IMPORTANT ASSUMPTIONS AND SIMPLIFICATIONS
TRANSPORT EQUATIONS FOR SOLIDIFICATION

• Only two phases present solid and liquid with
  the solid phase assumed to be
• stationary.
• The densities of both phases are assumed to be
  equal and constant except in the
• Boussinesq approximation term for
  thermosolutal buoyancy in the momentum
• equations.
• Interfacial resistance in the mushy zone modeled
  using Darcys law.
• The mushy zone permeability is assumed to vary
  only with the liquid volume fraction
• and is either isotropic or anisotropic.
• The solid is assumed to be stress free and pore
  formation is neglected.
• Material properties uniform (µ, k etc.) in an
  averaging volume dVk but can globally
• vary.

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IMPORTANT ASSUMPTIONS AND SIMPLIFICATIONS
MAGNETO-HYDRODYNAMIC (MHD) EQUATIONS

• Phenomenological cross effects galvomagnetic,
  thermoelectric and
• thermomagnetic are neglected.
• The induced magnetic field is negligible, only
  field is the externally applied
• field.
• Magnetic field assumed to be quasistatic.
• The current density is solenoidal.
• The external magnetic field is applied only in a
  single direction.
• The magnetic field is assumed to constant in
  space.
• Charge density is negligible

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GOVERNING EQUATIONS
Continuity equation
Intertial and advective terms
Viscous or diffusive terms
Pressure terms
Momentum equation
Darcy damping term
Thermosolutal buoyancy force term
Magnetic damping force
Ref Toshio and Tagawa (2002-04), Evans et al.
(2000), Ganapathysubramanian B. and Zabaras
(2004-05), Samanta and Zabaras, (2005)
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GOVERNING EQUATIONS
Energy equation
convective term
diffusive term
Latent heat term
Transient term
Solute equation
Convective term
diffusive term
Transient term
Electric Potential equation
where
Ref Toshio and Tagawa (2002-04), Evans et al.
(2000), Ganapathysubramanian B. and Zabaras
(2004-05), Samanta and Zabaras, (2005)
23
PERMEABILITY EXPRESSIONS IN ALLOY
SOLIDIFICATION
Isotropic permeability (empirical relation based
on Kozeny Carman relationship)
Kx Ky Kz fn(e,d)
d dendrite arm spacing important
microstructural parameter. e Volume fraction of
liquid phase.
Anisotropic permeability (obtained experimentally
and from regression analysis for directional
solidification of binary alloys, Heinrich et al.,
1997)
Kx Ky ? Kz Kx Ky fn1(e,d) Kz
fn2(e,d)
24
CLOSURE RELATIONSHIPS INVOLVING BINARY PHASE
DIAGRAM
Lever Rule (Infinite back-diffusion)
T
Scheil Rule (Zero back-diffusion)
Cl
C
(assumed constant for all problems)

• Phase diagram relationships depend on the state
  of the alloy solid,
• liquid or mushy.
• These relationships are used for obtaining mass
  fractions and solute
• concentrations of liquid and solid phases.
• Lead to strong coupling of the thermal and
  solutal problems.

25
FEM based numerical techniques
26
COMPUTATIONAL STRATEGY AND NUMERICAL TECHNIQUES

• In the presence of strong convection, standard
  FEM techniques lead to oscillations in solution.
• Inappropriate choice of interpolation functions
  for pressure and velocity can lead to
• oscillations in pressure.
• Stabilized FEM techniques 1) prevent
  oscillations 2) allow the use of same finite
  element
• spaces for interpolating pressure and velocity
  for the fluid flow problem.
• Stabilizing terms take into account the dominant
  underlying phenomena (convective, diffusive
• or Darcy flow regimes)

Convection stabilizing term
Thermal and species transport problems
Standard Galerkin FE formulation
Stabilized FE formulation

Stabilizing terms
Pressure stabilizing term
Fluid flow problem
Convection stabilizing term
Darcy drag stabilizing term
27
COMPUTATIONAL STRATEGY AND NUMERICAL TECHNIQUES

• For the thermal and solute sub-problems, SUPG
  technique used for discretization.
• The fluid flow sub-problem is discretized using
  a modified form of the SUPG-PSPG
• technique (Tezduyar et al.) incorporating the
  effects of Darcy drag force in the mushy zone
• (RefZabaras and Samanta 04,05).
• Both velocity and pressure and solved
  simultaneously and convergence rate is improved.
• Combination of direct and iterative solvers used
  to realize the transient solution.

