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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. Phenomena in Materials Processing.

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advanced scaling techniques for the modeling of materials processing

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

phenomena in materials processing
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
example weld pool at high currents
Example: Weld Pool at High Currents

gouging region

trailing region

rim

multiphysics in the weld pool
Driving forces in the weld pool (12)Multiphysics in the Weld Pool

electrode

arc

solidified metal

weld pool

substrate

multiphysics in the weld pool1
Driving forces in the weld pool (12)

Inertial forces

Multiphysics in the Weld Pool

electrode

arc

solidified metal

weld pool

substrate

multiphysics in the weld pool2
Driving forces in the weld pool (12)

Inertial forces

Viscous forces

Multiphysics in the Weld Pool

electrode

arc

solidified metal

weld pool

substrate

multiphysics in the weld pool3
Driving forces in the weld pool (12)

Inertial forces

Viscous forces

Hydrostatic

Multiphysics in the Weld Pool

electrode

arc

rgh

solidified metal

weld pool

substrate

multiphysics in the weld pool4
Driving forces in the weld pool (12)

Inertial forces

Viscous forces

Hydrostatic

Buoyancy

Multiphysics in the Weld Pool

electrode

arc

brghDT

solidified metal

weld pool

substrate

multiphysics in the weld pool5
Driving forces in the weld pool (12)

Inertial forces

Viscous forces

Hydrostatic

Buoyancy

Conduction

Multiphysics in the Weld Pool

electrode

arc

solidified metal

weld pool

substrate

multiphysics in the weld pool6
Driving forces in the weld pool (12)

Inertial forces

Viscous forces

Hydrostatic

Buoyancy

Conduction

Convection

Multiphysics in the Weld Pool

electrode

arc

solidified metal

weld pool

substrate

multiphysics in the weld pool7
Driving forces in the weld pool (12)

Inertial forces

Viscous forces

Hydrostatic

Buoyancy

Conduction

Convection

Electromagnetic

Multiphysics in the Weld Pool

electrode

arc

J

B

B

J×B

solidified metal

weld pool

substrate

multiphysics in the weld pool8
Driving forces in the weld pool (12)

Inertial forces

Viscous forces

Hydrostatic

Buoyancy

Conduction

Convection

Electromagnetic

Free surface

Multiphysics in the Weld Pool

electrode

arc

solidified metal

weld pool

substrate

multiphysics in the weld pool9
Driving forces in the weld pool (12)

Inertial forces

Viscous forces

Hydrostatic

Buoyancy

Conduction

Convection

Electromagnetic

Free surface

Gas shear

Multiphysics in the Weld Pool

electrode

arc

t

solidified metal

weld pool

substrate

multiphysics in the weld pool10
Driving forces in the weld pool (12)

Inertial forces

Viscous forces

Hydrostatic

Buoyancy

Conduction

Convection

Electromagnetic

Free surface

Gas shear

Arc pressure

Multiphysics in the Weld Pool

electrode

arc

solidified metal

weld pool

substrate

multiphysics in the weld pool11
Driving forces in the weld pool (12)

Inertial forces

Viscous forces

Hydrostatic

Buoyancy

Conduction

Convection

Electromagnetic

Free surface

Gas shear

Arc pressure

Marangoni

Multiphysics in the Weld Pool

electrode

arc

t

solidified metal

weld pool

substrate

multiphysics in the weld pool12
Driving forces in the weld pool (12)

Inertial forces

Viscous forces

Hydrostatic

Buoyancy

Conduction

Convection

Electromagnetic

Free surface

Gas shear

Arc pressure

Marangoni

Capillary

Multiphysics in the Weld Pool

electrode

arc

solidified metal

weld pool

substrate

multicoupling in the weld pool
Multicoupling in the Weld Pool
  • Inertial forces
  • Viscous forces

Capillary

Hydrostatic

Buoyancy

Marangoni

  • Conduction
  • Convection

Arc pressure

Gas shear

Electromagnetic

Free surface

multicoupling in the weld pool1
Multicoupling in the Weld Pool
  • Inertial forces
  • Viscous forces

Capillary

Hydrostatic

Buoyancy

Marangoni

  • Conduction
  • Convection

Arc pressure

Gas shear

Electromagnetic

Free surface

multicoupling in the weld pool2
Multicoupling in the Weld Pool
  • Inertial forces
  • Viscous forces

Capillary

Hydrostatic

Buoyancy

Marangoni

  • Conduction
  • Convection

Arc pressure

Gas shear

Electromagnetic

Free surface

multicoupling in the weld pool3
Multicoupling in the Weld Pool
  • Inertial forces
  • Viscous forces

Capillary

Hydrostatic

Buoyancy

Marangoni

  • Conduction
  • Convection

Arc pressure

Gas shear

Electromagnetic

Free surface

multicoupling in the weld pool4
Multicoupling in the Weld Pool
  • Inertial forces
  • Viscous forces

