Advanced Scaling Techniques for the Modeling of Materials Processing

1
Advanced Scaling Techniques for the Modeling of
Materials Processing

• Karem E. Tello
• Colorado School of Mines
• Ustun Duman
• Novelis
• Patricio F. Mendez
• Director, Canadian Centre for Welding and Joining
• University of Alberta

2
Phenomena in Materials Processing

• Transport processes play a central role
• Heat transfer
• Fluid Flow
• Diffusion
• Complex boundary conditions and volumetric
  factors
• Free surfaces
• Marangoni
• Vaporization
• Electromagnetics
• Chemical reactions
• Phase transformations
• Multiple phenomena are coupled

3
Example Weld Pool at High Currents
gouging region
trailing region
rim
4
Multiphysics in the Weld Pool

• Driving forces in the weld pool (12)

electrode
arc
solidified metal
weld pool
substrate
5
Multiphysics in the Weld Pool

• Driving forces in the weld pool (12)
• Inertial forces

electrode
arc
solidified metal
weld pool
substrate
6
Multiphysics in the Weld Pool

• Driving forces in the weld pool (12)
• Inertial forces
• Viscous forces

electrode
arc
solidified metal
weld pool
substrate
7
Multiphysics in the Weld Pool

• Driving forces in the weld pool (12)
• Inertial forces
• Viscous forces
• Hydrostatic

electrode
arc
rgh
solidified metal
weld pool
substrate
8
Multiphysics in the Weld Pool

• Driving forces in the weld pool (12)
• Inertial forces
• Viscous forces
• Hydrostatic
• Buoyancy

electrode
arc
brghDT
solidified metal
weld pool
substrate
9
Multiphysics in the Weld Pool

• Driving forces in the weld pool (12)
• Inertial forces
• Viscous forces
• Hydrostatic
• Buoyancy
• Conduction

electrode
arc
solidified metal
weld pool
substrate
10
Multiphysics in the Weld Pool

• Driving forces in the weld pool (12)
• Inertial forces
• Viscous forces
• Hydrostatic
• Buoyancy
• Conduction
• Convection

electrode
arc
solidified metal
weld pool
substrate
11
Multiphysics in the Weld Pool

• Driving forces in the weld pool (12)
• Inertial forces
• Viscous forces
• Hydrostatic
• Buoyancy
• Conduction
• Convection
• Electromagnetic

electrode
arc
J
B
B
JB
solidified metal
weld pool
substrate
12
Multiphysics in the Weld Pool

• Driving forces in the weld pool (12)
• Inertial forces
• Viscous forces
• Hydrostatic
• Buoyancy
• Conduction
• Convection
• Electromagnetic
• Free surface

electrode
arc
solidified metal
weld pool
substrate
13
Multiphysics in the Weld Pool

• Driving forces in the weld pool (12)
• Inertial forces
• Viscous forces
• Hydrostatic
• Buoyancy
• Conduction
• Convection
• Electromagnetic
• Free surface
• Gas shear

electrode
arc
t
solidified metal
weld pool
substrate
14
Multiphysics in the Weld Pool

• Driving forces in the weld pool (12)
• Inertial forces
• Viscous forces
• Hydrostatic
• Buoyancy
• Conduction
• Convection
• Electromagnetic
• Free surface
• Gas shear
• Arc pressure

electrode
arc
solidified metal
weld pool
substrate
15
Multiphysics in the Weld Pool

• Driving forces in the weld pool (12)
• Inertial forces
• Viscous forces
• Hydrostatic
• Buoyancy
• Conduction
• Convection
• Electromagnetic
• Free surface
• Gas shear
• Arc pressure
• Marangoni

electrode
arc
t
solidified metal
weld pool
substrate
16
Multiphysics in the Weld Pool