• Multistep Predictor Corrector method used for
  thermal and solute problems.
• Backward Euler fully implicit method for time
  discretization and Newton-Raphson method for
  solving fluid flow problem.
• Thermal and solutal transport problems along
  with the thermodynamic update scheme solved
  repeatedly in a inner loop in each time step.
• Fluid flow and electric potential problems
  decoupled from this iterative loop and solved
  only once in each time step.

28
COMPUTATIONAL STRATEGY AND NUMERICAL TECHNIQUES
SOLUTION ALGORITHM AT EACH TIME STEP
All fields known at time tn
n n 1
Solve for the induced electric potential (3D only)
Advance the time to tn1
Decoupled momentum solution only once in each
time step
Check if convergence satisfied
Solve for the temperature field
(energy equation)
Solve for velocity and pressure fields (momentum
equation)
(Ref Heinrich, et al.)
Inner iteration loop
Solve for the concentration field
(solute equation)
Yes
Is the error in
liquid concentration and liquid mass
fraction less than tolerance
Solve for liquid concentration, liquid
volume fraction (Thermodynamic relations)
Segregation model (Scheil rule)
No
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Numerical Examples 1)
Solidification of an aqueous binary alloy
effect of convection (no magnetic
fields). 2) Convection damping during
horizontal solidification of a Pb-Sn
alloy. 3) Convection damping during
directional solidification of a Pb-Sn
alloy.
30
SOLIDIFICATION OF AN AQUEOUS BINARY ALLOY
Initial and boundary conditions
(b)
(c)
(d)
(a)
RaT 1.938x107 RaC -2.514x107 Temperature
of hot wall, Thot 311 K Temperature of cold
wall, Tcold 223K Initial temperature, T0
311K Initial concentration, C0 0.7 Solutal flux
on all boundaries 0 (adiabatic flux
condition) vx vy 0.0 on all boundaries

• Velocity and mass fraction (b) isotherms
• (c) solute concentration (d) liquid solute
  concentration
• Thermal solutal convection is very strong and
  large scale
• solute distribution occurs
• Effect of thermosolutal convection seen in all
  other fields

31
DAMPING CONVECTION IN HORIZONTAL ALLOY
SOLIDIFICATION
g
SOLID
H 0.02 m
MELT
qs h(T Tamb)
Mushy zone
L 0.08 m

• Solidification of Pb 10 Sn alloy studied
  under the influence of magnetic
• fields (initial temperature 600 K) (RaT
  -2.678x107 RaC 4.941x108).
• This alloy is characterized by a large mushy
  zone and strong convection.
• Macrosegregation is severe and extent of
  segregated zone is large.
• A magnetic field of 5 T applied in the z
  direction.
• Lorentz force responsible for convection
  damping.
• Effect of Lorentz force on macrosegregation to
  be studied.

32
HORIZONTAL SOLIDIFICATION OF A METAL ALLOY (Pb
Sn)
(i)
(ii)
(iii)
(iv)
(b) A magnetic field of 5 T in the z direction
(a) No magnetic field
(i) Isotherms (ii) velocity vectors and
liquid mass fractions (iii) isochors of Sn
(iv) liquid solute concentration
33
FRECKLE SUPPRESSION IN 2D DIRECTIONAL
SOLIDIFICATION
?T/?z G
Important parameters
ux uz 0
L x B 0.04m x 0.007m C0 10 by weight Tin (Sn)
ux uz 0
ux uz 0
T(x,z,0) T0 Gz C(x,z,0) C0
?T/?x 0
g
?T/?x 0
Insulated boundaries on the rest of faces
Direction of solidification
?C/?x 0
?C/?x 0
(RaC 6.177x107)
B0
ux uz 0
?T/?t r

• Mushy zone permeability assumed to be
  anisotropic
• Formation of freckles and channels due to
• thermosolutal convection
• Lorentz force occurs once magnetic field is
  applied.