Capillary

Hydrostatic

Buoyancy

Marangoni

  • Conduction
  • Convection

Arc pressure

Gas shear

Electromagnetic

Free surface

multicoupling in the weld pool5
Multicoupling in the Weld Pool
  • Inertial forces
  • Viscous forces

Capillary

Hydrostatic

Buoyancy

Marangoni

  • Conduction
  • Convection

Arc pressure

Gas shear

Electromagnetic

Free surface

multicoupling in the weld pool6
Multicoupling in the Weld Pool
  • Inertial forces
  • Viscous forces

Capillary

Hydrostatic

Buoyancy

Marangoni

  • Conduction
  • Convection

Arc pressure

Gas shear

Electromagnetic

Free surface

disagreement about dominant mechanism
Experiments cannot show under the surface

Numerical simulations have convergence problems with a very deformed free surface

Disagreement about dominant mechanism
  • 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
state of the art in understanding of welding and materials processes
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?
scaling can help answer the difficult questions
Scaling can help answer the “difficult” questions
  • Dimensional Analysis
    • Buckingham’s “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)
typical ordering procedure
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.
typical ordering procedure1
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
typical ordering procedure2
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)
order of magnitude scaling oms
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.
oms of a high current weld pool
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?

velocity

thickness

1 governing equations1
Boundary Conditions:1. Governing Equations

at free surface

at solid-melt interface

far from weld

free surface

solid-melt interface

far from weld

1 governing equations2
Variables and Parameters

independent variables (2)

dependent variables (9)

parameters (18)

1. Governing Equations

with so many parameters Dimensional Analysis is not effective

from other models, experiments

2 normalization of variables
2. Normalization of variables

unknown characteristic values (9):

3 replace into governing equations1
3. Replace into governing equations

governing equation

scaled variables

OM(1)

4 normalize equations
output

input

input

4. Normalize equations

governing equation

scaled variables

OM(1)

normalized equation

5 solve for unknowns
output

input

input

5. Solve for unknowns

two possible balances

B1

5 solve for unknowns1
output

input

input

5. Solve for unknowns

two possible balances

B1

B2

5 solve for unknowns2
output

input

input

5. Solve for unknowns

two possible balances

balance B1 generates one algebraic equation:

B1

B2

5 solve for unknowns3
output

input

input

5. Solve for unknowns

two possible balances

balance B1 generates one algebraic equation:

balance B2 generates a different equation:

B1

B2

6 check for self consistency
output

input

input

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

shortcomings of manual approach
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!

shortcomings of manual approach1
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!

  • Each balance equation involves more than one unknown
shortcomings of manual approach2
Each balance equation involves more than one unknown

A system of equations involves many thousands of possible balances

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!

shortcomings of manual approach3
Shortcomings of manual approach

all coefficients are power laws

all terms in parenthesis expected to be OM(1)

shortcomings of manual approach4
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)
automating iterative process
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
matrix of coefficients
Matrix of Coefficients

one row for each term of the equation

9 equations

6 BCs

slide51
9 unknown charact. values

18 parameters

one row for each term of the equation

9 equations

6 BCs

solve for unknowns using matrices
Solve for unknowns using matrices

18 parameters

9 unknown charact. values

[No]S 9x9

[No]P’

solve for unknowns using matrices1
Solve for unknowns using matrices

Matrix [S]

18 parameters

9 unknowns

check for self consistency
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

scaling results
Tc

dc

Uc

Scaling results

dc=36 mm

scaling results1
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.^

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

Summary
properties of scaling laws
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

slide60
Calculation of a Balance
  • select 9 equations
  • select dom. input
slide61
Calculation of a Balance
  • select 9 equations
  • select dom. input
  • select dom. output
slide62
Calculation of a Balance
  • select 9 equations
  • select dom. input
  • select dom. output
  • build submatrix of selected normalized outputs

18 parameters

9 unknown charact. values

[No]S 9x9

[No]P’

scaling of fsw
Scaling of FSW

Crawford et al. STWJ 06

maximum temp?

shear rate?

thickness?

fsw limits of validity
FSW: Limits of validity

Va/a << 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

(<0.3)

Va<< wad

(0.01-.3)

d << a

(~0.1-0.3)

fsw comparison with literature
FSW: Comparison with literature

~1

flat trend

within limits

Stainless 304

Steel 1018

fsw comparison with literature1
FSW: Comparison with literature

Stainless 304

Steel 1018

Ti-6Al-4V

fsw comparison with literature2
FSW: Comparison with literature

Stainless 304

Steel 1018

C1 = 0.76

C2 = 0.33

C3 = -0.89

fsw comparison with literature3
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

other problems scaled
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
canadian centre for welding and joining
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
promising approaches to answer the difficult questions
Promising approaches to answer the “difficult”questions
  • 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

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