Advanced Scaling Techniques for the Modeling of Materials Processing

Advanced Scaling Techniques for the Modeling of Materials Processing

Karem E. TelloColorado School of Mines

Ustun DumanNovelis

Patricio F. MendezDirector, Canadian Centre for Welding and Joining

University of Alberta

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

2

Example: Weld Pool at High Currents

3

gouging regiontrailing region

rim

Multiphysics in the Weld Pool

4

• Driving forces in the weld pool (12)

weld pool

substrate

solidified metal

arc

electrode


5

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

weld pool

substrate

solidified metal

arc

electrode


6

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

weld pool

substrate

solidified metal

arc

electrode


7

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

weld pool

substrate

solidified metal

arc

electrode

gh


8

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

weld pool

substrate

solidified metal

arc

electrode

ghT


9

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

weld pool

substrate

solidified metal

arc

electrode


10

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

weld pool

substrate

solidified metal

arc

electrode


11

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

weld pool

substrate

solidified metal

arc

electrode

J

BB

J×B


12

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

weld pool

substrate

solidified metal

arc

electrode


13

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

weld pool

substrate

solidified metal

arc

electrode


14

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

weld pool

substrate

solidified metal

arc

electrode


15

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

weld pool

substrate

solidified metal

arc

electrode


16

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

substrate

solidified metal

arc

electrode

Multicoupling in the Weld Pool

17

Hydrostatic

Buoyancy

Electromagnetic

Free surface

Capillary

Gas shear

Arc pressure

Marangoni

Inertial forcesViscous forces

ConductionConvection


18

Hydrostatic

Buoyancy

Electromagnetic

Free surface

Capillary

Gas shear

Arc pressure

Marangoni




19

Hydrostatic

Buoyancy

Electromagnetic

Free surface

Capillary

Gas shear

Arc pressure

Marangoni




20

Hydrostatic

Buoyancy

Electromagnetic

Free surface

Capillary

Gas shear

Arc pressure

Marangoni




21

Hydrostatic

Buoyancy

Electromagnetic

Free surface

Capillary

Gas shear

Arc pressure

Marangoni




22

Hydrostatic

Buoyancy

Electromagnetic

Free surface

Capillary

Gas shear

Arc pressure

Marangoni




23

Hydrostatic

Buoyancy

Electromagnetic

Free surface

Capillary

Gas shear

Arc pressure

Marangoni



Disagreement about dominant mechanism

24

• 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

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?

25

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)

26

Typical ordering procedure1. Write governing equations2. Normalize the variables using their characteristic values.

• Some characteristic values might be unknown. • This step results in differential expressions based on the normalized

variables.

3. Replace normalized expressions into governing equations.4. Normalize equations using the dominant coefficient5. Solve for the unknown characteristic values

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

6. Verify that the terms not chosen are not larger than one.7. If any term is larger than one, normalize equations again

assuming different dominant terms.

27

Typical ordering procedure

• Limitations1. Approximation of differential expressions can be

grossly inaccurate

not true in important practical cases!– Higher order derivatives– Functions with high curvature

28

€

∂nu

∂x n≈

Δu

Δx( )n

Typical ordering procedure

• Limitations2. 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)

29

Order of Magnitude Scaling (OMS)

• Addresses the drawbacks1. Table of improved characteristic values2. 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.

30

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?

31

thickness

velocity

1. Governing Equations

32

U

z’

xz


33

• Boundary Conditions:at free surface at solid-melt interface

far from weld

free surface

solid-melt interfacefar from weld


34

• Variables and Parameters– independent variables (2)

– dependent variables (9)

– parameters (18)

from other models, experiments

with so many parameters Dimensional Analysis is not effective

2. Normalization of variables

35

unknown characteristic values (9):

3. Replace into governing equations

36

governing equation

3. Replace into governing equations

37

governing equation

scaled variables

OM(1)

4. Normalize equations

38

governing equation

scaled variables

OM(1)normalized equation

output inputinput

5. Solve for unknowns

39

output inputinput

two possible balances

B1


40

output inputinput


B1 B2


41

output inputinput


B1 B2

balance B1 generates one algebraic equation:


42

output inputinput


B1 B2


balance B2 generates a different equation:

6. Check for self-consistency

43

output inputinput


B1 B2



self-consistency: choose the balance that makes the neglected term less than 1

Shortcomings of manual approach

44





TWO BIG PROBLEMS FOR MATERIALS PROCESSES!