Constant magnetic field of 3.5 T applied in x
direction
34
FRECKLE SUPPRESSION IN 2D DIRECTIONAL
SOLIDIFICATION
(b)
(a)
(i) CSn
(ii) fl
(i) CSn
(ii) fl

• (a) No magnetic field
  (b) Magnetic
  field (3.5 T)
• Significant damping of convection throughout the
  cavity
• Freckle formation is totally suppressed ?
  homogeneous solute distribution
• (a) ?C Cmax Cmin 2.63 wt Sn (t 800 s)
  (b) ?C Cmax Cmin 1.3 wt Sn (t 800 s)

35
OPTIMIZATION PROBLEM WITH A TIME VARYING MAGNETIC
FIELD
g
SOLID
H
MELT
ql
qs
Mushy zone
L

• Growth under diffusion dominated conditions leads
  to
• A uniform solute concentration profile due to
• reduced convection.
• Reduction of defects and sites of fatigue
  cracking.
• Uniform properties in the final cast alloy.
• Reduction in rejection rate of cast alloy
  components

Micro-gravity based growth is purely diffusion
based Objective is to achieve some sort of
reduced gravity growth
36
OPTIMIZATION PROBLEM WITH A TIME VARYING MAGNETIC
FIELD
Temporal variations in thermosolutal convection
Time varying magnetic fields
Non-linear finite dimensional optimal control
problem to determine time variation
Design parameter set b b1 b2,,bn
Cost Functional
Measure of convection in the entire domain and
time interval considered
Minimization of this cost functional yields
design parameter set that leads to a growth
regime where convection is minimized.
37
OPTIMIZATION PROBLEM WITH A TIME VARYING MAGNETIC
FIELD
Find a quasi solution B (bk) such that
J(Bbk) ? J(Bb) ? b an optimum design
variable set bk sought
Define the inverse solidification problem as an
unconstrained spatio temporal optimization
problem
Non linear conjugate gradient method
Gradient of the cost functional
Sensitivity of velocity field
Gradient information
Obtained from sensitivity field
n sensitivity problems to be solved
Continuum sensitivity equations
Design differentiate with respect to
Direct Problem
38
OPTIMIZATION PROBLEM WITH A TIME VARYING MAGNETIC
FIELD
Differentiate with respect to design parameters
Discretize in space and time
Continuum sensitivity equations
Direct Problem Single domain volume averaged
equations for alloy solidification
Optimization problem
DESIGN OBJECTIVES Find the optimal magnetic field
B(t) in 0,tmaxdetermined by the set b
such that, in the presence of coupled
thermosolutal buoyancy, and electromagnetic
forces in the melt, diffusion dominated growth is
obtained leading to minimum macrosegregation in
the cast alloy
Sensitivity of each variable with respect to the
design parameters
Gradient information, step size in nonlinear CG
algorithm
39
CONVECTION DAMPING IN 2D HORIZONTAL SOLIDIFICATION
L 0.08 m
H 0.02 m
T Ti 580 K
C C0 10 by wt. Sn
Isotropic permeability (Kozeny Carman
relationship)
(t)
40
CONVECTION DAMPING IN 2D HORIZONTAL SOLIDIFICATION
Time varying optimal magnetic field
Cost functional
5 design variables (5 CSM problems
solved)
41
CONVECTION DAMPING IN 2D HORIZONTAL SOLIDIFICATION
(a)
(b)
(c)
(i) No magnetic field
(ii) Optimal magnetic field

Comparison of results at time t tmax 120 s
(a) Isotherms (b) Solute concentration
(c) Liquid mass fractions and velocity vectors
- Convection is almost fully damped throughout
the solidification process. - Significant
reduction in macrosegregation - Use of a
time varying optimal magnetic field results in a
near diffusion based growth - Near
homogeneous solute concentration profile obtained.
42
CONVECTION DAMPING IN 2D HORIZONTAL SOLIDIFICATION
Elemental length, h 1 x 10-3 m
Convection damping
Diffusion dominated growth regime
Macro- -segregation suppression
43
FRECKLE SUPPRESSION IN 3D DIRECTIONAL
SOLIDIFICATION
z
Important parameters
L x B x H 0.01m x 0.01m x 0.02m C0 10 by
weight Tin
?T/?z G
vx vy vz 0 on all surfaces A magnetic
field applied in x
Direction of solidification
g
Insulated boundaries on the rest of faces
x
y
?T/?t r
(RaC 7.721x106)

• Mushy zone permeability assumed to be
  anisotropic
• Formation of freckles and channels due to
  thermal solutal convection
• Lorentz force primary damping force once
  magnetic field is applied.

