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.

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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 equations

    1. Governing Equations

    z’

    x

    z

    w

    U


    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 equations

    3. Replace into governing equations

    governing equation


    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


    Advanced scaling techniques for the modeling of materials processing

    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


    Advanced scaling techniques for the modeling of materials processing

    Calculation of a Balance

    • select 9 equations

    • select dom. input


    Advanced scaling techniques for the modeling of materials processing

    Calculation of a Balance

    • select 9 equations

    • select dom. input

    • select dom. output


    Advanced scaling techniques for the modeling of materials processing

    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 scaling laws

    FSW: Scaling laws


    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


    Shortcomings of manual approach5

    Shortcomings of manual approach

    Boundary conditions


    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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