Stability Analysis of Non-Iterative Inverse Heat Conduction Algorithms

1. A method for analyzing the stability of non-iterative inverse heat conduction algorithms Xianwu Ling Russell Keanini Harish Cherukuri Department of Mechanical Engineering University of North Carolina at Charlotte Presented at the 2003 IPES Symposium

2. Acknowledgements • NSF • Alcoa Technical Center

3. Outline • Objective • Literature Review • Inverse Problem Statement • Direct Problem • Inverse Algorithm – Sequential Function Specification Method • Derivation of Error Propagation Equation • Stability Criterion Defined • Application to 1-D Problem • Summary and Conclusions

4. Objectives • Formulate a general, non-empirical approach for assessing the stability of Beck’s sequential function specification method

5. Literature Review • Maciag and Al-Khatih (2000). Int. J. Num. Meths. Heat & Fluid Flow. Used integral (Green’s function) solution and backward time differencing to obtain Convergence, as determined by spectral radius of B, determines stability.

6. Literature Review, cont’d • Liu (1996). J. Comp. Phys. Used Duhamel’s integral to obtain where d is a response function that depends on the measured data and where the set of coefficients X are used to determine stability:

7. Unknown surface heat fluxes (q) at  2 q Interpolated node Overview of IHCP • Inverse Heat Conduction Problem (IHCP) Known Initial conditions 1 Boundary conditions Known temperatures at   1 Known temperature measurements 2

8. Unknown heat fluxes to be solved   2 J: the total number of nodes on (J=5). Unknown heat fluxes actually solved K: the total number of chosen nodes from J (K=3). Inverse Problem Statement  1  2 q Known initial temperatures Known temperatures on  1 Interior temperature measurements I: the total number of measurement sites (I=6).

9. Direct Heat Conduction Problem

10. Example: Inverse Algorithm • Introduction to computational time steps Experimental time step, computational time step, and future time , R: the number of future temperatures used.

11. Inverse Algorithm • Objective function

12. Inverse Algorithm • Minimization of with respect to leads to:

13. q(t) Dt q n+1 q2 q1 Time 0 1 2 n n+1 Inverse Algorithm • Introduction to function specification method Idea: Assume a function form of the unknown, and convert IHCP into a problem in which the parameters for the function are solved for. Piecewise constant function: (1) q n+1 are solved for step by step; (2) For each step from n to n+1, an unknown constant is assumed for each future temperature time; the final resultant heat flux for the step is the average of the unknown constants in the strict least squares error sense.

14. Inverse Algorithm • Akey observation is a linear function of nodal heat fluxes at  2

15. Inverse Algorithm • Solve for computed temperatures at the measurement sites

16. Approximate methods: u fraction of two finite differences u governing sensitivity coefficient equation • Improvements: u Time efficiency u Accuracy Inverse Algorithm • Sensitivity coefficient matrix

17. Inverse Algorithm • Matrix normal equation

18. Characteristics uSequential u Non-iterative u FEM-based u future temperature regularization u explicit calculation of sensitivity coefficient matrix Inverse Algorithm • Inverse algorithm procedures (1) Given the temperatures at n, and the measured temperatures at some interior locations at some future times, the heat fluxes from n to n+1 can be solved using the matrix normal equation (together with the sensitivity coefficient matrix equation) (2) Given the heat fluxes from n to n+1, the temperature at the end of n+1 can be updated using (3) Go to the next time step

19. q/qc qc x 1 L 0 Time, t Numerical Tests 1. Step change in heat flux: A flat plate subjected to a constant heat flux qc at x=0 and insulated at x=L. Fictitious measurement site

20. (a) Results from the present method (b) Results from Beck’s function specification method The calculated surface heat flux for const qc input for a plate. . • Observations usmaller time step; ularge error suppression for large number of future temperatures; uNo early time damping. Numerical Tests • Results

21. qc x q+ L 0 Time, t Numerical Tests 2.Triangular heat flux: A flat plate subjected to a triangular heat flux at x=0 and insulated at x=L. Noise input temperatures data are simulated by (1) decimal truncating, (2) adding a random error component generated using a Gaussian probability distribution . Fictitious measurement site

22. The calculated heat flux. Decimal Truncating errors. =0.01. The calculated heat flux. Random errors. =0.06. • Observations usmaller time step; uless susceptible to input errors; Numerical Tests • Results

23. Application to quenching • Drayton Quenchalyer, Inconel 600 probe, Quenchant: oil. • Sampling Freq: 8 Hz, Duration: 60 S Typical temperature history at the center of the probe

24. Calculated heat fluxes vs. time Calculated temperature vs. time. Application to quenching • Burggraf’s analytical solution: • Results 1. Excellent agreement 2. Influence of small oscillations 3. Temperature comparison

25. Error Propagation Equation Globalstandard form equation yields computed temperatures at measurement sites where

26. Error Propagation Equation Matrix normal equation and global force vector then yield where

27. Error Propagation Equation Substitution of into standard form eqn. then gives where

28. Error Propagation Equation Letting be the computed global temperature the measured temperature vector, the error and propagation equation is finally obtained: where In linear problems

29. Solution Stability to An Input Error • One-dimensional axisymmetric problem uModel uGoverning temperature equation (with no future temperature regularization) where , and ,

30. Solution Stability to An Input Error uFrobenius norm analysis 1. Assumption and 2. Equations of temperature error propagations , 3. Temperature error propagation factors , 4. Convergence criterion 5. Frobenius norm

31. Observations: a) For the first time step, the deviation is very high for small time steps and deeply imbedded sensors; b) For small time steps, the errors are high, and suppressed slowly; for large time steps, the errors reduced, the suppression rate extremely high; c) As the sensor is far away from the surface, the initial errors increase, yet accompanied by much higher subsequent error suppression rates. Solution Stability to An Input Error 6. Results and discussions 1) Effect of measurement location and computational time step

32. uSpectral norm analysis 1. Governing temperature equation 2. Spectral norm 3. Convergence criterion 4. Results and discussions a) Clear indication of the allowable time steps; b) No hint of the error suppression rates. Solution Stability to An Input Error 2) Effect of number of elements a) Increasing the number of elements increases the error suppression rates; b) A choice of 20 element would be proper for the problem under study, as observed as J. Beck.

33. Questions