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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

- 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

gouging region

trailing region

rim

Driving forces in the weld pool (12)

electrode

arc

solidified metal

weld pool

substrate

Driving forces in the weld pool (12)

Inertial forces

electrode

arc

solidified metal

weld pool

substrate

Driving forces in the weld pool (12)

Inertial forces

Viscous forces

electrode

arc

solidified metal

weld pool

substrate

Driving forces in the weld pool (12)

Inertial forces

Viscous forces

Hydrostatic

electrode

arc

rgh

solidified metal

weld pool

substrate

Driving forces in the weld pool (12)

Inertial forces

Viscous forces

Hydrostatic

Buoyancy

electrode

arc

brghDT

solidified metal

weld pool

substrate

Driving forces in the weld pool (12)

Inertial forces

Viscous forces

Hydrostatic

Buoyancy

Conduction

electrode

arc

solidified metal

weld pool

substrate

Driving forces in the weld pool (12)

Inertial forces

Viscous forces

Hydrostatic

Buoyancy

Conduction

Convection

electrode

arc

solidified metal

weld pool

substrate

Driving forces in the weld pool (12)

Inertial forces

Viscous forces

Hydrostatic

Buoyancy

Conduction

Convection

Electromagnetic

electrode

arc

J

B

B

J×B

solidified metal

weld pool

substrate

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

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

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

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

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

- 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

- 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

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

- 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?

- 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)

- 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.

- 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

- 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

- 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.

- 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

z’

x

z

w

U

Boundary Conditions:

at free surface

at solid-melt interface

far from weld

free surface

solid-melt interface

far from weld

Variables and Parameters

independent variables (2)

dependent variables (9)

parameters (18)

with so many parameters Dimensional Analysis is not effective

from other models, experiments

unknown characteristic values (9):

governing equation

governing equation

scaled variables

OM(1)

output

input

input

governing equation

scaled variables

OM(1)

normalized equation

output

input

input

two possible balances

B1

output

input

input

two possible balances

B1

B2

output

input

input

two possible balances

balance B1 generates one algebraic equation:

B1

B2

output

input

input

two possible balances

balance B1 generates one algebraic equation:

balance B2 generates a different equation:

B1

B2

output

input

input

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

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!

?

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

A system of equations involves many thousands of possible balances

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!

all coefficients are power laws

all terms in parenthesis expected to be OM(1)

- 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)

- 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

one row for each term of the equation

9 equations

6 BCs

9 unknown charact. values

18 parameters

one row for each term of the equation

9 equations

6 BCs

18 parameters

9 unknown charact. values

[No]S 9x9

[No]P’

Matrix [S]

18 parameters

9 unknowns

- 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

Tc

dc

Uc

dc=36 mm

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

- 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

- select 9 equations
- select dom. input

Calculation of a Balance

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

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’

Crawford et al. STWJ 06

maximum temp?

shear rate?

thickness?

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)

~1

flat trend

within limits

Stainless 304

Steel 1018

Stainless 304

Steel 1018

Ti-6Al-4V

Stainless 304

Steel 1018

C1 = 0.76

C2 = 0.33

C3 = -0.89

Ti-6Al-4V

ferrous alloys

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

Aluminum alloys

- 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

- 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

Boundary conditions

- 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