1 Determination 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 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 e.g. maximum pressure

3 Scaling 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

4 Regressions 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

5 Example: Ceramic-metal joints Parameters: E c : elasticity of ceramic E m : elasticity of metal σ y : yield strength of metal r: cylinder radius ε T : thermal mismatch Goal: U: strain energy in ceramic Parameters: E c : elasticity of ceramic E m : elasticity of metal σ y : yield strength of metal r: cylinder radius ε T : thermal mismatch Goal: U: strain energy in ceramic ceramic metal

6 Input Data Can’t determine trends for radius independent parameters constant! dependent magnitude

7 Standard regression constant conflict! arbitrary exponent This formula CANNOT predict trends for r RSS=0.007

8 Homogeneous regression (constrained) 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… 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 First simplification: eliminate E m Scatter grows slightly Consistent units simpler formula RSS=0.015

11 Generation of dimensionless groups Homogeneous regression First constrained backwards elimination First dimensionless group Least influence of all possible dimensionless groups

12 Second simplification: no constant Scatter keeps growing Even simpler formula RSS=0.026

13 Second dimensionless group First constrained backwards elimination Second dimensionless group Simpler expression than previous Second constrained backwards elimination

14 Third simplification: eliminate ε T Scatter still grows slightly Formula keeps getting simpler RSS=0.258

15 Third dimensionless group Second constrained backwards elimination Third dimensionless group Keeps getting simpler Third constrained backwards elimination

16 Fourth simplification: eliminate E c Scatter increases significantly Simplest possible formula Order of magnitude is wrong: HUGE ERRORS RSS=535 (!!)

17 Fourth dimensionless group Third constrained backwards elimination Fourth dimensionless group Simplest Fourth constrained backwards elimination

18 Evolution of simplicity and error Simpler formulas Larger error

19 Relevance of dimensionless groups Simpler and more relevant

20 Physical interpretation Strain in ceramic + thermal strain (+ proportionality) + elasticity in metal

21 Output We can express the homogeneous regression as Where the dimensionless are ranked We can express the homogeneous regression as Where the dimensionless are ranked Homogeneous regression Scaling factor Correction factors Essential Lesser importance

22 Comparison with results using traditional methods Dimensionally constrained backwards elimination Maximum simplicity with reasonable results Using physical considerations and traditional scaling approach Very similar

23 Discussion 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! If we used scaling law for elasticity, RSS=3 much greater than 0.3 for our simplest reasonable model.

24 Next steps Orthogonal basis Currently Orthogonal Round exponents Currently Round Orthogonal basis Currently Orthogonal Round exponents Currently Round

25 Similarities 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 OMS OMSDCBE InputGoverning equations Empirical data RegimesOutputInput UnitsOutputInput

27

28 Dimensionless relationships ˆ rE U U U yc  