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Determination of Scaling Laws from Statistical Data

Determination of Scaling Laws from Statistical Data. Patricio F. Mendez (Exponent/MIT) Fernando Ord óñez (U. South California) pmendez@exponent.com. Scaling factors. Characteristic value of functions can give insight into the physics of a problem often power laws. numerical experimental.

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Determination of Scaling Laws from Statistical Data

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

  2. Scaling factors • 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

  4. Regressions in Engineering • Used to summarize experimental data • Fit input data well • Difficult to extract physical meaning • Difficult to simplify

  5. Example: 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 ceramic metal

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

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

  8. Homogeneous regression (constrained) RSS=0.008 A little more scatter Consistent units! This formula CAN predict trends for r Must know all parameters exponent determined by homogeneity (e.g. Vignaux)

  9. A step further… Backwards elimination with homogeneity constraint • Iterative method to eliminate parameters • Minimize error (traditional back. elim.) • Maintain homogeneity (new?) • Changing formula with homogeneity  new dimensionless groups

  10. First simplification: eliminate Em RSS=0.015 Scatter grows slightly Consistent units simpler formula

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

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

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

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

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

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

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

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

  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! • 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

  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

  26. Differences with OMS

  27. U U = ˆ . 045 2 . 045 -1 s 3 E r U 5 c y Dimensionless relationships

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