44
FRECKLE SUPPRESSION IN 3D DIRECTIONAL
SOLIDIFICATION
(a) Concentration of Sn (t 800 s)
(b) Liquid Mass fraction (t 800 s)
- Formation of freckles in the absence
of magnetic field (t 800 s). -
Thermosolutal convection is strong and leads to
large scale solute distribution (at t 400 s,
?C Cmax Cmin 5.97 wt. Sn at t
800 s, ?C Cmax Cmin 7.4 wt. Sn)
45
FRECKLE SUPPRESSION WITH OPTIMAL MAGNETIC FIELD
Time varying optimal magnetic field
Cost functional
4 design variables (4 CSM problems
solved)
46
FRECKLE SUPPRESSION WITH OPTIMAL MAGNETIC FIELD

• Concentration of Sn (t 800 s)

(b) Liquid mass fraction (t 800 s)
- Complete
suppression of freckles in the presence of
optimal time varying magnetic field. -
Thermosolutal convection causing convection is
totally suppressed. - Homogeneous solute
distribution in the solidifying alloy (at
t 400 s, ?C Cmax Cmin 0.4 wt. Sn
at t 800 s, ?C Cmax Cmin 1.65 wt. Sn)
47
Surface defect formation in Aluminum alloys
48
INTRODUCTION

• Aluminum industry relies on direct chill casting
  for aluminum ingots.
• Aluminum ingots are often characterized by
  defects in surface due to
• non-uniform heat extraction, improper contact
  at metal/mold interface,
• inverse segregation, air-gap formation and
  meniscus freezing etc.
• These surface defects are often removed by post
  casting process such
• as scalping/milling.
• Post-processing leads to substantial increase of
  cost, waste of material
• and energy.
• The purpose of this work is to reduce
  scalp-depth in castings.
• Detailed understanding of the highly coupled
  phenomenon in the early
• stages of solidification is required.

49
INTRODUCTION
Surface defects in casting (Ref. ALCOA corp.)
(b)
(a)

• Sub-surface liquation and crack formation on
• top surface of a cast
• (b) Non-uniform front and undesirable
• growth with non-uniform thickness (left)
• and non-uniform microstructure (right)
• (c) Ripple formation

(c)
50
INTRODUCTION
Engineered mold surface (Ref. ALCOA Corp.)
Uniform front growth (left) and uniform
microstructure (right) obtained using grooved
molds

• In industry, the mold surface is pre-machined to
  control heat extraction in directional
• solidification.
• This periodic groove surface topography allows
  multi-directional heat flow on the
• metal-mold interface.
• However, the wavelengths should be with the
  appropriate value to obtain anticipated
• benefits.

51
Numerical model for deformation of solidifying
alloys
52
PREVIOUS WORK

• Zabaras and Richmond (1990,91) hypoelastic
  rate-dependent small deformation model to
• study the deformation of solidifying body.
• Rappaz (1999), Mo (2004) deformation in mushy
  zone with a volume averaging model
• Continuum model for deformation of mushy zone
  in a solidifying alloy and development of a
• hot tearing criterion.
• Mo et al. (1995-98) Surface segregation and
  air gap formation in DC cast Aluminum alloys.
• Hector and Yigit (2000) semi analytical
  studies of air gap nucleation during
  solidification
• of pure metals using a hypoelastic perturbation
  theory.
• Hector and Barber (1994,95) Effect of strain
  rate relaxation on the stability of solid front
• growth morphology during solidification of pure
  metals.
• Chen et al. (1991 93), Heinrich et al.
  (1993,97) Inverse segregation caused by
  shrinkage
• driven flows or combined shrinkage and
  buoyancy driven flows during alloy
  solidification.
• A thermo-mechanical study of the effects of mold
  topography on the solidification of Al alloys

53
PROBLEM DEFINITION

• Solidification of Aluminum-copper alloys on
  sinusoidal mold surfaces.
• With growth of solid shell, air gaps form
  between the solid shell and mold due to imperfect
  contact which leads to variation in thermal
  boundary conditions.
• The solid shell undergoes plastic deformation
  and development of thermal and plastic strain
  occurs in the mushy zone also.
• Inverse segregation caused by shrinkage driven
  flow affects variation in air gap sizes, front
  unevenness and stresses developing in the casting.

