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Inverse Problems in Engineering
Symposium Tuscaloosa, June 2003
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.
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Scope of the Presentation

• Problem formulation
• Solution procedure
• - sensitivity analysis for geometry and
  boundary problems
• Numerical results
• Conclusions

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Continuous casting
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Problem Formulation
y
A
B
liquid
C
F
D

solid
E
O
x
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General Solution Strategy
make boundary problem well-posed
FDM FEM BEM
solve direct problem
best matching
modify assumed data
sensitivity coefficients
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Sensitivity Coefficient
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Main Set of Equations
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Solution Algorithm
direct problem formulation -assumption of
vector Yy1,..., yn, q1,..., qm
freezing of heat fluxes
freezing of front location
end of calculations
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Sensitivity Analysis - Boundary Part
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Sensitivity Analysis Geometry Part
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Sensitivity Analysis Geometry Part
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
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Sensor Locations
y
A
B
liquid
C
all sensors are used
F
D
all sensors are used

solid
E
O
x
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Numerical Results
phase change front location
heat flux distribution
measurements error 0.1 number of sensors 35
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Numerical Results
measurements error 0.1 number of sensors 35
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Sensor Locations
y
A
B
liquid
C
geometry sensors are used
F
D
geometry sensors are used

solid
E
O
x
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Numerical Results
phase change front location
heat flux distribution
measurements error 0.1 number of sensors
35 sensors separated
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Numerical Results
measurements error 0.1 number of sensors
35 sensors separated
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Numerical Results
phase change front location
heat flux distribution
measurements error 2.0 number of sensors
35 sensors separated
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Numerical Results
measurements error 2.0 number of sensors
35 sensors separated
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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
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Numerical Results
Measured and predicted temperature in sensor
locations
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Numerical Results
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Numerical Results
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Numerical Results
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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