1. Application of Bezier splines and sensitivity analysis in inverse geometry and boundary problems Iwona NOWAK*, Andrzej J. NOWAK** * Institute of Mathematics, ** Institute of Thermal Technology, Technical University of Silesia, Gliwice, Poland. Inverse Problems in Engineering Symposium Tuscaloosa, June 2003

2. Scope of the Presentation Problem formulation Solution procedure - sensitivity analysis for geometry and boundary problems Numerical results Conclusions

3. Continuous casting solid liquid v x x y F AB C D E O

4. Problem Formulation solid liquid x y F AB C D E O

5. General Solution Strategy make boundary problem well-posed solve direct problem modify assumed data sensitivity coefficients FDM FEM BEM best matching

6. Sensitivity Coefficient i measurement j estimated value ij i j

7. Main Set of Equations

8. Solution Algorithm direct problem formulation -assumption of vector Y * =[y 1 *,..., y n *, q 1 *,..., q m *] „freezing” of heat fluxes geometry problem - iterative solution „freezing” of front location boundary problem - single step convergence check end of calculations

9. Sensitivity Analysis - Boundary Part solid liquid v x x y F AB C D E O

10. Sensitivity Analysis –Geometry Part solid liquid v x x y F AB C D E O

11. Sensitivity Analysis –Geometry Part V0V0 V1V1 V2V2 V3V3 u u u V01V01 V11V11 V21V21 u u V02V02 V12V12 V 0 3 = P(u) u Bezier Spline Nowak I., Nowak A.J, Wrobel L.C: Identification of Phase Change Front by Bezier Splines and BEM, International Journal of Thermal Sciences, vol.41 (2002) Elsevier Science, pp.492-499

12. Sensor Locations geometry part - iterative solution boundary part - single step solid liquid x y F AB C D E O all sensors are used all sensors are used

13. Numerical Results measurements error : 0.1% number of sensors : 35 phase change front location heat flux distribution

14. Numerical Results measurements error : 0.1% number of sensors : 35

15. Sensor Locations geometry part - iterative solution boundary part - single step solid liquid x y F AB C D E O geometry sensors are used geometry sensors are used

16. Numerical Results phase change front location heat flux distribution measurements error : 0.1% number of sensors : 35 sensors separated

17. Numerical Results measurements error : 0.1% number of sensors : 35 sensors separated

18. Numerical Results phase change front location heat flux distribution measurements error : 2.0 % number of sensors : 35 sensors separated

19. Numerical Results measurements error : 2.0% number of sensors : 35 sensors separated

20. Experiment Drezet J.-M., Rappaz M., Grun G.-U.,Gremaud M., Determination of Thermophysical Properties and Boundary Conditions of Direct Chill-Cast Aluminium Alloys Using Inverse Methods, Metallurgical and Materials Transactions A, vol.31A, June 2000, pp.1627--1634

21. Numerical Results Measured and predicted temperature in sensor locations

22. Numerical Results

23. Temperature distribution along the surface 00.40.81.21.62 X 0 200 400 600 800 measurement calculations 00.040.080.12 X 0 200 400 600 800 Temperature

24. Numerical Results

25. Conclusions BEM based algorithms for solving inverse boundary and geometry problems application of Bézier function and sensitivity coefficients stabile results does not provide accurate solution sensors separation permits to obtain encouraging results even with experimental measurements