• Driving forces in the weld pool (12)
• Inertial forces
• Viscous forces
• Hydrostatic
• Buoyancy
• Conduction
• Convection
• Electromagnetic
• Free surface
• Gas shear
• Arc pressure
• Marangoni
• Capillary

electrode
arc
solidified metal
weld pool
substrate
17
Multicoupling in the Weld Pool
Inertial forces Viscous forces
Hydrostatic
Capillary
Buoyancy
Marangoni
Conduction Convection
Arc pressure
Electromagnetic
Gas shear
Free surface
18
Multicoupling in the Weld Pool
Inertial forces Viscous forces
Hydrostatic
Capillary
Buoyancy
Marangoni
Conduction Convection
Arc pressure
Electromagnetic
Gas shear
Free surface
19
Multicoupling in the Weld Pool
Inertial forces Viscous forces
Hydrostatic
Capillary
Buoyancy
Marangoni
Conduction Convection
Arc pressure
Electromagnetic
Gas shear
Free surface
20
Multicoupling in the Weld Pool
Inertial forces Viscous forces
Hydrostatic
Capillary
Buoyancy
Marangoni
Conduction Convection
Arc pressure
Electromagnetic
Gas shear
Free surface
21
Multicoupling in the Weld Pool
Inertial forces Viscous forces
Hydrostatic
Capillary
Buoyancy
Marangoni
Conduction Convection
Arc pressure
Electromagnetic
Gas shear
Free surface
22
Multicoupling in the Weld Pool
Inertial forces Viscous forces
Hydrostatic
Capillary
Buoyancy
Marangoni
Conduction Convection
Arc pressure
Electromagnetic
Gas shear
Free surface
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Multicoupling in the Weld Pool
Inertial forces Viscous forces
Hydrostatic
Capillary
Buoyancy
Marangoni
Conduction Convection
Arc pressure
Electromagnetic
Gas shear
Free surface
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Disagreement about dominant mechanism

• Experiments cannot show under the surface
• Numerical simulations have convergence problems
  with a very deformed free surface

• Proposed explanations for very deformed weld pool
• Ishizaki (1980) gas shear, experimental
• Oreper (1983) Marangoni, numerical
• Lin (1985) vortex, analytical
• Choo (1991) Arc pressure, gas shear, numerical
• Rokhlin (1993) electromagnetic, hydrodynamic,
  experimental
• Weiss (1996) arc pressure, numerical

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State of the Art in Understanding of Welding (and
Materials) Processes

• Questions that can be easily answered
• For a given current, gas, and geometry, what is
  the maximum velocity of the molten metal?
• For a given set of parameters, what are the
  temperatures, displacements, velocities, etc?
• Questions more difficult to answer
• What mechanism is dominant in determining metal
  velocity?
• If I am designing a weld, what current should I
  use to achieve a given penetration?
• Can I alter one parameter and compensate with
  other parameters to keep the same result?

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Scaling can help answer the difficult questions

• Dimensional Analysis
• Buckinghams Pi theorem
• Informed Dimensional Analysis
• dimensionless groups based on knowledge about
  system
• Inspectional Analysis
• dimensionless groups from normalized equations
• Ordering
• Scaling laws from dominant terms in governing
  equations (e.g. Bejan, M M Chen, Dantzig and
  Tucker, Kline, Denn, Deen, Sides, Astarita, and
  more)

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Typical ordering procedure

• Write governing equations
• Normalize the variables using their
  characteristic values.
• Some characteristic values might be unknown.
• This step results in differential expressions
  based on the normalized variables.
• Replace normalized expressions into governing
  equations.
• Normalize equations using the dominant
  coefficient
• Solve for the unknown characteristic values
• choose terms where they are present
• make their coefficients equal to 1.
• Verify that the terms not chosen are not larger
  than one.
• If any term is larger than one, normalize
  equations again assuming different dominant terms.

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Typical ordering procedure

• Limitations
• Approximation of differential expressions can be
  grossly inaccurate
• not true in important practical cases!
• Higher order derivatives
• Functions with high curvature

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Typical ordering procedure

• Limitations
• Cannot perform manually balances for coupled
  problems with many equations
• when making coefficients equal to 1, there maybe
  more than one unknown
• impractical to check manually for all balances
  (there is no guaranteed unicity in ordering)

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Order of Magnitude Scaling (OMS)

• Addresses the drawbacks
• Table of improved characteristic values
• Linear algebra treatment
• Mendez, P.F. Advanced Scaling Techniques for the
  Modeling of Materials Processing. Keynote paper
  in Sohn Symposium. August 27-31, 2006. San Diego,
  CA. p. 393-404.