54
SCHEMATIC OF THE HIGHLY COUPLED SYSTEM
Mold
Phase change and mushy zone evolution
Heat transfer
Solute transport
Casting domain
Fluid flow
Deformable or non-deformable mold
Heat transfer
Contact pressure/ air gap criterion
Inelastic deformation

• There is heat transfer and deformation in both
  mold and casting region interacting with the
  contact pressure or air gap size between mold and
  casting.
• The solidification, solute transport, fluid flow
  will also play important roles.

55
GOVERNING TRANSPORT EQUATIONS FOR SOLIDIFICATION
Continuity equation
Momentum equation
Energy equation
Solute equation
Initial conditions
Isotropic permeability
56
MODEL FOR DEFORMATION OF SOLIDIFYING ALLOY

• For deformation, we assume the total strain to
  be decomposed into three parts
• elastic strain, thermal strain and plastic
  strain.
• Elastic strain rate is related with stress rate
  through an hypo-elastic constitutive law
• Plastic strain evolution satisfy this creep law
  with its parameters determined from
• experiments (Strangeland et al. (2004)).
• The thermal strain evolution is determined from
  temperature decrease and shrinkage.

Strain measure
Elastic strain
Thermal strain
Plastic strain
57
MODELING DEFORMATION IN MUSHY ZONE

• Low solid fractions usually accompanied
• by melt feeding and no deformation due to
• weak or non existent dendrites ?
• leads to zero thermal strain.
• With increase in solid fraction, there is an
  increase in strength and bonding ability of
• dendrites ? to non zero thermal strain.
• The presence of a critical solid volume fraction
  is observed in experiment and varies for
  different alloys.

The parameter w is defined as

• Liquid or low solid fraction mush
• - zero thermal and plastic strains. (Without any
  strength)
• Solid or high solid fraction mush
• - thermal and plastic strains start developing
  gradually.

58
IMPORTANT PARAMETERS FOR DEFORMATION IN MUSHY ZONE
Critical solid fraction for different copper
concentrations in aluminum-copper alloy
Ref Mo et al.(2004)
Creep law for plastic deformation Ref.
Strangeland et al. (2004)
Strain-rate scaling factor
Al Cu 0.3 pct Mg
Stress scaling factor
Activation energy
Creep law exponent
Volumetric thermal expansion coefficient
Mushy zone softening parameter
Volumetric shrinkage coefficient
59
THERMAL RESISTANCE AT THE METAL-MOLD INTERFACE
Contact resistance
At very early stages, the solid shell is in
contact with the mold and the thermal resistance
between the shell and the mold is determined by
contact conditions

• Before gap nucleation, the thermal resistance
• is determined by pressure

• After gap nucleation, the thermal resistance
• is determined by the size of the gap

Example Aluminum-Ceramic Contact
Heat transfer retarded due to gap formation
Uneven contact condition generates an uneven
thermal stress development and accelerates
distortion or warping of the casting shell.
60
MOLD METAL BOUNDARY CONDITIONS
Consequently, heat flux at the mold metal
interface is a function of air gap size or
contact pressure
Air-gap size at the interface
Contact pressure at the interface

• The actual air gap sizes or contact pressure
  are determined from the contact sub problem.
• This modeling of heat transfer mechanism due to
  imperfect contact is very crucial for studying
  the non-uniform growth at early stages of
  solidification.

61
COMPUTATIONAL STRATEGY AND NUMERICAL TECHNIQUES

• The thermal problem is solved in a region
  consisting of both mold and casting to account
  for non-linear (contact pressure/air gap
  dependent) boundary conditions at the mold
  metal interface.
• Deformation problem is solved in both casting
  and mold (if mold deformable) or only the casting
  (if mold rigid, for most of our numerical
  studies).
• Solute and momentum transport equations is only
  solved in casting with multistep predictor
  Corrector method for solute problems, and
  Newton-Raphson method for solving heat transfer,
  fluid flow and deformation problems.
• Backward Euler fully implicit method is
  utilized for time discretization to make the
  numerical scheme unconditionally stable.
• The contact sub-problem is solved using
  augmentations (using the scheme introduced by
  Laursen in 2002).
• All the matrix computations for individual
  problems are performed using the parallel
  iterative Krylov solvers based on the PETSc
  library.

