1

Determination of Scaling Laws from Statistical DataDetermination of Scaling Laws from Statistical Data

Patricio F. Mendez (Exponent/MIT)

Fernando Ordóñez (U. South California)

pmendez@exponent.com

Patricio F. Mendez (Exponent/MIT)

Fernando Ordóñez (U. South California)

pmendez@exponent.com

2

Scaling factorsScaling factors• Characteristic value of functions

• can give insight into the physics of a problem

• often power laws

• Characteristic value of functions• can give insight into the physics of a

problem• often power laws

•numerical•experimental

•scaling factors

2202

1ˆCCS JRP

e.g. maximum pressure

2ˆCS JP

3

Scaling factorsScaling factors

• Non homogeneous:• Proportionality laws• The mismatch of units indicates missing

physics

• Homogeneous• Can potentially capture all physics• Often there are multiple possibilities• Last year: from equations• This year: from data

• Non homogeneous:• Proportionality laws• The mismatch of units indicates missing

physics

• Homogeneous• Can potentially capture all physics• Often there are multiple possibilities• Last year: from equations• This year: from data

2202

1ˆCCS JRP

2ˆCS JP

4

Regressions in EngineeringRegressions in Engineering

• Used to summarize experimental data

• Fit input data well

• Difficult to extract physical meaning

• Difficult to simplify

• Used to summarize experimental data

• Fit input data well

• Difficult to extract physical meaning

• Difficult to simplify

126.0153.2773.1194.0158.1424.01

ˆTymc rEEeU

5

Example: Ceramic-metal jointsExample: Ceramic-metal joints

Parameters:

• Ec: elasticity of ceramic

• Em: elasticity of metal

• σy: yield strength of metal

• r: cylinder radius

• εT: thermal mismatch

Goal:

• U: strain energy in ceramic

Parameters:

• Ec: elasticity of ceramic

• Em: elasticity of metal

• σy: yield strength of metal

• r: cylinder radius

• εT: thermal mismatch

Goal:

• U: strain energy in ceramic

cera

mic

met

al

6

Input DataInput Data

• Can’t determine trends for radius• Can’t determine trends for radius

independent parameters

constant!

Ec Em σy r εT U

[Pa] [Pa] [Pa] [m] [-] [Pa m3] Si3N4 Cu 3.04E+11 1.28E+11 7.58E+07 6.25E-03 6.85E-03 4.23E-03 Si3N4 Ni 3.04E+11 2.08E+11 1.48E+08 6.25E-03 5.15E-03 1.52E-02 Si3N4 Nb 3.04E+11 1.03E+11 2.40E+08 6.25E-03 2.10E-03 2.80E-02 Si3N4 Inc600 3.04E+11 2.06E+11 2.50E+08 6.25E-03 5.15E-03 3.78E-02 Si3N4 304SS 3.04E+11 2.06E+11 2.56E+08 6.25E-03 7.10E-03 3.88E-02 Si3N4 AISI316 3.04E+11 1.94E+11 2.90E+08 6.25E-03 7.00E-03 4.91E-02 Al2O3 Ti 3.58E+11 1.20E+11 1.72E+08 6.25E-03 5.05E-04 1.04E-02 Al2O3 Inc600 3.58E+11 2.06E+11 2.50E+08 6.25E-03 2.95E-03 3.00E-02 Al2O3 304SS 3.58E+11 2.00E+11 2.56E+08 6.25E-03 4.90E-03 3.16E-02

dependentmagnitude

7

Standard regressionStandard regression

R2 = 0.9968

0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.01 0.02 0.03 0.04 0.05 0.06

Model 1 [Pa0.8091m2.153]

Nu

mer

ical

cal

cula

tio

ns

[Pa

m3 ]

minimum scatter

inconsistent units

126.0153.2773.1194.0158.1424.01

ˆTymc rEEeU

constant conflict! arbitrary exponent

This formula CANNOT predict trends for r

RSS=0.007

8

Homogeneous regression (constrained)Homogeneous regression (constrained)

141.03776.1173.0949.0083.12

ˆTymc rEEeU

R2 = 0.9967

0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.01 0.02 0.03 0.04 0.05 0.06

Model 2 [Pa m3]

Nu

mer

ical

cal

cula

tio

ns

[Pa

m3]

A little more scatter

Consistent units!

exponent determined by homogeneity (e.g. Vignaux)

RSS=0.008

This formula CAN predict trends for r

Must know all parameters

9

A step further…A step further…

• Iterative method to eliminate parameters• Minimize error (traditional back. elim.)• Maintain homogeneity (new?)

• Changing formula with homogeneity new dimensionless groups

• Iterative method to eliminate parameters• Minimize error (traditional back. elim.)• Maintain homogeneity (new?)