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OMS of a high current weld pool

• Goals
• Estimate characteristic values
• velocity, thickness, temperature
• Relate results to process parameters
• materials properties, welding velocity, weld
  current
• Capture all physics, simplifications in the math
• Identify dominant phenomena
• gas shear? Marangoni? electromagnetic? arc
  pressure?

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1. Governing Equations
z
x
z
w
U
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1. Governing Equations

• Boundary Conditions

at free surface
at solid-melt interface
far from weld
free surface
solid-melt interface
far from weld
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1. Governing Equations

• Variables and Parameters
• independent variables (2)
• dependent variables (9)
• parameters (18)

with so many parameters Dimensional Analysis is
not effective
from other models, experiments
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2. Normalization of variables
unknown characteristic values (9)
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3. Replace into governing equations
governing equation
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3. Replace into governing equations
governing equation
scaled variables
OM(1)
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4. Normalize equations
governing equation
scaled variables
OM(1)
normalized equation
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5. Solve for unknowns
two possible balances
B1
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5. Solve for unknowns
two possible balances
B1
B2
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5. Solve for unknowns
two possible balances
balance B1 generates one algebraic equation
B1
B2
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5. Solve for unknowns
two possible balances
balance B1 generates one algebraic equation
balance B2 generates a different equation
B1
B2
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6. Check for self-consistency
two possible balances
balance B1 generates one algebraic equation
balance B2 generates a different equation
self-consistency choose the balance that makes
the neglected term less than 1
B1
B2
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Shortcomings of manual approach
two possible balances
balance B1 generates one algebraic equation
balance B2 generates a different equation
self-consistency choose the balance that makes
the neglected term less than 1
TWO BIG PROBLEMS FOR MATERIALS PROCESSES!
45
Shortcomings of manual approach
?
two possible balances
1 equation 2 unknowns
balance B1 generates one algebraic equation
?
?
?
1 equation 3 unknowns
balance B2 generates a different equation
?
self-consistency choose the balance that makes
the neglected term less than 1
TWO BIG PROBLEMS FOR MATERIALS PROCESSES!

1. Each balance equation involves more than one
   unknown

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Shortcomings of manual approach
two possible balances
balance B1 generates one algebraic equation
balance B2 generates a different equation
self-consistency choose the balance that makes
the neglected term less than 1
TWO BIG PROBLEMS FOR MATERIALS PROCESSES!

1. Each balance equation involves more than one
   unknown
2. A system of equations involves many thousands of
   possible balances

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Shortcomings of manual approach
all coefficients are power laws all terms in
parenthesis expected to be OM(1)
48
Shortcomings of manual approach

• Simple scaling approach involves 334098 possible
  combinations
• There are 116 self-consistent solutions
• there is no unicity of solution
• we cannot stop at first self-consistent solution
• self-consistent solutions are grouped into 55
  classes (1- 6 solutions per class)

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Automating iterative process

• Power-law coefficients can be transformed into
  linear expressions using logarithms
• Several power law equations can then be
  transformed into a linear system of equations
• Normalizing an equation consists of subtracting
  rows

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Matrix of Coefficients
one row for each term of the equation
9 equations
6 BCs
51
18 parameters
9 unknown charact. values
one row for each term of the equation
9 equations
6 BCs
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Solve for unknowns using matrices
18 parameters
9 unknown charact. values
NoS 9x9
NoP
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Solve for unknowns using matrices
Matrix S
18 parameters
9 unknowns
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Check for self-consistency

• can be checked using matrix approach
• checking the 334098 combinations took 72 seconds
  using Matlab on a Pentium M 1.4 GHz

submatrices of normalized secondary terms
secondary terms
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Scaling results
dc36 mm
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Scaling results
plasma shear causes crater
gas shear / viscous
inertial / viscous
electromagnetic / viscous
convection / conduction
Marangoni / gas shear
arc pressure / viscous
hydrostatic / viscous
buoyancy / viscous
capillary / viscous
diff./diff.
57
Summary