62
SOLUTION ALGORITHM AT EACH TIME STEP
Convergence criteria based on gap sizes or
contact pressure in iterations
All fields known at time tn
n n 1
Check if convergence satisfied
Advance the time to tn1
Contact pressure or air gap obtained from Contact
sub-problem
Solve for displacement and stresses in the
casting (Deformation problem)
Solve for the temperature field
(energy equation)
Decoupled momentum solver
Solve for velocity and pressure
fields (momentum equation)
Inner iteration loop
Solve for the concentration field
(solute equation)
(Ref Heinrich, et al.)
Yes
Is the error in
liquid concentration and liquid mass
fraction less than tolerance
Solve for liquid concentration, mass
fraction and density (Thermodynamic relations)
Segregation model (Scheil rule)
No
63
Numerical examples
64
SOLIDIFICATION OF Al-Cu ALLOY ON UNEVEN SURFACES

• Combined thermal, solutal and
• momentum transport in casting.
• Assume the mold is rigid.
• Imperfect contact and air gap
• formation at metal mold interface

Solidification problem
We carried out a parametric analysis by
changing these four parameters 1) Wavelength of
surfaces (?) 2) Solute concentration (CCu) 3)
Melt superheat (?Tmelt) 4) Mold material (Cu, Fe
and Pb)
Heat Transfer (Mold is rigid and non-deformable)
Deformation problem
Both the domain sizes are on the mm scale
65
SOLIDIFICATION COUPLED WITH DEFORMATION AND
AIR-GAP FORMATION
Important parameters 1) Mold material - Cu 2) CCu
8 wt. 3) ?Tmelt 0 oC Air gap is
magnified 200 times.

• Preferential formation of solid occurs at the
  crests and air gap formation occurs at the
  trough, which in turn causes re-melting.

• Because of plastic deformation, the gap formed
  initially will gradually decrease.
• As shown in the movies, a 1mm wavelength mold
  would lead to more uniform growth and less fluid
  flow.

66
TRANSIENT EVOLUTION OF IMPORTANT FIELDS (? 3 mm)

• Temperature
• Solute concentration
• Equivalent stress
• (d) Liquid mass fraction

Important parameters 1) Mold material - Cu 2) CCu
5 wt. 3) ?Tmelt 0 oC
(b)
(a)

• We take into account solute transport and the
  densities of solid and liquid phases are assumed
  to be different.
• Inverse segregation, caused by shrinkage driven
  flow, occurs at the casting bottom.This is
  observed in (b).

(d)
(c)
67
TRANSIENT EVOLUTION OF IMPORTANT FIELDS (? 5 mm)

• Temperature
• Solute concentration
• Equivalent stress
• (d) Liquid mass fraction

• For other wavelengths, similar result is
  observed
• (1) preferential formation of solid occurs at the
  crests
• (2) remelting at the trough due
• to the formation of air gap.
• For ? 3mm, the solid shell unevenness
  decreases faster
• than for ? 5mm.

(a)
(b)
(d)
(c)
68
VARIATION OF AIR-GAP SIZES AND MAX. EQUIVALENT
STRESS
?Tmelt 0 oC, CCu 5 wt., mold material Cu

• Max. equivalent stress seq variation with ?
• seq first increases and then decreases
• Initially, seq is higher for greater ?
• Later (t100 ms), stress is lowest for
• 5 mm wavelength.

• Air-gap size variation with wavelength ?
• Initially, air-gap sizes nearly same for
• different ?
• At later times, air-gap sizes increase
• with increasing ?

69
VARIATION OF AIR-GAP SIZES AND MAX. EQUIVALENT
STRESS
?Tmelt 0 oC, ? 5 mm, mold material Cu
Increase of solute concentration leads to
increase in air-gap sizes, but its effect on
stresses are small.

• seq first increases and then decreases
• Variation of seq with Cu concentration
• is negligible after initial times

• Air-gap sizes increase with time
• Increasing Cu concentration leads to
• increase in air-gap sizes

70
EFFECT OF INVERSE SEGREGATION AIR GAP SIZES
(a) With inverse segregation
(b) Without inverse segregation
inverse segregation actually plays an important
role in air-gap evolution.

