Addressing the Thermal Validation

1
Addressing the Thermal Validation Challenge
Problem Using Polynomial Regression
Models Anthony Giunta Validation Uncertainty
Quantification Processes Dept. (1533) Sandia
National Laboratories Albuquerque, NM Email
aagiunt_at_sandia.gov, Phone [phone] May
2006
Sandia is a multiprogram laboratory operated by
Sandia Corporation, a Lockheed Martin
Company,for the United States Department of
Energys National Nuclear Security
Administration under contract DE-AC04-94AL85000.
2
Technical Approach

• Generate low-order polynomial regression models
  to capture k-vs-T and rhoCp-vs-T trends.
• Use standard errors of regression model
  coefficients to capture uncertainty in the
  coefficients.
• Propagate this uncertainty through T(x,t)
  Equation via random sampling of k-vs-T and
  rhoCP-vs-T model terms.
• Tools JMP (stats) and Mathematica (math plots)
• Caveats due to time constraints
• Only used the high data-rich material
  characterization data (Nc30)
• Did not model systematic bias in validation
  accreditation data

3
Verification of T(x,t) Equation

• T(x,t) equation implemented in Mathematica

time (s) Temp (deg C) dT/d(k) dT/d(rhoCp)
Note Easy to get sensitivity info in Mathematica!
4
JMP Analysis of k-vs-T Data

• Note
• Higher order models checked.
• Residuals check via plot.

Polynomial Model Std. Errors of Coefficients
(follow a t-Distribution)
5
k-vs-T Polynomial Model Approach 1

• Model k k(T) (no residual term)
• k(T) b0 b1T
• b0 0.0503 t280.00151
• b1 2.479e-5 t282.466e-6
• t28 is a random draw from a t-distribution with
  28 DOF

10 Randomly Selected k-vs-T models
6
JMP Analysis of rhoCp-vs-T Data

• Note
• Higher order models checked.
• Residuals checked via plot.

Polynomial Model Std. Errors of Coefficients
(follow a t-Distribution)
7
rhoCp-vs-T Polynomial Model Approach 1

• Model rhoCp rhoCp(T)
• (no residual term)
• rhoCp(T) c0 c1T
• c0 385973.2 t2811715.4
• c1 15.7278 t2819.12917
• t28 is a random draw from a t-distribution with
  28 DOF

10 Randomly Selected rhoCp-vs-T models
8
Application Prediction(q3000 W/m2, L1.9 cm)
One 5,000-Sample Run PTgt900 C 0.0
Typical 1,000-Sample Run Data PTgt900 C 0.001
(1 failure) Failure Probability Estimate (20
1,000-Sample Runs) 8 failures out of 20,000 runs
Pfail 0.004)
Assessment Design Meets Requirement!
Requirement PTemp(x0,t1000 sec) gt 900 deg C
lt 0.01
9
Erratum/Addendum Slides(Post Workshop)

• Here is an alternate (more correct?) approach,
  where Ive included a random error term to model
  the residuals in my polynomial fits to k-vs-T and
  rhoCp-vs-T data.
• Take home message
• Including the residual error term changes the
  assessment!
• Without residual error Prob(Tgt900 deg C) 0.004
• With residual error Prob(Tgt900 deg C) 0.03 to
  0.05

10
k-vs-T Polynomial Model Approach 2

• k k(T) eps1
• (includes residual error term)
• k(T) b0 b1T
• eps1 N0,0.00462
• b0 0.0503 t280.00151
• b1 2.479e-5 t282.466e-6
• t28 is a random draw from a t-distribution with
  28 DOF

10 Randomly Selected k-vs-T models
11
rhoCp-vs-T Polynomial Model Approach 2

• rhoCp rhoCp(T) eps2
• (includes residual error term)
• rhoCp(T) c0 c1T
• eps2 N0,35821.59
• c0 385973.2 t2811715.4
• c1 15.7278 t2819.12917
• t28 is a random draw from a t-distribution with
  28 DOF

10 Randomly Selected rhoCp-vs-T models
12
Application Prediction(q3000 W/m2, L1.9 cm)
One 10,000-Sample Run PTgt900 C 0.0437
Typical 1,000-Sample Run Data PTgt900 C
0.047 Failure Probability Estimates (10
1,000-Sample Runs) Min 0.034, Max 0.052,
Median 0.041
Assessment Design Fails to Meet Requirements!
Requirement PTemp(x0,t1000 sec) gt 900 deg C
lt 0.01
13
Summary

• My initial analysis (w/o residual error term in
  polynomial models) predicts a probability of
  failure of 0.004 which is below the 0.010
  requirement.
• Initial assessment the design satisfies the
  requirement.
• But....
• I only used the data-rich case (Nc30).
• I didnt perform a serious comparison to the
  validation data or accreditation data (due to
  time constraints).
• My k-vs-T and rhoCp-vs-T regression models do not
  include a pure random error term. Is this a
  mistake?
• Did I miss anything else?
• If I include a random error term in the
  polynomial models, the probability of failure is
  approx. 0.03 to 0.05.
• The design fails the requirements!
• More investigation needed!
• Comments and suggestions are welcome!