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Advanced Scaling Techniques for the Modeling of Materials Processing

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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 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

Driving forces in the weld pool (12) Processing

Multiphysics in the Weld Poolelectrode

arc

solidified metal

weld pool

substrate

Driving forces in the weld pool (12) Processing

Inertial forces

Multiphysics in the Weld Poolelectrode

arc

solidified metal

weld pool

substrate

Driving forces in the weld pool (12) Processing

Inertial forces

Viscous forces

Multiphysics in the Weld Poolelectrode

arc

solidified metal

weld pool

substrate

Driving forces in the weld pool (12) Processing

Inertial forces

Viscous forces

Hydrostatic

Multiphysics in the Weld Poolelectrode

arc

rgh

solidified metal

weld pool

substrate

Driving forces in the weld pool (12) Processing

Inertial forces

Viscous forces

Hydrostatic

Buoyancy

Multiphysics in the Weld Poolelectrode

arc

brghDT

solidified metal

weld pool

substrate

Driving forces in the weld pool (12) Processing

Inertial forces

Viscous forces

Hydrostatic

Buoyancy

Conduction

Multiphysics in the Weld Poolelectrode

arc

solidified metal

weld pool

substrate

Driving forces in the weld pool (12) Processing

Inertial forces

Viscous forces

Hydrostatic

Buoyancy

Conduction

Convection

Multiphysics in the Weld Poolelectrode

arc

solidified metal

weld pool

substrate

Driving forces in the weld pool (12) Processing

Inertial forces

Viscous forces

Hydrostatic

Buoyancy

Conduction

Convection

Electromagnetic

Multiphysics in the Weld Poolelectrode

arc

J

B

B

J×B

solidified metal

weld pool

substrate

Driving forces in the weld pool (12) Processing

Inertial forces

Viscous forces

Hydrostatic

Buoyancy

Conduction

Convection

Electromagnetic

Free surface

Multiphysics in the Weld Poolelectrode

arc

solidified metal

weld pool

substrate

Driving forces in the weld pool (12) Processing

Inertial forces

Viscous forces

Hydrostatic

Buoyancy

Conduction

Convection

Electromagnetic

Free surface

Gas shear

Multiphysics in the Weld Poolelectrode

arc

t

solidified metal

weld pool

substrate

Driving forces in the weld pool (12) Processing

Inertial forces

Viscous forces

Hydrostatic

Buoyancy

Conduction

Convection

Electromagnetic

Free surface

Gas shear

Arc pressure

Multiphysics in the Weld Poolelectrode

arc

solidified metal

weld pool

substrate

Driving forces in the weld pool (12) Processing

Inertial forces

Viscous forces

Hydrostatic

Buoyancy

Conduction

Convection

Electromagnetic

Free surface

Gas shear

Arc pressure

Marangoni

Multiphysics in the Weld Poolelectrode

arc

t

solidified metal

weld pool

substrate

Driving forces in the weld pool (12) Processing

Inertial forces

Viscous forces

Hydrostatic

Buoyancy

Conduction

Convection

Electromagnetic

Free surface

Gas shear

Arc pressure

Marangoni

Capillary

Multiphysics in the Weld Poolelectrode

arc

solidified metal

weld pool

substrate

Multicoupling in the Weld Pool Processing

- Inertial forces
- Viscous forces

Capillary

Hydrostatic

Buoyancy

Marangoni

- Conduction
- Convection

Arc pressure

Gas shear

Electromagnetic

Free surface

Multicoupling in the Weld Pool Processing

- Inertial forces
- Viscous forces

Capillary

Hydrostatic

Buoyancy

Marangoni

- Conduction
- Convection

Arc pressure

Gas shear

Electromagnetic

Free surface

Multicoupling in the Weld Pool Processing

- Inertial forces
- Viscous forces

Capillary

Hydrostatic

Buoyancy

Marangoni

- Conduction
- Convection

Arc pressure

Gas shear

Electromagnetic

Free surface

- Inertial forces
- Viscous forces

Capillary

Hydrostatic

Buoyancy

Marangoni

- Conduction
- Convection

Arc pressure

Gas shear

Electromagnetic

Free surface

- Inertial forces
- Viscous forces

Capillary

Hydrostatic

Buoyancy

Marangoni

- Conduction
- Convection

Arc pressure

Gas shear

Electromagnetic

Free surface

- Inertial forces
- Viscous forces

Capillary

Hydrostatic

Buoyancy

Marangoni

- Conduction
- Convection

Arc pressure

Gas shear

Electromagnetic

Free surface

- Inertial forces
- Viscous forces

Capillary

Hydrostatic

Buoyancy

Marangoni

- Conduction
- Convection

Arc pressure

Gas shear

Electromagnetic

Free surface

Experiments cannot show under the surface Processing

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