• Materials processes are Multiphysics and
  Multicoupled
• Scaling helps understand the dominant forces in
  materials processes
• Several thousand iterations are necessary for
  scaling
• The Matrix of Coefficients and associate matrix
  relationships help automate scaling

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Properties of Scaling Laws

• Simple closed-form expressions
• Typically are exact solution of asymptotic cases
• Display explicitly the trends in a problem
• insightful (explicit variable dependences)
• generalize data, rules of thumb
• Power Laws
• Only way to combine units
• Everything plotted in log-log axes becomes a
  straight line
• Are valid for a family of problems (which can be
  reduced to a canonical problem)
• useful to interpolate / extrapolate, detect
  outliers
• Range of validity can be determined (Process
  maps)
• Provide accurate approximations
• can be used as benchmark for numerical models
• Useful for fast calculations
• massive amounts of data (materials informatics)
• meta-models, early stages of design
• control systems
• Reductionist (system answers can be build by
  understanding the elements individually)

Simple, Accurate, General, Fast
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Calculation of a Balance

1. select 9 equations
2. select dom. input

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Calculation of a Balance

1. select 9 equations
2. select dom. input
3. select dom. output

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Calculation of a Balance

1. select 9 equations
2. select dom. input
3. select dom. output
4. build submatrix of selected normalized outputs

18 parameters
9 unknown charact. values
NoP
NoS 9x9
63
Scaling of FSW
Crawford et al. STWJ 06
maximum temp? shear rate? thickness?
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FSW Scaling laws
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FSW Limits of validity
Va/a ltlt 1

• Slow moving heat source
• isotherms near the pin circular
• Slow mass input
• deformation around tool has radial symmetry
  concentric with the tool
• Thin shear layer
• the shear layer sees a flat (not cylindrical)
  tool

(lt0.3)
Va ltlt wad
(0.01-.3)
d ltlt a
(0.1-0.3)
66
FSW Comparison with literature
1 flat trend
within limits
Stainless 304
Steel 1018
67
FSW Comparison with literature
Stainless 304
Steel 1018
Ti-6Al-4V
68
FSW Comparison with literature
Stainless 304
Steel 1018
C1 0.76 C2 0.33 C3 -0.89
69
FSW Comparison with literature
Ti-6Al-4V
ferrous alloys

• Corrected using trend based on shear layer
  thickness
• Good for aluminum, steel and Ti
• Good beyond hypotheses

Aluminum alloys
70
Other problems scaled

• Weld pool recirculating flows
• Arc
• P.F. Mendez, M.A. Ramirez, G. Trapaga, and T.W.
  Eagar, Order of Magnitude Scaling of the Cathode
  Region in an Axisymmetric Transferred Electric
  Arc, Metallurgical Transactions B, 32B (2001)
  547-554
• Ceramic to metal bonding
• J.-W. Park, P.F. Mendez, and T.W. Eagar, Strain
  Energy Distribution in Ceramic to Metal Joints,
  Acta Materialia, 50 (2002) 883-899
• J.-W. Park, P.F. Mendez, and T.W. Eagar, Residual
  Stress Release in Ceramic-to-Metal Joints by
  Ductile Metal Interlayers, Scripta Materialia, 53
  (2005) 857-861
• Penetration at high currents
• Electrode melting
• RSW

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Canadian Centre for Welding and Joining

• Vision and Mission
• Ensure that Canada is a leader of welding and
  joining technologies through
• research and development
• education
• application
• The main focus of the Centre is meeting the needs
  of Canadian resource-based industries.

• Structure
• Weldco/Industry Chair in Welding and Joining 4M
• Metal products fabrication industry in Alberta
  4.8 billion in revenue in 2005, projected to
  7.5 billion by 2009.
• In oil sands, investment in major projects for
  the next 25 years exceed 200 billion with 86
  billion already committed for starts by 2011

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Shortcomings of manual approach
Boundary conditions
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Promising approaches to answer the
difficultquestions

• closed form solutions
• exact solutions
• asymptotics / perturbation
• dimensional analysis
• regressions
• not considered state of the art
• hold great promise
• numerical, experiments are state of the art

Applied mathematics
Scaling
Engineering