1
Determination of Scaling Laws from Statistical DataDetermination of Scaling Laws from Statistical Data
Patricio F. Mendez (Exponent/MIT)
Fernando Ordóñez (U. South California)
Patricio F. Mendez (Exponent/MIT)
Fernando Ordóñez (U. South California)
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