• Differences in air-gap sizes for different
  solute concentrations are more pronounced in the
  presence of inverse segregation.

71
VARIATION OF EQUIVALENT STRESSES AND FRONT
UNEVENNESS
Time t 100 ms

• Value of front unevenness and maximum equivalent
  stress for various wavelengths
• one cannot simultaneously reduce both stress and
  front unevenness
• when wavelength greater than 5mm, both
  unevenness and stress increase ? wavelength less
  than 5 mm is optimum

• Equivalent stress at dendrite roots
• The highest stress observed for 1.8 copper
  alloy suggests that aluminum copper alloy with
  1.8 copper is most susceptible to hot tearing
• Phenomenon is also observed experi-mentally
  Rappaz(99), Strangehold(04)

72
CONCLUSIONS AND OBSERVATIONS

• Magnetic fields are successfully used to damp
  convection during
• solidification of metallic alloys in
  terrestrial gravity conditions.
• Near homogeneous solute element distributions
  obtained.
• Suppression of freckle defects during
  directional solidification of alloys
• achieved.
• An optimization problem solved to determine time
  varying magnetic fields
• that damp convection and minimize
  macrosegregation in solidifying alloys.
• Optimal time varying magnetic fields take into
  account variations in
• thermosolutal convection superior to
  constant magnetic fields.
• Reduction in current and power requirements
  possible.

73
CONCLUSIONS AND OBSERVATIONS

• Early stage solidification of Al-Cu alloys
  significantly affected by non uniform boundary
• conditions at the metal mold interface.
• Variation in surface topography leads to
  variation in transport phenomena, air-gap sizes
• and equivalent stresses in the solidifying
  alloy.
• Air-gap nucleation and growth significantly
  affects heat transfer between metal and mold.
• Distribution of solute primarily caused by
  shrinkage driven flows and leads to inverse
• segregation at the casting bottom.
• Presence of inverse segregation leads to an
  increase in gap sizes and front unevenness.
• Effects of surface topography more pronounced
  for a mold with higher thermal conductivity
• Computation results suggest that an Al-Cu alloy
  with 1.8 Cu is the most susceptible to
• hot tearing defects. An optimum mold
  wavelength should be less than 5mm.

74
Suggestions for future research
75
MULTILENGTH SCALE SOLIDIFICATION MODELING
Importance of multi length scale modeling

• Ability to resolve morphology of microstructural
  entities like dendrites.
• Effect of various instabilities on the actual
  growth front morphology can be studied.
• Key to understanding effects of micro scale
  phenomenon on macro scale and
• vice versa.
• Can avoid very fine grids for macro scale
  simulations.
• Lay the foundations of multi length scale
  robust design.

Large scale casting metres
Evolution of microstructure
Small scale phenomena

• Macro defects
• Degrade quality of casting

• Dendritic growth

Multi length scale solidification model
Macroscopic transport phenomenon
76
MULTILENGTH SCALE SOLIDIFICATION MODELING
Macroscopic governing equations based on volume
averaging to compute fields like velocity,
temperature and solute concentration
Macro scale grid
Boundary conditions for micro problem
On each element micro scale grid
Multi- length scale direct problem
Microstructure evolution model
compute equivalent parameters (permeability)
1) Interface temperature condition 2) Thermal
Flux jump at the interface 3) Concentration flux
jump at the interface 4) Thermal, solutal and
momentum problems in the liquid phase 5)
Thermal problem in the solid phase. 6) Curvature
effects 7) Tracking the interface position
compute average of physical quantities on fine
scale
Averaging techniques
Allows variation of macro variables on both
scales
Upscaling methods based on homogenization
77
MULTILENGTH SCALE SOLIDIFICATION MODELING
Volume averaged continuum macro model
CSM based optimization method
Microstructure evolution model
Multi-length scale design problem
Multi-length scale direct problem
Averaging/ upscaling techniques
Design macro variables (magnetic field or
boundary heat flux to get a desired microstructure
Probabilistic nucleation model
Cast components with desired properties and
microstructure
78
MICROSTRUCTURE EVOLUTION DURING EARLY STAGE
SOLIDIFICATION
Combined effect of several phenomena on
microstructure evolution (during early stages of
solidification)
79
THANK YOU FOR YOUR ATTENTION