# Presentation: 950323

## 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@sandia.gov, Phone: 505/844-4280**
**22 May 2006**

- Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company,for the United States Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000.

## 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 (Nc=30)
- Did not model systematic bias in validation & accreditation data

## 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!

## JMP Analysis of k-vs-T Data

- Polynomial Model
- Std. Errors of Coefficients (follow a t-Distribution)

- Note:
- Higher order models checked.
- Residuals check via plot.

## k-vs-T Polynomial Model: Approach 1

- 10 Randomly Selected k-vs-T models

- _Model: k = k(T) __(no residual term)_
- k(T) = b0 + b1*T
- b0 = 0.0503 + t[28]*0.00151
- b1 = 2.479e-5 + t[28]*2.466e-6
- t[28] is a random draw from a t-distribution with 28 DOF

## JMP Analysis of rhoCp-vs-T Data

**JMP Analysis of rhoCp-vs-T Data**

- Polynomial Model
- Std. Errors of Coefficients (follow a t-Distribution)

- Note:
- Higher order models checked.
- Residuals checked via plot.

## rhoCp-vs-T Polynomial Model: Approach 1

- 10 Randomly Selected rhoCp-vs-T models

- _Model: rhoCp = rhoCp(T)_
- _(no residual term)_
- rhoCp(T) = c0 + c1*T
- c0 = 385973.2 + t[28]*11715.4
- c1 = 15.7278 + t[28]*19.12917
- t[28] is a random draw from a t-distribution with 28 DOF

## Application Prediction(q=3000 W/m^2, L=1.9 cm)

- _Typical 1,000-Sample Run Data:_
- P[T>900 C]= 0.001 (1 failure)
- _Failure Probability Estimate (20 1,000-Sample Runs):_
- 8 failures out of 20,000 runs = Pfail = 0.004)

- _One 5,000-Sample Run:_
- P[T>900 C] = 0.0

- _Requirement:_
- P[Temp(x=0,t=1000 sec) > 900 deg C] < 0.01

- _Assessment_:
- Design Meets Requirement!

## Erratum/Addendum Slides(Post Workshop)

- Here is an alternate (more correct?) approach, where I’ve 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(T>900 deg C) ~ 0.004
- With residual error: Prob(T>900 deg C) ~ 0.03 to 0.05

## k-vs-T Polynomial Model: Approach 2

- 10 Randomly Selected k-vs-T models

- _k = k(T) + eps1_
- _(includes residual error term)_
- k(T) = b0 + b1*T
- eps1 ~ N[0,0.00462]
- b0 = 0.0503 + t[28]*0.00151
- b1 = 2.479e-5 + t[28]*2.466e-6
- t[28] is a random draw from a t-distribution with 28 DOF

## rhoCp-vs-T Polynomial Model: Approach 2

- 10 Randomly Selected rhoCp-vs-T models

- _rhoCp = rhoCp(T) + eps2_
- _(includes residual error term)_
- rhoCp(T) = c0 + c1*T
- eps2 ~ N[0,35821.59]
- c0 = 385973.2 + t[28]*11715.4
- c1 = 15.7278 + t[28]*19.12917
- t[28] is a random draw from a t-distribution with 28 DOF

## Application Prediction(q=3000 W/m^2, L=1.9 cm)

**Application Prediction****(q=3000 W/m^2, L=1.9 cm)**

- _Typical 1,000-Sample Run Data:_
- P[T>900 C]= 0.047
- _Failure Probability Estimates (10 1,000-Sample Runs):_
- Min = 0.034, Max = 0.052, Median = 0.041

- _One 10,000-Sample Run:_
- P[T>900 C] = 0.0437

- _Requirement:_
- P[Temp(x=0,t=1000 sec) > 900 deg C] < 0.01

- _Assessment_:
- Design Fails to Meet Requirements!

## 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 (Nc=30).
- I didn’t 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!