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

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

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

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

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

Sensitivity Coefficient i measurement j estimated value ij i j

Main Set of Equations

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

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

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

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

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

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

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

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

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

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

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

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

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

Numerical Results Measured and predicted temperature in sensor locations

Numerical Results

Temperature distribution along the surface X measurement calculations X Temperature

Numerical Results

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