• Changing formula with homogeneity new dimensionless groups

Backwards elimination

with

homogeneity constraint

10

R2 = 0.9948

0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.01 0.02 0.03 0.04 0.05 0.06

Model 3 [Pa m3]

Nu

mer

ical

cal

cula

tio

ns

[Pa

m3]

First simplification: eliminate EmFirst simplification: eliminate Em

182.03816.1816.0669.03

ˆTyc rEeU

Scatter grows slightly

Consistent units

simpler formula

RSS=0.015

11

Generation of dimensionless groupsGeneration of dimensionless groups

141.03776.1173.0949.0083.12

ˆTymc rEEeU

182.03816.1816.0669.03

ˆTyc rEeU

Homogeneous regression

First constrained backwards elimination

041.0040.0173.0133.0415.01

Tymc EEe

First dimensionless group• Least influence of all possible dimensionless groups

12

R2 = 0.9915

0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.01 0.02 0.03 0.04 0.05 0.06

Model 4 [Pa m3]

Nu

mer

ical

cal

cula

tio

ns

[Pa

m3]

182.03816.1898.04

ˆTyc rEU

Second simplification: no constantSecond simplification: no constant

Scatter keeps growing

Even simpler formula

RSS=0.026

13

Second dimensionless groupSecond dimensionless group

182.03816.1816.0669.03

ˆTyc rEeU First constrained

backwards elimination

011.0082.0082.0669.02

TycEe

Second dimensionless group•Simpler expression than previous

182.03816.1898.04

ˆTyc rEU Second constrained

backwards elimination

14

R2 = 0.9683

0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.01 0.02 0.03 0.04 0.05 0.06Model 5 [Pa m3]

Nu

mer

ical

cal

cula

tio

ns

[Pa

m3 ]

3045.2045.15

ˆ rEU yc

Third simplification: eliminate εTThird simplification: eliminate εT

Scatter still grows slightly

Formula keeps getting simpler

RSS=0.258

15

Third dimensionless groupThird dimensionless group

Second constrained backwards elimination

194.0147.0147.03

TycE

Third dimensionless group•Keeps getting simpler

182.03816.1898.04

ˆTyc rEU

Third constrained backwards elimination

3045.2045.15

ˆ rEU yc

16

R2 = 0.7668

0

0.01

0.02

0.03

0.04

0.05

0.06

0 20 40 60 80

Model 6 [Pa m3]

Nu

mer

ical

cal

cula

tio

ns

[Pa

m3]

36

ˆ rU y

Fourth simplification: eliminate EcFourth simplification: eliminate Ec

Scatter increases significantly

Simplest possible formula

Order of magnitude is wrong: HUGE ERRORS

RSS=535 (!!)

17

Fourth dimensionless groupFourth dimensionless group

Third constrained backwards elimination

045.1045.14 ycE

Fourth dimensionless group•Simplest

Fourth constrained backwards elimination

3045.2045.15

ˆ rEU yc

36

ˆ rU y

18

Evolution of simplicity and errorEvolution of simplicity and error

0.007 0.008 0.015 0.026

0.258

535

6 6

5

4

3

2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Standardregression

Homogeneousregression

Firstsimplification

Secondsimplification

Thirdsimplification

Fourthsimplificatio n

Qu

ad

rati

c e

rro

r

0

1

2

3

4

5

6

7

Nu

mb

er

of

pa

ram

ete

rs

Simpler formulas

Larg

er e

rror

19

Relevance of dimensionless groupsRelevance of dimensionless groups

0.007 0.011

0.232

534.742

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

err

or

va

ria

tio

n

Simpler a

nd more

relevant

20

Physical interpretationPhysical interpretation

041.0040.0173.0133.0415.01

Tymc EEe

011.0082.0082.0669.02

TycEe

194.0147.0147.03

TycE

045.1045.14 ycE Strain in ceramic

+ thermal strain

(+ proportionality)

+ elasticity in metal

21

OutputOutput

• We can express the homogeneous regression as

• Where the dimensionless are ranked

• We can express the homogeneous regression as

• Where the dimensionless are ranked

123462ˆˆ UU

Homogeneous regression Scaling factor

Correction factors

Essential Lesser importance

22

Comparison with results using traditional methodsComparison with results using traditional methods

3045.2045.15

ˆ rEU yc Dimensionally constrained backwards elimination• Maximum simplicity with reasonable results

321ˆ rEU yc Using physical considerations and traditional scaling approach

Very similar

23

DiscussionDiscussion• Data must belong to the same

regime• Regime: range of conditions with the

same dominant input and output• Different scaling laws for different

regimes!

• Data must belong to the same regime• Regime: range of conditions with the

same dominant input and output• Different scaling laws for different

regimes!R2 = -0.2626

0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Elastic [Pa m3]

Nu

mer

ical

cal

cula

tio

ns

[Pa

m3 ] •If we used scaling

law for elasticity, RSS=3 much greater than 0.3 for our simplest reasonable model.

24

Next stepsNext steps

• Orthogonal basis• Currently

• Orthogonal

• Round exponents• Currently

• Round

• Orthogonal basis• Currently

• Orthogonal

• Round exponents• Currently

• Round

140.04

194.03 T

194.03 T

045.1045.14 ycE

ycE 14

25

Similarities with OMSSimilarities with OMS

• Generation of simple and accurate scaling laws

• Automatic generation of dimensionless groups

• Dimensionless groups ranked by relevance

• Need to know all parameters involved

• Relevance of regimes

• Generation of simple and accurate scaling laws

• Automatic generation of dimensionless groups

• Dimensionless groups ranked by relevance

• Need to know all parameters involved

• Relevance of regimes

26

Differences with OMSDifferences with OMS

OMS DCBE

Input Governing equations

Empirical data

Regimes Output Input

Units Output Input

27

28

Dimensionless relationshipsDimensionless relationships

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0.8 0.9 1 1.1 1.2 1.3

3

304

5.2

045

. -15ˆ

rE

U

UU

yc

194.0147.0147.03

TycE