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

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  is a response function that depends on the measured data and where the set of coefficients X are used to determine stability:

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

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

9. Direct Heat Conduction Problem

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

11. Objective function Inverse Algorithm

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

13. q(t) Time 0 12nn+1 tt q1q1 q2q2 q n+1 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. Inverse Algorithm

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

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

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

17. Matrix normal equation Inverse Algorithm

18. (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) Inverse algorithm procedures (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 Characteristics  Sequential  Non-iterative  FEM-based  future temperature regularization  explicit calculation of sensitivity coefficient matrix

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

20. Numerical Tests (a) Results from the present method The calculated surface heat flux for const q c input for a plate.. (b) Results from Beck’s function specification method Results  smaller time step;  large error suppression for large number of future temperatures;  No early time damping. Observations

21. Numerical Tests 2. Triangular heat flux: q+q+ Time, t 0 Fictitious measurement site 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. qcqc x L

22. Numerical Tests The calculated heat flux. Decimal Truncating errors. =0.01. The calculated heat flux. Random errors. =0.06. Results Observations  smaller time step;  less susceptible to input errors;

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

25. Error Propagation Equation Global  standard form equation where yields computed temperatures at measurement sites

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

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

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

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

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

31. 6. Results and discussions 1) Effect of measurement location and computational time step 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; Solution Stability to An Input Error c) As the sensor is far away from the surface, the initial errors increase, yet accompanied by much higher subsequent error suppression rates.

32. 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.  Spectral 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

33. Questions