- 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 Processes

- 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 Processes Replace normalized expressions into governing equations. Normalize equations using the dominant coefficient Solve for the unknown characteristic values 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.

- 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.

- choose terms where they are present
- make their coefficients equal to 1.

Typical ordering procedure Processes

- Limitations
- Approximation of differential expressions can be grossly inaccurate
not true in important practical cases!

- Higher order derivatives
- Functions with high curvature

- Approximation of differential expressions can be grossly inaccurate

Typical ordering procedure Processes

- 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)

- Cannot perform manually balances for coupled problems with many equations

Order of Magnitude Scaling (OMS) Processes

- 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 Processes

- 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?

- Estimate characteristic values:

velocity

thickness

Boundary Conditions: Processes

1. Governing Equationsat free surface

at solid-melt interface

far from weld

free surface

solid-melt interface

far from weld

Variables and Parameters Processes

independent variables (2)

dependent variables (9)

parameters (18)

1. Governing Equationswith so many parameters Dimensional Analysis is not effective

from other models, experiments

2. Normalization of variables Processes

unknown characteristic values (9):

3. Replace into governing equations Processes

governing equation

output Processes

input

input

4. Normalize equationsgoverning equation

scaled variables

OM(1)

normalized equation

output Processes

input

input

5. Solve for unknownstwo possible balances

balance B1 generates one algebraic equation:

B1

B2

output Processes

input

input

5. Solve for unknownstwo possible balances

balance B1 generates one algebraic equation:

balance B2 generates a different equation:

B1

B2

output Processes

input

input

6. Check for self-consistencytwo 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 Processes

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 approach Processes

?

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

Each balance equation involves more than one unknown Processes

A system of equations involves many thousands of possible balances

Shortcomings of manual approachtwo 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 approach Processes

all coefficients are power laws

all terms in parenthesis expected to be OM(1)

Shortcomings of manual approach Processes

- 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 Processes

- 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

9 unknown charact. values Processes

18 parameters

one row for each term of the equation

9 equations

6 BCs

Solve for unknowns using matrices Processes

18 parameters

9 unknown charact. values

[No]S 9x9

[No]P’

Check for self-consistency Processes

- 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 Processes

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.^

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

SummaryProperties of Scaling Laws “Multicoupled”

- 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

- insightful (explicit variable dependences)
- 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

Calculation of a Balance “Multicoupled”

- select 9 equations
- select dom. input

Calculation of a Balance “Multicoupled”

- select 9 equations
- select dom. input
- select dom. output

Calculation of a Balance “Multicoupled”

- 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’

FSW: Scaling laws “Multicoupled”

FSW: Limits of validity “Multicoupled”

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 “Multicoupled”

Stainless 304

Steel 1018

C1 = 0.76

C2 = 0.33

C3 = -0.89

FSW: Comparison with literature “Multicoupled”

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 “Multicoupled”

- 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 “Multicoupled”

- 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.

- Ensure that Canada is a leader of welding and joining technologies through

- 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

Shortcomings of manual approach “Multicoupled”

Boundary conditions

Promising approaches to answer the “difficult”questions “Multicoupled”

- 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