45






?

?? ?

?

1 equation2 unknowns

1 equation3 unknowns

1. Each balance equation involves more than one unknown


46

1. Each balance equation involves more than one unknown

2. A system of equations involves many thousands of possible balances







47

all coefficients are power lawsall 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)

48

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

49

Matrix of Coefficients

50

9 equations

6 BCs

one row for each term of the equation

51

9 equations

6 BCs

one row for each term of the equation

18 parameters 9 unknown charact. values

Solve for unknowns using matrices

52


[No]P’ [No]S 9x9

9 unknowns 18 parametersMatrix [S]

Solve for unknowns using matrices

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

54

secondary terms submatrices of normalizedsecondary terms

Scaling results

55

€

δc ≈ 50 μm

K100cT

€

Uc ≈1 m/s

δ c=

36

m

€

Tc = qcδc k

€

Uc = 2UD δc€

δc = 2μUD τ c( )1/2

Scaling results

56force dominant

force drivinggroups essdimensionl provide termsSecondary

1.00

0.34

0.08

0.07

0.06

0.03

0.03

0.03

7.E

-05

3.E

-04

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

arc

pres

sure

/ vi

scou

s

elec

trom

agne

tic

/ vis

cous

hydr

osta

tic

/ vis

cous

capi

llar

y / v

isco

us

Mar

ango

ni /

gas

shea

r

buoy

ancy

/ vi

scou

s

gas

shea

r / v

isco

us

conv

ecti

on /

cond

ucti

on

iner

tial

/ vi

scou

s

diff

.=/d

iff.

plasma shear causes crater

Summary

57

• 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

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)

59Simple, Accurate, General, Fast

60

Calculation of a Balance1. select 9 equations2. select dom. input

61

Calculation of a Balance1. select 9 equations2. select dom. input3. select dom. output

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Calculation of a Balance1. select 9 equations2. select dom. input3. select dom. output4. build submatrix of

selected normalized outputs


[No]P’ [No]S 9x9

63

Scaling of FSW

shoulder

pin

substrate

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

• “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

Va/ << 1

Va << aδ

δ << a

(<0.3)

(0.01-.3)

(~0.1-0.3)

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FSW: Comparison with literature

Stainless 304Steel 1018

~1flat trend

within limits

67


Stainless 304Steel 1018Ti-6Al-4V

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3ˆ1)ˆ( 21

C

aCCaf

δδStainless 304Steel 1018

C1 = 0.76C2 = 0.33C3 = -0.89

69


Aluminum alloys

ferrous alloys

Ti-6Al-4V

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

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

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.000 0.004 0.008 0.012 0.016 0.020 0.024 0.028 0.032((2p)1 /2*s

q*qmax)/(U**H) [m]

Dm

[m]

A36

AISI 304

1020 Al

5083 Alsp

HU

qD q max2

base metal

electrode

rim

gougingregion

rim

rim

electrode

curr

ent

forc

efo

rce

weld nugget

heat affected zone (HAZ)

steel sheets

solidification shrinkage

electrode

cool

ing

cool

ing

δ aδ ls

wire (solid)

convectionand dissipationthrough core(liquid)

thermalboundarylayers (liquid)

anode

arc

electronflow

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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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Boundary conditions

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”

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Appl

ied

mat

hem

atics

Engi

neer

ing Scaling

Advanced Scaling Techniques for the Modeling of Materials Processing

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